Question 5.5: A Two-Degree-of-Freedom Quarter-Car Model Consider a quarter...
A Two-Degree-of-Freedom Quarter-Car Model
Consider a quarter-car model shown in Figure 5.34a, where m_{1} is the mass of one-fourth of the car body and m_{2} is the mass of the wheel-tire-axle assembly. The spring k_{1} represents the elasticity of the suspension and the spring k_{2} represents the elasticity of the tire. z(t) is the displacement input due to the surface of the road. The actuator force, f, applied between the car body and the wheel-tire-axle assembly, is controlled by feedback and represents the active components of the suspension system.
a. Draw the necessary free-body diagrams and derive the differential equations of motion.
b. Determine the state-space representation. Assume that the displacements of the two masses, x_{1} and x_{2}, are the outputs and the state variables are x_{1}=x_{1}, x_{2}=x_{2}, x_{3}=\dot{x}_{1}, and x_{4}=\dot{x}_{2}.
c. The parameter values are m_{1}=290 \mathrm{~kg}, m_{2}=59 \mathrm{~kg}, b_{1}=1000 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}, k_{1}=16,182 \mathrm{~N} / \mathrm{m}, and k_{2}=19,000 \mathrm{~N} / \mathrm{m}. Use MATLAB commands to define the system in the state-space form and then convert it to the transfer function form. Assume that all the initial conditions are zero.
Our explanations are based on the best information we have, but they may not always be right or fit every situation.
Learn more on how do we answer questions.