A schematic view of 32-dofs biomechanical model of a seated human is presented in Fig. 2. The human body is imaginarily segmented and each segment is connected with an adjacent segment via stiffness and damping characteristics that are both direct and cross-coupled. Since the human body is symmetric about the sagittal plane. Hence, in the modeling the biomechanical properties (mass, stiffness and damping) are considered symmetric about the sagittal plane. In addition, the cross-coupled parameters are considered symmetric. The model consists of different segments (i = 1–16) that represents anatomy of head (m1), thorax (m2), abdomen (m3), pelvis (m4), upper arms (m5 & m8), forearms (m6 & m9), hands (m7 & m10), thighs (m11 & m14), legs (m12 & m15), feet (m13 & m16) (refer Fig. 2). To avoid complexity in representation only springs are shown in Fig. 2, whereas dampers can also be shown alongside springs. Direct and cross-coupled stiffness (Kij) and damping (Cij) qualities exist for each spring and damper (i = x, j = z; where ‘x’ and ‘z’ indicate fore-and-aft and vertical direction, respectively).
2.1. Governing equations
By applying Newton's second law to each section, the system's governing equations are formulated (refer to Fig. 2(c)). For brevity and completeness, the governing equation for ‘ith’ segment connected to ‘i + 1th ’ segment may present as:
$${m_i}{\ddot {x}_i}+c_{{xx}}^{i}({\dot {x}_i} - {\dot {x}_{i+1}})+c_{{xz}}^{i}({\dot {z}_i} - {\dot {z}_{i+1}})+k_{{xx}}^{i}({x_i} - {x_{i+1}})+k_{{xz}}^{i}({z_i} - {z_{i+1}})={f_{{x_i}}}$$
1
$${m_i}{\ddot {z}_i}+c_{{zx}}^{i}({\dot {x}_i} - {\dot {x}_{i+1}})+c_{{zz}}^{i}({\dot {z}_i} - {\dot {z}_{i+1}})+k_{{zx}}^{i}({x_i} - {x_{i+1}})+k_{{zz}}^{i}({z_i} - {z_{i+1}})={f_{{z_i}}}$$
2
The above Eqs. ((1) & (2)), may present in matrix form as,
$$[{M_i}]\left\{ {{{\ddot {\chi }}_i}} \right\}+[{C_i}]\left\{ {{{\dot {\chi }}_i} - {{\dot {\chi }}_{i+1}}} \right\}+[{K_i}]\left\{ {{\chi _i} - {\chi _{i+1}}} \right\}=\{ {f_i}\}$$
3
where
\([{M_i}]=\left[ {\begin{array}{*{20}{c}} {{m_i}}&0 \\ 0&{{m_i}} \end{array}} \right],[{C_i}]=\left[ {\begin{array}{*{20}{c}} {c_{{xx}}^{i}}&{c_{{xz}}^{i}} \\ {c_{{zx}}^{i}}&{c_{{zz}}^{i}} \end{array}} \right],[{K_i}]=\left[ {\begin{array}{*{20}{c}} {k_{{xx}}^{i}}&{k_{{xz}}^{i}} \\ {k_{{zx}}^{i}}&{k_{{zz}}^{i}} \end{array}} \right],\{ {f_i}\} =\left\{ {\begin{array}{*{20}{c}} {{f_{{x_i}}}} \\ {{f_{{z_i}}}} \end{array}} \right\},\{ {\chi _i}\} =\left\{ {\begin{array}{*{20}{c}} {{x_i}} \\ {{z_i}} \end{array}} \right\}\)
After developing equations of motion (EOMs) for each segment, the global EOM may express as,
$${[M]_{16 \times 16}}{\left\{ {\ddot {\chi }} \right\}_{16 \times 1}}+[C]{\left\{ {\dot {\chi }} \right\}_{16 \times 1}}+[K]{\left\{ \chi \right\}_{16 \times 1}}={\{ f\} _{16 \times 1}}$$
4
Now by substituting χ = χejωt and f = Fejωt in Eq. (4), the time series equation may convert into frequency series as,
$${( - {\omega ^2}M+j\omega C+K)_{16 \times 16}}{{\text{\varvec{\chi}}}_{16 \times 1}}={F_{16 \times 1}}$$
5
After solving Eq. (5), the biodynamic responses (STHT and AM) may acquire as (Marzbanrad and Afkar, 2013),
$$STHT=\frac{{{Z_1}\sin {\theta _1}}}{{{Z_0}}}$$
6
$$AM=\frac{{{F_4}}}{{{a_4}}}$$
7
Where, Z1 and Z0 are vertical displacements at head and seat (input), respectively. a4 and F4 are acceleration and force at the contact point between human and seat i.e., pelvis, respectively. θ1 is the backrest angle. For numerical simulation, its value is taken as θ1 = 240 (Wang et al., 2008).
STHT is a dimensional less quantity; it helps researchers and designers to investigate the amount and frequency of vibration passes to the human body through a vibrating medium (i.e., seat or floor). Whereas AM provides information about the mass of humans in a dynamic environment. In case of a rigid body AM is the mass of the system in a static state. However, in dynamic conditions at resonance, the apparent mass can quadruple (Liang and Chiang, 2008).
2.2. Model parameters estimation with firefly algorithm
In this section, the biomechanical parameters of the developed model is optimized with the help of the firefly algorithm (FA). The parameters are optimized by reducing the sum square error between the experimental (Wang et al., 2008) and analytical response. The objective function includes both magnitude and phase responses of STHT and AM as,
$${\text{Minimize }}({O_f})=\mathop \Sigma \limits_{{i=1}}^{p} ({\alpha _1}.{\lambda _1}+{\alpha _2}.{\lambda _2}+{\alpha _3}.{\lambda _3}+{\alpha _4}.{\lambda _4})$$
8
Here
\({\lambda _1}={[STH{T_E}({f_i}) - STH{T_A}({f_i})]^2}_{{Mag}}{\text{, }}{\lambda _2}={[STH{T_E}({f_i}) - STH{T_A}({f_i})]^2}_{{Pha}}\)
\({\lambda _3}={[A{M_E}({f_i}) - A{M_A}({f_i})]^2}_{{Mag}}{\text{, }}{\lambda _4}={[A{M_E}({f_i}) - A{M_A}({f_i})]^2}_{{Pha}}\)
The experimental and analytical readings are denoted by the subscripts 'E' and 'A', respectively. Subscripts 'Mag' and 'Pha' denote magnitude and phase responses, respectively. Symbols α1, α2, α3 are α4 denote weight functions. Equal importance (weight) is assigned to both the biodynamic responses i.e., (α1 = α2 = α3 = α4). While ‘p’ denotes the number of experimental data points. A flow chart (Fig. 3) represents the typical process followed to optimize the biomechanical parameters. To minimize the objective function and acquire optimized parameters of the human body, the following decision variables and constraints are applied in the analysis.
Decision variables (refer Fig. 2):
-
m 1 , m2,……m16 are the segmental mass of the model
-
K 1 , K2,…….K26 are the stiffness matrices of inbetween segments
-
C 1 , C2,…….C26 are the damping matrices of inbetween segments
Here direct and cross-coupled stiffness and damping parameters are contained in the stiffness and damping matrices, respectively.
Constraints:
$$\left\{ \begin{gathered} \sum\limits_{{i=1}}^{{16}} {{m_i}=77.3kg} \hfill \\ {m_5}={m_8},{m_6}={m_9},{m_7}={m_{10}} \hfill \\ {m_{11}}={m_{14}},{m_{12}}={m_{15}},{m_{13}}={m_{16}} \hfill \\ {k_{xz}}={k_{zx}},{c_{xz}}={c_{zx}} \hfill \\ 100N{m^{ - 1}}<[{k_{ii}}]<300000N{m^{ - 1}} \hfill \\ 100Ns{m^{ - 1}}<[{c_{ii}}]<300000Ns{m^{ - 1}} \hfill \\ \end{gathered} \right\}$$
9
m i , kij and cij are mean weight, lower and upper bounds of stiffness and damping coefficients of the proposed model, respectively. To achieve the desired accuracy of objective function decision parameters are selected as total number of variables = 224, swarm size = 100, and the number of iterations = 50. By minimizing the sum square error in Eq. (8) under mentioned constraints in Eq. (9), the model parameters are tuned till the desired accuracy is achieved.
The experimental data are referred from Wang et al., (2008) to minimize the objective function. (Wang et al., 2008) conducted experiments on 12 healthy male individuals under random vibration conditions. The magnitude of vibration in the vertical direction was set as 1ms− 2 r.m.s n the frequency range of 0.5 to 15 Hz. The goodness of fit (GOF) is determined as (Guruguntla and Lal, 2022c),
$$\varepsilon =1 - \frac{{\sqrt {\Sigma {{({\tau _e} - {\tau _a})}^2}/(N - 2)} }}{{\Sigma {\tau _e}/N}}$$
10
The experimental and analytical responses are denoted by ‘τe’ and ‘τa’ respectively. The total number of data points chosen for analysis is referred to as ‘N’. The ‘ɛ’ reflects/mimics a good model that might be used instead of an experimental investigation. If the value is 1, it means that the analytical model and experimental responses are identical. It signifies that ‘ɛ’ makes the model effectiveness. Table 1 shows the best-optimized parameters for the suggested model with the highest GOF value.
Table 1
The optimized mass, stiffness, and damping coefficients with FA.
Mass (kg) |
m1 | m2 | m3 | m4 | m5 | m6 | m7 | m8 | m9 | m10 |
6.13 | 14.39 | 10.18 | 11.22 | 2.26 | 1.34 | 0.77 | 2.26 | 1.34 | 0.77 |
m11 | m12 | m13 | m14 | m15 | m16 | – | – | – | – |
8.15 | 3.14 | 1.24 | 8.15 | 3.14 | 1.24 | – | – | – | – |
Stiffness parameters (Nm-1, ×105) |
\(k_{{xx}}^{1}\) | 1.62 | \(k_{{xx}}^{2}\) | 1.72 | \(k_{{xx}}^{3}\) | 1.94 | \(k_{{xx}}^{4}\) | 2.24 | \(k_{{xx}}^{5}\) | 1.34 |
\(k_{{xz}}^{1}\) | 0.74 | \(k_{{xz}}^{2}\) | 0.49 | \(k_{{xz}}^{3}\) | 0.64 | \(k_{{xz}}^{4}\) | 1.26 | \(k_{{xz}}^{5}\) | 0.61 |
\(k_{{zx}}^{1}\) | 0.74 | \(k_{{zx}}^{2}\) | 0.49 | \(k_{{zx}}^{3}\) | 0.64 | \(k_{{zx}}^{4}\) | 1.26 | \(k_{{zx}}^{5}\) | 0.61 |
\(k_{{zz}}^{1}\) | 2.39 | \(k_{{zz}}^{2}\) | 2.45 | \(k_{{zz}}^{3}\) | 2.38 | \(k_{{zz}}^{4}\) | 2.51 | \(k_{{zz}}^{5}\) | 1.67 |
\(k_{{xx}}^{6}\) | 1.67 | \(k_{{xx}}^{7}\) | 1.17 | \(k_{{xx}}^{8}\) | 1.34 | \(k_{{xx}}^{9}\) | 1.67 | \(k_{{xx}}^{{10}}\) | 1.17 |
\(k_{{xz}}^{6}\) | 0.35 | \(k_{{xz}}^{7}\) | 0.54 | \(k_{{xz}}^{8}\) | 0.61 | \(k_{{xz}}^{9}\) | 0.35 | \(k_{{xz}}^{{10}}\) | 0.54 |
\(k_{{zx}}^{6}\) | 0.35 | \(k_{{zx}}^{7}\) | 0.54 | \(k_{{zx}}^{8}\) | 0.61 | \(k_{{zx}}^{9}\) | 0.35 | \(k_{{zx}}^{{10}}\) | 0.54 |
\(k_{{zz}}^{6}\) | 2.07 | \(k_{{zz}}^{7}\) | 1.85 | \(k_{{zz}}^{8}\) | 1.67 | \(k_{{zz}}^{9}\) | 2.07 | \(k_{{zz}}^{{10}}\) | 1.85 |
\(k_{{xx}}^{{11}}\) | 1.27 | \(k_{{xx}}^{{12}}\) | 1.92 | \(k_{{xx}}^{{13}}\) | 1.72 | \(k_{{xx}}^{{14}}\) | 1.62 | \(k_{{xx}}^{{15}}\) | 1.72 |
\(k_{{xz}}^{{11}}\) | 0.85 | \(k_{{xz}}^{{12}}\) | 1.45 | \(k_{{xz}}^{{13}}\) | 0.46 | \(k_{{xz}}^{{14}}\) | 0.42 | \(k_{{xz}}^{{15}}\) | 0.46 |
\(k_{{zx}}^{{11}}\) | 0.85 | \(k_{{zx}}^{{12}}\) | 1.45 | \(k_{{zx}}^{{13}}\) | 0.46 | \(k_{{zx}}^{{14}}\) | 0.42 | \(k_{{zx}}^{{15}}\) | 0.46 |
\(k_{{zz}}^{{11}}\) | 1.64 | \(k_{{zz}}^{{12}}\) | 2.56 | \(k_{{zz}}^{{13}}\) | 2.19 | \(k_{{zz}}^{{14}}\) | 1.86 | \(k_{{zz}}^{{15}}\) | 2.19 |
\(k_{{xx}}^{{16}}\) | 1.62 | \(k_{{xx}}^{{17}}\) | 1.76 | \(k_{{xx}}^{{18}}\) | 1.26 | \(k_{{xx}}^{{19}}\) | 1.09 | \(k_{{xx}}^{{20}}\) | 1.13 |
\(k_{{xz}}^{{16}}\) | 0.42 | \(k_{{xz}}^{{17}}\) | 0.34 | \(k_{{xz}}^{{18}}\) | 0.64 | \(k_{{xz}}^{{19}}\) | 0.71 | \(k_{{xz}}^{{20}}\) | 0.32 |
\(k_{{zx}}^{{16}}\) | 0.42 | \(k_{{zx}}^{{17}}\) | 0.34 | \(k_{{zx}}^{{18}}\) | 0.64 | \(k_{{zx}}^{{19}}\) | 0.71 | \(k_{{zx}}^{{20}}\) | 0.32 |
\(k_{{zz}}^{{16}}\) | 1.86 | \(k_{{zz}}^{{17}}\) | 2.37 | \(k_{{zz}}^{{18}}\) | 1.84 | \(k_{{zx}}^{{19}}\) | 1.76 | \(k_{{zx}}^{{20}}\) | 1.75 |
\(k_{{xx}}^{{21}}\) | 1.64 | \(k_{{xx}}^{{22}}\) | 1.76 | \(k_{{xx}}^{{23}}\) | 1.26 | \(k_{{xx}}^{{24}}\) | 1.09 | \(k_{{xx}}^{{25}}\) | 1.13 |
\(k_{{xz}}^{{21}}\) | 0.51 | \(k_{{xz}}^{{22}}\) | 0.34 | \(k_{{xz}}^{{23}}\) | 0.64 | \(k_{{xz}}^{{24}}\) | 0.71 | \(k_{{xz}}^{{25}}\) | 0.32 |
\(k_{{zx}}^{{21}}\) | 0.51 | \(k_{{zx}}^{{22}}\) | 0.34 | \(k_{{zx}}^{{23}}\) | 0.64 | \(k_{{zx}}^{{24}}\) | 0.71 | \(k_{{zx}}^{{25}}\) | 0.32 |
\(k_{{zz}}^{{21}}\) | 2.13 | \(k_{{zz}}^{{22}}\) | 2.37 | \(k_{{zz}}^{{23}}\) | 1.84 | \(k_{{zz}}^{{24}}\) | 1.76 | \(k_{{zz}}^{{25}}\) | 1.75 |
\(k_{{zz}}^{{26}}\) | 1.64 | \(k_{{zz}}^{{26}}\) | 0.51 | \(k_{{zz}}^{{26}}\) | 0.51 | \(k_{{zz}}^{{26}}\) | 2.13 | | |
Damping parameters (Nsm-1, ×103) |
\(c_{{xx}}^{1}\) | 1.21 | \(c_{{xx}}^{2}\) | 1.24 | \(c_{{xx}}^{3}\) | 1.06 | \(c_{{xx}}^{4}\) | 1.44 | \(c_{{xx}}^{5}\) | 1.13 |
\(c_{{xz}}^{1}\) | 0.45 | \(c_{{xz}}^{2}\) | 0.26 | \(c_{{xz}}^{3}\) | 0.48 | \(c_{{xz}}^{4}\) | 0.67 | \(c_{{xz}}^{5}\) | 0.72 |
\(c_{{zx}}^{1}\) | 0.45 | \(c_{{zx}}^{2}\) | 0.26 | \(c_{{zx}}^{3}\) | 0.48 | \(c_{{zx}}^{4}\) | 0.67 | \(c_{{zx}}^{5}\) | 0.72 |
\(c_{{zz}}^{1}\) | 2.06 | \(c_{{zz}}^{2}\) | 1.86 | \(c_{{zz}}^{3}\) | 1.26 | \(c_{{zz}}^{4}\) | 2.19 | \(c_{{zz}}^{5}\) | 1.84 |
\(c_{{xx}}^{6}\) | 1.34 | \(c_{{xx}}^{7}\) | 1.46 | \(c_{{xx}}^{8}\) | 1.13 | \(c_{{xx}}^{9}\) | 1.34 | \(c_{{xx}}^{{10}}\) | 1.46 |
\(c_{{xz}}^{6}\) | 0.57 | \(c_{{xz}}^{7}\) | 0.61 | \(c_{{xz}}^{8}\) | 0.72 | \(c_{{xz}}^{9}\) | 0.57 | \(c_{{xz}}^{{10}}\) | 0.61 |
\(c_{{zx}}^{6}\) | 0.57 | \(c_{{zx}}^{7}\) | 0.61 | \(c_{{zx}}^{8}\) | 0.72 | \(c_{{zx}}^{9}\) | 0.57 | \(c_{{zx}}^{{10}}\) | 0.61 |
\(c_{{zz}}^{6}\) | 1.94 | \(c_{{zz}}^{7}\) | 2.01 | \(c_{{zz}}^{8}\) | 1.84 | \(c_{{zz}}^{9}\) | 1.94 | \(c_{{zz}}^{{10}}\) | 2.01 |
\(c_{{xx}}^{{11}}\) | 1.89 | \(c_{{xx}}^{{12}}\) | 2.26 | \(c_{{xx}}^{{13}}\) | 1.17 | \(c_{{xx}}^{{14}}\) | 1.24 | \(c_{{xx}}^{{15}}\) | 1.17 |
\(c_{{xz}}^{{11}}\) | 1.17 | \(c_{{xz}}^{{12}}\) | 1.54 | \(c_{{xz}}^{{13}}\) | 0.84 | \(c_{{xz}}^{{14}}\) | 0.61 | \(c_{{xz}}^{{15}}\) | 0.84 |
\(c_{{zx}}^{{11}}\) | 1.17 | \(c_{{zx}}^{{12}}\) | 1.54 | \(c_{{zx}}^{{13}}\) | 0.84 | \(c_{{zx}}^{{14}}\) | 0.61 | \(c_{{zx}}^{{15}}\) | 0.84 |
\(c_{{zz}}^{{11}}\) | 2.28 | \(c_{{zz}}^{{12}}\) | 2.61 | \(c_{{zz}}^{{13}}\) | 2.30 | \(c_{{zz}}^{{14}}\) | 2.13 | \(c_{{zz}}^{{15}}\) | 2.30 |
\(c_{{xx}}^{{16}}\) | 1.24 | \(c_{{xx}}^{{17}}\) | 1.15 | \(c_{{xx}}^{{18}}\) | 1.26 | \(c_{{xx}}^{{19}}\) | 1.37 | \(c_{{xx}}^{{20}}\) | 0.72 |
\(c_{{xz}}^{{16}}\) | 0.61 | \(c_{{xz}}^{{17}}\) | 0.65 | \(c_{{xz}}^{{18}}\) | 0.51 | \(c_{{xz}}^{{19}}\) | 0.65 | \(c_{{xz}}^{{20}}\) | 0.26 |
\(c_{{zx}}^{{16}}\) | 0.61 | \(c_{{zx}}^{{17}}\) | 0.65 | \(c_{{zx}}^{{18}}\) | 0.51 | \(c_{{zx}}^{{19}}\) | 0.65 | \(c_{{zx}}^{{20}}\) | 0.26 |
\(c_{{zz}}^{{16}}\) | 2.13 | \(c_{{zz}}^{{17}}\) | 2.14 | \(c_{{zz}}^{{18}}\) | 1.96 | \(c_{{zx}}^{{19}}\) | 2.12 | \(c_{{zx}}^{{20}}\) | 1.51 |
\(c_{{xx}}^{{21}}\) | 1.15 | \(c_{{xx}}^{{22}}\) | 1.15 | \(c_{{xx}}^{{23}}\) | 1.26 | \(c_{{xx}}^{{24}}\) | 1.37 | \(c_{{xx}}^{{25}}\) | 0.72 |
\(c_{{xz}}^{{21}}\) | 0.64 | \(c_{{xz}}^{{22}}\) | 0.65 | \(c_{{xz}}^{{23}}\) | 0.51 | \(c_{{xz}}^{{24}}\) | 0.65 | \(c_{{xz}}^{{25}}\) | 0.26 |
\(c_{{zx}}^{{21}}\) | 0.64 | \(c_{{zx}}^{{22}}\) | 0.65 | \(c_{{zx}}^{{23}}\) | 0.51 | \(c_{{zx}}^{{24}}\) | 0.65 | \(c_{{zx}}^{{25}}\) | 0.26 |
\(c_{{zz}}^{{21}}\) | 2.14 | \(c_{{zz}}^{{22}}\) | 2.14 | \(c_{{zz}}^{{23}}\) | 1.96 | \(c_{{zz}}^{{24}}\) | 2.12 | \(c_{{zz}}^{{25}}\) | 1.51 |
\(c_{{zz}}^{{26}}\) | 1.15 | \(c_{{zz}}^{{26}}\) | 0.64 | \(c_{{zz}}^{{26}}\) | 0.64 | \(c_{{zz}}^{{26}}\) | 2.14 | | |
2.3. Biodynamic responses
The biodynamic responses (magnitude & phase) of STHT and AM are depicted in Fig. 4. For automotive passengers, (Judic et al. 1993) proposed comfortable backrest inclination angles as 180, 210 and 240. In the present article, along with these three angles, the vertical backrest (θ = 00) is added to the numerical analysis and compared with the experimental biodynamic responses (Wang et al., 2008). Figure 4 represents the good agreement between the experimental response and the proposed model with 210 backrest angle both in magnitude and phase. Some deviation in the responses may observe particularly at high frequencies (> 12 Hz). From Fig. 4 it may notice that both STHT and AM responses are deviating more at vertical backrest (θ = 00) conditions as compared to inclined backrest conditions. The obtained overall goodness of fit (GOF) value is (ε = 95.10%).