Modeling and Optimization of Passive Seat Suspension

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Figure 2. Dynamic model of the seat (case B)

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Coordinate u represents the specified kinematic excitation of the seat, excitated from the working machine cabin floor, coordinate x (the Figure 1) and x₂ (the Figure 2) define the position of the seat squab, x₁ defines the position of the point of connection between the damper c and the additional spring k_t.

Equation of motion for the case A (the Figure 1) can be written as follows

(1)

and for the case B

(2)

2.2. Dynamic and Mathematical Model of the Seat with the Dynamic Vibration Absorber (Case C)

The dynamic model of the seat with the dynamic vibration absorber is schematically illustrated in the Figure 3. The dynamic vibration absorber is of mass m₂, stiffness of the elastic beam (placed under the seat squab) is k₃. The meaning of the other parameters is obvious.

Figure 3. Dynamic model of the seat with the dynamic absorber (case C)

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Equations of motion for the case C can be written in the form

(3)

2.3. Kinematic Excitation of the Seat

Experimentally obtained excitation will be used. It was measured experimentally inside the driver’s cabin (on its floor under the driver’s seat in vertical direction) in the bucket-wheel excavator Schrs 1320. For the predesign of the parameters of the dynamic vibration absorber it is beneficial to do the frequency analysis of the seat excitation for determination of the interval comprising the dominant frequencies.

The displacement time response of the excitation is shown in the Figure 4. In the Figure 5 its frequency spectrum is shown.

Figure 4. Displacement time response in the Schrs 1320 cabin

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Figure 5. Frequency spectrum of excitation

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From the frequency spectrum flows that the most distinguished frequency components lie in the interval from 1 to 3 Hz, but the dominant component, which is assigned by the peak, is located at approximately 0.5 Hz.

3. Optimization of the Seat Suspension Parameters

In the chapter the optimization problem is formulated and optimization results are given for all three suspension systems A, B and C.

3.1. Formulation of the Optimization Problem

For all cases of suspension A, B and C the objective function will be assumed the same

(4)

It is a multiobjective optimization, where w is the weighting coefficient (gaining values from 0 to 1), the first component is characterized by a_ef which defines the effective acceleration of the seat squab ( in the case A, or in both cases B and C) and x_rel,ef= x – u in the case A, or x_rel,ef= x₂– u in the cases B a C, define the relative displacement effective value. For this quantities the following equations hold

(5)

and

(6)

Utilization of optimization methods is needed, because the minimization of a_ef usually leads to its decreasing (increase in comfort), but also increasing of x_rel,ef(decreasing the ability of clear sightedness of the driver) and vice versa.

Both effective values mentioned above are in equation (4) devided by their nominal values (defined by the mean values of the optimization parameters in their search intervals), because values of a_ef and x_rel,efare not commensurable, which could cause problems in the numerical optimization.

In the cases A and B the optimization paramaters (optimization variables) are the spring stiffness k and the coefficient of damping c. Their search intervals are: (N/m), (N.s/m). The value of the coeficient of stiffness k₂ in the case B is 76800 N/m ^[3].

3.2. Optimization Results

For optimization the Matlab Optimization Toolbox was used ^[15]. After the numerical optimization it could be possible to draw in the coordinate system composed of axes a_ef and x_rel,efa curve that would correspond to values of the weighting coefficient w within the interval from 0 to 1. It is clear that if w = 1 we get a so called “soft” suspension of the seat, that prefers comfort of the driver, value w = 0 represents a so called “hard” suspension, which prefers the ability of clear sightedness, and value w = 0.5 a so called “medium” suspension with a balanced emphases on both components of the objective function under consideration.

For the case A with the value of w = 1 were obtained the following optimum values of optimization parameters: k_op = 3300 N/m, c_op = 500 N.s/m. The corresponding value of the effective acceleration of the seat is a_ef= 0.1598 m/s² and the value of the effective relative displacement is x_rel,ef= 0.0018 m. The relative displacementand accelerationtime responses of the seat squab are shown in the Figure 6 and Figure 7.

Figure 6. Time response of the seat relative displacement for the case A

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Figure 7. Time response of the seat acceleration

for the case A

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For the case B the optimal “soft” suspension (w = 1) is characterized again by the following values: k_op = 3300 N/m, c_op = 500 N.s/m. The value of the seat effective acceleration is a_ef = 0.1710 m/s² and the value of the effective relative displacement is x_rel,ef= 0.0019 m. The relative displacement and acceleration time responses of the seat are shown in the Figure 8 and Figure 9.

From the comparison of the values of the effective acceleration a_ef for the cases A and B flows, that there is an increase in the value of the effective acceleration by 7%. The increase in the value of x_rel,efis negligible. It is obvious, that the influence of the additional spring serially ordered with the viscous damper is not significant, see Figure 6 and Figure 8 and also Figure 7 and Figure 9. On the other side the 7% decrease in the value of the effective acceleration is not negligible.

In the case C with the dynamic vibration absorber optimization was realized in such a way, that the parameters taken from the “soft” suspension (the case B) were considered, and for the selected weight of the dynamic vibration absorber m₂= 6 kg the minimum value of a_ef for the case C was obtained by 1–dimensional optimization, where the only optimization variable was the stiffness coefficient k₃. For the optimum value of this coefficient k_3op = 1760.52 N/m there was determined the value of a_ef = 0.142780 m/s² and x_rel,ef= 0.0019 m. Comparing values of the effective acceleration in the cases B and C it is possible to allege the decrease in the value of the effective acceleration by 16.5 %. The value of the effective relative displacement was unchanged.

The Figure 10 shows the time responses of the seat relative displacement and the Figure 11 the time responses of the seat acceleration of the seat squab for the cases B and C.

Figure 8. Time response of the seat relative displacement for the case B

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Figure 9. Time response of the seat acceleration

for the case B

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Figure 10. Time responses of the seat relative displacement for the cases B and C

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Figure 11. Time responses of the seat acceleration

for the cases B and C

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