$$\require{cancel}$$

# 4: Oscillations

• • Johan Wevers
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Harmonic oscillation

The general form of a harmonic oscillation is: $$\Psi(t)=\hat{\Psi}{\rm e}^{i(\omega t\pm\varphi)}\equiv\hat{\Psi}\cos(\omega t\pm\varphi)$$,

where $$\hat{\Psi}$$ is the amplitude. A superposition of several harmonic oscillations with the same frequency results in another harmonic oscillation: $\sum_i \hat{\Psi}_i\cos(\alpha_i\pm\omega t)=\hat{\Phi}\cos(\beta\pm\omega t)$ with:

$\tan(\beta)=\frac{\sum\limits_i\hat{\Psi}_i\sin(\alpha_i)}{\sum\limits_i\hat{\Psi}_i\cos(\alpha_i)}~~~\mbox{and}~~~ \hat{\Phi}^2=\sum_i\hat{\Psi}^2_i+2\sum_{j>i}\sum_i\hat{\Psi}_i\hat{\Psi}_j\cos(\alpha_i-\alpha_j)$

For harmonic oscillations: $$\displaystyle\int x(t)dt=\frac{x(t)}{i\omega}$$ and $$\displaystyle\frac{d^nx(t)}{dt^n}=(i\omega)^n x(t)$$.

## Mechanic oscillation

For a spring with constant $$C$$ and damping $$k$$ which is connected to a mass $$M$$, to which a periodic force $$F(t)=\hat{F}\cos(\omega t)$$ is applied the equation of motion is $$m\ddot{x}=F(t)-k\dot{x}-Cx$$. With complex amplitudes, this becomes $$-m\omega^2 x=F-Cx-ik\omega x$$. With $$\omega_0^2=C/m$$ it follows that:

$x=\frac{F}{m(\omega_0^2-\omega^2)+ik\omega}~~,\mbox{and for the velocity:}~~ \dot{x}=\frac{F}{i\sqrt{Cm}\delta+k}$

where $$\displaystyle\delta=\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega}$$. The quantity $$Z=F/\dot{x}$$ is called the impedance of the system. The quality of the system is given by $$\displaystyle Q=\frac{\sqrt{Cm}}{k}$$.

The frequency with minimal $$|Z|$$ is called the velocity resonance frequency. This is equal to $$\omega_0$$. In the resonance curve $$|Z|/\sqrt{Cm}$$ is plotted against $$\omega/\omega_0$$. The width of this curve is characterized by the points where $$|Z(\omega)|=|Z(\omega_0)|\sqrt{2}$$. At these points: $$R=X$$ and $$\delta=\pm Q^{-1}$$, and the width is $$2\Delta\omega_{\rm B}=\omega_0/Q$$.

The stiffness of an oscillating system is given by $$F/x$$. The amplitude resonance frequency $$\omega_{\rm A}$$ is the frequency where $$i\omega Z$$ is a minimum. This is the case for $$\omega_{\rm A}=\omega_0 \sqrt{1-\frac{1}{2} Q^2}$$.

The damping frequency $$\omega_{\rm D}$$ is a measure for the time in which an oscillating system comes to rest. It is given by $$\displaystyle\omega_{\rm D}=\omega_0\sqrt{1-\frac{1}{4Q^2}}$$. A weak damped oscillation $$(k^2<4mC)$$ dies out after $$T_{\rm D}=2\pi/\omega_{\rm D}$$. For a critically damped oscillation $$(k^2=4mC)$$ $$\omega_{\rm D}=0$$. A strong damped oscillation $$(k^2>4mC)$$ decays like (if $$k^2\gg 4mC$$) $$x(t)\approx x_0\exp(-t/\tau)$$.

## Electric oscillations

The impedance is given by: $$Z=R+iX$$. The phase angle is $$\varphi:=\arctan(X/R)$$. The impedance of a resistor is $$R$$, of a capacitor $$1/i\omega C$$ and of a self inductor $$i\omega L$$. The quality of a coil is $$Q=\omega L/R$$. The total impedance in case several elements are connected is given by:

1. Series connection: $$V=IZ$$,
$Z_{\rm tot}=\sum_i Z_i~,~~L_{\rm tot}=\sum_i L_i~,~~ \frac{1}{C_{\rm tot}}=\sum_i\frac{1}{C_i}~,~~Q=\frac{Z_0}{R}~,~~ Z=R(1+iQ\delta)$
2. Parallel connection: $$V=IZ$$,
$\frac{1}{Z_{\rm tot}}=\sum_i\frac{1}{Z_i}~,~~ \frac{1}{L_{\rm tot}}=\sum_i\frac{1}{L_i}~,~~ C_{\rm tot}=\sum_i C_i~,~~Q=\frac{R}{Z_0}~,~~ Z=\frac{R}{1+iQ\delta}$ Here, $$\displaystyle Z_0=\sqrt{\frac{L}{C}}$$ and $$\displaystyle\omega_0=\frac{1}{\sqrt{LC}}$$.

The power from a source is given by $$P(t)=V(t)\cdot I(t)$$, so $$\left\langle P \right\rangle_t=\hat{V}_{\rm eff}\hat{I}_{\rm eff}\cos(\Delta\phi)$$
$$= \frac{1}{2} \hat{V}\hat{I}\cos(\phi_v-\phi_i)= \frac{1}{2} \hat{I}^2{\rm Re}(Z)= \frac{1}{2} \hat{V}^2{\rm Re}(1/Z)$$, where $$\cos(\Delta\phi)$$ is the work factor.

## Waves in long conductors

If cables are used for signal transfer, e.g. coax cables then: $$\displaystyle Z_0=\sqrt{\frac{dL}{dx}\frac{dx}{dC}}$$.
The transmission velocity is given by $$\displaystyle v=\sqrt{\frac{dx}{dL}\frac{dx}{dC}}$$.

## Coupled conductors and transformers

For two coils enclosing each others flux if $$\Phi_{12}$$ is the part of the flux originating from $$I_2$$ through coil 2 which is enclosed by coil 1, then $$\Phi_{12}=M_{12}I_2$$, $$\Phi_{21}=M_{21}I_1$$. The coefficients of mutual induction $$M_{ij}$$ is given by:

$M_{12}=M_{21}:=M=k\sqrt{L_1L_2}=\frac{N_1\Phi_1}{I_2}=\frac{N_2\Phi_2}{I_1}\sim N_1N_2$

where $$0\leq k\leq1$$ is the coupling factor. For a transformer  $$k\approx1$$. At full load:

$\frac{V_1}{V_2}=\frac{I_2}{I_1}=-\frac{i\omega M}{i\omega L_2+R_{\rm load}}\approx-\sqrt{\frac{L_1}{L_2}}=-\frac{N_1}{N_2}$

## Pendulums

The oscillation time $$T=1/f$$, for different types of pendulums is given by:

• Oscillating spring: $$T=2\pi\sqrt{m/C}$$ if the spring force is given by $$F=C\cdot\Delta l$$.
• Physical pendulum: $$T=2\pi\sqrt{I/\tau}$$ with $$\tau$$ the moment of force and $$I$$ the moment of inertia.
• Torsion pendulum: $$T=2\pi\sqrt{I/\kappa}$$ where $$\displaystyle\kappa=\frac{2lm}{\pi r^4\Delta\varphi}$$ is the constant of torsion and $$I$$ the moment of inertia.
• Mathematical pendulum: $$T=2\pi\sqrt{l/g}$$ with $$g$$ the acceleration of gravity and $$l$$ the length of the pendulum.
Template:HypTest

4: Oscillations is shared under a CC BY license and was authored, remixed, and/or curated by Johan Wevers.