303_Vibration of Mass-Spring-Damper System (4)

Vibration of Mass-Spring-Damper System (4)

\[ \left\{ \begin{array}{l} \begin{align} \ddot{x} + \dfrac{c}{m} \dot{x} + \dfrac{k}{m} x &= \dfrac{f(t)}{m}\tag{1}\\ \ddot{x} + 2\zeta \omega_{n} \dot{x} + \omega_{n}^2 x &= \dfrac{f(t)}{m} \tag{2} \\ \end{align} \end{array} \right. \]

Summary of the analytical solutions of the mass-spring damper system

(1) Time responses to the unit step input \(f(t) = 1\)

Transfer function
\[\dfrac{X(s)}{F(s)} = \dfrac{1}{ms^2 + cs + k} = \dfrac{1}{m} \cdot \dfrac{1}{s^2 + 2\zeta\omega_{n}s + \omega_{n}^2}\]		Normalised Transfer function
\[\dfrac{X(s)}{F(s)} = \dfrac{k}{ms^2 + cs + k} = \dfrac{\omega_{n}^2}{s^2 + 2\zeta\omega_{n}s + \omega_{n}^2}\]
(i) if \(0<\zeta <1\) (underdamped)\[x(t) = \dfrac{1}{k} \Bigl[ 1 \;- \dfrac{e^{-\zeta\omega_{n}t}}{\sqrt{1-\zeta^2}} \sin{(\omega_{d}t + \theta)} \Bigr]
\]
where
\[
\theta = \tan^{-1}{\dfrac{\omega_{n}\sqrt{1-\zeta^2}}{\zeta\omega_{n}}}
\]		\[x(t) = 1 \;- \dfrac{e^{-\zeta\omega_{n}t}}{\sqrt{1-\zeta^2}} \sin{(\omega_{d}t + \theta)}
\]
where
\[
\theta = \tan^{-1}{\dfrac{\omega_{n}\sqrt{1-\zeta^2}}{\zeta\omega_{n}}}
\]
(ii) if \(\zeta =1\) (critically damped)\[
x(t) = \dfrac{1}{k} \Bigl[ 1 \; – (1+\omega_{n}t)e^{-\omega_{n}t} \Bigr] \
\]		\[
x(t) = 1 \; – (1+\omega_{n}t)e^{-\omega_{n}t} \
\]
(iii) if \(\zeta >1\) ( overdamped)\[x(t) = \dfrac{1}{\omega_{n}^2} \Bigl(1- \dfrac{\alpha_{1}e^{-\alpha_{2}t} – \alpha_{2}e^{-\alpha_{1}t} }{\alpha_{1}-\alpha_{2}} \Bigr) \]
where
\[
\left\{
\begin{array}{l} \begin{align}
\alpha_{1} &= \zeta\omega_{n} + \omega_{n} \sqrt{\zeta^2-1} \\
\alpha_{2} &= \zeta\omega_{n} \;- \omega_{n} \sqrt{\zeta^2-1} \\
\end{align} \end{array}
\right.
\]		\[x(t) = \dfrac{k}{\omega_{n}^2} \Bigl(1- \dfrac{\alpha_{1}e^{-\alpha_{2}t} – \alpha_{2}e^{-\alpha_{1}t} }{\alpha_{1}-\alpha_{2}} \Bigr) \]
where
\[
\left\{
\begin{array}{l} \begin{align}
\alpha_{1} &= \zeta\omega_{n} + \omega_{n} \sqrt{\zeta^2-1} \\
\alpha_{2} &= \zeta\omega_{n} \;- \omega_{n} \sqrt{\zeta^2-1} \\
\end{align} \end{array}
\right.
\]
The normalised transfer function is formed by the multiplication of the gain of \(k = m \omega_{n}^2 \) to the original transfer function.

(2) Time responses to the impulse input \(f(t) = \delta(t)\)

Transfer function
\[\dfrac{X(s)}{F(s)} = \dfrac{1}{ms^2 + cs + k} = \dfrac{1}{m} \cdot \dfrac{1}{s^2 + 2\zeta\omega_{n}s + \omega_{n}^2}\]		Normalised Transfer function
\[\dfrac{X(s)}{F(s)} = \dfrac{k}{ms^2 + cs + k} = \dfrac{\omega_{n}^2}{s^2 + 2\zeta\omega_{n}s + \omega_{n}^2}\]
(i) if \(0<\zeta <1\) (underdamped)\[
x(t) = \dfrac{1}{m\omega_{d}} e^{-\zeta\omega_{n}t}\sin{\omega_{d}t}
\]		\[
x(t) = \dfrac{\omega_{n}}{\sqrt{1-\zeta^2}} e^{-\zeta\omega_{n}t}\sin{\omega_{d}t}
\]
(ii) if \(\zeta =1\) (critically damped)\[
x(t) = \dfrac{1}{m}te^{-\omega_{n}t}
\]		\[
x(t) = \omega_{n}^2 te^{-\omega_{n}t}
\]
(iii) if \(\zeta >1\) ( overdamped)\[
x(t) = \dfrac{1}{2\omega_{n}\sqrt{\zeta^2-1}}
\Bigl[ e^{-\omega_{n}(\zeta-\sqrt{\zeta^2-1})t} \; – e^{-\omega_{n}(\zeta+\sqrt{\zeta^2-1})t} \Bigr]
\]		\[
x(t) = \dfrac{m\omega_{n}}{2\sqrt{\zeta^2-1}} \Bigl[ e^{-\omega_{n}(\zeta-\sqrt{\zeta^2-1})t} \; – e^{-\omega_{n}(\zeta+\sqrt{\zeta^2-1})t} \Bigr]
\]
The normalised transfer function is formed by the multiplication of the gain of \(k = m \omega_{n}^2 \) to the original transfer function.