Nonlinearity in the localised apparent masses of the seated human body exposed to vertical vibration

Mechanical Systems and Signal Processing 135 (2020) 106394

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Nonlinearity in the localised apparent masses of the seated human body exposed to vertical vibration Chi Liu, Yi Qiu ⇑ Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, United Kingdom

a r t i c l e

i n f o

Article history: Received 11 April 2019 Received in revised form 20 September 2019 Accepted 24 September 2019 Available online 11 October 2019 Keywords: Apparent mass Whole-body vibration Nonlinearity

a b s t r a c t The apparent mass of the seated human body exposed to vertical whole-body vibration has shown nonlinear characteristics with the principal resonance frequency decreasing with increasing vibration magnitude. Experimental studies have been designed to investigate the cause of nonlinearity but there seems no consistent conclusion on the role of body parts in the nonlinearity of the apparent mass. Biodynamic modelling provides insight into effect of each model parameter on the dynamics of the system so that the contribution of individual body parts on the nonlinearity can be investigated. In this paper, a multi-body dynamic model of seated human body that can represent the dynamic forces over the local area of contact beneath the ischial tuberosities and thighs was developed. Model parameters were determined for 14 individuals who were exposed to vertical whole-body vibration at 0.25, 0.5, and 1.0 m/s2 r.m.s. in a previous study. With the model parameters determined for 14 individuals and 3 vibration magnitudes, statistical analysis was performed to confirm the effect of vibration magnitude on model parameters. The model developed in this study contained six rigid bodies representing upper-torso, lower-torso, pelvis, thighs, legs, and viscera of the body, interconnected by linear rotational springs and dampers. The geometry parameters of the model were initially derived from a finite-element model of spine and were later scaled to each of the 14 individuals. It was found as vibration magnitude increased, the vertical stiffness beneath the ischial tuberosities, the middle thighs, and the front thighs, the stiffness of viscera, the stiffness of the lower thoracic joint, and the vertical damping beneath the ischial tuberosities decreased significantly. For the vertical apparent mass, reducing the vertical stiffness beneath the ischial tuberosities and thighs and the stiffness of viscera reduced the resonance frequency, while reducing the other stiffness parameters did not see the similar effect. For the foreand-aft cross-axis apparent mass, reducing the fore-and-aft stiffness beneath the ischial tuberosities and thighs reduced the resonance frequency, while reducing the other parameters of stiffness did not show the similar effect. It was concluded that the soft tissues beneath the ischial tuberosities and thighs and the viscera are the parts principally causing the nonlinearity observed in the apparent masses of the human body. Ó 2019 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, England, United Kingdom. E-mail address: [email protected] (Y. Qiu). https://doi.org/10.1016/j.ymssp.2019.106394 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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1. Introduction Biodynamic responses of the human body to whole-body vibration are widely described by the transfer function between the dynamic force acting on the human body and the acceleration of the seat (i.e., apparent mass) (e.g., [1–7]). With vertical vibration excitation of a constant acceleration spectrum between 0 and 15 Hz at a magnitude between 0.25 and 1.5 ms
2 r.m. s., the coherence between the measured force and the acceleration is close to unity (e.g., [1,2]), which suggests that the human body can be simplified as a linear system for vibration at a certain magnitude. However, with increasing vibration magnitude, nonlinearity has been reported in the apparent mass of human body with the resonance frequency decreasing [3–7]. Similar nonlinear phenomenon has also been found in the transmissibilities from vibration at the seat pan to that at the head, various locations along spine, and the pelvis [4]. The dynamic responses of human body to whole-body vibration have also been analysed using the reversed path multiinput-single-output (reMISO) method [8,9] to examine the uncorrelated relationship between the physical input and outputs. Huang and Ferguson [8] applied the reMISO method to study the ordinary and the conditioned partial coherence between the reversed inputs (vertical and fore-and-aft forces) and output (seat acceleration). It was found at both vibration magnitudes of 0.125 and 1.0 m/s2 r.m.s., the partial coherence of the vertical force obtained by the recursive conditioning procedure improved the overall multiple coherence at frequencies between 8 and 20 Hz but showed marginal effect at frequencies less than 8 Hz [8], which suggested that at low frequencies with a certain vibration magnitude, the body responses seem to be dominated by linear responses. Both the frequencies of the peak in the ordinary frequency response function and in the conditioned frequency response function decreased with increasing vibration magnitude [8]. There are nonlinear phenomena associated with the properties of the soft tissues and musculoskeletal structure of the human body [7,10–13]. The soft tissues of the lower limbs show a nonlinear stress-strain relationship [10]. The increase of activities of muscles is disproportionate with increasing vibration magnitude [11]. The soft tissues or the musculoskeletal structure of the body have a thixotropic behaviour in which stiffness of the structure decreases with movements [7,12,13]. However, not all the body parts that have nonlinear properties are necessarily the cause of the nonlinearity in the apparent mass shown with different vibration magnitudes. There seems no consistent conclusion on the role of body parts in the nonlinearity of the apparent mass. Different parts have been suggested to be associated with the nonlinearity [5,7,14–17]. With the muscles at the buttocks and abdomen area being tensed, the nonlinearity in the apparent mass was reduced, and the buttocks and abdomen were therefore suggested to be the parts causing the nonlinearity [14]. Biodynamic models have shown that the frequency of principal resonance in the apparent mass is most sensitive to the stiffness of the vertical spring beneath the ischial tuberosities and the stiffness of viscera [15–17], suggesting the buttocks and abdomen being the parts causing the nonlinearity. Huang and Griffin [5] found that the periodical voluntary activities of the erector muscle group greatly reduced the nonlinearity in the apparent mass. The phasic activities of the back muscles in response to vibration were expected to be reduced by the voluntary movements, and the back muscles were therefore suggested to contribute to the nonlinearity. In the study of Huang and Griffin [7], human bodies in a supine posture, where vibration excitation was applied at the soft tissues of back rather than at the buttocks, also showed nonlinearity in the apparent masses with different vibration magnitudes. The authors suggested that the thixotropic behaviour of the musculoskeletal structure of the whole body results in the nonlinearity. It has been challenging to understand the mechanism of biodynamics associated with the nonlinearity in the apparent mass. The main difficulty is that the variables used in experimental studies (e.g., [5,7,14]), such as sitting postures, voluntary tensions of specific muscles and voluntary body movements, would influence the stiffness and damping of more than one body part so that the contribution from a single part cannot be distinguished. To advance the understanding of contribution of specific body parts to the nonlinearity in the apparent mass, biodynamic models can be further used as the effect of each model parameter on the biodynamic responses can be identified. Linear biodynamic models were used to model the apparent mass of the human body exposed to vibration excitation with a constant acceleration spectrum between 0 and 15 Hz, based on the fact that the coherence between the input acceleration and the dynamic force acting on the human body is close to unity [1,5,15–17]. Linear lumped-parameter models with single or two degrees of freedom were also used to study the nonlinearity in the apparent mass, by determining the model parameters to represent the apparent mass for each vibration magnitude and then identifying how model parameters differ between vibration magnitudes [5]. The process of parameter determination was often done for a group of individuals so as to allow statistical analysis, which is needed due to large inter-subject variabilities between individuals and intrasubject variabilities within an individual [1-7,14]. Using this method it has been shown that the effective stiffness of the body decreases with increasing vibration magnitude [5]. However since those simple models do not have anatomical representations of the body, they cannot further indicate the location of the body that contributes to the nonlinearity. This method is expected to be extended to improve understanding if the model is able to represent the various body parts and able to represent dynamic responses of a group of individuals. Multi-body dynamic models can be developed with anatomical representations of the body to represent the body motion during vibration. These models usually consist of rigid bodies representing the pelvis and thighs and one or two rigid bodies representing the torso, interconnected by rotational springs and dampers (e.g., [16,17]). The buttocks and thighs segment in the model of Matsumoto and Griffin [16] was restrained in the vertical direction, and thus the fore-and-aft forces at the seat cannot be calculated. Moreover, a single pair of vertical spring and damper was used in this model to represent the soft

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tissues of the buttocks and thighs and the role of each in contribution to the nonlinearity in the apparent mass cannot be distinguished. The model of Zheng et al. [17] did not include the viscera which may have influence on the resonances in the vertical apparent mass. A model with separate representation of the soft tissues of the thighs and the buttocks as well as the viscera is needed. In addition to the apparent mass of the human body, transfer functions between the vertical acceleration of the seat and the dynamic forces measured over a local area of contact (e.g., area beneath ischial tuberosities and beneath the thighs) have also been quantified, defined as the ‘localised apparent mass’ at the corresponding area [18]. For clarity, the apparent mass of the human body is called ‘overall apparent mass’ in this paper. Biodynamic models that can represent the localised apparent masses have not been seen in literature and are expected to reflect the dynamic properties of the human body more closely than the models only representing the overall apparent mass. The main objective of this study is to investigate the effect of vibration magnitudes on the properties of body parts. This would require to define a biodynamic model that has the necessary representations of the body and can represent the localised apparent masses at the ischial tuberosities and the thighs. Development of the model is the secondary objective of this study. It was hypothesised that the stiffness beneath the ischial tuberosities and thighs was the principal cause of the nonlinearity in the apparent mass shown with different vibration magnitudes. 2. Method The influence of vibration magnitude on the effective stiffness and damping of the body was investigated preciously based on linear biodynamic modelling [5]. This method was further developed in the current study by proposing a more representative model of various body parts that can reflect both the overall and localised biodynamic responses. The parameters of the model were identified with the measured overall and localised apparent masses, for 14 individuals and for three different vibration magnitudes. This generated a set of values for each model parameter across individuals with each vibration magnitude, allowing to identify the influence of vibration magnitudes on model parameters statistically. 2.1. Biodynamic modelling of apparent mass of seated human body A linear eight degree-of-freedom multi-body dynamic model was developed with six rigid bodies representing the uppertorso (from head to the eighth thoracic vertebra (T8)), lower-torso (from T9 to the fifth lumbar vertebra (L5)), pelvis, thighs, legs and viscera of the body (Fig. 1). The upper-torso and lower-torso were connected via a revolute joint at the location of the spinal disc between T8 and T9, assigned with rotational stiffness and damping. This joint was named the ‘lower thoracic joint (T8)’ in this study. Revolution joints with rotational stiffness and damping were also defined between the lower-torso and pelvis, between pelvis and thighs, and between thighs and legs, referred to as the ‘lower lumbar joint (L5)’, ‘hip joint’, and ‘knee joint’, respectively. Translational springs and dampers in vertical and fore-and-aft directions were used to represent the soft tissues beneath the ischial tuberosities, the middles thighs, and the front thighs. Definition of the parameters was listed in Tables A1 and A2 Appendix A. Initial geometry and inertia parameters of the model (i.e., the length, mass, and moment inertia of each rigid body) were derived from a dynamic model of the spine of a subject with a sitting mass of 63 kg and a sitting height of 83 cm [15]. Scaling was then performed for each subject, following the method used in [16]. The angles between rigid bodies were estimated from the normal upright sitting posture of subjects adopted in the experiment [18]. Derivation of geometry parameters were listed in Table A2, Appendix A. Equations of motion were derived using Lagrange formulation, as detailed in Appendix B. Laplace transformation was then applied to derive the apparent masses in the frequency domain. 2.2. Identification of model parameters The model was to represent the apparent masses of 14 subjects measured in a previous study [18]. In that experimental study, the subjects sat on a rigid seat with their upper-body in a normal upright posture without contacting a backrest and with their feet unsupported while exposed to vertical whole-body vibration between 0.2 and 20 Hz at 0.25, 0.5, and 1.0 ms
2 r.m.s., each with a duration of 60 s. Duration vibration vertical and fore-and-aft dynamic forces were measured over three areas of contact (i.e., beneath the ischial tuberosities, the middle part of the thighs, and the front part of the thighs during vibration). For each subject, the model parameters were identified by fitting the model responses to eight quantities measured in [18], explained in Table 1. During the model calibration, all the geometry and inertia parameters were fixed, while all the stiffness and damping parameters were treated unknown and to be determined. An error function, E, was defined as the root of the sum of the squared differences between the measured and modelled values:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )ffi u 8 ( N h uX X  m 2  p 2 i m m p p t E¼ W M 1 ðiÞ
M0 ðiÞ þ W M1 ðiÞ
M 0 ðiÞ j¼1

i¼1

j

ð1Þ

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m6 kj3, cj3 m2

m3 kj2, cj2

m1

kj1, cj1

kmx, cmx kfx, cfx m4

kj4, cj4

kjT, cjT m5

ktx, cjx

kfz, cfz

ktz, cjz kfx, cfx L6 r6

α06

L3

α03

r3 L4 r4

αp

α04

L1 r 1

α01

α05

rm

r5 rf

L5

Fig. 1. Structure of the multibody dynamic model (parameters described in Tables A1 and A2, Appendix A).

Table 1 Definition of quantities used in model parameter identification (adopted from [18]). Quantity

Definition

Overall vertical in-line apparent mass

Transfer function between the total vertical force over the seat pan and the vertical acceleration of the seat pan Transfer function between the total fore-and-aft force over the seat pan and the vertical acceleration of the seat pan Transfer function between the vertical force beneath area A* and the vertical acceleration of the seat pan Transfer function between the fore-and-aft force beneath area A* and the vertical acceleration of the seat pan

Overall fore-and-aft cross-axis apparent mass Localised vertical in-line apparent mass at area A* Localised fore-and-aft cross-axis mass at area A* *

A refers to the contact area beneath the ischial tuberosities, the middle part of thighs, or the front part of thighs, as defined in [18].

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p p m th where M m 1 ðiÞ and M 1 ðiÞ are the modelled modulus and phase of an apparent mass at i frequency point; M 0 ðiÞ and M 0 ðiÞ are the measured modulus and phase of an apparent mass at ith frequency point; j is the index of the eight measured apparent masses to be fitted; Wm and Wp are the weightings for modulus and phase. The weighting, Wp, made the phase into a similar level as the modulus, which made the optimisation process more effective. The stiffness and damping parameters were optimized to minimise the error, using the optimisation algorithm Complex [19]. The algorithm has been used is previous study to model the apparent mass of the human body and the transmissibility of a seat [20]. The error function was derived based on the overall apparent mass and the localised apparent masses in both vertical and fore-and-aft directions. Fitting to multiple localised responses introduced more constraints in optimisation than fitting to only the overall apparent mass, which is expected to delivery more reasonable parameters of the model. Through trials it was found the phase of apparent masses must be included in the error function to achieve good fit to both the modulus and the phase. For the frequency range, the error function was calculated over 0.25–15 Hz with a step of 0.25 Hz for the vertical apparent masses and over 1.5–15 Hz for the fore-and-aft cross-axis apparent masses. The fore-and-aft cross-axis apparent masses at frequencies less than 1.5 Hz were excluded in the error function because the coherence between the measured fore-and-aft forces at the seat surface and the vertical seat acceleration was low [18]. The algorithm Complex requires to define boundaries within which the optimal parameters are searched to achieve minimum error function. Global optimum may not be found if boundaries are not defined properly. Usually inappropriate definition of boundaries would result in the optimisation procedure to terminate with some parameters lying on their boundaries. It was assumed that the global optimum is achieved if the optimisation procedure terminates with all the parameters obtained within the boundaries. As the structure of the model in the current study is extended based on previous models [16,17,21], the initial parameters for the upper body and viscera were decided with reference to [16], while those for the buttocks and thighs were based on [17,21], as listed in Table 2. In a previous modelling study [22], for 24 male subjects with the weight support by the seat surface between 43.9 and 108 kg, the stiffness of the single degree of freedom model of the apparent mass varied between 46% and 121% relative to the stiffness obtained from fitting the mean apparent mass of the 24 male subjects. Therefore in this study, the lower and upper boundary of the search domain for each parameter was defined as half and double of its initial value. To assist in defining proper boundaries, an extra iterative procedure was introduced on top of the optimisation algorithm Complex, as shown in Fig. 2. The Complex algorithm was executed with the first set of initial values in Table 2 and with the lower and the upper boundaries defined as the half and double of the initial values respectively. Once the execution finished, the obtained parameters were checked. If any of the obtained parameters was at the lower or upper boundary, the obtained value was used as a new initial value for that parameter and new lower and upper boundaries for that parameter were re-set by half and double of the new initial value and Complex algorithm was called again. The iterative procedure was repeated till

Table 2 Initial parameters used in optimisation.

*

Parameters

Initial value

Unit

Reference

Vertical stiffness beneath the ischial tuberosities Vertical stiffness beneath the middle thighs Vertical stiffness beneath the front thighs Viscera stiffness Fore-and-aft stiffness beneath the ischial tuberosities Fore-and-aft stiffness beneath the middle thighs Fore-and-aft stiffness beneath the front thighs Vertical damping beneath the ischial tuberosities Vertical damping beneath the middle thighs Vertical damping beneath the front thighs Viscera damping Fore-and-aft damping beneath the ischial tuberosities Fore-and-aft damping beneath the middle thighs Fore-and-aft damping beneath the front thighs Pitch stiffness at the ischial tuberosities Pitch stiffness at the hip joints Pitch stiffness at the lower lumbar joint (L5) Pitch stiffness at the lower thoracic joint (T8) Pitch stiffness at the knee joints Pitch damping at the ischial tuberosities Pitch damping at the hip joints Pitch damping at the lower lumbar joint (L5) Pitch damping at the lower thoracic joint (T8) Pitch damping at the knee joints

125,424 13,734 13,734 27,400 11,100 11,100 11,100 1010 80 80 195 659 45 45 477 477 1040 1210 10 6.72 6.72 54.4 8.17 5

N/m N/m N/m N/m N/m N/m N/m N/m Ns/m Ns/m Ns/m Ns/m Ns/m Ns/m Nm/rad Nm/rad Nm/rad Nm/rad Nm/rad Nms/rad Nms/rad Nms/rad Nms/rad Nms/rad

[17]

[16] [21]*

[17]

[16] [17]

[16]

Estimated [16]

Estimated

Only a single spring with stiffness of 33,355?N/m was modelled at the human-seat interface in [21], so the value was divided by 3 as the initial values in the current model.

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Adjust boundaries

Adjust initial values

Initial values

Upper and lower boundaries

Complex algorism

Yes

Optimised values ptimised value reach ach boundaries? boundarie No Optimised values accepted Fig. 2. Iteration process of optimisation.

all obtained parameters were within the boundaries, or was forced to stop after three repetitions, designed based on trial. The parameters obtained from the last optimisation were adopted as the optimised values. 2.3. Analysis Having the calibrated model parameters for each of the 14 subjects and for each of the three vibration magnitudes, effect of vibration magnitude on the stiffness and damping parameters was identified by non-parametric statistical analysis, including Friedman two-way analysis of variance for a k-sample case and Wilcoxon matched-pairs signed ranks test for a two-sample case. The reported p-values have not been corrected for multiple comparisons. Three analysis cases were further designed to identify the location of body that is producing nonlinearity in the apparent mass. In the analysis of Case 1, the vertical stiffness beneath the ischial tuberosities, middle thighs and front thighs together with the stiffness of the viscera was reduced by 30% while the other parameters were fixed. In the analysis of Case 2, the foreand-aft stiffness beneath the ischial tuberosities, middle thighs and front thighs was reduced by 30% while the others were fixed. In the analysis of Case 3, the rotational stiffness of all the joints was reduce by 30% while the others were fixed. All the three analyses were performed for each subject and statistical analysis was then performed. 3. Results 3.1. Modelling of localised apparent masses for each vibration magnitude Comparisons between the modelled and the measured apparent masses of one subject (Subject 1) are shown in Figs. 3–5 for different vibration magnitudes. Comparisons for all subjects are shown in Figures C1, C2 and C3, Appendix C. For each vibration magnitude, the linear model provided reasonable fits to the vertical apparent masses and the fore-andaft cross-axis apparent masses measured at all the locations. The phenomenon that the vertical apparent mass measured at the front thighs has two resonances with the primary resonance around 6–9 Hz was well presented. Discrepancies were observed in the phase of the fore-and-aft cross-axis apparent masses at frequencies less than 3 Hz, but the coherence between the force and acceleration in the measurements was also low (<0.5) at these frequencies for all the subjects [18]. Discrepancy was also seen around 8 and 10 Hz in the cross-axis apparent mass at the front thighs, where the peak from the model was not obvious. Possible reasons for the discrepancies were discussed in Section 4.1. As the vibration magnitude increased there were significant decreases in the principal resonance frequencies in all the modelled apparent masses (both the vertical in-line and fore-and-aft cross-axis apparent masses) (Friedman, p < 0.05). 3.2. Effect of vibration magnitude on the model parameters As the vibration magnitude increased (i.e., from 0.25 to 0.5 ms
2 r.m.s., from 0.25 to 1.0 ms
2 r.m.s., and from 0.5 to 1.0 ms
2 r.m.s.), there were significant decreases in the vertical stiffness beneath the ischial tuberosities, the middle thighs, and front thighs, the viscera stiffness, the lower thoracic (T8) stiffness, and the vertical damping beneath the ischial tuberosities (Wilcoxon, p < 0.05; Table 3). Of the above three magnitude increments, two showed reduction in the fore-and-aft stiffness at middle thighs and front thighs, and the pitch stiffness at the ischial tuberosities (Wilcoxon, p < 0.05; Table 3). The median of the fore-and-aft stiffness at the ischial tuberosities decreased as vibration magnitude increased, but no statistical significance was found (Wilcoxon, p > 0.05; Table 3).

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Fig. 3. Comparison between the modelled and measured vertical apparent masses and fore-and-aft cross-axis apparent masses with vibration magnitude at 0.25 ms
2 r.m.s. values: ––– modelled; - - - - measured. Individual data (Subject 1) are shown here; comparisons for all subjects are shown in Appendix C.

3.3. Relation of body location with the nonlinearity in the apparent masses For all the vertical apparent masses, reducing the vertical stiffness beneath the ischial tuberosities, middle thighs, and front thighs, and the viscera stiffness (Case 1 as defined in Section 2.3) significantly reduced the resonance frequencies (Fig. 6a; Wilcoxon, p < 0.001), while reducing the force-and-aft stiffness beneath the ischial tuberosities and thighs (Case 2) did not significantly alter the resonance frequencies (Fig. 6b; Wilcoxon, p > 0.05). Reducing the stiffness of all the joints (Case 3) significantly reduced the resonance frequencies in the vertical apparent mass at the ischial tuberosities and the front thighs (Fig. 6c; Wilcoxon, p < 0.001), but did not significantly affect the resonance frequencies in the overall vertical apparent mass and the apparent mass at the middle thighs (Wilcoxon, p > 0.05). The reduction in the resonance frequencies caused by reducing the vertical stiffness beneath the ischial tuberosities, middle thighs and front thighs, and the stiffness of viscera (Case 1) was greater than that by reducing the stiffness of the joints (Wilcoxon, p < 0.01). For all the fore-and-aft cross-axis apparent masses, reducing the vertical stiffness of the ischial tuberosities and thighs and the viscera (Case 1), and reducing the fore-and-aft stiffness at the ischial tuberosities and thighs (Case 2) reduced the resonance frequencies (Fig. 7a and b; Wilcoxon, p < 0.05). Reducing the stiffness of all joints did not significantly affect the resonance frequencies (Fig. 7c; Wilcoxon, p > 0.05). The reduction in the resonance frequency caused by reducing the fore-and-aft stiffness of the ischial tuberosities, middle thighs, and front thighs (Case 2) was greater than that due to reducing the vertical stiffness at the three locations (Wilcoxon, p < 0.05). 4. Discussion 4.1. The multi-body dynamic model of the seated human body Nonlinearity in the apparent mass of the body has been suggested to be possibly caused by combination of factors, including the softening property of passive soft tissues and skeleton [1,7], the different active muscular forces at different magnitude of vibration [5,14], and the ‘inverted pendulum’ geometric nonlinear effect [3]. There has been no certain conclusion which factor is the main cause of nonlinearity. Rotational motion of the upper body in the sagittal plane during whole body vibration could introduce geometric nonlinearity, as an inverted pendulum. This effect is expected to be significantly

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Fig. 4. Comparison between the modelled and measured vertical apparent masses and fore-and-aft cross-axis apparent masses with vibration magnitude at 0.5 ms
2 r.m.s. values: ––– modelled; - - - - measured. Individual data (Subject 1) are shown here; comparisons for all subjects are shown in Appendix C.

reduced when the body sits against a backrest or is in a supine posture. The fact that the apparent mass showed nonlinearity with the body sitting against a backrest at various angles of 0, 10, 20, 30 degrees from vertical [23] and with the body being in the supine posture [7] suggests that the geometric nonlinearity might not be the primary cause of nonlinearity. Based on the above discussion, although rotational joints were used in the current model, linearization (i.e., sin(h)  h, where h is the rotational oscillation about the equilibrium position) was carried out assuming that there was only small oscillation during vibration and thus the geometric nonlinearity was not included in the model. This linearization treatment has commonly been conducted in the biodynamic models with rotational joints [16,17,20,21]. The linearization seems reasonable, supported by the linear model being able to represent the body transmissibilities [16]. Modal analysis of the spine using a finite-element model indicated that during vibration at frequencies less than 10 Hz the body acted as a beam representing the spine supported by springs and dampers representing the soft tissues of the buttocks and thighs [15]. The model suggested seven modes at frequencies less than 10 Hz, but in terms of the spine bending (from lumber to head), only the first two orders of bending modes were seen. To reflect these bending modes in a multi-body model, the torso should at least contain two rigid bodies connected with a rotational joint, as shown in the current model. The rotation mode of the pelvis was shown around 9 Hz in [15], and therefore a rigid body must be required in the multibody model to represent the pelvis. The spine model suggested two modes of viscera bouncing in-phase and out-of-phase with the bouncing of the pelvis [15]. Therefore, modelling of the viscera using a rigid body supported by a spring and a damper, as shown in the current model, was sufficient to represent the contribution of viscera to the biodynamics of the body. Using the three pairs of springs and dampers to represent the soft tissues beneath the ischial tuberosities, the middle thighs, and the front thighs is the simplest structure to represent the localised apparent masses at the three locations. Based on the above discussion, the structure of the current model appeared to be simplified suitable structure that can represent the body modes as well as the localised apparent masses at frequencies less than 10 Hz. A more complex model of the torso with more degrees of freedom than the current model might improve to some extent the fitting to the measured apparent masses but it seemed unnecessary within the scope of the current study as they are not sensitive to the apparent mass. A simpler model of the torso than the current model, on the other hand, would not be able to represent the required modes of the spine at frequencies less than 10 Hz. The human body has shown nonlinear responses with vibration at different magnitude [1–7]. However for each vibration magnitude, it has been consistently shown the coherence between the vertical force at the seat and the acceleration is close

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Fig. 5. Comparison between the modelled and measured vertical apparent masses and fore-and-aft cross-axis apparent masses with vibration magnitude at 1.0 ms
2 r.m.s. values: ––– modelled; - - - - measured. Individual data (Subject 1) are shown here; comparisons for all subjects are shown in Appendix C.

to unity at frequencies between 0 and 15 Hz. (e.g., [1,2]). This suggested that for a certain vibration magnitude, the body could be treated as a linear system. This allowed the responses of the human body to be modelled with linear models for a certain vibration magnitude, as in Refs. [15–17,22] and allowed the nonlinearity to be investigated by comparing the model parameters between vibration magnitudes. By using a linear two degrees of freedom model to fit the apparent masses of 12 subjects exposed to vertical vibration at 0.125, 0.25, 0.5, 0.75, and 1.0 ms
2 r.m.s, between vibration magnitudes, there was no significant difference in any of the sprung masses of the model while there were significant differences in each of the stiffness parameters [5]. This seems to suggest that the mass properties were less affected by the vibration magnitude than the stiffness, and thus in the current model only the stiffness and damping were to be identified in optimization.

4.2. Discrepancies between the modelled and measured apparent masses Main differences between the modelled and measured apparent masses were observed in the phase of the fore-and-aft apparent masses at frequencies less than 3 Hz (Figs. 3–5). At these frequencies, the coherence between the measured foreand-aft forces and the acceleration was low [18]. The low coherence arose because the fore-and-aft forces at the seat pan were low and can be affected by factors, such as sitting postures and muscle activities [18]. The model showed the two resonances in the vertical apparent mass at the front thighs, but did not show the second peak in the fore-and-aft cross-axis apparent mass at the front thighs as obvious in the measurements (Figs. 3–5). In the measurements, the top front edge of the plate supporting the front thighs was rounded (see [18]) so as to prevent a high concentration of stress in the soft tissue of the front thighs. The rounded surface would have resulted in forces in the fore-and-aft direction at the front thighs due to vertical vibration of the thighs [18], which, however, was not included in the model.

4.3. Effect of vibration magnitude on model parameters As vibration magnitude increased, there was significant reduction in the vertical stiffness at the ischial tuberosities, the middle thighs, the front thighs, and the viscera, in the fore-and-aft stiffness at the middle thighs and front thighs, and the

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Table 3 Statistical significance of effect of vibration magnitude on the model parameters (Wilcoxon matched-pairs signed-ranks test). Parameters

Vertical stiffness beneath the ischial tuberosities Vertical stiffness beneath the middle thighs Vertical stiffness beneath the front thighs Viscera stiffness Fore-and-aft stiffness beneath the ischial tuberosities Fore-and-aft stiffness beneath the middle thighs Fore-and-aft stiffness beneath the front thighs Vertical damping beneath the ischial tuberosities Vertical damping beneath the middle thighs Vertical damping beneath the front thighs Viscera damping Fore-and-aft damping beneath the ischial tuberosities Fore-and-aft damping beneath the middle thighs Fore-and-aft damping beneath the front thighs Pitch stiffness at the ischial tuberosities Pitch stiffness at the hip joints Pitch stiffness at the lower lumbar joint (L5) Pitch stiffness at the lower thoracic joint (T8) Pitch stiffness at the knee joints Pitch damping at the ischial tuberosities Pitch damping at the hip joints Pitch damping at the lower lumbar joint (L5) Pitch damping at the lower thoracic joint (T8) Pitch damping at the knee joints

Comparison between vibration magnitude, Unit of vibration magnitude (ms
2 r.m.s.) 0.25 vs 0.5

0.25 vs 1.0

0.5 vs 1.0

****; ****; ****; ***;
;
;
; *;
"
"
;
"
;
" *;
;
" **; *; *;
;
;
;
"

****; ****; ****; ****;
; *; *; ***;
"
"
;
"
;
"
;
;
" ***;
;
;
;
;
;
;

****; ****; **; ****;
; *; **; **; ;
"
;
"
;
" *;
;
; *;
"
;
;
;
"
"

*: p < 0.05; **: p < 0.01; ***: p < 0.005; ****: p < 0.001; -: p > 0.05. ;: Lower median with greater vibration magnitude. ": Greater median with greater vibration magnitude.

Fig. 6. Effect on the vertical in-line apparent masses due to 30% reduction in the vertical stiffness beneath the ischial tuberosities, middle thighs, and front thighs and the viscera stiffness (a, Case 1), due to 30% reduction in the fore-and-aft stiffness beneath the ischial tuberosities, middle thighs, and front thighs (b, Case 2), or due to 30% reduction in the joints stiffness (c, Case 3): ––– apparent mass for each of the three cases; - - - - apparent mass without changing the stiffness. Median of the 14 modelled apparent masses.

C. Liu, Y. Qiu / Mechanical Systems and Signal Processing 135 (2020) 106394

11

Fig. 7. Effect on the fore-and-aft cross apparent masses due to 30% reduction in the vertical stiffness beneath the ischial tuberosities, middle thighs, and front thighs and the viscera stiffness (a, Case 1), due to 30% reduction in the fore-and-aft stiffness beneath the ischial tuberosities, middle thighs, and front thighs (b, Case 2), or due to 30% reduction in the joints stiffness (c, Case 3): ––– apparent mass for each of the three cases; - - - - apparent mass without changing the stiffness. Median of 14 the modelled apparent masses.

vertical damping at the ischial tuberosities (Wilcoxon, p < 0.05; Table 2). This indicates that the above body parts show different properties with different vibration magnitudes. With the joints representing the properties of the back in the current model, no significant differences were found in the stiffness and damping of all the joints as vibration magnitude increased, except the stiffness of lower thoracic joint (Table 2). The marginal effect of vibration magnitude on the joint stiffness and damping arises from the apparent mass being not sensitive to the properties of the joints, consistent with previous studies [16,17].

4.4. Relation of body location with the nonlinearity in the apparent mass For all the vertical in-line apparent masses, reducing the vertical stiffness beneath ischial tuberosities, middle thighs and front thighs, and the stiffness of the viscera reduced resonance frequencies (Fig. 6a). However, reducing the fore-and-aft stiffness at the ischial tuberosities, middle thighs, and front thighs or stiffness of all joints had less effect (Fig. 6b and c). This suggests that with different vibration magnitudes the changes in the vertical stiffness at the viscera, buttocks and thighs are more related to the nonlinearity in the vertical apparent mass than the changes in stiffness in other directions or at other locations of the body. This is consistent with the nonlinearity in the vertical apparent mass being more affected by tensing the muscles of the abdomen and buttocks than of the other locations [14]. For all the fore-and-aft cross-axis apparent masses, reducing the fore-and-aft stiffness beneath ischial tuberosities, middle thighs and front thighs reduced resonance frequencies, while reducing the vertical stiffness of the buttocks and thighs or the stiffness of all the joints showed less effect (Fig. 7). This suggests that with changing vibration magnitude the nonlinearity in the fore-and-aft cross-axis apparent mass is more related to the change in the fore-and-aft stiffness of the soft tissues beneath the ischial tuberosities and thighs than the changes in stiffness in other directions or at other locations of the body. However, Matsumoto and Griffin [14] found that voluntary tension of the buttocks tissues did not reduce the nonlinearity in the fore-and-aft cross-axis apparent mass, which seems different from the observation of this study. This might be explained by the voluntary tension mainly influencing the properties of muscles while the fore-and-aft stiffness of the buttocks and thighs tissues being attributed to the combination of the properties of muscles, fat and skins of the buttocks and thighs. Voluntary back-abdomen bending movement during whole-body vibration has been found to reduce the nonlinearity in the vertical apparent mass, which suggests that the back muscles are related to the nonlinearity [5]. Influence of back mus-

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C. Liu, Y. Qiu / Mechanical Systems and Signal Processing 135 (2020) 106394

cles was reflected by changing of the stiffness of the joints in the model. These changes showed little effect on the resonance frequency in the vertical apparent mass (Fig. 6c), which seems different from the findings of Huang and Griffin [5]. This may be because the voluntary back-abdomen bending movement used in [5] may also affect other properties of the body. The movement mainly involves the contraction and relaxation of the erector spinae muscles and the activities of psoas major, producing pitch motions of the pelvis. The pitch motions of the pelvis can stretch the muscles and other soft tissues in the buttocks region and alter the pressures beneath the ischial tuberosities, affecting the properties of the soft tissues beneath the ischial tuberosities. Changing joint stiffness in the model does not completely reflect all the influence of back muscles induced by the voluntary back-abdomen bending movement. 4.5. Mechanism associated with the reduced stiffness of the soft tissues with increasing vibration magnitudes The current modelling study indicated that reducing the vertical stiffness and the fore-and-aft stiffness beneath the ischial tuberosities, middle thighs and front thighs as well as the stiffness of the viscera reduces the principal resonance in the overall and localised apparent masses (Figs. 6a and 7b). Introducing nonlinear springs that have softening loaddeflection relations is expected to be able to reflect the ‘softening’ phenomenon. To demonstrate this, nonlinear springs with quadratic stiffness with ‘softening’ characteristic were applied to all the vertical and fore-and-aft springs at the human-seat interface, and to the viscera stiffness. The expression of the quadratic stiffness was as Eq. (2), where fs is the force provided by the nonlinear springs; k0 is a coefficient for the linear part; kn is for the nonlinear part; z is the deflection of the spring.

f s ðzÞ ¼ k0 z
kn zjzj

ð2Þ

The values of k0 and kn were listed in Table 4. The values for k0 are the same as in the linear model with vibration at 0.25 ms
2 r.m.s. The values for kn were chosen manually to fit the vertical apparent mass with vibration at 0.5 and 1.0 ms
2 r.m.s. Other model parameters were the same as in the linear model with vibration 0.25 ms
2 r.m.s. The model was solved in time domain using the 4th order Runge-Kutta method. Random vibration with a constant acceleration spectrum over 0.25–30 Hz at 0.25, 0.5 and 1.0 ms
2 r.m.s. respectively with a duration of 60 s was used as excitations in the vertical direction. Transfer functions between the acceleration and the total forces at the human-seat interface were calculated using CSD method [24]:

MðxÞ ¼ Saf ðxÞ=Saa ðxÞ

ð3Þ

where M(x) is the apparent mass, Saf(x) is the cross-spectral density between the acceleration and the force, and Saa(x) is the power spectral density of the acceleration. All transfer functions were obtained with a frequency resolution of 0.25 Hz. Results showed that the nonlinearity in the apparent masses can be reflected by the nonlinear model (Fig. 8). The chosen quadratic stiffness (or other forms of stiffness with softening effect) can provide the reduced stiffness matching the nonlinearity in the apparent mass shown with different vibration magnitudes. However, measurements of the stressstrain curve of the soft tissues, both in vivo and in vitro, have suggested ‘hardening’ characteristics rather than ‘softening’ [10,25–28]. This may be because the reported ‘hardening characteristics’ were obtained from quasi-static measurements where there is no vibration applied, while ‘softening’ caused by the thixotropy in muscle tissues, which has been considered as one of the mechanisms associated with the nonlinearity in the apparent mass [7], would need motions applied to the muscles to breakdown the bonds between actin and myosin in muscles [29]. In addition, the reported ‘hardening characteristics’ were obtained when the level of muscle activities was unchanged throughout the tests, for example in vitro, or fully relaxed in vivo. However, the non-proportional change of muscle activities due to the change of the vibration magnitude has been considered as a mechanism associated with the nonlinearity [5,11,14]. Neither the effect of thixotropy nor the effect of muscle activities on muscle stiffness would be shown in such quasi-static tests under a constant muscle activity. Application of softening stiffness may be a quantitative representation of the mechanism associated with the nonlinearity in the apparent mass.

Table 4 Parameters for the quadratic stiffness in the nonlinear model. Parameters

k0 [N/m]

kn [N/m2]

Vertical stiffness at the ischial tuberosities Vertical stiffness at the middle thighs Vertical stiffness at the front thighs Fore-and-aft stiffness at the ischial tuberosities Fore-and-aft stiffness at the middle thighs Fore-and-aft stiffness at the front thighs Viscera stiffness

86,423 18,873 64,537 23,217 18,328 20,052 27,216

5.18  107 5.66  106 3.87  107 6.97  106 1.10  107 1.20  107 1.63  107

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13

Fig. 8. Overall vertical in-line apparent mass and overall fore-and-aft cross-axis apparent mass with vibration at various magnitudes predicted by the nonlinear model: ––– modelled; - - - - measured.

5. Conclusions In this paper, a multi-body biodynamic model of seated human body that is able to represent both the overall and localised apparent masses of 14 individuals was developed. Using the model, it was found that the vertical stiffness beneath the ischial tuberosities, the middle thighs, and the front thighs, the stiffness of viscera, the stiffness of the lower thoracic joint, and the vertical damping beneath the ischial tuberosities decrease significantly with increasing vibration magnitudes. There was a trend in which the fore-and-aft stiffness beneath the middle thighs and front thighs decrease with increasing vibration magnitude. The soft tissues beneath the ischial tuberosities and thighs and the viscera were suggested to be the primary body parts causing the nonlinearity observed in the vertical in-line and the fore-and-aft cross-axis apparent masses of the human body with different vibration magnitudes.

Appendix A Parameters of the model.

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C. Liu, Y. Qiu / Mechanical Systems and Signal Processing 135 (2020) 106394 Table A1 Stiffness and damping parameters of the model. Parameter

Definition

ktz kmz kfz kv ktx kmx kfx kjt kj1 kj2 kj3 kj4 ctz cmz cfz cv ctx cmx cfx cjt cj1 cj2 cj3 cj4

Vertical stiffness of ischial tuberosity tissues Vertical stiffness of middle thigh tissues Vertical stiffness of front thigh tissues Stiffness of viscera tissues Fore-aft stiffness of ischial tuberosity tissues Fore-aft stiffness of middle thigh tissues Fore-aft stiffness of front thigh tissues Rotational stiffness at ischial tuberosities Rotational stiffness at hip joint Rotational stiffness at L5 Rotational stiffness at T8 Rotational stiffness at knee joint Vertical damping of ischial tuberosity tissues Vertical damping of middle thigh tissues Vertical damping of front thigh tissues Damping of viscera tissues Fore-aft damping of ischial tuberosity tissues Fore-aft damping of middle thigh tissues Fore-aft damping of front thigh tissues Rotational damping at ischial tuberosity Rotational damping at hip joint Rotational damping at L5 Rotational damping at T8 Rotational damping at knee joint

Table A2 Geometry of the model. Parameter

Definition

Value

H L1 [m] L3 [m] L4 [m] L5 [m] L6 [m] Lb [m] r1 [m] rj1 [m] r3 [m] r4 [m] r5 [m] r6 [m] rm [m] rf [m] a10 [degree] a30 [degree] a40 [degree] a50 [degree] a60 [degree] ap [degree] ab [degree] aB [degree]

Sitting height Length of pelvis (ischial tuberosity – L5 disc) Length of lower torso (L5 disc - T8 disc) Length of thigh (hip joint – knee joint) Length of leg Length of upper torso (head - T8 disc) Length between contact point and T8 disc Length between ischial tuberosity and centre of mass of pelvis projected to L1 Length between ischial tuberosity and hip joint projected to L1 Length between L5 disc and centre of mass of lower torso Length between hip joint and centre of mass of thigh Length between knee joint and centre of mass of leg Length between T8 disc and centre of mass of upper torso Length between hip joint and location of middle thigh stiffness Length between hip joint and location of front thigh stiffness Angle of pelvis (respect to horizontal plane) Angle of lower torso (respect to horizontal plane) Angle of thigh (respect to horizontal plane) Angle of leg (respect to horizontal plane) Angle of upper torso (respect to horizontal plane) Angle between L1 and line connecting ischial tuberosity and centre of mass of pelvis Angle between L6 and line connecting T8 disc and centre of mass of upper-torso Angle of backrest (respect to vertical plane)

Ref. [15]

r6  0.10 L1  0.71 L1  0.30 L3  0.55 L4  0.43 L5  0.43 L6  0.50 L4  0.30 L4  0.60 85 90
5 280 85 15 5 0

Appendix B Equations of motion The model has eight degrees of freedom, ½ q1 q2 q3 q4 q5 q6 q7 q8 , whereq1 is the fore-and-aft displacement of ischial tuberosity; q2 is the vertical displacement of ischial tuberosity; q3 is the translational displacement of viscera; q4 is the pitch of pelvis; q5 is the pitch of lower torso; q6 is the pitch of thighs; q7 is the pitch of legs; q8 is the pitch of upper torso. The fore-and-aft and vertical displacements of the centre of gravity of the pelvis and its pitch can be written:

x1 ¼ q1
r j1 sinða10 Þq4

ðB:1Þ

C. Liu, Y. Qiu / Mechanical Systems and Signal Processing 135 (2020) 106394

15

z1 ¼ q2 þ r j1 cosða10 Þq4

ðB:2Þ

a1 ¼ q4

ðB:3Þ

where x1 is the fore-and-aft displacement of the centre of the pelvis; z1 is the vertical displacement of the centre of the pelvis; a1 is the pitch of the pelvis; a10 is the initial angle of the pelvis with respect to the horizontal plane as shown in Fig. 1; rj1 is the length between ischial tuberosity and hip joint projected to the line between the ischial tuberosity and the joint between pelvis and low torso. For the viscera, the translational displacements are

  x2 ¼ q1
L1 cos ap sinða10 Þq4 þ q3 cosða10 Þ

ðB:4Þ

  z2 ¼ q2 þ L1 cos ap cosða10 Þq4 þ q3 sinða10 Þ

ðB:5Þ

where x2 is the fore-and-aft displacement of the viscera; z2 is the vertical displacement of the viscera; ap is the angle between L1 and the line connecting ischial tuberosity and centre of mass of pelvis. For the lower torso, the fore-and-aft and vertical displacements of the centre of gravity and its pitch are

  x3 ¼ q1
L1 sin ap þ a10 q4
r 3 sinða30 Þq5

ðB:6Þ

  z3 ¼ q2 þ L1 cos ap þ a10 q4 þ r 3 cosða30 Þq5

ðB:7Þ

a3 ¼ q5

ðB:8Þ

where x3 is the fore-and-aft displacement of the lower torso; z3 is the vertical displacement of the lower torso; a3 is the pitch of the lower torso; a30 is the initial angle of the lower torso with respect to the horizontal plane as shown in Fig. 1; r3 is the length between the lower lumbar joint and centre of mass of lower torso. For the thigh the fore-and-aft and vertical displacements of the centre of gravity and its pitch are

x4 ¼ q1
r j1 sinða10 Þq4
r4 sinða40 Þq6

ðB:9Þ

z4 ¼ q2 þ r j1 cosða10 Þq4 þ r4 cosða40 Þq6

ðB:10Þ

a4 ¼ q6

ðB:11Þ

where x4 is the fore-and-aft displacement of the thigh; z4 is the vertical displacement of the thigh; a4 is the pitch of the thigh; a40 is the initial angle of the thigh with respect to the horizontal plane as shown in Fig. 1; r4 is the length between the hip joint and centre of mass of the thigh. For the leg the fore-and-aft and vertical displacements of the centre of gravity and its pitch are

x5 ¼ q1
r j1 sinða10 Þq4
L4 sinða40 Þq6
r 5 sinða50 Þq7

ðB:12Þ

z5 ¼ q2 þ r j1 cosða10 Þq4 þ L4 cosða40 Þq6 þ r 5 cosða50 Þq7

ðB:13Þ

a5 ¼ q7

ðB:14Þ

where x5 is the fore-and-aft displacement of the leg; z5 is the vertical displacement of the leg; a5 is the pitch of the leg; a50 is the initial angle of the leg with respect to the horizontal plane as shown in Fig. 1; L4 is the length of the thigh; r5 is the length between the knee joint and centre of mass of the leg. For the upper torso, the fore-and-aft and vertical displacements of the centre of gravity and its pitch are

  x6 ¼ q1
L1 sin ap þ a10 q4
L3 sinða30 Þq5
r 6 sinða60 Þq8

ðB:15Þ

  z6 ¼ q2 þ L1 cos ap þ a10 q4 þ L3 cosða30 Þq5 þ r6 cosða60 Þq8

ðB:16Þ

a6 ¼ q8

ðB:17Þ

where x6 is the fore-and-aft displacement of the upper torso; z6 is the vertical displacement of the upper torso; a6 is the pitch of the leg; a60 is the initial angle of the upper torso with respect to the horizontal plane as shown in Fig. 1; L3 is the length of the lower torso; r6 is the length between the lower thoracic joint and centre of mass of the upper torso. Displacements of each contact point are:

xm ¼ q1
r j1 sinða10 Þq4
rm sinða40 Þq6

ðB:18Þ

zm ¼ q2 þ r j1 cosða10 Þq4 þ r m cosða40 Þq6

ðB:19Þ

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C. Liu, Y. Qiu / Mechanical Systems and Signal Processing 135 (2020) 106394

xf ¼ q1
r j1 sinða10 Þq4
rf sinða40 Þq6

ðB:20Þ

zf ¼ q2 þ r j1 cosða10 Þq4 þ r f cosða40 Þq6

ðB:21Þ

where xm and xf are the fore-and-aft displacements of the contact points at the middle thighs and the front thighs; zm and zf are the vertical displacements of the contact points at the middle thighs and the front thighs; rm is the length between the hip joint and the contact point at the middle thigh; rf is the length between the hip joint and the contact point at the front thigh. The equations of motion of the model were derived using Lagrange formulation. The kinetic energy, T, potential energy, U, dissipation energy, D, satisfy:

  d @T @U @D þ þ ¼0 dt @ q_ i @qi @ q_ i

ðB:22Þ

where i is the index of the degree of freedom. The kinetic energy is as Eq. (B.23):

T¼

6   1 1X 2 2 mi x_ i þ z_ i þ 2 i¼1 2

X

Ii a_ i

2

ðB:23Þ

i¼1;3;4;5;6

where i is the index of the rigid body. The potential energy is as Eq. (B.24):

Fig. C1. Comparison between the modelled apparent mass and the measured apparent mass for each individual (upper: vertical in-line apparent mass; lower: fore-and-aft cross-axis apparent mass) with vibration at 0.25 ms
2 r.m.s.: ––– modelled; - - - measured.

C. Liu, Y. Qiu / Mechanical Systems and Signal Processing 135 (2020) 106394

U¼

 1  2  1 1 ktx x2t þ kmx x2m þ kfx x2f þ ktz ðzt
q0 Þ2 þ kmz ðzm
q0 Þ2 þ kfz zf
q0 þ kj1 ða1
a3 Þ2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 þ kj1 ða1
a3 Þ þ kj2 ða6
a3 Þ þ kj3 ða1
a4 Þ þ kj4 ða4
a5 Þ þ kjt a1 þ kv q23 2 2 2 2 2 2

17

ðB:24Þ

The dissipation energy is as Eq. (B.25):

D¼

 1  2  1 1 2 2 2 2 2 2 ctx x_ t þ cmx x_ m þ cfx x_ f þ ctz ðz_ t
q_ 0 Þ þ cmz ðz_ m
q_ 0 Þ þ cfz z_ f
q_ 0 þ cj1 ða_ 1
a_ 3 Þ 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2 þ cj1 ða_ 1
a_ 3 Þ þ cj2 ða_ 6
a_ 3 Þ þ cj3 ða_ 1
a_ 4 Þ þ cj4 ða_ 4
a_ 5 Þ þ cjt a_ 1 þ cv q_ 3 2 2 2 2 2 2

ðB:25Þ

The equations of motion were obtained by substituting Eqs. ((B.1)–(B.21)) into Eqs. ((B.22)–(B.25)). Laplace transformation was then performed. The ratio of the Laplace transforms of qi, where i is the index of degree of freedom, and the Laplace transforms of displacement of the vibration input were then derived, and denoted as Qi0(s). The vertical apparent mass at the ischial tuberosity, AMzzt, is derived as Eq. (B.26):

AMzzt ðsÞ ¼ ðktz þ ctz sÞð1
Q 20 Þ=s2

ðB:26Þ

The vertical apparent mass at the middle thighs, AMzzm, is derived as Eq. (B.27):

  AMzzm ðsÞ ¼ ðkmz þ cmz sÞ 1
Q 20
rj1 cosa10 Q 40
rm cosa40 Q 60 =s2

ðB:27Þ

The vertical apparent mass at the front thighs, AMzzf, is derived as Eq. (B.28):

Fig. C2. Comparison between the modelled apparent mass and the measured apparent mass for each individual (upper: vertical in-line apparent mass; lower: fore-and-aft cross-axis apparent mass) with vibration at 0.5 ms
2 r.m.s.: ––– modelled; - - - measured.

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C. Liu, Y. Qiu / Mechanical Systems and Signal Processing 135 (2020) 106394

   AMzzf ðsÞ ¼ kfz þ cfz s 1
Q 20
r j1 cosa10 Q 40
rf cosa40 Q 60 =s2

ðB:28Þ

The overall vertical apparent mass at the seat pan, AMzz, is the sum of vertical apparent masses at the ischial tuberosities, middle thighs, and front thighs. The fore-and-aft apparent mass at the ischial tuberosity, AMzxt, is derived as Eq. (B.29):

AMzxt ðsÞ ¼ ðktx þ ctx sÞð
Q 10 Þ=s2

ðB:29Þ

The fore-and-aft cross-axis apparent mass at the middle thighs, AMzxm, is derived as Eq. (B.30):

  AMzxm ðsÞ ¼ ðkmx þ cmx sÞ
Q 10 þ r j1 sina10 Q 40 þ r m sina40 Q 60 =s2

ðB:30Þ

The fore-and-aft cross-axis apparent mass at the front thighs, AMzxf, is derived as Eq. (B.31):

   AMzxf ðsÞ ¼ kfx þ cfx s
Q 10 þ r j1 sina10 Q 40 þ rf sina40 Q 60 =s2

ðB:31Þ

The overall fore-and-aft cross-axis apparent mass at the seat pan, AMzx, is the sum of fore-and-aft cross-axis apparent masses at the ischial tuberosities, middle thighs, and front thighs.

Appendix C Comparison of individual apparent masses between modelling and measurements (See Figs. C1–C3).

Fig. C3. Comparison between the modelled apparent mass and the measured apparent mass for each individual (upper: vertical in-line apparent mass; lower: fore-and-aft cross-axis apparent mass) with vibration at 1.0 ms
2 r.m.s.: ––– modelled; - - - measured.

C. Liu, Y. Qiu / Mechanical Systems and Signal Processing 135 (2020) 106394

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