Forced Oscillations and Resonance — Question 8

Question 8

Moving the support is different from applying a force to the mass

A mass $m$ is connected to a support by a spring $k$ and viscous damper $b$ in parallel. The support moves as $y=Y\cos\Omega t$. Let $x$ be the absolute mass displacement. Both spring extension and damper velocity depend on relative motion.

1. Derive the equation for $x$ and the steady amplitude ratio $X/Y$.

2. Find the condition for $X/Y<1$. Explain why the high-frequency response cannot be obtained by simply setting $F_0=kY$ in the fixed-support force-driven formula when $b>0$.

Question 8 - Solution

Moving the support is different from applying a force to the mass

(a) Derivation and calculation

$$m\ddot x=-k(x-y)-b(\dot x-\dot y).$$$$\boxed{m\ddot x+b\dot x+kx=ky+b\dot y}.$$$$y=Y\cos\Omega t\Rightarrow F_{\mathrm{input}}=kY\cos\Omega t-b\Omega Y\sin\Omega t.$$$$\boxed{\frac XY=\sqrt{\frac{k^2+b^2\Omega^2}{(k-m\Omega^2)^2+b^2\Omega^2}}}.$$

(b) Result and interpretation

For $\Omega>0$,$$X/Y<1\iff(k-m\Omega^2)^2>k^2\iff\boxed{\Omega>\sqrt{2k/m}}.$$$$\Omega\to\infty,\ b>0:\quad\boxed{X/Y\sim b/(m\Omega)}.$$$$b=0:\quad X/Y\sim k/(m\Omega^2).$$Using $F_0=kY$ omits the velocity-dependent support input and gives the wrong high-frequency behavior for $b>0$.

Physical check

At zero frequency $X/Y\to1$: the mass follows a slowly displaced support.