Lumped Model of a Human Body

Several mass–spring–damper models have been developed to study the response of a human body. In such models, the lumped elements represent the mass of different body parts, and stiffness and damping properties of various tissues.

In this example, a lumped model of a human body having five degrees of freedom is analyzed. The model includes the shoe-ground interaction with the human body. The , , and nodes of the Lumped Mechanical System interface are used to model the body including shoe and ground. First, an eigenfrequency study is performed to find out the natural frequencies of the system and then a frequency response analysis is performed to compute the system response for a specified base excitation. The data for this model is taken from Ref. 1.

Model Definition

Figure 1: The lumped 5-dof model of a human body including shoe-ground interaction. The five degrees of freedom of the system as well as the node numbers of the systems are also shown.

The lumped model of the human body is shown in Figure 1. This model has three main components:

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Human body (4 dofs: u4, u2, u3, u4)

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Shoe sole (1 dof: us)

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Ground (0 dof)

Human body modeling

The four-body model, also called LN model, is one of most commonly used lumped representation of a human body. This model consists of four masses (m1, m2, m3, m4), five springs (k1, k2, k3, k4, k5), and three dampers (c1, c2, c4). In this model, the entire human body is divided into four parts:

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Lower rigid mass (m1)

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Lower wobbling mass (m2)

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Upper rigid mass (m3)

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Upper wobbling mass (m4)

Here the wobbling mass includes the mass of all nonrigid parts such as muscles, skin, blood vessels, and so on.

Shoe Modeling

A shoe is modeled as a one-body model having mass (ms), spring (ks), and damper (cs) elements. In practice the force between the shoe and foot is a nonlinear function of spring deformation. In this model however it is assumed to be a linear function.

Ground Modeling

The ground is modeled with a spring (kg) element representing the stiffness of the soil. Three different values of ground stiffness is used in this model to compare the response of the system:

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Soft soil (99 kN/m)

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Hard soil (359 kN/m)

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Very hard soil (880 kN/m)

For the frequency response analysis, a base excitation (ub) is prescribed at the ground.

Vibration transmissibility

A vertical displacement transmissibility is defined as the ratio of displacement at one of the masses to the base excitation. As an example, the displacement transmissibility at the second mass can be defined as:

The transmissibility is computed in the frequency response analysis and can be plotted as a function of frequency.

Model parameters

The parameters used in the model are given in the table below:

Table 1: Model parameters
Description	name	Expression
Mass 1	m1	6.15 kg
Mass 2	m2	6 kg
Mass 3	m3	12.58 kg
Mass 4	m4	50.34 kg
Spring constant, spring 1	k1	6 kN/m
Spring constant, spring 2	k2	6 kN/m
Spring constant, spring 3	k3	10 kN/m
Spring constant, spring 4	k4	10 kN/m
Spring constant, spring 5	k5	18 kN/m
Damping coefficient, damper 1	c1	0.3 kN-s/m
Damping coefficient, damper 2	c2	0.65 kN-s/m
Damping coefficient, damper 4	c4	1.9 kN-s/m
Mass, shoe sole	ms	0.3 kg
Spring constant, shoe sole	ks	403 kN/m
Damping coefficient, shoe sole	cs	2170 kN-s/m
Spring constant, ground	kg	880 kN/m
Base excitation	ub	10 mm

Results and Discussion

The model considered here has five degrees of freedom and hence in total it can predict five natural frequencies. The first two natural frequencies of the system are given in Table 2:

Table 2: Eigenfrequencies
Eigenfrequency	value
First	1.765+0.611i Hz
Second	55.4+11.024i Hz

Figure 2 and Figure 3 show the displacement amplitude and phase of various masses at different frequencies. It can be seen that some of the masses have higher displacement near the two natural frequencies computed in the eigenfrequency analysis. It can also be noticed that the lower body masses (m1, m2) have comparatively higher displacement at the second natural frequency whereas the upper body masses (m3, m4) have higher displacement at the first natural frequency.

Figure 2: Variation of displacement amplitude with frequency.

Figure 3: Variation of displacement phases with frequency.

Figure 4: Variation of amplitude of various spring forces with frequency.

Figure 4 and Figure 5 show the amplitude and phase of various spring forces at different frequencies. It can be noticed that amplitude of the spring forces are much higher at higher frequencies compared to lower frequencies.

Figure 6 shows the frequency spectrum of the transmissibility of the second mass. The plot compares the transmissibility for three different soils having different hardness values. It is seen that the first peak corresponding to the first natural frequency is almost unaffected. The second peak, however, shifts toward lower frequencies as the soil stiffness is reduced.