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Oscillations: Resonant Response

CLMECH-AR4XNB

When a simple mass-spring system is excited by a piston that pushes and pulls on the spring in a sinusoidal manner, the oscillatory response of the of the mass-spring system can be described by the equation:

$$\frac{d^2x}{dt^2}+b\,\frac{dx}{dt}+\omega_o\,x=X(t)$$

where $x(t)$ is the position of the mass, $b$ is the damping coefficient, $\omega_o$ is the natural frequency of oscillation and $X(t)$ is the position of the piston that drives the system. Let the motion of the piston be of the form:

$$X(t)=X_o\,cos(\omega\,t)$$

The response of the mass-spring system to the driving force of the piston depends strongly on the frequency of oscillation, $\omega$, of the piston motion, and is also strongly affected by the value of the damping coefficient $b$.

The response can be characterized by a dimensionless parameter $\xi$ equal to the ratio of the amplitude of oscillation, $x_o$ to the amplitude of the piston motion, $X_o$:

$$\xi=\frac{x_o}{X_o}$$

Relative to the natural frequency of oscillation, $\omega_o$, at what frequency, $\omega$, should the piston oscillate to maximize $\xi$?

A

$\omega=\omega_o$

B

$\omega=\frac{\omega_o}{\sqrt{2}}$

C

$\omega=\sqrt{2}\,\omega_o$

D

$\omega=2\,\omega_o$