Let's break down the solution step by step for the **quarter-car model** as described. --- ### **Given:** - \( M_1 \): Automobile body mass = 2500 kg - \( M_2 \): Wheel and suspension mass = 320 kg - \( K_1 \): Suspension spring constant = 80,000 N/m - \( K_2 \): Tire spring constant = 500,000 N/m - \( B \): Damping constant = 350 N·s/m Let: - \( x_1(t) \): Displacement of automobile body - \( x_2(t) \): Displacement of wheel and suspension - \( x_0(t) \): Road displacement input (excitation) - \( F(t) \): External force (can be set as \( x_0(t) \) for road input) --- ## **Step 1: Write the Equations of Motion** Using Newton's 2nd Law for each mass: ### **For \( M_1 \):** \[ M_1 \ddot{x}_1 = -K_1(x_1 - x_2) - B(\dot{x}_1 - \dot{x}_2) \] --- ### **For \( M_2 \):** \[ M_2 \ddot{x}_2 = K_1(x_1 - x_2) + B(\dot{x}_1 - \dot{x}_2) - K_2(x_2 - x_0) \] --- ## **Step 2: Take Laplace Transforms** Let \( X_1(s) \), \( X_2(s) \), and \( X_0(s) \) be Laplace transforms of \( x_1 \), \( x_2 \), and \( x_0 \). ### **Equation 1:** \[ M_1 s^2 X_1(s) = -K_1 (X_1(s) - X_2(s)) - B (sX_1(s) - sX_2(s)) \] \[ M_1 s^2 X_1(s) + K_1 (X_1(s) - X_2(s)) + B s (X_1(s) - X_2(s)) = 0 \] \[ (M_1 s^2 + K_1 + B s) X_1(s) - (K_1 + B s) X_2(s) = 0 \] --- ### **Equation 2:** \[ M_2 s^2 X_2(s) = K_1 (X_1(s) - X_2(s)) + B (sX_1(s) - sX_2(s)) - K_2 (X_2(s) - X_0(s)) \] \[ M_2 s^2 X_2(s) + K_1 (X_2(s) - X_1(s)) + B s (X_2(s) - X_1(s)) + K_2 (X_2(s) - X_0(s)) = 0 \] \[ (M_2 s^2 + K_1 + K_2 + B s) X_2(s) - (K_1 + B s) X_1(s) - K_2 X_0(s) = 0 \] --- ## **Step 3: Write in Matrix Form** \[ \begin{bmatrix} M_1 s^2 + K_1 + Bs & -(K_1 + Bs) \\ -(K_1 + Bs) & M_2 s^2 + K_1 + K_2 + Bs \end{bmatrix} \begin{bmatrix} X_1(s) \\ X_2(s) \end{bmatrix} = \begin{bmatrix} 0 \\ K_2 X_0(s) \end{bmatrix} \] --- ## **Step 4: Solve for Transfer Functions** Let’s denote: - \( A = M_1 s^2 + K_1 + Bs \) - \( B = -(K_1 + Bs) \) - \( C = M_2 s^2 + K_1 + K_2 + Bs \) So, \[ \begin{bmatrix} A & B \\ B & C \end{bmatrix} \begin{bmatrix} X_1(s) \\ X_2(s) \end{bmatrix} = \begin{bmatrix} 0 \\ K_2 X_0(s) \end{bmatrix} \] From the first equation: \[ A X_1 + B X_2 = 0 \implies X_1 = -\frac{B}{A} X_2 \] Plug into the second equation: \[ B X_1 + C X_2 = K_2 X_0 \] \[ B \left( -\frac{B}{A} X_2 \right) + C X_2 = K_2 X_0 \] \[ - \frac{B^2}{A} X_2 + C X_2 = K_2 X_0 \] \[ \left( C - \frac{B^2}{A} \right) X_2 = K_2 X_0 \] \[ X_2(s) = \frac{K_2 A}{A C - B^2} X_0(s) \] Recall, \( X_1(s) = -\frac{B}{A} X_2(s) \): \[ X_1(s) = -\frac{B}{A} \left( \frac{K_2 A}{A C - B^2} X_0(s) \right) = -\frac{B K_2}{A C - B^2} X_0(s) \] --- ### **So, the transfer functions are:** \[ T_1(s) = \frac{X_1(s)}{X_0(s)} = -\frac{B K_2}{A C - B^2} \] \[ T_2(s) = \frac{X_2(s)}{X_0(s)} = \frac{K_2 A}{A C - B^2} \] Where: - \( A = M_1 s^2 + K_1 + Bs \) - \( B = K_1 + Bs \) - \( C = M_2 s^2 + K_1 + K_2 + Bs \) --- ## **Step 5: Substitute Numerical Values** Given: - \( M_1 = 2500 \) - \( M_2 = 320 \) - \( K_1 = 80,000 \) - \( K_2 = 500,000 \) - \( B = 350 \) So: - \( A = 2500 s^2 + 80,000 + 350s \) - \( B = 80,000 + 350s \) - \( C = 320 s^2 + 80,000 + 500,000 + 350s = 320 s^2 + 580,000 + 350s \) Plug these into the transfer functions: \[ T_1(s) = -\frac{(80,000 + 350s) \cdot 500,000}{[2500 s^2 + 80,000 + 350s][320 s^2 + 580,000 + 350s] - (80,000 + 350s)^2} \] \[ T_2(s) = \frac{500,000 [2500 s^2 + 80,000 + 350s]}{[2500 s^2 + 80,000 + 350s][320 s^2 + 580,000 + 350s] - (80,000 + 350s)^2} \] --- ### **Summary Table** | Transfer Function | General Form | Numerical Form | |-------------------|-------------|---------------| | \( T_1(s) \) | \( -\frac{B K_2}{AC - B^2} \) | \( -\frac{(80,000 + 350s) \cdot 500,000}{[2500 s^2 + 80,000 + 350s][320 s^2 + 580,000 + 350s] - (80,000 + 350s)^2} \) | | \( T_2(s) \) | \( \frac{K_2 A}{AC - B^2} \) | \( \frac{500,000 [2500 s^2 + 80,000 + 350s]}{[2500 s^2 + 80,000 + 350s][320 s^2 + 580,000 + 350s] - (80,000 + 350s)^2} \) | --- ## **Final Answers:** ### **(a) General Transfer Functions:** \[ T_1(s) = \frac{X_1(s)}{X_0(s)} = -\frac{(K_1 + Bs) K_2}{[M_1 s^2 + K_1 + Bs][M_2 s^2 + K_1 + K_2 + Bs] - (K_1 + Bs)^2} \] \[ T_2(s) = \frac{X_2(s)}{X_0(s)} = \frac{K_2 [M_1 s^2 + K_1 + Bs]}{[M_1 s^2 + K_1 + Bs][M_2 s^2 + K_1 + K_2 + Bs] - (K_1 + Bs)^2} \] ### **(b) With Numerical Values:** \[ T_1(s) = -\frac{(80,000 + 350s) \cdot 500,000}{[2500 s^2 + 80,000 + 350s][320 s^2 + 580,000 + 350s] - (80,000 + 350s)^2} \] \[ T_2(s) = \frac{500,000 [2500 s^2 + 80,000 + 350s]}{[2500 s^2 + 80,000 + 350s][320 s^2 + 580,000 + 350s] - (80,000 + 350s)^2} \] --- **Let me know if you need further simplification or step-by-step calculation for specific values!**