The human body is a complex dynamic system which has mechanical properties that change from person to another and from time to time. Many mathematical models have been developed on the basis of diverse field of measurements to describe the biodynamic responses of human beings [14] .

Based on different modeling methods, the human body model can be composed of lumped-parameter models. The lumped-parameter model is simple system consists of concentrated mass connected internally with springs and dampers. These models are simple construction and easier to deal with for analysis, mathematically solved and simulation.

The model used in this work is linear model as shown in Figure 1. The values of the parameters are specific constant values and are summarized in Table 1 and Table 2.

Linear and bond graphs are one of the modeling techniques used in order to find mathematical equations for mechanical skeleton. The whole system’s structure represented in linear and bond graphs are shown in Figure 2 and Figure 3, respectively.

The proposed seated-human body model consists of 12 degree of freedoms. In order to find the mathematical equations for each segment, free body diagram should be drawn and then Newton’s 2^nd law should be applied.

The equations of a general n-degree-of-freedom system are divided into n equations of the following form:

Rewrite the above equation, you will obtain:

For mass 1 (Seat)

For mass 2 (Pelvis)

For mass 3 (Abdomen)

For mass 4 (Lumbar spine)

For mass 5 (Diaphragm)

For mass 6 (Thoracic spine)

For mass 7 (Thorax)

For mass 8 (Cervical spine)

For mass 9 (Torso)

For mass 10 (Head)

For mass 11 (Upper Arm)

For mass 12 (Lower Arm)

The first simulation model is built to include all the mass segments of the human body together as shown in Figure 4.

The second simulation model of the human body is built by drawing the 2^nd ODE for each mass of the body segments separately using Simulink/MATLAB as shown in Figures 5-16.

On this work we only care about the head and lumber spine segments as they are taking the most damages. In order to find their transfer function, MATLAB’s control and estimation manager (CEM) toolbox is used. This is done by using the State Space equation and the Bode plot for each mass of the human body as shown in Figures 17-23 and as follows.

1) Select the inputs of the system.

2) Select the outputs of the system.

3) Open the control and estimation tools manager and linearize the system in order to get the transfer function, state space equations, and other characteristics of the system.

4) Obtaining the Bode plot for the seat, head and lumbar spine segments.

5) Obtaining the transfer functions.

Equations of motion of a multi-degree-of-freedom system under external forces are given by the following equation:

where [m], [B], [K] and [f] for our system are explained as the following;

Vertical input force is any input force applied on the system such as sinusoidal wave signal, step-function signal, ramp-function signal, impulse-function signal, etc. All the variables in the above equations are numerically defined and set in Table 1 and Table 2.

The State-Space method is useful for modelling multi-DOF systems. It could be used to represent the entire states of the system at any given time.

General form representation:

where A, B, C and D are the system, input, output and feed-forward matrices, and x, y and u are the state, output and input vectors.

For human body model in previous sections, the equations could be rewritten in state-space representation form in order to model the whole system. It is assumed that:

According to that, equations of the human body are as the following:

There are several methods to solve the state-space equations. In this work, the Laplace inverse method has been used in order to get the transition matrix

Then,

lsim(sys, u, t) function produces a plot of the time response of the dynamic system model sys to the input time history t, u. The vector t specifies the time samples for the simulation (in system time units, specified in the Time Unit property of sys), and consists of regularly spaced time samples.

Figure 20 represents the transfer function (gain and phase shift) of the human segments

(seat, head and lumber spin) subjected to vertical sinusoidal force with amplitude of 15 N. Head and lumbar spine segments are most segments that suffer from the vibration.

The gain represents the amplitude ration between the output and input inertia. Positive gain indicates that the relationship will have the same phase. Conversely, negative gain is reflected by the opposite phases.

Resonance can cause large oscillation when the frequency of excitation coincides with the natural frequency of the relevant subject. Harmful mechanical effect of vibration occurs because of induced strain on different tissues, caused by motion and deformation within the body.

The mechanical energy due to vibration is absorbed by tissues and organs, when the vibrations are attenuated in the relevant body segment. Consequently, vibration leads to muscle contractions (Voluntary and involuntary) which can cause local muscle fatigue, particularly when the body vibrated at the resonant frequency level.

Two simulation models have been designed to represent the human body subjected to vertical vibration by using mechanical parameters obtained from [15] . Simulink/ MATLAB software has been used to develop and implement the models obtained using the model analysis and state-space methods. The results from these two approaches confirm that the head and lumbar suffer the most from vibration in terms of forces. In addition, it is found that the rest of the organs may become strained leading to pain and fatigue, depending on the magnitude of the vibrations.