Nonlinear dynamic behavior of the human knee joint Part I:
Postmortem frequency domain analyses
Citation for published version (APA):
Dortmans, L. J. M. G., Jans, H., Sauren, A. A. H. J., & Huson, A. (1991). Nonlinear dynamic behavior of the
human knee joint Part I: Postmortem frequency domain analyses. Journal of Biomechanical Engineering :
Transactions of the ASME, 113(4), 387-391. https://doi.org/10.1115/1.2895416
DOI:
10.1115/1.2895416
Document status and date:
Published: 01/01/1991
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L. Dortmans
Eindhoven University of Technology,Faculty of Mechanical Engineering, Eindhoven, The Netherlands; Presently, Centre for Technical Ceramics, Eindhoven, The Netherlands
H. Jans
A. Sauren
A. Huson
Eindhoven University of Technology,Faculty of Mechanical Engineering, Eindhoven, The Netherlands
Nonlinear Dynamic Behavior of the
Human Knee Joint—Part I:
Postmortem Frequency Domain
Analyses
Characteristic results of postmortem experiments on five knee-joint specimens are
reported. The experiments were performed to investigate the applicability of a local
linearization technique that would make it possible to describe the dynamic behavior
of the joint in terms of transfer functions. The results indicate that the stiffness of
the bracing wires, attached to muscle tendons to create a static equilibrium position,
can be accounted for when determining the stiffness of the joint. Besides the static
equilibrium configuration, the magnitude of the dynamic load and the type of
dynamic load applied to the joint can be shown to have their influence. As the
influence of the dynamic load is significant, it has to be concluded that in essence
the knee joint has to be regarded as a nonlinear system, making application of a
Local Linearization Technique questionable. However, when the magnitude of the
dynamic load is included as an additional measurement parameter, an indication
can be obtained about the behavior of the joint and the degree of nonlinearity.
1 Introduction
Although in many human activities the loading of the human
knee joint is of a dynamic nature, little fundamental research
has been done on the dynamic behavior of this joint (Hefzy
and Grood, 1988). The presence of highly incongruent load
bearing joint surfaces together with menisci, ligaments,
cap-sule, and muscles make the knee one of the most complex
joints in the musculoskeletal system. Apart from their
impor-tance for maintenance of joint stability, the different joint
elements may also influence the transmission of dynamic loads.
The work reported in this paper is part of a long-term research
project aiming at the development of an experimentally
vali-dated model of the dynamic behavior of the human knee joint,
with parameters that can be interpreted in terms of the
prop-erties and functions of the joint elements and interactions
between the elements.
After reviewing literature on this subject one cannot but
endorse the statement made by Hefzy and Grood (1988) that
" . . . the lack of experimental data needed in the determination
of model system parameters and in the validation of the model,
is the main reason why mathematical knee joint modelling has
not progressed further."
Because of the limited amount of experimental data and
validated models for the dynamic behavior of the human knee
joint an attempt has been made to gain more insight into the
behavior of the joint by means of experiments. It was decided
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENOINEERING. Manuscript received by the Bioengineering Division March 1, 1990; revised manuscript received January 15, 1991.to start an explorative experimental investigation into the
me-chanical characeristics of the human knee joint under dynamic
loading. It should be recognized that in general the dynamic
behavior of the joint must be described by means of 1)
rela-tionships between loads exerted on the joint (loads due to
muscular activity, and loads due to inertial and gravitational
effects), 2) the 3-dimensional position and orientation of the
tibia relative to the femur and 3) deformations of and stress
distributions in the joint elements, which are related by their
individual constitutive behavior. All these relationships will in
general be time-dependent and nonlinearly coupled, where it
can be assumed that the knee joint behaves as a nonlinear
system with nonlinear and time-dependent stiffness and
damp-ing characteristics. The nonlinearities can be of a geometrical
as well as of a physical nature due to the large movements of
the femur with respect to the tibia and the nonlinear material
characteristics of the joint elements.
In an explorative experimental investigation the dynamic
behavior described above is much too complicated to deal with,
due to a lack of experimental techniques to quantify all
pa-rameters and quantities governing the relationships mentioned.
Therefore some restrictions had to be made. For a start it was
assumed that a Local Linearization Technique (LLT) might
yield an appropriate experimental procedure. First results
ob-tained from this procedure have been reported in an earlier
paper (Jans et al., 1988). The LLT basically consists of two
steps. The first step involves the creation of a stable, static
equilibrium configuration of the joint by means of a static
load. To include one important effect of muscular activity this
Journal of Biomechanical Engineering NOVEMBER 1991, Vol. 113 / 387
Copyright © 1991 by ASME
L.
Dortmans
Nonlinear Dynamic Behavior of the
Human Knee Joint-Part I:
Eindhoven University of Technology, Faculty of Mechanical Engineering, Eindhoven, The Netherlands; Presently, Centre for Technical Ceramics, Eindhoven, The Netherlands
Postmortem Frequency Domain
Analyses
H. Jans
A. Samen
A. Huson
Eindhoven University of Technology, Faculty of Mechanical Engineering, Eindhoven, The NetherlandsCharacteristic results of postmortem experiments on five knee-joint specimens are reported. The experiments were performed to investigate the applicability of a local linearization technique that would make it possible to describe the dynamic behavior of the joint in terms of transfer functions. The results indicate that the stiffness of the bracing wires, attached to muscle tendons to create a static equilibrium position, can be accounted for when determining the stiffness of the joint. Besides the static equilibrium configuration, the magnitude of the dynamic load and the type of dynamic load applied to the joint can be shown to have their influence. As the
influence of the dynamic load is significant, it has to be concluded that in essence
the knee joint has to be regarded as a nonlinear system, making application of a Local Linearization Technique questionable. However, when the magnitude of the
dynamic load is included as an additional measurement parameter, an indication
can be obtained about the behavior of the joint and the degree of nonlinearity.
1 Introduction
Although in many human activities the loading of the human knee joint is of a dynamic nature, little fundamental research has been done on the dynamic behavior of this joint (Hefzy and Grood, 1988). The presence of highly incongruent load bearing joint surfaces together with menisci, ligaments, cap-sule, and muscles make the knee one of the most complex joints in the musculoskeletal system. Apart from their impor-tance for maintenance of joint stability, the different joint elements may also influence the transmission of dynamic loads. The work reported in this paper is part of a long-term research project aiming at the development of an experimentally vali-dated model of the dynamic behavior of the human knee joint, with parameters that can be interpreted in terms of the prop-erties and functions of the joint elements and interactions between the elements.
After reviewing literature on this subject one cannot but endorse the statement made by Hefzy and Grood (1988) that " ... the lack of experimental data needed in the determination of model system parameters and in the validation of the model, is the main reason why mathematical knee joint modelling has not progressed further."
Because of the limited amount of experimental data and validated models for the dynamic behavior of the human knee joint an attempt has been made to gain more insight into the behavior of the joint by means of experiments. It was decided
Contributed by the Bioengineering Division for publication in the JOURNAL OF BJOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division March 1, 1990; revised manuscript received January 15, 1991.
Journal of Biomechanical Engineering
to start an explorative experimental investigation into the me-chanical characeristics of the human knee joint under dynamic loading. It should be recognized that in general the dynamic behavior of the joint must be described by means of 1) rela-tionships between loads exerted on the joint (loads due to muscular activity, and loads due to inertial and gravitational effects), 2) the 3-dimensional position and orientation of the tibia relative to the femur and 3) deformations of and stress distributions in the joint elements, which are related by their individual constitutive behavior. All these relationships will in general be time-dependent and nonlinearly coupled, where it can be assumed that the knee joint behaves as a nonlinear system with nonlinear and time-dependent stiffness and damp-ing characteristics. The nonlinearities can be of a geometrical as well as of a physical nature due to the large movements of the femur with respect to the tibia and the nonlinear material characteristics of the joint elements.
In an explorative experimental investigation the dynamic behavior described above is much too complicated to deal with, due to a lack of experimental techniques to quantify all pa-rameters and quantities governing the relationships mentioned. Therefore some restrictions had to be made. For a start it was assumed that a Local Linearization Technique (LL T) might yield an appropriate experimental procedure. First results ob-tained from this procedure have been reported in an earlier paper (Jans et aI., 1988). The LLT basically consists of two steps. The first step involves the creation of a stable, static equilibrium configuration of the joint by means of a static load. To include one important effect of muscular activity this
static load can also be used to.generate a compressive preload, which may have considerable effect on the transmission of dynamic forces through the joint. The second step in the LLT is the application of dynamic loads such that onlyrelatively small changes in the joint configuration will occur. It is as-sumed that these changes are small enough to obtain a dynamic behavior that corresponds to the behavior of a linear system with constant mass, damping and stiffness characteristics. These characteristics may depend on the static equilibrium configuration and the static load exerted, but it is essential that they are constant for'a certain range of the magnitude of the applied dynamic loads. This range must be determined experimentally.
This approach is attractive because an experimental tool, the so-called Transfer Function Analysis (TFA) , is available for the analysis of linear systems, yielding a full description of their dynamic behavior (Bendat and Piersol, 1980).
It must be realized that this approach may fail for two reasons. First it is possible that the behavior of the joint is essentially nonlinear such that nonlinearities arising from non-linear damping or nonnon-linear static load-displacement charac-teristics cannot be linearized. Beforehand this is not expected to occur as the friction in the knee joint is found to be very small (Radin and Paul, 1972) and because a LLT has been applied by Crowninshield et al. (1976), Moffat et al. (1969) and Pope et al. (1976), to describe joint behavior without giving indications for possible failure.
Second, generally available measurement techniques provide a threshold for the minimal magnitude of dynamic loads to be applied as below this threshold the signal-to-noise ratio considerably decreases. This finite measurement accuracy may makeitimpossible to apply the small dynamic loads required to allow for the local linearization. Because of the lack of knowledge of the dynamic behavior of the joint, at this juncture
itwas not possible to judge whether these factors would play a disturbing role. Consequently it was decided to start with the experimental procedure described above bearing in mind the possible reasons for failure mentioned.
2
Experimental Procedure
An extensive description of the experimental setup, used in the experiments and shown in Fig. 1, has been given already (J ans et aI., 1988). Figure 1 shows the specimen mounted in this setup.
An electro-mechanical shaker(1)was used to apply a small dynamic load to the tibia. The accelerations of the tibia were measured by means of accelerometers mounted on a cylinder in which the tibia is cast. The dynamic load was chosen to be a zero-mean random signal generated by means of a comput-erized data-acquisition system (Jans et aI., 1988). Its variance
Utcould not be set lower than approximately 1 N2because of a poor signal-to-noise ratio in the measured accelerations. Un-der the small dynamic load the displacements of the tibia were small too: they were hardly observable by eye and therefore were not likely to result in geometrical nonlinearities (which are to be avoided when using the LLT).
Following a standard procedure the experiments started with the preparation of the joint specimen the day before the actual measurements were done. This preparation took about 4 hours. The description given below is valid for all specimen used in' the experiments described. All knee joint specimens were ex-cised from macroscopically intact human cadavera, freshly frozen at - 80
'c.
Each specimen had an overall length of 0.4 m in extension, with an equal length of the tibia and the femur of 0.2 m. The day prior to the experiments the specimen was thawed in approximately 4 hours in water of 20°C. The prep-aration started with removal of the skin and subcutis. Sub-sequently the heads of the muscles were removed, such that the relevant muscle tendons were mobilised but left intact (the388 IVol. 113, NOVEMBER 1991
Fig. 1 The upper picture shows the electro·mechanical shaker (1), sus· pended In the hook (2) of a travelling crane, connected to the tibia by means of a flexible connection. This connection Is shown in detail in the lower picture: a
=
force transducer, b=
flexible string, c=
flexible hose, d=
magnet, e = head of the electro·mechanical shaker, R =stainless steel cylinder Into which the distal part of the tibia is fixed, g
=
gravity.tendon of the rectus femoris, biceps femoris, gracilis, sartorius, and semitendinosus muscles). During preparation care was taken to keep the joint capsule and extra-capsular ligaments intact. To be able to exert the static load on the muscle tendons by means of bracing wires a minimum length of approximately
0.1m of the tendons was required. After preparation the spec-imen was stored overnight at SoC in Ringer's solution and used in the experiments the day after.
3
Experimental Results
Measurements were carried out on five knee joint specimens (denoted KNEEl through KNEES), varying the static equilib-rium position (controlled by means of the static preload), the magnitude of the static preload, the magnitude of the dynamic load and the direction in which the dynamic load was applied. Because of autolysis the total measurement time was limited to two days. Therefore not each of the influencing parameters mentioned above could be analyzed for each knee joint spec-imen. In the sequel a number of results are presented which are considered of prime importance regarding I) the question whether the LLT is a valid procedure and 2) to find essential characteristics of the knee joint. Results for experiments on KNEEl, KNEE3, KNEE4, and KNEES are described in the subsequent sections, focusing on these two key topics. There-fore a discussion on variability between different knee joint specimens is omitted as this was not the prime objective of the experiments carried out. KNEE2 was used for experiments in which selected joint elements were damaged. An extensive dis-cussion on the influence of these operations on the dynamic behavior of the joint is omitted here because this is described in more detail in a separate paper (Dortmans et aI., 1991).
The static forces acting on the muscle tendons are denoted by Fro Fb , and Fa representing the force on the rectus femoris
muscle, biceps femoris muscle, and the pes anserinus, respec-tively. The flexion angle of the joint is denoted by<jJ (Jans et aI., 1988). The results are given in terms of transfer-functions
H llfbetween the applied loadfe and the displacement of the tibiau (u = X, Y, orZ)in a specific direction relative to the static equilibrium configuration:Hllf{f) = u (f)Ife(f),where
f
is the frequency in Hz (J ans et aI., 1988). As shown in a previous paper (Jans et aI., 1988) the frequency interval ofTransactions of the ASME
Downloaded 20 Nov 2008 to 131.155.151.52. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm static load can also be used to.generate a compressive preload,
which may have considerable effect on the transmission of dynamic forces through the joint. The second step in the LL T is the application of dynamic loads such that only relatively small changes in the joint configuration will occur. It is
as-sumed that these changes are small enough to obtain a dynamic behavior that corresponds to the behavior of a linear system with constant mass, damping and stiffness characteristics. These characteristics may depend on the static equilibrium
configuration and the static load exerted, but it is essential that they are constant for'a certain range of the magnitude of the applied dynamic loads, This range must be determined experimentally.
This approach is attractive because an experimental tool, the so-called Transfer Function Analysis (TFA), is available
for the analysis of linear systems, yielding a full description of their dynamic behavior (Bendat and Piersol, 1980).
It must be realized that this approach may fail for two reasons. First it is possible that the behavior of the joint is essentially nonlinear such that nonlinearities arising from non-linear damping or nonnon-linear static load-displacement charac-teristics cannot be linearized. Beforehand this is not expected
to occur as the friction in the knee joint is found to be very
small (Radin and Paul, 1972) and because a LL T has been applied by Crowninshield et al. (1976), Moffat et al. (1969) and Pope et al. (1976), to describe joint behavior without giving
indications for possible failure.
Second, generally available measurement techniques provide
a threshold for the minimal magnitude of dynamic loads to be applied as below this threshold the signal-to-noise ratio considerably decreases. This finite measurement accuracy may make it impossible to apply the small dynamic loads required to allow for the local linearization. Because of the lack of knowledge of the dynamic behavior of the joint, at this juncture
it was not possible to judge whether these factors would play
a disturbing role. Consequently it was decided to start with the experimental procedure described above bearing in mind the possible reasons for failure mentioned.
2 Experimental Procedure
An extensive description of the experimental setup, used in
the experiments and shown in Fig. 1, has been given already (lans et aI., 1988). Figure 1 shows the specimen mounted in this setup.
An electro-mechanical shaker (1) was used to apply a small dynamic load to the tibia. The accelerations of the tibia were measured by means of accelerometers mounted on a cylinder in which the tibia is cast. The dynamic load was chosen to be a zero-mean random signal generated by means of a
comput-erized data-acquisition system (Jans et aI., 1988). Its variance
uJ could not be set lower than approximately 1 N2 because of
a poor signal-to-noise ratio in the measured accelerations. Un-der the small dynamic load the displacements of the tibia were
small too: they were hardly observable by eye and therefore were not likely to result in geometrical nonlinearities (which
are to be avoided when using the LL T).
Following a standard procedure the experiments started with the preparation of the joint specimen the day before the actual
measurements were done. This preparation took about 4 hours. The description given below is valid for all specimen used in'
the experiments described. All knee joint specimens were
ex-cised from macroscopically intact human cadavera, freshly frozen at - 80
'c.
Each specimen had an overall length of 0.4 m in extension, with an equal length of the tibia and the femurof 0.2 m. The day prior to the experiments the specimen was thawed in approximately 4 hours in water of 20°C. The prep-aration started with removal of the skin and subcutis.
Sub-sequently the heads of the muscles were removed, such that the relevant muscle tendons were mobilised but left intact (the
388 I Vol. 113, NOVEMBER 1991
Fig, 1 The upper picture shows the electro·mechanical shaker (1), sus·
pended In the hook (2) of a travelling crane, connected to the tibia by means of a flexible connection. This connection is shown in detail in the lower picture: a
=
force transducer, b=
flexible string, c=
flexible hose, d=
magnet, e = head of the electro·mechanical shaker, R = stainless steel cylinder into which the distal part of the tibia is fixed,g
=
gravity.tendon of the rectus femoris, biceps femoris, gracilis, sartorius, and semitendinosus muscles). During preparation care was taken to keep the joint capsule and extra-capsular ligaments intact. To be able to exert the static load on the muscle tendons by means of bracing wires a minimum length of approximately
0.1 m of the tendons was required. After preparation the spec-imen was stored overnight at SoC in Ringer's solution and used in the experiments the day after.
3 Experimental Results
Measurements were carried out on five knee joint specimens
(denoted KNEEl through KNEES), varying the static
equilib-rium position (controlled by means of the static preload), the magnitude of the static preload, the magnitude of the dynamic load and the direction in which the dynamic load was applied. Because of autolysis the total measurement time was limited to two days. Therefore not each of the influencing parameters mentioned above could be analyzed for each knee joint
spec-imen. In the sequel a number of results are presented which
are considered of prime importance regarding 1) the question whether the LL T is a valid procedure and 2) to find essential characteristics of the knee joint. Results for experiments on KNEEl, KNEE3, KNEE4, and KNEES are described in the
subsequent sections, focusing on these two key topics. There-fore a discussion on variability between different knee joint
specimens is omitted as this was not the prime objective of the experiments carried out. KNEE2 was used for experiments in which selected joint elements were damaged. An extensive dis -cussion on the influence of these operations on the dynamic behavior of the joint is omitted here because this is described in more detail in a separate paper (Dortmans et aI., 1991).
The static forces acting on the muscle tendons are denoted by Fro Fb , and Fa representing the force on the rectus femoris muscle, biceps femoris muscle, and the pes anserinus,
respec-tively. The flexion angle of the joint is denoted by ¢ (lans et aI., 1988). The results are given in terms of transfer-functions HIIJ between the applied load fe and the displacement of the tibia u (u = X, Y, or Z) in a specific direction relative to the
static equilibrium configuration: HIIJ(f) = u (f) Ife (f), where
f
is the frequency in Hz (Jans et aI., 1988). As shown in aprevious paper (Jans et aI., 1988) the frequency interval of
Fig. 2 Influence function Hxl(f)
0 0.75 mm I 1.00 mm
— f [Hz]
of the diameter of the bracing wires upon the transfer 60 deg; Fb = 131; F, 60 deg; Fb = 131; F, KNEE3: a, = 200; F, = KNEE3: a, = 200; Fa = = 20; 0 133 = 20; 0 133 - M [Hz]
Fig. 4 Influence of the flexion angle of the joint upon the transfer function Hxl(f) KNEE3: a, = 20; 0 = 20 deg; Fb = 131; Fr = 200; F„ = 133 KNEE3: a, = 20; <j> = 30 deg; F„ = 131; Fr = 200; Fa = 133 KNEE3: a, = 20; 0 = 60 deg; Fb = 131; Fr = 200; Fa = 133 5E-4- 1E-5-, — ~ f [Hz] Fig. 3 Influence of the diameter of the bracing wires upon the transfer function Hyi(f)
0 0.75 mm KNEE3: a, = 6; <A = 30 deg; Fb = 120; F, = 210;
Fa = 133
0 1.00 mm KNEE3: a, = 6; 4, = 30 deg; Fb = 120; Fr = 210; F, = 133
interest is from 5 to 50 Hz. In this frequency range two distinct
resonance frequencies of approximately 20 and 28 Hz could
be detected which correspond to two vibration modes of the
tibia denoted mode I and mode II, respectively.
3.1 Influence of the Bracing Wires. Figures 2 and 3 give
results for experiments on KNEE3 in which the diameter of
the bracing wires was changed from 1.0 to 0.75 mm. From
Fig. 2 follows that vibration mode II is hardly affected by the
stiffness of the bracing wires, whereas Fig. 3 indicates the
opposite for mode I. For mode I the modal stiffness K (the
combined stiffness of the joint and the bracing wires) was
determined for both cases by means of a curve-fit procedure
(Mergeay, 1980). The modal stiffness for a diameter of 0.75
and 1.0 mm, K
QJSand K
L0, was calculated as 14500 N/m and
10470 N/m, respectively. Because the transversal stiffness of
the bracing wires is negligible, only their longitudinal stiffness
is relevant. If this stiffness would dominate the intrinsic
stiff-ness of the joint,
KQJSand K
l0would be proportional to the
square of the diameter of the wires. From the values given
above, it is concluded that this does not apply. Hence it is
concluded that the stiffness of the bracing wires cannot be
neglected, but on the other hand is such that the contribution
of the joint to the modal stiffness can be determined. In a
numerical model this influence can be incorporated and
there-fore no modifications were introduced.
3.2 Influence of the Static Equilibrium Position. Various
experiments have been conducted to study the influence of the
static equilibrium configuration. Figures 4 and 5 show some
typical results for KNEE3. It is observed that changes in the
static equilibrium position result in measurable changes in the
transfer functions. For both mode I and mode II, increase of
IHI [m/N]
t
- * f [Hz]
Fig. 5 Influence of the flexion angle of the joint upon the transfer function Hz,(f)
KNEE3: u, = 20; <t> = 20 deg; Fb = 131; F, = 200; F„ = 133 KNEE3: a, = 20; <j> = 30 deg; Fb = 131; F, = 200; Fa = 133
— . — .—. KNEE3: u, = 20; <|> = 60 deg; Fb = 131; F, = 200; Fs = 133
the flexion angle leads to a decrease in the resonance frequency,
indicating a reduction in the resistance of the joint against
applied forces.
3.3 Influence of the Static Preload. Another parameter
influencing the dynamic behavior of the knee joint is the
mag-nitude of the static preload exerted by the bracing wires. Figure
6 shows the transfer function H
yf(f) for KNEE1 for three
levels of the static preload. Obviously an increase of the forces
acting on the muscle tendons results in an increase of the
resistance of the joint such that the resonance frequency for
mode I increases.
3.4 Influence of the Dynamic Load. A key factor to judge
the applicability of the transfer functions given in the preceding
sections is to which extent the magnitude of the dynamic load
influences the results. In a previous paper (Jans et al., 1988)
results of preliminary experiments were reported, suggesting
that the magnitude of the dynamic load had only a minor
influence. More extensive research has been devoted to this
particular point in the experiments on KNEE3 through KNEE5.
Figures 7 and 8 give the transfer functions for two experiments
on KNEE4. Although the coherence function, corresponding
to these transfer functions, always attained values between 0.95
and 1.0, these results show that the influence of the magnitude
of the dynamic load cannot always be neglected. Such findings
make use of a LLT questionable. To verify that the results are
not caused by the non-deterministic nature of the loading
pat-tern (random excitation), additional experiments were carried
out on KNEE5 with 4> = 30 deg using sinusoidal excitation
with varying amplitude (Fig. 9). The results from these
ex-periments confirm the results discussed before (both in a
qual-itative and a quantqual-itative sense). It may therefore be concluded
that the nonlinearities observed are not bound to a particular
excitation signal.
Journal of Biomechanical Engineering NOVEMBER 1991, Vol. 113 / 389
4E-4 IHI
[miN]
5
Fig. 2 Influence of the diameter of the bracing wires upon the transfer function Hx/(f)
o
0.75 mm _ _ _ KNEE3: (I/ = 20; <I>=
01.00 mm 5E-4 IHI [miN]
t
= 200; F. = 133 _ _ _ KNEE3: (I/ = 20; <I>= 200; F. = 133 o
t----5 60 deg; Fb = 131; F, 60 deg; Fb = 131; F, 5'0 - f [Hz]Fig. 3 Influence of the diameter of the bracing wires upon the transfer function Hy/(f)
00.75 mm _ _ _ ~~:E~~~/
=
6;<1>=
30deg; Fb=
120; F,=
210; 01.00 mm _ _ _ KNEE3: (I/ = 6; <I> = 30 deg; Fb = 120; F, =210; F.
=
133interest is from 5 to 50 Hz. In this frequency range two distinct resonance frequencies of approximately 20 and 28 Hz could be detected which correspond to two vibration modes of the tibia denoted mode I and mode II, respectively.
3.1 Influence of the Bracing Wires. Figures 2 and 3 give results for experiments on KNEE3 in which the diameter of the bracing wires was changed from 1.0 to 0.75 mm. From Fig. 2 follows that vibration mode II is hardly affected by the stiffness of the bracing wires, whereas Fig. 3 indicates the opposite for mode 1. For mode I the modal stiffness K (the combined stiffness of the joint and the bracing wires) was determined for both cases by means of a curve-fit procedure (Mergeay, 1980). The modal stiffness for a diameter of 0.75 and 1.0 mm, KO.75 and Kl.o, was calculated as 14500 N/m and
10470 N/m, respectively. Because the transversal stiffness of the bracing wires is negligible, only their longitudinal stiffness is relevant. If this stiffness would dominate the intrinsic stiff-ness of the joint, KO.75 and Kl.o would be proportional to the
square of the diameter of the wires. From the values given above, it is concluded that this does not apply. Hence it is concluded that the stiffness of the bracing wires cannot be neglected, but on the other hand is such that the contribution of the joint to the modal stiffness can be determined. In a numerical model this influence can be incorporated and there-fore no modifications were introduced.
3.2 Influence of the Static Equilibrium Position. Various experiments have been conducted to study the influence of the static equilibrium configuration. Figures 4 and 5 show some typical results for KNEE3. It is observed that changes in the static equilibrium position result in measurable changes in the transfer functions. For both mode I and mode II, increase of
Journal of Biomechanical Engineering
4E-4
l
IHI [miN],I~
J"\
l::::=: .
\",>-~"--0 5 ' 5'0 -f[Hz]Fig. 4 Influence of the flexion angle of the joint upon the transfer function HxtU)
_ _ _ KNEE3: (I/
=
20; <I>=
20 deg; Fb=
131; F,=
200; Fa=
133 KNEE3: (I/ = 20; <I> - 30 deg; Fb=
131; F,=
200; Fa=
133 :-:-. ~.-:-: KNEE3: (II=
20; <I> - 60 deg; Fb = 131; Fr=
200; F.=
133lE-5 IHI [mIN]
t
o+-~---,--,---~~~~~r=~==~ 5 50 - f [Hz]Fig. 5 Influence of the flexion angle of the joint upon the transfer function Hzl(f)
_ _ _ KNEE3: (I/
=
20; <I>=
20 deg; Fb=
131; F,=
200; F.=
133 KNEE3: (I/ = 20; <I> - 30 deg; Fb = 131; Fr=
200; F.=
133 :-:-. ~.-:-: KNEE3: (I/ = 20; <I> - 60 deg; Fb = 131; Fr = 200; F. = 133the flexion angle leads to a decrease in the resonance frequency, indicating a reduction in the resistance of the joint against applied forces.
3.3 Influence of the Static Preload. Another parameter influencing the dynamic behavior of the knee joint is the mag-nitude of the static preload exerted by the bracing wires. Figure 6 shows the transfer function HyJ(j) for KNEEl for three levels of the static preload. Obviously an increase of the forces acting on the muscle tendons results in an increase of the resistance of the joint such that the resonance frequency for mode I increases.
3.4 Influence of the Dynamic Load. A key factor to judge the applicability of the transfer functions given in the preceding sections is to which extent the magnitude of the dynamic load influences the results. In a previous paper (Jans et aI., 1988) results of preliminary experiments were reported, suggesting that the magnitude of the dynamic load had only a minor influence. More extensive research has been devoted to this particular point in the experiments on KNEE3 through KNEES. Figures 7 and 8 give the transfer functions for two experiments on KNEE4. Although the coherence function, corresponding to these transfer functions, always attained values between 0.95 and 1.0, these results show that the influence of the magnitude of the dynamic load cannot always be neglected. Such findings make use of a LL T questionable. To verify that the results are not caused by the non-deterministic nature of the loading pat-tern (random excitation), additional experiments were carried out on KNEES with 1> = 30 deg using sinusoidal excitation with varying amplitude (Fig. 9). The results from these ex-periments confirm the results discussed before (both in a qual-itative and a quantqual-itative sense). It may therefore be concluded that the nonlinearities observed are not bound to a particular excitation signal.
7E-4 IHI -!m/N]
f [Hz 50
Fig. 6 Influence of the static preload upon the transfer function HyAI)
KNEE1: a, = 20; 0 = 20 deg; F„ = 40; Fr = 90; F. = 60 KNEE1: a, •• K N E E 1 : < T , 20 deg; F„ = 84; Fr = 210; F, 90 2; (6 2; <f = 20 deg; Fb = 224; F, = 396; F„ = 244 [m/N]
Fig. 7 Influence of the dynamic load applied to the tibia upon the transfer function H„,(f) KNEE4: a, = 7.5; <j> = 30 deg; F„ = 118; F, = 240; Fa = 97 KNEE4: a, = 15; 4> = 30 deg; Fb = 118; F, = 240; F, = 97 ~ KNEE4: a, = 25; 0 = 30 deg; F6 = 118; F, = 240; F„ = 97 4E-4 IHI [m/N] . 5$; A I
rf
i
I ' I a F I ^ 5 ^f
\i\ |i';
3
/ \ V
_ _ ^ ^ ^ V ^ _ u'fe * ^ **»»-_____ fi 50 - * - f [HzFig. 8 Influence of the dynamic load applied to the tibia upon the
transfer function Hr,{f)
KNEE4: a, = 7.5; 0 = 30 deg; Fb = 118; F, = 240; F„ = 97
KNEE4: a, = 15; 0 = 30 deg; F„ = 118; Fr = 240; Fa = 97
— . — — KNEE4:a, = 25; 0 = 30 deg; Fb = 118; Fr = 240; Fa = 97
4 Discussion of the Results
In this paper a number of aspects of the dynamic behavior
of the knee joint have been discussed. Using the LLT proposed,
the dynamic behavior of the joint can be quantified, although
it turns out that linearization of the behavior of the joint is
not allowed for the range of the dynamic loads applied to the
tibia. The transfer functions for various load levels show a
marked shift, indicating a decrease of the stiffness of the joint
with increasing load level. The value of the results is that they
essentially give a good insight in the behavior of the joint.
Also an order of magnitude for stiffness and damping
param-eters can be obtained.
An important observation is that the results for various knee
4E-4-IHI [m/N]
1 0 2 0 3 0 4 0
- * ~ f [Hz]
Fig. 9 The transfer function Hy^f) obtained for experiments with
sin-usoidal loading on KNEE5 with 0 = 30 deg.
• I / , I = 1 . 8 N ; + l /0l = 2 . 4 N ; * l/„l=3.0 N
joint specimens may well be compared. Also the results for
random and sinusoidal loading agree fairly well. It must be
kept in mind however that the obtained results depend on the
magnitude of the applied dynamic load.
Due to the use of Transfer Function Analysis (which is an
averaging process) and random excitation, the transfer
func-tions must be seen as to give an "average" behavior of the
joint for the loads applied. The nonlinearities of the behavior
of the joint are translated into a kind of effective stiffness and
damping. As the coherence function for all measurements was
almost unity, the transfer functions seem to give an acceptable
description for the behavior of the joint, despite the
nonlin-earities involved. This may partly be due to the use of random
excitation which is considered as a rather "gentle" excitation
(compared to impulse- or step-like excitation). A description
of the behavior of the joint using the transfer functions
ob-tained may therefore be considered. As was done in the work
of van Heck (1984), who encountered similar difficulties
an-alysing the strongly nonlinear dynamic behavior of slideways,
in this case the stiffness and damping characteristics of the
joint can be calculated for various levels of the dynamic load
applied. Such an approach may yield valuable information on
the degree of non-linearity. It is noticed however that in this
case no direct insight is gained, as the nonlinearities may partly
be smoothed due to the use of spectral analysis and random
excitation. This disadvantage may to a certain extent be avoided
by application of a time-domain analysis technique. Using
measured time domain signals, the best fitting linear system
can be determined to obtain stiffness and damping values for
various levels of the dynamic load applied. Evidently such an
approach may result in discrepancies between the measured
and best fitting signals, which can only be resolved by use of
a non-linear dynamic model of the knee joint. Such a model
is not available, however, and must be focussed on in future
research. Therefore the use of a best fitting linear model is
considered in the end as an inevitable alternative, that
never-theless may provide valuable information for the development
of a nonlinear dynamic model of the knee joint.
Acknowledgments
These investigations were supported by the Netherlands
Foundation of Technical Research (S.T.W.), future Technical
Science Branch/Division of the Netherlands Organization for
the Advancement of Pure Research (Z.W.O.) (Grant:
STW-EWT25.0259).
390 / Vol. 113, NOVEMBER 1991 Transactions of the ASME
Downloaded 20 Nov 2008 to 131.155.151.52. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
7E-4 IHI [mIN]
t
... -.... 50 - f [Hz)Fig. 6 Influence of the static preload upon the transfer function Hy,(f) _ _ _ KNEE1:
U,
= 20; '" = 20 deg; Fb = 40; F, = 90; F. = 60 KNEE1: Ut = 2; '" = 20 deg; Fb - 84; F, - 210; F. - 90 :-:-. -:-:- --:-: KNEE1:U,
= 2; '" = 20 deg; Fb = 224; F, = 396; F. = 244 8E-4 IHI [miN)t
50 - f [Hz]Fig. 7 Influence of the dynamic load applied to the tibia upon the transfer function Hx,(f) _ _ _ KNEE4: Ut = 7.5; '" = 30 deg; Fb = 118; F, = 240; F. = 97 KNEE4: Ut = 15; '" = 30 deg; Fb = 118; F, = 240; F. = 97 :-:-.:-:- .-:-: KNEE4: Uf = 25; t/> = 30 deg; Fb = 118; F, = 240; F. = 97 4E·4 IHI [mIN]
t
50 - f [Hz]Fig. 8 Influence of the dynamic load applied to the tibia upon the transfer function Hyt(f)
KNEE4: Ut = 7.5; '" = 30 deg; Fb = 118; F, = 240; F. = 97 - - - KNEE4: Uf = 15; '" = 30 deg; Fb = 118; F, = 240; F. = 97 :-:-.:-:-.-:-: KNEE4: Uf = 25; '" = 30 deg; Fb = 118; F, = 240; F. = 97
4 Discussion of the Results
In this paper a number of aspects of the dynamic behavior of the knee joint have been discussed. Using the LL T proposed, the dynamic behavior of the joint can be quantified, although, it turns out that linearization of the behavior of the joint is not allowed for the range of the dynamic loads applied to the tibia. The transfer functions for various load levels show a marked shift, indicating a decrease of the stiffness of the joint with increasing load level. The value of the results is that they essentially give a good insight in the behavior of the joint. Also an order of magnitude for stiffness and damping param-eters can be obtained.
An important observation is that the results for various knee
390 I Vol. 113, NOVEMBER 1991 4E-4
IHI
[mIN]t
+ +.
O~---r---r---'----'-o
10 20 30 40- - f
[Hz]Fig.9 The transfer function Hy,(f) obtained for experiments with sin· usoidal loading on KNEE5 with",
=
30 deg.. If. I = 1.8 N; + 1f.1 = 2.4 N; • If. I = 3.0 N
joint specimens may well be compared. Also the results for random and sinusoidal loading agree fairly well. It must be kept in mind however that the obtained results depend on the magnitude of the applied dynamic load.
Due to the use of Transfer Function Analysis (which is an averaging process) and random excitation, the transfer func-tions must be seen as to give an "average" behavior of the joint for the loads applied. The nonlinearities of the behavior of the joint are translated into a kind of effective stiffness and damping. As the coherence function for all measurements was almost unity, the transfer functions seem to give an acceptable description for the behavior of the joint, despite the nonlin-earities involved. This may partly be due to the use of random excitation which is considered as a rather "gentle" excitation (compared to impulse- or step-like excitation). A description of the behavior of the joint using the transfer functions ob-tained may therefore be considered. As was done in the work of van Heck (1984), who encountered similar difficulties an-alysing the strongly nonlinear dynamic behavior of slideways, in this case the stiffness and damping characteristics of the joint can be calculated for various levels of the dynamic load applied. Such an approach may yield valuable information on the degree of non-linearity. It is noticed however that in this case no direct insight is gained, as the nonlinearities may partly be smoothed due to the use of spectral analysis and random excitation. This disadvantage may to a certain extent be avoided by application of a time-domain analysis technique. Using measured time domain signals, the best fitting linear system can be determined to obtain stiffness and damping values for various levels of the dynamic load applied. Evidently such an approach may result in discrepancies between the measured and best fitting signals, which can only be resolved by use of a non-linear dynamic model of the knee joint. Such a model is not available, however, and must be focussed on in future research. Therefore the use of a best fitting linear model is considered in the end as an inevitable alternative, that never-theless may provide valuable information for the development of a nonlinear dynamic model of the knee joint.
Acknowledgments
These investigations were supported by the Netherlands Foundation of Technical Research (S. T. W .), future Technical Science Branch/Division of the Netherlands Organization for the Advancement of Pure Research (Z.W.O.) (Grant: STW-EWT25.0259).
References
Jans, H., Dortmans, L., Sauren, A., and Huson, A., 1988, "An Experimental Approach to Evaluate the Dynamic Behavior of the Human Knee," ASME
JOURNAL OF BIOMECHANICAL ENGINEERING, Vol. 110, pp. 69-73.
Hefzy, H., and Grood, C , 1988, "Review of Knee Models," Applied Me-chanics Reviews, Vol. 41, no. 1, pp. 1—43.
Bendat, J., Piersol, A., 1980, Engineering Applications of Correlation and Spectral Analysis, Wiley-Interscience.
Crowninshield, R., Pope, M., Johnson, M., and Miller, R., 1976, "The Impedance of the Human Knee," Journal of Biomechanics, Vol. 9, pp. 529-535.
Moffatt, C , Harris, E., and Haslam, E., 1969, "An Experimental and An-alytic Study of the Dynamic Properties of the Leg," Journal of Biomechanics, Vol. 2, pp. 373-387.
Announcement and Call for Papers
International Symposium on the Three-Dimensional Scoliotic Deformities combined with the
Vllth International Symposium on Spinal Deformity and Surface Topography June 27-30, 1992, Hotel Meridien, Montreal, Quebec, Canada
The "International Symposium on the Three-Dimensional (3-D) Scoliotic Deformities" will be a scientific and technical forum allowing the opportunity for all researchers and clinicians, working in this field, to exchange knowledge, to establish contact between investigators and to generate further communication and discussion specifically on the 3-D aspect of the scoliotic spine and thorax.
SCIENTIFIC PROGRAM:
The symposium will be divided in six main themes; each theme will be introduced and discussed by an invited keynote speaker who has a specific expertise. These main themes are:
1) 3-D acquisition, reconstruction and modeling techniques 2) 3-D biomechanical analysis
3) 3-D deformity measurement methods and terminology 4) 3-D etiological and prognostic aspects
5) 3-D treatment of scoliosis
6) Surface topography vs. internal 3-D spinal and/or trunk anatomy
LANGUAGES:
The official languages of the symposium are English and French. No simultaneous translation will be available.
REGISTRATION FEE:
The fee for the Symposium is $420 prior to April 1st, 1992 and $460 afterwards. The registration fee includes: meeting material, one copy of the proceedings, welcoming reception, two lunches, banquet, refreshments and another special social activity. The fee is due upon acceptance of paper. All cancellations received by April 1st, 1992 will be assessed a $100 non refundable charge. After this date, there will be NO REFUNDS.
DEADLINES:
• Abstract submission: December 1, 1991 • Notification to the authors: January 15, 1992
• Complete paper submission (4 to 8 pages): March 1, 1992
PLEASE FORWARD REQUESTS FOR INFORMATION AND AUTHOR'S KIT TO:
Dr. Jean Dansereau, Ph.D. Biomedical Engineering Institute Ecole Polytechnique
P.O. Box 6079, Station " A " Montreal, Quebec, Canada H3C 2A7
FAX: (514) 340-4611
Pope, M., Crowninshield, R., Miller, R.,and Johnsson, R., 1976, "The Static and Dynamic Behavior of the Human Knee in Vivo," Journal of Biomechanics, Vol. 9, pp. 449-452.
Mergeay, M., 1980, "Theoretical Background of Curve-Fitting Methods used by Modal Analysis," Seminar on Modal Analysis, University of Leuven, Bel-gium.
Heck, J. van, 1984, "On the Dynamic Characteristics of Slideways," Thesis Eindhoven University of Technology, Eindhoven, The Netherlands.
Radin, E. L., and Paul, I. L., 1977, " A Consolidated Concept of Joint Lubrication," Journal of Bone and Joint Surgery, Vol. 54-A, pp. 607-616.
Dortmans, L., Jans, H., Sauren, A., and Huson, A., 1991, "Nonlinear Dy-namic Behavior of the Human Knee Joint. Part II: Time-Domain Analyses: Effects of Structural Damage in Postmortem Experiments," published in this issue, pp. 392-396.
NAME: __ ADRESS:
TELEPHONE: FAX: NUMBER OF AUTHOR'S KIT:
Journal of Biomechanical Engineering NOVEMBER 1991, Vol. 113 / 391
References
Jans, H., Dortmans, L., Sauren, A., and Huson, A., 1988, "An Experimental Approach to Evaluate the Dynamic Behavior of the Human Knee," ASME JOURNAL OF BIOMECHANICAL ENGINEERING, Vol. 110, pp. 69-73.
Hefzy, H., and Grood, C., 1988, "Review of Knee Models," Applied Me-chanicsReviews, Vol. 41, no. I, pp. 1-43.
Bendat, J., Piersol, A., 1980, Engineering Applications of Correlation and Spectral Analysis, Wiley-Interscience.
Crowninshield, R., Pope, M., Johnson, M., and Miller, R., 1976, "The Impedance of the Human Knee," Journal of Biomechanics, Vol. 9, pp. 529-535.
Moffatt, c., Harris, E., and Haslam, E., 1969, "An Experimental and An-alytic Study of the Dynamic Properties of the Leg," Journal of Biomechanics, Vol. 2, pp. 373-387.
Journal of Biomechanical Engineering
Pope, M., Crowninshield, R., Miller, R., and Johnsson, R., 1976, "The Static and Dynamic Behavior of the Human Knee in Vivo," Journal of Biomechanics, Vol. 9, pp. 449-452.
Mergeay, M., 1980, "Theoretical Background of Curve-Fitting Methods used by Modal Analysis," Seminar on Modal Analysis, University of Leuven, Bel-gium.
Heck, J. van, 1984, "On the Dynamic Characteristics of Slideways," Thesis Eindhoven University of Technology, Eindhoven, The Netherlands.
Radin, E. L., and Paul, I. L., 1977, "A Consolidated Concept of Joint Lubrication," Journal of Bone and Joint Surgery, Vol. 54-A, pp. 607-616.
Dortmans, L., Jans, H., Sauren, A., and Huson, A., 1991, "Nonlinear Dy-namic Behavior of the Human Knee Joint. Part II: Time-Domain Analyses: Effects of Structural Damage in Postmortem Experiments," published in this issue, pp. 392-396.
