The figure above shows a model tor a car suspension with an ideal actuator. The

Question

The figure above shows a model tor a car suspension with an ideal actuator. The tire is modeled as a combination of a spring K_1 and viscous damper B_t. A spring K_s is connected in parallel with the actuator between the axle and the car body. To minimize the energy requirements the average actuator force f_a(t) is ideally maintained at zero as shown. The actuator between the axle the car body To minimize the energy requirements. The average actuator force f_a(t) is ideally maintained at zero as shown. The actuator force f_a(t) is proportional to the control current i(t). Note that the relative velocity and displacement of each end of the actuator mechanism are dependent on the interaction of the applied force f_a(t) with the other system forces. In the figure x_r(t) is the road displacement v_r(t) is the rate of change of x_r(i), x_r(t) is the displacement of the wheel and axle, x_0(t) is the displacement of the body. Write the equation(s) of motion for the system in time domain and in s-domain.

Explanation / Answer

solution:

1)here are three variables so we have to adjust variable xw and xr as single variable by inducing relativity between them and now we will have two variables and two equations of motion ,one for each variable

z=xw-xr

z'=xw'-xr'

z''=xw''-xr''

and xb.

2)here equation of motion for mass mw as

mw*z''(t)+(Kt+Ks)*z(t)-Ks*xb(t)+Bt*z'(t)+Fa(t)=0

3)where equation of motion for mass mb as

mb*xb''(t)+Ks*xb(t)-Ks*z(t)=Fa(t)

4)in this way two equation of motion in time domain as

mw*z''(t)+(Kt+Ks)*z(t)-Ks*xb(t)+Bt*z'(t)+Fa(t)=0

mb*xb''(t)+Ks*xb(t)-Ks*z(t)=Fa(t)

5)above equation can be transform to s domain by applying laplace transform to both equations assuming zero initial conditions as follows

[mwS^2+Bt*S+(Kt+Ks)]Z(s)-Ks*xb(s)+Fa(s)=0

and second equation as

[mb*S^2+Ks]xb(s)-Ks*Z(s)=Fa(s)

where

Z(s)=(xw(s)-xr(s))

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