Consider a quarter-car model shown in Figure 5.123, where \(m_{1}\) is the mass of the seats including passengers, \(m_{2}\) is the mass of one-fourth of the car body, and \(m_{3}\) is the mass of the wheel-tire-axle assembly. The spring \(k_{1}\) represents the elasticity of the seat supports, \(k_{2}\) represents the elasticity of the suspension, and \(k_{3}\) represents the elasticity of the tire. \(z(t)\) is the displacement input due to the surface of the road. Draw the necessary free-body diagrams and derive the differential equations of motion. Write the differential equations of motion in the second-order matrix form.

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FIGURE 5.123 Problem 5. k3 m m2 m3 b x2 b x3