Differential Equations Quiz: Forced Oscillations And Resonance

A forced, damped mass-spring system is described by the differential equation $2y'' + 8y' + 32y = 4\cos(\omega t)$. At which of the following driving angular frequencies $\omega$ is the amplitude of the steady-state solution maximized?

The solution to an initial value problem for a forced, undamped mass-spring system, $y'' + \omega_0^2 y = F(t)$, is found to be $y(t) = 8\sin(0.5t)\sin(10.5t)$. Which of the following can be inferred about the forcing function $F(t)$ and the natural angular frequency $\omega_0$?

A system is described by $y'' + \gamma y' + 25y = \cos(\omega t)$, where $\gamma>0$ is a small damping coefficient. The steady-state solution can be written as $y_p(t) = A(\omega) \cos(\omega t - \delta(\omega))$. What value does the phase shift $\delta(\omega)$ approach as the driving frequency $\omega$ approaches the natural frequency $\omega_0 = 5$?

A damped mechanical system is modeled by $2y'' + \gamma y' + 18y = F(t)$. It is observed that the amplitude of the steady-state response is maximized when the driving angular frequency is $\omega = 2$ rad/s. What is the value of the damping coefficient $\gamma$?

Consider two distinct mass-spring systems, System 1 and System 2, both driven by the same forcing function $F(t) = F_0 \cos(\omega t)$. System 1: $y_1'' + 0.5 y_1' + 4y_1 = F_0 \cos(\omega t)$ System 2: $y_2'' + 0.2 y_2' + 4y_2 = F_0 \cos(\omega t)$ Let $A_1(\omega)$ and $A_2(\omega)$ be the amplitudes of the steady-state solutions for System 1 and System 2, respectively. Which statement is true for all driving frequencies $\omega > 0$?

An RLC circuit is modeled by $L Q'' + R Q' + \frac{1}{C} Q = E_0 \sin(\omega t)$. Given $L=1$ H, $R=3$ $\Omega$, and $C=1/2$ F, what is the amplitude of the steady-state current $I(t) = Q'(t)$ if the input voltage has amplitude $E_0=10$ V and angular frequency $\omega=2$ rad/s?

An undamped oscillator with a natural angular frequency of $\omega_0 = 100$ rad/s is subjected to a forcing function $F(t) = F_0\cos(102t)$. The system is initially at rest. The resulting motion exhibits beats. What is the angular frequency of the slow oscillation (the envelope)?

A mass of $m=1$ kg is attached to a spring with constant $k=100$ N/m. The system is damped with coefficient $\gamma$ and driven by a force $F(t) = \cos(\omega t)$. The driving frequency $\omega$ is fixed at a value that is not the natural frequency. If the damping coefficient $\gamma$ is decreased from a large value towards zero (but remains positive), what is the effect on the amplitude of the steady-state oscillation?

A mass-spring system with forcing is modeled by $my'' + \gamma y' + ky = F_0 \cos(\omega t)$, where $m, k, F_0$ are fixed positive constants. Assume the damping coefficient $\gamma$ is small enough that a resonant peak exists. Let $\omega_r(\gamma)$ be the resonant frequency and $A_{max}(\gamma)$ be the maximum amplitude of the steady-state solution. If $\gamma$ is increased, what is the effect on $\omega_r(\gamma)$ and $A_{max}(\gamma)$?

Consider a forced oscillator described by $\ddot{y} + 3\dot{y} + 2y = 4\cos(\omega t) + 3\sin(\omega t)$. The driving force can be written as $F(t) = R\cos(\omega t - \alpha)$. What are the values of $R$ and $\alpha$, and how does this affect the steady-state solution compared to a simple cosine drive?

For the equation $\ddot{x} + 4\dot{x} + 13x = 26\cos(3t - \pi/6)$, the steady-state solution is $x_{ss}(t) = A\cos(3t - \pi/6 + \phi)$ where $\phi$ is the phase lag. If the driving force phase is changed to $26\cos(3t + \pi/3)$, how does the new steady-state solution $x_{new}(t)$ relate to the original?

A mass attached to a spring and dashpot is driven by $F(t) = F_0[\cos(\omega_1 t) + \cos(\omega_2 t)]$ where $\omega_1$ and $\omega_2$ are close but not equal. The system parameters give resonance frequency $\omega_{res}$ such that $\omega_1 < \omega_{res} < \omega_2$. Which statement best describes the long-term behavior?

The response amplitude $A(\omega)$ of a driven damped oscillator is given by $A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\gamma\omega)^2}}$. If $\omega_0 = 3$ and $\gamma = 0.6$, at which frequency is the rate of change of amplitude with respect to frequency, $dA/d\omega$, equal to zero?

A forced harmonic oscillator has the equation $x'' + \gamma x' + \omega_0^2 x = F_0\cos(\omega t)$ where $\gamma = 0.5$, $\omega_0 = 2$, and $F_0 = 8$. At what driving frequency $\omega$ will the power absorbed by the system be exactly half of the maximum possible power absorption?

A mass-spring system with damping is driven by an external force $F(t) = 4\cos(\omega t)$. The equation of motion is $x'' + 2x' + 5x = 4\cos(\omega t)$. If the system exhibits resonance, what is the driving frequency $\omega$ and what characterizes the steady-state amplitude behavior?

Consider the differential equation $y'' + \gamma y' + 9y = \sin(\omega t)$. For which combination of non-negative parameter values $\gamma$ and $\omega$ will solutions be unbounded as $t \to \infty$?

Consider the initial value problem $y'' + 0.1y' + 4y = 2\cos(2t)$, with $y(0)=10$ and $y'(0)=0$. Which of the following best describes the behavior of the solution $y(t)$ for large values of $t$?