Impulse response of second order system which is not underdampled

Nasser Abbasi

Abstract:The impulse response $h\left( t\right) $ for second order single degree of freedom system which is underdamped is well known. In this note, the derivation to the impulse response of critically dampled and overdamped systems are given.

Impulse response for overdamped system

Given the systemMATH Where MATH 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 MATH 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 speedMATH Hence, the system will now have initial conditions of MATH and MATH. Now, the response of (1), when $\xi>1$ is known and given byMATH Apply MATH, we obtain that $0=A+B$ or $B=-A$. NowMATH Apply MATH to the above, we obtainMATH But $B=-A$, hence MATH or MATH

Hence (2) becomesMATH When the magnitude of the impulse is unity, i.e. a unit impulse, hence $\hat{F}=1$, then we obtain the unit impulse responseMATH

Impulse response for critically damped system

The response of (1), when $\xi=1$ is given byMATH Apply MATH, we obtain that $0=A$ NowMATH Apply MATH to the above, we obtainMATH Hence (3) becomesMATH When the magnitude of the impulse is unity, i.e. a unit impulse, hence $\hat{F}=1$, then we obtain the unit impulse responseMATH