Optimization of Semi-active Seat Suspension

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Figure 2. Vertical excitation in the cabin of the bucket wheel excavator

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The sky-hook control algorithm will be considered. The task of the damper is to generate the same force, with the same direction and magnitude as the fictitious damper on the seat would generate, the so called sky-hook damper placed between the seat and a point, which location in the vertical direction does not change. In the case that the semi-active damper could not generate the same force as the force developed by the fictitious sky-hook damper () the semi-active damper will be in the “off” state. In the case that the semi-active damper will be capable of generating such a force, its continuously variable coefficient of damping c_sa will be defined from the equality of the sky-hook and semi-active force

(2)

and for the coefficient of damping the following equation holds

(3)

The force in the semi-active damper is

(4)

This force can be realized in the same direction as the force in the sky-hook damper F_sky, only in the case that the direction of the relative velocity between the seat and the working machine floor and the absolute velocity of the seat are of the same direction. The condition can be written in the following form

(5)

In the case that this condition is met, the force in the semi-active damper is given according to equation (4). If this condition is not met, the semi-active damper will be in the “off” state. On the magnitude of the force F_sa will be applied an upper limit, so this damper would not generate higher forces than a real magnetorheological damper can do. In the “off” state of the damper a zero force F_sa will be considered, although in real (for example magnetorheological dampers) this force cannot be a zero force (moreover it is dependent on the relative velocity ). However, in practice this force practically does not affect the numerical results, if an appropriate magnetorheological damper is chosen. This condition is actually one of the criteria for the right choice of the magnetorheological damper. The advantage of the friction semi-active dampers is that they can easy generate the zero force in the “off” state, but in practice the magneto-rheological dampers are more used. These cannot give the zero force in the “off” state.

For the purpose of optimization we will consider the coefficient of the spring stiffness k and also the coefficient of damping of the sky-hook damper c_sky continuously variable in their search intervals, which will be shown in the following parts of the paper.

2.2. Dynamic and Mathematical Models of the Semi-active Suspension with the Dynamic Absorber

The dynamic model of the semi-active suspension with the dynamic vibration absorber is shown in the Figure 3. In this case there is an added mass m₂ connected to the seat (under it) by a spring with stiffness coefficient k₂ compared to the model without the dynamic vibration absorber shown in the Figure 1. In practice the realization of the dynamic vibration absorber is by an elastic beam, fixed to the seat frame, with a possibility to adjust the weight along it, which enables to adjust the parameter of the dynamic vibration absorber (its stiffness).

Mathematical model in this case can be written in the form

(6)

For the semi-active damping force F_sa are valid the same conditions as were described above. The kinematic excitation will be considered with the same time response as it is shown in the Figure 2.

Figure 3. Dynamic model of the seat with the dynamic vibration absorber

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3. Suspension Parameters Optimization

A multi-objective function is considered in the form

(7)

where w is the weighting factor (gaining values from 0 to 1), a_ef is the effective acceleration of the seat described as

(8)

where the acceleration of the seat a(t) is in the case without the dynamic vibration absorber, the Figure 1, or in the case with the dynamic absorber, the Figure 3. The minimization of the effective acceleration a_efleads to the maximization of the comfort of driver seated on the seat.

The second component of the objective function (7) defines the effective value of the relative displacement x_rel= x–u for the case without the dynamic absorber, the Figure 1, and x_rel= x₁–u in the case with the dynamic vibration absorber, see the Figure 3. Similarly to equation (8) the following equation holds for the effective relative displacement

(9)

T in the equations (8) and (9) represents the time of integration. The integration has to be sufficiently long in order to capture the dynamics of the system completely. Both the effective values a_efand x_rel,ef are in the equation (7) devided by their nominal values (defined for the mean values of the optimization variables c_sky and k in their search intervals, because the values of a_efand x_rel,ef are not commensurable. That would disable the numerical optimization.

3.1. Semi-active Suspension without the Dynamic Vibration Absorber

Because of the experimentally obtained kinematic excitation represented by displacement u(t) the optimiza-tion will be realized in the time domain. The Optimization Toolbox of Matlab ^[6] will be used. First it is necessary to transform the equation (1) into a system of the first order ordinary differential equations. If we use the transformation equations

(10)

then the system of the first order differential equations has the form

(11)

where for force F_sa holds (after applying transformation equations (10) and their substitution to the equation (4))

(12)

3.2. Semi-active Suspension with the Dynamic Absorber

For the transformation of the mathematical model (6) into a system of the first order ordinary differential equations we will use transformation equations as follows

(13)

We obtain a system of the four first order differential equations. Substituting equations (13) to equations (6) we aquire the following equations

(14)

4. Optimization Results

Two optimization variables are considered: the spring stiffness coefficient k and the sky-hook damping coefficient c_sky. Their search intervals are as follows:

Table 1 and Table 2 describe the optimization results for the case without the dynamic vibration absorber in dependence on the weighting factor w decreasing from value 1 to value 0.

From Table 2 flows that the values of the effective acceleration a_ef significantly change in dependence on the weighting coefficient w. On the other side the values of the effective relative displacement x_rel,efchange only slightly. This leads to the conclusion that in this case the optimal solution is the one that completely prefers comfort (w = 1). The time responses of the relative displacement x_rel(t) and accerelation of the seat are presented in the Figures 4 and 5.

Table 1. Optimum stiffness and damping coefficients for the case without the dynamic vibration absorber

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Table 2. The effective acceleration and effective relative displacement for the case without the dynamic vibration absorber

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Figure 4. Relative displacement time response x_rel(t) for the case without the vibration absorber

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Figure 5. Acceleration time response

for the case without the vibration absorber

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In the case of the semi-active suspension with the dynamic vibration absorber was according to the optimization results from the case without the dynamic vibration absorber considered only the weighting coefficient w = 1. For the chosen value of the dynamic vibration absorber mass m₂ = 6 kg by the 1-dimensional optimization the stiffness k_2op = 1825.54 N/m of the dynamic vibration absorber spring was obtained. For the corresponding case without the dynamic vibration absorber (w = 1) the following optimum values were used: k_1op = k_op and c_sky,op. This allowed to determine the value of a_ef = 0.054978 m/s² and x_rel,ef= 0.001833 m. Comparing the values of the effective acceleration of the seat in both numerically simulated cases it is possible to allege that the decrease of the effective acceleration value of the seat in the case with the dynamic vibration absorber was about 9.6 %. The value of the effective relative displacement remained practically unchanged. The figure 6 shows the time responses of the seat displacement x₁(t) for both cases (with and without the dynamic vibration absorber) and the Figure 7 shows the time responses of the seat acceleration for both cases.

Figure 6. The seat relative displacement time responses

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Figure 7. The seat acceleration time responses

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