91Ó°ÊÓ

For a damped, driven oscillator, show that the average kinetic energy is the same at a frequency of a given number of octaves* above the kinetic energy resonance as at a frequency of the same number of octaves below resonance.

For a lightly damped oscillator, show that \(Q \cong \omega_{0} / \Delta \omega\) (Equation 3.65 ).

Use a computer to produce a phase space diagram similar to Figure \(3-11\) for the case of critical damping. Show analytically that the equation of the line that the phase paths approach asymptotically is \(\dot{x}=-\beta x\). Show the phase paths for at least three initial positions above and below the line.

Figure 3-B illustrates a mass \(m_{1}\) driven by a sinusoidal force whose frequency is \(\omega\) The mass \(m_{1}\) is attached to a rigid support by a spring of force constant \(k\) and slides on a second mass \(m_{2}\). The frictional force between \(m_{1}\) and \(m_{2}\) is represented by the damping parameter \(b_{1}\), and the frictional force between \(m_{2}\) and the support is represented by \(b_{2}\). Construct the electrical analog of this system and calculate the impedance.

Obtain the Fourier representation of the output of a full-wave rectifier. Plot the first three terms of the expansion and compare with the exact function.

A damped linear oscillator, originally at rest in its equilibrium position, is subjected to a forcing function given by $$ \frac{F(t)}{m}=\left\\{\begin{array}{ll} 0, & t<0 \\ a \times(t / \tau), & 0\tau \end{array}\right. $$ Find the response function. Allow \(\tau \rightarrow 0\) and show that the solution becomes that for a step function.

Obtain the response of a linear oscillator to a step function and to an impulse function (in the limit \(\tau \rightarrow 0\) ) for overdamping. Sketch the response functions.

Obtain the response of a linear oscillator to the forcing function $$ \frac{F(t)}{m}=\left\\{\begin{array}{ll} 0, & t<0 \\ a \sin \omega t, & 0\pi / \omega \end{array}\right. $$

Derive an expression for the displacement of a linear oscillator analogous to Equation 3.110 but for the initial conditions \(x\left(t_{0}\right)=x_{0}\) and \(\dot{x}\left(t_{0}\right)=\dot{x}_{0}.\)

Use Green's method to obtain the response of a damped oscillator to a forcing function of the form $$ F(t)=\left\\{\begin{array}{ll} 0 & t<0 \\ F_{0} e^{-\gamma t} \sin \omega t & t>0 \end{array}\right. $$