User:Egm4313.s12.team8.tclamb/R1

=R1.1 - Equation of Motion for a Parallel Spring-Dashpot System=

Find
Derive the equation of motion of a spring-dashpot system in parallel, with a mass and applied force $$f(t)$$.

Solution
Solved on my own

Kinematics

 * $$\displaystyle\begin{align}

y = y_k = y_c \end{align}$$

Kinetics

 * $$\displaystyle m y'' + f_k + f_c = f(t)$$

Constitutive Relations

 * $$\displaystyle\begin{align}

f_k &= k y_k \\ f_c &= c y_c' \end{align}$$

Therefore

 * $$\displaystyle m y''(t) + c y'(t) + k y(t) = f(t)$$

The general solution of the above ODE is:


 * $$\displaystyle\begin{align}

y_g (t) &= C_1 \; e^{\frac{-c - \sqrt{c^2 - 4 m k}}{2 m} t} \\ &+ C_2 \; e^{\frac{-c + \sqrt{c^2 - 4 m k}}{2 m} t} \\ &- \frac{\int e^{\frac{\left( c + \sqrt{c^2 + 4 m k} \right) t}{2 m}} f(t) \,dt}{\sqrt{c^2 - 4 m k}} \; e^{\frac{-c - \sqrt{c^2 - 4 m k}}{2 m} t} \\ &+ \frac{\int e^{\frac{\left( c - \sqrt{c^2 + 4 m k} \right) t}{2 m}} f(t) \,dt}{\sqrt{c^2 - 4 m k}} \; e^{\frac{-c + \sqrt{c^2 - 4 m k}}{2 m} t} \end{align}$$