Modeling of biomechanical human body model for seat to head transmissibility analysis

Abstract

Under low-frequency excitation, the human subject is highly sensitive to Whole Body Vibrations (WBV). In the present paper, a biodynamic human model is seated posture without backrest support and also with backrest support at different angles (i.e., 0°, 15° and 30°) is formulated to analyse the influences of the frequency and the magnitude of the vibration, backseat support and the angles of inclination of the backseat support on Seat To Head Transmissibility (STHT) of the biodynamic human subject in seating position exposed to WBV. The model parameters such as damping, mass and stiffness of the mathematical model considered in this paper are derived from an alternate methodology which is based on anthropometric data. The STHT is computed analytically with the help of MATLAB software for all the considered cases. The effect of vibration and excitation frequencies on STHT is analysed. The results of STHT evaluated from this work are justified by comparing the same with the mean STHT characteristics reported in past studies and the results of the present analysis come out to be in good agreement with past published research.

Keywords

Anthropometric biodynamic model stiffness ellipsoidal segments seat to head transmissibility

Introduction

Ride quality and comfort is the key performance parameter in railway vehicle dynamics. This performance parameter is mostly affected by different factors like vertical and lateral vibrations produced by random track irregularities, vehicle tilt on the curves, sway due to hunting, flanges contact with the rails, noise, thermal factors, humidity, design of the seat and its location point, sudden acceleration and deceleration, and other running and environmental aspects. The major source of passenger’s discomfort is the vibrations which are transmitted from the track to the carbody. These transmitted vibrations from the track to the carbody are mostly compensated by the primary and secondary suspensions. Generally, the severity of passenger’s discomfort is defined by the car-body acceleration.^1,2 The human body is an extremely complex network of rigid bodies and formulation of a mathematical biomechanical human structure model and analysing the dynamics response of the same is a demanding and exciting exercise. A huge amount of experimental data has been reported in past. The human body sensitivity is different for different frequencies of excitation and the peak of the discomfort level is generally obtained in the lower range (i.e., 0.5–20) of the excitation frequency.^3,4,5 The vibrations transmitted from the track due to random track irregularities to the passenger mostly depend on the design of bogie and suspension system, profile of the wheels, profile of the rail, speed of the vehicle, track design, track irregularities and operating conditions. In different countries, different standards are followed to categorize passenger comfort like ISO 2631,^6,7,8 ENV 12299,⁹ UIC 513R (International Union of Railway)¹⁰ and Sperling index.¹¹ In Japan as well as in South Korea, ISO 2631 and RMS (Root mean square) acceleration¹² are used to define the passenger’s comfort level. The equations of motion (EoM) for the biodynamic human body model have been formulated in past.¹³ In past, a LP (Lumped Parameter) biodynamic human body model in sitting posture is developed^14,15,16 and formulated its differential EoM. This model consists of six rigid bodies namely, leg, pelvic, viscera, upper torso, vertebral column and head. These isolated masses are considered to be joined using non-linear springs-dampers combinations, which show properties like stiffness and damping constants of the system.

Yu et al.¹⁷ presented an uncoupled spatial seated human multi-body dynamic model with 48 DoF to determine the comfort level. The model consists of feet, legs, thighs, trunk, head, and neck and it incorporates the uncoupled longitudinal, lateral and vertical translational motions and pitch, roll and yaw rotational motions for Whole Body Vibration (WBV). Guruguntla and Lal¹⁸ presented a ten DoF model of the seated human subject and divided it into parts i.e., head, thorax, abdomen, pelvis, left upper arm, left forearm, left hand, right upper arm, right forearm, and right hand. After the optimization of the parameters, the biodynamic responses: SHT, apparent mass and driving point mechanical impedance has been evaluated for the proposed model and compared with previous reports.

Dong et al.¹⁹ developed a detailed three-dimensional (3D) finite element (FE) model of the body-seat system to determine the influence of sitting posture and seat on the biodynamic responses of human subject subjected to vertical white noise excitation. Desai et al.²⁰ formulated twelve DoF seated human body model with inclined backrest support with rotational and translational springs-dampers and contact with an inclined backrest. A biodynamic human body model, consisting of different lumped masses which are connected by linear spring and dampers combinations, is modelled to evaluate the dynamic response to whole-body vertical vibrations and the vibration transmissibility for seated human body posture.²¹ Two multi-body human body models are formulated by Liang and Chiang²² in which one is without backrest support and the other one is with the same. The proper modifications have been done in these two multi-body models to represent the several automotive postures with or without back seat support. These models were validated by a huge existing experimental data from past published research for the same postures. Several LP mathematical models were developed in past.^23,24 These models consisting anatomical segments of a human body for representing different motions of the body related to resonance frequencies of the upright masses and length-wise cross-axis masses of a biodynamic model for five standing positions. The anthropometric data published in the past research is used to determine the inertial and geometric parameters for these models. The model responses were compared with the existing experimental data from the past research to obtain the stiffness and damping parameters values. A property-based method is proposed to derive the response of a human body model and this response is directly compared with the arithmetically averaged responses.^25,26 The researchers^27,28 have analysed the dynamic behavior for both the standing as well as sitting biomechanical human models. The apparent mass of the six locations along the spine, pelvis and head along with transmissibilities have been calculated for eight male subjects exposed to WBV. In both positions (i.e., standing and sitting) the key resonance for apparent masses is obtained in low-frequency range (5–6 Hz). Some WBV experiments for human body subjects exposed to vertical vibration are performed to analyse the effects of backseat support inclination on STHT.²⁹ These experiments are performed on 12 male human subjects under vertical vibration to characterize their biodynamic response parameters exposed to 3 different inclination angles of backseat, with upper body supported at vertical backseat, and also inclined at two different angles (i.e., 10° and 30°) with vertical. As the angle of inclination of backseat is increased, total calculated absorption of power particularly in the frequency range of 1–20 Hz is decreased. The apparent mass responses of human model in sitting posture on elastic and rigid seat with and without backseat support is also measured in existing literature.^30,31 A seat pressure sensor is used to sense the pressure of seat exposed to 3 levels of vertical vibration i.e., 0.25, 0.50 and 0.75 m/s² acceleration in the frequency range of 0.50–20 Hz. The responses of human body model on elastic seat in sitting posture showed that the relative frequency and its peak magnitude are significantly lower than that of those measured while rigid seat is used, regardless of backseat support. An experimental study was conducted by Bhiwapurkar et al.³² to define the effects of accelerations produced due to the motion of the rail vehicle on passenger’s comfort. This study analyses the influences of posture variations and vibration magnitude variations on head movement in 3 different directions i.e., vertical, lateral, and fore-and-aft. The STHT response is evaluated when exposed to two sitting positions such as back seat as well as forward lean. The STHT response attains peak head acceleration in lateral direction of having single peak at frequency 2 Hz in both the sitting positions. The dynamic behavior also registered an associated peak about at 6 Hz in backseat and forward lean postures. In the present work, the anthropometric data of Indian male human subjects is utilized to formulate the biodynamic human body model. Different angles of backrest inclinations for rigid backrest support are considered in this article. The STHT values are measured with the help of analytical modelling for different excitation frequencies and vibration magnitudes.

Modelling of biodynamic human body model

In the present study, the anthropometric data for different parameters of Indian male subjects is used from the past researches³³ to model the biodynamic human model in sitting posture. A 9 DoF LP model was developed in past by researchers³⁴ to analyse the response of human biomechanical body model in sitting position and exposed to WBV (vertical and horizontal). Five different body segments of masses are considered in this LP model, which are connected through two horizontal as well as four vertical springs in a way that feet and seat are fixed with the vehicle floor through two horizontal as well as two vertical springs and the arms rested in the lap. This LP model is altered suitably according to the demand of the present work. The main difference between the present model and the above-said model developed in past³⁴ is the clubbing of different masses for generating such a biodynamic human body model which is practically near to reality. In the present work, the clubbing of lower legs and feet is done to represent mass $m_{e}$ as both the segments vibrate simultaneously as a single mass when exposed to external excitations. Moreover, the mass of the lower torso and the thighs is clubbed to represent mass of pelvic $m_{d}$ as both the segments vibrate simultaneously as a single mass due to vibrations transmitted through seat. In this work, the human body (Figure 1) is modelled in seatting posture without backrest support with its head towards the direction of motion of the vehicle. It is divided into five parts i.e., head & neck, upper torso, viscera, pelvic and leg. The vertical and longitudinal motion is assigned to each part except head & neck which have only vertical motion.

Figure 1.

Human body model in seated posture without backrest support.

For calculation of the stiffness of various human model parts, it is considered that the ellipsoids are truncated at the ends³⁵ (Figure 2). As the ends of the ellipsoids are truncated a non-dimensional factor namely the truncation factor comes into the picture and this factor is the ratio of two lengths i.e., ( $d_{i}$ ) and ( $c_{i}$ ), where $c_{i}$ is the ellipsoidal segment’s original length, whereas $d_{i}$ is the truncated length.

Figure 2.

(a) Ellipsoidal segment (b) Truncated ellipsoid.³⁵

Truncating the ellipsoid is a mathematical need to evaluate the Integral factor ( $I_{i}$ ) which is used to compute springs stiffness and found in equation (2).

Generally, the smaller value of truncation factor gives a more accurate compromise among the mathematical need and overlapping of the interconnected ellipsoidal segments. In past, some researchers³⁵ used 5% truncation factor in their work, however, in this work only 1% truncation factor is used for best fit.

Mass segments evaluation

The anthropometric statistics for male subjects from India (50th percentile)³³ is considered as shown in Table 1. The volume of an individual segment is dependent on the semi-axes of the ellipsoid related to that segment. The mass of a particular body segment is directly proportional to the volume of the same segment. This mass of a particular segment may be expressed as a fraction of the mass of the complete body. The mass (

m_{i}

) of

i^{t h}

ellipsoid can be calculated with the help of equation (1).¹⁷

m_{i} = \frac{m a_{i} b_{i} c_{i}}{(\sum_{i = 1}^{n} a_{i} b_{i} c_{i})}

(1)

Table 1.

Seated Indian male subject’s anthropometric data (50th Percentile).³³

Dimensions		Measurement (mm)
$L_{1}$	Standing height	1648
$L_{2}$	Shoulder height	576
$L_{3}$	Armpit height	417
$L_{4}$	Waist height	213
$L_{5}$	Seated height	837
$L_{6}$	Head depth	187
$L_{7}$	Head breadth	147
$L_{8}$	Head to chin	243
$L_{9}$	Neck breadth	112
$L_{10}$	Shoulder breadth	380
$L_{11}$	Chest depth	209
$L_{12}$	Chest breadth	298
$L_{13}$	Waist depth	192
$L_{14}$	Waist breadth	259
$L_{15}$	Buttock circum	957
$L_{16}$	Hip breadth	331
$L_{17}$	Shoulder to elbow	312
$L_{18}$	Forearm length	243
$L_{19}$	Biceps circum	244
$L_{20}$	Elbow circum	273
$L_{21}$	Forearm circum	228
$L_{22}$	Wrist circum	156
$L_{23}$	Knee height	519
$L_{24}$	Thigh breadth	129
$L_{25}$	Upper leg length	127
$L_{26}$	Knee circum	351
$L_{27}$	Calf circum	319
$L_{28}$	Ankle circum	239
$L_{29}$	Ankle height	61
$L_{30}$	Foot breadth	94
$L_{31}$	Foot depth	248
$L_{32}$	Neck height	43
$L_{33}$	Upper leg length	558
$L_{34}$	Neck length	86
$L_{35}$	Lower lumbar length	101
$L_{36}$	Abdomen depth	217
$L_{37}$	Elbow to elbow	409
$L_{38}$	Abdomen breadth	260
$L_{39}$	Elbow rest	188
$L_{40}$	Lower leg height	425
$L_{41}$	Foot height	61
$L_{42}$	Head circum	544
$L_{43}$	Heat height	219
$L_{44}$	Hand length	180

Meanwhile,

m_{i}

i^{t h}

segment’s mass,

n

is total number of ellipsoid segments,

m

is the total body mass and

a_{i}

b_{i}

, and

c_{i}

are the

i^{t h}

ellipsoid’s semi-axes (Figure 2). Tables 2 and 3 depicts some formulae derived in past³³ and used to calculate physical dimensions

a_{i}

b_{i}

, and

c_{i}

as well as all ellipsoidal masses of a 15 DoF biomechanical human body model. Table 4 shows the value of the masses of all ellipsoidal segments by clubbing of masses of a 9 DoF model.

Table 2.

Formulae for physical dimensions of the ellipsoids representing body segments.³³

S. n	Body segment	Mass	Depth $(a_{i})$		Breadth $(b_{i})$		Height $(c_{i})$
S. n	Body segment	Mass	Formula	Value	Formula	Value	Formula	Value
1	Head	$m_{1}$	$L_{6} / 2$	93.50	$L_{7} / 2$	73.50	$L_{43} / 2$	106.50
2	Neck	$m_{2}$	$L_{9} / 2$	56.00	$L_{9} / 2$	56.00	$L_{5} - L_{2} - L_{43} / 2$	24.00
3	Upper torso	$m_{3}$	$L_{11} / 2$	104.50	$L_{12} / 2$	149.00	$L_{17} / 2$	156.00
4	Central torso	$m_{8}$	$L_{36} / 2$	108.50	$L_{38} / 2$	13.00	$L_{17} + L_{18} + L_{44} / 4$	183.75
5	Lower torso	$m_{9}$	$L_{15} / 2 π$	152.30	$L_{16} / 2$	165.50	$L_{18} + L_{44} / 4$	105.75
6	Upper arm	$m_{4} m_{5}$	$L_{19} / 2 π$	38.80	$L_{19} / 2 π$	38.80	$L_{17} / 2$	156.00
7	Lower arm	$m_{6} m_{7}$	$L_{21} / 2 π$	38.80	$L_{21} / 2 π$	38.80	$L_{18} + L_{44} / 2$	211.50
8	Upper leg	$m_{10} m_{11}$	$L_{33} / 2$	279.00	$L_{25} / 2$	63.50	$L_{25} / 2$	63.50
9	Lower leg	$m_{12} m_{13}$	$L_{27} / 2 π$	50.77	$L_{27} / 2 π$	50.77	$L_{40} / 2$	212.50
10	Feet	$m_{14} m_{15}$	$L_{31} / 2$	124.00	$L_{30} / 2$	47.00	$L_{41} / 2$	30.50

Table 3.

Different body part masses of 15 DoF human body model.³³

Sr no.	Human body parts	Symbol	Mass (kg)
1	Head	$m_{1}$	2.97
2	Neck	$m_{2}$	0.29
3	Upper torso	$m_{3}$	9.86
4	Upper arm (right)	$m_{4}$	0.95
5	Upper arm (left)	$m_{5}$	0.95
6	Lower arm (right)	$m_{6}$	1.29
7	Lower arm (left)	$m_{7}$	1.29
8	Central torso	$m_{8}$	10.52
9	Lower torso	$m_{9}$	10.82
10	Upper leg (right)	$m_{10}$	4.57
11	Upper leg (left)	$m_{11}$	4.57
12	Lower leg (right)	$m_{12}$	2.23
13	Lower leg (left)	$m_{13}$	2.23
14	Foot (right)	$m_{14}$	0.72
15	Foot (left)	$m_{15}$	0.72

Table 4.

Mass of ellipsoidal segments for a 9 DoF model considered in the present work.

S. no.	Body parts	Mass element	Formula	Mass in kg
1	Head & neck	$m_{a}$	$m_{1} + m_{2}$	3.26
2	Upper torso	$m_{b}$	$m_{3} + m_{4} + m_{5} + m_{6} + m_{7}$	14.35
3	Viscera	$m_{c}$	$m_{8}$	10.52
4	Pelvic	$m_{d}$	$m_{9} + m_{10} + m_{11}$	19.96
5	Leg	$m_{e}$	$m_{12} + m_{13} + m_{14} + m_{15}$	5.90

Rigid mass assumption

Although the human body is a complex structure, the mass segments are assumed to be rigid elements as the longitudinal wave velocity ${(E / ρ)}^{1 / 2}$ through a human body model exposed to vertical acceleration is 110 m/s in which modulus of elasticity different body components³⁵ is $E = {(E_{b} E_{t})}^{1 / 2} = 13 M N / m^{2}$ meanwhile, modulus of elasticity of human bones is $E_{b} = 22.6 G N / m^{2}$ and the same for tissues³⁶ is $E_{t} = 7.5 k N / m^{2}$ and bulk average density of all human body parts³⁷ is $ρ = 1.062 \times 10^{3} k g / m^{3}$ . Vibration frequencies lower than 100 Hz are related to a wavelength of over 1 m which may generally over a mean height of a human body without showing any vibratory effects.

Evaluation of the stiffness of different body segments ( $S_{i}^{x, z}$ )

The individual ( $i^{t h}$ ) segment’s stiffness may be shown by.

S_{i}^{x, z} = \frac{π E a_{i} b_{i}}{(c_{i} I_{i})} (\frac{k N}{m})

(2)

With the help of equation (2) we can calculate $S_{i}^{z}$ and $S_{i}^{x}$ by modifying above equation as:

S_{i}^{z} = \frac{π E a_{i} b_{i}}{(c_{i} I_{i})} (\frac{k N}{m})

(3)

S_{i}^{x} = \frac{π E b_{i} c_{i}}{(a_{i} I_{i})} (\frac{k N}{m})

(4)

Meanwhile, $a_{i}$ , $b_{i}$ , and $c_{i}$ = semi axes of the ellipse E = Modulus of elasticity of the same ellipse (Figure 2) and $I_{i}$ (integral factor) is then calculated as:

\begin{array}{l} I_{i} = c_{i} \int_{- d_{i}}^{d_{i}} \frac{d z}{(c_{i}^{2} - z^{2})} = \frac{\log (c_{i} + d_{i})}{(c_{i} - d_{i})} \\ = \frac{\log (2 - t_{r}^{i})}{(t_{r}^{i})} \end{array}

(5)

Meanwhile, truncation factor is as follows:

t_{r}^{i} = 1 - \frac{d_{i}}{c_{i}}

(6)

In this equation, $d_{i}$ is the half of the length of the truncated ellipsoid.

To evaluate vertical

(S_{i}^{z})

& horizontal

(S_{i}^{x})

stiffness for different body parts of a 9 DoF model considered in this study, it is assumed that all the human body elements are physically connected in series. The values of the vertical and horizontal stiffness (i.e.,

S_{i}^{z}

S_{i}^{x}

) for different body elements of 15 DoF model used in past³³ and the 9 DoF model used in this study is given in Tables 5 to 7, respectively.

Table 5.

$S_{i}^{z}$ & $S_{i}^{x}$ of different segments of human model³³ having 15 DoF.

Sr no.	Stiffness
Sr no.	$S_{i}^{z}$ (kN/m)		$S_{i}^{x}$ (kN/m)
1	$S_{1}^{z}$	498.16	$S_{1}^{x}$	646.31
2	$S_{2}^{z}$	1052.60	$S_{2}^{x}$	117.56
3	$S_{3}^{z}$	770.54	$S_{3}^{x}$	1717.16
4	$S_{4}^{z}$	74.49	$S_{4}^{x}$	1204.32
5	$S_{5}^{z}$	74.49	$S_{5}^{x}$	1204.32
6	$S_{6}^{z}$	54.95	$S_{6}^{x}$	1632.78
7	$S_{7}^{z}$	54.95	$S_{7}^{x}$	1632.78
8	$S_{8}^{z}$	592.60	$S_{8}^{x}$	1699.64
9	$S_{9}^{z}$	1840.07	$S_{9}^{x}$	887.15
10	$S_{10}^{z}$	2153.88	$S_{10}^{x}$	111.57
11	$S_{11}^{z}$	2153.88	$S_{11}^{x}$	111.57
12	$S_{12}^{z}$	93.75	$S_{12}^{x}$	1640.50
13	$S_{13}^{z}$	93.75	$S_{13}^{x}$	1640.50
14	$S_{14}^{z}$	1475.15	$S_{14}^{x}$	89.25
15	$S_{15}^{z}$	1475.15	$S_{15}^{x}$	89.25

Table 6.

Different body segments vertical stiffness of 9 DoF model used in this study.

S. no.	( $S_{i}^{z}$ ) (kN/m)
1	$S_{a}^{z}$	$(S_{1}^{z} \times S_{2}^{z}) / (S_{1}^{z} + S_{2}^{z})$	338.13
2	$S_{b}^{z}$	$\begin{array}{l} (1 / S_{4}^{z}) = (1 / S_{3}^{z}) + (1 / p) + (1 / q) \\ a n d \\ p = (S_{4}^{z} \times S_{6}^{z}) / (S_{4}^{z} + S_{6}^{z}), q = (S_{5}^{z} \times S_{7}^{z}) / (S_{5}^{z} + S_{7}^{z}) \end{array}$	15.49
3	$S_{c}^{z}$	$S_{3}^{z}$	770.54
4	$S_{d}^{z}$	$(1 / S_{2}^{z}) = (1 / S_{9}^{z}) + (1 / S_{10}^{z}) + (1 / S_{11}^{z})$	679.34
5	$S_{e}^{z}$	$\begin{array}{l} (p \times q) / (p + q) \\ a n d \\ p = (S_{12}^{z} \times S_{14}^{z}) / (S_{12}^{z} + S_{14}^{z}), q = (S_{13}^{z} \times S_{15}^{z}) / (S_{13}^{z} + S_{15}^{z}) \end{array}$	44.07

Table 7.

Different body segments horizontal stiffness of 9 DoF model used in this study.

S. no.	Segment stiffness ( $S_{i}^{x}$ ) (kN/m)
1	$S_{a}^{x}$	$(S_{1}^{x} \times S_{2}^{x}) / (S_{1}^{x} + S_{2}^{x})$	139.29
2	$S_{b}^{x}$	$\begin{array}{l} (1 / S_{4}^{x}) = (1 / S_{3}^{x}) + (1 / p) + (1 / q) \\ a n d \\ p = (S_{4}^{x} \times S_{6}^{x}) / (S_{4}^{x} + S_{6}^{x}), q = (S_{5}^{x} \times S_{7}^{x}) / (S_{5}^{x} + S_{7}^{x}) \end{array}$	288.35
3	$S_{c}^{x}$	$S_{3}^{x}$	1717.16
4	$S_{d}^{x}$	$(1 / S_{2}^{x}) = (1 / S_{9}^{x}) + (1 / S_{10}^{x}) + (1 / S_{11}^{x})$	52.48
5	$S_{e}^{x}$	$\begin{array}{l} (p \times q) / (p + q) \\ a n d \\ p = (S_{12}^{x} \times S_{14}^{x}) / (S_{12}^{x} + S_{14}^{x}), q = (S_{13}^{x} \times S_{15}^{x}) / (S_{13}^{x} + S_{15}^{x}) \end{array}$	42.32

Evaluation of stiffness of connecting springs ( $k_{i}^{x, z}$ )

The different masses $(m_{a}, m_{b}, m_{c}, m_{d}, m_{e})$ of the body segments of a 9 DoF LP model are interconnected with the help of some springs and dampers. The vertical and horizontal stiffness.

(

k_{i}^{x, z}

) of these springs may be evaluated from the vertical and horizontal stiffness of different body segments (i.e.,

S_{i}^{z}

and

S_{i}^{x}

) of a 9 DoF model and are shown in Table 8.

Table 8.

Stiffness of different connecting spring of 9 DoF LP model.

S. n.	Spring element	Stiffness ( $k_{i}^{x, z}$ ) (kN/m)
1	$k_{a b}^{z} = (S_{a}^{z} \times S_{b}^{z}) / (S_{a}^{z} + S_{b}^{z})$	14.81
2	$k_{b c}^{z} = (S_{b}^{z} \times S_{c}^{z}) / (S_{b}^{z} + S_{c}^{z})$	15.19
3	$k_{b d}^{z} = (S_{b}^{z} \times S_{d}^{z}) / (S_{b}^{z} + S_{d}^{z})$	15.14
4	$k_{c d}^{z} = (S_{c}^{z} \times S_{d}^{z}) / (S_{c}^{z} + S_{d}^{z})$	361.04
5	$k_{d}^{z} = S_{d}^{z}$	679.34
6	$k_{e}^{z} = S_{e}^{z}$	44.07
7	$k_{b c}^{x} = (S_{b}^{x} \times S_{c}^{x}) / (S_{b}^{x} + S_{c}^{x})$	246.89
8	$k_{d}^{x} = S_{d}^{x}$	52.49
9	$k_{d e}^{x} = (S_{d}^{x} \times S_{e}^{x}) / (S_{d}^{x} + S_{e}^{x})$	23.43
10	$k_{e}^{x} = S_{e}^{x}$	42.32

Evaluation of damping of different body segments ( $C_{i}^{x, z}$ )

In the present work, the values of the damping ratio (ξ) are obtained from the past researches³⁸ and are iterated to get a best fit response. The damping constant of different body segments may be calculated with the help of.

C_{i} = 2 ξ \sqrt{S_{i} m_{i}}

(7)

Meanwhile, $C_{i}$ is damping constant, $m_{i}$ is mass, $S_{i}$ is stiffness and $ξ$ is damping ratio of different human body segments.

Vertical damping constant

(C_{i}^{z})

and horizontal damping constant

(C_{i}^{x})

may be calculated with the help above relation for all human body (ellipsoidal) segments and are shown in Table 9.

Table 9.

Vertical $(C_{i}^{z})$ and Horizontal $(C_{i}^{x})$ damping constants of different body segments.

S. No.	Vertical damping of segments ( $C_{i}^{z}$ ) (kNs/m)		Horizontal damping of segments ( $C_{i}^{x}$ ) (kNs/m)
1	$C_{a}^{z}$	0.19	$C_{a}^{x}$	0.12
2	$C_{b}^{z}$	0.04	$C_{b}^{x}$	0.39
3	$C_{c}^{z}$	3.35	$C_{c}^{x}$	4.05
4	$C_{d}^{z}$	27.15	$C_{d}^{x}$	25.49
5	$C_{e}^{z}$	15.06	$C_{e}^{x}$	10.29

Evaluation of damping of connecting dashpots ( $c_{i}^{x, z}$ )

The different masses

(m_{a}, m_{b}, m_{c}, m_{d}, m_{e})

of the body segments of a 9 DoF LP model are interconnected with the help of some springs and dashpots. The vertical and horizontal damping constants (

c_{i}^{x, z}

) of these dashpots may be calculated from the vertical and horizontal damping constants of different body segments (i.e.,

C_{i}^{z}

and

C_{i}^{x}

) of a 9 DoF model (Table 9). The same methodology is applied to measure the damping constants of connecting dashpots also as in the case of spring stiffness and is shown in Table 10.

Table 10.

Damping constants of connecting dashpots (without backrest support).

S. no.	Damping constant (kNs/m)
1	$c_{a b}^{z}$	0.11
2	$c_{b c}^{z}$	0.04
3	$c_{b d}^{z}$	0.58
4	$c_{c d}^{z}$	3.15
5	$c_{d}^{z}$	27.15
6	$c_{e}^{z}$	15.06
7	$c_{b c}^{x}$	0.34
8	$c_{d}^{x}$	25.49
9	$c_{d e}^{x}$	7.98
10	$c_{e}^{x}$	10.29

Derivation of EoM

Figure 3 shows the free body diagram of 9 DoF biodynamic human model in sitting position without backrest support.

Figure 3.

Free body diagram of human body model in sitting posture without backseat support.

The Newton second law of motion is utilized for developing the EoM which describes the system dynamic behavior of the present model. These EoM are expressed as follows:

[M] {{\ddot{z}}_{j}} + [C] {{\dot{z}}_{j}} + [K] {z_{j}} = {F_{j}}

(8)

Meanwhile, [M] is the mass matrix, [C] is the damping matrix and [K] is the stiffness matrix. { ${\ddot{z}}_{j}}$ is the acceleration vector ${{\dot{z}}_{j}}$ is the velocity vector, ${z_{j}}$ is the displacement vector of each segment. {Fj} is excitation vector and j is number of DoF.

All the EoM for the different rigid bodies of present model can be expressed as follows:

m_{e} {\ddot{z}}_{e} + c_{e}^{z} \dot{z_{e}} + k_{e}^{z} z_{e} = c_{e}^{z} \dot{z_{2}} + k_{e}^{z} z_{2}

(9)

m_{e} {\ddot{x}}_{e} + (c_{d e}^{x} + c_{e}^{x}) \dot{x_{e}} - c_{d e}^{x} \dot{x_{d}} + (k_{d e}^{x} + k_{e}^{x}) x_{e} - k_{d e}^{x} x_{d} = c_{e}^{z} \dot{x_{2}} + k_{e}^{z} x_{2}

10)

m_{d} {\ddot{z}}_{d} - (c_{d}^{z} + c_{b d}^{z} + c_{c d}^{z}) {\dot{z}}_{d} - c_{c d}^{z} {\dot{z}}_{c} - c_{b d}^{z} {\dot{z}}_{b} + (k_{d}^{z} + k_{b d}^{z} + k_{c d}^{z}) z_{d} - k_{c d}^{z} z_{c} - k_{b d}^{z} z_{b} = c_{d}^{z} {\dot{z}}_{2} + k_{d}^{z} z_{2}

(11)

m_{d} {\ddot{x}}_{d} - c_{d e}^{x} {\dot{x}}_{e} + (c_{d e}^{x} + c_{d}^{x}) {\dot{x}}_{d} - k_{d e}^{x} x_{e} + (k_{d e}^{x} + k_{d}^{x}) x_{d} = c_{d}^{x} \dot{x_{2}} + k_{d}^{x} x_{2}

(12)

m_{c} {\ddot{z}}_{c} - c_{c d}^{z} {\dot{z}}_{d} + (c_{c d}^{z} + c_{b c}^{z}) {\dot{z}}_{c} - c_{b c}^{z} {\dot{z}}_{b} - k_{c d}^{z} z_{d} + (k_{c d}^{z} + k_{b c}^{z}) z_{c} - k_{b c}^{z} z_{b} = 0

(13)

m_{c} \ddot{x_{c}} + c_{c b}^{x} \dot{x_{c}} - c_{c b}^{x} \dot{x_{b}} + k_{c b}^{x} x_{c} - k_{c b}^{x} x_{b} = 0

(14)

m_{b} \ddot{z_{b}} - c_{b d}^{z} \dot{z_{d}} - c_{b c}^{z} \dot{z_{c}} + (c_{b d}^{z} + c_{a b}^{z} + c_{b c}^{z}) \dot{z_{b}} - c_{a b}^{z} \dot{z_{a}} - k_{b d}^{z} z_{d} - k_{b c}^{z} z_{c} + (k_{a b}^{z} + k_{b d}^{z} + k_{b c}^{z}) z_{b} - k_{a b}^{z} z_{a} = 0

(15)

m_{b} \ddot{x_{b}} - c_{c b}^{x} \dot{x_{c}} + c_{c b}^{x} \dot{x_{b}} - k_{c b}^{x} x_{c} + k_{c b}^{x} x_{b} = 0

(16)

m_{a} \ddot{z_{a}} - c_{a b}^{z} \dot{z_{b}} + c_{a b}^{z} \dot{z_{a}} - k_{a b}^{z} z_{b} + k_{a b}^{z} z_{a} = 0

(17)

Seat to head transmissibility evaluation for the seated human model without backrest

The vibration flowing through the human body is measured in terms of STHT. The analysis of human response, when subjected to vibration in sitting posture, is related to the mode of transportation where the passenger is exposed to whole-body vibration. The crucial aspects in this context are the vibration characteristic of seat, vibration environment and human body vibrational response. A human body may be modelled in several ways during train travel. In seated posture, it may be modelled in different ways e.g., erect without backrest, with backrest, forward-leaning on table and erected with leg fold. A particular model may not consider all the factors but may describe the aspect of the system. The transmissibility analysis is restricted to a frequency range from 1 to 20 Hz as the eigenvalues of human body parts and resonating frequencies remain in this zone. STHT of the considered model may be expressed as:

S T H T = \frac{z_{a} (f)}{z_{2} (f)}

(18)

Meanwhile, $z_{a} (f)$ is human head and neck vertical output response.

$z_{2} (f)$ is the vertical input response at the seat as a function of temporal frequency.

The complex ratio of output response to input response is called STHT, where output means vertical acceleration at head and vertical acceleration at the seat is termed input response which is determined analytically in MATLAB environment. MATLAB simulation for analytical measurement of STHT is done by using different parameters of the present model computed above and the EoM as input. The plot obtained from the simulation is shown in Figure 4.

Figure 4.

Seated LP human body model’s STHT without backrest.

For a LP model sitting without backrest support, STHT values are determined analytically when exposed to WBV. In the vibration magnitudes range between 0.5 to 1.5 m/s², the values of STHT are identical. The change in absolute values of acceleration evaluated as output at the head have similar pattern for all the input responses provided to the seat in terms of excitation. Hence, as the magnitude of the vibration at the seat is elevated the vibration magnitude is also increased in the almost similar pattern at the head. Because the net result comes almost the same for the input and output, hence, the ratio of two would always be the same. As a result, the plots for STHT obtained in the present study overlap with each other for all the three vibration magnitudes assumed.

Determination of STHT for sitting position with backrest support

A seated biodynamic LP human body model in vertical (i.e., inclined at 0° with the vertical axis) position with backrest support is shown in Figure 5. Further, an extra set of spring and dashpot is added in both directions (vertical and horizontal) to the present model as now the backseat support is present. All the parameters of the model are same except that the effect of $k_{b}^{z}$ , $k_{b}^{x}$ , $c_{b}^{z}$ and $c_{b}^{x}$ is considered and suitable modifications are made in the respective EoM. The values of parameters of these two new spring-dashpot combinations are $k_{b}^{z} = 15.67$ kN/m, $k_{b}^{x} = 321.23$ kN/m, $c_{b}^{z} = 0.05$ kNs/m and $c_{b}^{x} = 0.41$ kNs/m.

Figure 5.

A 9 DoF human body model in seated posture with backrest support.

The variation of STHT with respect to frequency of seated biodynamic human body LP model with vertical backrest support is depicted in Figure 6. The magnitude of the vibration in 1–20 Hz frequency range affects the STHT almost identically as affected the same in case of human body model without backrest support. The peak value of STHT (i.e., 1.83) is observed at 5.5 Hz. This frequency lies in the frequency range of human interest. Decrement in STHT is noticed above 10 Hz as frequency increases. The value of STHT for the model using backrest support is significantly lower than that of the model without backrest. Figures 4 and 6 revealed that the value of STHT is almost 13.56% less when backrest is used.

Figure 6.

Seated LP human body model’s STHT with vertical backrest.

Seat to head transmissibility evaluation of human model in sitting posture using inclined backrest

A seated biodynamic LP human body model using inclined backrest support is shown in Figure 7. The inclination angle to the backrest support is provided in the anticlockwise direction from the vertical axis. Moreover, same inclination angle in the same direction to the set of the springs and dashpots (vertical and horizontal) are provided. The vertical and horizontal components of the stiffness ( $k$ ) and damping coefficient ( $c$ ) are now become $k \cos^{2} β$ and $\cos β$ in the direction of motion. In the present work, the backrest support is assumed to be inclined with the vertical at the inclination angles of 15° and 30°. The basis of selecting these two backrest support inclination angles is due to the fact that in India most of the vehicles have a range of 0° to 30° for the backrest support inclination angle. The different Mathematical human body models developed by several researchers assumed that the set of spring and dashpot (i.e., $k_{b}^{x, z}$ and $c_{b}^{x, z}$ ) attached directly among the upper torso ( $m_{b}$ ) and backrest support would be inclined only. However, the mathematical model considered in this paper with inclined backrest support considers that the spring and dashpot combinations attached to the head & neck ( $m_{a}$ ), upper torso ( $m_{b}$ ) and viscera ( $m_{c}$ ) would be inclined for providing more realistic model. Hence, needful modifications are incorporated to derive the EoM for inclined model. With angle of inclination β for backseat support with vertical axis, the mass $m_{c}$ , $m_{b}$ and $m_{a}$ are inclined at the same angle β so the corresponding values of spring-dashpot combinations are measured (Table 11). Variation in STHT with respect to frequency for a rigid seat with backseat support inclined at two different angles i.e., 15° and 30° to the vertical axis is depicted in Figure 8.

Figure 7.

A 9 DoF LP model for sitting posture with an inclination (β) in backrest.

Table 11.

Various parameters of seated biodynamic model with backrest inclination of 15° and 30° from vertical.

S. no.	Stiffness of springs $(k N / m)$			Damping constant $(k N s / m)$
S. no.	Symbol	β = 15°	β = 30°	Symbol	β = 15°	β = 30°
1	$k_{a b}^{z}$	15.70	12.62	$c_{a b}^{z}$	0.12	0.12
2	$k_{b c}^{z}$	16.12	12.96	$c_{b c}^{z}$	0.05	0.04
3	$k_{b d}^{z}$	16.06	12.90	$c_{b d}^{z}$	0.50	0.50
4	$k_{c d}^{z}$	408.70	328.54	$c_{c d}^{z}$	3.21	2.88
5	$k_{d}^{z}$	679.34	679.34	$c_{d}^{z}$	27.15	27.15
6	$k_{e}^{z}$	44.07	44.07	$c_{e}^{z}$	15.06	15.06
7	$k_{b c}^{x}$	262.24	210.80	$c_{b c}^{x}$	0.37	0.34
8	$k_{d}^{x}$	52.49	52.49	$c_{d}^{x}$	25.47	25.47
9	$k_{d e}^{x}$	23.43	23.43	$c_{d e}^{x}$	7.98	7.98
10	$k_{e}^{x}$	42.32	42.32	$c_{e}^{x}$	10.29	10.29
11	$k_{b}^{z}$	16.41	13.19	$c_{b}^{z}$	0.05	0.04
12	$k_{b}^{x}$	307.05	246.82	$c_{b}^{x}$	0.42	0.37

Figure 8.

Seated LP human body model’s STHT with inclined backrest.

For backrest inclination angle of 15°, STHT peak value of 1.58 is obtained at a frequency of 4.5 Hz and for backrest inclination angle of 30°, STHT peak value of 1.55 is obtained, both at a frequency of 5 Hz which corresponds to the primary natural frequency of human body. Attenuation in STHT is observed beyond 8.5 Hz and 7.5 Hz in case of backrest inclination of 15° and 30°, respectively with a progressive decrease in STHT as frequency increases. For a rigid seat, when the seat inclination was increased from 0° to 15° STHT decreased by 13.66% and with the increment in seat inclination from 0° to 30° STHT reduced by 15.3%. STHT higher for 0° inclination as compared with 15° and 30° inclination of backrest support irrespective of magnitudes of vibration. The results suggest that the comfort level is higher in case of backrest inclined at 30° to the vertical.

The results of STHT obtained from the present analysis are validated by comparing the curve obtained in Figures 4, 6, and 8, with curves of mean STHT characteristics (Figure 9) obtained in different reported studies.³⁹ The STHT characteristics obtained from past research reflect significant variations which are attributed to differences in factors i.e., measurement & analysis methods, experiment designs, subject anthropometry and nature of whole-body vibration. Peak transmissibility values in seating position with erect and forward lean postures are higher than the same obtained in seating backrest posture. From the majority of past research, peak transmissibility value is obtained as 1.5–2.5 and is observed in the frequency range of 3–8 Hz (Figure 9). Therefore, the results STHT of present analysis appear to be in good agreement with past reported research.

Figure 9.

Variations in mean STHT characteristics (synthesized data) obtained from different past research.³⁹

Conclusions

In the present work, a biodynamic human model is modelled in seated posture without backrest support and also with backrest support at different angle (i.e., 0°, 15° and 30°) to analyse the effects of the magnitude of accelerations, frequency of excitation, backrest support as well as different angles of inclination of backrest support on STHT of seated subject exposed to WBV. The model parameters such as damping, mass and stiffness of the mathematical model considered in this paper are derived from an alternate methodology which is based on anthropometric data. The STHT is evaluated analytically with the help of MATLAB programming platform for all the cases. From the analysis it is found that STHT increases as the vibration magnitude increased irrespective of the position. The slight increment in frequency affects the STHT in terms of the effective peak in the close vicinity of the first resonance; also, attenuation in STHT is visible as frequency increases. The STHT values decreased about 18.95%–22.85% at different vibration magnitudes for the human model sitting with backrest support as compared with whom sitting without backrest support, which indicates that higher comfort level is achieved with sitting using backrest support as compared with sitting without backrest support.

The comfort level of human subjects is improved as the inclination angle increased from 0° to 30° as the STHT is decreasing with inclination angle of the backrest support irrespective of the vibration magnitudes. Higher comfort level is achieved in case of backrest inclined at 30° compared with the vertical posture. The results of STHT obtained from the present analysis are validated by comparing with the mean STHT characteristics obtained in different reported studies and the results of present analysis appear to be in good agreement with past reported research.

Footnotes

Author contributions

1. Dr Sono Bhardawaj: Simulation, Resultsf

2. Dr Rakesh Chandmal Sharma: Modelling,

3. Dr Sunil Kumar Sharma: Simulation,

4. Dr L.V.V. Gopala Rao: Equations of Motion

5. Amit Vashist: Literature survey, formating, revisions

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Rakesh Chandmal Sharma

Sunil Kumar Sharma

L.V.V. Gopala Rao

References

Sharma

, et al. Analysis of generalized force and its influence on ride and stability of railway vehicle. Noise Vib Worldw 2020; 51(6): 95–109, DOI: 10.1177/0957456520923125

Sharma

. Sensitivity analysis of three-wheel vehicle’s suspension parameters influencing ride behavior. Noise Vib Worldw 2018; 49(7–8): 272–280, doi:10.1177/0957456518796846

Sharma

Palli

Koona

. Stress and vibrational analysis of an Indian railway RCF bogie. Int J Veh Struct Systems 2017; 9(5): 296–302, DOI: 10.4273/ijvss.9.5.06

Bhardawaj

Sharma

. Ride Analysis of track-vehicle-human body interaction subjected to random excitation. J Chin Soc Mech Eng 2020; 41(2): 227–236.

Sharma

. Ride, eigenvalue and stability analysis of three-wheel vehicle using Lagrangian dynamics. Int J Veh Noise Vib 2017; 13(1): 13–25. DOI: 10.1504/IJVNV.2017.086021

ISO 2631(E) . Guide for the evaluation of human exposure to whole-body vibration. Geneva, Switzerland: International Organization for Standardization, 1974.

ISO 2631/1 . Evaluation of human exposure to whole- body vibration. Part 1. General requirements. Geneva, Switzerland: International Organization for Standardization, 1985.

ISO/DIS 2631-1.2 . Mechanical vibration and shock- evaluation of human exposure to whole-body vibration. Part 1: general Requirements. Geneva, Switzerland: International Organization for Standardization, 1995.

ENV 12299 . Railway applications-ride comfort for passengers-measurement and evaluation. Draft, Brussels: European Committee for Standardization, 1999.

10.

UIC 513R . Guidelines for evaluating passenger comfort in relation to vibration in railway vehicles. 1st ed. Paris: International Union of Railways, 1994.

11.

Garg

Dukkipati

. Dynamics of railway vehicle systems. Canada: Academic Press, 1984.

12.

Kim

Kwon

Kim

, et al. Correlation of ride comfort evaluation methods for railway vehicles. Proc Inst Mech Eng F: J Rail Rapid Transit 2003; 217(2): 73–88.

13.

Sharma

. Linear and Nonlinear Analysis of Ride and Stability of a Three-Wheeled Vehicle Subjected to Random and Bump Inputs Using Bond Graph and Simulink Methodology. SAE Int J Commer Veh 2022; 15(1), doi:10.4271/02-15-01-0001

14.

Sharma

, et al. Analysis of bio-dynamic model of seated human subject and optimization of the passenger ride comfort for three-wheel vehicle using random search technique. Proc Inst Mech Eng K: J Multi-Body Dynamics 2021; 235(1): 106–121. DOI: 10.1177/1464419320983711

15.

Sharma

. Ride analysis of road surface-three-wheeled vehicle-human subject interactions subjected to random excitation. SAE Int J Commer Veh 2022; 15(3): 2022. DOI: 10.4271/02-15-03-0017

16.

Sharma

Singh

, et al. Modeling and simulation of human body-vehicle-track system for the investigation of ride comfort. SAE Int J Commer Veh 2021; 15(2). DOI: 10.4271/02-15-02-0008

17.

Zhao

Yang

, et al. Uncoupled spatial biodynamic model for seated humans exposed to vibration-development and validation. Int J Ind Ergon 2021; 85: 103171.

18.

Guruguntla

Lal

. An improved biomechanical model to optimize biodynamic responses under vibrating medium. J Vib Eng Technol 2021; 9: 675–685.

19.

Dong

R-c

, et al. Effect of sitting posture and seat on biodynamic responses of internal human body simulated by finite element modeling of body-seat system. J Sound Vib 2019; 438(6): 543–554. DOI: 10.1016/j.jsv.2018.09.012

20.

Desai

Guha

Seshu

, Multibody modelling of the human body for vibration induced direct and cross-axis seat to head transmissibility. Proc Inst Mech Eng C J Mech Eng Sci 2021; 235(17), 3146–3161. doi:10.1177/0954406220967957.

21.

Kim

Yoon

. Development of a biomechanical model of the human body in a sitting posture with vibration transmissibility in the vertical direction. Int J Indust Ergon 2005; 35: 817–829.

22.

Liang

Chiang

. Modelling of a seated human body exposed to vertical vibrations in various automotive postures. Ind Health 2008; 46: 125–137.

23.

Subashi

Matsumoto

Griffin

. Modelling resonances of the standing body exposed to vertical whole-body vibration: effects of posture. J Sound Vib 2008; 317: 400–418.

24.

Subashi

Nawayseh

Matsumoto

, et al. Nonlinear subjective and dynamic responses of seated subjects exposed to horizontal whole-body vibration. J Sound Vib 2009; 321: 416–434.

25.

Dong

, et al. Modeling of biodynamic responses distributed at the fingers and the palm of the human hand-arm system. J Biomech 2007; 40: 2335–2340.

26.

Dong

Welcome

McDowell

, et al. Methods for deriving a representative biodynamic response of the hand-arm system to vibration. J Sound Vib 2009; 325(4): 1047–1061.

27.

Matsumoto

Griffin

. Modelling of the dynamic mechanisms associated with the principal resonance of the seated human body. Clin Biomech 2001; 16(1): S31–S44.

28.

Bhardawaj

Sharma

, et al. Development of multibody dynamical using MR damper based semi-active bio-inspired chaotic fruit fly and fuzzy logic hybrid suspension control for rail vehicle system. Proc Inst Mech Eng K: J Multi-Body Dynamics 2020; 234(4): 723–744, doi:10.1177/1464419320953685

29.

Shibata

Maeda

. Determination of backrest inclination based on biodynamic response study for prevention of low back pain. J Med Eng Phys 2010; 32(6): 577–583.

30.

Dewangan

Rakheja

Marcotte

, et al. Comparisons of apparent mass responses of human subjects seated on rigid and elastic seats under vertical vibration. Ergonomics 2013; 56(12): 1806–1822.

31.

Dewangan

Shahmir

Rakheja

, et al. Seated body apparent mass responseto vertical whole body vibration: gender and anthropometric effects. Int J Indust Ergon 2013; 43(4): 375–391.

32.

Bhiwapurkar Mahesh

Saran

Harsha

. Effects of vibration magnitude and posture on seat-to-head-transmissibility responses of seated occupants exposed to lateral vibration. Int J Veh Noise Vib 2016; 12(1): 42–59.

33.

Chakrabarti

. Indian anthropometric dimensions: for ergonomic design practice. Ahmedabad, Gujarat: National Institute of Design Ahmedabad, 1997, pp. 30–93.

34.

Harsha

Desta

Prashanth

, et al. Measurement and biodynamic model development of seated human subjects exposed to low frequency vibration environment. Int J Veh Noise Vib 2014; 10(1/2): 1–24.

35.

Nigam

Malik

. A study on a vibratory model of a human body. J Biomech Eng 1987; 109(2): 148–153.

36.

Goldman

Von Gierke

. Harris’ shock and vibration handbook. NewYork: McGraw-Hill, 1961.

37.

Shafer

Siders

Johnson

, et al. Body density estimates from upper-body skinfold thicknesses compared to air-displacement plethysmography. Clin Nutr 2010; 29: 249–254.

38.

Gupta

. Identification and experimental validation of damping ratios of different human body segments through anthropometric vibratory model in standing posture. J Biomech Eng 2007; 129(4): 566–574.

39.

Wang

Rakheja

Boileau

P-E

, et al. Effect of back support condition on seat to head transmissibilities of seated occupants under vertical vibration. J Low Freq Noise VibActive 2006; 25(4): 239–259.

Modeling of biomechanical human body model for seat to head transmissibility analysis

Abstract

Keywords

Introduction

Modelling of biodynamic human body model

Mass segments evaluation

Rigid mass assumption

Evaluation of the stiffness of different body segments ( S i x , z )

Evaluation of stiffness of connecting springs ( k i x , z )

Evaluation of damping of different body segments ( C i x , z )

Evaluation of damping of connecting dashpots ( c i x , z )

Derivation of EoM

Seat to head transmissibility evaluation for the seated human model without backrest

Determination of STHT for sitting position with backrest support

Seat to head transmissibility evaluation of human model in sitting posture using inclined backrest

Conclusions

Footnotes

Author contributions

Declaration of conflicting interests

Funding

ORCID iDs

References

Evaluation of the stiffness of different body segments ( $S_{i}^{x, z}$ )

Evaluation of stiffness of connecting springs ( $k_{i}^{x, z}$ )

Evaluation of damping of different body segments ( $C_{i}^{x, z}$ )

Evaluation of damping of connecting dashpots ( $c_{i}^{x, z}$ )