Forced Oscillations and Resonance — Question 10

Question 10

A drive phase reversal creates a transient even at resonance

A damped oscillator with $\beta<\omega_0$ has reached resonant steady motion $x=X\sin\omega_0t$ under force $F_0\cos\omega_0t$, where $X=F_0/(b\omega_0)$. At $t=0$, the drive reverses sign abruptly to $-F_0\cos\omega_0t$, with no impulse on the mass.

1. Find the motion for $t\ge0$ by adding a decaying free response to the new steady response.

2. Check continuity of displacement and velocity at the reversal. Explain why the amplitude cannot jump immediately to the new steady motion.

Question 10 - Solution

A drive phase reversal creates a transient even at resonance

(a) Derivation and calculation

For $t\ge0$, superpose the new particular solution and free motion:$$x=-X\sin\omega_0t+e^{-\beta t}(C\cos\omega_dt+D\sin\omega_dt).$$$$x(0)=0\Rightarrow C=0,\quad v(0)=-X\omega_0+D\omega_d=X\omega_0.$$$$\boxed{D=2X\omega_0/\omega_d},\qquad\omega_d=\sqrt{\omega_0^2-\beta^2}.$$

(b) Result and interpretation

$$\boxed{x=-X\sin\omega_0t+\frac{2X\omega_0}{\omega_d}e^{-\beta t}\sin\omega_dt}.$$$$x(0^+)=0=x(0^-),\quad v(0^+)=-X\omega_0+2X\omega_0=X\omega_0.$$$$\Delta a=\Delta F/m=-2F_0/m.$$A finite force step changes acceleration without an instantaneous velocity jump. The transient matches the old state to the new steady response.

Physical check

As $t\to\infty$, the free term vanishes and the oscillator approaches the reversed steady motion.