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Vibration phenomena

Part 2 · Functions · Chapter 6 · lecture notes by Fabio Furini · Chapter PDF

1. Vibration phenomena

  • We have seen that the sine and cosine functions are periodic with period \(2\: \pi\). We will use the variable \(t\) to denote time.

    Figure 1

  • Recall that \(\sin (t + \varphi)\) corresponds to a phase shift \(\varphi\) (horizontal translation).

    Figure 2

The functions:

\[\begin{equation} \label{trig_vibr} t \mapsto a\: \sin \:(\omega \: t), \qquad\qquad t \mapsto b\: \cos \:(\omega \: t) \end{equation}\]

where \(a\), \(b\) and \(\omega\) are positive real numbers, are called elementary vibrations. They are periodic functions with

\[ {\rm \textbf{period}~~~} T = \frac{2\:\pi}{\omega} \]

Example 1: period

For example, considering \(t' = t + \frac{2\: \pi}{\omega}\):

\[ \sin \left[ \omega \underbrace{\left( t + \frac{2\: \pi}{\omega} \right)}_{t'}\right] = \sin (\omega \: t + 2\: \pi) = \sin \omega \: t \]

  • Clearly we have:

    \[ |a\: \sin \:(\omega \: t)| \le a \qquad{\rm ~~and~~}\qquad |b\: \cos \:(\omega \: t)| \le b \]

The functions \(\eqref{trig_vibr}\) describe elementary vibrations characterized by

\[ {\rm \textbf{amplitude}~~} a {\rm ~~and~~} b {\rm ~~~~(respectively)} \]
\[ {\rm \textbf{angular frequency}~~} \omega = \frac{2\: \pi}{T} \]

also called angular velocity, which indicates how many periods there are in an interval of length \(2\: \pi\). Moreover, they are characterized by the

\[ {\rm \textbf{frequency}~~} \nu = \frac{\omega}{2\:\pi} \]

which indicates how many times the function repeats itself in an interval of length \(1\)

Example 2: of amplitude, angular frequency and frequency

\[ t \mapsto 2\: \sin \:\left( \frac{3}{2} \: t \right) \qquad a=2,~ \omega=\frac{3}{2},~ T=\frac{4\:\pi}{3} \]

Figure 3

The angular frequency \(\omega=\frac{3}{2}\) means that there are 1.5 periods in an interval of length \(2 \: \pi\). The frequency \(\nu\) is \(\frac{3}{4\:\pi}\).

  • The function

    \[\begin{equation} \label{EEE} h(t) = a\: \sin \:(\omega \: t) + b\: \cos \:(\omega \: t) \end{equation}\]

    describes the superposition of the two elementary vibrations with the same angular frequency \(\omega\). The latter is again an elementary vibration, phase-shifted with respect to the previous ones.

  • Indeed, setting

    \[ A= \sqrt{a^2 + b^2} \]

    we can write

    \[\begin{equation} \label{EEEE} h(t) = A \: \underbrace{\frac{a}{ \sqrt{a^2 + b^2}}}_{\alpha} \: \sin \:(\omega \: t) + A \: \underbrace{\frac{b}{ \sqrt{a^2 + b^2}}}_{\beta} \: \cos \:(\omega \: t) \end{equation}\]

  • Now observe that the numbers

    \[ \alpha=\frac{a}{ \sqrt{a^2 + b^2}} {\rm ~~and~~} \beta=\frac{b}{ \sqrt{a^2 + b^2}} \]

    satisfy the conditions

    \[ -1 \le \alpha \le 1 \qquad -1 \le \beta \le 1 \qquad \alpha^2 + \beta^2=1 \]

    Hence there exists a unique angle \(\varphi\) such that

    \[ \cos \varphi = \alpha \qquad \sin \varphi = \beta \]

    so that \(\eqref{EEEE}\) can be rewritten in the following form

    \[\begin{align} \label{EEEEE} h(t) & = A \: \cos \varphi \: \sin (\omega \:t) + A \: \sin \varphi \: \cos (\omega \:t)\nonumber\\[2ex] & = A \: \sin (\omega\:t + \varphi) \end{align}\]

    Therefore \(h(t)\) represents an elementary vibration with amplitude \(A\), angular frequency \(\omega\), phase-shifted by an angle \(\varphi\).

    Figure 4

In summary:

\[\begin{align*} a\: \sin \:(\omega \: t) + b\: \cos \:(\omega \: t) = A \: \sin (\omega\:t + \varphi)\\[2ex] A = \sqrt{a^2 + b^2} \qquad \begin{cases} a= A \: \cos \varphi \\ b= A \: \sin \varphi \end{cases} \end{align*}\]
  • Under fairly general conditions, a periodic natural phenomenon can be written as the superposition of a finite or infinite number of elementary vibrations with different frequencies (Fourier series).

Try it — the interactive graph below shows what you have just read: move the sliders.

2. Damping and amplification effects

  • By multiplying an elementary vibration by powers or exponentials, we can model damping or amplification effects.

First example

  • For example, consider the function

    \[ h(t) = t \: \sin \:(\omega \: t) \]

    which models an amplified vibration.

  • Since \(-1 \le \sin \:(\omega \: t) \le 1\), we have

    \[ -t \le t \: \sin \:(\omega \: t) \le t \]

    and hence the graph of \(h(t)\) lies between the graphs of the lines with equations \(y = -t\), \(y = t\).

  • At the points where

    \[ \sin \:(\omega\:t) = 1 \]

    i.e.,

    \[ t = \frac{\pi}{2 \: \omega} + k \: \frac{2\:\pi}{\omega} \qquad (k=0,1,2,\dots) \]

    the graph of \(h(t)\) touches that of \(y=t\).

  • At the points where

    \[ \sin \:(\omega\:t) = -1 \]

    i.e.,

    \[ t = \frac{3\: \pi}{2 \: \omega} + k \: \frac{2\:\pi}{\omega} \qquad (k=0,1,2,\dots) \]

    the graph of \(h(t)\) touches that of \(y=-t\).

    Figure 5

  • From the graph we can see that multiplication by \(t\) has the effect of amplifying the vibration as \(t\) increases.

Figure 6

Figure 7

Second example

  • For example, consider the function

    \[ k(t) = e^{-\alpha\:t} \: \sin \:(\omega \: t) \qquad (\alpha > 0) \]

    which models a damped vibration.

  • Recalling that

    \[ e^{-\alpha\:t} = \left(\frac{1}{e^{\alpha}}\right)^t \]

    is an exponential with base less than \(1\), considerations analogous to those made for the function \(h(t)\) show that the graph of \(k\) lies between the graphs of the functions

    \[ y_1 = e^{-\alpha\:t} {\rm ~~~~and~~~~} y_2 = -e^{-\alpha\:t} \]

    as shown in the figure:

Figure 8

Figure 9