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Vibrational characteristics of the embalmed human femur
Journal ofSound and Vibration (1981) 75(3), 417-436
VIBRATIONAL
CHARACTERISTICS
EMBALMED T. B. KHALIL,
HUMAN
D. C. VIANO
AND
OF THE
FEMUR L. A. TABER
Biomedical Science Department, General Motors Research Laboratories, Warren, Michigan 48090. U.S.A. (Received 6 May 1980, and in revised form 6 October 1980)
The resonant frequencies and mode shapes of contralateral femurs have been identified by experimental and analytical procedures. Also, the cross-sectional area, centroid, and principal moments of inertia were computed throughout the femur length for both compact and cancellous bone. The resonant frequencies of freely vibrating specimens were identified from transfer function measurements by using a Fourier analyzer. Twenty frequencies were noted in a frequency range of 20 Hz-8 kHz. A mathematical model of the femur consisting of 59 joined uniform segments, with each composed of compact and/or cancellous bone, was analyzed by using a transfer matrix technique. Results of the model enabled classification of the experimental resonances into deformations corresponding to flexure (about principal planes of inertia), torsion, and longitudinal extension with fundamental frequencies at 250, 308, 557, and 2138 Hz, respectively. Generalized nondimensional resonant frequencies were computed based on femur geometry averaged over its length and compared with those predicted by simple beam models. This analysis provided further understanding of the vibrational behavior of the femur.
1. INTRODUCTION The incidence of femoral fracture accompanied with complex failure patterns has prompted numerous investigations of femur mechanics. The femur is the longest and strongest bone of the musculoskeletal system. Femoral fracture generally results from excessive mechanical loads transmitted through the articulating joints. Much of the currently published work is concerned with two basic questions: (1) what is the non-traumatic load carrying capacity of the femur, and (2) what are the fracture patterns associated with traumatic loads? Femoral studies fall into one of two categories: theoretical studies and experimental studies in which cadaveric specimens are used. Among the theoretical studies is the work of Koch [l] who published the results of a comprehensive study of the structural mechanics of the femur. Using elementary beam theories, Koch determined how equi-stress contours lead to the remarkable geometrical adaptation of the femur to physiological mechanical loads. He also concluded that the femur is an optimum structural member (i.e., minimum weight for maximum strength) for load transmission between the acetabulum and the tibia. The earliest interest in femur mechanics, pointed out by Koch, ddtes back to 1638 and is accredited to Galileo [2]. Toridas [3] extended Koch’s work by modeling the femur as a system of joined three-dimensional beams. Femoral stresses are presented for both body weight and muscle force loads. Piotrowski and Wilcox [4] discussed a “stress program” that was used to determine the stress distribution in long bones. Other theoretical investigations of the femur include the work of Rybicki et al. [5] and of Advani et al. [6]. 417 @ 19X1Academic Press Inc. (London) Limited 0022-460X/81/070417+20 $02.00/O
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T. B. KHALII..
D. C‘. VIANO
AND
I.
A. TABER
The investigations cited have been conducted for static loading of the femur. In a typical car crash, the femur is subjected to dynamic loads of duration on the order of 50-60 ms. Accordingly, Viano and Khalil [7] investigated the dynamic response of a planar finite element model of the femur. Dynamic loads of varying duration (3-75 msj were applied to the lateral and medial condyles, respectively. The deformation response of the femur depends significantly on the load duration. First and second flexural vibration modes corresponding to the fixed-free beam can be excited. An extension of that study [S] provides evidence that the structural response characteristics of the femur play a significant role in increasing its load carrying ability for short duration impact. Dynamic experimental studies are, however, of recent origin. In these studies primarily human cadaver tissue has been utilized to assess the impact response of the femur and load transfer characteristics of the lower extremity musculoskeletal system. Typical of the laboratory experiments is the work of Melvin er al. [9] and Powell et al. [lo]. Viano and Stalnaker [ 1 l] subjected seated unembalmed human cadavers to knee impact aligned with the axis of the femur shaft. Strain gages bonded to the femur midshaft showed significant femoral bending during axial compressive loading due to the complex anatomical configuration of the knee-thigh-hip. Variations in test specimens and experimental conditions resulted in inconsistent force levels at fracture. It has been generally concluded that the peak applied load is not sufficient to predict femur fracture. Viano [12] summarized the available biomechanical data concerning femoral fracture and concluded that much of the scatter in experimental data is attributable to the load-time dependence for the initiation of fracture when the load duration is below 20 ms. Studies of the impact response and fracture of the femur are incomplete without basic knowledge of its structural dynamic characteristics. Information about femur resonant frequencies and associated mode shapes is invaluable to understanding femoral deformation patterns excited by various dynamic loads [7]. Also, this information is useful in the development of improved analytical models. In spite of the noted need for dynamic data, few studies have been conducted to delineate the vibrational characteristics of the femur. Campbell and Jurist [13] conducted preliminary tests on excised femurs to determine the mechanical impedance in a frequency band 20-4000 Hz. Since the authors were primarily interested in changes in mechanical impedance caused by femoral head fractures, they concentrated only on identifying the lowest resonance at about 70 Hz. Although the existence of numerous higher order resonances was noted, no attempt was made to identify or explain them. Melvin [9] conducted mechanical impedance tests by vibrating the knee joint of seated unembalmed cadavers. He noted a pronounced resonance in the range of 150-500 Hz. the resonant frequencies and mode shapes of macerated Viano et al. [14] measured femur shafts using a stabilized Michelson interferometer. Sixteen resonant frequencies were identified for the central shaft of the femur in a frequency range of 1 Hz-20 kHz. The lowest resonance was 1670 Hz. Classical beam theories were numerically adapted to account for the varying geometric characteristics of the compact bone along the femoral shaft. By iteratively matching analytical predictions with the experimental data for flexural, torsional, and longitudinal resonances, the elastic moduli of compact bone were evaluated for eight specimens. The study described in what follows here was concerned with experimental and analytical identification of the resonant frequencies of contralateral embalmed human femurs in a frequency range of 20 Hz-8 kHz. The resonant frequencies were determined from measurements of the transfer function of the femur by using Fourier analysis techniques. A similar test procedure was successfully applied to determine the resonant frequencies and mode shapes of the human skull [15]. The measured frequencies are
VIBRATIONAL
CHARACTERISTICS
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FEMUR
419
compared with results from an analytical model of the femur based on non-uniform beam vibrations [16]. By using the model one can calculate the vibratory motion in flexure, torsion, and longitudinal extension of a structure consisting of joined uniform beam segments-with each composed of compact and/or cancellous bone. By matching experimental and analytical resonances, the flexural, torsional, and extensional vibrations of an isolated whole femur were determined.
2. FEMUR TRANSFER
FUNCTION
In this study, femoral resonances were determined by digital Fourier analysis [17]. The non-destructive test procedure involves computation of repeated impact responses of the femur to identify its structural transfer function. The dynamic response of the femur is assumed to be linear and described by a second order differential equation. For such a system, the transfer function (frequency response function) defines the response in the frequency domain at a remote point due to a Dirac delta function, s(t), force applied to the structure. The transfer function is complex valued and can theoretically be determined by simultaneous monitoring, fast Fourier transforming (FFT), and dividing the acceleration response by the input force. It may be written as G(iw) = 9[A(t)]/F[F(t)] where 9 is a Fourier transform operator, A(t) is the output acceleration signal, and F(t) is the input force signal. In practice, G(iw) is determined from the quotient of an average of the input-output cross power spectra and input power spectrum of several repeat tests to minimize the influence of extraneous noise. The transfer function is measured experimentally in the frequency band of interest by use of a digital Fourier analyzer (HewlettPackard 545 1C). The resonant frequencies can be identified by inspection of the imaginary component and polar plot of the transfer function [17]. Coincidence of the peak of the imaginary component with a location on the polar plot where the rate of change of arc length is a maximum with frequency variations indicates the existence of a structural resonance. The theoretical basis of this technique has been given by Kennedy and Pancu [18].
3. TEST SPECIMENS
Two embalmed human femurs were obtained and tested at the Gurdjian-Lissner Biomechanics Laboratory, Wayne State University, Detroit, Michigan. The contralateral femurs were from a 74-year-old male who was 1.85 m in height and 81 kg in weight at the time of death. The cause of death was heart disease and respiratory failure. The exterior appearance of the femurs was excellent with no obvious sign of osteoporosis. The embalmed femurs weighed 865 g and had an overall length of 515 mm.
4. EXPERIMENTAL
METHOD
A schematic of the experimental set-up is shown in Figure 1. The isolated femur was supported on 50 mm of soft foam to approximate a free support condition. The femur was excited by a transient force F(t), delivered by lightly tapping on the femur surface by a hand-held hammer. The force was monitored by a load cell (Briiel and Kjaer #8200)
420
T.
R. KHALIL,
D. C. VIANO
Founer
mlifiers,
AND
L. A. TAbER
analyzer
Load
cell n-”
Figure
1. Schematic
of experimental
set-up.
attached to the hammer head which had a 12.5 mm diameter (steel) flat surface. The frequency response of the load cell is flat (* 1.0%) in the frequency range 20 Hz-70 kHz. Preliminary tests were conducted to ensure that sufficient energy was available in the force signal to excite the femur in the frequency band of interest, 20 Hz-8 kHz. The transient output signal A(t)was monitored by a transducer consisting of three miniature accelerometers (Endevco #2222B) glued to the femur surface. The accelerometer pack weighed 5 g and exhibited a flat frequency response (*2*5%) in the frequency range of 20 Hz-10 kHz. The femur was excited by tapping along three perpendicular directions, X (superior-inferior), Y (medial-lateral), or 2 (posterior-anterior) to selectively enhance the three primary modes of vibrational response: flexure in the plane of maximum and minimum inertia, torsion, and longitudinal extension. The input force and the corresponding acceleration component were amplified and passed through matched anti-aliasing filters. The signals were then fed into the Fourier analyzer to determine the transfer function. Response linearity was assured by (a) increasing the input force level and observing a proportional increase in acceleration, and (b) verifying Betti’s reciprocal theorem for elastic structures by measuring identical transfer functions for interchanged input/output locations. This finding is not surprising in view of the linear elastic behavior of compact and cancellous bone [19, 201. An ensemble average of 20 impacts was used to minimize the potential effects of spurious noise on the numerical value of the transfer function, particularly at higher frequencies where a poor signal-to-noise ratio may exist. Measuring the transfer function from various impact locations ensures that all vibrational modes of the structure are captured; also, using various accelerometer positions guards against missing a resonance by inadvertent placement of the accelerometer at a vibration node. Supporting the femur on a very soft rubber foam pad provided free boundary conditions at the extremities.
VIBRATIONAL
CHARACTERISTICS
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421
5, EXPERIMENTAL RESULTS The input force F(t), output acceleration A(t), and the associated Fourier transform are shown for a typical tapping in Figure 2. In this test, the force was applied normal to the femur surface along the Y direction (see Figure 1) and the acceleration was monitored along the same axis. Similar input/output signals were obtained for tests involving the other two, X and 2, directions, which indicated that a complex three-dimensional motion is stimulated by local impact.
Figure
2. Typical
input/output
signal and their Fourier
transforms.
Though the force signal is of short duration (0.5 ms), it provided sufficient excitatory energy over the frequency range of interest (20 Hz-8 kHz). The acceleration signal shows significant ringing which lasted well beyond the duration of the input load. It was also noted that the acceleration response exhibits a “beat phenomenon” which is typically encountered when resonances of a two degree of freednm system are close in value. The proximity of resonant frequencies which correspond to flexural vibration about minimum and maximum planes of inertia (as will be shown later) is probably the cause of the “beat phenomenon”. The Fourier transform of the acceleration signal demonstrates the complex vibratory characteristic of the femur. Figure 3 is typical of the measured transfer function in a frequency band, 20 Hz-8 kHz. The magnitude, real part, and imaginary part of the transfer function are presented in addition to the coherence function. A coherence value of less than unity indicates that the output signal may be contaminated with extraneous noise and the frequency measurements may not be a reliable measure of the structural resonance. Such situations were remedied by taking more response averages. Twenty experimental resonances (see Figure 3) were identified for the right femur. The resonances were identified from the imaginary component of the transfer function and its associated polar plot (Figure 4). Since only 12 frequencies could be identified in this example (listed in Figure 4), it is clear that a single experimental transfer function may not exhibit all resonant frequencies of a complex structure. Therefore, the femur was tapped at
422
T.
B. KHALIL.
D.
C. VIANO
AND
L. A.
TABER
Log mognltude
Real part
Imagmory port
Coherence function
Figure 3. Typical femur transfer function (frequency band 20 Hz-8 kHz). Resonant frequencies (Hz): 244, 317, 561, 830, 879, 1563, 1612, 1882, 2125, 2344, 2564, 3052, 3370, 3697.4151, 4445.4712.4907, 5372, 7300.
different locations (femur head, condyles, and shaft) and along different directions to ensure capture of all resonances. The imaginary components of two additional transfer functions, which were obtained by tapping on the medial condyle and the femur head, respectively, are shown in Figures 5 and 6. Inspection of resonance values indicates small variations in frequency (*4%) between these two experimental runs. This is partly attributable to the frequency resolution associated with the digital Fourier analyzer. Also, when independent modes of vibration have resonances in close proximity to one another, there may exist a vibrational interaction which can shift the apparent resonance [ 171. The use of multiple test conditions minimizes this possible artifact by varying the distribution of energy among the modes of vibration in a particular frequency range. The experimental resonant frequencies are presented in ascending sequential order in Table 1 for both the right femur (RF) and the left femur (LF). At this stage, the resonant frequencies correspond to a combination of flexural, torsional, and longitudinal vibrations. Later on, the frequency spectrum is partitioned to identify the resonances corresponding to the three principal motions. As may be observed in Table 1, the resonant spectra of the contralateral femurs are not identical. However, the average difference in frequency for all modes was 3-3 f 3.5%. Resonance data is presented only to a maximum frequency of 8 kHz since the spectral coherence decays significantly below unity beyond that frequency. Although resonant
VIBRATIONAL
I I
CHARACTERISTICS
OF THE
423
FEMUR
I 2
3
4
5
Frequency
6
7
8
CkHz)
Figure 4. Imaginary part of transfer function and its polar plot (left femur). Axial tap on femur head (X-direction). Resonant frequencies (Hz): 250, 546, 750, 1450, 1900, 2350, 2867, 3050, 3678, 4330, 5290, 7239.
I
0
I
I
I
I
I
I
I
I
2
3
4 Frequency
I 5
I
I
I
I 7
6
(kHz)
Figure 5. Imaginary part of transfer function. Force applied along X-axis on medial condyle. Accelerometer was mounted near the distal end of the shaft (left femur). Resonant frequencies (Hz): 250,555,745,1445, 1907, 2160,2850,3304,4050,4600,5050.
424
T. B. KHALIL.
0
I I
D. (‘. VIANO
I 2
I 3
AND
I 4
Frequency
I.. A. TABER
I
I
I
5
6
7
Figure 6. Imaginary part of transfer function. Force applied along Z-direction was mounted at midshaft (left femur). Resonant frequencies (Hz): 250,300,810, 3285,3660,4365,5060,5300.
TABLE
Experimental
6
CkHz)
1
resonant
frequencies
0-W Right femur
Left femur
(RF)
(LF)
250 315 563 825 879 1563 1607 1832 2125 2344 2572 3060 3365 3680 4146 4440 4712 4907 5375 7350
250 300 550 750 810 1450 1508 1900 2150 2345 2865 3050 3300 3670 4050 4350 4600 5050 5300 7239
on femur head. Accelerometer 1450,1508,2150,2345,3050,
VIBRATIONAL
CHARACTERISTICS
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FEMUR
425
frequencies above 8 kHz can be identified by increasing the amount of energy delivered to excite the femur, this was not pursued since the frequency content of impact loads suspected of initiating femur fracture is well below the cut-off frequency of 8 kHz [12]. The experimental frequency spectrum for each femur represents resonances corresponding to a complex three-dimensional motion of the femur. It is desirable from a structural response point of view to partition the frequency spectrum into three parts corresponding to flexural, torsional, and longitudinal vibration. This can be accomplished experimentally by the Fourier analyzer to identify the mode shapes associated with the resonant frequencies. Unfortunately, our Fourier analysis system does not, as yet, include a mode-shape package. Accordingly, we developed a mathematical model of the femur which allows us to sort the frequencies into their three corresponding motions. A description of the model and the identification of resonances corresponding to flexural, torsional, and longitudinal vibrations follow. 6. MATHEMATICAL
MODELING
OF FEMUR VIBRATIONS
6.1. FEMUR OSTEOMETRY To construct a mathemtical model of the femur, it is necessary to determine the varying cross-sectional geometric properties of the compact and cancellous bone. Although the
Figure 7. Posterior bone.
view of femur and cross-sections.
The difference
in color is due to the process of cutting
the
426
T. B. KHALIL.
D. C. VIANO
AND
I_. A. TABER
femur has a complicated three-dimensional configuration, it structurally resembles a long column (shaft) whose proximal epiphesis (head and neck) forms an angle of about 125” with the axis of the shaft. The distal epiphesis (condyles) is somewhat expanded and articulates with the tibia. Both extremities consist of cancellous bone covered by a thin shell of compact bone. The shaft may be regarded as a thick-walled cylinder of compact bone. Cross-sectional properties of the femur were determined from 59 transverse sections cut normal to the shaft axis. Because of large variations in section properties, slices were made every 6.4 mm (0.25 in) at the extremities. In the midshaft region, sections were made every 12.7 mm (0.5 in). Figure 7 shows the posterior view of the femur and a selected number of representative cross-sectional photographs. The location of the neutral axis is also shown. For each section, the area (A), the neutral axis (y, z), and the principal moments of inertia (Ii, Iz) with the orientation of the principal axes (13)were computed for both the compact and cancellous parts of the bone [ 141. The distribution of bone area and the centroid locations are presented in Figure 8 as a function of the femoral length. A lateral view of the neutral axis, along with the principal moments of inertia are presented in Figure 9. Cross-sectional area and moments of inertia from previous studies [l, 141 are presented in Figures 8 and 9 for comparison. The femur space centroid trajectory (Figure 10) is a complex three-dimensional curve.
0
100
200
300
400
I
I
500
X (mm) 40 (
I
I
, 220
”
X (mm) Figure 8. Cross-sectional area and centroid locations. bone; ZIeT Koch [l]; --C-, Viano etal. [14].
Present
study: -
-, cancellous
bone; -e,
compact
100
300
200 X
2.5
I
0
100
I
6.2.
MATERIAL
501
400
500
-l
200
300 X
Figure 9. Cross-sectional
400
(mm)
(mm)
moments of intertia. - - -, Viano et al. [ 141; - - -,
Koch [I]
PROPERTIES
The elastic moduli and density of compact bone (E = 22.0 x lo6 kPa, G = 8.6 x 10” kPa, p = 2090 kg/m3) were selected from the extensive literature on the subject (see, for example, references [14, 191). Similar properties of cancellous bone (E = 0.38 x lo6 kPa, G = O-15 x 10” kPa, p = 890 kg/m31 were taken from the work of Ducheyne et al. [20]. Bone anisotropy [21] and inhomogeneity [20] have been disregarded in this study to simplify the analysis. 6.3.
ANALYTICAL
MODEL
To sort the experimental resonant frequencies according to vibratory motion of a non-uniform bimaterial beam in flexure, torsion, or longitudinal extension, we utilize an extension [16] of a mathematical model developed by Viano et al. [14]. The non-uniform
428
T. B. KHALIL.
D. c‘. VIANO
Figure 10. Three-dimensional
AND
1.. A. TABER
femur centroid trajectory.
femur structure is conceptually divided into a set of 59 one-dimensional uniform beam segments. Each segment may consist of two elastic materials (compact and cancellous bone) with known uniform geometric characteristics. The general linear motion of each beam segment is described by a superposition of bending in two principal planes, torsion, and longitudinal extension. The following equations [22] describe the motion of each uniform beam segment:
(a) flexure
(according
to Timoshenko
/!?(J4W(X, t)/ex4)+(y/cy)[S(a4W(X,
beam theory):
t)/at4)-p(a4w(X,
?)/c?X2~f2)
+cr@*W(x, t)/at2)]-6(a4W(X, (b) torsion
(according
to simple &*4(x,
(c) longitudinal
torsion
of a non-circular
t)/&t’)
(1)
=o;
cylinder);
t)/ax2) - k’7(a2C$(x, t)/at*) = 0;
(2)
extension: zQ2U(X, t)/ax2) - K(&4(X, t)/ar2) = 0.
In equations
(3)
(1) to (3),
a = klG,Al+ K =PIAI+PzA~,
kzG2A2,
P = El11 + ~5212, v =
E,A, + E2A2,
Y =
PIAI
+ ~2A2,
T = G,JI + G,+Jz,
6 = p1I1+
CL= PIJIf
P212,
PZJZ,
u is longitudinal displacement, C$ is the angle of twist, w is the transverse deflection, x is position along the longitudinal axis, t is time, pi is material density (i = 1 for compact bone and 2 for cancellous bone), Ai is cross-sectional area, Ei is the elastic modulus, Gi is the shear modulus, Ji is the polar moment of inertia, Ii is the area moment of inertia, ki is
VIBRATIONAL
CHARACTERISTICS
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FEMUR
429
the Timoshenko shear coefficient (kl = kz - @55), and k’ is the torsional rigidity constant (k’ = O-36). The generalized longitudinal and shear wave speeds are given by CO= JV/K and C, = Jp/k’T, respectively. For each of the three independent vibratory motions of the femur (flexure, about the principal axis system with minimum and maximum moments of inertia, torsion, and longitudinal extension) the discretized equations of motion are written in matrix form. The spatial eigenvalue problem is solved by an iterative procedure previously described [ 141 to yield the analytical resonant frequencies. Free-free boundary conditions are assumed at both ends of the femur and continuity of deformation and force balance are insured between adjacent uniform sections. In this formulation no account is taken of curvature of the neutral axis of the femur, and therefore it provides a solution in which axial and bending vibrations are uncoupled. Because of the mild curvature of the femur, we expect only a slight effect on the resonances. In a similar analysis, Hight et al. [23] found that exclusion of curvature effects changed the resonant frequencies of the tibia by less than 5%. 7. ANALYTICAL RESULTS The frequency spectra predicted by the mathematical model of the femur for three types of vibrations, flexure, torsion and longitudinal extension, are presented in Table 2. The computations were carried out in a frequency range of O-8 kHz. The mode shapes of the vibrating neutral axis were also computed. Figure 11 shows the first two modes corresponding to flexure (about the minimum inertial axis), torsion, and longitudinal extension. TABLE
2
Analytical resonant frequencies (Hz)
Flexure Mode no. 1 2 3 4 5 6
W
Longitudinal
mm
max
Torsion
extension
245 771 1507 2373 3358 4412
289 859 1672 2635 3744 4880
501 1810 3149 4281 5315
2118 4407 7264
Inspection of the resonant frequencies reveals that the lowest resonant frequency (245 Hz) corresponds to flexural vibration. Also, it is noted that the paired flexural resonances are close to each other, which may explain the “beat phenomenon” observed in the acceleration response (see Figure 2). The lowest torsional resonant frequency is 501 Hz. Higher torsional frequencies, in contrast with those given by simple beam theories, are not simple integer harmonics. After detailed analysis we have concluded that the femur vibration, particularly in torsion, resembles that of a flexible shaft with two rigid masses attached at the extremities. This point is confirmed by observing the relative lack of motion of the femur extremities in the mode shapes (see Figure 11). The validity of representing the femur by simple mathematical models will be discussed in section 9. For the mathematical model the lowest resonant frequency in longitudinal extension predicted is 2118 Hz.
430
T. B.
KHALIL.
D.
(‘. VIANO
AND
I_. A. TABER
Flexure
501
Hz
2118 Hz
1810 Hz
Torsion
Longitudinal
4907
Hz
Figure 11. Femur mode shapes.
Variation in the elasticity coefficient (E) of cancellous bone by a factor of f 10 times, to account for potential bone inhomogeneity and the scatter in published data [20], did not change the resonant frequencies by more than &12%, with the lower modes being less affected than higher modes. Frequencies below 3000 Hz were not changed by more than *4%. It appears that the distributed mass of cancellous bone, in contrast with its stiffness, has more influence on femur vibrations. 8. COMPARISON
OF ANALYTICAL
AND
EXPERIMENTAL
RESULTS
In a frequency band of 20 Hz-8 kHz, 20 analytical and experimental resonances were identified for the femur. Based on the mathematical analysis, the experimental spectrum was partitioned into three independent (principal) resonance motions. This resulted in identfication of the six lowest mode pairs of flexural vibration about the minimum and maximum planes of cross-sectional inertia, five torsional, and three longitudinal resonances. Good agreement is noted between the experimental data and the mathematical model prediction as shown in Figure 12. In typical impact conditions, where the load duration is in excess of 2 ms, the predominant frequency of the approximate half-sine load is less than 250 Hz. Assuming that the actual in uivo boundary conditions of the femur are somewhere between “free-free” and “free-fixed”, the two lowest llexural modes and/or the first torsional mode are most likely to be excited.
9. DISCUSSION
9.1. ON USING SIMPLE BEAM MODELS TO SIMULATE FEMUR VIBRATIONS The femur is a complex three-dimensional structure. An understanding of its basic vibratory motion is fundamental to comprehending its structural response to mechanical
VIBRATIONAL
CHARACTERISTICS
8
OF THE
431
FEMUR
l-
I
7-
Longltudlnol
6-
5-
N 5 G4 5 i+ z 3-
2-
I-
I
2
Figure 12. Resonant O-0, theoretical data.
frequency
spectrum.
4
3 Mode
5
number
?? , Experimental
(right
femur):
A, experimental
(left femur);
impact. The analysis presented, though providing complete information about femur vibrations, is limited to the tested specimens. It is desirable to extend this information to determine the resonant frequencies of other femurs of different osteometry. This can be achieved by establishing generalized non-dimensional eigenvalues (A,,) of the femur. Following Fliigge’s convention [22], the eigenvalues of the frequency equations (l), (2) and (3) corresponding to flexural, torsional, and longitudinal vibrations, respectively, can be written as follows: (a) flexure with respect to minimum and maximum axes of inertia:
(b) torsion:
(c) longitudinal extension: ,-Y
432
7’. B. KHALIL, D. C. VIANO
AND
L. A. TAHEK
Here A,, is a constant to be determined for various mode numbers o (n = 1.2,3, . .). L is the femur length, and N is the number of femur segments. All other constants are as defined previously. The values of A for flexure, torsion, and longitudinal vibration as determined from both theoretical and experimental data are presented in Tables 3,4 and 5, respectively. Values in parentheses are normalized with respect to the geometry at the midshaft. Small variations (less than 3%) were observed in calculating the value of A for minimum and maximum frequencies at a particular mode number. This variation is possibly due to oversimplification in computing C, from equation (5) at the midshaft only. Also, A‘s of
TABLET
Values of A for flexural vibration
A”
\ n
Theory
\
1 2 3 4 5 6
3.77 6.60 9.21 11.56 13.77 15.75
Experiment
(4.17) (7.31) (10.01) (12.41) (15.25) (17.44)
3.85 (4.13) 6.61 (7.09) 9.05 (9.72) 11.60 (12.46) 13.66 (14.67) 15.79 (16.96)
Free-free shaft with concentrated masses at the ends Q = 0.2 4.27 7.32 10.40 1351 16.62 19.75
Simple free-free uniform beam 4.73 7.85 10.99 14.14 17.28 20.42
TABLET
Values of A for torsional vibration
Theory 1 2 3 4 5
0.33 1.18 2.05 2.79 3.46
(0.25) (0.90) (1.56) (2.13) (2.64)
Experiment 0.36 1.22 1.98 2.59 3.43
(0.27) (0.93) (1.51) (1.97) (2.61)
Free-free shaft with concentrated masses at the ends Q=2.8 0.27 1.08 2.04 3.02 4.04
Simple free-free uniform beam 1.0 2.0 3.0 4.0 5.0
TABLET
Values of A for longitudinal vibration
A,
n \
Theory 1 2 3
0.83 (0.67) 1.73 (1.40) 3.77 (2.24)
Experiment 0.82 (0.66) 1.72 (1.39) 2.77 (2.24)
Free-free shaft with concentrated masses at the ends Q=Os2 0.75 1.54 2.45
Simple free-free uniform beam 1.0 2.0 3.0
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CHARACTERISTICS
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433
FEMIJR
simple free-free beam vibrations [22] are included in the tables. In addition, we have included h’s obtained from vibratory solutions of flexible shafts with two rigid masses attached to their extremities, with Q being the ratio of the end mass to the shaft mass. The resonances of these models explain the vibrational behavior of the femur as will be discussed later. A close inspection of Tables 3, 4 and 5 reveal the following: (a) With use of midshaft femoral properties (C,, equation (5)), simple beam theories [22] provide reasonable predictions of the flexural resonant frequencies (see Table 3), particularly for the lower modes. Resonances of the first three modes are higher by about 9% ; however, higher mode frequencies are off by as much as 15%. Higher mode frequencies predicted by simple beam theories are progressively in error due to neglect of shear deformation effects. The analysis provided earlier (based on Timoshenko beam theory) accounts for shear deformation across the beam section through the introduction of the shear coefficients ki. (b) Inspection of Table 4 reveals that simple free-free beam analysis does not provide reasonable predictions of torsional frequencies. The analysis yields n (n = 1,2,3, . . .) as the value of A,,. However, the experimental data of Table 4 appear to exhibit an extra frequency. The appearance of this extra frequency will be discussed in more detail later. For now, it is clear that the simple beam theory does not provide a good model for femur torsional vibrations. (c) For longitudinal vibrations, the simple beam theory predicts that the frequencies are integer multiples of n. Although the experimental data (see Table 5), appear to exhibit a similar trend to the simple beam theory data, the experimental frequencies are lower by approximately 20%. A similar situation to the torsional case, i.e., the appearance of an extra frequency, may occur for longitudinal vibration. (d) Simple beam theories overestimate the value of higher order resonant frequencies due to the neglect of cross-sectional shear deformations. For practical situations, where only lower modes are excited, simple beam theories may provide acceptable estimates of femur resonant frequencies. 9.2.
FEMUR
REPRESENTATION
BY
A FLEXIBLE
SHAFT
WITH
END
MASSES
The inadequacy of simple free-free beam theories to provide good approximations of the resonant frequencies, particularly for torsional vibrations, is obvious. Our aim is to seek a simple, yet representative, model of the femur. To achieve our goal, we inspected the vibration mode shapes predicted by the mathematical model. In torsion, they revealed a relative lack of motion of the femur extremities for all modes higher than the first. This behavior indicates significant inertial effects at the femur extremities. Accordingly, we proceeded to investigate a femur model consisting of a shaft of length L (the overall length of the femur, 51.5 cm) with two concentrated weights of mass MO and mass moment of inertia Jo each located at one extremity. The shaft mass is denoted by M, and its polar moment of inertia by J,. The solution for such a system in flexural motion has been provided by Laura et al. [24] for different mass ratios, 0 = MO/M,. Since this model consists of a uniform beam with end masses, it is consistent here to compare results with the h’s normalized with respect to the midshaft. Resonant frequencies for Q = 0.2 are noted to be in good agreement with our experimental data, particularly for lower modes. The non-dimensional frequencies are presented in Table 3. Inclusion of shear deformation would bring the higher frequencies into closer agreement. The equations of motion for torsional and longitudinal vibration are the same as equations (2) and (3), respectively. The frequency equation can be obtained by imposing
434
T. R. KHAL.IL.
the proper
boundary
conditions
D. (‘.
VIANO
AND
at the shaft extremities. tan
A’=2QA’/[(QA’)‘-
1
-2. TABER
It may be written
as
11,
110~
with Q = Jo/J, for torsional vibrations Q = MO/M, for longitudinal vibrations and A ’= Arr. The roots of equation (lo), which may be solved numerically or graphically, are the resonant frequencies of the system. For Q = 0, which is the case of a simple free-free beam, the roots A, are multiples of n (n = 1,2,3, . . .). For Q # 0, an extra root appears at the branch which intersects the asymptote located at A ’ = l/Q. This extra root corresponds to the lower resonant frequency in torsion. A value of Q = 2.8 provides a simple model which well represents the torsional vibrations of the isolated embalmed femur. The non-dimensional A’s predicted by this model are listed in Table 4. Note the prediction of the “extra” fundamental mode. As in the flexural case, a value of Q = 0.2 provides a model with resonances in general agreement with the experimental data (Table 5) for longitudinal vibrations. The above analysis demonstrated that the flexible shaft with end masses model exhibited similar vibrational behavior to that of the isolated femur. Whether this model or another can depict the vibrations of an in situ or in viva case is beyond the scope of this study. Several factors may influence the selection of a model, among which are the distribution of soft tissue mass and its mechanical impedance and vibrational coupling to the femur. Also, delineation of representative boundary condition at the articulating joints of the femur remains to be investigated. 9.3.
COMPARISON
OF RESULTS
OF THIS
STUDY
WITH
OTHER
INVESTIGATIONS
The lowest resonant frequency measured in this study (250 Hz) is in the same band (150-500 Hz) where Melvin [9] noted “a pronounced resonance” of the in situ femur under forced vibration. Melvin’s mechanical impedance data show a resonance of about 180 Hz. Campbell and Jurist [13] found the lowest resonant frequency of the femur to be 70 Hz. The reason that our lowest frequency is higher than that given by other workers is attributable to the complicated boundary conditions, which are not specifically “freefree”, encountered in in situ experimentation and to the added mass associated with the shaker head attachment to the knee joint. Also, Melvin [9] tested a cadaver leg, and obviously the added mass of soft tissue would additionally lower the resonant frequencies. Viano [14] measured the resonant frequencies of macerated femur shafts which had an average length of 22.5 cm. He reported that the lowest resonant frequencies for minimum flexure, maximum flexure, torsion, and longitudinal vibrations are 1.67, 1.73, 3.9 and 7.78 kHz, respectively. Scaling the average resonance data for the two femurs in this study, based on change of femur length to 54.6 cm (length of neutral axis), yields the corresponding resonances of 1.48, 1.82,1*22 and 5.20 kHz, which are in close agreement for flexure only. The large discrepancy in the torsional and longitudinal resonant frequencies confirms our earlier suggestion that the whole femur vibrates in a manner similar to that of a flexible shaft with rigid masses attached to its extremities. Knowledge of the embalmed femur resonant frequencies and the associated mode shapes provides an understanding of its impact response in cadaver tests. The frequency content of the input force and the associated energy in conjunction’with the dynamic properties of the femur determine its mode of deformation in dynamic load environments and, accordingly, its failure patterns. Viano and Khalil [7] demonstrated that the deformation pattern of a femur model depends on the load duration. Two bending modes (fixed-free beam) of 10 ms and 3.3 ms periods, respectively, were predicted. Our lowest measured flexural resonances of approximately 250 and 800 Hz scaled to a fixed-free beam condition yield resonance frequencies of 125 and 400 Hz with periods of 8 and
VIBRATIONAL
CHARACTERISTICS
OF THE FEMUR
435
2.5 ms, respectively, which is in close agreement with those predicted from the impact response of a planar finite element femur model of unembalmed bone [7]. The difference must be attributable to representing the femur by a two-dimensional model and/or to small differences between the unembalmed and embalmed bone material properties. 10. SUMMARY
AND CONCLUSIONS
1. Transfer function measurements of contralateral embalmed human femurs in a frequency band of 20 Hz-8 kHz provided 20 resonant frequencies for the “free-free” bone. The lowest frequencies in flexure (about principal planes of inertia), torsion, and longitudinal extension were 250, 308, 557 and 2138 Hz, respectively. 2. With use of an extension of classical beam theories to accommodate non-uniform bimaterial beam characteristics, a mathematical model was developed consisting of 59 joined uniform beam segments. Each beam segment may contain compact and/or cancellous bone. An iteration technique was used to solve the eigenvalue problem for “free-free” vibration for three principal types of vibration (flexure in two principal planes of inertia, torsion, and longitudinal extension). With use of published data for compact bone (E = 22 x lo6 kPa, G = 8.6 x lo6 kPa, p = 2080 kg/m” [14]) and cancellous bone (E = 0.38 x lo6 kPa, G = 0.15 x lo6 kPa, p = 890 kg/m3 [20]), the lowest predicted resonant frequencies for the three types of vibrations were 245, 289, 501 and 2118 Hz, respectively. Flexural, torsional, and longitudinal mode shapes for the first two harmonics, based on the mathematical model, have been presented. 3. By matching the measured frequency spectrum against the mathematical model prediction, it was possible to identify six pairs of flexural, five torsional, and three longitudinal resonances in the frequency range of 20 Hz-8 kHz. Exceptional agreement was noted between the theoretical and experimental resonant frequencies (within 3% on the average) for the 20 lowest modes. 4. Non-dimensional resonant frequencies A, corresponding to the three principal types of vibration were identified. These “femur constants” may be used to predict other femur resonances once their geometric and material properties have been determined. The experimentally determined h,‘s for the first three vibrational modes compare well with those of a simple uniform shaft with average geometric and material properties and two rigid masses attached at the extremities. The length of the shaft equals the length of the neutral axis of the femur. This finding allows for accurate predictions of femur resonances for other than free-free boundary conditions. 5. The measured resonant frequencies are in reasonable agreement with those reported by other investigators. However, the experimental technique with use of Fourier analysis ensures a more accurate determination of structural response by eliminating external mass-coupling effects and artificial boundary conditions. 6. Osteometric characteristics of the femur (cross-sectional area, centroid, principal moments of inertia) were determined throughout its entire length. The unique 3-D curve representing the neutral axis can be used to partition an axial load applied at the knee into flexural, torsional, and longitudinal components. This partition aids our understanding of femoral fracture mechanisms associated with axial impact.
REFERENCES 1. J. C. KOCH 1917 American Journal of Anatomy 21, 177-293. The laws of bone architecture, 2. G. GALILEO 1638 Dialog I. Mechanic. 3. T. G. TORIDAS 1969 Journal of Biomechunics 2, 163-l 74. Stress analysis of the femur.
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T. B. KHALIL,
D. C. VIANO
AND L. A. TABER
4. G. PIOTROWSKI and G. A. WILCOX 1971 Journal of Biomechanics 4, 497-506.
The stress program: a computer program for the analysis of stresses in long bones. 5. E. F. RYBICKI, F. H. SIMONEN and E. B. WEIS, Jr 1972 JournalofBiomechanics 5, 203-215. On the mathematical analysis of stress in the human femur, 6. S. H. ADVANI, H. V. S. GANGARAO, R. B. MARHIA and H. Y. CHANG, 1975 American Society of Civil Engineers National Structural Engineering Convention, New Orleans, Louisiana,
Structural response of human femurs to axial loads. 7. D. C. VIANO and T. B. KHALIL 1976 Proceedings of the Fourth New England Bioengineering Conference, New Haven, Connecticut. Plane strain analysis of a femur mid-section. 8. D. C. VIANO and T. B. KHALIL 1976 Mathematical Modeling-Biodynamic Response to Impact, Society of Automotive Engineers National Automobile and Manufacturing
Meeting.
Investigation of impact response and fracture of the human femur by finite element modeling. 9. J. W. MELVIN, R. L. STALNAKER, N. M. ALEM, J. B. BENSON and D. MOHAN 1975 Proceedings of the 19th Stapp Car Crash Conference, San Diego, California. Impact response and tolerance of the lower extremities. 10. W. R. POWELL, S. J. OJALA, S. H. ADVANI and R. B. MARTIN, 1975 Proceedingsof the 19th Stapp Car Crash Conference, San Diego, California. Cadaver femur responses to longitudinal impact. 11. D. C. VIANO and R. L. STALNAKER 1980 Journal of Biomechanics 13,701-7 15. Mechanisms of femoral fracture. 12. D. C. VIANO 1977 Proceedings of the 21st Stapp Car Crash Conference, New Orleans, Louisiana. Considerations for a femur injury criterion. 13. J. N. CAMPBELL and J. N. JURIST 1971 Journal of Biomechanics 4, 319-322. Mechanical impedance of the femur: a preliminary report. 14. D. C. VIANO, U. HELFENSTEIN, M. ANLIKER and P. R~~EGSEGGER 1976 Journal of Biomechanics 9, 703-710. Elastic properties of cortical bone in female human femurs. 15. T. B. KHALIL, D. C. VIANO and D. L. SMITH 1979 Journal of Sound and Vibration 63, 351-376. Experimental analysis of the vibrational characteristics of the human skull. 16. L. A. TABER and D. C. VIANO 1980 (submitted) Journalof Sound and Vibration. An analytical and experimental study of free vibration of composite beams of variable cross section. 17. M. RICHARDSON 1975 Seminar on Understanding Digital Control Analysis in Vibration Test Systems, Shock and Vibration Information Center Publication, 43-64. Modal analysis using digital test systems. 18. C. C. KENNEDY and C. D. P. PANCU 1947 JournalofAeronauticaZSciences 14,603-625. Use of vectors in vibration measurement and analysis. 19. H. YAMADA and F. G. EVANS 1970 Strength of BiologicalMaterials. Baltimore: Williams and Wilkins Company. 20. P. DUCHEYNE, L. HEYMANS, M. MARTENS and E. AERNOUDT 1977 Journal of Biomechanics 10,747-762. The mechanical behaviour of intracondylar cancellous bone of the femur at different loading rates. 21. H. S. YOON and J. L. KATZ 1976 Journal of Biomechanics 9, 459-464. Ultrasonic wave propagation in human cortical bone. II. Measurements of elastic properties and microhardness. 22. W. FL~~GGE 1962 Handbook of Engineering Mechanics. New York: McGraw-Hill Book Company. 23. T. K. HIGHT, R. L. PIZIALI and D. A. NAGEL 1980 Journal of Biomechanics 13, 139-147. Natural frequency analysis of a human tibia. 24. P. A. A. LAURA, J. L. POMBO and E. A. SUSEMIHL 1974 Journal of Sound and Vibration 37, 161-168. A note on the vibrations of a clamped-free beam with a mass at the free end.
VIBRATIONAL
CHARACTERISTICS
EMBALMED T. B. KHALIL,
HUMAN
D. C. VIANO
AND
OF THE
FEMUR L. A. TABER
Biomedical Science Department, General Motors Research Laboratories, Warren, Michigan 48090. U.S.A. (Received 6 May 1980, and in revised form 6 October 1980)
The resonant frequencies and mode shapes of contralateral femurs have been identified by experimental and analytical procedures. Also, the cross-sectional area, centroid, and principal moments of inertia were computed throughout the femur length for both compact and cancellous bone. The resonant frequencies of freely vibrating specimens were identified from transfer function measurements by using a Fourier analyzer. Twenty frequencies were noted in a frequency range of 20 Hz-8 kHz. A mathematical model of the femur consisting of 59 joined uniform segments, with each composed of compact and/or cancellous bone, was analyzed by using a transfer matrix technique. Results of the model enabled classification of the experimental resonances into deformations corresponding to flexure (about principal planes of inertia), torsion, and longitudinal extension with fundamental frequencies at 250, 308, 557, and 2138 Hz, respectively. Generalized nondimensional resonant frequencies were computed based on femur geometry averaged over its length and compared with those predicted by simple beam models. This analysis provided further understanding of the vibrational behavior of the femur.
1. INTRODUCTION The incidence of femoral fracture accompanied with complex failure patterns has prompted numerous investigations of femur mechanics. The femur is the longest and strongest bone of the musculoskeletal system. Femoral fracture generally results from excessive mechanical loads transmitted through the articulating joints. Much of the currently published work is concerned with two basic questions: (1) what is the non-traumatic load carrying capacity of the femur, and (2) what are the fracture patterns associated with traumatic loads? Femoral studies fall into one of two categories: theoretical studies and experimental studies in which cadaveric specimens are used. Among the theoretical studies is the work of Koch [l] who published the results of a comprehensive study of the structural mechanics of the femur. Using elementary beam theories, Koch determined how equi-stress contours lead to the remarkable geometrical adaptation of the femur to physiological mechanical loads. He also concluded that the femur is an optimum structural member (i.e., minimum weight for maximum strength) for load transmission between the acetabulum and the tibia. The earliest interest in femur mechanics, pointed out by Koch, ddtes back to 1638 and is accredited to Galileo [2]. Toridas [3] extended Koch’s work by modeling the femur as a system of joined three-dimensional beams. Femoral stresses are presented for both body weight and muscle force loads. Piotrowski and Wilcox [4] discussed a “stress program” that was used to determine the stress distribution in long bones. Other theoretical investigations of the femur include the work of Rybicki et al. [5] and of Advani et al. [6]. 417 @ 19X1Academic Press Inc. (London) Limited 0022-460X/81/070417+20 $02.00/O
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T. B. KHALII..
D. C‘. VIANO
AND
I.
A. TABER
The investigations cited have been conducted for static loading of the femur. In a typical car crash, the femur is subjected to dynamic loads of duration on the order of 50-60 ms. Accordingly, Viano and Khalil [7] investigated the dynamic response of a planar finite element model of the femur. Dynamic loads of varying duration (3-75 msj were applied to the lateral and medial condyles, respectively. The deformation response of the femur depends significantly on the load duration. First and second flexural vibration modes corresponding to the fixed-free beam can be excited. An extension of that study [S] provides evidence that the structural response characteristics of the femur play a significant role in increasing its load carrying ability for short duration impact. Dynamic experimental studies are, however, of recent origin. In these studies primarily human cadaver tissue has been utilized to assess the impact response of the femur and load transfer characteristics of the lower extremity musculoskeletal system. Typical of the laboratory experiments is the work of Melvin er al. [9] and Powell et al. [lo]. Viano and Stalnaker [ 1 l] subjected seated unembalmed human cadavers to knee impact aligned with the axis of the femur shaft. Strain gages bonded to the femur midshaft showed significant femoral bending during axial compressive loading due to the complex anatomical configuration of the knee-thigh-hip. Variations in test specimens and experimental conditions resulted in inconsistent force levels at fracture. It has been generally concluded that the peak applied load is not sufficient to predict femur fracture. Viano [12] summarized the available biomechanical data concerning femoral fracture and concluded that much of the scatter in experimental data is attributable to the load-time dependence for the initiation of fracture when the load duration is below 20 ms. Studies of the impact response and fracture of the femur are incomplete without basic knowledge of its structural dynamic characteristics. Information about femur resonant frequencies and associated mode shapes is invaluable to understanding femoral deformation patterns excited by various dynamic loads [7]. Also, this information is useful in the development of improved analytical models. In spite of the noted need for dynamic data, few studies have been conducted to delineate the vibrational characteristics of the femur. Campbell and Jurist [13] conducted preliminary tests on excised femurs to determine the mechanical impedance in a frequency band 20-4000 Hz. Since the authors were primarily interested in changes in mechanical impedance caused by femoral head fractures, they concentrated only on identifying the lowest resonance at about 70 Hz. Although the existence of numerous higher order resonances was noted, no attempt was made to identify or explain them. Melvin [9] conducted mechanical impedance tests by vibrating the knee joint of seated unembalmed cadavers. He noted a pronounced resonance in the range of 150-500 Hz. the resonant frequencies and mode shapes of macerated Viano et al. [14] measured femur shafts using a stabilized Michelson interferometer. Sixteen resonant frequencies were identified for the central shaft of the femur in a frequency range of 1 Hz-20 kHz. The lowest resonance was 1670 Hz. Classical beam theories were numerically adapted to account for the varying geometric characteristics of the compact bone along the femoral shaft. By iteratively matching analytical predictions with the experimental data for flexural, torsional, and longitudinal resonances, the elastic moduli of compact bone were evaluated for eight specimens. The study described in what follows here was concerned with experimental and analytical identification of the resonant frequencies of contralateral embalmed human femurs in a frequency range of 20 Hz-8 kHz. The resonant frequencies were determined from measurements of the transfer function of the femur by using Fourier analysis techniques. A similar test procedure was successfully applied to determine the resonant frequencies and mode shapes of the human skull [15]. The measured frequencies are
VIBRATIONAL
CHARACTERISTICS
OF THE
FEMUR
419
compared with results from an analytical model of the femur based on non-uniform beam vibrations [16]. By using the model one can calculate the vibratory motion in flexure, torsion, and longitudinal extension of a structure consisting of joined uniform beam segments-with each composed of compact and/or cancellous bone. By matching experimental and analytical resonances, the flexural, torsional, and extensional vibrations of an isolated whole femur were determined.
2. FEMUR TRANSFER
FUNCTION
In this study, femoral resonances were determined by digital Fourier analysis [17]. The non-destructive test procedure involves computation of repeated impact responses of the femur to identify its structural transfer function. The dynamic response of the femur is assumed to be linear and described by a second order differential equation. For such a system, the transfer function (frequency response function) defines the response in the frequency domain at a remote point due to a Dirac delta function, s(t), force applied to the structure. The transfer function is complex valued and can theoretically be determined by simultaneous monitoring, fast Fourier transforming (FFT), and dividing the acceleration response by the input force. It may be written as G(iw) = 9[A(t)]/F[F(t)] where 9 is a Fourier transform operator, A(t) is the output acceleration signal, and F(t) is the input force signal. In practice, G(iw) is determined from the quotient of an average of the input-output cross power spectra and input power spectrum of several repeat tests to minimize the influence of extraneous noise. The transfer function is measured experimentally in the frequency band of interest by use of a digital Fourier analyzer (HewlettPackard 545 1C). The resonant frequencies can be identified by inspection of the imaginary component and polar plot of the transfer function [17]. Coincidence of the peak of the imaginary component with a location on the polar plot where the rate of change of arc length is a maximum with frequency variations indicates the existence of a structural resonance. The theoretical basis of this technique has been given by Kennedy and Pancu [18].
3. TEST SPECIMENS
Two embalmed human femurs were obtained and tested at the Gurdjian-Lissner Biomechanics Laboratory, Wayne State University, Detroit, Michigan. The contralateral femurs were from a 74-year-old male who was 1.85 m in height and 81 kg in weight at the time of death. The cause of death was heart disease and respiratory failure. The exterior appearance of the femurs was excellent with no obvious sign of osteoporosis. The embalmed femurs weighed 865 g and had an overall length of 515 mm.
4. EXPERIMENTAL
METHOD
A schematic of the experimental set-up is shown in Figure 1. The isolated femur was supported on 50 mm of soft foam to approximate a free support condition. The femur was excited by a transient force F(t), delivered by lightly tapping on the femur surface by a hand-held hammer. The force was monitored by a load cell (Briiel and Kjaer #8200)
420
T.
R. KHALIL,
D. C. VIANO
Founer
mlifiers,
AND
L. A. TAbER
analyzer
Load
cell n-”
Figure
1. Schematic
of experimental
set-up.
attached to the hammer head which had a 12.5 mm diameter (steel) flat surface. The frequency response of the load cell is flat (* 1.0%) in the frequency range 20 Hz-70 kHz. Preliminary tests were conducted to ensure that sufficient energy was available in the force signal to excite the femur in the frequency band of interest, 20 Hz-8 kHz. The transient output signal A(t)was monitored by a transducer consisting of three miniature accelerometers (Endevco #2222B) glued to the femur surface. The accelerometer pack weighed 5 g and exhibited a flat frequency response (*2*5%) in the frequency range of 20 Hz-10 kHz. The femur was excited by tapping along three perpendicular directions, X (superior-inferior), Y (medial-lateral), or 2 (posterior-anterior) to selectively enhance the three primary modes of vibrational response: flexure in the plane of maximum and minimum inertia, torsion, and longitudinal extension. The input force and the corresponding acceleration component were amplified and passed through matched anti-aliasing filters. The signals were then fed into the Fourier analyzer to determine the transfer function. Response linearity was assured by (a) increasing the input force level and observing a proportional increase in acceleration, and (b) verifying Betti’s reciprocal theorem for elastic structures by measuring identical transfer functions for interchanged input/output locations. This finding is not surprising in view of the linear elastic behavior of compact and cancellous bone [19, 201. An ensemble average of 20 impacts was used to minimize the potential effects of spurious noise on the numerical value of the transfer function, particularly at higher frequencies where a poor signal-to-noise ratio may exist. Measuring the transfer function from various impact locations ensures that all vibrational modes of the structure are captured; also, using various accelerometer positions guards against missing a resonance by inadvertent placement of the accelerometer at a vibration node. Supporting the femur on a very soft rubber foam pad provided free boundary conditions at the extremities.
VIBRATIONAL
CHARACTERISTICS
OF THE
FEMUR
421
5, EXPERIMENTAL RESULTS The input force F(t), output acceleration A(t), and the associated Fourier transform are shown for a typical tapping in Figure 2. In this test, the force was applied normal to the femur surface along the Y direction (see Figure 1) and the acceleration was monitored along the same axis. Similar input/output signals were obtained for tests involving the other two, X and 2, directions, which indicated that a complex three-dimensional motion is stimulated by local impact.
Figure
2. Typical
input/output
signal and their Fourier
transforms.
Though the force signal is of short duration (0.5 ms), it provided sufficient excitatory energy over the frequency range of interest (20 Hz-8 kHz). The acceleration signal shows significant ringing which lasted well beyond the duration of the input load. It was also noted that the acceleration response exhibits a “beat phenomenon” which is typically encountered when resonances of a two degree of freednm system are close in value. The proximity of resonant frequencies which correspond to flexural vibration about minimum and maximum planes of inertia (as will be shown later) is probably the cause of the “beat phenomenon”. The Fourier transform of the acceleration signal demonstrates the complex vibratory characteristic of the femur. Figure 3 is typical of the measured transfer function in a frequency band, 20 Hz-8 kHz. The magnitude, real part, and imaginary part of the transfer function are presented in addition to the coherence function. A coherence value of less than unity indicates that the output signal may be contaminated with extraneous noise and the frequency measurements may not be a reliable measure of the structural resonance. Such situations were remedied by taking more response averages. Twenty experimental resonances (see Figure 3) were identified for the right femur. The resonances were identified from the imaginary component of the transfer function and its associated polar plot (Figure 4). Since only 12 frequencies could be identified in this example (listed in Figure 4), it is clear that a single experimental transfer function may not exhibit all resonant frequencies of a complex structure. Therefore, the femur was tapped at
422
T.
B. KHALIL.
D.
C. VIANO
AND
L. A.
TABER
Log mognltude
Real part
Imagmory port
Coherence function
Figure 3. Typical femur transfer function (frequency band 20 Hz-8 kHz). Resonant frequencies (Hz): 244, 317, 561, 830, 879, 1563, 1612, 1882, 2125, 2344, 2564, 3052, 3370, 3697.4151, 4445.4712.4907, 5372, 7300.
different locations (femur head, condyles, and shaft) and along different directions to ensure capture of all resonances. The imaginary components of two additional transfer functions, which were obtained by tapping on the medial condyle and the femur head, respectively, are shown in Figures 5 and 6. Inspection of resonance values indicates small variations in frequency (*4%) between these two experimental runs. This is partly attributable to the frequency resolution associated with the digital Fourier analyzer. Also, when independent modes of vibration have resonances in close proximity to one another, there may exist a vibrational interaction which can shift the apparent resonance [ 171. The use of multiple test conditions minimizes this possible artifact by varying the distribution of energy among the modes of vibration in a particular frequency range. The experimental resonant frequencies are presented in ascending sequential order in Table 1 for both the right femur (RF) and the left femur (LF). At this stage, the resonant frequencies correspond to a combination of flexural, torsional, and longitudinal vibrations. Later on, the frequency spectrum is partitioned to identify the resonances corresponding to the three principal motions. As may be observed in Table 1, the resonant spectra of the contralateral femurs are not identical. However, the average difference in frequency for all modes was 3-3 f 3.5%. Resonance data is presented only to a maximum frequency of 8 kHz since the spectral coherence decays significantly below unity beyond that frequency. Although resonant
VIBRATIONAL
I I
CHARACTERISTICS
OF THE
423
FEMUR
I 2
3
4
5
Frequency
6
7
8
CkHz)
Figure 4. Imaginary part of transfer function and its polar plot (left femur). Axial tap on femur head (X-direction). Resonant frequencies (Hz): 250, 546, 750, 1450, 1900, 2350, 2867, 3050, 3678, 4330, 5290, 7239.
I
0
I
I
I
I
I
I
I
I
2
3
4 Frequency
I 5
I
I
I
I 7
6
(kHz)
Figure 5. Imaginary part of transfer function. Force applied along X-axis on medial condyle. Accelerometer was mounted near the distal end of the shaft (left femur). Resonant frequencies (Hz): 250,555,745,1445, 1907, 2160,2850,3304,4050,4600,5050.
424
T. B. KHALIL.
0
I I
D. (‘. VIANO
I 2
I 3
AND
I 4
Frequency
I.. A. TABER
I
I
I
5
6
7
Figure 6. Imaginary part of transfer function. Force applied along Z-direction was mounted at midshaft (left femur). Resonant frequencies (Hz): 250,300,810, 3285,3660,4365,5060,5300.
TABLE
Experimental
6
CkHz)
1
resonant
frequencies
0-W Right femur
Left femur
(RF)
(LF)
250 315 563 825 879 1563 1607 1832 2125 2344 2572 3060 3365 3680 4146 4440 4712 4907 5375 7350
250 300 550 750 810 1450 1508 1900 2150 2345 2865 3050 3300 3670 4050 4350 4600 5050 5300 7239
on femur head. Accelerometer 1450,1508,2150,2345,3050,
VIBRATIONAL
CHARACTERISTICS
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FEMUR
425
frequencies above 8 kHz can be identified by increasing the amount of energy delivered to excite the femur, this was not pursued since the frequency content of impact loads suspected of initiating femur fracture is well below the cut-off frequency of 8 kHz [12]. The experimental frequency spectrum for each femur represents resonances corresponding to a complex three-dimensional motion of the femur. It is desirable from a structural response point of view to partition the frequency spectrum into three parts corresponding to flexural, torsional, and longitudinal vibration. This can be accomplished experimentally by the Fourier analyzer to identify the mode shapes associated with the resonant frequencies. Unfortunately, our Fourier analysis system does not, as yet, include a mode-shape package. Accordingly, we developed a mathematical model of the femur which allows us to sort the frequencies into their three corresponding motions. A description of the model and the identification of resonances corresponding to flexural, torsional, and longitudinal vibrations follow. 6. MATHEMATICAL
MODELING
OF FEMUR VIBRATIONS
6.1. FEMUR OSTEOMETRY To construct a mathemtical model of the femur, it is necessary to determine the varying cross-sectional geometric properties of the compact and cancellous bone. Although the
Figure 7. Posterior bone.
view of femur and cross-sections.
The difference
in color is due to the process of cutting
the
426
T. B. KHALIL.
D. C. VIANO
AND
I_. A. TABER
femur has a complicated three-dimensional configuration, it structurally resembles a long column (shaft) whose proximal epiphesis (head and neck) forms an angle of about 125” with the axis of the shaft. The distal epiphesis (condyles) is somewhat expanded and articulates with the tibia. Both extremities consist of cancellous bone covered by a thin shell of compact bone. The shaft may be regarded as a thick-walled cylinder of compact bone. Cross-sectional properties of the femur were determined from 59 transverse sections cut normal to the shaft axis. Because of large variations in section properties, slices were made every 6.4 mm (0.25 in) at the extremities. In the midshaft region, sections were made every 12.7 mm (0.5 in). Figure 7 shows the posterior view of the femur and a selected number of representative cross-sectional photographs. The location of the neutral axis is also shown. For each section, the area (A), the neutral axis (y, z), and the principal moments of inertia (Ii, Iz) with the orientation of the principal axes (13)were computed for both the compact and cancellous parts of the bone [ 141. The distribution of bone area and the centroid locations are presented in Figure 8 as a function of the femoral length. A lateral view of the neutral axis, along with the principal moments of inertia are presented in Figure 9. Cross-sectional area and moments of inertia from previous studies [l, 141 are presented in Figures 8 and 9 for comparison. The femur space centroid trajectory (Figure 10) is a complex three-dimensional curve.
0
100
200
300
400
I
I
500
X (mm) 40 (
I
I
, 220
”
X (mm) Figure 8. Cross-sectional area and centroid locations. bone; ZIeT Koch [l]; --C-, Viano etal. [14].
Present
study: -
-, cancellous
bone; -e,
compact
100
300
200 X
2.5
I
0
100
I
6.2.
MATERIAL
501
400
500
-l
200
300 X
Figure 9. Cross-sectional
400
(mm)
(mm)
moments of intertia. - - -, Viano et al. [ 141; - - -,
Koch [I]
PROPERTIES
The elastic moduli and density of compact bone (E = 22.0 x lo6 kPa, G = 8.6 x 10” kPa, p = 2090 kg/m3) were selected from the extensive literature on the subject (see, for example, references [14, 191). Similar properties of cancellous bone (E = 0.38 x lo6 kPa, G = O-15 x 10” kPa, p = 890 kg/m31 were taken from the work of Ducheyne et al. [20]. Bone anisotropy [21] and inhomogeneity [20] have been disregarded in this study to simplify the analysis. 6.3.
ANALYTICAL
MODEL
To sort the experimental resonant frequencies according to vibratory motion of a non-uniform bimaterial beam in flexure, torsion, or longitudinal extension, we utilize an extension [16] of a mathematical model developed by Viano et al. [14]. The non-uniform
428
T. B. KHALIL.
D. c‘. VIANO
Figure 10. Three-dimensional
AND
1.. A. TABER
femur centroid trajectory.
femur structure is conceptually divided into a set of 59 one-dimensional uniform beam segments. Each segment may consist of two elastic materials (compact and cancellous bone) with known uniform geometric characteristics. The general linear motion of each beam segment is described by a superposition of bending in two principal planes, torsion, and longitudinal extension. The following equations [22] describe the motion of each uniform beam segment:
(a) flexure
(according
to Timoshenko
/!?(J4W(X, t)/ex4)+(y/cy)[S(a4W(X,
beam theory):
t)/at4)-p(a4w(X,
?)/c?X2~f2)
+cr@*W(x, t)/at2)]-6(a4W(X, (b) torsion
(according
to simple &*4(x,
(c) longitudinal
torsion
of a non-circular
t)/&t’)
(1)
=o;
cylinder);
t)/ax2) - k’7(a2C$(x, t)/at*) = 0;
(2)
extension: zQ2U(X, t)/ax2) - K(&4(X, t)/ar2) = 0.
In equations
(3)
(1) to (3),
a = klG,Al+ K =PIAI+PzA~,
kzG2A2,
P = El11 + ~5212, v =
E,A, + E2A2,
Y =
PIAI
+ ~2A2,
T = G,JI + G,+Jz,
6 = p1I1+
CL= PIJIf
P212,
PZJZ,
u is longitudinal displacement, C$ is the angle of twist, w is the transverse deflection, x is position along the longitudinal axis, t is time, pi is material density (i = 1 for compact bone and 2 for cancellous bone), Ai is cross-sectional area, Ei is the elastic modulus, Gi is the shear modulus, Ji is the polar moment of inertia, Ii is the area moment of inertia, ki is
VIBRATIONAL
CHARACTERISTICS
OF THE
FEMUR
429
the Timoshenko shear coefficient (kl = kz - @55), and k’ is the torsional rigidity constant (k’ = O-36). The generalized longitudinal and shear wave speeds are given by CO= JV/K and C, = Jp/k’T, respectively. For each of the three independent vibratory motions of the femur (flexure, about the principal axis system with minimum and maximum moments of inertia, torsion, and longitudinal extension) the discretized equations of motion are written in matrix form. The spatial eigenvalue problem is solved by an iterative procedure previously described [ 141 to yield the analytical resonant frequencies. Free-free boundary conditions are assumed at both ends of the femur and continuity of deformation and force balance are insured between adjacent uniform sections. In this formulation no account is taken of curvature of the neutral axis of the femur, and therefore it provides a solution in which axial and bending vibrations are uncoupled. Because of the mild curvature of the femur, we expect only a slight effect on the resonances. In a similar analysis, Hight et al. [23] found that exclusion of curvature effects changed the resonant frequencies of the tibia by less than 5%. 7. ANALYTICAL RESULTS The frequency spectra predicted by the mathematical model of the femur for three types of vibrations, flexure, torsion and longitudinal extension, are presented in Table 2. The computations were carried out in a frequency range of O-8 kHz. The mode shapes of the vibrating neutral axis were also computed. Figure 11 shows the first two modes corresponding to flexure (about the minimum inertial axis), torsion, and longitudinal extension. TABLE
2
Analytical resonant frequencies (Hz)
Flexure Mode no. 1 2 3 4 5 6
W
Longitudinal
mm
max
Torsion
extension
245 771 1507 2373 3358 4412
289 859 1672 2635 3744 4880
501 1810 3149 4281 5315
2118 4407 7264
Inspection of the resonant frequencies reveals that the lowest resonant frequency (245 Hz) corresponds to flexural vibration. Also, it is noted that the paired flexural resonances are close to each other, which may explain the “beat phenomenon” observed in the acceleration response (see Figure 2). The lowest torsional resonant frequency is 501 Hz. Higher torsional frequencies, in contrast with those given by simple beam theories, are not simple integer harmonics. After detailed analysis we have concluded that the femur vibration, particularly in torsion, resembles that of a flexible shaft with two rigid masses attached at the extremities. This point is confirmed by observing the relative lack of motion of the femur extremities in the mode shapes (see Figure 11). The validity of representing the femur by simple mathematical models will be discussed in section 9. For the mathematical model the lowest resonant frequency in longitudinal extension predicted is 2118 Hz.
430
T. B.
KHALIL.
D.
(‘. VIANO
AND
I_. A. TABER
Flexure
501
Hz
2118 Hz
1810 Hz
Torsion
Longitudinal
4907
Hz
Figure 11. Femur mode shapes.
Variation in the elasticity coefficient (E) of cancellous bone by a factor of f 10 times, to account for potential bone inhomogeneity and the scatter in published data [20], did not change the resonant frequencies by more than &12%, with the lower modes being less affected than higher modes. Frequencies below 3000 Hz were not changed by more than *4%. It appears that the distributed mass of cancellous bone, in contrast with its stiffness, has more influence on femur vibrations. 8. COMPARISON
OF ANALYTICAL
AND
EXPERIMENTAL
RESULTS
In a frequency band of 20 Hz-8 kHz, 20 analytical and experimental resonances were identified for the femur. Based on the mathematical analysis, the experimental spectrum was partitioned into three independent (principal) resonance motions. This resulted in identfication of the six lowest mode pairs of flexural vibration about the minimum and maximum planes of cross-sectional inertia, five torsional, and three longitudinal resonances. Good agreement is noted between the experimental data and the mathematical model prediction as shown in Figure 12. In typical impact conditions, where the load duration is in excess of 2 ms, the predominant frequency of the approximate half-sine load is less than 250 Hz. Assuming that the actual in uivo boundary conditions of the femur are somewhere between “free-free” and “free-fixed”, the two lowest llexural modes and/or the first torsional mode are most likely to be excited.
9. DISCUSSION
9.1. ON USING SIMPLE BEAM MODELS TO SIMULATE FEMUR VIBRATIONS The femur is a complex three-dimensional structure. An understanding of its basic vibratory motion is fundamental to comprehending its structural response to mechanical
VIBRATIONAL
CHARACTERISTICS
8
OF THE
431
FEMUR
l-
I
7-
Longltudlnol
6-
5-
N 5 G4 5 i+ z 3-
2-
I-
I
2
Figure 12. Resonant O-0, theoretical data.
frequency
spectrum.
4
3 Mode
5
number
?? , Experimental
(right
femur):
A, experimental
(left femur);
impact. The analysis presented, though providing complete information about femur vibrations, is limited to the tested specimens. It is desirable to extend this information to determine the resonant frequencies of other femurs of different osteometry. This can be achieved by establishing generalized non-dimensional eigenvalues (A,,) of the femur. Following Fliigge’s convention [22], the eigenvalues of the frequency equations (l), (2) and (3) corresponding to flexural, torsional, and longitudinal vibrations, respectively, can be written as follows: (a) flexure with respect to minimum and maximum axes of inertia:
(b) torsion:
(c) longitudinal extension: ,-Y
432
7’. B. KHALIL, D. C. VIANO
AND
L. A. TAHEK
Here A,, is a constant to be determined for various mode numbers o (n = 1.2,3, . .). L is the femur length, and N is the number of femur segments. All other constants are as defined previously. The values of A for flexure, torsion, and longitudinal vibration as determined from both theoretical and experimental data are presented in Tables 3,4 and 5, respectively. Values in parentheses are normalized with respect to the geometry at the midshaft. Small variations (less than 3%) were observed in calculating the value of A for minimum and maximum frequencies at a particular mode number. This variation is possibly due to oversimplification in computing C, from equation (5) at the midshaft only. Also, A‘s of
TABLET
Values of A for flexural vibration
A”
\ n
Theory
\
1 2 3 4 5 6
3.77 6.60 9.21 11.56 13.77 15.75
Experiment
(4.17) (7.31) (10.01) (12.41) (15.25) (17.44)
3.85 (4.13) 6.61 (7.09) 9.05 (9.72) 11.60 (12.46) 13.66 (14.67) 15.79 (16.96)
Free-free shaft with concentrated masses at the ends Q = 0.2 4.27 7.32 10.40 1351 16.62 19.75
Simple free-free uniform beam 4.73 7.85 10.99 14.14 17.28 20.42
TABLET
Values of A for torsional vibration
Theory 1 2 3 4 5
0.33 1.18 2.05 2.79 3.46
(0.25) (0.90) (1.56) (2.13) (2.64)
Experiment 0.36 1.22 1.98 2.59 3.43
(0.27) (0.93) (1.51) (1.97) (2.61)
Free-free shaft with concentrated masses at the ends Q=2.8 0.27 1.08 2.04 3.02 4.04
Simple free-free uniform beam 1.0 2.0 3.0 4.0 5.0
TABLET
Values of A for longitudinal vibration
A,
n \
Theory 1 2 3
0.83 (0.67) 1.73 (1.40) 3.77 (2.24)
Experiment 0.82 (0.66) 1.72 (1.39) 2.77 (2.24)
Free-free shaft with concentrated masses at the ends Q=Os2 0.75 1.54 2.45
Simple free-free uniform beam 1.0 2.0 3.0
VIBRATIONAL
CHARACTERISTICS
OF
THE
433
FEMIJR
simple free-free beam vibrations [22] are included in the tables. In addition, we have included h’s obtained from vibratory solutions of flexible shafts with two rigid masses attached to their extremities, with Q being the ratio of the end mass to the shaft mass. The resonances of these models explain the vibrational behavior of the femur as will be discussed later. A close inspection of Tables 3, 4 and 5 reveal the following: (a) With use of midshaft femoral properties (C,, equation (5)), simple beam theories [22] provide reasonable predictions of the flexural resonant frequencies (see Table 3), particularly for the lower modes. Resonances of the first three modes are higher by about 9% ; however, higher mode frequencies are off by as much as 15%. Higher mode frequencies predicted by simple beam theories are progressively in error due to neglect of shear deformation effects. The analysis provided earlier (based on Timoshenko beam theory) accounts for shear deformation across the beam section through the introduction of the shear coefficients ki. (b) Inspection of Table 4 reveals that simple free-free beam analysis does not provide reasonable predictions of torsional frequencies. The analysis yields n (n = 1,2,3, . . .) as the value of A,,. However, the experimental data of Table 4 appear to exhibit an extra frequency. The appearance of this extra frequency will be discussed in more detail later. For now, it is clear that the simple beam theory does not provide a good model for femur torsional vibrations. (c) For longitudinal vibrations, the simple beam theory predicts that the frequencies are integer multiples of n. Although the experimental data (see Table 5), appear to exhibit a similar trend to the simple beam theory data, the experimental frequencies are lower by approximately 20%. A similar situation to the torsional case, i.e., the appearance of an extra frequency, may occur for longitudinal vibration. (d) Simple beam theories overestimate the value of higher order resonant frequencies due to the neglect of cross-sectional shear deformations. For practical situations, where only lower modes are excited, simple beam theories may provide acceptable estimates of femur resonant frequencies. 9.2.
FEMUR
REPRESENTATION
BY
A FLEXIBLE
SHAFT
WITH
END
MASSES
The inadequacy of simple free-free beam theories to provide good approximations of the resonant frequencies, particularly for torsional vibrations, is obvious. Our aim is to seek a simple, yet representative, model of the femur. To achieve our goal, we inspected the vibration mode shapes predicted by the mathematical model. In torsion, they revealed a relative lack of motion of the femur extremities for all modes higher than the first. This behavior indicates significant inertial effects at the femur extremities. Accordingly, we proceeded to investigate a femur model consisting of a shaft of length L (the overall length of the femur, 51.5 cm) with two concentrated weights of mass MO and mass moment of inertia Jo each located at one extremity. The shaft mass is denoted by M, and its polar moment of inertia by J,. The solution for such a system in flexural motion has been provided by Laura et al. [24] for different mass ratios, 0 = MO/M,. Since this model consists of a uniform beam with end masses, it is consistent here to compare results with the h’s normalized with respect to the midshaft. Resonant frequencies for Q = 0.2 are noted to be in good agreement with our experimental data, particularly for lower modes. The non-dimensional frequencies are presented in Table 3. Inclusion of shear deformation would bring the higher frequencies into closer agreement. The equations of motion for torsional and longitudinal vibration are the same as equations (2) and (3), respectively. The frequency equation can be obtained by imposing
434
T. R. KHAL.IL.
the proper
boundary
conditions
D. (‘.
VIANO
AND
at the shaft extremities. tan
A’=2QA’/[(QA’)‘-
1
-2. TABER
It may be written
as
11,
110~
with Q = Jo/J, for torsional vibrations Q = MO/M, for longitudinal vibrations and A ’= Arr. The roots of equation (lo), which may be solved numerically or graphically, are the resonant frequencies of the system. For Q = 0, which is the case of a simple free-free beam, the roots A, are multiples of n (n = 1,2,3, . . .). For Q # 0, an extra root appears at the branch which intersects the asymptote located at A ’ = l/Q. This extra root corresponds to the lower resonant frequency in torsion. A value of Q = 2.8 provides a simple model which well represents the torsional vibrations of the isolated embalmed femur. The non-dimensional A’s predicted by this model are listed in Table 4. Note the prediction of the “extra” fundamental mode. As in the flexural case, a value of Q = 0.2 provides a model with resonances in general agreement with the experimental data (Table 5) for longitudinal vibrations. The above analysis demonstrated that the flexible shaft with end masses model exhibited similar vibrational behavior to that of the isolated femur. Whether this model or another can depict the vibrations of an in situ or in viva case is beyond the scope of this study. Several factors may influence the selection of a model, among which are the distribution of soft tissue mass and its mechanical impedance and vibrational coupling to the femur. Also, delineation of representative boundary condition at the articulating joints of the femur remains to be investigated. 9.3.
COMPARISON
OF RESULTS
OF THIS
STUDY
WITH
OTHER
INVESTIGATIONS
The lowest resonant frequency measured in this study (250 Hz) is in the same band (150-500 Hz) where Melvin [9] noted “a pronounced resonance” of the in situ femur under forced vibration. Melvin’s mechanical impedance data show a resonance of about 180 Hz. Campbell and Jurist [13] found the lowest resonant frequency of the femur to be 70 Hz. The reason that our lowest frequency is higher than that given by other workers is attributable to the complicated boundary conditions, which are not specifically “freefree”, encountered in in situ experimentation and to the added mass associated with the shaker head attachment to the knee joint. Also, Melvin [9] tested a cadaver leg, and obviously the added mass of soft tissue would additionally lower the resonant frequencies. Viano [14] measured the resonant frequencies of macerated femur shafts which had an average length of 22.5 cm. He reported that the lowest resonant frequencies for minimum flexure, maximum flexure, torsion, and longitudinal vibrations are 1.67, 1.73, 3.9 and 7.78 kHz, respectively. Scaling the average resonance data for the two femurs in this study, based on change of femur length to 54.6 cm (length of neutral axis), yields the corresponding resonances of 1.48, 1.82,1*22 and 5.20 kHz, which are in close agreement for flexure only. The large discrepancy in the torsional and longitudinal resonant frequencies confirms our earlier suggestion that the whole femur vibrates in a manner similar to that of a flexible shaft with rigid masses attached to its extremities. Knowledge of the embalmed femur resonant frequencies and the associated mode shapes provides an understanding of its impact response in cadaver tests. The frequency content of the input force and the associated energy in conjunction’with the dynamic properties of the femur determine its mode of deformation in dynamic load environments and, accordingly, its failure patterns. Viano and Khalil [7] demonstrated that the deformation pattern of a femur model depends on the load duration. Two bending modes (fixed-free beam) of 10 ms and 3.3 ms periods, respectively, were predicted. Our lowest measured flexural resonances of approximately 250 and 800 Hz scaled to a fixed-free beam condition yield resonance frequencies of 125 and 400 Hz with periods of 8 and
VIBRATIONAL
CHARACTERISTICS
OF THE FEMUR
435
2.5 ms, respectively, which is in close agreement with those predicted from the impact response of a planar finite element femur model of unembalmed bone [7]. The difference must be attributable to representing the femur by a two-dimensional model and/or to small differences between the unembalmed and embalmed bone material properties. 10. SUMMARY
AND CONCLUSIONS
1. Transfer function measurements of contralateral embalmed human femurs in a frequency band of 20 Hz-8 kHz provided 20 resonant frequencies for the “free-free” bone. The lowest frequencies in flexure (about principal planes of inertia), torsion, and longitudinal extension were 250, 308, 557 and 2138 Hz, respectively. 2. With use of an extension of classical beam theories to accommodate non-uniform bimaterial beam characteristics, a mathematical model was developed consisting of 59 joined uniform beam segments. Each beam segment may contain compact and/or cancellous bone. An iteration technique was used to solve the eigenvalue problem for “free-free” vibration for three principal types of vibration (flexure in two principal planes of inertia, torsion, and longitudinal extension). With use of published data for compact bone (E = 22 x lo6 kPa, G = 8.6 x lo6 kPa, p = 2080 kg/m” [14]) and cancellous bone (E = 0.38 x lo6 kPa, G = 0.15 x lo6 kPa, p = 890 kg/m3 [20]), the lowest predicted resonant frequencies for the three types of vibrations were 245, 289, 501 and 2118 Hz, respectively. Flexural, torsional, and longitudinal mode shapes for the first two harmonics, based on the mathematical model, have been presented. 3. By matching the measured frequency spectrum against the mathematical model prediction, it was possible to identify six pairs of flexural, five torsional, and three longitudinal resonances in the frequency range of 20 Hz-8 kHz. Exceptional agreement was noted between the theoretical and experimental resonant frequencies (within 3% on the average) for the 20 lowest modes. 4. Non-dimensional resonant frequencies A, corresponding to the three principal types of vibration were identified. These “femur constants” may be used to predict other femur resonances once their geometric and material properties have been determined. The experimentally determined h,‘s for the first three vibrational modes compare well with those of a simple uniform shaft with average geometric and material properties and two rigid masses attached at the extremities. The length of the shaft equals the length of the neutral axis of the femur. This finding allows for accurate predictions of femur resonances for other than free-free boundary conditions. 5. The measured resonant frequencies are in reasonable agreement with those reported by other investigators. However, the experimental technique with use of Fourier analysis ensures a more accurate determination of structural response by eliminating external mass-coupling effects and artificial boundary conditions. 6. Osteometric characteristics of the femur (cross-sectional area, centroid, principal moments of inertia) were determined throughout its entire length. The unique 3-D curve representing the neutral axis can be used to partition an axial load applied at the knee into flexural, torsional, and longitudinal components. This partition aids our understanding of femoral fracture mechanisms associated with axial impact.
REFERENCES 1. J. C. KOCH 1917 American Journal of Anatomy 21, 177-293. The laws of bone architecture, 2. G. GALILEO 1638 Dialog I. Mechanic. 3. T. G. TORIDAS 1969 Journal of Biomechunics 2, 163-l 74. Stress analysis of the femur.
436
T. B. KHALIL,
D. C. VIANO
AND L. A. TABER
4. G. PIOTROWSKI and G. A. WILCOX 1971 Journal of Biomechanics 4, 497-506.
The stress program: a computer program for the analysis of stresses in long bones. 5. E. F. RYBICKI, F. H. SIMONEN and E. B. WEIS, Jr 1972 JournalofBiomechanics 5, 203-215. On the mathematical analysis of stress in the human femur, 6. S. H. ADVANI, H. V. S. GANGARAO, R. B. MARHIA and H. Y. CHANG, 1975 American Society of Civil Engineers National Structural Engineering Convention, New Orleans, Louisiana,
Structural response of human femurs to axial loads. 7. D. C. VIANO and T. B. KHALIL 1976 Proceedings of the Fourth New England Bioengineering Conference, New Haven, Connecticut. Plane strain analysis of a femur mid-section. 8. D. C. VIANO and T. B. KHALIL 1976 Mathematical Modeling-Biodynamic Response to Impact, Society of Automotive Engineers National Automobile and Manufacturing
Meeting.
Investigation of impact response and fracture of the human femur by finite element modeling. 9. J. W. MELVIN, R. L. STALNAKER, N. M. ALEM, J. B. BENSON and D. MOHAN 1975 Proceedings of the 19th Stapp Car Crash Conference, San Diego, California. Impact response and tolerance of the lower extremities. 10. W. R. POWELL, S. J. OJALA, S. H. ADVANI and R. B. MARTIN, 1975 Proceedingsof the 19th Stapp Car Crash Conference, San Diego, California. Cadaver femur responses to longitudinal impact. 11. D. C. VIANO and R. L. STALNAKER 1980 Journal of Biomechanics 13,701-7 15. Mechanisms of femoral fracture. 12. D. C. VIANO 1977 Proceedings of the 21st Stapp Car Crash Conference, New Orleans, Louisiana. Considerations for a femur injury criterion. 13. J. N. CAMPBELL and J. N. JURIST 1971 Journal of Biomechanics 4, 319-322. Mechanical impedance of the femur: a preliminary report. 14. D. C. VIANO, U. HELFENSTEIN, M. ANLIKER and P. R~~EGSEGGER 1976 Journal of Biomechanics 9, 703-710. Elastic properties of cortical bone in female human femurs. 15. T. B. KHALIL, D. C. VIANO and D. L. SMITH 1979 Journal of Sound and Vibration 63, 351-376. Experimental analysis of the vibrational characteristics of the human skull. 16. L. A. TABER and D. C. VIANO 1980 (submitted) Journalof Sound and Vibration. An analytical and experimental study of free vibration of composite beams of variable cross section. 17. M. RICHARDSON 1975 Seminar on Understanding Digital Control Analysis in Vibration Test Systems, Shock and Vibration Information Center Publication, 43-64. Modal analysis using digital test systems. 18. C. C. KENNEDY and C. D. P. PANCU 1947 JournalofAeronauticaZSciences 14,603-625. Use of vectors in vibration measurement and analysis. 19. H. YAMADA and F. G. EVANS 1970 Strength of BiologicalMaterials. Baltimore: Williams and Wilkins Company. 20. P. DUCHEYNE, L. HEYMANS, M. MARTENS and E. AERNOUDT 1977 Journal of Biomechanics 10,747-762. The mechanical behaviour of intracondylar cancellous bone of the femur at different loading rates. 21. H. S. YOON and J. L. KATZ 1976 Journal of Biomechanics 9, 459-464. Ultrasonic wave propagation in human cortical bone. II. Measurements of elastic properties and microhardness. 22. W. FL~~GGE 1962 Handbook of Engineering Mechanics. New York: McGraw-Hill Book Company. 23. T. K. HIGHT, R. L. PIZIALI and D. A. NAGEL 1980 Journal of Biomechanics 13, 139-147. Natural frequency analysis of a human tibia. 24. P. A. A. LAURA, J. L. POMBO and E. A. SUSEMIHL 1974 Journal of Sound and Vibration 37, 161-168. A note on the vibrations of a clamped-free beam with a mass at the free end.