Vibrations and Waves

French A.P

Chapter 4

Forced vibrations and resonance - all with Video Answers

Educators

Chapter Questions

Problem 1

Construct a table, covering as wide a range as possible, of resonant systems occurring in nature. Indicate the order of magnitude of (a) the physical size of each system, and (b) its resonant frequency.

Ajay Singhal

Ajay Singhal

Numerade Educator

Problem 2

Consider how to solve the steady-state motion of a forced oscillator if the driving force is of the form $F=F_{0} \sin \omega t$ instead of $F_{0} \cos$ wt.

Anand Jangid

Numerade Educator

Problem 3

An object of mass $0.2 \mathrm{~kg}$ is hung from a spring whose spring constant is $80 \mathrm{~N} / \mathrm{m}$. The body is subject to a resistive force given by $-b v$, where $v$ is its velocity $(\mathrm{m} / \mathrm{sec})$ and $b=4 \mathrm{~N}-\mathrm{m}^{-1} \mathrm{sec}$.
(a) Set up the differential equation of motion for free oscillations of the system, and find the period of such oscillations.
(b) The object is subjected to a sinusoidal driving force given by $F(t)=F_{0} \sin \omega t$, where $F_{0}=2 \mathrm{~N}$ and $\omega=30 \mathrm{sec}^{-1}$. In the steady
state, what is the amplitude of the forced oscillation?

James Kiss

Numerade Educator

Problem 4

A block of mass $m$ is connected to a spring, the other end of which is fixed. There is also a viscous damping mechanism. The following observations have been made on this system:
(1) If the block is pushed horizontally with a force equal to $m g$, the static compression of the spring is equal to $h$.
(2) The viscous resistive force is equal to $m g$ if the block moves with a certain known speed $u$.
(a) For this complete system (including both spring and damper) write the differential equation governing horizontal oscillations of the mass in terms of $m, g, h$, and $u$. Answer the following for the case that $u=3 \sqrt{g h}$ :
(b) What is the angular frequency of the damped oscillations?
(c) After what time, expressed as a multiple of $\sqrt{h / g}$, is the energy down by a factor $1 / e ?$
(d) What is the $Q$ of this oscillator?
(c) This oscillator, initially in its rest position, is suddenly set into motion at $t=0$ by a bullet of negligible mass but nonnegligible momentum traveling in the positive $x$ direction. Find the value of the phase angle $\delta$ in the equation $x=A e^{-y+2} \cos (\omega t-\delta)$ that describes the subsequent motion, and sketch $x$ versus $t$ for the first few cycles.
(f) If the oscillator is driven with a force $m g \cos \omega t$, where $\omega=\sqrt{2 g / h}$, what is the amplitude of the steady-state response?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 5

A simple pendulum has a length ( $I$, of $1 \mathrm{~m}$. In free vibration the amplitude of its swings falls off by a factor $e$ in 50 swings. The pendulum is set into forced vibration by moving its point of suspension horizontally in SHM with an amplitude of $1 \mathrm{~mm}$.
(a) Show that if the horizontal displacement of the pendulum bob is $x$, and the horizontal displacement of the support is $\xi$, the equation of motion of the bob for small oscillations is
$$
\frac{d^{2} x}{d t^{2}}+\gamma \frac{d x}{d t}+\frac{g}{l} x=\frac{g}{l} \xi
$$
Solve this equation for steady-state motion, if $\xi=\xi_{0} \cos \omega t .$ (Put $\left.\omega_{0}^{2}=g / l .\right)$
(b) At exact resonance, what is the amplitude of the motion of the pendulum bob? (First, use the given information to find $Q .$ )
(c) At what angular frequencies is the amplitude half of its resonant value?

Penny Riley

Numerade Educator

Problem 6

Imagine a simple seismograph consisting of a mass $M$ hung from a spring on a rigid framework attached to the earth, as shown. The spring force and the damping force depend on the displacement and velocity relative to the earth's surface, but the dynamically significant acceleration is the acceleration of $M$ relative to the fixed stars.
(a) Using $y$ to denote the displacement of $M$ relative to the earth and $\eta$ to denote the displacement of the earth's surface itself, show that the equation of motion is
$$
\frac{d^{2} y}{d t^{2}}+\gamma \frac{d y}{d t}+\omega_{0}^{2} y=-\frac{d^{2} \eta}{d t^{2}}
$$
(b) Solve for $y$ (steady-state vibration) if $\eta=C$ cos w.
(c) Sketch a graph of the amplitude $A$ of the displacement $y$ as a function of $\omega$ (supposing $C$ the same for all $\omega$ ).
(d) A typical long-period seismometer has a period of about $30 \mathrm{sec}$ and a $Q$ of about 2 . As the result of a violent earthquake the earth's surface may oscillate with a period of about $20 \mathrm{~min}$ and with an amplitude such that the maximum acceleration is about $10^{-9} \mathrm{~m} / \mathrm{sec}^{2} .$ How small a value of $A$ must be observable if this is to be detected?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

Consider a system with a damping force undergoing forced oscillations at an angular frequency $\omega$.
(a) What is the instantaneous kinetic energy of the system?
(b) What is the instantaneous potential energy of the system?
(c) What is the ratio of the average kinetic energy to the average potential energy? Express the answer in terms of the ratio $\omega / \omega_{0} .$
(d) For what value(s) of $\omega$ are the average kinetic energy and the average potential energy equal? What is the total energy of the system under these conditions?
(e) How does the total energy of the system vary with time for an arbitrary value of $\omega$ ? For what value(s) of $\omega$ is the total energy constant in time?

Penny Riley

Numerade Educator

Problem 8

A mass $m$ is subject to a resistive force $-b v$ but no springlike restoring force.
(a) Show that its displacement as a function of time is of the form
$$
x=C-\frac{v_{0}}{\gamma} e^{-\gamma t}
$$
where $\gamma=b / m .$
(b) At $t=0$ the mass is at rest at $x=0 .$ At this instant a driving force $F=F_{0}$ cos $\omega t$ is switched on. Find the values of $A$ and $\delta$ in the steady-state solution $x=A \cos (\omega t-\delta)$.
(c) Write down the general solution [the sum of parts (a) and
(b)] and find the values of $C$ and $v_{0}$ from the conditions that $x=0$ and $d x / d t=0$ at $t=0 .$ Sketch $x$ as a function of $t$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

(a) A forced damped oscillator of mass $m$ has a displacement varying with time given by $x=A \sin \omega t .$ The resistive force is $-b v$. From this information calculate how much work is done against the resistive force during one cycle of oscillation.
(b) For a driving frequency w less than the natural frequency $\omega_{0}$, sketch graphs of potential energy, kinetic energy, and total energy for the oscillator over one complete cycle. Be sure to label important turning points and intersections with their values of energy and time.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 10

The power input to maintain forced vibrations can be calculated by recognizing that this power is the mean rate of doing work against the resistive force $-b v$.
(a) Satisfy yourself that the instantaneous rate of doing work against this force is equal to $b v^{2}$.
(b) Using $x=A \cos (\omega t-\delta)$, show that the mean rate of doing work is $b \omega^{2} A^{2} / 2$
(c) Substitute the value of $A$ at any arbitrary frequency and hence obtain the expression for $\bar{P}$ as given in Eq. (4-23).

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

Consider a damped oscillator with $m=0.2 \mathrm{~kg}, b=4 \mathrm{~N}-\mathrm{m}^{-1}$ sec and $k=80 \mathrm{~N} / \mathrm{m}$. Suppose that this oscillator is driven by a force $F=F_{0} \cos \omega t$, where $F_{0}=2 \mathrm{~N}$ and $\omega=30 \mathrm{sec}^{-1}$.
(a) What are the values $A$ and $\delta$ of the steady-state response described by $x=A \cos (\omega t-\delta) ?$
(b) How much energy is dissipated against the resistive force in one cycle?
(c) What is the mean power input?

Ajay Singhal

Numerade Educator

Problem 12

An object of mass $2 \mathrm{~kg}$ hangs from a spring of negligible mass. The spring is extended by $2.5 \mathrm{~cm}$ when the object is attached. The top end of the spring is oscillated up and down in SHM with an amplitude of $1 \mathrm{~mm}$. The $Q$ of the system is 15 .
(a) What is $\omega_{0}$ for this system?
(b) What is the amplitude of forced oscillation at $\omega=\omega_{0}$ ?
(c) What is the mean power input to maintain the forced oscillation at a frequency $2 \%$ greater than $\omega_{0}$ ? [Use of the approximate formula, Eq. (4-26), is justified.]

James Kiss

Numerade Educator

Problem 13

The graph shows the power resonance curve of a certain mechanical system when driven by a force $F_{0} \sin \omega t$, where $F_{0}=$ constant and $\omega$ is variable.
(a) Find the numerical values of $\omega_{0}$ and $Q$ for this system.
(b) The driving force is turned off. After how many cycles of free oscillation is the energy of the system down to $1 / e^{5}$ of its initial value? $(e=2.718 .)$ (To a good approximation, the period of free oscillation can be set equal to $2 \pi / \omega_{0}$.)

Ajay Singhal

Numerade Educator

Problem 14

The figure shows the mean power input $\bar{P}$ as a function of driving frequency for a mass on a spring with damping. (Driving force $=$
$F_{0} \sin \omega t$, where $F_{0}$ is held constant and $\omega$ is varied.) The $Q$ is high enough so that the mean power input, which is maximum at $\omega_{0}$, falls to half-maximum at the frequencies $0.98 \omega_{0}$ and $1.02 \omega_{0}$.
(a) What is the numerical value of $Q$ ?
(b) If the driving force is removed, the energy decreases according to the equation
$$
\boldsymbol{E}=\boldsymbol{E}_{0} e^{-\mathbf{q}}
$$
What is the value of $\gamma$ ?
(c) If the driving force is removed, what fraction of the energy is lost per cycle?

A new system is made in which the spring constant is doubled, but the mass and viscous medium are unchanged, and the same driving force $F_{0}$ sin $\omega t$ is applied. In terms of the corresponding quantities for the original system, find the values of the following:
(d) The new resonant frequency $\omega_{0}^{\prime}$.
(e) The new quality factor $Q^{\prime}$.
(f) The maximum mean power input $\bar{P}_{m}^{\prime} .$
(g) The total energy of the system at resonance, $E_{0^{\prime}}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

The free oscillations of a mechanical system are observed to have a certain angular frequency $\omega_{1}$. The same system, when driven by a force $F_{0} \cos \omega t$ (where $F_{0}=$ const. and $\omega$ is variable), has a power resonance curve whose angular frequency width, at half-maximum power, is $\omega_{1} / 5$.
(a) At what angular frequency does the maximum power input occur?
(b) What is the $Q$ of the system?
(c) The system consists of a mass $m$ on a spring of spring constant
k. In terms of $m$ and $k$, what is the value of the constant $b$ in the resistive term $-b v$ ?
(d) Sketch the amplitude response curve, marking a few characteristic points on the curve.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

For the electrical system in the figure, find
(a) The resonant frequency, $\omega_{0}$.
(b) The resonance width, $\gamma$.
(c) The power absorbed at resonance.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

The graph shows the mean power absorbed by an oscillator when driven by a force of constant magnitude but variable angular frequency $\omega$.
(a) At exact resonance, how much work per cycle is being done against the resistive force? (Period = $2 \pi / \omega$.)
(b) At exact resonance, what is the total mechanical energy $E_{0}$ of the oscillator?
(c) If the driving force is turned off, how many seconds does it take before the energy decreases to a value $E=E_{0} e^{-1}$ ?

Ajay Singhal

Numerade Educator