4.1 Impulse response of second order system which is not under-damped
The impulse response \(h\left ( t\right ) \) for second order single degree of freedom system which is under-damped
is well known. In this note, the derivation to the impulse response of critically damped and
over-damped systems are given.
4.1.1 Impulse response for over-damped system
Given the system
\begin{equation} \ddot {x}\left ( t\right ) +2\xi \omega _{n}\dot {x}\left ( t\right ) +\omega _{n}^{2}x\left ( t\right ) =\delta \left ( t\right ) \tag {1}\end{equation}
Where \(\delta \left ( t\right ) \) is an impulse. We seek to find \(x\left ( t\right ) \), the response of the above system to this
impulse.
Assume the system is initially at rest. Due to the action of this impulse, the system will obtain an
initial speed which is found as follows. Let \(\delta \left ( t\right ) \equiv \hat {F}=F\Delta t\) where \(\Delta t\) is the duration of the impulse and \(F\) is the magnitude
(in Newtons) of the impulse (hence units of \(\hat {F}\) is \(N~\sec \)). This impulse will impart a momentum on the mass
being hit which we use to determine the initial speed
\begin{align*} \hat {F} & =mv_{0}\\ v_{0} & =\frac {\hat {F}}{m}\end{align*}
Hence, the system will now have initial conditions of \(x\left ( 0\right ) =0\) and \(\dot {x}\left ( 0\right ) =v_{0}=\frac {\hat {F}}{m}\). Now, the response of (1), when \(\xi >1\) is known
and given by
\begin{equation} x\left ( t\right ) =e^{-\xi \omega _{n}t}\left ( Ae^{\omega _{n}\sqrt {\xi ^{2}-1}t}+Be^{-\omega _{n}\sqrt {\xi ^{2}-1}t}\right ) \tag {2}\end{equation}
Apply \(x\left ( 0\right ) =0\), we obtain that \(0=A+B\) or \(B=-A\). Now
\begin{align*} \dot {x}\left ( t\right ) & =-\xi \omega _{n}e^{-\xi \omega _{n}t}\left ( Ae^{\omega _{n}\sqrt {\xi ^{2}-1}t}+Be^{-\omega _{n}\sqrt {\xi ^{2}-1}t}\right ) \\ & +e^{-\xi \omega _{n}t}\left ( A\omega _{n}\sqrt {\xi ^{2}-1}e^{\omega _{n}\sqrt {\xi ^{2}-1}t}-B\omega _{n}\sqrt {\xi ^{2}-1}e^{-\omega _{n}\sqrt {\xi ^{2}-1}t}\right ) \end{align*}
Apply \(\dot {x}\left ( 0\right ) =\frac {\hat {F}}{m}\) to the above, we obtain
\[ \frac {\hat {F}}{m}=\left ( A\omega _{n}\sqrt {\xi ^{2}-1}-B\omega _{n}\sqrt {\xi ^{2}-1}\right ) \]
But \(B=-A\), hence \(\frac {\hat {F}}{m}=2A\omega _{n}\sqrt {\xi ^{2}-1}\) or \(A=\frac {\hat {F}}{2m\omega _{n}\sqrt {\xi ^{2}-1}}\)
Hence (2) becomes
\begin{align*} x\left ( t\right ) & =e^{-\xi \omega _{n}t}\left ( \frac {\hat {F}}{2m\omega _{n}\sqrt {\xi ^{2}-1}}e^{\omega _{n}\sqrt {\xi ^{2}-1}t}-\frac {\hat {F}}{2m\omega _{n}\sqrt {\xi ^{2}-1}}e^{-\omega _{n}\sqrt {\xi ^{2}-1}t}\right ) \\ & =\frac {\hat {F}}{2m\omega _{n}\sqrt {\xi ^{2}-1}}e^{-\xi \omega _{n}t}\left ( e^{\omega _{n}\sqrt {\xi ^{2}-1}t}-e^{-\omega _{n}\sqrt {\xi ^{2}-1}t}\right ) \end{align*}
When the magnitude of the impulse is unity, i.e. a unit impulse, hence \(\hat {F}=1\), then we obtain the unit
impulse response
\[ \fbox {$h\left ( t\right ) =\frac {1}{2m\omega _{n}\sqrt {\xi ^{2}-1}}e^{-\xi \omega _{n}t}\left ( e^{\omega _{n}\sqrt {\xi ^{2}-1}t}-e^{-\omega _{n}\sqrt {\xi ^{2}-1}t}\right ) $}\]
4.1.2 Impulse response for critically damped system
The response of (1), when \(\xi =1\) is given by
\begin{equation} x\left ( t\right ) =Ae^{-\xi \omega _{n}t}+Bte^{-\xi \omega _{n}t} \tag {3}\end{equation}
Apply \(x\left ( 0\right ) =0\), we obtain that \(0=A\) Now
\[ \dot {x}\left ( t\right ) =Be^{-\xi \omega _{n}t}-\xi \omega _{n}Bte^{-\xi \omega _{n}t}\]
Apply \(\dot {x}\left ( 0\right ) =\frac {\hat {F}}{m}\) to the above, we obtain
\[ \frac {\hat {F}}{m}=B \]
Hence (3) becomes
\[ x\left ( t\right ) =\frac {\hat {F}}{m}te^{-\xi \omega _{n}t}\]
When the magnitude of the impulse is unity, i.e. a unit impulse, hence \(\hat {F}=1\), then we
obtain the unit impulse response
\[ h\left ( t\right ) =\frac {1}{m}te^{-\xi \omega _{n}t} \]