The following assumptions/limitations are considered in this study:
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a.
Since the most pronounced and dominant direction of vibration due to road geometry (humps, uneven surfaces, curved roads, etc.) is vertical, only the influence of different biomechanical reactions and vertical vibrations on different segments will be taken into account. It will also be recognized that any suspension system a car has is meant to compensate for this vertical vibration.
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b.
Additionally, most standards such as the International Organization for Standardization (ISO-2631), British Standards (BS-6841), and European Standards EN-12,299 recognize that the human body is more sensitive and susceptible to vertical vibrations than the front. I am. Longitudinal direction.
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c.
Additionally, simultaneous multidirectional estimation has practical limitations in the number of sensors and measurement locations. As a result, more complex measurement equipment, data acquisition systems, and data analysis techniques are required.
With these assumptions, the schematic diagram for 32 looks like this:Degree of freedom A biomechanical model of a seated human is shown in Figure 2. The human body is virtually segmented, and each segment is connected to adjacent segments through direct and interconnected stiffness and damping properties. Because the human body is symmetrical about the sagittal plane. Therefore, the modeling assumes that the biomechanical properties (mass, stiffness, damping) are symmetrical with respect to the sagittal plane. Additionally, cross-coupled parameters are assumed to be symmetric. The model is divided into various segments (I= 1–16) represents the head anatomy (meters1), chest (meters2),abdomen(meters3),pelvis(metersFour), upper arm (metersFive and meters8), forearm (meters6 and meters9),hand(meters7 and metersTen), thighs (meters11 and meters14), legs (meters12 and meters15), feet (meters13 and meters16) (see Figure 2). To avoid presentation complexity, only the spring is shown in Figure 2, but the damper could also be shown alongside the spring. Direct and cross-coupled stiffness (Kij) and damping (Cij) Quality exists for each spring and damper (I= X ,j =z; where ‘X’and ‘z’indicate the longitudinal and vertical directions, respectively).
governing equation
By applying Newton’s second law to each section, the governing equations of the system are formulated (see Figure 2c). For simplicity and completeness, the governing equations for are: IThe th segment is connected to ‘ I+ The first segment appears as follows:
$$m_{i} \ddot{x}_{i} + c_{xx}^{i} (\dot{x}_{i} – \dot{x}_{i + 1} ) + c_{ xz}^{i} (\dot{z}_{i} – \dot{z}_{i + 1} ) + k_{xx}^{i} (x_{i} – x_{i + 1} ) + k_{xz}^{i} (z_{i} – z_{i + 1} ) = f_{{x_{i} }}$$
(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)
In the above equation, (1, 2) is expressed in matrix form as follows:
$$[M_{i} ]\left\{ {\ddot{\chi }_{i} } \right\} + [C_{i} ]\left\{ {\dot{\chi }_{i} – \dot{\chi }_{i + 1} } \right\} + [K_{i} ]\left\{ {\chi_{i} – \chi_{i + 1} } \right\} = \{ f_{i} \}$$
(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 creating the equation of motion (EOM) for each segment, the global EOM can be expressed as:
$$[M]_{16 \times 16} \left\{ {\ddot{\chi }} \right\}_{16 \times 1} + [C]\left\{ {\dot{\chi }} \right\}_{16 \times 1} + [K]\left\{ \chi \right\}_{16 \times 1} = \{ f\}_{16 \times 1}$$
(Four)
now by replacing χ=χejωt and f = ironjωt In Equation (4), the time series equation can be converted to a frequency series as follows.
$$( – \omega^{2} M + j\omega C + K)_{16 \times 16} {\upchi }_{16 \times 1} = F_{16 \times 1}$$
(Five)
After solving equation (5), the biomechanical responses (STHT and AM) can be obtained as follows.
$$STHT = \frac{{Z_{1} \sin \theta_{1} }}{{Z_{0} }}$$
(6)
$$AM = \frac{{F_{4} }}{{a_{4} }}$$
(7)
where Z1 and Z0 are the vertical displacements of the head and seat (input), respectively. beFour and FFour Acceleration and force at the point of contact between the person and the seat.In other words,Each pelvis.θ1 This is the angle of the backrest. For numerical simulation, its value is obtained as follows.θ1= 24°.
STHT is a dimensionless quantity. It helps researchers and designers investigate the amount and frequency of vibrations transmitted to the human body through vibrating media (such as seats and floors). On the other hand, AM provides information about human populations in dynamic environments. For rigid bodies, AM is the mass of the system in the static state. However, under dynamic conditions at resonance, the apparent mass can be increased by a factor of four. 16.
Model parameter estimation using Firefly algorithm
In this section, the biomechanical parameters of the developed model are optimized with the help of Firefly algorithm (FA). Parameters are optimized by reducing the sum-of-squares error between the experimental and analytical responses. The objective function includes both the STHT and AM amplitude and phase responses.
$${\text{Minimize }}(O_{f} ) = \mathop \Sigma \limits_{i = 1}^{p} (\alpha_{1} .\lambda_{1} + \alpha_{2} . \lambda_{2} + \alpha_{3} .\lambda_{3} + \alpha_{4} .\lambda_{4} )$$
(8)
here
$$\begin{align} & \lambda_{1} = [STHT_{E} (f_{i} ) – STHT_{A} (f_{i} )]^{2}_{Mag} {,}\quad \lambda_{2} = [STHT_{E} (f_{i} ) – STHT_{A} (f_{i} )]^{2}_{Pha} \\ & \lambda_{3} = [AM_{E} (f_{i} ) – AM_{A} (f_{i} )]^{2}_{Mag} {,}\quad \, \lambda_{4} = [AM_{E} (f_{i} ) – AM_{A} (f_{i} )]^{2}_{Pha} \\ \end{Alignment}$$
Experimental and analytical readings are indicated with a subscript ‘.E ‘ and ‘ a‘, Each. subscript ‘ mug‘ and ‘ fur‘ denote the amplitude and phase responses, respectively.symbol α1, α2,α3 teethαFour indicates the weight function. Both biomechanical responses are assigned the same importance (weight). In other words,(α1= α2=α3 = αFour ). meanwhile ‘p‘ indicates the number of experimental data points. The flowchart (Figure 3) depicts a typical process performed to optimize biomechanical parameters. The following decision variables and constraints are applied in the analysis to minimize the objective function and obtain the optimized parameters of the human body.
Decision variables (see Figure 2):
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(1)
meters1, meters2,… meters16 is the segment mass of the model
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(2)
K1,K2,….K26 is the stiffness matrix between segments
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(3)
C1 ,C2 ,….C26is the attenuation matrix between segments
Here, the direct and interconnected stiffness and damping parameters are included in the stiffness and damping matrices, respectively.
Constraints:
$$\left\{ \begin{gathered} \sum\limits_{i = 1}^{16} {m_{i} = 77.3\;{\text{kg}}} \hfill \\ m_{5} = m_{8} , m_{6} = m_{9} , m_{7} = m_{10} \hfill \\ m_{11} = m_{14} , m_{12} = m_{15} , m_{ 13} = m_{16} \hfill \\ k_{xz} = k_{zx} ,c_{xz} = c_{zx} \hfill \\ 100\;{\text{Nm}}^{ – 1} [k_{ii} ] < 300000\;{\text{N}}\;{\text{m}}^{ - 1} \hfill \\ 100\;{\text{Nsm}}^{ - 1} [c_{ii} ] < 300000\;{\text{N}}\;{\text{s}}\;{\text{m}}^{ - 1} \hfill \\ \end{gathered} \right\}$$
(9)
metersI,kijandcijare the average weight, lower and upper stiffness bounds, and damping coefficient of the proposed model, respectively. As far as stiffness values are concerned, the limit values are taken from the literature.25,26. They performed compression tests on different segments and drew load-deflection curves to obtain the stiffness of different segments. Meanwhile, the attenuation values were obtained from the National Institutes of Health (USA).27. They performed free vibration tests to obtain damping values for each segment. During the optimization process, the sum of squared errors (SSE) is calculated to obtain the desired accuracy of the optimized parameters. The SSE value was set to 0.000001 to stop the iteration. Other decision parameters chosen were: total number of variables = 224, group size = 100, and number of replicates = 50. Under the constraints stated in equation (8), model parameters are adjusted in (9) until the desired accuracy is achieved. The experimental data was referred to below.8 Minimize the objective function.in8, the authors conducted an experiment under random vibration conditions on 12 healthy men.Set vertical vibration magnitude to 1ms−2 RMS value in the frequency range 0.5 to 15 Hz. The goodness of fit (GOF) is determined as follows.
$$\varepsilon = 1 – \frac{{\sqrt {\Sigma (\tau_{e} – \tau_{a} )^{2} /(N – 2)} }}{{\Sigma \tau_{e } /N}}$$
(Ten)
Experimental and analytical responses are indicated by “”.τe‘ and ‘ τbe‘ Each. The total number of data points selected for analysis is referred to as “.N’.‘α‘ reflects/imitates a good model that can be used in place of experimental research. A value of 1 means that the analytical model and experimental response are identical. It means “.”α‘ enables the model. Table 1 shows the optimized parameters of the recommended model with the highest GOF value.
biomechanical response
The biomechanical responses (magnitude and phase) of STHT and AM are shown in Figure 4.28 We proposed comfortable backrest inclination angles of 18°, 21°, and 24°. In this article, in addition to these three angles, we will discuss the vertical backrest (θ= 0°) is added to the numerical analysis and compared with the experimental biomechanical response. In Figure 4, the biomechanical response is plotted with and without the inclusion of the thigh, leg, and foot. We also compare these responses with experimental studies to visualize lower limb inclusion effects. The subscripts “ex” and “in” used in the legend of Figure 4 represent lower extremity exclusion and lower extremity inclusion, respectively. From Figure 4, the influence of the lower limbs (thigh, leg, and foot) on both biomechanical responses is observed. The primary resonant frequency of the transmission rate from the seat to the head is approximately 4 Hz when the lower limbs are excluded, and approximately 5 Hz when the lower limbs are included. A similar phenomenon may be observed in the apparent mass response.
Figure 4 shows that the experimental response and the proposed model for a backrest angle of 21° are in good agreement both in magnitude and phase. Some deviations in response may be observed, especially at high frequencies (> 12 Hz). From Figure 4, it may be noticed that both the STHT and AM responses deviate more at the vertical backrest (θ= 0°) with the backrest tilted. The obtained overall goodness of fit (GOF) value is (ε = 95.10%) for the backrest angle.θ1= 24°. Also, the deviations between the experimental and simulated responses at different backrest angles were obtained as 8.14%, 5.62%, 4.68%, and 4.90% for backrest angles of 0°, 18°, 21°, and 24°, respectively. Masu. The deviation is calculated by the following formula: δ = (100 − ε)%, where “δ” is the deviation and “ε” is the goodness of fit.