Imagine a damped driven mass spring system with a time-dependent damping coefficient
\begin{equation} m\frac{d^2x}{dt^2} + b(t)\frac{dx}{dt} + kx = F_0 cos(wt) \end{equation} \begin{equation} b(t) = \begin{cases} b_1 & t< t_0\\ b_2 & t\geq t_0 \end{cases} \end{equation}

I am trying to understand how the system would respond to the change in damping coefficient at $t = t_0$. Can we talk about a transient response in that case?

The concept of transient response is often explained with respect to initial conditions of a system and its input signal. I never saw an example where the system's response to a change in internal parameter is investigated.


1 Answer 1


Can we talk about a transient response in that case?
Yes, as in effect you are considering a new system with initial conditions set at $t=t_0$.

Just think about what would happen if you set $b_2=0$.
Then you would expect the system to undergo undamped simple harmonic motion with the initial conditions set at $t=t_0$.

  • $\begingroup$ I see. Then, the transient response will be determined by the new system's parameters. $\endgroup$
    – Krlngc
    Dec 22, 2021 at 14:05

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