Question 10.4.1: A structure of one degree of freedom is acted on by a pulsat...
A structure of one degree of freedom is acted on by a pulsating force P = P_0 \cos \omega _lt. It is known that the damping coefficient may be taken as zero. Show that, if the circular frequency \omega _l of the force is equal to the natural circular frequency \omega of the structure, very large displacements can occur, but such displacements always remain finite as long as the time t remains finite.
For c = 0, Eqn 10.4-7
x=\exp \left(-ct/2M\right)\left(A\sin \omega _dt+B\cos \omega _dt \right)+\frac{P_0\cos \left(\omega _lt-\phi \right) }{\left[\left(k-M\omega ^{2}_{l} \right)^2+c^2\omega ^{2}_{l}\right]^{1/2} } (10.4-7)
becomes:
x=A\sin \omega t+B\cos \omega t+\frac{P_0\cos \omega _lt}{k-M\omega ^{2}_{l} }
=A\sin \omega t+B\cos \omega t+\frac{P_0\cos \omega _lt}{\omega ^2M\left[\left(k/\omega ^2M\right)-\left(\omega ^{2}_{l}/\omega ^2\right) \right]}
=A\sin \omega t+B\cos \omega t+\frac{P_0\cos \omega _lt}{\omega ^2M\left[1-\left(\omega ^{2}_{l}/\omega ^2\right) \right]} (10.4-36)
(since k/M = \omega ^2 from Eqn 10.3-17).
\begin{matrix} m_1 \\ m_2 \end{matrix}=\pm i√\frac{k}{M}=\pm i\omega (10.3-17)
If the initial conditions are:
at t=0 \begin{cases} x=x_0 & \quad \\ \frac{dx}{dt}=\dot{x} _0 \end{cases}
then the reader should verify that
A=\frac{\dot{x} _0}{\omega }; B=x_0-\frac{P_0}{\omega ^2M\left[1-\omega ^{2}_{l}/\omega ^2 \right] }
i.e. Eqn 10.4-36 becomes:
x=\frac{\dot{x} _0}{\omega }\sin \omega t+x_0\cos \omega t+\frac{P_0(\cos \omega _lt-\cos\omega t)}{\omega ^2M\left[1-\left(\omega ^{2}_{l}/\omega ^2\right) \right]}
\underset{\omega _l\rightarrow \omega }{\lim }x=\frac{\dot{x} _0}{\omega }\sin \omega t+x_0\cos \omega t+\frac{0}{0}
Using L’Hospital’s rule, we take derivatives of the numerator and denominator of the last term with respect to \omega _l, and then let \omega _l\rightarrow \omega :
\underset{\omega _l\rightarrow \omega }{\lim }x=\frac{\dot{x} _0}{\omega }\sin \omega t+x_0\cos \omega t+\frac{P_0\left(-t\sin \omega t-0\right) }{\omega ^2M\left[0-\left(2\omega /\omega ^2\right) \right] }
=\frac{\dot{x} _0}{\omega }\sin\omega t+x_0\cos \omega t+\frac{P_0t\sin \omega t}{2\omega M} (10.4-37)
Equation 10.4-37 shows that, as t increases, x becomes progressively larger, but that it remains finite as long as t remains finite; that is, infinitely large displacements do not occur unless the pulsating load acts for an infinitely long time. (The reader should note, however, that if the load acts for a sufficiently long time for the displacement to become so large that the structure is no longer linearly elastic, then Eqn 10.4-7 would no longer apply.)