opus.bath.ac.ukopus.bath.ac.uk/40168/1/Journal_paper_2013_11_12.docx · Web view1Centre for Power Transmission and Motion Control, Department of Mechanical Engineering, University

111Equation Chapter 1 Section 1

Biodynamic modelling of seated human subjects exposed to whole-body vibration in both vertical and fore-and-aft directions

Zengkang Gan1*, Andrew J. Hillis1, Jocelyn Darling1

1Centre for Power Transmission and Motion Control, Department of Mechanical Engineering, University of Bath, Bath, United Kingdom

{z.gan, a.j.hillis, j.darling}@bath.ac.uk

Keywords: Biodynamics, Seated human subjects, Whole-body vibration, Lumped-parameter models.

Abstract: In this article, the biodynamic responses of seated human subjects (SHS) exposed to whole-body vibration (WBV) in both vertical and fore-and-aft directions are modelled. The mathematical model can be used to obtain a better insight into the mechanisms and biodynamic behaviour of the SHS system. The main limitation of some previous SHS models is that they were derived to satisfy a single biodynamic response function. Such an approach may provide a reasonable fit with the function data being considered but uncertain matches with the others. The model presented in this study is based on all three types of biodynamic response functions: seat-to-head transmissibility (STHT), driving-point mechanical impedance (DPMI) and apparent mass (APM). The objective of this work is to match all three functions and to represent the biodynamic behaviour of SHS in a more comprehensive way. Three sets of synthesized experimental data from published literature are selected as the target values for each of the three transfer functions. A curve fitting method is used in the parameter identification process which involves the solution of a multivariable optimization function comprising the root mean square errors between the computed values using the model and those target values measured experimentally. Finally, a numerical simulation of the frequency response of the model in terms of all three biodynamic functions has been carried out. The results show that an improved fit is achieved comparing with the existing models.

Zengkang Gan, Andrew J. Hillis, Jocelyn Darling

1 INTRODUCTION

The harmful effects on human performance and health caused by low frequency whole-body vibration are of increasing concern. It is well know that the transmission of unwanted vibration to the human body may lead to fatigue and discomfort. In addition, severe vibration may cause neck and spine injuries. Much evidence has shown that there is a positive relationship between lower back pain (LBP) and continuous exposure to whole-body vibration [1-5].

Backman [6] conducted a health survey comprising 633 male drivers showed that the most common complaints among the drivers were shoulder and neck pain and back trouble. The study also found that 40 per cent of bus drivers had LBP with prevalence increasing with age. Bovenzi et al. [7] carried out a study of LBP in port machinery operators and it was found that the prevalence of low back symptoms increased with the increase of WBV exposure expressed as duration of exposure (driving years), equivalent vibration magnitude, or cumulative vibration exposure. A cross-sectional study by Rehn et al. [8] showed that occupational drivers of various categories of all-terrain vehicles (ATVs), such as forest machines, snowmobiles and snowgroomers, exhibited significantly increased risks of symptoms of musculoskeletal disorders primarily in the neck and shoulder regions. From the study it was suggested that the increased risk was related to exposure to whole-body vibration and shock. Seidel et al. [9] also concluded that people who sit in a vibrating environment that is close to or exceeds the ISO Exposure Limit [10, 11], place their musculoskeletal system at risk, particularly for LBP.

In order to gain a better understanding of seated human subject biodynamic response and adverse effects under low frequency whole-body vibration, a variety of statistical and analytical studies have been carried out by various researchers. Statistical studies usually involve measuring the kinetic and biodynamic responses of human subjects. There are three types of generalized biodynamic responses functions - seat-to-head transmissibility (STHT), driving-point mechanical impedance (DPMI) and apparent mass (APM), which are widely used to characterize biodynamic response of the seated human subject under most commonly encountered vibration environment. The STHT function is defined as the complex ratio of the output vibration level on the head to the input vibration level on the seat in the frequency range of interest [12]. This can be obtained in different forms by examining different response quantities, such as displacement, velocity and acceleration. The DPMI function is defined as the complex ratio between the transmitted dynamic force to which the subject is exposed and the input driving-point velocity [13]. The APM function is defined similarly to the DPMI function. It specifies the complex ratio of driving force to the driving-point acceleration. While the STHT function can provide indications on the dynamic behaviour of human body parts which are distant from the driving-point, the DPMI and APM functions can show the biodynamic characteristics of the human body load at the input point. All three functions can be evaluated by calculating the magnitude and phase responses in the frequency range of interest.

The measured biodynamic responses data from the statistical study can be used to identify mechanical-equivalent properties of the human body and to help in developing and validating mathematical models for analytical study in order to obtain a better insight of the human body behaviour under vibration. These mathematical models can be further used to help in developing anthropodynamic manikins for vibration assessment and to design anti-vibration seats and devices.

In the past few decades, a number of mathematical models have been developed based upon diverse measurements. Suggs et al. [14] developed a two-DOF damped spring-mass

2


model based on measurements which closely approximated the major dynamic characteristics of a seated man to vertical modes of vibration below 10 Hz. The model was used to build a standardised vehicle seat testing procedure. Wei and Griffin [15] suggested a two-DOF human and seat model to predict car seat vibration transmissibility. It was found that the predicted seat transmissibilities were close to those measured in a group of eight subjects over the entire frequency range. Boileau and Rakheja [13] proposed a four-DOF human body model considered the typical vehicle driving positions, such as erect without backrest support (ENS) posture, feet supported and low frequency excitation below 4 m/s2. The model parameters were estimated by attempting to match the magnitude and phase characteristics of the vertical DPMI function. Rosen and Arcan [16] constructed a multi-DOF lumped-parameter model incorporating global dynamics in terms of apparent mass and local dynamics of the body/seat contact interaction. The model parameters were optimised based on the APM experimental data.

Some other human subject models have also been developed by Muskian and Nash [17], Patil and Palanichamy [18], Qassem and Othman [19], Wan and Schimmels [20], Qiu and Griffin [21], and Stein et al. [22]. However, a concern regarding the majority of human subject models is raised by the fact that they were derived to satisfy a single biodynamic response function. Such an approach may provide a reasonable fit with the function data being considered but uncertain matches with the others. Since the biodynamic behaviour of the seated human body is equally characterized by all the three types of functions, it would be more useful for a mathematical model to fit all three biodynamic functions.

The objective of this work is to develop a mathematical model which can portray the essential biodynamic behaviour of seated human subject exposed to low frequency WBV in a more comprehensive way. In the present paper a synthesis of experimental data of seated human subject responses in both vertical and fore-and-aft directions in terms of STHT, DPMI and APM is generated. The mean values of the synthesized are selected as the target values for the model parameters identification. Model equations of motion are derived and the solutions in frequency domain are given. In the end, a numerical simulation of the magnitude and phase responses of the STHT, DPMI and APM biodynamic functions are carried out. A comparison with the existing models is made and the results are analysed and discussed.

2 MEASUREMENT DATA OF SEATED HUMAN SUBJECTS

The biodynamic responses data of seated human subjects was obtained from a variety of field and experimental measurements which were carried out under widely varying test conditions [12-13, 23]. The variation of test conditions for individual measurements may involve both intrinsic and extrinsic variables, such as subjects mass and populations, seat postures, feet and hand positions, vibration excitation types and levels, seat backrest angle and measurement locations on the subjects. In order to avoid significant discrepancies among the measurements data associated with the above variable conditions, the following requirements were specified for the synthesis of the biodynamic characteristics of seated human subjects [13, 23]:

(a) Studies presenting measurements results based on at least six subjects; (b) The measured subjects are considered to be sitting erect or upright posture, irrespective

of the hands’ position; (c) Feet are supported and vibrated on the same excitation base; (d) Subject mass are limited in the range of 45-100 kg;

3


(e) Excitation levels are below 5m/s2 and magnitude and phase data are reported in the 0-20 Hz frequency range;

(f) Either sinusoidal or random vibration excitation is used in the measurements.Based upon the above requirements, the following published measurement data were

selected for the synthesis of seated human body biodynamic properties: Paddan and Griffin [12], Boileau and Rakheja [13] and Hinz et al. [24] for the seat-to-head transmissibility (STHT) data; Boileau and Rakheja [13], Fairley and Griffin [25], Hinz and Seidel [26] and Holmlund and Lundström [27] for the driving-point mechanical impedance (DPMI) data; Mansfield and Griffin [28], Toward and Griffin [29], Fairley and Griffin [30] and Qiu and Griffin [31] for the apparent mass (APM) data. The synthesized data, shown in Table 1 and 2, is derived by averaging the above data sets and applying proper smoothing within the frequency range of interest.

Frequency(Hz)

STHT (abs) DPMI (N*s/m) APM (kg)Magnitude Phase (deg) Magnitud

ePhase (deg) Magnitud

e Phase (deg)

0.5 1.01 -0.2 95 89.5 59 -2.20.75 1.00 -0.7 175 89.0 60 -2.31.0 1.01 -0.8 310 88.5 60 -3.52.0 1.10 -6.0 754 87.5 61 -4.53.0 1.16 -10.0 1255 82 71 -104.0 1.28 -17.5 2252 66 81 -154.5 1.37 -29 2704 45 80 -235.0 1.45 -40 2605 31 76 -315.5 1.43 -50 2254 23 67 -436.0 1.30 -61 2105 23 53 -556.5 1.18 -62 1865 20 48 -607.0 1.09 -60 1892 22 44 -648.0 0.99 -62 1998 21 39 -689.0 0.94 -70 2002 20 36 -7010.0 0.95 -76 2015 16 32 -7212.0 0.86 -85 1905 17 31 -8014.0 0.76 -97 1770 18 25 -8316.0 0.67 -105 1625 19 18 -8218.0 0.60 -113 1585 20 14 -8120.0 0.56 -121 1605 20 11 -81

Table 1: Synthesized data of STHT, DPMI and APM mean values in the vertical direction.

4


Frequency(Hz)

STHT (abs) DPMI (N*s/m) APM (kg)Magnitude Phase (deg) Magnitud

ePhase (deg) Magnitude Phase (deg)

0.5 1.26 -1 55 83.0 53.0 -8.00.75 1.44 -16 148 80.8 57.0 -10.61.0 1.62 -40 257 76.6 59.5 -12.81.25 1.59 -64 392 71.55 63.1 -16.01.5 1.41 -85 515 65.4 67.2 -20.21.75 1.23 -103 655 59.5 69.6 -24.02.0 1.10 -119 850 48.4 70.2 -28.52.5 0.86 -135 1010 33.6 62.5 -39.42.75 0.71 -144 1068 25.0 58.6 -46.03.0 0.56 -159 1095 17.1 54.0 -53.53.5 0.41 -168 1083 1.8 45.1 -62.54.0 0.29 -175 1061 -7.5 37.7 -75.04.5 0.26 -189 1028 -15.5 35.3 -80.55.0 0.23 -198 974 -21.5 30.6 -81.06.0 0.18 -211 912 -25.8 20.5 -86.27.0 0.13 -231 853 -29.0 17.4 -91.08.0 0.10 -247 746 -29.8 12.9 -90.29.0 0.08 -253 647 -29.3 10.3 -87.010.0 0.08 -255 555 -28.0 7.8 -85.011.0 0.08 -254 521 -27.2 7.2 -84.012.0 0.08 -252 487 -25.7 6.0 -84.0

Table 2: Synthesized data of STHT, DPMI and APM mean values in the fore-and-aft direction.

The lower and upper limits of each data are not included in the above tables. However, they are shown in Table 6. It is noted that sufficient measurement data of STHT, DPMI and APM responses in the fore-and-aft direction is only available up to 12 Hz.

3 MATHEMATICAL MODEL DEVELOPMENT

3.1 Model description

The proposed model is a lumped-parameter linear spring and damper system. The model includes segments representing appropriate anatomical parts of the body and is capable of accommodating translational and rotational (head and neck joint) movements of these segments, which enable it to represent the measured STHT, DPMI and APM function data under low frequency whole-body vibration.

This model is composed of two sub-models: the vertical model and the fore-and-aft model, as shown in Figure 1. The vertical model consists of five segments: head and neck (m5), upper torso (m4), arms (m3), viscera (m2), and lower torso (m1). The spring (k41) and damper (c41) connecting the upper and lower torsos represent the body spine. In the fore-and-aft model, the main body mass (m1+ m2+ m3+ m4) is treated as a single limped mass. The head and neck (m5) and the main body are connected by a rotational degree. These rigid masses are coupled by linear elastic and damping elements. The masses of the lower legs and the feet are not incorporated in the model representation, assuming their negligible contributions to the

5

Seat -m0

z5(t)

z4(t)

z3(t)

z2(t)

z1(t)

Spine

k3 c3

k1 c1

k5

k41 c41

z0(t)

c5

kr, cr

k4 c4

k2 c2

Lower torso -m1

Upper torso -m4

Arms -m3

Viscera -m2

Head and neck-m5

x1(t)

kb

cbθ(t)

x0(t)


whole-body biodynamic response. This assumption is in agreement with the evidence that the contribution of the supported legs to the whole-body vibration is relatively insignificant when both the seat and the footrest are vibrated under the same excitation.

Figure 1: Schematic of model for seated human subjects in both vertical and fore-and-aft directions.

3.2 Equations of motion (EOMs)

EOMs of the model were derived from the free-body diagram of each part. The vertical model has five degrees of freedom in total: the vertical displacement of each segment { z1(t), z2(t), z3(t), z4(t ), z5(t)}. The vertical model EOMs can be expressed by the following coupled differential equations:

m1 z1+c1 ( z1− z0 )+k1 ( z1−z0 )−c2 ( z2− z1 )−k2 ( z2−z1 )

−c41 ( z4− z1 )−k41 ( z 4−z1 )=0 (1)

m2 z2+c2 ( z2− z1 )+k2 ( z2−z1 )−c4 ( z4− z2 )−k4 ( z4−z2 )=0 (2)

m3 z3−c3 ( z4− z3 )−k 3 ( z4−z3 )=0 (3)

m4 z4+c3 ( z4− z3 )+k3 ( z 4−z3 )+c4 ( z4− z2)+k4 ( z4−z2)

+c41 ( z4− z1 )+k 41 ( z4−z1 )−c5 ( z5− z4 )−k5 ( z5−z4 )=0 (4)

m5 z5+c5 ( z5− z4 )+k5 ( z5−z4 )=0 (5)

The above differential equations can be expressed in matrix form:

[ M ] { z }+[ C ] { z }+ [ K ] { z }={ f n } (6)

6


where [ M ], [ C ] and [ K ] are mass, damping and stiffness matrices with a size of 5 × 5, respectively; { z }, { z } and { z } are acceleration, velocity and displacement vectors, respectively, with a size of 5 × 1; { f n } is an 5 × 1 excitation force vector. All the above matrices and vectors can be expressed as follows:

[ M ] = [m1 0 0 0 00 m2 0 0 00 0 m3 0 00 0 0 m4 00 0 0 0 m5

] ;[ C ] = [

c1+c2+c41 −c2 0 −c41 0−c2 c2+c4 0 −c4 0

0 0 c3 −c3 0−c41 −c4 −c3 c3+c4+c41+c5 −c5

0 0 0 −c5 c5

] ;[ K ] = [

k1+k2+k 41 −k2 0 −k 41 0−k2 k2+k 4 0 −k 4 0

0 0 k3 −k3 0−k 41 −k 4 −k3 k 3+k4+k41+k5 −k5

0 0 0 −k5 k5

] ;{ z } ={

z1

z2

z3

z4

z5

}; { z } ={z1

z2

z3

z4

z5

} ; { z } ={z1

z2

z3

z4

z5

} ; { f n } ={c1 z0+k1 z0

0000

} .The fore-and-aft model has two degrees of freedom in total: the fore-and-aft displacement

of the main body part {x1(t)} and the head and neck rotational degree {θ ¿)}. The EOMs of the fore-and-aft model can be expressed by the following differential equations:

M x1+m5ln θ cosθ−m5ln θ2 sinθ+kb ( x1+x0 )+cb ( x1+ x0 )=0 (7)

m5ln2 θ+m5ln x1cosθ−m5ln g sin θ+k t θ+c t θ=0 (8)

where M is the mass of the whole body (M= m1+ m2+ m3+ m4+ m5), ln is the average distance between the shoulder and the gravity centre of the head, g is the acceleration due to gravity (g=9.81 m/ s2). k t and c t are the rotational spring and damper coefficients of the neck, k b and cb are the spring and damper coefficients between the main body and the backrest, respectively.

7


3.3 EOMs solution in the frequency domain

By taking Fourier transforms of the above EOMs the models can be analysed in the frequency domain. The Fourier transform of the vertical model EOMs (Eq. (6)) results in:

{Z ( jω)}=[−ω2 [ M ]+ jω [C ]+ [ K ] ]−1{Fn( jω)} (9)

where { Z ( jω)} and {Fn( jω)} are the complex Fourier transform vectors of { z } and { f n }, respectively, j is the imaginary unit and ω is the angular frequency. The vector {Z ( jω)} contains the complex displacement responses of the 5 mass segments as a function of angular frequency, and they can be represented by {Z1 ( jω ) , Z2 ( jω ) , Z3 ( jω ) , Z4 ( jω ) , Z5( jω)}. {Fn( jω)} contains the complex excitation forces as a function of angular frequency as well, which is {( k1+ jω c1 ) Z0 ( jω ) ,0 ,0 , 0 , 0}, where Z0 ( jω ) is the complex displacement of excitation. The EOMs of the fore-and-aft model contain some nonlinear terms −m5ln θ cosθ, −m5 ln θ2 sin θ, m5ln x1cosθ, −m5ln g sin θ . Small oscillations were assumed (i.e. around θ=0), and the following linearization were used: cosθ=1, sin θ=θ, θ2 sin θ=0. The Fourier transform of the linearized equations can be expressed as follows:

−ω2 M X1 ( jω )−ω2m5 lnθ ( jω )+(kb+ jω cb)(X 1 ( jω )−X0 ( jω ))=0 (10)

−ω2m5 ln2 θ ( jω )−ω2 m5ln X1 ( jω )+m5 ln g θ ( jω )+(k t + jω c t)θ ( jω )=0 (11)

Based on the preceding definitions, the STHT, DPMI and APM biodynamic functions for the vertical model can be derived as follows:

STHT v=Z5 ( jω )Z0 ( jω )

(12 )

DPMI v=|(k1+ jω c1 ) [Z0 ( jω )−Z1( jω)]jω Z0( jω) |(13)

APMv=|DPMI v

jω |=|(k1+ jωc1 ) [Z0 ( jω )−Z1( jω)]−ω2 Z0( jω) |(14)

Considering the Eqs. (10) and (11) of the fore-and-aft model in a similar manner, the STHT, DPMI and APM biodynamic functions for the fore-and-aft model can be derived as follows:

STHT f =ln θ ( jω )+X1 ( j ω)

X0 ( jω )(15 )

DPMI f=¿

APM f =|DPMI f

jω |=¿

4 MODEL PARAMETER IDENTIFICATION

Model parameters were identified using curve fitting methods formulated in Matlab (version 2011b). The Least Absolute Residual (LAR) method and the 'Trust-Region'

8


algorithm are used. The fitting process involves the solution of a multivariable optimization function comprising the root mean square errors between the computed values using the model and those target values measured experimentally (Tables 1 and 2). In the vertical model there are 17 unknown parameters in total, which can be represented in a vector as: pv=¿[c1, c2

, c3, c4, c41, c5, k1, k 2, k3, k 4, k 41, k5, m1, m2, m3, m4, m5]T. Since the mass (m1, m2, m3, m4, m5) are shared parameters, in the fore-and-aft model the unknown parameters vector can be expressed as: pf =¿[cb, c t, k b,k t, ln]T. The vectors pv and pf were identified separately by fitting the biodynamic functions in Eqs. (12-17).

In order to make the fitting procedure more effective, a set of initial, upper and lower limits values was estimated by referring to previously published studies. 73.6% (percentage of body mass supported by the seat for erect seating posture) of the whole body weight (75 kg) is used for the total model mass (i.e. 73.6% of 75kg =55.2kg). The damping and stiffness coefficients of the human body segments are not known precisely, therefore, the ranges are relatively large. The estimated initial, upper and lower limits values are listed in Table 3. The identified model parameters are listed in Table 4.

Model parameters Initial values Lower limits Upper limitsHead and neck mass m5(kg) 5.5 5 7Upper torso mass m4 (kg) 22 20 25Arms mass m3 (kg) 6 5 8Viscera mass m2 (kg) 10.2 8 12Lower torso mass m1 (kg) 11.7 10 15Average distance ln (m) 0.16 0.13 0.19Damping coefficient c1 (Ns/m) 2000 500 5000Damping coefficient c2 (Ns/m) 1000 400 5000Damping coefficient c3 (Ns/m) 300 100 2000Damping coefficient c4 (Ns/m) 4000 400 5000Damping coefficient c41 (Ns/m) 4000 500 5000Damping coefficient c5 (Ns/m) 400 300 2000Damping coefficient cb (Ns/m) 200 10 3000Damping coefficient c t (Ns/m) 200 10 2000Stiffness coefficient k1(N/m) 120000 5000 200000Stiffness coefficient k 2(N/m) 6000 5000 100000Stiffness coefficient k3 (N/m) 10000 5000 200000Stiffness coefficient k 4 (N/m) 7000 5000 100000Stiffness coefficient k 41 (N/m) 160000 5000 250000Stiffness coefficient k5 (N/m) 300000 5000 500000Stiffness coefficient k b (N/m) 10000 500 200000Stiffness coefficient k t (N/m) 1000 500 200000

Table 3: Estimated initial, upper and lower limits values for model parameter identification.

9


Model parameters Identifiedvalues Model parameters Identified

valuesHead and neck mass m5(kg) 5.6 Damping coefficient c5 (Ns/m) 977.4Upper torso mass m4 (kg) 20.3 Damping coefficient cb (Ns/m) 621.9Arms mass m3 (kg) 8.0 Damping coefficient c t (Ns/m) 18.9Viscera mass m2 (kg) 9.2 Stiffness coefficient k1(N/m) 120123.3Lower torso mass m1 (kg) 10.0 Stiffness coefficient k 2(N/m) 5300.3Average distance ln (m) 0.19 Stiffness coefficient k3 (N/m) 13177.7Damping coefficient c1 (Ns/m) 2376.4 Stiffness coefficient k 4 (N/m) 9151.1Damping coefficient c2 (Ns/m) 675.8 Stiffness coefficient k 41 (N/m) 128198.6Damping coefficient c3 (Ns/m) 145.8 Stiffness coefficient k5 (N/m) 292010.0Damping coefficient c4 (Ns/m) 1797.7 Stiffness coefficient k b (N/m) 9925.7Damping coefficient c41 (Ns/m) 4023.2 Stiffness coefficient k t (N/m) 772.4

Table 4: Identified parameter values for the vertical and fore-and-aft human body models.

5 SIMULATION AND COMPARISON RESULTS

After all the model parameters have been identified, the magnitude and phase responses of the STHT, DPMI and APM biodynamic functions were simulated in Matlab. The simulation results are listed in Table 6. To evaluate the goodness-of-fit (GOF) of the presented models, the ratio of the root-mean-square error to the mean value was calculated using the following equation [23]:

GOF=1−√∑( ym− yc )2/ ( N−2 )

∑ ym/ N(18)

where ym and yc are the measured target data and calculated value, respectively. N is the number of the measured target data points. The GOF statistic can take on any value less than or equal to 1, with a value closer to 1 indicating a better fit. The GOF values of the seated human subject models in the comparison are summarized in Table 7. In table 6 and 7, any GOF value less than 0 is marked by ~.

As shown in the simulation results, four previous seated human subject models were selected for comparison: a four-DOF linear vertical model developed by Wan and Schimmels [20], which has been found providing the highest average of goodness-of-fit in [23] and a four-DOF vehicle driver model proposed by Boileau and Rakheja [13] were chose for the vertical model comparison; a two-DOF fore-and-aft model developed by Stein et al. [22] and a four-DOF fore-and-aft apparent mass model presented by Qiu and Griffin [21] were selected for the fore-and-aft model comparison. The schematics and parameters of the comparison models are listed in Table 5.

10


Model Name

Model parameters Schematic of modelMass(kg)

Stiffness (N/m)

Damping (Ns/m)

Wan and Schimmels [20]

m1 36.0 k1 49340 c1 2475.0

Seat -m0

z4(t)

z2(t)

z1(t)k1 c1

k5

k41 c41

z0(t)

c5

k4 c4

k2 c2

Lower torso -m1

Upper torso -m3

Viscera -m2

Head and neck -m4

z3(t)m2 5.5 k2 20000 c2 330.0

m3 15.0 k41 192000 c41 909.1

m4 4.17 k4 10000 c4 200.0

k5 134400 c5 250.0

Boileau and Rakheja [13]

m1 12.78 k1 90000 c1 2064

Seat -m0

z4(t)

z2(t)

z1(t)k1 c1

k4

z0(t)

c4

k3 c3

k2 c2

Thighs & pelvis -m1

Chest & upper torso -m3

Lower torso -m2

Head & neck -m4

z3(t)m2 8.62 k2 162800 c2 4585

m3 28.49 k3 183000 c3 4750

m4 5.31 k4 310000 c4 400

Stein et al. [22]

m1 54 k1 39322 c1 465.9

c1

k2

c2

k1

m2

m1

k3

c3

x1(t) x2(t) x3(t)

m2 10.4 k2 9 c2 8.0

k3 1054 c3 113.1

Qiu and Griffin [21]

mb 8 k1 39886 c1 359

c1

k2

c2

k1

m2

m1

mb

ms

kt, ct

kb

cb

x3(t) x2(t)

x1(t)

cs

ks

ms 10 k2 10924 c2 542

m1 20 kb 24610 cb 0.0

m2 35 kt 10 ct 112

ks 26646 cs 0.0

Table 5: Schematics and parameters of comparison models of seated human subject.

11


Presented model (Vertical) Wan’s model (Vertical)

Fig.2-1(a)

0 5 10 15 20-150

-100

-50

0

50

Frequency (Hz)

Pha

se (

deg)

0 5 10 15 200

0.5

1

1.5

2 STHT (Seat-to-head transmissibility)

STH

T (a

bs)

Fig.2-2(a)

0 5 10 15 20-200

-100

0

100

Frequency (Hz)

Pha

se (

deg)

0 5 10 15 200

0.5

1

1.5


STH

T (a

bs)

Fig.2-1(b)

5 10 15 200

20

40

60

80

Frequency (Hz)

Pha

se (

deg)

0 5 10 15 200

1000

2000

3000

4000 DPMI (Driving-point mechanical impedance)

DPM

I (N

*s/m

)

Fig.2-2(b)

5 10 15 20

20

40

60

80

Frequency (Hz)

Pha

se (

deg)

0 5 10 15 200

1000

2000

3000


DPM

I (N

*s/m

)

Fig.2-1(c)

0 5 10 15 20-100

-50

0

Frequency (Hz)

Pha

se (

deg)

0 5 10 15 200

50

100APM (Apparent mass)

APM

(Kg)

Fig.2-2(c)

0 5 10 15 20-100

-50

0

Frequency (Hz)

Pha

se (

deg)

0 5 10 15 200

50


APM

(Kg)

Upper limits, Lower limits, Mean values.

Table 6: Simulation and comparison results of the seated human subject models.

Presented model GOF=90.84%






Wan’s model GOF=89.27%






12


Boileau’s model (Vertical) Presented model (Fore-and-aft)

Fig.2-3(a)

0 5 10 15 20-150

-100

-50

0

50

Frequency (Hz)

Pha

se (

deg)

0 5 10 15 200

0.5

1

1.5


STH

T (a

bs)

Fig.2-4(a)0 2 4 6 8 10 12

-300

-200

-100

0

100

Frequency (Hz)

Pha

se (

deg)

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

STH

T (a

bs)

Fig.2-3(b)

5 10 15 20

0

50

100

Frequency (Hz)

Pha

se (

deg)

0 5 10 15 200

1000

2000

3000


DPM

I (N

*s/m

)

Fig.2-4(b)

2 4 6 8 10 12

-50

0

50

100

Frequency (Hz)

Pha

se (

deg)

0 2 4 6 8 10 120

500

1000

1500

DPM

I (N

*s/m

)

Fig.2-3(c)

0 5 10 15 20-100

-50

0

Frequency (Hz)

Pha

se (

deg)

0 5 10 15 200

50


APM

(Kg)

Fig.2-4(c)

0 2 4 6 8 10 12-150

-100

-50

0

Frequency (Hz)

Pha

se (

deg)

0 2 4 6 8 10 120

50

100

APM

(Kg)


Table 6 (continued)

Boileau’s model GOF=80.19%












13


Stein’s model (Fore-and-aft) Qiu’s model (Fore-and-aft)

Fig.2-5(a)

0 2 4 6 8 10 12-300

-200

-100

0

100

Frequency (Hz)

Pha

se (

deg)

0 2 4 6 8 10 120

1

2S

THT

(abs

)

Fig.2-6(a)0 2 4 6 8 10 12

-300

-200

-100

0

100

Frequency (Hz)

Pha

se (

deg)

0 2 4 6 8 10 120

0.5

1

1.5

22.5

STH

T (a

bs)

Fig.2-5(b)

2 4 6 8 10 12

-50

0

50

100

Frequency (Hz)

Pha

se (

deg)

0 2 4 6 8 10 120

2000

4000

DPM

I (N

*s/m

)

Fig.2-6(b)

2 4 6 8 10 12-100

-50

0

50

100

Frequency (Hz)

Pha

se (

deg)

0 2 4 6 8 10 120

500

1000

1500

DPM

I (N

*s/m

)

Fig.2-5(c)

0 2 4 6 8 10 12-150

-100

-50

0

Frequency (Hz)

Pha

se (

deg)

0 2 4 6 8 10 120

50

100

150

APM

(Kg)

Fig.2-6(c)

0 2 4 6 8 10 12-150

-100

-50

0

Frequency (Hz)

Pha

se (

deg)

0 2 4 6 8 10 120

50

100

APM

(Kg)


Table 6 (continued)

Stein’s model GOF ~

Stein’s model GOF=29.86%



Qiu’s model GOF=80.85%





Qiu’s model GOF ~

Qiu’s model GOF ~


14


Direction Model NameSTHT GOF (%) DPMI GOF (%) APM GOF (%)

Magnitude Phase Magnitude Phase Magnitude Phase

Vertical

Presented model 90.84 82.71 81.33 78.33 87.80 86.85

Wan’s model 89.27 77.48 76.10 74.87 84.91 86.11

Boileau’s model 75.30 65.98 79.36 68.81 80.19 73.57

Fore-and-aft

Presented model 37.09 36.57 80.92 52.68 91.99 81.33

Stein’s model ~ 29.86 ~ 30.32 ~ 42.53

Qiu’s model ~ 39.53 79.59 ~ 80.85 61.80

Table 7: Summary of comparison results of the seated human subject models.

6 DISCUSSION

Since there is some variation between the synthesized target data used in these models, the calculated GOF of the comparison models may not be the original presented values. The measurement condition of the target data used in Stein’s model varies somewhat from those of the synthesized target data in the fore-and-aft direction. However, the target data used in Wan, Boileau and Qiu’s models are very close to the synthesized target data in this study.

From the simulation results, the vertical seated human models show a higher average goodness-of-fit than the fore-and-aft models in both the magnitude and phase responses of the STHT, DPMI and APM functions. The presented vertical model provides 90.84% GOF and 82.71% GOF for the STHT magnitude and phase responses, 81.33% GOF and 78.33% GOF for the DPMI magnitude and phase responses , 87.80% GOF and 86.85% GOF for the APM magnitude and phase responses, respectively. This indicates that a better overall GOF is achieved for predicting the above biodynamic functions for the seated human subject under vertical vibration. The results also show that very close peaks occur at about 5 Hz in the magnitude responses of all the three functions, which indicates the reliabilities of predicting identical primary resonant frequencies are validated by each other. In addition, the presented vertical model predicts a second resonant frequency around 8 Hz which is observed in the target data. Wan’s model has been found providing a generally good fit of STHT and APM functions. However, the fit of DPMI function is relatively poor, with 76.10% GOF for magnitude response and 74.87% GOF for phase response. The peak values occur at about 4 Hz for the STHT and APM functions while the peak value is around 7.5 Hz for the DPMI function. Besides, both Wan’s and Boileau’s models fail to predict the second resonant frequency.

The fore-and-aft model simulation results show that a relatively large deviation is exhibited. The presented model provides the highest match for the APM function, with 91.99% GOF for magnitude response and 81.33% GOF for phase response. The GOF values for the DPMI function are relatively lower, with 80.92% for magnitude response and 52.68% for phase response. However, these values still outperform the comparison models. The prediction for the STHT function is the poorest for all the three models. The GOF values for both magnitude and phase responses are below 40%. Because of the variation between the

15


measurement data, Stein’s model shows poor matches for all the three functions. Qiu’s model provides a reasonably good fit for the APM function, but poor fits for the other two functions. One of the reasons is that the model was developed based only on the APM measurement data.

It is noted that the quantity of reported experimental data for the seated human subject responses in the fore-and-aft direction is considerably less than the data in the vertical direction. More measurement data is needed to guide and validate the human body modelling in the fore-and-aft direction. It is also noted that the phase responses of the three biodynamic functions are usually measured in experimental studies; however, they are rarely evaluated and analysed in human body modelling studies. The phase responses of the biodynamic functions are evaluated in this study since the phase responses can be equally as important as the magnitude responses, if not more so, when it comes to human body vibration cancelation.

7 CONCLUSION

A lumped-parameter biodynamic model of a seated human subject exposed to low frequency whole-body vibration in both the vertical and fore-and-aft directions is developed. Model parameters were identified using curve fitting methods and the STHT, DPMI and APM biodynamic magnitude and phase response functions are simulated in Matlab. The goodness-of-fit of the presented model is evaluated graphically and statistically, a comparison with the existing models is carried out and the results show that an improved fit with the synthesized experimental data is achieved. Through the model, the biodynamic behaviour of the seated human subjects can be observed in a more comprehensive way.

REFERENCES

[1] M.H. Pope, D.G. Wilder, M.L. Magnusson, A review of studies on seated whole body vibration and low back pain. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 213-435, 1999.

[2] C. Hulshof, B. Veldhuijzen van Zanten, Whole-body vibration and low back pain- A review of epidemiologic studies. International Archives of Occupational and Environmental Health, 59, 205-220, 1987.

[3] A. Burdorf, G. Sorock, Positive and negative evidence on risk factors for back disorders. Scandinavian Journal of Work, Environment and Health, 23, 243-256, 1997.

[4] P.M. Bongers, H.C. Boshuizen, C. Hulshof, A.C. Koemeester, Back disorders in crane operators exposed to whole-body vibration. International Archives of Occupational and Environmental Health, 60, 129-137, 1988.

[5] M. Bovenzi, A. Betta, Low-back disorders in agricultural tractor drivers exposed to whole-body vibration and postural stress. Applied Ergonomics, 25, 231-241, 1994.

[6] A.L. Backman, Health survey of professional drivers. Scandinavian Journal of Work, Environment & Health, 9(1), 30-35, 1983.

[7] M. Bovenzi, I. Pinto, N. Stacchini, Low back pain in port machinery operators. Journal of Sound and Vibration, 253(1), 3-20, 2002.

16


[8] B. Rehn, I.A. Bergdahl, R. Lundström, C. Ahlgren, C. From, B. Järvholm, R. Lundström, T. Nilsson, G. Sundelin, Musculoskeletal symptoms among drivers of all-terrain vehicles. Journal of Sound and Vibration, 253(1): 21-29, 2002.

[9] H. Seidel, R. Bastek, D. Brauer, C.H. Buchholz, A. Meister, A.M. Metz, R. Rothe, On human response to prolonged repeated whole-body vibration. Ergonomics, 23(3), 191-211, 1980.

[10] ISO 2631 (E), Guide for the Evaluation of Human Exposure to Whole Body Vibration (International Organization for Standardization), 1978.

[11] ISO 2631 (E), Evaluation of Human Response to Whole Body Vibration, (International Organization for Standardization), 1985.

[12] G.S. Paddan, M. J. Griffin, A review of the transmission of translational seat vibration to the head. Journal of Sound and Vibration, 215(4), 863-882, 1998.

[13] P.E. Boileau, S. Rakheja, Whole-body vertical biodynamic response characteristics of the seated vehicle driver measurement and model development. International Journal of Industrial Ergonomics, 22, 449-472, 1998.

[14] C.W. Suggs, C.F. Abrams, L.F. Stikeleather, Application of a damped spring-mass human vibration simulator in vibration testing of vehicle seats. Ergonomics, 12, 79-90, 1969.

[15] L. Wei, M.J. Griffin, The prediction of seat transmissibility from measures of seat impedance. Journal of Sound and Vibration, 212, 855-874, 1998.

[16] J. Rosen, M. Arcan, Modelling the human body-Seat system in a vibration environment. Journal of Biomechanical Engineering, 125, 223-231, 2003.

[17] R. Muskian, C.D. Nash, A model for the response of seated humans to sinusoidal displacements of the seat. Journal of Biomechanics, 7, 209-215, 1974.

[18] M.K. Patil, M.S. Palanichamy, A mathematical model of tractor-occupant system with a new seat suspension for minimization of vibration response. Applied Mathematical Modelling, 12, 63-71, 1988.

[19] W. Qassem, M.O. Othman, Vibration effects on sitting pregnant women- subjects of various masses. Journal of Biomechanics, 29 (4), 493-501, 1996.

[20] Y. Wan, J.M. Schimmels, A simple model that captures the essential dynamics of a seated human exposed to whole body vibration. Advances in Bioengineering, ASME, BED, 31, 333-334, 1995.

[21] Y. Qiu and M.J. Griffin, Modelling the fore-and-aft apparent mass of the human body and the transmissibility of seat backrests. Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 49(5), 703-722, 2011.

[22] G.J. Stein, P. Múčka, R. Chmúrny, B. Hinzb, R. Blüthner, Measurement and modelling of x-direction apparent mass of the seated human body–cushioned seat system. Journal of Biomechanics, 40, 1493-1503, 2007.

[23] C.-C. Liang, C.-F. Chiang, A study on biodynamic models of seated human subjects exposed to vertical vibration. International Journal of Industrial Ergonomics, 36, 869-890, 2006.

17


[24] B. Hinz, G. Menzel, R. Blüthner and H. Seidel, Seat-to-head transfer function of seated men-determination with single and three axis excitations at different magnitudes. Industrial Health, 48, 565-583, 2010.

[25] T.E. Fairley, M.J. Griffin, A test method for the prediction of seat transmissibility. Society of Automotive Engineers International Congress and Exhibition Paper, 860047, 1986.

[26] B. Hinz, H. Seidel, The nonlinearity of the human body’s dynamic response during sinusoidal whole body vibration. Industrial Health, 25, 169-181, 1987.

[27] P. Holmlund, R. Lundström, Mechanical impedance of the human body in the horizontal direction. Journal of Sound and Vibration, 215(4), 801-812, 1998.

[28] N.J. Mansfield, M.J. Griffin, Effects of posture and vibration magnitude on apparent mass and pelvis rotation during exposure to whole-body vertical vibration. Journal of Sound and Vibration, 253, 93-107, 2002.

[29] M.G. Toward, M.J. Griffin, Apparent mass of the human body in the vertical direction: effect of a footrest and a steering wheel. Journal of Sound and Vibration, 329, 1586-1596, 2010.

[30] T.E. Fairley, M.J. Griffin, The apparent mass of the seated human body: vertical vibration. Journal of Biomechanical Engineering, 109, 148-153, 1987.

[31] Y. Qiu, M.J. Griffin, Biodynamic responses of the seated human body to single-axis and dual-axis vibration. Industrial Health, 48, 615-627, 2009.

18

Documents

opus.bath.ac.ukopus.bath.ac.uk/40168/1/Journal_paper_2013_11_12.docx · Web view1Centre for Power Transmission and Motion Control, Department of Mechanical Engineering, University