Let's solve **part (a): Rotational Mechanical System 1** step-by-step. --- ## 1. **Free-Body Diagram and Torques** We have two rotating inertias \( J_1 \) and \( J_2 \), coupled by springs and dampers, with an applied torque \( T_a \) on \( J_2 \): - \( J_1 \) rotates by \( \theta_1(t) \) - \( J_2 \) rotates by \( \theta_2(t) \) - \( k_1 \), \( b_1 \): spring and damper connected to ground on \( J_1 \) - \( k_2 \), \( b_2 \): spring and damper between \( J_1 \) and \( J_2 \) - \( T_a \): applied to \( J_2 \) --- ### **Torques acting on \( J_1 \):** - **Spring \( k_1 \):** \( -k_1 \theta_1 \) (restoring torque) - **Damper \( b_1 \):** \( -b_1 \dot{\theta}_1 \) (resistive torque) - **Spring \( k_2 \):** \( -k_2 (\theta_1 - \theta_2) \) (opposes relative motion) - **Damper \( b_2 \):** \( -b_2 (\dot{\theta}_1 - \dot{\theta}_2) \) (opposes relative angular speed) #### **Equation for \( J_1 \):** \[ J_1 \ddot{\theta}_1 = -k_1 \theta_1 - b_1 \dot{\theta}_1 - k_2 (\theta_1 - \theta_2) - b_2 (\dot{\theta}_1 - \dot{\theta}_2) \] Simplify: \[ J_1 \ddot{\theta}_1 + (b_1 + b_2)\dot{\theta}_1 + (k_1 + k_2)\theta_1 - b_2 \dot{\theta}_2 - k_2 \theta_2 = \] --- ### **Torques acting on \( J_2 \):** - **Spring \( k_2 \):** \( +k_2 (\theta_1 - \theta_2) \) - **Damper \( b_2 \):** \( +b_2 (\dot{\theta}_1 - \dot{\theta}_2) \) - **Applied torque:** \( +T_a \) #### **Equation for \( J_2 \):** \[ J_2 \ddot{\theta}_2 = k_2 (\theta_1 - \theta_2) + b_2 (\dot{\theta}_1 - \dot{\theta}_2) + T_a \] Simplify: \[ J_2 \ddot{\theta}_2 + b_2 \dot{\theta}_2 + k_2 \theta_2 = b_2 \dot{\theta}_1 + k_2 \theta_1 + T_a \] --- ## 2. **Final Equations of Motion** ### **For \( J_1 \):** \[ \boxed{ J_1 \ddot{\theta}_1 + (b_1 + b_2)\dot{\theta}_1 + (k_1 + k_2)\theta_1 - b_2 \dot{\theta}_2 - k_2 \theta_2 = } \] ### **For \( J_2 \):** \[ \boxed{ J_2 \ddot{\theta}_2 + b_2 \dot{\theta}_2 + k_2 \theta_2 = b_2 \dot{\theta}_1 + k_2 \theta_1 + T_a } \] --- ## 3. **Summary** - The system is described by the coupled 2nd-order ODEs above. - The variables \( \theta_1(t) \) and \( \theta_2(t) \) describe the angular positions of the inertias as a function of the applied torque \( T_a(t) \). --- If you want the equations for part (b) as well, please let me know!

# Step-by-Step Solution for Rotational Mechanical Systems ## Part (a): Rotational Mechanical System 1 ### 1. **Free-Body Diagram and Torques** For the system with two inertias \( J_1 \) and \( J_2 \): - **Inertia \( J_1 \)** rotates by \( \theta_1(t) \) - **Inertia \( J_2 \)** rotates by \( \theta_2(t) \) - Applied torque \( T_a \) acts on \( J_2 \) - Springs and dampers are connected as follows: - \( k_1, b_1 \): connected to ground on \( J_1 \) - \( k_2, b_2 \): between \( J_1 \) and \( J_2 \) ### 2. **Torques Acting on \( J_1 \)** - **Restoring torque from spring \( k_1 \):** \( -k_1 \theta_1 \) - **Resistive torque from damper \( b_1 \):** \( -b_1 \dot{\theta}_1 \) - **Restoring torque from spring \( k_2 \):** \( -k_2 (\theta_1 - \theta_2) \) - **Resistive torque from damper \( b_2 \):** \( -b_2 (\dot{\theta}_1 - \dot{\theta}_2) \) #### **Equation for \( J_1 \)** \[ J_1 \ddot{\theta}_1 = -k_1 \theta_1 - b_1 \dot{\theta}_1 - k_2 (\theta_1 - \theta_2) - b_2 (\dot{\theta}_1 - \dot{\theta}_2) \] Simplifying: \[ J_1 \ddot{\theta}_1 + (b_1 + b_2)\dot{\theta}_1 + (k_1 + k_2)\theta_1 - b_2 \dot{\theta}_2 - k_2 \theta_2 = 0 \] --- ### 3. **Torques Acting on \( J_2 \)** - **Restoring torque from spring \( k_2 \):** \( +k_2 (\theta_1 - \theta_2) \) - **Resistive torque from damper \( b_2 \):** \( +b_2 (\dot{\theta}_1 - \dot{\theta}_2) \) - **Applied torque:** \( +T_a \) #### **Equation for \( J_2 \)** \[ J_2 \ddot{\theta}_2 = k_2 (\theta_1 - \theta_2) + b_2 (\dot{\theta}_1 - \dot{\theta}_2) + T_a \] Simplifying: \[ J_2 \ddot{\theta}_2 + b_2 \dot{\theta}_2 + k_2 \theta_2 = b_2 \dot{\theta}_1 + k_2 \theta_1 + T_a \] --- ## 4. **Final Equations of Motion** ### **For \( J_1 \):** \[ \boxed{ J_1 \ddot{\theta}_1 + (b_1 + b_2)\dot{\theta}_1 + (k_1 + k_2)\theta_1 - b_2 \dot{\theta}_2 - k_2 \theta_2 = 0 } \] ### **For \( J_2 \):** \[ \boxed{ J_2 \ddot{\theta}_2 + b_2 \dot{\theta}_2 + k_2 \theta_2 = b_2 \dot{\theta}_1 + k_2 \theta_1 + T_a } \] --- ## Part (b): Rotational Mechanical System 2 ### 1. **System Description** - Assume a massless, rigid pendulum with angles \( \theta_1 \) and \( \theta_2 \) being small. - The spring remains approximately horizontal. ### 2. **Free-Body Diagram and Forces** - **Forces acting on the pendulum:** - Gravitational force - Spring force - Viscous damping ### 3. **Equations of Motion** For small angles, the equations of motion can be derived based on the following considerations: - **Torque due to gravity:** \( -mgL \sin(\theta) \approx -mgL \theta \) for small \( \theta \) - **Restoring torque from spring:** \( -k\theta \) - **Damping torque:** \( -b\dot{\theta} \) #### **Equation for Pendulum Motion:** Combining these effects: \[ J \ddot{\theta} + b \dot{\theta} + k \theta = 0 \] --- ## 4. **Final Equation of Motion for System 2** \[ \boxed{ J \ddot{\theta} + b \dot{\theta} + k \theta = 0 } \] --- ## Summary Both systems are described by second-order differential equations that relate the angular positions and velocities to the applied torques, springs, and damping forces. The equations can be solved for the angular positions \( \theta_1(t) \) and \( \theta_2(t) \) as functions of time given specific initial conditions.