Response to Arbitrary Excitation

In many practical situations the dynamic excitation is neither harmonic nor periodic. Thus we are interested in studying the dynamic response of SDF systems to excitations varying arbitrarily with time.

1.General view

Like the study process of the response of harmonic and periodic excitation, we are also interested in the solution (which named EOV) of EOM including its derivation and its observation.

The arbitrary excitation can be extended to several excitations, such as step excitation(with finite rise time or not), linearly excitation, pulse excitation. This is because the study of arbitrary excitation can offer us another powerful method Duhamel Integral which enables us to solve the EOV besides the classic method & superposition method.

There is no doubt that the study of sequence of infinitesimally short impulses comes first.

The mind mapping as follow shows the train of thought clearly.

2.The response to unit impulse

For convenience, we`d better give the def. as follows:

Magnitude of the Impulse(冲量): The cumulative effect of a force that changes over time, which is \( I_m = \int _{t_1}^{t_2} p(t)dt \)

Impulsive force(脉冲力): which is \( p(t) = \frac {I_m}{\Delta t}\)

With the definition of impulsive force, we can easily define the unit impulsive force: \( p(t) = \frac{1}{\varepsilon}\) with unit impulse \( I_m = 1 \)

Observing the expression above, we better understand the characteristics of impulse and unit impulse:
\[
\varepsilon \to 0, p(t) \to \infty
\\ \int _\tau ^{\tau + \varepsilon} p(t)dt = 1
\]

Now we write the EOM by Newton`s second law, with zero initial condition. \( u(0) = 0; \dot{u}(0) = 0 \)

For convenience, we specify zero initial states. This is proved to be resonable because the functions with nonzero initial states can be obtained by superposition as we will see in the following subsections.

\[
p(t) - ku(t) -c\dot{u}(t) = m \ddot{u}(t)
\\ ∵t \to 0
\\ ∴u(t) \to u(0) = 0
\\ \dot{u}(t) \to \dot{u}(0) = 0
\\ ∴p(t) = m \ddot{u}(0)
\\ hence, I_m = \int_{t_1}^{t_2} p(t)dt = m[\dot{u}(t_2)-\dot{u}(t_1)]
\]

The conclusion above is vital as it indicates that impulse equals the change of momentum. With this rule, we can get \( \dot{u}(\tau) \) at the same way.
\[
\int_0^\tau p(t)dt = m \dot{u}(\tau)
\\ \dot{u}(\tau) = \frac{I_m}{m}
\\ ∵\tau \to 0
\\ ∴\dot{u}(0^+) = \frac{I_m}{m}
\]

Substitute it into the EOV of viscous damping SDOF system, we can easily give the expression:
\[
u(t) = e^{-\zeta w_nt}[\overbrace {u(0)}^{=0}cosw_Dt + \frac{\overbrace{\dot{u}(0)}^{=0} + \zeta w_n \overbrace{u(0)}^{=0}}{w_D}sinw_Dt]…①
\\ \Rightarrow u(t) = I_m \overbrace{[\frac{1}{mw_D}e^{-\zeta w_nt}sin(w_Dt)]}^{h(t)}
\\ \Rightarrow h(t) = \frac{u(t)}{I_m}
\]
Apparently, \(h(t)\) represents the response to unit impulse \(I_m = 1\), so it is called unit impulse-response function. Of course together with the zero initial condition. Actually from ①, we can find that the non-zero initial condition only add a free vibration response( \(u(0)cosw_Dt\) ) to the zero initial condition, which means it could be linearly superposition. It explains why we dare to assume that the system is initially at rest.

Now that we’re at this point, why don’t we extend the conclusion to more general terms? So, if we consider the random moment \(t = \tau\) of a SDOF system subjected to an impulse \(I_m\) with \(\Delta t → 0\) at \(t=\tau\) with zero initial condition.

The derivation is as same as above, the only work should we do is change \(t\) to \(t-\tau\):
\[
u(t) = I_m h(t-\tau)
\\ h(t-\tau) = \frac{1}{mw_D}e^{-\zeta w_n(t-\tau)}sin[w_D(t-\tau)]
\]

3.The response to arbitrary force: \(\sum\)responses to unit impulses

As the mind mapping shows, we viewed the arbitrary force as a sequence of infinitesimally short impulses. So we can say:
$$
du(t) = [p(\tau)d\tau]h(t-\tau)
$$
The response of the system at time \(t\) is the sum of the responses to all impulses up to that time. Thus
\[
u(t) = \int_0^t p(\tau)h(t-\tau)d\tau
\\ = \frac{1}{mw_D}\int_0^t p(t)e^{-\zeta w_n(t-\tau)}sin[w_D(t-\tau)]d\tau
\]
This is known as the convolution integral, a general result that applies to any linear dynamic system.

It is necessary to emphasize again that the solution is applicable for zero initial condition. If the initial condition is not at rest, then the solution will be only added a free vibration response:
$$
u(t) = e^{-\zeta w_nt}[u(0)cosw_Dt + \frac{\dot{u}(0) + \zeta w_n u(0){w_D}}{sinw_Dt}]
$$

4.Conclusion

Finally, let`s get back to the mind mapping given in the top of this article. We can fill the blanket now.