(SANITIZED)BALLISTOCARDIOGRAPHY IN RELATION TO MEDICINE, CARDIOVASCULAR PHYSIOLOGY, AND BIOPHYSICS(SANITIZED)
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July 17, 1958
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ARDC TECHNICAL REPORT TR 5872
AVIA DOCUMENT No. 158301
? PHYSICAL PRINCIPLES
OF VECTOR
BALLISTOCARDIOGRAPHIC MEASUREMENT
STAT
So A. Talbot
The Johns Hopkins University
June 1938
AIR RESEARCH AND DEVELOPMENT COMMAND
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STAT
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CONTENTS
Preface: pattern of contract and report.
Chapter 1: Introduction.
Relation to medicine, physiology, biophysics.
Theory, methodology and significance of BCG information,
General aims of this program.
Specific aims and procedures.
History: force ballistocardiography.
displacement ballistocardiography.
II: Concepts  the quantitative basis of ballistocardiography.
Dynamical assumptions.
Other concepts and assumptions from biophysical view.
III: Dynamics of the supportbody system
General analytical approach.
Reduced equations in 6 dimensions.
Relation of translation and rotation through line of
resistance.
Derivation of equations of motion in xzp plane.
Experimental determination of coefficients: inertias.
Determination of spring and damping coefficients.
Solution of equations of motion: relation between planes.
Equations in detail for xzp plane.
Solution by Reeves computer.
Solution by LCR analog.
Critique of solutions.
Chapter IV: Design of practical support systems for recumbent human
subject  detailed analysis of structure.
Analysis of suspension: stiffness and frequency in
6 dimensions.
Damping in 6 dimensions (translatior)
(Rotation) yaw, pitch and roll.
V: Mercury bed dynamics.
VI: Frequency analysis of the BCG.
VII: Conclusions.
Chapter
Chapter
99
103
114 Chapter
127 Chapter
133 Chapter
137 References.
140 Appendices.
?
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Preface:
The purposes of this contract were stated:
(1) To improve the technique of recording and measuring the
internal force patterns of the heart and great vessels.
(2) To establish relationships between ballistocardiographic
data, and the normal and abnormal physiology of the cardiovascular system.
It developed first that the technique of getting an external view of
the internal forces generated by the moving heart and blood, was barred in
every mode of motion by extraneous forces.
These reaction artefacts on the
body from its necessary supports to earth, had been recognized in part and
remedies devised for the headfoot BCG, prior to our contract. We have
deepened and extended the understanding of the nature and coupling of these
extraneous forces, in all six dimensions (3 translation, 3 rotation) of
bodyballistic motion. We also have devised and constructed practical means
INF
everceming them. This hes led to problems of designing structures with
maximum rigidity and minimum weight, requiring the detailed application of
aerodynamic and airframe principles, analysis and methods. It also led to
devising and testing various solutions to the problem of sixdimensional
seism support; taking account also of the special biophysics of the human
physique. The final solutions employed are given in this report. The steps
intermediate thereto were given in progress reports previously submitted.
The study of relationships between BCG data and cardiovascular forces,
was limited to the biophysical aspects of body dynamics. We did not reach
the stage of investigating controlled hemodynamic alterations, as to their
BCG manifestation. But we have investigated certain biological matters
which underly the interpretation of all such physiological experiments.
Such matters are the transmission of cardiovascular force information, from
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its diverse sources to the local transducer signals which define the BCG
in practice. Without understanding this detailed coupling of the observa
tion to the biological activity, we cannot hope to realize detailed bio
logical meaning of BCG observations. These relationships are discussed in
our first two chapteLs.
This writing is therefore as much a Technical Report on Physical Prin
ciples of BCG Measurement, as it is a final report on our contract under
taking. As such, it is phrased to reach the understanding of those whose
use it may best serve, namely the physiologists, biophysicists, engineers
and mathematicians associated with scientifically advancing our understand
ing of hemodynamics in humans.
The research reported herein is the result of extensive teamwork.
Wolf W. vonWittern contributed much to the early conceptual basis and the
study of low frequency suspensions. Sidney Roedel and James Harrison did
the exploratory developmental work. After v.Wittern's departure to Karls
ruhe, Dr. Paul A. Crafton helped with the engineering analysis and on
physical models. The detailed work on structural stress and the generalized
analysis in Chap. III, IV and V, were carried out by Mr. Joseph M. Gwinn III,
assouisted with the G. L. Martin Company. Fabrication of the rugged but
lightweight structure was by them, as was assistance in the analog computa
tions. Thomas Englar and Robert Cramer explored the simpler LCR method of
computing. To Mr. John G. Kummell the author owes much for detailing,
supervision and development of successive structures, and to Mr. Stanley
Przyborowsky for its execution. Joseph H. Condon contributed analytical in
sight to the differential pendulum, negative spring and other problems. Mr.
Peter Hume devised the ingenious modification of Fourier Analysis presented;
which was subsequently developed by Mr. G. N. Webb and Robert N. Glackin.
Finally, the competent handling of this difficult typescript should be credited
to Mrs. R. B. Garrett,
1
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 I 
Physical Principles of Vector Ballistocardiographic Measurement
Introduction:
A report under this title should interest two types of readers:
those dealing with ballistocardiography (BCG) for clinical or physio
logical information, and those concerned with medical methodology or
with the physics of biological systems. Unfortunately these groups
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which describe multidimensional systems in several modes of motion,
are central to engineering and physical training, but are absent from
medical or biological training. Although these principles are widely
applicable in biology, and essential to understanding the methods and
rationale of BCG measurements, a survey of basic mechanics would be
out of place in this report. We choose therefore to aim at the under
standing of engineers and physicists who may be called to work in this
field; and of those biologists and clinicians whose special concern
with ballistocardiography gives them some prior grasp(1)of the mechanics
involved.
We will include a review of these mechanical principles, but for
readers untrained in biology, will first show from the physical point
of view, why ballistocardiography is needed, what it consists of,
justification of the physical assumptions (model) used, and what has
been done so far.
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Chap. 1.
SETTING and PHILOsOPHY:
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1. Perspectives of Ballistocardiography in relation to medicine,
cardiovascular physiology, and biophysics.
Present knowledge of the cardiovascular system rests on a rather
limited range of information; we lack a whole area of data, which prevents
our forming clear concepts of how the heart and vascular system performs
as a whole, or of defects in that performance. This situation may be
described in nonmedical terms.
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cardiography (EKG) and from auscultation. The EKG gives information on
what may be called the ignition, trigger or firing system. There are
various ways in which firing can be faulty: parts of the musculature that
cannot fire (infarcts), incorrect order of firing, irregularity, delayed
timing, weak or overdeveloped firing. Insofar as the magnitude of the
EKG reflects the thickness of the heart muscle, and its extent in the
chest, the electrical information does tell something about the potential
strength of the muscular pump. But in no way does the EKG foretell the
mechanical performance, which only begins after firing: namely, the force,
velocity pattern or volume of the stroke, nor the character of the
systemic flow after ejection. Neither does the EKG give indication of
impending failure, or of how far the mechanism is reaching into its
reserve capacity just to satisfy the ordinary demands of the body.
Though the pump's ignition may be operating, the EKG does not reveal that
the conduits are clogged, or their walls strained to the limit; whether
the gas lines (coronaries) are half closed, the motor idling or straining*
*Small changes in the recovery wave of the "ignition system" between
firing indicates but does not measure strain.
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under load or damage. The EKG measures cardiac status and performance
very inadequately, and vascular not at all; and so must be supplemented
by many other tests.
Failing "ignition" is, of course, important. While some of the
cellular electrophysiology is understood, much of the coordinated
mechanism of the EKG is still controversial. The ambiguity is being at
tacked by several new analytical approaches(2,3,4i5) so that misfiring
sequences should be more accurately described.
Auscultation (stethoscopy) gives mechanical or performance in
formation to a limited and equivocal degree, about certain restricted
parts of the system.
The intake and output performance of all four heart
valves can be characterized as normal, opening imperfectly, or closing
imperfectly. The anatomical character of such defects can be inferred,
but not accurately. Many other sounds (murmurs) are thought related to
degree of defect, but with low reliability. As regards detailed physical
bases of these sounds, it can be said they are completely unknown as yet.
That is, the hydromechanical mechanisms by which turbulent blood produces
murmur or clicks in this kind of an environment, are entirely unexplained:
more is known about automobile valves and knocks by auscultation, than
those of the heart. Although the basic problems are now being attacked,
this source of cardiovascular information has progressed but little in
two centuries; and at best is restricted in its purview, in relation to
both capacity and performance of the normal or the subclinically weakened
cardiovascular astern under load.
Blood pressure data taken by catheter provides most of our current
information on the dynamical or performance characteristics of the heart
and vascular system. Experiments with dogs (and recently with humans) at
rest or exercised, normal and diseased, have given much information of how
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this system which supplies all the others, responds to stress or demand.
However, taking pressures is by no means a simple, nondangerous procedure,
especially near the left heart; nor does the information so gained, really
determine the mechanical action desired.
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in order to characterize performance of a system primarily designed to
provide blood flow. Completely knowing all the pressures in the system
throughout the cardiac cycle, does not suffice to specify the flow. In
principle, if also one knew all imsansstl flow would be known; but we
do not yet know the nature of these impedances, nor can we measure them.
Indeed other than the En22/11t1T1 relation there is no direct measure
of impedance: just as in the case of electric current networks. Like
active electrical networks and systems, these impedances change rapidly,
to provide general and local control of the flow: so that the temporal
behavior of the impedance pattern must be known also,
to understand
the rationale of cardiovascular performance. The problem of externally
estimating cardiac and vascular status and performance, (at rest, under
normal load, or sustained stress, or sickness) therefore reduces to find
ing and developing the interpretation of measurement of flow: the thing
for which the cardiovascular system is designed. Flow is the main study
of ballistocardiography, and ultimately of this research,
Direct measures of flow are just now coming into existence, but
until recently all flowmeters required procedures which either changed
the flow or measured it erroneously. Among these the best were the
orifice(6) and bristle mythods(7) a Preliminary models exist of flowmeters
which can be sewn into animals, to measure flow correctly and without
changing what they measure. But these (magnetic and supersonic) methods
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cannot be used to study normal humans or patients without surgery.
A catheter method of promise" has appeared.
This may serve well
for research in the hands of experts, and in a few locations, but is
unsafe for routine or group studies
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cardiovascular conditions; and
too stringent for many subjects. The catheter method itself can psycho
somatically and by stimulation of the arterial inner wall, seriously
alter the cardiovascular function. While this will not prevent eventually
accumulating important and necessary data, it is a barrier
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and general use, either as a scientific or a practical test.
In sum, because of these defects of observation by EKG, stetho
scope and pulse pressure, we have now and in prospect no better methods
of assessing the functional status of cardiac and vascular performance
as to blood flow, than those of plethysmograph y and ballistics.
Plethysmography (measurement of blood flow from swelling) gives
information more local than general; i.e. best when related to limbs.
It may give valuable information on the state
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function of the peripheral
vascular system, if one can distinguish between deep and superficial flow
(which have separate controls). The measurement of central flow by
(43)
electrical "impedance plethysmography" is still too unspecific to be
successful.
The ballistic method of evaluating cardiovascular function to
which this report refers, has moved ahead considerably in the last fifteen
years.
The rather superficial method of correlations (which in early
stages of medical science is alone available), has sustained medical
interest by a few outstanding successes and a general positive relation
ship to heart and artery disease, and to the athletic heart. But through
out medical science, such statistics eventually give place to the study of
functional relations in both the physiological and mathematical sense.
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So in ballistocardiography (BCG) (the study of motions of the body during
the cardiac cycle), the correlation of BCG waveforms with normality vs.
disease, is aow declining in importance. Emphasis has shifted to more
scientific questions. To what degree do individual cardiac and vascular
mechanisms, account for each detail of the observed bodymotion? How do
the observed displacement, velocity, acceleration of the body relate to
those of the blood at rest and in specified stress? How do known alter
ations of the cardiac action, and of the (complex) impedance of the vas
cular network with age or drugs, alter the blood flow in detail?
In
particular, to what extent are these measurable externally,by measuring
the vector components of the body motions? Clearly, to answer such
questions is to acquire important practical information for medicine
and for physical fitness, as well as basic undei:standing for physiology
and biophysics.
Modern methods of recording these body motions reveal such rich
and characteristic detail, that such crude categories as "normal vs.
abnormal" become too complex to have scientific or medical meaning.
But scientists close to BCG development, find physiological relationships
emerging v and measurement problems yielding steadily and rationally; so
that this approach to external and objective specification of cardiovascular
functions, offers increasing encouragement.
In interpreting the BCG, it turns out that we must give
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tion to individual differences, and to distinguish vascular from cardiac
dynamics. The statistical approach common in medicine, of correlating
recorded details with normality and disease, worked poorly enough with the
EKG, but is worse with the BCG. Instead of one, there are four major
cycles of activity to interpret, all varying with the individual. Without
the functional (mechanistic) approach to supplement the usual normalabnormal
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rubrics, the very richness of physiological information in the BCG
defeats statistical methods. This has discouraged the correlational
school of physicians. To compound the problem, vascular hemodynamics
is still an infant science, and cardiac dynamics as well, lacking
measurements of flew,
is gt411 but little understood. So in this mid
epoch, the demand for practical interpretation of the BCG finds scant
satisfaction: much as was true with virus or allergies before their
mechanisms became known.
Indeed the present situation is selfdefeating, (1) in that ex
ternal measurements of cardiovascular dynamics by this method are so
poor technically, that only statistics instead of mechanisms are applicable.
(2) Internal measurements on normal hemodynamics are not yet possible
technically. (3) The low correlation of observations with individual
cardiovascular status, caused by inhomogeneity of subjects,by artefact
and by irrelevance of component information (arising from ignorance of
mechanisms), has reduced the interest of the medical profession in this
field. In this situation, we here attack the sector of reducing artefact
and irrelevance in the record itself. The problem of cardiovascular
dynamic mechanisms, is being attacked also by our laboratory.
(8)
To a considerable accuracy' the flow of blood in the body is
actually measured by the body motion it produces. But the principles of
measuring and interpreting the dynamic aspects of blood flow sensed
through the body motion, are still being developed. At any instant
the observations include multiple actions, and a vector summation of
flow information in several directions at once. Were it not that the
organization of this flow is quite different in the three major axes of
the body, one might despair of separating the variables. Also, cardiovas
cular events are separated not only in direction but also in time, which
encourages the hope that reliable and informative differential relations
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can be found, from adequate multidimensional data showing phase relations.
The principles and methods of getting these data are a main objective of
this report.
In interpreting bodyballistic measurements, one encounters strong
individual differences among normals, such that the use of BCG wave ampli
tude ratios as in the rKa is inadequate as a criterion. The whole normal
vs. abnormal view for the population as a whole, breaks down except in ex
treme or advanced cases. In this situation, an obvious remedy is to select
subgroups identified by other criteria,
and to use dynamic as well as
static descriptions: such groups are sex, age, relative size of heart,
elastic status of vessels (phystoloeical age), constitutional type. All
these may be used as starting areas for systematic work on dynamic (stress
strain) analysis, of the successive complexes in the BCG cycle:
A second remedy is to separate the cardiac from the vascular factors
in analysing cardiovascular function. This procedure is harder, involving
as it does the selective use of shortacting drugs (both natural and synthetic)
to create standard changes, and a clearer understanding of the typical ways
in which the heart normally responds to standard changes in its hydro
mechanical load. It is encouraging to realize, however, that we now have an
observing method which can ask and answer such questions on physiology and
biophysics.
Thirdly, in cardiovascular mechanics in contrast to the present EKG,
there are three kinds of information, whose intercorrelation may be used.
The moment, momentum and force of the heart and blood (displacement, velocity,
acceleration of the body),  though related as derivatives and by frequency
content  give quite diverse information about cardiovascular properties
and behavior. This focuses the analysis on the physiology of the individual,
by introducing mechanical principles which hold for all individuals. The
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validity of this attack has a of course, which has yet to be
determined.
In summarizing our perspective, we may say that contrasted with the
information content of auscultation and the electrocardiogram, the
ballistocardiogram intrinsically carries much more discriminable and
significant data about the functional status and performance of the
cardiovascular system. Physiologically and medically, the problems are
to obtain reproducible data free of artefact and needless irrelevance;
to establish bases of classifying and analysing these data; to relate
them to other facts of physiology and pathology in a rational way.
This whole program is progressing, but is still in an early stage of
understanding.
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2. Theory, methodology and the significance of BCG information.
The union of physics with biological science emphasizes that the way
of observing things often decides the value of the observation. With the
EKG the use of limbleads for decades prevented the differential diagnosis
of infarction. Reliance on the ear with the stethoscope, for generations
prevented any detailed objective analysis of heart sounds. In the case of
cardiovascular performance, restriction to bloodpressure data has misled
and confused our understanding; the naive methods of recording the BCG now
in clinical use, have obscured and discredited the information contained
therein.
In a new field, one should design methods of observation which minimize
irrelevancies and artefacts. This implies foreknowledge and criteria of
what is relevant and true: an absurd position. But the simpler problem of
electrocardiography (another field of vector dynamics) showed that advances
in 1119.21x, have each time brought advances in method. The planar analysis
of EKG limbleads followed the theories of Waller and Einthoven. The
spherical approximation and potential theory introduced by Wilson, at once
stimulated the clinical study of the newer chest and dorsal leads. Burger's
theory of leadspace, extended by Frank, led in turn to new methods which
reduce the artefact arising from heartposition and bodyconfiguration, and
so let us view the heartvector more directly.
Similarly in studying cardiovascular dynamics by the BCG, the theoreti
cal analysis of body motions by Curtis and Nickerson(10)(perceived by Gordon
in 1877) was further explored and verified experimentally for acceleration
by v.Wittern(12)and for displacement by Burger(9).
Without expecting that
treating the body as a rigidmass in this way will ultimately provide
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basic understanding of the ballistic effects, this simple concept of
the BCG* stands alongside Wilson's potential theory, in cutting away
the ad hoc clinical methods which went before. The theoretical advances
by Burger and v.Wittern generated new methods and revealed details which
at once superseded the obscure criteria of normal vs. abnormal that clogged
the clinical literature. So we may legitimately expect that this interplay
of theory and method, which we extend in this report from the headfoot to
the other vector components of the
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recording and further eliminate current obscurities. However, we are not
entitled to expect that for threedimensional body dynamics, the basic
theory will be simple: any more than was Wilson's for the threedimensional
EKG. Luckily it happens that public interest in the dynamics of missies
and planes, at least makes the technical vocabulary of our analysis familiar
to many.
Certainly the theory of rigid (as opposed to elastic) bodies in vibra
tion, is only a first approximation; so that as new aspects are revealed,
the BCG recording methods which result from these rigidbody concepts
likewise will become outmoded. (1) The modern methods'and lines of progress,
result from asking certain physical questions about bloodflow in the body;
but the answers given by these methods are colored by the rigidbody theory
of the measurement. (2) Because of the questions asked, information arising
from other forces (tissue vibration, action of joints, etc.) may rightly be
regarded as irrelevant, artefactual, and to be avoided. (3) Scientific
methodology is concerned with separating variables? to make information
from various aspects as independent (orthogonal) as possible. Consequently
*The name "Ballistocardiogram' given in 1940 to the record of wholebody
motions under cardiovascular forces, has since proved almost a misnomer.
Very little of the information recorded is attributable uniquely to the
neaLL.,in the sense th2t the EKG is.
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this whole physical approach differs basically from the biological approach.
The latter presumes that the observations relate to the biological events,
in an intrinsic way, which is not really determined either by the observer's
method or his conceptual approach. However, we use here the more physical
strategy which emphasizes dependence of observations on method, to attack
the problem of determining the cardiovascular dynamics from body motion.
Furthermore, this information is considered significant for our purposes,
if it brings out various independent aspects of the displacement,
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acceleration of the blood flow, with a minimum of background or distortion
from other factors. The ultimate test of this "significance" is indeed a
crosscorrelation, but not with other information of the same kind as in
statistics, nor with disease symptoms or pathology; rather with direct in
ternal observation of blood flow in relation to controlled variables of
cardiac function.
Consequently, BCG methodology itself  our concern at the
moment  cannot even be developed without intimate knowledge of the physics
of bloodflow, seen in the detailed anatomical context of the human body, in
a way specifically related to the particular BCG methods proposed. However,
as a problem in biophysics, at each stage we design not for the actual com
plexities of the body, but for our abstraction: a reduced or simplified model
which includes only as much at this stage as we choose to. The real art is
to include enough features in the abstraction, to make the results useful as
well as interesting.
In summary, we may attribute the slow growth of understanding in ballisto
cardiography in the last fifteen years, to lack of an adequate theory, as with
the EKG. The rigidbody theory of the body, plus conservation of momentum
(as developed by Curtis, Burger and v.Wittern) admittedly does not conform to
the bodily detail, but does separate the variables for clarity by introducing
a model. When finally this must be related to details of blood flow, the
model probably must be modified.
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B. GENERAL AIMS OP THIS THEORETICAL AND EXPERIMENTAL PROGRAM.
The present program attempts (a) to clarify the physical problems aris
ing in vector ballistocardiography, and (b) to spotlight the kinds of in
formation available. With this information clearly in view, we attack (c) the
practical problem of inventing devices and procedures to observe it correctly.
We concentrate on the pla.s_ital_EET.plla.s_ of threedimensional mechanics as
the most important at this stage, for several reasons: (a) the rotational
motion clearly affects the translational motion used to define cardiovascular
force. This coupling must be reduced, before one can deal intelligently with
the translational vector components of the cardiac hemodynamics. (b) In the
anteriorposterior motion and in pitchrotation, one should find unique views
of the cardiac output (stroke volume); which in other directions of motion
(e.g. headfoot) may be mixed with other information (e.g. arterial elasticity).
Due to this mixing, the vector summation may not always be desirable. (c) The
structural difference of the body along its three axes, raises new problems of
passive bodymechanics. (d) The ultimate aim (stresstesting the cardiac per
formance) for hemodynamic reasons requires a sitting posture, which in turn
requires solving similar problems of experimental design (e.g. a vertical
suspension of "zero" frequency) mentioned in (b) above.
The physical information we seek is of three kinds. (a) An effective
record of the vector components of body motion, separated in such a way
(orthogonal and informationally homogeneous) that vector composition becomes
permissible*. (b) A study of this vector motion in its various dynamical aspects
* Some BCG vector composition which is not permissiblein this sense, has
$
been done for several years.(201 41) 22, 24, 15 55).
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(displacement of c.g.1 momentum, and force); these have separate significance
both physically and physiologically. (c) A study of each of the six motional
components (3 translation, 3 rotation) with relation to its distinct physio
logical meaning; eliminating those data which are least useful.
Besides clarifying and formulating the detailed physical observables to
be recorded and the parameters involved, there is finally an equally demanding
practical problem of creating a method to get this information. That is, even with
improved understanding of what
wain.
"
'Velma
from ballistocardiography in
the headfoot axis, methods must be worked out in practice, de novo, for the
five other axes of motion. The first two years of this "hardware" part.of
the task, were spent in exploring various practical means that suggested them
selves, for (1) observing translation, free of rotational component (2) fabri
cating bodysuspensions which did in fact fulfill dynamical requirements for
frequency and stiffness not previously achieved in applied mechanisms.
Nullfrequency suspensions in three rotational dimensions were being worked
on (secretly), under the name of inertial guidance systems. Our problem
however, though dynamically similar, is in six dimensions, requiring isolation
from the surround in translation as well as in rotation. This resembles in
essence designing an artificial cloud or flying carpet, strong enough for the
subject to lie or sit upon.
A qualitative review of trial solutions rejected and accepted has been
given in previous progress reports*,
will be presented in this final report
The quantitative results found practical
Accompanying these trial solutions,
of course, is the mass of detailed design, calculations, drawings, mechanical
Investigation of the Unloaded Internal Ballistocardiogram: Its iophysica
and Physiological Relation to Cardiac Performance. Contract #AF18(600)11n7
9/22/56.
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work and and outbugging which go with each proposition tested. These cannot
(a)
be detailed here. However, we will present/the precise dynamical basis and
values which emerged as acceptable for this problem, and their limitations in
general and in particular. We will present (b) experimental procedures which
can be used to put this theory in action for: (1) determination of parameters,
(2) interaction between degrees of freedom, (3) spectral range of fidelity and
(4) evidence of practical validity of this approach, from records of humans.
C. SPECIFIC AIMS AND PROCEDURES,
To design dynamical measurements on a biological system one must start
with designing specific hardware and experimental procedures, which intrinsical
ly determine what is observed. If one chooses to observe motions of the whole
body, one must decide upon what set of concepts are to govern the design details.
These concepts have already been subjected to discussion by the author(1) and
others(9) and will be further criticized here. Furthermore, these basic con
cepts once adopted imply a physical model; which then invokes the corresponding
physical theory and laws. Most of this theory is quite old, but has been re
developed in convenient form, for the motion of missiles and stresstesting
aircraft. It is the theory of two rigid bodies, coupled in vector rotation and
translation in free space. The range of validity of this theory for our purpose
will be discussed.
After arguing the conceptual basis, we will apply rigid body mechanics
to the special interaxial couplings and body dimensional peculiarities of
ballistocardiography, which differ somewhat for the various kinds of body
supports used. The dynamical and physiological interpretation of the records
in turn depend on these supports. A theoretical solution of the differential
equations in certain planes (by analog computer) will be shown vs. frequenu
the boundary conditions being certain intrinsic frequencies seen in BCG records.
1
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Our experience from headfoot recording is transferred to the five other modes
of motion, to get orderofmagnitude values for the individual's bodyparameters
in these other axes.
We aim to show a practical way how to record the translational motions in
dependently of the rotational, especially difficult as regards body roll, This
way minimizes mechanical coupling between these basically independent aspects,
without requiring routine use of complex computing equipment to separate the
variables. This aim is specially important because the data of multidimensional
or vector recording can easily get out of hand. Separating the rotation vector
from the translation spacevector, enables us (for the first time) to display
correctly the phase relations of the cardiovascular events in each plane of
projection. While we have not yet carried this aim so far as to correct thor
oughly for orthogonality, standardization and directional transmission of the
body (as has recently been done for the EKG) this elimination of (rottrans)
interaction, does take a step nearer to sensing the "intrinsic" or essential
geometry of cardiovascular dynamics. Since observation of the cardiovascular
system through bodydynamics, is only one of several important aspects, it is
important to reduce it to essentials. The "spacevector" presentation probably
better uses the synthetic powers of the observer's mind, and exhibits certain
details in a form more striking than does the older BCG record against time;
but both enhance understanding.
In summary, for this aspect of the problem of cardiovascular dynamics,
we aim
1. To establish a physical basis for data in several degrees of
freedom, whose assumptions, range and degree of validity are clear.
2. To design rationally and to execute structures which produce
such data, in a way which separates the variables.
3. To discuss the limitations and interpretation of these data, in
each degree of freedom:
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a) Graphically, in its aspects of frequency spectrum, the time
axis, and phased vectors (Lissajous complexes) both in plane projection
on the body axes and in space view (intrinsic axes).
b) Physically, as to the meaning of the displacement moments,
momentum and force, derivable from motions of the body.
Our procedures have been:
a) To improve the conceptual foundation of ballistocardiography as a
vector problem.
b) To formulate and carry out the sixdimensional analysis entailed
by the conceptual basis chosen.
c) To devise and construct mechanical means of taking the measurements
required.
d) To display in several forms, criticize and interpret these measure
ments as meaningful for the cardiovascular system.
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D. HISTORY: Prior development of the theory and practice of the ballistic
measurement of cardiovascular dynamics.
Understanding of this problem has progressed in two parallel streams
(11)(12)
or dynamic views: the force aspect (expositions of Starr and v.Wittern)
and the displacement aspect (Burger's exposition)(13)(9). Lying between is
the momentum aspect (whose integral is the displacement, and whose derivative
the force). This third aspect is best shown by the data of Nickerson(14)
His method, by virtue of a suspension having broad mechanical resonance,
records mainly t,,,Aly_vpinr4+v
7 7 
/4?
proportional to cardiovascular MOMentUMkJ)
and so exhibits the best correlation with cardiac output. However, the
records of this method, like those of Starr and Dock, are shifted in phase,
distorted and confused by resonance of the bodysupport coupling, which un
Airranm4,4
recognizedly intermixed the 01,11CL t.tylawm
aCnar
ts. W. v:Wittern resolved
this confusion about force recording, by taking the acceleration record
from pendular suspensions like Burger's [similar to those of Gordon (1877)
and Henderson (1905)]. As a result, we now have available bodies of data
based on two complementary views of the bloodbody dynamics: the displacement(16)
and the force in the headfoot direction. These two views conflict in no way,
but supplement each other's information, and contribute to each other's under
standing.
1. Force ballistocardiography.
Records of headfoot bodyacceleration, insofar as the body accelerates
as a unit, measure the forces exerted thereon by the cardiovascular system.
For frequencies up to 8 cis or so, this unitary motion of the body in the headfoot
(12,1,17)
direction has been corroborated roughly by several workers  using an
indirect method (external shakers).
Firstly, the proof is rough, because it refers only to the impedance of
the system as seen from the shaking position. Local recordings from the body
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itself resemble each other somewhat in displacement(12)up to 8 c/s, but
poorly in acceleration. This results from local resonances (head, shoul
ders, etc.) and local forces (apex beat).
Secondly, the shakertable evidence of body unity is indirect. Shaking
the body from without is by no means comparable to shaking it from within as
is done by the cardiovascular system. Shaking by dorsal contact (v.Witternts
method) actuates the coupling springs in a particular stiffness configuration
and drives them inphase Thus the constituent masses of the body are made
to vibrate in a particular amplitude distribution at each frequency. This is
not the same amplitude distribution with which these submasses vibrate,. when
driven by the feet at various footboard pressures (Cunningham's method), or
when driven by the heart and aorta via the skeleton. These differences of
effective massdistribution vs. frequency, are greatest in the vicinity of
the natural frequency of the supine body (in response to a step displacement).
As a result, the shape of the BCG record for frequency components above 4 c/s,
depends appreciably on the use of footboards (Scarborough), anklesupports
(18) (12)
(Smith) 1 shoulder braces, etc.'
In sum, we can readily measure acceleration of support; but to multiply
this by a unitary bodymass to arrive at "force" on body, we assume an ap
proximation which shows increasing error above about 8 c/s (Cunningham).
Consequently we cannot regard wave details of the acceleration BCG in the
1230 c/s band, as representing "forces" which are on the same scale, or in
phase, with the slower force components. It is therefore better to scale
such records frankly in acceleration units rather than force units(19); and
not refer in any quantitative sense to the "force BCG". Qualitatively, how
ever, the "force" interpretation is legitimate and helpful for relating the
observations to cardiovascular dynamics, in contrast to the "momentum" and
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"displacement" aspects which also operate on the same assumptions of body
unity.
This report, concerned with improving the dynamical foundation of these
cardiovascular measurements, must omit the growing body of literature on
physiological interpretation, as to details of the "force" record and the
evidence for it.
We turn now to review the work done on recognition of the vector nature
,r
OA Liitau forces. Lateral components of the BCG arise from the oblique pos
ture of the heart in ejection (and bilateral nonsymmetry of vessels as well)
which varies with individuals, body habitus, respiratory phase, and age.
It is found that persons having quite disorderly headfoot records, may have
quite regular lateral components, which increase systematically as the heart
tilts with age.(20)
In most cases the lateral (RL) record looks more
oscillatory, especially in diastole.
As with the headfoot "force" record,
we must first scrutinize the method and its assumptions, before we accept
such lateral records as valid or try to interpret them physiologically.
(1) In the lateral mode the bodyunity assumption rests on much shakier
uounds. Transverse rigidity of the body is far less than headfoot; in
fact, most of what transverse rigidity the supine body has, comes from the
support or bed rather than the skeleton. Because of the smaller mass
(thoracic cage) engaged in lateral motion, the smaller lateral forces give
R,L. accelerations large enough to compare with the headfoot. This scale
factor needed for the vector sum has not been recognized in the literature.
(2) The lateral record is more likely to contain "local BCG" aspects
than the headfoot, because "averaging t'y support" transverse to the spine
is poorer. The role of a platform in summing or averaging forces from the
body is important in the headfoot direction but more so laterally. The HP
forcepattern entering the platform varies with the subject's pressure on
the footboard.(13)
In this way, the contribution via spine + legs + pedal
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compression becomes more or less strong, compared with the contribution
via shearofdorsaltissues. The pattern of HF motion varies accordingly,
mainly in smaller details. This factor has not been studied systematically,
although v.Wittern has attempted to accentuate the HF forces of thoracic
origin by coupling a footstrap to the bed at hiplevel.
(3) As to method, the lateral motion is currently recorded by con
straining rotation (yaw) of the bodysupport, at the foot end. Thus, by
moving the axis of rotation (yaw) from the body's e.g. to beyond the feet,
the inertia seen by the driving force; innrp2sed, and the motion is re
duced. But it is simplified also, in that there is no phasereversal of
the lateral motion at the center of rotation. However, this yaw artifact
introduces another scalefactor in the lateral (RL) record; partly deter
minable, but varying by twentyfive percent between individuals.
In the lateral (RL) record it is plain that the motion observed depends
strongly on where the body transmits force to the platform. The spine acts
as a flexible shockabsorber, for lateral internal impacts originating be
tween shoulders and hips. Since the latter points lie on opposite sides
of the c.g. of support, this factor may even reverse the phase of the out
put (for higher frequencies), depending simply on where the dorsal contacts
are, which contribute to the lateral summation. In other words, there is
local relative motion between the body and the support, which only forms
the average observed. This averaging is inevitable to some degree, since
even in headfoot the right and left sides of the body do not displace
equally to the apex beat and the aorta is not strictly axial. Consequently,
lateral chocking of the body must in some way be standardized(21). Clearly
also, although the hips do contribute part of the HF drive to the platform,
they do not assist the RL drive and so become mainly an inertial load in the
cntoro
lateral
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As a result of these three factors (flexure, rotation, local effects)
the standardization of a method for lateral recording becomes especially
important, if one is to speak of "the" lateral component of "the" cardio
vascular force cycle. Various workers have used several quite different
methods. Braunstein(22) used a platform of considerable inertia mounted on
stiff springs, which (like the Starr platform in the HP mode) follows the
body in RL motion only below about 3 cis, above which it gets out of phase
(both laterally and in yaw), at frequencies low in the BCG spectrum. Scar
borough and Talbot( 21)
used a lighter platform mounted transversely to a
. Starr bed (stiff to earth), with hip and shoulder boards to pick up the.
.LIIC
p Dock(2 3,24)
L.uuLauit a.44.L.ivA.m moves laterally
lateral bodymotion.
against stiff metal springs, with respect to which (again like the Starr
bed) the body oscillates on its own tissuesprings, both in translation and
roll (independently). Both these oscillations appear in the record, often
as dominating artefacts, v.Wittern(25) and Honig(20) used a quite light
pendular platform pivoted at feet. Since this is constrained to move in a
horizontal plane, the body rolls somewhat: which shows in the record. The
mercury bed (Talbot and Deuchar)(26) is mechanically similar and produces
the same artefact.
Indeed it can be said that no lateral BCG record has been published
so far, which is acceptable physically. The errors can probably not be
characterized in detail until good records have been taken for comparison,
but there are at least two and sometimes three basic artefacts in current
(1)
records, which prevent quantitative interpretation . With the highfrequency
supports of Braunstein, S.carborough and Talbot, and Dock, there is firstly, a lateral
resonance in translation, corresponding to the headfoot resonance at 45 c/s
seen in the Dock and Starr headfoot BCG(2)
With the ultralow frequency
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supports of v.Wittern, Honig, Talbot and Deuchar, Scarborough and Soper(27) ,
the translational ..oftness of the suspension prevents such resonance in
lateral translation. Secondly, because the body oscillates in roll (depending
on the aboveaxis impact of the heart and roll constraint, e.g. flatness
of back) a resonance in rotation also emerges.(3)
Thirdly, there is the
coupling in recording the transverse motion of platform, between roll and
lateral motion:
that is, the "true rollBCG" (as distinguished from roll
resonance) exerts its tangential forces on the bed. These forces combine
with the true lateral ones, to give a mixed or composite RL record, contain
ing both translation and rotation (with the latter not proportionate to
former) and two characteristic resonance frequencies.
Both these latter errors in the BCG have been demonstrated by Scar
(28)
borough ', by taking RI, records from n subject first prone, then supine.
If roll artefact is absent, the record appears inverted. If excessive,
the records look alike; all gradations between occur.
It becomes clear that any body support which fails to follow the body
freely in "roll" (i.e., is aperiodic in roll) provides the conditions for
rollresonance, and also for a mixed rolllateral. BCG., Any relative motion
between body and support then creates (torsional) restoring forces about
the body's long axis (0, when the body rolls under the transverse offaxis
impacts of the heart. This error occurs with all BCor's currently in use:
they are all maintained horizontal, i.e., stiff about this p axis. Secondly,
even if the platform were mounted in compliant gimbals about this axis, one
must still devise a way to sense the lateral BCG alone, sans roll BCG.
It results that a true "frontalplane BCG" has not yet been published,
either in components against time, or in vector form:. although many authors
(cited above) have thought to do so. Research programs even exist here and
abroad specifically for "vector" BCG studies, whose equipment is intrinsi
cally incapable of recording true lateral components (not to mention the
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vertical). The nature of this error is clarified by considering how radically
the shape and timiTof the HF record changed when its resonances were
removed around 1955. A main activity of this contract has been to under
stand and remedy these defects of the frontalplane BCG.
Anteriorposterior (AP) or vertical component of the "force" BCG.
It is possible to mount a platform on a set of springs which have high and
equal stiffness in all three axes(29)(24). Here again the AP record is
found to be distorted by translational resonance, as well as admixture
of the resonance in roll. But although the RI, (x) and HP (y) motions are
relatively easy to make so compliant as to be aperiodic(26), away to do
this has not heretofore been found simultaneously for the vertical (z)
motion.
The literature of mechanics actually provides no ready solutions
for this practical problem; althqugh suspensions of very low vertical
frequency have been achieved by air springs for busses; and by coil springs and
graVity for seismography(30)and for calibrating lowfrequency accelerometers.
Previous progress reports on this contract, have outlined our experience in
using both these approaches. Later in this report we will describe a suc
cessful quantitative solution applicable for ballistocardiography by use
of nonlinear negative springs.
There exists in principle(31), a different solution to the problem
of recording the xyz components of body motion, unmixed with "roll" in
formation. This is to construct a suspension which provides a system of
"torque rods" (as commonly used in automobiles) to prevent roll and pitch
of the body. If the body is fastened down to such a table so stiffly that
roll vibrations can occur only at very high frequency (15 to 20 c/s)1 with
respect to the table, and the torque rods are coupled rigidly enough to
prevent rotation of the table to the same degree, then the frequencies
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unn
below this in the records of xyz' and xyz, will have no rotational artefact.
Such a table has been designed and constructed in this project. It results
that to combine the practical requirements of extreme compliance in 3 trans
lations, with extreme stiffness in 3 rotations, with errors in the .0001"
range, demands an extraordinary amount of fine machine work. We have con
cluded that such a suspension is impractical to advocate for general use,
and that tying down the subject to this degree inflicts excessive discom
fort. This last has been shown to alter the HCG(32)
FurthPrmore the use
of tight body constraint to earth in roll produces an underdamped r?
snnnnrp
condition which is still well within the upper BCG range of frequencies.
In principle, again, one can construct a BCG to measure rotational
components directly as?Ernsthausen has done( 33). Here the body rests on
a tubular bar, whose rotation may be defined and measured about any single
axis (vertical, horizontal or oblique) passing throup it. If then one
records the rotations and determines the three inertias, one can in prin
ciple compute the forces (as well as torques) producing them. At the time
this work was published, attention had not been focussed on the emergence
in practice of a resonance between body and support, in the case that the
latter is stiffly coupled to earth. The subsequent demonstration of this
factor by Burger(13) and by v.Wittern(12), proved at once the practical
impossibility of correctly recording cardiovascular forces by this "tor
sional BCG" approach. The reasons in rotation are precisely the same,
as account for the large errors in phase and amplitude of the Starr and
Dock methods in translation. This torsional BCG is also dynamically im
pure_ or "mixed", in the sense of the following paragraph.
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Theory of "force BCG".
Theoretical cevelopment in this field has proceeded along two lines.
The first explored the consequences of considering the body a rigid mass
for motion in the HF (y) axis. The analytical solution of this problem
by the author(1), its graphical interpretation and practical limitations,
were almost simultaneously confirmed by Burger, et al.(9). The author's
further exposition(1) of the "mixed" dynamical content of existing BCG
(9)
"fnrro"
ror.nrric
?
urnc mlcn aA 11v, Dt74.rvess.0
? LUC oi.aLL
(and Dock) reco:ds can legitimately be interpreted as "force" only below
the body resonance frequency. At and around this resonance (45 c/s) the
(jerk)
record becomes the'aerivative of force' and the finer details ( > 6 c/s)
theusecond derivative of force".
Secondly, the basis of this whole analysis was challenged by Cunning
(17)
ham (Griswold and Cunningham) as a result of their experiments. This ana
lytical approach showed that the motion of the body when shaken via the
feet, followed a law (in amplitude and phase) resembling above 10 c/s that
of acoustic vibrations propagated in an elastic medium. Below 10 c/s a
number of separate dampedresonators showed; suggesting that at quite low
frequency the body is uncoupled from its unity into separate oscillators,
and above this to an acoustic continuum:
This serious challenge to any
use at all of the unified bodymass concept, requires rebuttal. Since
these newer data and concepts conflict with the experimental results of
Wittern and of Talbot and Harrison, they must be explained by the differ
ence in technique, and some reason suggested for a choice between the two
views.
We may suggest that the disagreement results from how the body is
driven: In their case, by an external sinusoidal force acting through a
series of leg joints designed for compression only. Such a force must
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pass through tendons; which having the properties of collagen and elastin
known to be viscoelastic(34) can be expected to show phaselag increasing
with frequency. However, when the actual ballistic drive is from inside
the body, we suggest that the strong longitudinal tethering of arteries,
and of the diaphragmatic foramina, respond better dynamically than do
joints seen in pressurerelease, transmitting via tendons in parallel.
Further work is needed; clearly, on the colligative properties of
the body in small scale vibrations. However, this should be done with due
regard to prestressing:
which is the normal relation of joints, and with
us has proved a basic consideration in transferring body mn+4n.,c
port, for dynamic measurement.
Another analysis treating the body as a nonunitary structure, has
been offered by 13urger(35). This differs less radically from our first
order approximation of body unity than Cunningham's. He evaluated
v.Wittern's(12) suggestion that heart mass be regarded separately as
mechanical element. The calculations show that the mass of the heart
made less difference, than the spring and damper couplings from heart to
body. That is, if underdamped, a strongly selective response at the
frequency of the heart as a passive oscillator, should result. Since no
such transient occurs, Burger concluded that the heart's coupling must be
overdamped; which seems reasonable dynamically in the closed and inflated
chest, and corresponds with direct xray observation, Burger's argument
applies only insofar as the heart's suspension participates in the pattern
of forces acting on the body. However, this participation is limited to
the events at isometric contraction and early ejection, which cover only a
interval of
BC0. cycle; beyond this time the heart inertial reaction no
small
longer enters the driving force. Subsequently such forces come from changes
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of blood momentum other than by the heart; so that Burger':, analysis
refers only to features of the record in an initial ten percent of the
cycle. That is to say, in his basic equation,
mh xh
It
h(xh s
x) = m x = F *
SC
h = heart
S = rest of subject
c = center of gravity
of rest of body
the driving force on the right can indeed be identified with heart force.
But this equation expresses the total driving force F* only during this
short period. Immediately thereafter, one should insert in the equation
also, the inertiaelastic reaction [due to "waveguide" impedance (Van der
Tweel)(36) ] of the aorta_ Snlif there occurs at all a heartresonance as
predicted by the values used in Burger's calculation, it should show up
as differences in the HI segment of the BCG only. For this reason, the
name BCG is not wellchosen.
The nature of the bodydriving force was considered qualitatively
by Honig(37). He did not attempt to subdivide the body masses, nor to
propose any dynamic model which is exact or simple enough to formulate
mathematically. Honig focussed on the later events in systole and early
diastole, rather than the motion of the heart itself, confined to early
systole. His model emphasizes first the role of ejection in stretching
arterial walls, whereby the cardiac work is progressively stored and then
discharged as a wave of kinetic energy. While it is true that a relation
exists between heart work and body motion, it is not the simple one Honig
assumed, i.e., that energy of body motion is proportional to combined
potential and kinetic energy of blood ejection. What is conserved between
* The notation of this paper, is confusing. Contrary to this notation,
the motion of the whole center of gravity (Mc)=0 at all times. The ex
pression NM refers to the driving force on the body or moving cardio
vascular mases.
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the two is momentum, rather than energy; so that the energy of body motion
is a minute fraction of the kinetic energy of the blood. This fraction
varies continuously, according to the mass of blood instantaneously in motion.
The true relation involves the time derivative of the energy, in Lagrange's
equation, of which Newton's law of conserved momentum is a. special case. It
results that Honig's conclusions as to the irrelevancy of the BOG to cardiac
strength, are based on erroneous physics.
Secondly, emphasis ;3 laid on reflection at vessel junctions, as gener
ating counter forces which give
resonance used by physiologists
nrIrl
15?./0 WP.vCoo
4 ?
L.
Tk4c
Vs1n:4.1 4111:1
eXpa.a.1.11 ie14%,
is based on assumptions of
pulse wave. This interpretation,
(38)
while still under discussion, loses force on quantitative examination, and as
other explanations of the pulseform come to light. The BCG record is indeed
sensitive to limb position and limblessness, which suggests a reflection com
ponent under these special circumstances. However, the circulatory dynamics
also changes, so that new changes of momentum enter even when arms or legs are
bent.
Several physiologists, like Honig, are interested in establishing the
relation between changes in cardiovascular mechanisms and changes in the 11CG
record. In the author's estimation this can only be done by further hemo
dynaMic understanding, based on actual measurement of local blood flow veloci
ties in chronic animals. The attempts to account for the force record dn the
basis of simple localized action, like cardiac ejection pattern (Starr), heart
energetics and vascular resonance (Honig), or mass displacements of blood,
do not take account of the summations of local dynamic action involved. The
? concept of a series of discreteeventg'accounting for the separate waves
(Starr, v.Wittern) is probably a good first approximation, but the theory of
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a hydrodynamic network (Noordergraaf) better admits the pertinent variables.
Pending fuller development of this theory, the interpretation of the body
ballistic record, will rest on the deceptive ground of correlation.
At pres
ent, such correlation in one dimension is possible when the heart is idling,
as on a bed; but the other components of the force vector, and the action of
the heart under normal and heavy loads, which is of greatest interest, cannot
be studied without further advances in methodology such as those in this
reporI.
One may summarize the progress in understanding the "force" aspects of
tile external ballistics of blood flow, as follows:
(1) By making the rude approximation that the body moves as a whole, in
response to the sequence of composite forces, one can more clearly spotlight
certain other large factors in the force picture. These are: (a) the ex
traneous forces arising from springcoupling to earth in translations and
rotations; and (b) the role of the support, both as to its inertia, its
coupling stiffness and damping, and as to its averaging action in defining what
I& being measured. Better understanding and control of these factors, in turn
helps one take better practical account of departures from the unitarybody
as
(2) Coupling the body ?to earth via stiff supports has been shown to
produce several effects: (a) strong resonant oscillation of the body on its
own (dorsal) tissues, which distorted and confused the record; (b) mechanical
differentiation and integration of the record (on either side of the resonance
frequency), which makes erroneous any interpretation of the record as one of
"force"; (c) external reactions from the ground, which enter the body via
various contacts, act on the body in ways dynamically different from internal
forces of the cardiovascular system. Consequently it has proved(1) .
impos
sible practically or theoretically, to simply subtract their resonancepro
ducing effect from the record, to get the "undistorted" or true record.
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(d) Interaction forces between the axes of translation and rotation are
created by grounding, so that one cannot simply combine the components of
translation. When the headfoot coupling to ground was removed (Burger,
v.Wittern, Talbot) it then became clear that interaction attributable to
remaining coupling to ground (e.g., RL, AP, roll, yaw), still remained to
distort the other components of motion.
(3) Once the stiffness of support to earth was so reduced that any
resonant distortion from this cause was pushed downward out of the BCG
(1)
spectrum, then the weight of the platform was shown  to remain as an
important source of error. Due to its inertia, a platform even as light
as 1/10 the body weight, gets out of phase with the bodymotion at BCG
frequencies as low as 10 c/s. Although this relative motion never produces
much resonance (since light platforms are overdamped by the viscosity of
bodytissues) important phase and amplitude losses occur(1). In other words,
these errors from poor coupling between body and support, formerly at 4 c/s
with stiff supports, are now moved up to 10 c/s or so, but are still well
within the spectrum. One may reduce this decoupling error appreciably by
lightening the support(39) but its weight must be 7 lbs. or more (for an
adult bed) to keep adequate flexural, lateral and torsional rigidity, as we
will show. Further improvement was had by v.Wittern's procedure, of coupling
the body better to its suspension. This has complications, if one forgets
the averaging property of a bed. A shouldercontact for instance, enhances
the contribution of local vibrations frpm the loose shoulderstructures.
v.Wittern(31) advocates a footpressure derived from a springrope (rather than a
foottoard)in'effect; this restricts the 'high frequency BCG information (in HF
direction),
the thoracic region.
In sum, by adopting temporarily this theory of the unitary body, we
have beea (1) led to practical changes in method which have brought out a
new characteristic form for the headfoot BCG record; and (2) we have dis.
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covered similar errors to be overcome in the other components of BCG
response. If then we pass to the theory of the nonunitary body (Burger,
Cunningham), our attention focusses on stiffness of coupling between body
parts, and the frequencies at which they "break loose" under shaking by the
heart. This invalidates (a) the ascription of all resonance artefacts to
the bodybed coupling alone and (b) the use of [platform acceleration x whole
bodymass] as a'measure of force on body, by which one can legitimately call
the record a "force" BCG. One must therefore speak warily of the bodyaccel
erati
on Bra record in the y direction as measuring a y"force" in all its details;
and must examine this usage even more carefully in the roll (0, lateral (x),
and anteriorposterior (z) modes. In passing ultimately to using the body in
the seated position (as we shall propose), these questions of localresonance
and of massunity vs. frequency must all be raised again.
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Displacement ballistocardiography.
The simplest physical picture of ballistocardiography, is that of the dis
placement of blood in an isolated system: the body. With no external force,
the center of gravity cannot move, so that at any instant, the sum of the
moments (displacement x mass) of the moving blood volumes must equal the change
in moment of the rest of the body (displacement x mass). The time derivatives
of this express the conservation of momentum, and the second derivative, the
Newton's law of action. This view emphasizing displacement, is easily con
ceived, and was the basis of physical interpretation by Gordon(44), Henderson(45),
Burger and Noordergraaf. In contrast have been the force or impact interpre
(
tation developed by Starr 11), Curtis and Nickerson(14)1 v.Wittern(12), and
Ta1bot(1). Physically, the latter is better suited to visualizing the case
when other external forces (from earth or suspensions) act on the body. The
"moment" (mass x displacement) and its derivatives (the relative momenta and
accelerations) aid more in understanding the internal dynamics.
acting internally on the body are very complex. Thus, at the J peak, there
has developed a maximum flow velocity (not a headon impact) around the aortic
arch, yielding a maximum pressure on its outer circumference, a maximum tug
on the tethering of the aorta, mainly to spine and secondarily on pleural
mediastinum and pericardial membranes, (transmitting forces to sternum,
diaphragm and ribs). To trace these forces requires better knowledge of
structural connections and stiffness as well as blood acceleration patterns
than we yet have. Starr has worked for years to identify the force pattern
at the heart itself, in a situation overcomplicated by other forces.
However, the summation of blood displacements in detail, has proved feas
ible, though laborious. Noordergraaf, taking account of the measured pulsatile
propagation, has constructed the contribution of headfoot moments from all
the great arteries, and shown(8) the sum to equal the displacement BCG.
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If this is done accurately, the time derivatives should equal the velocity
and acceleration BCG's taken on a light bed: which indeed they
This displacement BCG (called "the BKG" in Europe) or "whole body
plythysmogram" (Nyboer)(43) has great rational appeal, showing clearly how
each of the vessels gives part of the record. However, the displacement
record itself contains so little detail, that clinicians are hard put to
associate it with local physiological or pathological changes. It has a
range of forms associated with body habitus and other factors;
ferential dynamical aspects of heart action are too small to
I,. 4+
the ail
came. nrs 011R
.waa
record. Since these aspects are also faster, they are considerably enhanced
in the derivative; so that the bloodvelocity record really begins to ex
hibit alterations in cardiac physiology or function.
In our problem, we have felt that the dynamical changes in the cardio
vascular system were of most interest, and so have centered the analysis
on the accelerations of the body and how to observe them. The approach to
the latter depends largely on the concepts used: next to be discussed.
?
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211241Lt Conceptst IlitARELIELINe basis of balliqamolimmtly.
Three physical principles have been invoked as the conceptual basis
of studying the status and operation of the cardiovascular system thru
body vibrations: (1) conservation of center of gravity, by which a body
freely suspended is displaced equally and oppositely to the displacements
of all cardiovascular material within,
= 0 or 2: m x
I c.v. c.v. raj,
#
(2) Conservation of momentum in a free system, by which the velocity of
the body follows equally and oppositely the summed momentum of the blood
and vascular masses,
46.1m1v1 x 0
(3) Equality of action and reaction, by which a body freely suspended
accelerates equally and oppositely to the summed momentum champs. (forces)
inside it,
d
:Fozmi = 0 or mv = 0 since m and v vary.
dt
These three types of information, although related mathematically,
are not dimensionally the same, and have separate and distinct physical,
physiological and clinical significance. Which is most "important" is not
yet known; indeed importance may depend mainly on usefulness or intelligi
bility. But technically, the potential information is proportional to the
frequency spectrum which is always greater in the acceleration record.
However; the technical requirements for rer....in of the three aspects
of motion, though the same in kind are not in degree. As with heart sounds:
the high frequency components of the motions seen clearly in velocity and
force records, are much smaller in the BCG than are the components of dis.
placement at heart frequency or lower; so that to sense them well enough for
low noise in the second derivative, the free suspension mentioned above must
be designed primarily for the force record.
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Consequently we we will confine our discussion and theory to the prob
:I
lem of observing MB and9 as the component aspects of translation
jk ijk
and rotation to be made available in the respective axes.
To apply analysis in force ballistocardiography, one must carefully
define what masses and what accelerations are to be observed. This we
will discuss under "assumptions".
Dynamical assumptions.
We will assume:
114?
kJ/
.1.144.h4v% the hAdv are
transmitted to the suspension. PrimN
(2) The body moves as a unit, i.e. may be treated as rigid in all
A4rections and about all axes, to a sufficient approximation.
(3) The acceleration of the suspension in all axes measures the
corresponding accelerations of body, and so gives thefirce on the body,
taking its mass as a whole, i.e. PB
?
(4) The suspension also moves esseatially as a. unit. The justification
for such assumptions, as regards the headfoot (HP or y) directions has
been described by the author( 1)
in some detail. However, the introduction of
the frontal planar and 3D vector components, demands a second look; as
well as recent papersC))4?on theory of body ballistic suspensions.
We will examine these assumptions in order.
1. Transmission of body forctti.2221.
In a. biophysical system, the coupling of information from the living
Material to the measuring device becomes of utmost importance.. Several
questions arise: (a) how quantitatively do forces from accelerating material
in the cardiovascular system, reach the bed or suspension? (b) Over how
wide a frequency range? (c) And to what degree is this transmission of
force information alike for all subjects?
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a. Outward transmission and couplim. Prom the softness of
inner suspensory tissues and the relatively heavy body shell, one would
expect most of the force to be shunted off into motion of internal parts
at heart frequency, leaving little for the body. The fact remains, that
the slower force components 14 HP do move the whole body with almost the exact
00 a)
acceleration calculated from blood motion, and with frequency com
ponents measurable to 30 or 40 c/s '. (The amount of attenuation and shift
of the faster components must await further knowledge of the internal
dynamics.) Mechanistically it is reasonable for this to be true in the
HF direction, from the stiffness of aortic tethering to the spine and the strong
mediastinum dorsad. The transmission laterad should be pporer and more
variable; nevertheless lateral ballistic force records do show
low
frequency detail calculable to first order. But dorsad, the transmission
of force information out to the support should again be good, though
poorer ventrad, Consequently it is worthwhile to have the suspension
sense forces in all three directions, provided one can couple the forces
well through the body in the lateral (RL) direction. This transverse
internal coupling in the chest, being weak, varies with respiration and
organic differences more than in the HP or AP directions,
Although we have little control of the transmission of forces inside
the body, we can do jnuch to assure transmission from the outside to the
platform. The HP (y) and AP (anteroposterior) (z) transmission can be
made good to probably 20 c/s(1) (of support following body) by prestress.
ing the dorsal tissuestwith the couplings now used.. This also applies to
pitch (X axis). Yaw (l) and roll () coupling are related to the lateral
fixation, and depend on chocking methods and strapping which will be dis
cussed, under the various supports. The lateral (RL or x) coupling from
body to support is the most critical; variations in it can radically change
?
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the higher frequency components of R.L. forces (accelerations) observed.
With suitable methods, distinct 20 cis components of lateral motion may
reasonably be expected in the record.
It is fashionable now on theoretical grounds, to view dimly the
possibility of sensing externally the distinguishable details of internal
cardiovascular forces, momenta and displacement, just as of flying fifty
years ago. But the few laboratories now working to improve methods and
principles of observation, do not find this to be so. Every change toward
better coupling of body and support has improved reliability, increased
detail and intelligibility. It has become clear that we are moving toward
reproducible and highly detailed BCG information. While this does not
automatically give illIEEELtILLIal it is prerequisite. The assumption
that cardiovascular forces can "get out" into the support with good fre
quency range and in a systematic way, is justified by experience.
b. The fag.t.....yienc range for which provision must be made in a
good vibrationmeasuring system, has two aspects; (1) The support must not
evoke resonances (if undamped)  or phase distortion (if well damped) 
which can be driven by any component of the emerging force. (2) motional coupling
exists between the modes of a solid structure excited off its elastic
center, so that the highest driving frequency in any one direction governs
the design. Since in the HF direction bodydriven frequencies up to 40 c/s
(49)
in the force record have been demonstrated, our design criterion will be
this rigidity in all fundamental modes of support vibration. However, the
high degree of damping from the body, suggests this is excessive.
c. Transmission of force information to the support, differs be
tween individuals. It depends on tissue coupling, which varies in elas
ticity and stiffness; on the area of body contacts; and on their location
1
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 39 
referred to the main internal generators of cardiovascular force, i.e.
the dorsal thorax. Other parts of the body probably act as passive
dampers (e.g. buttocks) and may reduce faster components. Such questions
are important in a detailed way; but can all be studied later on a clini
cal basis. This matter of positive and negative transmission of body
forces must eventually be studied, as part of the examination of the
"integrating function of the support".
d. Definiteness of designcriteria would be improved by
statistical analysis of individual variability in body conformation.
For the present stage of exploring principles, we have designed on an
average basis, with small safetyfactors. The internal variability of
individuals (size of heart, impedance of joints, bones, etc.) may affect
^4 An+a, but is not taken into account as
the biophysical interpretation 1,0wwv.,..
yet in design of methods.
a. Our assumption that the body moves as a unit (i.e. a rigid body)
is the basis of the analysis and (analog) computation in this report.
This should be taken as a firstorder approximation, sufficiently valid
at frequencies up to about 8 c/s in the headfoot HP (y) direction, and
much less in the RL (x) and AP (z) directions.
The strength and purpose of this assumption is that it lets us pro
ceed to calculations of body suspension systems, as to the rigidity (stiff
ness or high frequency response) which will be sufficient to avoid resonance
artifacts. That is, if the body oscillates (a) is an assembly of coupled
masses or (b) as a propagative continuum (or both)
then the structural stiff
nesses in the support found sufficient to respond at high natural frequency
to the body taken as a whole, will yield still higher frequency toward the body
parts. This results from the lower masses of the parts and the higher
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 40 
vibrational modes of the suspension excited. Whether these stiffnesses
are necessary will vary a great deal with the 2211121im between our sup
port and the body. If this were uniform (body "poured in") the damping
of the structure by the body would be so great, that no resonances could
occur, and phase shift with frequency would be very gradual.
In essence, some definite assumption must be made about the rigidity
of the bodymass, in order to calculate the inertia and damping terms of
the support under design. The acousticvibration propagation theory
[proposed by Cunninghamb while reasonable at first sight (but see be
low) for small vibrations in the longitudinal direction, cannot hold in
the transverse, anteriorposterior, pitch, roll or yaw modes because of
(a) the small dimensions of the body in relation to wavelength of such
waves, and (b) the strong damping of propagated waves by tissues Nest
,
reicher02) j. In general neither the allrigid nor allacoustic theories
of force transmission probably holds through the BCG vibrational spectrum,
but are superimposed along with more complex coupling of body parts at
intermediate frequencies and in the several directions.
More specifically, in the headfoot direction, the recent work of
Noordergraaf in predicting the observed acceleration BCG of the whole
body to over 10 c/s, is sufficient validation of our assumption for this
axis alone. But the full implications of this assumption for the other
axes, should be noted. Although we may be justified in assuming a rigidity
in the headfoot direction somewhat exceeding that demonstrated (Chap.')
the rigidities transverse to the spine (lateral and anteriorposterior)
are clearly much less. The body below the hips, as well as the arms,
contains no net component of blood motion away from the spine, and so is
passive in this region. The masses in these portions of the body mass mB
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do not enter the equation2,111B2i 0or mB Elofrom which forces on the
B
body are to be calculated. Similarly in roll, tE IBOB does not contain
1611C
rollinertia of the hips, thighs, and legs; since these portions suf
fer no rolltorque from the cardiovascular drive. Finally, it must be
admitted that the bodyinertia in yaw and pitch) standsin similar relation
to the confinement of driving torques to the thoracic cage, which comprises
perhaps 1/3 of the average body mass.
In the face of these elementary facts of anatomy, we propose simply
to ignore (in thexznRvayec) the inertial AffPr+ of the nacc4ue
(extra thoracic) parts of the bodymass in the calculations. To a first
approximation, they load the driving forces indirectly (via the support), and may
be interpreted as increasing in a complex way the effective mass of the support
in these axes. It will be the task of future work in this field of "external
ballistocardiography", to devise means of overcoming this crude approxima
tion as to body inertia.
[We have made a beginning in this direction by constructing an ultra
lowfrequency 3D suspension for the thorax alone (Chap. I). This, when
modified for freedom of rotations this report shows necessary, should be
a clear advance toward a BCG that is more quantitative in the xi z, a,
and y axes,]
We conclude that the strategy in this
a2mo
4 c
c4rc+
.11.41.4.41V
4*".
SPV
OnAmet...
LUC
problem in terms of the very low frequency BCG components (4( 6 c/s) com
prising the dominant features of the wholebody BCG pattern. The process
of removing artefacts of earthcoupling and rotational admixture by
developing suitable wholebody suspensions, results in methods modifyable
in detail, which let us better evaluate the bodily subvibrations and so
to reduce errors of rendering them. We would emphasize the need of suc
cessive approximations, and of carrying along simpler concepts like the
MN
unitarybody assumptions, to their acceptable limit.
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3. The acceleration of the suspension
measures that ol_1121.1.)2.ca.
This third assumption implies not only that the whole body has
unitary acceleration vectors (rotation and translation) as above, but
that the support is so connected to it, that it executes the same motions:
at low frequencies, anyhow. [Part of our task is to calculate, and measure just
how, at higher frequency, the inertial properties of the support make 4t
tost from the body motion in the various fundamental modes of motion:
xo yo zoo(' l.If only the body were actually rigid but with a soft
viscoelastic envelope, this assumption about the support would present
small difficulty. In practice, the motion_of "the boodyslial
mania apart from that measured on the support. If one attaches light ac
celerometers to various parts of the body surface to compare the motions
or read "the" body motion, one encounters a biophysical tangle. Firstly,
each accelerometer generates a spurious
1^nQi
AWysibj,
natural frequency due to its
own inertia and stiffness of surface attachment. Secondly the different
local parts of the body surface are differently attached (and under dif
fering prestress) to the spine and to other routes of momentum transmission.
There is a "local surface BCG" for every square inch of the body, with
large local differences over the thorax especially. The only way a body
volume BCG can be defined at all, is in terms of some "standard" system
of bodycontacts which summates and so smooths the local surface differ
ences, and minimizes local "bounces" like those of the chest and sides
near the heart (and shoulders, neck, feet, etc.) which are loosely or
flexibly coupled to the spine and in &quite variable way.
Of logical necessity, in this way the support itself defines the
turns out to have
"body" motion; so that "relative motion of body and support" VP no highly
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precise meaning. Piny engineer realizes this is also quite generally
true, e.g. the "point of insertion of a beam", etc.]. We therefore
assume here explicitly, that the body "has" an acceleration 'Which is
that of the support (other forces being absent) as seen at low frequency,
and that any concept of the 122111.2121i22 can only be had in terms of
some integrating support (or observing platform) rendered weightless.
It follows that the need for this kind of extrapolative definition
of theumntinn of the bothras a whole, at once affects our interpretation
of the highfrequency details or bloodmotion in terms of bodymotion via'
the laws, of momentum. In addition, the operational meaning of such con
cepts as "propagative transmission" of heart impacts, is rendered vague.
Real accelerometers (linear) put on the body are subject to transient
local tilts and jigglesowhich can produce as large local artefactual dif
ferences in phase as those being verified. Only by largearea summation
can reproducible highfrequency details be recorded usefully. So a clear
measure of the degree of failure ot the "unitary body" theory, is hard to
demonstrate . though not impossible.
4. The suspension moves as a unit (is rigid). Clearly this fourth
assumption is necessary, if accelerometers attached to the suspension are
to define the BCG. In the headfoot (y) direction, any simple tubular
(39)
frame  fulfils this criterion. But to record lateral (x) accelerations,
such simple hammocks
or even honeycomb sheets are not stiff enough.
The former vibrates in lateral flexure, and both in vertical flexure.
Trussing to kill the latter still leaves compliance in torsion toward
the Strong forces in roll [rotation (p) about the body axis]. But because
there exists x3 coupling (Chap. III) the suspension must also be sufficiently
stiff (unitary) in torsion. When this requirement is filled, we can say the
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suspension moves as a unit for the purposes of this problem. It is a
major problem in mechanical design, to satisfy this requirement for
torsional rigidity along with the lightness (1/10 the body weight) needed,
with a mechanical assembly of body size.
Other concepts and assumptions needed for physical analysis of the
BCG.
Simply to treat the motion of the body under cardiovascular forces,
as a problem of rigid body or even coupledsystem dynamics, overlooks
some dominating principles concerning information.
Unlike the dynamic
assumptions stated above, these additional simplifications are dictated
by the viewpoint of medical physics, rather than by physics.
This
1. An acceptable solution must result in separation of variables.
is the conditional of orthogonality, implied in the use of the tetra
hedral lead system in electrocardiography. Applied to ballistocardiography,
this requires not only that we seek natural axes of the system for trans.
lation, but also that rotational data be not mixed with translational.
This turns out to imply the assumption of small angular velocities, so
that centrifugal gyroscopic and coriolis forces are negligible (which
is true).
In principle, if enough is known about the coupling between modes
of motion, for a system of rigid bodies,' one can readily compute the
pure axial motions, in spite of interaction. For two reasons, this
possibility is excluded. (a) Unlike the ECG, where simple resistive
networks are now often used for this very purpose', the BCG 3D
computation involves complex reactive terms, which require a rather large
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analog computer of forbidding cost. (b) A BCG design which does not
succeed in separating (uncoupling) the translational and rotational
modes, introduces body artefacts so large as to obscure, even reverse
the appearance of some components. If the system were indeed a pair
of coupled rigid bodies (body and support), these spurious resonances
could readily be rejected by filters. But in the rotational modes,
with a real body, this assumption proves too simple, and the filter
remedy will not work. There appears no choice but to uncouple these
modes from the beginning, and avoid computers.
2. With systems as complex as those in biology, one must choose
what information is most valuable, and not simply record every mani
festation that can be measured. In the y axis (headfoot) alone, the
displacement, velocity and acceleration BCG yield quite different bio
logical information. When we add X and '7df4tn to we must raise the
whole question of the value of retaining all three derivatives in all
axes: a total of nine traces (apart from the necessary BCG, respiration
and phonogram). Clearly to record the three rotational components and
their derivatives as well, would require strong biophysical justifica
tion on purely informational grounds, however equivalent these axes may
be physically.
For this reason, in the analysis given here, we will concentrate
cation is dictated not only by the general intractibility of excess
information, but also by the special relation of blood displacement
to the physiology of the heart muscle and vascular structure. The
on recording the translational motion, solving but not studying in de
tail the rotations or how to read them from supports, This simplifi
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body rotations depend on features of body architecture which contribute
little added understanding of normal blood flow. If reasons should arise
from pathology to study the body's rotations (e.g. pitch) these will ac
quire informational value. Thirdly, we mention a need of unity and adhering
to a restricted field, to penetrate further in this research.
3. Errors. To distinguish between "true" and erroneous data, label
ing the latter "artefact" or "noise" is mainly an expostfacto art. Never
theless in biological measurements, there is generally some sk?'ine
activity unrelated to the problem at hand. The measurement problel in
biology often consists of so devising the
method
that only pertinent in
formation is seen, and other information rejected. For example in blood
flow measurement (magnetically) the ECG may come through to mix with the
flow information, and must be rejected.
So with the BCG we find both extraneous and internal "noise",
quite apart from misinformation mentioned above, where one mode of
mechanical motion leaks through into another mode. A very practical
requirement of BCG design, is to reduce couplings to extraneous systems.
Though no mechanical system (in a gravitational field) can be studied
completely free of coupling to earth, this simplification will be as
sumed in our analysis. Actually, this coupling proves a major problem
and source of error, both in practical BCG supports, and in the analog
computers used to solve the dynamic equations.
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.47
Chap. III. DYNAMICS OP THE SUPPORT SYSTEM
In order to obtain a repeatable and meaningful record of accelera
tion, it is necessary to secure the subject to a rigid frame and measure
the motions of the frame. It would be ideal if this frame would follow
the body vibrations without distortion, at
least up to 40 c/s, the band
width of interest for cardiovascular performance. However; the spring
and damping of the body tissue between body and frame stand against this
requirement. Consequently a knowledge of the phase shift and amplitude
change as between body and frame is needed, to interpret the data taken
from the frame.
This knowledge is expressed to a first approximation in the equations
of motion for the frame and body, regarded as a twomass system in rela
tive motion both of translation and rotation. These equations are derived
here, experimental means of getting numerical values of the coefficients
are suggested, and the nature of the solutions computed and shown. The
complexity of the equations (12 simultaneous second order differential
equations) require the analogue computer for solutions. Passive (LCR)
analogs gave fair solutions, but the difficulties of varying parameters
in rotational modes led us to use active (operational mnplifier) com.
puters (Reeves) for the larger networks.
For this the Martin Company
gave generously of time and assistance in their computer laboratory.
Solving the equations of motion in this way simulates the actual
problem in a simplified form, it stimulates and points up physical think
ing
and observation with real subjects.
The computer program has given
a clearer physical understanding if the genesis sand interaction of forces
producing translational and rotational motion. Its solutions to the
equations of motion are needed to interpret the acceleration record for
cardiovascular performance analysis, and to provide data fcr improving
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the design of the frame and suspension relations, by varying the
parameters embedded in the coefficients.
Apart from the practical problem of improving BCG methods, there
is the biophysical problem of enunciating principles by which the BCG
record may be rationally understood. The computer program here serves
to evaluate the range of validity for the theories of the BCG that are
under discussion. Experience accumulated in this way should lead to
better solutions with live subjects and real instrumentation.
AL. EQUATIONS OP MOTION
The most general equations of motion for a twomass system in 12
degrees of freedom with static coupling are excessively complicated,
and were whittled down considerably before a solution was considered.
To get an idea of the complication and the simplifications introduced,
degrees(50) of
consider the equations of motion of a Luis mass in six
freedom Figure 1
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Coriolis where
m (m.41 +4) = X hx Act Ef
m (y.ia.41 f) = Y h =  Di  Fa
Y
m i) =
h? h )1,+h p = L
x y z
hh f = m
z x
117 h i+h a = N
z
gyroscopic
and A, A, C are moments of inertial
defined by A. = J11(y24.z2)dm
while D E F are products of inertial
defined by D = fh yz, dm
The forces on the left are inertia forces; those on the right are external
forces (and torques) such as the drive. If now a second mass is coupled
to this mass, the forces arising from the stiffness and damping of this
coupling would appear on the righthand side as shown here, although
these
also would involve the dependent variables (relative dis
placements and velocities). Before considering these coupling forces,
in our case we may simplify considerably the expression for the inertia
forces. This follows from two observations regarding our system:
(1) Displacements and velocities are assumed small enough so
that their squares and products can be neglected; (see Appendix A).
This allows us to set the Coriolis accelerations [consisting of products
of a linear and an angular velocity] and the gyroscopic moments [consist
ing of the products of angular momentum and angular velocity] equal to
zero.
(2) The body exhibits a plane of LassIa, the YZ saggital plane.
Hence, products of inertia DEF are zero except in this plane of sym
metry, i.e., D :# 0, E = F = 0. So in our case, the above equations
reduce drastically and separate the forces and torques:
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X = mx
Ym
Z mz
50
= A CI
L
0 4
M ? B P*1 Dy
?
? C y f.? pp
However, we are still left with dynamic coupling of rotational
reactions, introduced by the productofinertia D about the transverse
body (X) axis. Because the summation in D (JII)m dm) does not contain
a squarelthe t contributions on either side of the body's xy plane tend to
cancel one another. So the productofinertia D will be an order of mag
nitude or so less than the moments of inertia, B and C. Hence, the terms
containing D can also be dropped, and the equations become simply those
of a completely symmetrical body for small motion:
irdt =X
my = Y
(C)
We turn our attention next to the righthand sides of these equa
tions, which involve the motion of both frame and body$ because the
stiffness and damping forces depend upon the relative motion of the
frame with respect to the body. The plane of body symmetry helps here
because with no dynamic coupling, motions in the plane of symmetry do not
excite motions out Of that plane. Hence, the equations can be separated
into symmetric and asymmetric as follows:
In plane of saggital or YZ plane
my
= y, ria = Z', It'a = L (D)
Out of plane of body's s mmetr transverse xz and frontal xy plane
(E)
B13 = MI Cy11 = N
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This means that torques in pitch such as L create only y, 2, a motion;
while torques in roll or yaw such as M or N create only xt and y motion.
The asymmetric equation will be a little more involved because of
static coupling between forces and moments in the transverse xv and frontal
xy planes.
While the derivation of detailed expressions for Y, Z will be
left to specific designs'
(the human bed, mercury bed and chair) the equations of motion in one
plane (the transverse xz) will be derived here to demonstrate certain
general features for all equations, and later to develop experimental
methods of coefficient determination.
1. Assumptions and geometry
Fig. 2
The above diagram is an abstract of the transverse plane of the body
on a bed seen by an observer looking footward along the spine. The con
ventions of direction are standard in BCG usage49). The body and bed are
treated here as rigid bodies coupled together through the damping and
spring characteristics of the body tissue. This coupling is assured be
cause the forces from the heart are not great enough in size to separate
the body from the bed or to slide the body with respect to the bed.
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While dependent nonlinearly with tissue stress, they are linear for the
small motions (.001) we are concerned with. That is, in the equations
that follow, spring and damping forces k and r are taken as linearly pro
portional to displacement and velocity.
"Centerofresistance" is defined by the axis of a static force
whose application causes no rotation. By way of illustration, consider
how the centerofresistance is determined for the xdirection. Apply
a force in the xdirection on the body; if the body rotates in addition
to translating) change the point of application. Keep changipg until a
point is found at which a force produces translation, without rotation.
The axis through this point in the xdirection is the center of xresis
tance. In the same way, the centers of y and z..resistance can be obtained.
The
concept of centerofresistance allows rotational moments due to
translational displacements and velocities, to be distinguished from
rotational moments due to angular displacements and velocities. The
driving (heart and blood) forces and moments are noted by Fx, Fz and
Mp, which cause the motion we are trying to measure.
Since the xz plane is not one of symmetry: the equations of motion
in this plane, as derived earlier, are
mU = X, q = M
for both support and body, yielding a total of 4 equations, Note that
the equation for vertical motion (mz = Z) is not involved in this motion
because it is in the (saggital) plane of symmetry which, in the figure,
Is a plane perpendicular to the paper through the Zaxis. That is, z forces
being symmetrical cause no roll motion.
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2. Derivation of Equations of Motion
Substituting the equivalent of X and M in terms of the known physical
properties of the body and support, we have from Fig. 2 for the equations
of motion* of the pody, in translation:
B s ss ,x B B. B s hSpS) = FX (1)
and in rotation:
I 4
BB + r (pnp ) k (p )
BIB p D S
?
fA 4
Lr fri B s"hs s x 0
x B B p)4k(xhBpBxs.
and for the supports, in translation:
?
in x +r (A 4.41 p A 441 p )41 (x +h
ssxsssBBB xss
and in rotation:
(*
h [r 4141 p A
S x s ss
S *PM +11 F h
z x F
B013);=
(2)
(3)
P )+k (x +11 p = (4)
X S SS DD
All distances have signs. A distance is positive when measured in the
positive direction of the coordinate.
For example, the distance to the
centerofresistance hn is negative because it is in the negative zdirec
tion from the body c.g., the origin of the body coordinate axes.
The various forces or moments and their signs are derived by imagining
a force or moment applied to produce a positive displacement. The reactions
are summed up on the lefthand side, being opposite in direction and hence
sign from the driving force.
* Rewritten from III (D) in terms of displacemepts and for each mass.
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54.
Consider the terms of equations (1). Application of a force Fx
is instantaneously reacted by the inertia force MBxB. As time passes,
velocity is developed, giving rise to the damping force
xBBBsss
The damping force depends only on the relative motion between body and
support. Owing to the coupling of body and support, angular and linear
velocities are built up for both support and body. The expression for
damping accounts for these various motions in developing a relative
Eve.ntuailv the velocity leads to displacements, bringing in
a reaction from the stiffness of body tissue, given by kx(xehA s h p ).
ss
with damping, the spring force depends on relative motion only, hence
the similarity between the damping and stiffness terms.
Equation (2) expresses the equilibrium of moments acting on the body.
The applied momentFzFthFFx+m (M being a pure couple) is reacted first
by the inertia moment,then by damping moments, followed by spring moments.
These latter two moments are due to pure relative rotation expressed by
t
r,,(pupd for damping torque and kp(pB0s) for stiffness. Additional
F 
rotational reactions are suffered if a relative linear motion exists, and
is obtained by multiplying the force due to linear motion by the appro
priate distance to the centerofresistance, as shown above.
The signs on these last terms will be explained to illustrate our
sign convention. Looking at equation (2) for body moment equilibriums
note that application of a positive moment F xh is reacted by a positive
linear displacement of the body greater than that of the support. The
linear displacement of the Locji at the contact with the support is xB+h p
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The displacement of the support at the contact isx+hp,the positive
s s s
Sign being required because although hs is negative, p being positive
counterclockwise, gives a negative displacement at the contact point.
Taking the difference, assuming the body displacement larger, we have
for the relative linear displacement between body and support
k 1 4. BPB) xs + h5 135 ))
multiplication of which by kx gives the translational spring force.
The damping force is obtained by replacing displacement by velocity,
and multiplying' by rx.
The ToTaltranslational force reacting the
. 4
plied rotational moment is
r [(x + h + ' '1 'kx[( +h p )(x R ))
x B B B? ?"s. B BB s s' s
s s
This must be multiplied by 114 to obtain this reacting moment.
nn=
No term representing the coupling to ground has been included in
the equations. It has been assumed that the very soft suspension,
1/3 c/s cutoff, will keep the influence of ground to an unobservable
size.
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4
 56 
B. EXPERIMENTAL DETERMINATION .OF COEFFICIENTS
Numerical values of the coefficients in the equations of motion are,
of course, needed for applications involving these equations. Although
estimates of certain coefficients are possible from theoretical consider
ations, much easier and more reliable are estimates based on experimental
procedures. We shall describe here the procedures required to measure all
coefficients excepting those associated with specific shapes of the support,
leaving this information to those sections dealing with support design.
1. Moments of Inertia
Moments of inertia are conveniently determined by oscillating the item
In some kind of pendulum, or springmass combination. The period of oscil
lation is related to the radius of gyration by a relation depending on the
type of restoring force. We shall next derive these relations for the
pendulums and springmass combinations used here. A detailed description
of procedures will be taken up in the sections on support design.
In this analysis, since the subject and support are treated as rigid
bodies, the inertias must be found at very low test frequencies: not
higher than one cps or so.
a. The Bifilar Pendulum
To determine inertia about a vertical axis. (72)
11.1??????????
L
................_*......_
, , 44.1as....,,n4....A.1.441./Ulor???*4
??....................,............
r
...........,......?....._, ....
....?0?.... ,..z.,,........?;__.
mai
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The bifilar pendulum is useful for determining moments of inertia when the
swing must be about a vertical axis, such as a subject sitting in a chair.
As seen in the figure, a bifilar pendulum is simply a twostrand pendulum
with the restoring moment FL being provided by the angular displacementes
of the strands. We shall derive the period of the bifilar with strands of
equal length, because these are the most convenient to use. This requires
that the cog, be midway between the strands if all the energy of the oscil
lation is to be confined to motion about the vertical
a,ic. Unequal c.g.
spacing would lead to oscillation about a horizontal axis too. From the
expression for F, we see immediately that the torsion spring rate is
.,2
FL = k a = a? = torque/unit angle
a 4.e
The frequency is computed from the usual relationship
1 lig L2 g
= 2 n
where t) is the radius of gyration in the expression I =mf
Solving forr the required quantity
TT in terms of T, L and A.
(2 =
4n
(5)
The test procedure is simple. Choose the strand spacing L and strand
length A , so as to get a period greater than one second but no greater than
five seconds or friction will influence the results. (For a recumbent adult,
if S = 6 ft., L 3: 40"). Adjustments after tests of the length,?( and spacing
L may be necessary to get repeatable results.
Set the pendulum in motion; making sure that the amplitude is small
enough to insure a linear oscillation. Keeping 6/k < .2 during the timing
will suffice. Also, make sure the oscillation is about an axis through the
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c.g., by making the initial rotation about this axis. Repeat until the
resulting motion is the required one. Time the oscillation over five
complete oscillations and obtain an average period T. Repeat several
times and average again, The value of r is available from equation (5)
knowing L, g and T.
?
The inertiafkbout theaxis of the body is determined similarly,
with Or 12" for a standing adult.
b. Wheelbarrow
rbre
Moments of inertia about a horizontal axis)4 for long slender shapes
such as a body on a bed, can readily be obtained by resting one end on a
knifeedge0 and the other end on a vertical spring of stiffness lc, shown
in the adjacent figure. The restoring moment about an
axis through the
knifeedge equals the force at
the spring kee times the
length 2 , from which is obtained
the torque/unit angle k k,e 2
0
From the relation for frequency
1. 
k.
= .1._AwL.
2n 1
T
...
follows the expression for radius of gyration
2n i fb2m
p
,,)
T h l o ki x.
=
10
in terms of period To:
Applying the momentofinertia transfer equation, we obtain the moment of
inertia about the horizontal axis through the center of gravity:
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41
1???????
?????
1477,2..
IA, me'
In the test, adjust k and ,(4' to obtain a period To of between one and
five seconds for reasons already discussed. (For a recumbent adult,
k = 2.5 lb/in). Make sure the oscillation is in the vertical plane only.
Leakage of energy to other degrees of freedom will reduce the accuracy
of estimating r by altering To.
59
s.1.0. 49,4
Z
431.,
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2. Determination of body spring and damping coefficients.
As far as we know, no one has ever made a serious attempt to obtain
complete information on the spring and damping characteristics of the body
tissue in this kind of motion. Some time ago, a measurement of head/foot
characteristics was obtained here by restraining the bed in this direction
by a very stiff spring and then exciting the head/foot motion of the body
by a blow on the bed. Decay rates and frequencies of the lightly damped
oscillation of the semifixed bed provided the basic data from which body
damping and spring characteristics were calculated.
The schemes for obtaining these coefficients described below is based
on the transient response of the bed referred to above. We do not know now
that this technique will give consistent results; it is advanced here as
theoretically feasible but not necessarily practical.
For illustrative purposes, determination of the stiffness and damping
coefficients by transient response of the bed will be discussed for motion
in one degree of freedom. We shall then give a detailed account of how this
method will be applied to estimating coefficients controlling the motion of
I the bed in the sagittal plane.
For free motion in one degree of freedom of the bed
1
4 + ri, + ky = 0 (1)
1 has for a characteristic equation
m.)
+ a + k = 0 where X. may be regarded as a (complex) "
frequency.
14
. whose roots Ai give the exponents in the general solution to (1): y
II
These roots are easily calculated as
II
 r +.rf'.7.... ,? 1 i IL02_y,2co2
(2)
1 2m m 2m
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For a lightly damped oscillation, the record will look as in the sketch.
The measurements to be made are shown.
The time 6 is chosen large enough to
make the ratiol02/y1, considerably
less than one in order to be insensi
tive to error in measurements of yi
and y2. Damping, r, and spring, k,
are related the test data by
00.0 0 ? ?????????? ??? .? ? ? 0.0.. ? ? ? ?
y1 2
...........
2n
OA.
????
lob
(3)
Knowing the mass (m) of the bed, k and r can be extracted from these relations.
Sagittal coefficients
First we introduce stiff springs K from bed to ground, so that the
natural frequency of the bed relative to ground is high compared to that
of the bed to body. In this case, the amplitude of the body oscillation
will be small compared to the amplitude of the bed, so that the bed displace
ment is relative to ground, which we can measure, is nearly equal to the dis
placement of the bed relative to the body. The equations of bed motion for
this configuration are:
rsor)?./1 di::: 4 k f4A .4.1* r icRx
// t' 7737?777;7.94:7;
{7779..eu.,;_
/
Ltrisx]
r
(,)
(4)
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 62 
In
n this analysis, spring and damping from shear have been neglected
because it is expected that they will be small for sagittal motion. Putting
in effective values of k' and r',? we have
2
141 *LI L * + (17 L + K)y = 0 (5)
for the equation of vertical motion. Assuming the pickoffs are acceler
ometers of output a1 and a2, this equation becomes in terms of the sum of
the output 
I.
m(a, + ao) + 7?L (a + a2) + (rOL + K)(a1 + a2) = 0
4 fa
Observe that (6) is similar to (1).Mence. from (3)
.1
r L(y )
1 t
2m 
6
(6)
(7)
where the quantities yl, y2, 6 and T are to be taken from a decay record of
the sum of the accelerometer outputs. How the bed is excited does not matter
in general, but if the frequency spectrum is scanned for maximum response to
sinusoidal drive, the period T is easily determined as the frequency of maxi
mum amplitude. Then by removing the drive, the decay over a time 6 yields
the ratio y1/y2.
What we have done to this point is to determine effective values of
damping and stiffness.
We have now to determine the position x where these effective values act.
This is determined from the moment equation by replacing the integrals by
effective values and expressing the equation in terms of the difference of
accelerometer readings  thus:
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.2
(a1  a2)+ r L x (a1 a2 ) +
63
2
+ c L R )(a1a2) = 0
(8)
I 
From the decay record of al  a2, one measures yi and y2 and 6, giving x:
2
r L x Y1/Y2 (9)
21 
No frequency (period) measurement is necessary since only one quantity
is unknown. The damping relation was chosen for its higher sensitivity.
2n
However, a frequency () measurement might be needed as an aid to getting
good accuracy in the measurement of 1c because it appears in combination
with a large term  the K spring. It may even be necessary to take several
1
readings in computing k and then apply the methods of statistics to get a
_1
meaningful estimate of k
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?
*.
 64 
C. SOLUTION OF THE EQUATIONS OF MOTION
As mentioned in Chapter I, the equations of motion of the body
support system have been solved(1)(9) for the simple case of headfoot
(y) motion, which is relatively free of coupling with rotation. However,
the lateral (x) motion of the body couples strongly with rotation in both
yaw and roll, because the cardiovascular driving forces do not pass
through the body's center of gravity.
In the case of yaw [motion about the y or AP axis in the xy plane]
the cardiovascular yforces are so nearly along the body axis, and the
inertia is so large about an AP axis through the navel (approx. cg), that
the contribution of yforces to rotation in yaw, is small. However, trans
verse (x) forces in the body's frontal plane are large because of the tilt
of the heart. Consequently the body tends to rotate counterclockwise in
yaw with a motion:
A
{where is the instantaneous
distance of transverse force X
from the cg, and M is the pure
couple exerted by the blood in
rounding the aortic arch]
However, a pendular body support rotates at quite low frequency in
yaw (see bifilar calculation above) so that the coupling of x and y is
compliant enough to be negligible if the x motion is not constrained.
Moreover in this xy plane the coupling between body and support may be
made relatively stiff, so their relative motion about the y axis is
negligible.
Hence there is little need to examine the xyy system of equations
for interaction between body and support. Moreover informationwise, we
do not need to solve for M (in the above equation) because the cardio
vascular "news" so obtained would be trivial. The value of x (transverse
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component of cardiovascular force) however, is so strongly involved in roll,
that the xyy system of force equations alone, will not define it.
In the case of roll (forces in xzp plane, transverse to the body), we
find a major difficulty in decoupling rotation from translation. This is
because (a) the roll inertia 10 is much less than are Ia and I
?[/A2 = (13/40)2= 1/10 for the radii of gyration] and (b) the cardio
vascular forces at beginning and end of ejection and of filling, act on the
body quite ventrad (3"6") to the body e.g..
As a result with this moment
arm, a rightward forceX tends to roll a light platform under the body to
ward the left, the while it translates the body (and platform) to the right.
If now there is spring coupling with resonance(111 in roll, a roll BCG not
only adds to the lateral, but has a highly variable phase and amplitude
relation.
In sum, the yaw BOG does not interfere with the lateral BCG as does
the roll response, because (1) the body inertia I is much larger; (2) the
)(force js on the same side of the body c.g. (i.e. headward) as is the best
coupling to the support; (3) the support can be free to rotate with the
body, avoiding excitation of springs k ; (4) the stiffness of coupling k
from body to support is naturally much higher than is kp.
For these reasons we will confine our analysis of the interaction
error due to rotatory coupling in measuring the BOG, to the motions seen
in the transverse (xpl)jolin.s. We will show analytically, that for sup
ports which are not free to roll with the body, the coupling from body to
support in roll acts as a highcut filter to the lateral BCG: (both as to
resonance and cutoff). Further, for deepchested individuals (high rota
tional moment of lateral forces) a doubleresonance occurs. This creates
a frequency band in midspectrum, where support and body act in phase
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opposition, and so suppresses important details in the lateral BCG.
These relations result when physiological values of body stiffness are
inserted in the' xzp equations, as follows,
Particular BCG Equations of Motion for the xzp Rlane and their solutions.
Equations 1  6 (Table 1) show the relations between the forces and
motions, as well as the torques and rotations defined in Pig. 2. Implicit
are the assumptions that through the frequency spectrum, there exist con
stant coefficients (parameters) of stiffness and damping (k's and r's) for
each mode when the other motions are zero. Thus, with locked support, a
pure couple will excite a roll frequency
6p .11117p7i;:(rp/2I13)2 [where kp is the equivalent shear
stiffness seen by a pure rotation]
rm?dows
= toy
and a damping factor rp [which
governs the p motion of the body
alone]
Similarly c7ox is the natural frequency of x vibration when rotation is locked;
although physically
U ic
A (.4
ei stiffness mcnmr44, ra: 41tcz ;r1,1/tissuesril tiSSPS ex
(.4.wkik;..6 VA 11,As?.
X
cited in a different way than for kr.
In practice one cannot measure the force coefficients k and r, but
only (7) and the frequency and damping of the single (locked) modes of
vibration. Experience shows that referred to the support, whole body fre
quencies are > 3 c/s or 1 18 with dam'ping
so that the damping term amounts to; =
2m
r
factor < .25 where
2mm 't
The complex frequency is: close to monotonic w2
The decrement en (y /y ) = 5/3
1 2 2m
... 5/3
shows damping Y2/Y1 ,..11 which is quite strong.
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or
5
?
=
Mg r W>50 lb.
1
= 2 in terms of
2
resonance ratio
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In order order to solve these six equations, one must calculate the k
and r from observed values C7) and assuming suitable masses or inertias.
Table 1. Equations of Motion (x 'z' plane)
rewriting with moment arms positive, and rearranging for computer connections
Forces on body and support:
(B)
rx [(k14410
+ (hB13B+hsk)]
[(X Xs) 4 (hBPJ + Os)] =
(S) mM
ss rx [(XBXs)
kx
Fx (drive)
sk)1 ri X' (ground)
x s
[(XEcXs) + (h0B+hsps)) kiX
X S
Torques on body and support:
(B) IB + rp(134s) + hBrx
kp(pBQs) + hBk,
ril(3134s)  h rx
?0
(S) p
s s
k p, s
) hkx
p B
[(xBx) +
? ?
[(xlitxs)
z, Forces on body and support:
(B) mJ' + rz(1BZs) + kz(zBzs)
416 O
($) mz rz(zBzs) kz(rrizs)
s s
where
**
mB = .375 inch slugs
ms = .025 inch slugs
**
T= .833
assumed 8"x20"
assumed 1"x20"
(ground) =
0
(hBpB+hsps)I
(b0B+hsps)] = Fzwf
(hBil3B+hjis)]
(hBPB411sPs)) =
(drive)
0
4
r z +k z (ground)
zs zs
Ph
kx = 375 lb/in kz = 1500 lb/in
rx = 5 lb.sec./in rz = 11 lb.sec./in
k, = 21,600 lb.in/rad hB= 2"
=4',
= 200 lb.in.sec./rad hs= 2" wf = 2"
rp
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C 1. Solution_ty2atlE_EmIlt_EL(4yferential anal set).
Discussion:
We have seen that the forces and torques in the transverse (xzp) plane
prove particularly interesting, because of (a) the small inertia (In) of the
recumbent human body in roll, and (b) the application of cardiovascular force
on the body, on the (anterior) side of the p axis (thru c.g.) away from the
contact with support. Qualitatively, the resulting lateral (x) response of
the body to such forces, is bucked out by the body's a...Ilia motion about its
c.g.. This unfavorable situation does not exist for any other rotational axis,
and so becomes the most important dynamical relation to solve for.
In Table I are gathered the xzp equations for body and support, as
previously developed. These equations show the following features, which
stand out clearly in the analog connections (Fig. 213).
(a) In every equation there occurs the term in square brackets
[(xBxs)+(hdB+hsps)]
TU4q
is a force in shear proportional to the relative motion of body and sup
port, whether displacement or velocity. Note that the translational term
depends on the difference of the motions, but the rotational term on the sum,
because of the body and support are tangent.
(b) This composite force in shear operates on both body and support,
to oppose the relative translation and rotation of both: the
1.14401 in
nrn?
r
portion to the distances of the c.g.'s from the interface of shear motion,
hB and hs' [Since this interface is curved, one must establish experimentally
an equivalent straight line of "resistance" to which h is measured]. while hs
Is constant, the dimension hB varies considerably with a subject's "thickness";
so that the coupling between lateral and roll BCG will vary similarly.
(c) There also occur spring and damping forces in "pure" relative roll
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A
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 69 
(131113s). For this to happen without relative motion in shear, i.e.
[NBxs).1.(hBilifhsps)] = 0
there must be a relative vibration of both bodies in pure roll (013p5), so
as to produce alternating ..._Nmpsss!tI!..aIILILL522sl on both sides of the midline.
This compressional vibration thus sees mainly the compressional tissue stiffness
in the z direction but partly also stiffness in shear. The natural frequency
of this vibration if excited separately, should be higher than that of tissue
in shear, due partly to greater stiffness in compression and to lower inertia
in roll.
(d) A theoretical dynamic system set up on an analog computer,
actually needs light coupling to "earth" just like a real system. This is
shown in Fig. 2B for both translation and rotation, using both "springs" k'
and "dampers" r'. During operation the integrated amplIfier drifts will
move the output off scale, if equivalent springs K' to earth are not added.
With such springs, the output will slowly oscillate till damped out by the
added coefficient r'.
So the
analog clearly parallels real Systems designed
to have zero force to earth, such as the "ideal BCG". This is because such
real systems also show drift with the least wind, so that practical suspensions
must have some "ultralowfrequency" springing rather than be completely
(e.g., mercury bed).
(e) The particular dynamical problem whose solution is shown in Fig. 2B
involves locking the platform rotationally, without stiffening the translation.
The analog then acts like current clinical BCG beds of ultra low frequency.
One accomplishes this in the computer, by grounding at p and p. When these
terms drop out of all the equations the "relative" rotations become simply I3B:
body roll on the support, as a recumbent subject on a table.
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?
6.
vvonowevasemoosomamaw,
1
ar . 1.r 0 WW.wM?ftlaftnio.dirmiegettej
1 R A \Ir.),
116. U %.? r OMPLYTag,PNAleCiroaral OFALp poomics
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Results of  70 
Computer solution of motion in Xql_ptlne.
The lateral motion ;s of the support when driven by a force on the body
Fx, appears in figure 2C as a function of frequency. The roll p of the body
IS shown in figure 2D. The dotted curves show the response when support is
free to roll with the body; while the solid curves show the resonances generated
when the support is not permitted to roll, as with current BCG practice.
The line T = 0 would be the response were the heart forces directed thru
the roll axis. The lines T show the change due to the torque resulting from
the anterior position of the heart. 2T is for normal heart position. When
platform is free to roll (dashed lines), the lateral amplitude xs decreases
somewhat at all frequencies as the heart force moves forward, due to energy
going into roll. This enhanced roll motion appears in Fig. 2D (line 2T).
However, the body would roll somewhat even if the heart force passes through
the roll axisl due to inertia reactions of platform.
The freely rolling platform resonance shows translational resonance at the
frequency Icx of the support on its coupling to the body. This resonance cor
responds exactly to the cutoff seen in the headfoot (ys) frequencyresponse
curves of ULF support(/); it is caused by finite mass of support ms and its
coupling kx to the body, and so inevitable. For convenience this cutoff was
set at 10 c/s, corresponding to a platform 'heavy" in translation.
With support not free to roll with the body (a) a strongly resonant peak
len
 2.5) appears at a frequency co_ where bodyrollresonance is in phase
, op
with translation; (b) just beyond this is a sharp trough (attenuation t
about 1/5 amplitude) where the roll goes out of phase with the lateral motion;
and finally (c) an increase to the undistorted (dotted) amplitude, exceeded
LAJI"...! because the body's roll inertia adds. to its trails
somewhat with zero 4.,.....nito.
I . of. coupled resonant systems.
lational inertia. The dual peaking from phase interference is characteristic
?..........ma
111_
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 71 
Our analog also describes (fig. 21), solid lines) the rollmotion of the
? body itself on a platform that won't roll. Even without impressed torque
from the heart (T = 0), the platform translation causes a strong bodyroll.
But with the normal anterior cardiac drive, the energy goes increasingly into
roll at the higher BCG frequencies, and less into lateral motion of the platform.
In net effect, the roll constraint on the platform introduces a lateral
BCG resonance followed by cutoff at and beyond the natural frequency of body
roll on the support. Since the dorsum is rounded the ribs and shoulders are
compliant when chocked, the stiffness in roll is intrinsically low; so with
the low inertia about the p axis, the rollresonance is hard to raise above
c/s. Such a periodicity is indeed seen commonly in lateral BCG records, as
well as the predicted absence of sharp waveforms (containing higher frequencies).
The analog computer, using the simple twomass model of Fig. 2A, therefore
has accounted for the major artefacts of the (lateral) RL(ULF)BCG. It also
predicts that these artefacts vanish when the support is free to roll.
The case of the 0114)BCG is not shown here, but again exhibits complex
resonance in the RL record, due to coupling between the resonances (7)13x and wq;
this time with higher Q and lower frequency ((7)Bx = 6 c/s).
The frequency at which body is resonant in roll can be shifted upward
(on the computer) to imitate tetheringstraps fastened across the chest, and
I.
a formfitting support. Since these will also raise w (the lateral resonance)
sx
the BCG passband can be increased. If instead the support is free to roll, r
the resistive torques between body and platform k[hpB+hsps) and k[peps] are
not forced into action, and such tethering becomes unnecessary.
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4.
11
xJ
F\t
?
?
?
i_or?tked
411.% ???? 011? 0" ?
U0144:Y?:. le A
000
oP
Urtioc.,
,e1.1*1
4
G , C
4 ...........r114**41/164MAWOR004~.. 4
Fre 4i.)1C'41 C/IS
ftewarporom
F u Exx
Fig. 2C Reeves computer results:
Lateral acceleration of support by lateral force in body. [solid lines]
(0): force through ca. (platform not free to roll), showing resonances due to
inertial reaction of support, tangential to body. (T) and (2T): lateral force
more anterior (in position of heart), showing enhanced rollresonance and cut
off resulting. (Dotted lines): (0) and (2T) lateral heart force acting through
c.g., and anterior to it (platform free to roll); showing simpler cutoff, due
only to translational bodycoupling to support (of 1/10 body mass).
Fig. 2D: Roll acceleration of body by lateral force in body [Solid lines]:
Support not free to roll  showing resonant bodyroll so created. [Dotted
lines]: Support free to roll  curve (0) force thru c.g.; body roll ceases,
above frequency of platform decoupling. Curve (2T) force in heart position;
bodyroll (driven) persists without transmission to support.
29
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72
C2. Passive analog solution. IL C R circuit]
Just as in pure y or z motion f a twomass system, one can set up a
simple mechanical analog
whose forces are:
? X, e? Its
fa,
13
re, i
1
m x + (x x )+k (x x )=F ti/P41i4
BBBBS BBS 1,
111 13 ii $ , n1/4
equivalent electrically to: & oNdl E
I
\
.00
where
where
i = A A
B S
Similarly there is a passive circuit equivalent to the xp equations of Table 1.
This circuit can be realized by multiplying the rotational quantities by a dis
tance h, which makes them linear and so able to combine with the linear equations.
[In fact the linear networks of the active analog, receive rotational couplings
by this same process.]
A peculiarity of the passive (LRC) model is that cLmponents required by
certain electrical analogs (resulting from particular mechanical assumptions)
are unobtainable, e.g. variable transformers. However, it was found(47)that a
more cpPr4A14zed model than Pig. 2, with
mechanical parameters lumped in a particu
lar way would express the same mechanics.
This is shown in Pig.
For the body:
yn+(r,+r,$)(1/110+(k,+k,')(knx
I 11 fir Bs)+k (xB xs )] h
BB x x
, ? $
+Irze(zBzs)+kzqz8zs)]ult Pxh +P
Y f
Por the support, similarly:
mS MS (r xx +e)BS xx+10)B(x )=0
i; +(r w(i )+r X )]
SS zBSxBS
+[k' w(r )+kh(xBxS)) =0
z B x
where K = torque/angle
px
= force .h/dist/h
force 2
.h
dist
= ke h2
SP
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This particular mechanical
hookup may be analyzed rather
simply, electrically [at far
less cost than pn active (opera
tional) analog computer] with
passive (LCR) elements: Fig. 3B.
73
? ,
tfrslEr
)
Experimentally the values of the inductors* (the most costly elements)
determine the impedances throughout the circuit. These are set at an arbitrary
level. The frequency level (machine time) is chosen at 100 x real frequency,
to put the performance spectrum in the range 5  5,000 r/s, where the inductors
have least error from intrinsic R and C. This puts the real spectrum at the
BCG range [.0550 c/s]. The L and C = 1/K values are calculated (as for the
active network) from the assumptions for realtime resonances and damping:
,
WPB = 5 cis.=IKP/1P
Q13 = 2
p
X
2
t"513x = 10 cis =1"6"1
Lw
qBx =2 R +R '
X X
To simulate constraining the roll of support (ps= 0), sever the circuit
at (x) so that body rollpB is excited via lc( and k;. There results a lateral
motion of the support Ms (lateral BCG spectrum) shown in Fig. 3C. It is clear
that the bodyroll strongly distorts the lateral BCG, especially when the
cardiovascular driving force creates (as usual) a clockwise moment. This
moment opposes platform motion Ms below wBp and assists above this roll
resonance. There results a pair of systemfrequencies f1f2 (displaced from
the locked natural frequencies (Ifit) which dominate the lateral BCG spectrum
seen with the usual horizontal (nonroll) platform. This agrees with the
"active analog" analysis above, confirming the passive
*Hycor type EM6.
oria1nrr
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 74 
Results of passive analog solution: response spectrum of the transverse BCG.
A. Effect of body coupling and platform mass.
Before investigating the offprt of adding BCG motion in the roll mode,
the purely translatory solutions were reexamined. This was done to tie in
with previous theory(1) , and to specify quantitatively the amount of coupling
("tethering") needed between body and support in any mode, in terms of the
"hatural frequency" of that coupling referred to a "locked" support. This
is the method of determining coefficients examined theoretically in Chap. III.
In Pig. 3C the solid lines show the influence of relative mass of sup
port with very stiff bodycoupling (77.)B= 10 c/s). In dotted lines are shown
heavy and light supports, with very weak bodycoupling (Wp= 2 c/s).
It is clear that the support could indeed be rather heavy, provided the
coupling is extremely tight [Klo as= 25 x 1( c/s]; but that if no particular
strapping or stressing of the body is used, the mass or inertia of the sup
port must be very much less than that of the body, to obtain satisfactory
responses to the sharp (20 c/s) components of the BCG. This is particularly
pertinent in the roll mode. In Table I the ratio of body inertia about the
roll axis to that of the support is IB /I S ]p = 17.5 According to Fig, 11)
with such a ratio no fastening in roll should be required, even on a flat
surfaced support.
Be Effect of body roll on lateral response.
When the "rotational" meshes of the circuit (Fig. 3B) are added, the
important region 310 c/s acquires a dual resonance, with striking effects
on the 51s /P spectrum.
x
The circled line in Fig. 3D shows the response to a lateral driving
force FBx acting directly through the axis (back alongside the dorsal
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Fig. 3C: Computation on LCR analog.
Effect of bodycoupling comnliance compared with effect of heavy
platform on lateral BCG spectrum:
relative mass (ms/mB) lateral stiffness (1b/in)
A 1/4 75
1/20 45
1/4 3
1/20 3
Showing resonance with tight coupling and heavy support, and loose coupling
sufficient with light support.
9?000 "eseoweola?oowea OOOOO *wow* Following page:
Fig, 3D: LCR analog computations. Effect of nonrolling support on lateral
BCG. [Solid lines] resonance and absorption due to enforced body roll
(Ti 2T, 3T increasing anterior heart position) [dotted lines) resonance
artefact predicted for counterclockwise torque (2T) exerted by heart in
dextrocardia.
Fig. 3B: Roll effects on lateral BCG without torque on body from blood flow:
A, Neither body nor support can roll, both move laterally.
B. Support cannot roll, body can roll, both move laterally.
C6 Both body and support can roll, and move laterally.
(roll resonance absent; lateral response reduced)
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 75 .
aorta). The roll resonance resembles that in Fig. 2C for the same case,
except that here .(Sp is set at 3.5 cis: somewhat less tethering in roll.
Now when the drive is moved anteriorly to the natural position of the
heart (curve T in Fig. 3D) a torque T is produced by F . A similar
Bx
peaking and sharp attenuation result in the active analog (Fig. 2C, curve T).
As this torque is increased (2T and 3T) (as with deep chests), resonance
in roll so increases as to obliterate the trough by phase opposition:
The dotted lines show the change in roll artefact which would occur,
were the
crnrn
^4^^441^11.1
???? ? ?'jag
j???? ?ty es ?
counterclockwise (seen footward).
LW WEk,WAIIV
[This could happen with an upright aorta ordextrocardia.] In this case the
S.shaped distortion of response begins at lower frequency, suppressing the
important 3 c/s BCG component; while a broad resonance 410 cis appears .
a bandpass effect.
Finally, the passive computer was caused to show the motion of the
system when the support was free to roll with the bodyforces. The spectrum
Pig 3E (C:dotdash) is the lateral motion of support reacting to body roll,
but now free of roll resonance. Mechanically, the springs responsible for
this resonance are not excited by such a support. This result agrees with
Fig. 2C. (dotted lines)
These responsespectrum studies of the transverse BCG have been mainly
exploratory, to gain control of the analytical method and models used. The
resonance phenomena shown are produced by these models, and from the parameters
(stiffness and damping) selected for the various modes of motion. The damping
factors were purposely cut down to exaggerate the amplitude (Q) of the resonan
ces, and to seek their physical basis.
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The rotational resonances clearly depended on certain rotatory
moments exerted by the cardiovascular force and the dynamic reactions
to it: (a) about the axis of the body (hf); (b) from the bodyaxis to
the surface of the support ( B' )? and (c) from this surface to the axis
(c.g.) of the support (hs). These important measures differ appreciably
among individuals and for variously designed supports, and must be con
sidered in interpreting the usual lateral riCa.
As in both BCG and ECG, one considers the heartaxis in interpret
ing the record, so in BCG, the effective thickness of chest (hf+hB)
matters whenever a suspension ?is used where bodyroll affects the lateral
*.ar.nrri.
This complication in BCG interpretation can be avoided, if one
gimbals the support to exclude the admixture of roll.
Our comparison of active computer and passive analog methods, in
practice, shows that they serve different purposes. The LCR circuit
is quite adequate to demonstrate the qualitative relations, and explore
the frequency and damping aspects of the coupling between body and support.
The differential analyzer lets one vary separate factors (such as the his)
more readily, and study the phase relations among all the motions and
their derivatives. This was helpful in understanding in detail the
sharp slopes of the response curves.
The analog computer has some disadvantages in practice. It is
slower for getting simple response curves, because at each frequency
several variables were recorded and later measured. With onepercent
precision in the calculator one takes more careful settings. A large
network (Fig. 28) runs into difficulties (not found on the LCR) as to
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signal level, and as to circuit instability from cumulative phaseshift.
This slowed down the problem to 1/10 real time vs. 100 x real time in
the passive analog. The connections are much more numerous and complex,
and require frequent checking. The main advantage is that the computer
imposes less limitation on the model. Thus the LCR model required in
troducing compression springs (capacitors) to simulate the "pure" roll
or stiffness; while the computer executed the corresponding mathematical
.1M.40.+ ....??????????????.f,
operation K (013ps) with a simple voltage divider (coefficient potenti
[ti
aPr1 ile al.
computers can avoid specifying a particular physical
mechanism, when this is unknown or unnecessary. More complex mechanics
(such as interacting rotations in our case) requiring threedimensional
networks, may involve couplings which can not be realized by practically
feasible LCR models.
Further study is needed of the dynamics of rotationtranslation
coupling. For example, the yaw BCG also participates in the lateral
motion; and in a more complex way, since the bodymass is distributed
less symmetrically about the yaw axis, than about the roll axis. It
must be verified experimentally, that a support with low frequency in
yaw, does not intermix the yaw BCG with the lateral BCG. In this case,
due to the several momentarms involved, the accompanying analysis re
quires an analog computer rather than an LCR model.
1
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t
1 L ?
\
/3
yray46)"171114?Ti4r11 T rbr
Fig. 4; A. Lightweight, torsion and yibrationfree support of max, strength
weight ratio.
B, Housing for 3D ultralow frequency suspension.
C. Relation of positive and negative springs of vertical suspension
to u.l.f. leg (see fig. 6).
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^ft
" 10 Nip
IV. DESIGN OF PRACTICAL SUPPORT SYSTEMS: RECUMBENT HUMAN SUBJECT
DETAILED ANALYSIS OF STRUCTURE
Since a great deal of data has been obtained on a recumbent human subject,
the bed was chosen as a support so that a comparison can be made between data
obtained from our design and others. This comparison is expected to resolve
much of the discussion concerning artefact since presumably our design has
reduced artefacts of the suspension to an unobservable value. Even if it
doesn't, the frequency response test of the support/patient combination will
reveal any resonances present which might alter the ballisto records With
this information, such artefacts can be eliminated from the record.
Our objective here is to present the details of the analysis used to
establish the design. Figure A shows the general arrangement of bed, sus
pension and housing. The bed consists of a sheet of aluminum (S) hung from
two parallel side rails (R), held apart by a series of bulkheads (P). The
bulkheads take very little of the patient's weight, most of it being diaphragmed
to the rails. The rail loads are delivered to the truss panels T, from which
the load is fed to earth at the ends via the two vertical legs L. Note the
high torsional rigidity provided by the torque box consisting of the sheet and
two side trusses.
Vertical and pitch frequencies are controlled by the vertical coil springs S
(Fig. 40) at each end. Alone, these springs produce a frequency of liDcps.
Horizontal tension T on the leg in opposition to the Aframes (Fig.4C) produces
a negative (toggle) spring effect, which reduces their stiffness sufficient to
bring the vertical frequency down to 1/3 cps or so.
Roll frequency is controlled by additional vertical Springs on either side
of the leg (not shown) acting between bed and base of outer leg.
Lateral frequencies (including headfoot, side and yaw motion) are controlled
by the legs at each end, each equivalent to a 20' pendular suspension. As will be
described in detail later, the leg contains a positive and negative pendulum,
whose combined stiffness yields a 1/3 cps natural frequency in linear motion.
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 79 R
The housing H is required to protect the bed from everyday abuse and
to provide a loading platform frOm which the patient can transfer his weight
to the bed with a minimum of concentrated and, dynamic loading on the bed
itself. The extreme lightness of bed construction required to get the low
mass makes it important to mount the patient without damage to the bed.
. This housing must establish rigid references for :Settilag the vertical
suspension (legs) plumb to high accuracy (.010). It must deflect little
under loading, tominimize the technicians concern with mechanical adjustments,
A, STRUCTURE OF BED ALONE
To achieve a bed of very low mass required detailed attention to stress
and 'rigidity. First, however, design criteria must be established.
1. Design criteria and loads*
A design load of 300 lbs. was used for static deflection and stress cal
culations. Temporary pressures from this load of 30 iblin2 are possible from
one hand. Test pressures are much lower for the recumbent patient, estimated
at 1 lb/in2
To insure no artefacts in the record from the bed, the stiffness should
be great enough to provide a minimum natural frequency of 40 cps. This is
not an easily applied criterion because the mass entering into oscillation
of this high a frequency is not that of the entire body, upon which the
criterion is based, but something much less. Probably only 20 to 30 pounds
mass  the tissue between the thorax and bed  are affected. To be definite.
25 lb. mass will be used in all stiffness calculations involving the 40 cps
requirement*
This natural frequency requirement will be applied to the first force
free mode of bending and of torsion. The bending axes will be chosen as
vertical and horizontal*
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2. Vertical frequency.
Vertical and lateral stiffness come from the tubular frames. The
bulkheads and sheet serve only to maintain the geometry and to deliver load
to the frames, as far as vertical frequency is concerned. The sheet, of
course, completes the torque box, so necessary to torsional rigidity.
Shown below is one of the truss frames, loaded by unit force, 1/2 lb at each bay:
0
Oteem????????....41011,........A.1110
The stiffn
01.?
.011 60.1,61 eh, ? 14 0414.? J.', I. Mr.. .? ? ? 4, .'.Mt
/ 0 ?
or.???????
?????????????????
ss is computed from the formula for deflection, 21, at the center:
=
LI12
AB
L =
A =
=
length of truss member
area of truss member
load in truss member
deflection
Young's modulus
Horizontals are all lux.03" dural tubes.
dural tubes. End diagonals are 3/8"x.030".
Center diagonals are 5/.8 ix.030
From the above formula:
EA . 2 (1.1)2 (29) 2 (.5)2(13) 4 (1.0)2(26)
.625 (.03)n .625 (.03)n n (.03)
Since this
= 3300 lb/in
is due to a one pound load, stiffness
1
3300
A
= 3000 lb/in
in the plane of the
cr`AmP
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The vertical vertical stiffness of the bed depends on the stiffness of both frames
and the geometry. The dotted outline in the sketch shows the deflection of an
imaginary center bulkhead under unit frame loads, from the zero load, the solid
The vertical load on the bed = 2 sin 6;
the vertical deflection of the bed under this load = 6 = A/sin 6.
Hence, the stiffness of the bed,
2 sin2 0 2
= 2 (9/13) 3000
2900 lb/in in the vertical plane.
The frequency of the bed referred to support is
f=
1 k 1 2900
27t c 2n 125/386
= 34 c/s = 40 c/s
1 Horizontal Frequenci
1
the horizontal horizontal stiffness
is
?????
cos2 8
10
() 3000
12
= 3500 1h/in
from which is computed the frequency
1 3500
2nJ 25/386
?
Side force = 2 cos
0
Horizontal deflection
= 6 = 41/cos 0, Hence,
= 37 c/s = 40 c/s
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immommft?
82
4. Torsional frequency
The torsional rigidity is determined by the torque box formed by the two
trusses and the sheet. For convenience, the shear rigidity of the trusses are
assumed at least equal to the sheet. The shear rigidity of the curved sheet
is assumed equal to that of a flat sheet for low stress levels. In this case,
we can use a simple formula to compute torsional rigidity:
shear modulus
4G A2t A = area enclosed by tube
= length of torque tube
Substituting in numerical values .
4(4x106) (60)2( 015).
kt =
48(78)
= .23x106
t = sheet thickness
U = perimeter of tube section
from which is computed the torsional frequency
rftabrimeirmal
1 kt
2n
.23x10
25 (3Y/386
= 100 C/S > 40 c/s allowed
Local Frequem
The frequency
of
the truss members must themselves be above the critical
frequency. The diagonals have the lowest frequency, being the longest and
thinnest of the members. The expression for frequency of these tubes of
diameter D and length L is:
,
f 11X106 )
2
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?83e.
For the diagonal of the truss under consideration here ?
f = (11x106)
375
(29)2
= 49 c/s 40 c/s allowable
6. Truss stress
330:1k
fuorrYstereave???????????10????????1?01111,????*1?4111.?????????????11.04.1
?
ijow044.Am... vimmilo*.mwo,Vmw
The 330 lb truss load arises from assuming a 300 lb subject placing his
entire weight at that station.
Direct stress on diagonal members
= 490/.375 (.03,0)n = 13,800 lb/in2
Z, 50,000 lb/in2 allowable
Direct stress on longitudinal members
= 2(440)/1 (.030)n = 9,400 lb/in2 e, 50,000 lb/in2
in the lower longitudinal member which carries load from both frames.
Diagonal Euler Load
P =
2n2 ku n 2 (10) (.3l)3(.03)
L2
(29)2
tor
660 lb allowable
) 245 lb actual
Horizontal Euler Load
2n2 e 2n2(102) n(.5)3 (.03)
T2
(26)2
= 3500 lb
400 lb actual
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7. Bulkhead stress
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41ht.
84?
wlz.
\\_,/yrz:
The function of the bulkhead is to
maintain the geometry of the bed.
If the sheet stresses Fs were parallel
(p=o) to the truss reactance FT, the
loading in the plane of the bulkhead
would be essentially zero, except for
some direct loading on the top due
to deflection of the sheet itself.
Loading_LLIalmisj
Static equilibrium requires
FT sin 0 = Fs sin a W/2
since Fs is due to a load in the center directly above the bulkhead. This will
be the critical loading for the bulkhead with W equal to the entire patient weight,
taken here as 300 lb.
The net squeezing load Fl? is the difference between the horizontal components
of Fs and FT
Fi3
For
= Fs cos a ? FT cos 0
= (cos a ? cos 0)
o and a = 30o
45
= 110 lb.
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p . stress at section aa perpendicular to center line
(This section probably has the highest stress)
L
Bending moment
M = 5(110)=550 in lb.
Moment of inertia
Lr ..otgb agkse?
9f2,'
Bending stress is, therefore:
^ MC 550(2.5)
= =  4200 lb/in2
b I .326
Direct Stress
110
= p/A =
6 .(.02)
920 lb/in2
53
4. 2(l/2)(2.5)2)(.020)
= .326 in4
Buckling of flange and web
Stresses are low so that no buckling is expected.
Total stress
0060
Direct bending = 5000 lb/in2 total which is well under allowable
tensile stress, but is probably near ?the allowable buckling stress.
1
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114 SUSPENSION: STIFFNESS
Considerable care was required in getting a suspension of the correct
frequency and damping characteristics. Analysis establishing the design
is contained in this section.
he damping is to be very small so that frequency criteria are based
on undamped natural frequencies.
1. Vertical stiffness.
The design frequency of the vertical springs without negative compensa
tion is 1 c/s, being a practical lower limit. To keep the springs below the
deck requires a length no greater than 20".
spring extended to 20" is 10".
Varying patient weights are accommodated by using several 25 lb.
springs with one at each end being adjustable for leveling and positioning
A practical solid height fora
1 LA
tne oeu esi
conditions:
1
f=l
2n
from which 
tm_,m0.1r1m,%11.. ra rnavieQ0
ly %,LawoCui vA
Z 25
k = (2n) 70
2.5 lb/in
meet the frequency
Extension of this spring to 10 inches requires 25 lb., showing that the
spring does have the required frequency of 1 c/s.
Spring Design
From reference (40),
The stress relation:
nrirm(Hk
The stiffness relation:
HC3k
d = 1
The surge frequency relation:
fs = 14,000/HC2
we have the following relations for the coil sprin
d = wire diameter
r = coil radius
H = solid height (10")
k = spring stiffness (2.5 lb/in)
P = spring max. load (25 lb)
C m 2 rid index of curvature
K = function of C'accounting for
stress concentration due to
curvature
 1.2 for C> 4.0
G m shear modulus (11x10 lb/in2)
lc a working shear stress (75,000 lb/iti)..
f m surge frequency
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87
Substituting in the equation for C
25 (1.2) (11x10 )
n (75,000) (10) 2.5)
= 7.5
The index C must be greater than this to satisfy the stress attainable.
Choosing C to be this minimum value, we have for the wire diameter
8 (10) (7.5)32.5
11 x 106
d
? .0875"
The coil radius r is determined from the definition of the index C
C = 7.6 = 2 r/d
Solving for coil radius,
r = 7.5d/2
c 7.5 (.0875)/2
= .328"
The surge frequency
fs = 14,000/10 (7.5)2
= 25 c/s, a minimum value, so that C cannot be larger.
This is needed also to determine the ratio of maximum surgeforce on the
support, to maximum BCG force on the support.
From appendix C , this ratio is shown to be
R = ,;Ic
(7):171
= 2.5
1(2n25)(.5)
= .032 negligible)
Design conclusion:
Use
where w =
65" mean diameter coil with .0875" diameter wire heattreated
to 250,000 lb/in2.
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Ne tive spring for vertical motion
"re to s
Ts,
????????.0.....~..4%,?......, ?
Above
Above is a sketch showing how the positive spring K is compensated
to achieve the very low frequency If 1/3 c/s. The toggle tod i$ under com
pression by the wire E attached to the spring K' under initial tension To due
to its initial deflection V. This arrangement produces a destabilizing
force P which subtracts from the supporting force Kv, to yield a net force
?(1)
P = K, P
Differentiating with respect to yields the net spring rate
Y = Yo
The equilibrium position yo due to the misalignment
condition
= 0 = K yo
a, Unstable Force %.? P
Taking moments about (A) gives the relation
P L cos $1
T (u sin 62
By geometry, for small angles
sin e1 = y/L
cos 0 =
[2]
s given by the
[3)
el.% a m 1.2.4?
p L
cos 02 1 1 ( Y 441
2
[4
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U = (p.m +./.
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Because of the sliding contact at the spring K', the tension is
computed as
2
2 + (Y+ / 1
= IOW  y/21, 12 pv
Combining equations 4, 5 and 6, we obtain for the unstable force 
[6]
x (p.m 4271 [(L
t
Ieft OL2. 2 '`Iii
(VIL)2) . 211 Li T2 rd)2. p
2 L?L I
V td.2
p L
p L 2
[7]
In order to extend the linearity of P' with respect to vertical dis
placement x we may choose the initial displacement x.! of the K' spring
so that for 4 = 0, the coefficients of y2 in the first curly bracket add
to zero. This requires
. where p = 2 L = 8"
L [8]
xf = L/3 = 2.7"
This assumption resembles setting the curvature (second derivative) of K
to zero at the origin, except the above includes the
A an 1 earnatiii A
which affects the curvature of K.
Simplifying by putting [8) i [7):
2:1 (
p P 2
K'I p . [(1.114)
r +J. or
,}
for the residual force about the zero position.
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1.31_12E1111.
The coefficient of y in [9] gives the first order spring rate of
the unstable force. Combining this with the positive spring, gives the
overall stiffness of the vertical spring system:
k
c, Equilibrium position y
Putting [9) into [3] with [10), we obtain the dependence of zero
position on leveling:
R T0/21, in this case [10)
To
W T.'
pi)
For all practical purposes k
in the negative spring wire Is:
T = K
o?p?si
2(8) 10
= = 160 lb. max. [10 = stiffness of 4 vertical' springs]
1
more accurately, k = 1 at f = 1/3 cis for 150 lbs
Hence, from [10] the tension
?
To =(K .. k.) 36 = (104) 16 = 144 lbs.
Substituting m [11) with W = 175 lb/2, we find for sensitivity to
leveling:
V A
Of 1"4
where
= 10 p 2
td = 2 rad/sec
= 101,
Second order forces
The presence of out oflevel ei, as just shown, very seriously affects
equilibrium position. The question
here Is how much outoflevel can the
design tolerate without affecting the
spring rate. The second order forces
????????????????ftslilliallar.
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( 0%, y2) are small compared to either positive or negative forces
(proportional to y), but not so small compared to their difference*
If we could operate the leg in the displaced position, no difficulty
would arise. However, the design requires that the equilibrium
position be held at y=o by cranking the patient up and down with the
positive spring. This subjects us to changes in spring rate due to second
order forces.
The condition for judging how much 4 can be tolerated is when the
ratio (R) of the forces involved are greater than some design value, say
10 percent. From [lb [10] and [11], the relation between yo and this
ratio RI is
Yo 2BR
ft'vd"1". where R s s 6K
01
For a 10 percent restriction,
y0=10
2 (2)(.1)
(21)
[12]
a 6.3 in,
so A = .63 in, out of level to change spring rate by 10 percent
or frequency by 5 percent.
5. Results.
*.?????.wartor?Nry.ar.somOSNA
Although the equilibrium position is
quite sensitive to level of the
tension wire, a given position can be maintained by cranking the positive
spring without fear of changing stability. Hence the tension wire need
not be releveled for practical operation of this suspension.
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prinnArInr,,,
nn A
?
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2. Horizontal stiffness.
The legs, one at each end of the bed, provide horizontal restoration(Fig .6).
As indicated earlier, these legs embody a differential (negative and posi
tive) pendulum which is equivalent to a pendulum adjustable from 8 to 20 ft.
long. The frequency of linear motion in a horizontal plane is, according to
the wellknown relation: f = 1/2n fir.
A pendulum restoring force is more convenient than springs because the
frequency is independent of the subject weight, a widely varying quantity in
this application.
Geneal_t12sau.
The essential features of the differential pendulum leg are illustrated
in the accompanying sketch, where the displacements have been greatly exag
gerated for clarity.
?"..`?
Fl
C
The outer tube T has a bridge at A
from which is suspended the positive
pendulum of length a. At its ex
tremity, the point B, is hung the
inner tube (b+c), the negative
pendulum, which is constrained to
vertical motion only at C (by the
outer tube). The bed weight rests
on the upper end D of the inner tube,
and is free to swing around in any
horizontal direction (circular
Pendulum). Actually, the wire a runs
down the inner tube (b+c) via a slide
clamp at B and receives the load at C.
As will be seen later, the frequency
is controlled by sliding the clamp at
B along the inner tube (b+c).
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The, point C is supposed to be on a plumb line through A, but any looseness
in the fit at C or rocking of the outer leg T can cause a departure from plumb,
indicated by A, In the scale showniei and x would be so small that a and
(b+c) would appear on a single line.
A displacement x is restored by the force in the wire a, but is aided by
the leverage of the inner tube against the side of the outer tube at C.
ELITIETEX?
The frequency of horizontal motions x, y, y (yaw) with this suspension
depends mainly on the geometry (as with simple pendulums) and not on the load.
If we neglect the stiffness of the wire it is shown (appendix D) the
horizontal frequency is that of a simple pendulum, multiplied by a difference
of two small quantities easil adjusted to near zero:
. 1. gla c2ab
2n , "172Ur when plumb (see below for "out of plumb")
That is, instead of a simple pendulum of frequency OFT we have a differ
ential pendulum adjustable to zero frequency (infinite period when c2 = ab).
Frequency control in this design, results from moving a slide at point B
in the previous figure, a distance along the inner tube (c+b). (Set Pig. 6)
If B is the center position of (b+c),
we have b+y = cy
So that
b
With y replacing et:, the expression for frequency in Appendix D becomes
?1
4na
when f = 0
) 9: PY7i.
4
yo ?.187"
when I 1/3 c/s: y = .96"
where ri= ba = .75"
0
which indicates substantial stability.
It is concluded that if the wire loop is clamped to the outer leg,
no special sensitivity to alignment is expected.
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a
 99 *6
Ce SUSPENSION: DAMPING
A small amount of damping is desirable to settle the system out follow
ing the many and varied external dlsturbances that will occur. Without some
damping, the patient and support would be in a continual state of low fre
quency vibration.
1. Formulation of
The damping coefficients in each axis of the system are related to
the respective mass and spring characteristics as follows:
for linear motion:
= 2 in wit along any axis i
for angular motion: ri = 2 Iiwit about any axis i
where w. (with ittaia) denotes the natural frequency associated with this
axis.
(a) The inertial toefficients
are determIned as fnllnwg!
In = mass of bed plus patient
= 175/384 = .45 slugs"
= moment of inertia
[(inch) slugs]
For the transverse axes, considering the bed and patient as a
long, uniform rod, the inertia would be
I = in L2/12 where L is the length of the bed. Based on this
Ia =
211L
I = :AL, /16 = .45
07212/16 = 170 slug in2.
[Division by 16 instead of 12 accounts for the ununiformity of mass dis
tribution along the bed, it being denser near the c.g. than at the ends].
For roll axis, the inertia is more or less a guess at the radius of
gyration '0.) in the basic formula,
2
rat)
= .45 (5)2 = 11.3 slug in2
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 100 
(b)The(undamped)naturalies(w.)are found as follows:
1
For the linear motions our design criterion was a linear natural frequency
2 rad/see,
ixyz
stiffness (k) giving this frequency was provided by the legs at each end
of the bed.
For angular motions, the natural frequencies may be found as followst
The legs resist an angular displacement (a) of the bed about the pitch (x)
axis by a torque kzL2 a/4. Substituting the general relation for angular
natural frequency about any axis 0 (corresponding to k
Then abo7rt
2 torque/8
w =
0
this axis
2
CA) = (K L2/4) (mL2/16)
a z
w 2w
a z
Thus, wa = wy
since kz = mw,
= 4 rad/sec.
That is, the natural frequency in pitch and yaw are twice that in translation.
The roll frequency wp is determined separately, by coil springs designed
to give w = 2 rad/sec.
(c) Damping coefficient assumed.
The damping ratio is chosen as .25 for each axis to keep the phase
shift low at 1 c/s when the cutoff frequency is 1/3 c/s = 2 rad/sec.
Substituting these values in the relations for the linear damping coefficients
required:
??
X
 rz = 2 (.45) 2) .25)
.45 lb/in/sec.
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I
1
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101
For the rotational damping coefficients required
= r = 2 (170) (4) (.25)
a
= 340 in/lb/rad/sec.
rp = 2 (11.3) (2) (.25)
= 11.3 in/lb/rad/sec.
Individual damping elements must be sized and positioned to meet
these conditions on damping.
2. T pe of damping element used
The greatest energy absorption per unit volume is obtained from
Couette flow, characterized by two close plates separated by a thin viscous
fluid. The resistance of one plate relative to another is given by the re
(52)
lation, as per page 621 of Lamb.
where
/ //t) //