4: Oscillations

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.