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Mechanical vibrations cause forces that affect human bodies. One of the most common positions for human bodies is the seated position. In this work, mathematical models of the seated human body are investigated and simulated in a Simulink/ MATLAB environment. In addition, segments of the human body are studied and models are developed and built by using Simulink/MATLAB. As part of this work, model analysis and state-space methods are used in order to check and validate the results obtained from the simulations. Two types of forces are used to test the whole seated human body under low frequency citation. The first is a sinusoidal wave signal based on literature, and the second is an impulse function. The effects of mechanical vibration on the head and lumbar are studied as these parts of the human body are usually the most effected areas. Kinematic states of the head segment and lumbar are considered. The characteristics of the vibration response on the two segments are also obtained. In addition to the me-chanical vibrations study, this paper is a resource for the development and implementation of models in the Simulink/MATLAB environment.

Our bodies are highly sensitive to the natural activities around us. One of the most important phenomena that always affect all human body and its organisms is the vibration. Humans are sensitive to vibrations under low frequency, and the problem becomes worse when they expose to vibrations continuously [

Experimental evidence has shown that a human body can be injured by vibrations. It was reported that about 12 million workers in the USA were affected by vibrations [

Most seated workers are exposure to vibrations effects; these effects are very harmful and in some cases lead to permanent pains, i.e. back pain. Vehicle drivers are exposed to this type of pain because they are driving their vehicle and interacting with vibration more than other people.

In order to study changes on the human body professionally, we should build a model that mimics the dynamics of the human body. For modeling the human body, it can be done by using mathematical equations derived by using one of the modeling methods, and then you can either find the analytical solutions in order to define the frequencies that have effects on the body, or use simulation software in order to build the mechanical structure of the human body and then examine it under different frequencies. In this work, several modeling methods are used to build the mathematical model of the human body in order to study the vertical vibrations on the seated human body. Then the results of these methods are compared to the results obtained from the simulated mechanical system of the human body by using MATLAB/Simulink Software.

The biodynamic study of the human body return back to 1918, when Hamilton define the vibration white finger syndrome as a result of the vibrating hand tools [

Zheng, Qiu and Griffin [

There are many factors could affect the response of human bodies to the vibrations and they differ from person to another. The experimental tests in [

Nawayseh and Griffin in [

There are many biodynamic models have been built to mimic the movement of the movement of human body. According to [

The objective of this work is to build two models of the human body and then study the effect of the vertical vibration on the head segment and lumbar spine using MATLAB software.

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 [

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

Seated human body segments | Mass in kg |
---|---|

Seat | 12 |

Lower arm | 5.297 |

Upper arm | 5.47 |

Torso | 32.697 |

Thoracic spine | 4.806 |

Thorax | 13.626 |

Cervical spine | 1.084 |

Head | 5.445 |

Lumbar spine | 2.002 |

Diaphragm | 0.454 |

Abdomen | 5.906 |

Pelvis | 27.174 |

Item no. | Stiffness (kg∙fm^{−1}) | Damping constant (kg∙fm^{−1}) |
---|---|---|

a | 89.41 | 29.8 |

b | 5364 | 365.1 |

c | 8941 | 29.8 |

d | 5364 | 365.1 |

e | 8941 | 29.8 |

f | 5364 | 365.1 |

g | 5364 | 365.1 |

h | 5364 | 365.1 |

i | 5364 | 365.1 |

j | 6885 | 365.1 |

k | 6885 | 365.1 |

l | 2550 | 37.8 |

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

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)

There are several methods to solve the n-DOF system; n equations of second order differential equation for each DOF. Solution could be achieved analytically or numerically by using one of the engineering Software. In this work, they will be solved using three techniques:

1) MATLAB/Simulink: 2 simulation models are used in order to validate results

2) Modal Analysis; this method was explained in previous section

3) State space Representation

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

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

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.

The velocity and the position for each of the head and lumber spine segments of the human body with respect to the sinusoidal wave input signal are shown in Figures 24-27, respectively.

The velocity and the position for each of the head and lumber spine segments of the human body with respect to the impulse-function input signal are shown in Figures 28-31, respectively.

The acceleration, velocity and position for both of the head and lumber spine segments of the human body with respect to the sinusoidal wave input signal are shown in Figures 32-37, respectively.

The acceleration, velocity and the position for each of the head and lumber spine segments of the human body with respect to the impulse-function input signal are shown in Figures 38-43, respectively.

The velocity and position for both of the head and lumber spine segments of the human body with respect to the sinusoidal wave input signal are shown in Figures 44-47, respectively.

The velocity and the position for each of the head and lumber spine segments of the human body with respect to the impulse-function input signal are shown in Figures 48-51, respectively.

(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

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 [

AlShabi, M., Araydah, W., ElShatarat, H., Othman, M., Younis, M.B. and Gadsden, S.A. (2016) Effect of Mechanical Vibrations on Human Body. World Journal of Mechanics, 6, 273- 304. http://dx.doi.org/10.4236/wjm.2016.69022