- Email: [email protected]
Resonant frequency of a pin-accelerometer system mounted in bone
Pergamon
J. Biomechanics, Vol. 28, No. 5, pp. 625-629, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0021-9290/95 $9.50 + 00
0021-9290(94)00090-5
TECHNICAL
RESONANT
NOTE
FREQUENCY OF A PIN-ACCELEROMETER MOUNTED IN BONE
SYSTEM
M. Rostedt,* H. Bromant and T. Hansson* *Department of Orthopaedics, Sahlgren Hospital, S-413 45 Giiteborg, Sweden; and tDepartment of Applied Electronics, Chalmers University of Technology, S-412 96 Goteborg, Sweden Abstract-Invasive measurements of spinal motion using intraosseous metal pins have become common. For this reason, the resonant frequency of intraosseous pins attached with accelerometers was determined using two different methods. It was concluded that plucking the pin is a reliable method for determining the resonant frequency and, in order to accurately measure bone movement at frequencies up to 32 Hz, the pin diameter should be 2.0 mm or more. With a mass of the accelerometer assembly equal to 27 g, the total pin length should not exceed 80 mm with a bone-accelerometer distance of 25 mm and a pin diameter of 2.4&m.
INTRODUCTION
Vibration is believed to be a risk factor for low back pain. The response of the human body is known to be dependent on the frequency of the vibration exposure. In order to investigate the effects of vibration on the spine, it is thus necessary to measure the response at various frequencies. The relevant information from this kind of testing is contained in the transfer function (transmissibility) of the actual system. The transfer function of a system is defined as the ratio of the Fourier transform of the output signal to the Fourier transform of the input signal. For sinusoidal inputs, the transfer function gives the gain and the phase-shift of the systemat the frequency of the sinusoid. As acceleration has a direction, it is also necessary to specify the directions of the stimulus and response.. Transfer function estimation by the impact technique is a fast and reproducible method for investigating the effects of vibration on the human body (e.g. fee-spine, buttock-spine). The test consists of placing the subject on a vertically movable platform and, while impacting the platform, simultaneously measuring the accelerations at the platform and at a location on the spine. Pope et cl. (1986) showed that, in order to accurately measure the acceleration of vertebrae, the transducers should be mounted on pins rigidly attached to bone. Accelerometers mounted on intraosseous pins have been used in many experiments (e.g. to determine transfer functions) with human subjects in various situations. (Christ and Dupuis, 1963,1966; Hagena er al., 1986, 1985; Krause, 1963; Lange and Coermann, 1965; Pope et al., 1989a,b, 1990, 1986, 1987). In this type of test, it is important that the resonant frequency (natural frequency) of the pin-accelerometer system be considerably higher than the frequencies of interest in the experiment. The dependency of the resonant frequency on pin length and accelerometer mounting has not been reported (it has been tested without any data presented by Hagena et al., 1985). In Cornelissen et al. (1984) the resonant
Received in final form 13 June 1994. Address correspondence to: Mats Rostedt, Yrkesortopeden, Sahlgrenska Sjukhuset, S-413 45 Giiteborg, Sweden.
frequency as a function of the rigitidy of the pin fixation was investigated. However, no values for the resonant frequency or pin length were presented. Although a convenient way to determine the resonant frequency is to pluck the pin and examine the peaks in the spectral density of the signal, determining the transfer function from where the pin is fixed to the accelerometer is of greater value because it holds information as to which extent the total transfer function (e.g. buttock-accelerometer) becomes distorted by the measurement system. This study addresses the following questions: (1) How should the accelerometers and pins be mounted and what pin diameter is most suitable in order to make the resonant frequency appropriately high? (2) Could relevant information about resonance be obtained from simply plucking the pin? (3) Does bone mounting significantly lower the resonant frequency as compared to rigid mounting in aluminum? MATERIALS
AND
METHODS
The experimental setup, similar to that used in human experiments conducted in our laboratory, consisted of piezoresistive accelerometers (Endevco 7265A-HS, San Juan Capistrano, CA, U.S.A.). mounted on steel pins (Steinman pin, Richards Medical Co.. Memnhis. U.S.A.). Three different diameters were tested: 1.4,2.0 and 2.4 mm. The mass per unit length of each pin was 35, 24 and 13mgmm-’ for the 2.4, 2.0 and 1.4 mm pins, respectively. The pins were either tapped into a wooden block or tapped into the spinous processes (approximately 20 mm) of cadaveric lumbar vertebrae in the sagittal plane or clamped in an aluminum block, see Fig. 1. The pins were inserted parallel to the horizontal plane so that the sensitive axis of the accelerometers could be closely aligned (in the sagittal plane) to the vertical and horizontal directions respectively. Most soft tissue was excised from each vertebra which was then plotted in Plastic Padding (Type Hard, AB Plastic Padding, Giiteborg, Sweden) in an aluminum dish bolted to an U-beam, see Fig. 2. The aluminum block and wooden block were also bolted directly to the U-beam together with the input signal accelerometer. The U-beam was clamped to the platform of the whole-body impactor described by Pope et al. (1987). The
625
Technical Note
626
Fig. 1. Accelerometer and pin assembly. Vertebra /Tic
Padding
Fig. 2. Mounting of the vertebra. impactor consists of a vertically movable platform supported by attenuating springs, and a pendulum that makes it possible to strike the platform in a highly reproducible manner. The accelerometer signals were amplified so that the peak value was about 5 V, as well as low-pass filtered at 3OOHz. The gravitational acceleration was subtracted so that the signal level was approximately zero before each impact. Data were sampled at 1024 Hz, and the sampling time was 2 s, resulting in a frequency resolution of 0.5 Hz. The mass of the accelerometer assembly was 27 g. The parameters which were varied were the following: pin diameter, d, the distance from the vertebra to the accelerometers, y, and the length of the pin except the part inserted into the vertebra, L, see Fig. 1. Two vertebrae were tested and one impact was done for each parameter combination since a pilot test demonstrated negligible difference between trials. As a check during the main tests, some repeat trials were performed and the differences in resonant frequency were always less than 1 Hz. The wooden block was used in the tests where transfer function estimation and pin plucking were contrasted. In the pin plucking tests,the horizontal and vertical accelerometer yielded the same value for the (lowest) resonant frequency but the vertical was always used. DATA
PROCESSING
All calculations were made by a program developed primarily to estimate transfer functions from this type of experiment and to plot the corresponding phase and magnitude. The program facilitates in magnifying parts of the curves and obtaining position readout from a cursor. The calculations performed were the following. First, any remaining offset was removed form the input (platform) and output (pin) signals by subtracting the mean (before the impact). A Fourier transformation was then performed on each of the signals. The cross power density was estimated as the
smoothed product of the output Fourier transform and the complex conjugate of the input Fourier transform. The input (output) power density was calculated in the same manner with output replaced by input (input replaced by output). Smoothing was done utilizing five data points and a triangular weighting function. The transfer function was then estimated as the quotient of the cross power density and the input power density. Only the output power density was calculated in the tests where the pin was plucked. The resonant frequency was determined by localizing the frequency of the largest peak of the magnitude of the transfer function (or the output power density), see Figs 3 and 4. THEORY
Previous tests at our laboratory have shown that very little energy passes into the spine for frequencies above 32 Hz due to soft tissue attenuation. Furthermore, in most occupational situations where vibrations are involved (e.g. motor vehicle driving), the main part of the energy is contained in frequencies lower than this limit. The aim is therefore to find a lower limit for the resonant frequency which will yield reliable data up to 32 Hz. To find this limit for the resonant frequency, one can look at a second-order system consisting of a mass, a damper and a spring (the squared magnitude of its transfer function is shown in the equation below). The reason for selecting this systemis that it is the simplest physically relevant system that can show resonant behavior: k2 + $02 IW412
=
(k - mwz)z
+ t+w3
With no damping (9 =0) and at l/3 of its resonant frequency, the magnitude will be 1.125 (1.02 dB), implying a measurement error of 12.5%. When damping is introduced, the
Technical Note 8-
Input signal acceleration
621
(m/s*)
8.
-2. -41 0
1
0.5
1. *. *. Output signal acceleration
84. 6.
Time
.
* (m/s*)
*
1.5 -
*
-.
-
*
(s)
-.
c 2 L
r
4.
-2. Time (s)
-4 7
l
1
0.5
0
1.5
2
25 3
+ Magnitude
(df3)
20. 15. 10. 5.
-5 . -1oi
Frequency (Hz) I 16
32
64
r 128
Fig. 3. From top to bottom: input signal, output signal and the corresponding transfer function magnitude for one impact. Bone mounted pin with L = I25 mm, y = 20 mm and d = 2.0 mm.
magnitude will decrease (for frequencies below J2 times the resonant frequency). By choosing the minimum allowable resonant frequency as 100 Hz and assuming a nonzero damping constant, one can be sure that the transfer function will not deviate from 1 by more than approximately 10% (~0.8 dB) at 32 Hz and thus yield a reliable estimate for the total transfer function with a deviation of 1 dB in the worst case. RESULTS
According to the reasons above, 100 Hz has been chosen as the low frequency limit for the resonant frequency. To reach this limit, the 1.4 mm pin needs to be unrealistically short (for in uiw situations) so the data from this pin are not presented in the figures. Figure 3 shows an example of an input and output signal and the corresponding transfer function estimation for one impact.
An example of an input signal and the corresponding spectral density estimate for a pin plucking test can be seen in Fig. 4. The resonant frequency for various L and y values can be seen in Figs 5 and 6, with data from the 2.0 and 2.4 mm pin, respectively. The difference in estimated resonant frequency from transfer function and plucking of the pin (wood block test) was always much less than the peak width (it was actually on the order of the frequency resolution, 0.5 Hz.) Therefore, the resonant frequency could be determined from plucking the pin without any loss of precision. The difference between aluminum and bone mounting can be seen in Figs 5 and 6. In the most interesting region (around 108 Hz for this study) the resonant frequency is about lo-20 Hz higher with aluminum mounting than with bone mounting for the 2.4 mm pin. Therefore, if one measures a resonant frequency of 115 Hz with aluminum mounting, it will be approximately 100 Hz in bone (for the same L and y).
628
Technical Note
-I
Spectral Density (dB)
-10
-20.
: r 60 100 120 140 Fig. 4. From top to. bottom: signal and the corresponding spectral density from a pin plucking test. The spectral density is normalized so that its peak value is OdB. Pin hammered into a wood block with L=90mi, y=20mm and 6=2.0mm. -60~0
.
Frequency (Hz)
20
40
60
2.0 mm pin
-Q-
Aluminum
fixed: ~120 mm
-a-
Aluminum
fixed: y-25
mm
+
Aluminum
fixed: ~40
mm
-+-
Bone fixed: y=20 mm
--C
Bone fixed: y=25mm
+
Bone fixed: y-30
L (mm) Fig. 5. The resonant frequency as a function of the parameters L and y for the 2.0 mm pin.
mm
Technical Note 2.4
629
mm pin
H
-o-
Aluminum
fixed: y-20
mm
-o-
Aluminum
fixed: y-25
mm
+
Aluminum
fixed: y=30 mm
+
Bone fixed: y-20
mm
-m-
Bone fixed: y-25
mm
-+
Bone fixed: y-30
mm
L (mm) Fig. 6. The resonant frequency as a function of the parameters L and y for the 2.4 mm pin.
For the 2.0 min pin, the difference is negligible considering the difficulties in measuring the length y in uiuo.
also wish to thank Allison Kaigle, MS, for her help in preparing the manuscript.
DISCUSSION REFERENCES
The appropriate pin diameter, pin length and boneaccelerometer distance can be seen in Figs 5 and 6. For the 2.0 mm pin the pin length (L) has to be 60 mm or below and the bone-accelerometer distance (y) cannot be larger than 20 mm. Regarding the question of reliability of pin plucking to determine the resonant frequency, it has been shown that the difference between values determined from transfer function estimates is negligible. The question about the attachment has to be answered: for the tested conditions, the behavior of the measurement system is mainly determined by the pin-accelerometer system and the elasticity of the bone, as compared to aluminum plays only a minor role. The tests also show that it is possible to reliably measure the acceleration of the vertebra (up to at least 32 Hz) with intraosseous pins and accelerometers if the pin diameter and length are chosen properly. Nevertheless, it may be important than the pins are tapped far enough in and become wedged between the cortical bone of the spinous process. It is also worth mentioning that the difference in resonant frequency between aluminum and bone mounting was bigger with larger pin diameter (at the same frequency). This was to be expected since the pin bending stiffness is proportional to the pin diameter raised to the fourth power while the bending stiffnessof the pin-bone interface is only proportional to the pin diameter, making the attachment more important for larger pins. This also indicates that the resonant frequency does not necessarily increase when the pin diameter increases. It is even possible that it will in fact decrease due to the greater mass of the measurement system.For this reason, the present results cannot be extrapolated to larger pin diameters. The skin and surrounding tissue may also change the resonant frequency due to its attenuating effect. The distance from the process to the skin may also be larger than the values tested here, resulting in a too low resonant frequency. Nevertheless, the most important conclusion drawn from this study is that, measuring the resonant frequency by plucking the pin and analyzing the signal, one will always be aware of the basic characteristics of the system. Acknowledgements-The authors wish to acknowledge the support from the Swedish Work Environmental Fund. We
Christ, W. and Dupuis, H. (1963) Der EinfluD vertikaler Schwingung auf Wirbelslule und Magen. Zentralbl f Arbeitsschutz
13, 4-9.
Christ, W. and Dupuis, H. (1966) iiber die Beanspruchung der Wirbelsiule unter dem Einflug sinusformiger und stochastischer Schwingungen. Int. Z. Angew. Physiol. Arbeitsphysiol.
22, 258-278.
Cornelissen,M., Burny, F., Van der Perre, G. and Donkerwolcke, M. (1984) Standardized method to measure the fixation quality of a pin. Orthopedics I, 623-626. Hagena, F. W., Piehler, J., Wirth, C. J., Hofmann, G. 0. and Zwingers, Th. (1986) The dynamic response of the human spine to sinusoidal G,-vibration. In-vioo-experiments. Neuro.
Orthop.
2, 29-33.
Hagena, F. W., Piehler, C. J.,Wirth, C. J., Piehler, J., Plitz, W., Hofmann, G. 0. and Zwingers, Th. (1985) AGARD Conf Proc. No. 378, p. 16. Krause, H. (1963) Das schwingungsmechanische Verhalten der Wirbelslule. Int. Z. Angew. Physiol. Einschl. Arbeitsphysiol. 20, 125-155. Lange, W. and Coermann, R. (1965) Relativbewegungen benachbarter Wirbel unter schwingungsbelastung. Inc. Z. Angew.
Physiol.
Einschl.
Arbeitsphysiol.
21, 326-334.
Pope, M. H. Broman, H. and Hansson, T. (1989a) The dynamic response of a subject seated on various cushions. Ergonomics 32, 1155-1166. Pope, M. H., Broman, H. and Hansson, T. (1989b) Impact response of the standing subject-a feasibility study. Clin. Biomech. 4, 195-200. Pope, M. H., Broman, H. and Hansson, T. (1990) Factors affecting the dynamic response of the seated subject. J. Spinal Disorders 3, 135-142. Pope, M. H., Svensson, M., Broman, H. and Andersson, G. B. J. (1986) Mounting of the transducers in measurement of segmental motion of the spine. J. Biomechanics 19, 675-677.
Pope, M. H., Wilder, D. G., Jomeus, L., Broman, H., Svensson, M. and Andersson, G. B. J. (1987) The response of iithe seated human to sinusoidal vibration and impact. .I. biomech. Engng 109,279-284.
J. Biomechanics, Vol. 28, No. 5, pp. 625-629, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0021-9290/95 $9.50 + 00
0021-9290(94)00090-5
TECHNICAL
RESONANT
NOTE
FREQUENCY OF A PIN-ACCELEROMETER MOUNTED IN BONE
SYSTEM
M. Rostedt,* H. Bromant and T. Hansson* *Department of Orthopaedics, Sahlgren Hospital, S-413 45 Giiteborg, Sweden; and tDepartment of Applied Electronics, Chalmers University of Technology, S-412 96 Goteborg, Sweden Abstract-Invasive measurements of spinal motion using intraosseous metal pins have become common. For this reason, the resonant frequency of intraosseous pins attached with accelerometers was determined using two different methods. It was concluded that plucking the pin is a reliable method for determining the resonant frequency and, in order to accurately measure bone movement at frequencies up to 32 Hz, the pin diameter should be 2.0 mm or more. With a mass of the accelerometer assembly equal to 27 g, the total pin length should not exceed 80 mm with a bone-accelerometer distance of 25 mm and a pin diameter of 2.4&m.
INTRODUCTION
Vibration is believed to be a risk factor for low back pain. The response of the human body is known to be dependent on the frequency of the vibration exposure. In order to investigate the effects of vibration on the spine, it is thus necessary to measure the response at various frequencies. The relevant information from this kind of testing is contained in the transfer function (transmissibility) of the actual system. The transfer function of a system is defined as the ratio of the Fourier transform of the output signal to the Fourier transform of the input signal. For sinusoidal inputs, the transfer function gives the gain and the phase-shift of the systemat the frequency of the sinusoid. As acceleration has a direction, it is also necessary to specify the directions of the stimulus and response.. Transfer function estimation by the impact technique is a fast and reproducible method for investigating the effects of vibration on the human body (e.g. fee-spine, buttock-spine). The test consists of placing the subject on a vertically movable platform and, while impacting the platform, simultaneously measuring the accelerations at the platform and at a location on the spine. Pope et cl. (1986) showed that, in order to accurately measure the acceleration of vertebrae, the transducers should be mounted on pins rigidly attached to bone. Accelerometers mounted on intraosseous pins have been used in many experiments (e.g. to determine transfer functions) with human subjects in various situations. (Christ and Dupuis, 1963,1966; Hagena er al., 1986, 1985; Krause, 1963; Lange and Coermann, 1965; Pope et al., 1989a,b, 1990, 1986, 1987). In this type of test, it is important that the resonant frequency (natural frequency) of the pin-accelerometer system be considerably higher than the frequencies of interest in the experiment. The dependency of the resonant frequency on pin length and accelerometer mounting has not been reported (it has been tested without any data presented by Hagena et al., 1985). In Cornelissen et al. (1984) the resonant
Received in final form 13 June 1994. Address correspondence to: Mats Rostedt, Yrkesortopeden, Sahlgrenska Sjukhuset, S-413 45 Giiteborg, Sweden.
frequency as a function of the rigitidy of the pin fixation was investigated. However, no values for the resonant frequency or pin length were presented. Although a convenient way to determine the resonant frequency is to pluck the pin and examine the peaks in the spectral density of the signal, determining the transfer function from where the pin is fixed to the accelerometer is of greater value because it holds information as to which extent the total transfer function (e.g. buttock-accelerometer) becomes distorted by the measurement system. This study addresses the following questions: (1) How should the accelerometers and pins be mounted and what pin diameter is most suitable in order to make the resonant frequency appropriately high? (2) Could relevant information about resonance be obtained from simply plucking the pin? (3) Does bone mounting significantly lower the resonant frequency as compared to rigid mounting in aluminum? MATERIALS
AND
METHODS
The experimental setup, similar to that used in human experiments conducted in our laboratory, consisted of piezoresistive accelerometers (Endevco 7265A-HS, San Juan Capistrano, CA, U.S.A.). mounted on steel pins (Steinman pin, Richards Medical Co.. Memnhis. U.S.A.). Three different diameters were tested: 1.4,2.0 and 2.4 mm. The mass per unit length of each pin was 35, 24 and 13mgmm-’ for the 2.4, 2.0 and 1.4 mm pins, respectively. The pins were either tapped into a wooden block or tapped into the spinous processes (approximately 20 mm) of cadaveric lumbar vertebrae in the sagittal plane or clamped in an aluminum block, see Fig. 1. The pins were inserted parallel to the horizontal plane so that the sensitive axis of the accelerometers could be closely aligned (in the sagittal plane) to the vertical and horizontal directions respectively. Most soft tissue was excised from each vertebra which was then plotted in Plastic Padding (Type Hard, AB Plastic Padding, Giiteborg, Sweden) in an aluminum dish bolted to an U-beam, see Fig. 2. The aluminum block and wooden block were also bolted directly to the U-beam together with the input signal accelerometer. The U-beam was clamped to the platform of the whole-body impactor described by Pope et al. (1987). The
625
Technical Note
626
Fig. 1. Accelerometer and pin assembly. Vertebra /Tic
Padding
Fig. 2. Mounting of the vertebra. impactor consists of a vertically movable platform supported by attenuating springs, and a pendulum that makes it possible to strike the platform in a highly reproducible manner. The accelerometer signals were amplified so that the peak value was about 5 V, as well as low-pass filtered at 3OOHz. The gravitational acceleration was subtracted so that the signal level was approximately zero before each impact. Data were sampled at 1024 Hz, and the sampling time was 2 s, resulting in a frequency resolution of 0.5 Hz. The mass of the accelerometer assembly was 27 g. The parameters which were varied were the following: pin diameter, d, the distance from the vertebra to the accelerometers, y, and the length of the pin except the part inserted into the vertebra, L, see Fig. 1. Two vertebrae were tested and one impact was done for each parameter combination since a pilot test demonstrated negligible difference between trials. As a check during the main tests, some repeat trials were performed and the differences in resonant frequency were always less than 1 Hz. The wooden block was used in the tests where transfer function estimation and pin plucking were contrasted. In the pin plucking tests,the horizontal and vertical accelerometer yielded the same value for the (lowest) resonant frequency but the vertical was always used. DATA
PROCESSING
All calculations were made by a program developed primarily to estimate transfer functions from this type of experiment and to plot the corresponding phase and magnitude. The program facilitates in magnifying parts of the curves and obtaining position readout from a cursor. The calculations performed were the following. First, any remaining offset was removed form the input (platform) and output (pin) signals by subtracting the mean (before the impact). A Fourier transformation was then performed on each of the signals. The cross power density was estimated as the
smoothed product of the output Fourier transform and the complex conjugate of the input Fourier transform. The input (output) power density was calculated in the same manner with output replaced by input (input replaced by output). Smoothing was done utilizing five data points and a triangular weighting function. The transfer function was then estimated as the quotient of the cross power density and the input power density. Only the output power density was calculated in the tests where the pin was plucked. The resonant frequency was determined by localizing the frequency of the largest peak of the magnitude of the transfer function (or the output power density), see Figs 3 and 4. THEORY
Previous tests at our laboratory have shown that very little energy passes into the spine for frequencies above 32 Hz due to soft tissue attenuation. Furthermore, in most occupational situations where vibrations are involved (e.g. motor vehicle driving), the main part of the energy is contained in frequencies lower than this limit. The aim is therefore to find a lower limit for the resonant frequency which will yield reliable data up to 32 Hz. To find this limit for the resonant frequency, one can look at a second-order system consisting of a mass, a damper and a spring (the squared magnitude of its transfer function is shown in the equation below). The reason for selecting this systemis that it is the simplest physically relevant system that can show resonant behavior: k2 + $02 IW412
=
(k - mwz)z
+ t+w3
With no damping (9 =0) and at l/3 of its resonant frequency, the magnitude will be 1.125 (1.02 dB), implying a measurement error of 12.5%. When damping is introduced, the
Technical Note 8-
Input signal acceleration
621
(m/s*)
8.
-2. -41 0
1
0.5
1. *. *. Output signal acceleration
84. 6.
Time
.
* (m/s*)
*
1.5 -
*
-.
-
*
(s)
-.
c 2 L
r
4.
-2. Time (s)
-4 7
l
1
0.5
0
1.5
2
25 3
+ Magnitude
(df3)
20. 15. 10. 5.
-5 . -1oi
Frequency (Hz) I 16
32
64
r 128
Fig. 3. From top to bottom: input signal, output signal and the corresponding transfer function magnitude for one impact. Bone mounted pin with L = I25 mm, y = 20 mm and d = 2.0 mm.
magnitude will decrease (for frequencies below J2 times the resonant frequency). By choosing the minimum allowable resonant frequency as 100 Hz and assuming a nonzero damping constant, one can be sure that the transfer function will not deviate from 1 by more than approximately 10% (~0.8 dB) at 32 Hz and thus yield a reliable estimate for the total transfer function with a deviation of 1 dB in the worst case. RESULTS
According to the reasons above, 100 Hz has been chosen as the low frequency limit for the resonant frequency. To reach this limit, the 1.4 mm pin needs to be unrealistically short (for in uiw situations) so the data from this pin are not presented in the figures. Figure 3 shows an example of an input and output signal and the corresponding transfer function estimation for one impact.
An example of an input signal and the corresponding spectral density estimate for a pin plucking test can be seen in Fig. 4. The resonant frequency for various L and y values can be seen in Figs 5 and 6, with data from the 2.0 and 2.4 mm pin, respectively. The difference in estimated resonant frequency from transfer function and plucking of the pin (wood block test) was always much less than the peak width (it was actually on the order of the frequency resolution, 0.5 Hz.) Therefore, the resonant frequency could be determined from plucking the pin without any loss of precision. The difference between aluminum and bone mounting can be seen in Figs 5 and 6. In the most interesting region (around 108 Hz for this study) the resonant frequency is about lo-20 Hz higher with aluminum mounting than with bone mounting for the 2.4 mm pin. Therefore, if one measures a resonant frequency of 115 Hz with aluminum mounting, it will be approximately 100 Hz in bone (for the same L and y).
628
Technical Note
-I
Spectral Density (dB)
-10
-20.
: r 60 100 120 140 Fig. 4. From top to. bottom: signal and the corresponding spectral density from a pin plucking test. The spectral density is normalized so that its peak value is OdB. Pin hammered into a wood block with L=90mi, y=20mm and 6=2.0mm. -60~0
.
Frequency (Hz)
20
40
60
2.0 mm pin
-Q-
Aluminum
fixed: ~120 mm
-a-
Aluminum
fixed: y-25
mm
+
Aluminum
fixed: ~40
mm
-+-
Bone fixed: y=20 mm
--C
Bone fixed: y=25mm
+
Bone fixed: y-30
L (mm) Fig. 5. The resonant frequency as a function of the parameters L and y for the 2.0 mm pin.
mm
Technical Note 2.4
629
mm pin
H
-o-
Aluminum
fixed: y-20
mm
-o-
Aluminum
fixed: y-25
mm
+
Aluminum
fixed: y=30 mm
+
Bone fixed: y-20
mm
-m-
Bone fixed: y-25
mm
-+
Bone fixed: y-30
mm
L (mm) Fig. 6. The resonant frequency as a function of the parameters L and y for the 2.4 mm pin.
For the 2.0 min pin, the difference is negligible considering the difficulties in measuring the length y in uiuo.
also wish to thank Allison Kaigle, MS, for her help in preparing the manuscript.
DISCUSSION REFERENCES
The appropriate pin diameter, pin length and boneaccelerometer distance can be seen in Figs 5 and 6. For the 2.0 mm pin the pin length (L) has to be 60 mm or below and the bone-accelerometer distance (y) cannot be larger than 20 mm. Regarding the question of reliability of pin plucking to determine the resonant frequency, it has been shown that the difference between values determined from transfer function estimates is negligible. The question about the attachment has to be answered: for the tested conditions, the behavior of the measurement system is mainly determined by the pin-accelerometer system and the elasticity of the bone, as compared to aluminum plays only a minor role. The tests also show that it is possible to reliably measure the acceleration of the vertebra (up to at least 32 Hz) with intraosseous pins and accelerometers if the pin diameter and length are chosen properly. Nevertheless, it may be important than the pins are tapped far enough in and become wedged between the cortical bone of the spinous process. It is also worth mentioning that the difference in resonant frequency between aluminum and bone mounting was bigger with larger pin diameter (at the same frequency). This was to be expected since the pin bending stiffness is proportional to the pin diameter raised to the fourth power while the bending stiffnessof the pin-bone interface is only proportional to the pin diameter, making the attachment more important for larger pins. This also indicates that the resonant frequency does not necessarily increase when the pin diameter increases. It is even possible that it will in fact decrease due to the greater mass of the measurement system.For this reason, the present results cannot be extrapolated to larger pin diameters. The skin and surrounding tissue may also change the resonant frequency due to its attenuating effect. The distance from the process to the skin may also be larger than the values tested here, resulting in a too low resonant frequency. Nevertheless, the most important conclusion drawn from this study is that, measuring the resonant frequency by plucking the pin and analyzing the signal, one will always be aware of the basic characteristics of the system. Acknowledgements-The authors wish to acknowledge the support from the Swedish Work Environmental Fund. We
Christ, W. and Dupuis, H. (1963) Der EinfluD vertikaler Schwingung auf Wirbelslule und Magen. Zentralbl f Arbeitsschutz
13, 4-9.
Christ, W. and Dupuis, H. (1966) iiber die Beanspruchung der Wirbelsiule unter dem Einflug sinusformiger und stochastischer Schwingungen. Int. Z. Angew. Physiol. Arbeitsphysiol.
22, 258-278.
Cornelissen,M., Burny, F., Van der Perre, G. and Donkerwolcke, M. (1984) Standardized method to measure the fixation quality of a pin. Orthopedics I, 623-626. Hagena, F. W., Piehler, J., Wirth, C. J., Hofmann, G. 0. and Zwingers, Th. (1986) The dynamic response of the human spine to sinusoidal G,-vibration. In-vioo-experiments. Neuro.
Orthop.
2, 29-33.
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