Question #42638

A damped harmonic oscillator consists of a mass $m$, a spring of constant $k$, and a damping medium providing a damping force $-b\dot{x}$. It is driven by a time-varying force $F(t) = F_0 \cos(\theta(t))$, where the instantaneous driving frequency $\omega(t) = \frac{d\theta}{dt}$ is slowly swept linearly as $\omega(t) = \omega_0 + \beta t$. Here $\omega_0 = \sqrt{k/m}$ is the undamped natural frequency, and $\beta$ is a small positive constant. Assuming the system continuously adapts to a quasi-steady state, its displacement is $x(t) = A(t) \cos(\theta(t) - \delta(t))$, where $\delta(t)$ is the instantaneous phase lag. At $t=0$, the driving frequency passes through resonance ($\omega = \omega_0$). What is the difference between the instantaneous frequency of the driving force and the instantaneous frequency of the displacement at $t=0$?