Chapter 10

Vibrations

1.
The study of vibrations involves the study of $\rule{1.4in}{.01in}$motions.



Conservative Systems

1.
Note that a ``one-degree-of-freedom'' system is one in which the motion is described by one coordinate (e.g., x). Examples
2.
Sketch Figure 10.1(a).










3.
Derive the equation

\begin{displaymath}\frac{d^2x}{dt^2} + \frac{k}{m}x = 0\end{displaymath}

below, using Newton's second law. (Remember to first draw the free body diagram.)















4.
Notice that in this chapter the acceleration is represented by $\mbox{${\ddot x}$}$ as opposed to simply a. This is because we desire the differential equation describing the motion at any time t as opposed to simply determining the acceleration at a particular instant in time.
5.
Sketch Figure 10.1(c).













6.
Derive the equation

\begin{displaymath}\frac{d^2x}{dt^2} + \frac{k}{m}x = g\end{displaymath}

below, using Newton's second law.















7.
Make the change of variable $\mbox{${\tilde x}$}=x-mg/k$ and show that $ \frac{d^2x}{dt^2} + \frac{k}{m}x = g$ is equivalent to

\begin{displaymath}\frac{d^2\mbox{${\tilde x}$}}{dt^2} + \frac{k}{m}\mbox{${\tilde x}$}= 0.\end{displaymath}

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Solutions

8.
Consider the homogeneous, linear, ordinary differential equation

\begin{displaymath}\frac{d^2x}{dt^2} + \omega^2x = 0.\end{displaymath}

. Why is this equation considered ordinary?





Why is this equation considered linear?





Why is this equations considered homogeneous?





9.
For the above spring-mass systems, what is $\omega$?



10.
By substitution, show that $\frac{d^2x}{dt^2} + \omega^2x = 0$ is satisfied by

\begin{displaymath}x = A\sin\omega t + B\cos\omega t\end{displaymath}

where A and B are arbitrary constants.













11.
Show that $x = A\sin\omega t + B\cos\omega t$ is equivalent to

\begin{displaymath}x = E\sin(\omega t - \phi)\end{displaymath}

where E and $\phi$ are constants, by using a standard trigonometric identity to rewrite $\sin(\omega t - \phi)$.












12.
Write A and B in terms of E and $\phi$.
13.
Note that $ x = E\sin(\omega t - \phi)$ is useful for plotting x as a function of time t. This expression clearly demonstrates the oscillatory nature of the motion. This motion is called $\rule{1.0in}{.01in}$ $\rule{1.0in}{.01in}$ $\rule{1.0in}{.01in}$.
14.
Sketch x vs. t and clearly label E and $\phi$ on the plot.












15.
The constant $\phi$ is called the $\rule{1.0in}{.01in}$ of the response, while E is called the $\rule{1.0in}{.01in}$ of the response.
16.
Note that $\omega$ is the natural frequency of the system in units of radians per sec. The natural frequency in units of cycles per sec is denoted by f where

\begin{displaymath}f = \frac{\omega}{2\pi}.\end{displaymath}

The period of vibration in units of seconds is denoted by $\tau$ where

\begin{displaymath}\tau = \frac{1}{f} = \frac{\omega}{2\pi}.\end{displaymath}

(The book does not call $\omega$ a natural frequency so this discussion differs slightly from that in the book.)

Damped Vibrations

This section may be covered in a class lecture.


Forced Vibrations

This section may be covered in a class lecture.