# **Spring-Mass-Damper System: Free Response Solution** Let's break down the problem step-by-step! --- ## **1. System Equation** The equation of motion for the spring-mass-damper system is: \[ m\ddot{x}(t) + c\dot{x}(t) + kx(t) = f(t) \] For the free response (no external force), \( f(t) = 0 \): \[ m\ddot{x}(t) + c\dot{x}(t) + kx(t) = 0 \] --- ## **2. Initial Conditions** - Initial position: \( x(0) = x_0 \) - Initial velocity: \( \dot{x}(0) = 0 \) --- ## **3. Characteristic Equation** Rewrite the equation as: \[ \ddot{x}(t) + 2\zeta \omega_n \dot{x}(t) + \omega_n^2 x(t) = 0 \] Where: - **Natural frequency:** \( \omega_n = \sqrt{\frac{k}{m}} \) - **Damping ratio:** \( \zeta = \frac{c}{2\sqrt{km}} \) The characteristic equation is: \[ r^2 + 2\zeta \omega_n r + \omega_n^2 = 0 \] --- ## **4. Solution Form** ### **Case 1: Underdamped (\( \zeta < 1 \))** Most common case, so let's focus on this: \[ x(t) = e^{-\zeta \omega_n t} [A \cos(\omega_d t) + B \sin(\omega_d t)] \] where: - \( \omega_d = \omega_n \sqrt{1 - \zeta^2} \) (damped natural frequency) --- ## **5. Apply Initial Conditions** At \( t = 0 \): - \( x(0) = x_0 \) - \( \dot{x}(0) = 0 \) Plug into general solution: ### \( x(0) = x_0 \): \[ x(0) = e^{0} [A \cdot 1 + B \cdot 0] = A = x_0 \] ### \( \dot{x}(0) = 0 \): Differentiate \( x(t) \): \[ \dot{x}(t) = e^{-\zeta \omega_n t} [-\zeta\omega_n A \cos(\omega_d t) - \zeta\omega_n B \sin(\omega_d t) - A \omega_d \sin(\omega_d t) + B \omega_d \cos(\omega_d t)] \] At \( t = 0 \): \[ \dot{x}(0) = [-\zeta\omega_n A] + [B \omega_d] = 0 \] Plug \( A = x_0 \): \[ -\zeta\omega_n x_0 + B \omega_d = 0 \implies B = \frac{\zeta\omega_n x_0}{\omega_d} \] --- ## **6. Final Free Response Solution** \[ \boxed{ x(t) = e^{-\zeta \omega_n t} \left[ x_0 \cos(\omega_d t) + \frac{\zeta\omega_n x_0}{\omega_d}\sin(\omega_d t) \right] } \] - \( \omega_n = \sqrt{\frac{k}{m}} \) - \( \zeta = \frac{c}{2\sqrt{km}} \) - \( \omega_d = \omega_n \sqrt{1-\zeta^2} \) --- ## **Summary Table** | Quantity | Formula | |--------------------|------------------------------------------| | Natural frequency | \( \omega_n = \sqrt{\frac{k}{m}} \) | | Damping ratio | \( \zeta = \frac{c}{2\sqrt{km}} \) | | Damped frequency | \( \omega_d = \omega_n\sqrt{1-\zeta^2} \)| | Free response | See boxed formula above | --- **Let me know if you want the solution for the overdamped (\( \zeta > 1 \)) or critically damped (\( \zeta = 1 \)) cases!**