Types of Oscillation

 $F_R$ $\mathrm{N}$ repelling force $F_D$ $\mathrm{N}$ damping force, frictional force $F_E$ $\mathrm{N}$ exciting force $k$ $\mathrm{N/m}$ spring constant $m$ $\mathrm{kg}$ mass $a$ $\mathrm{m/s^2}$ acceleration $v$ $\mathrm{m/s}$ velocity $t$ $\mathrm{s}$ time $x$ $\mathrm{m}$ deflection, elongation $\varphi$ $\mathrm{rad}$ excursion angle $\omega_0$ $\mathrm{s^{-1}}$ angular eigenfrequency $\omega_d$ $\mathrm{s^{-1}}$ eigenfrequency of damped oscillation $\omega_E$ $\mathrm{s^{-1}}$ exciting frequency, eigenfrequency in steady state $\omega_{RES}$ $\mathrm{s^{-1}}$ resonance frequency $T_d$ $\mathrm{s}$ period of damped oscillation $\mu$ friction coefficient $b$ $\mathrm{kg/s}$ damping constant $\delta$ $\mathrm{s^{-1}}$ decay coefficient $D$ damping ratio $d$ dissipation factor $G$ quality, resonance peak $\Lambda$ logarithmic decrement $\alpha$ $\mathrm{rad}$ phase delay of resonator with respect to exciter $x_{RES}$ $\mathrm{m}$ resonance amplitude $x_R$ $\mathrm{m}$ resulting amplitude in superposition $x_{STAT}$ $\mathrm{m}$ static deflection at constant force $f_S$ $\mathrm{Hz}$ beat frequency

Free Damped Oscillation

$$F_R = m \, a \qquad F_R = – k \, x – – F_D$$

Friction independent on velocity

$$F_D = \mu \, F_N$$

Differential equation:

$$m \, \ddot{x} + \mu \, F_N + k \, x = 0$$

$$x = \left( \hat{x} + x_0 \right) \, \cos\left( \omega_0 \, t \varphi_0 \right) – x_0$$

Friction dependent on velocity, viscous friction

$$F_D = b \, v$$

Differential equation:

$$m \, \ddot{x} + b \, \dot{x} + k \, x = 0$$

$$\ddot{x} + 2 \,\delta\, \dot{x} + \omega_0^2 \, x = 0 \qquad \ddot{x} + 2 \, D \, \omega_0 \, \dot{x} + \omega_0^2 \, x = 0$$

$$\omega_0 = \sqrt{\frac{k}{m}} \qquad \delta = \frac{b}{2 \, m} \qquad D = \frac{\delta}{\omega_0} \qquad G = \frac{1}{D}$$

$$d = 2 \, D \qquad d = \frac{b}{m \, \omega_0} \qquad d = \frac{b}{ \sqrt{ m \, k }}$$

Underdamped Case

Low damping

 Conditions: $\delta < \omega_0$ $D < 1$

$$x(t) = \hat{x}_0 \, \mathrm{e}^{-\delta\,t} \, \cos\left( \omega_d \, t + \varphi_0 \right)$$

$$\omega_d = \sqrt{ \frac{k}{m} – \frac{b^2}{4 \, m^2} } \qquad \omega_d = \sqrt{ \omega_0^2 – \delta^2 } \qquad \omega_d = \omega_0^2 \, \sqrt{1 – D^2}$$

$$\frac{\hat{x}_i}{\hat{x}_{i+1}} = \mathrm{e}^{-\delta\,T_d} \qquad \Lambda = \ln \frac{\hat{x}_i}{\hat{x}_{i+1}} \qquad \Lambda = \delta \, T_d$$

Aperiodic Borderline Case

Medium damping

 Conditions: $\delta = \omega_0$ $D = 1$

$$x(t) = \left( \hat{x}_0 + \hat{x}_1 \, t \right) \, \mathrm{e}^{-\delta\,t} \qquad x(t) = \hat{x}_0 \, \left( 1 + \delta \, t \right) \, \mathrm{e}^{-\delta\,t}$$

$$b = 2 \, \sqrt{ m \, k }$$

Overdamped Case

Strong damping

 Conditions: $\delta > \omega_0$ $D > 1$

$$x(t) = \hat{x}_0 \, \mathrm{e}^{-\delta\,t} \, \cosh ( \omega_d’ \, t )$$

Eigenfrequency becomes imaginary:

$$\omega_d’ = j \, \sqrt{ \delta^2 – \omega^2 }$$

$$b > 2 \, \sqrt{ m \, k }$$

Forced Oscillation

 Condition: $\sum$ external forces $= 0$

$$F_E + F_R + F_D = m \, \ddot{x}$$

$$F_E = \hat{F}_E \, \cos ( \omega_E \, t ) \qquad F_R = – k \, x$$

$$F_D = – b \, \dot{x} \qquad F_D = – b \, v$$

Differential equation:

$$m \, \ddot{x} + b \, \dot{x} + k \, x = F_E(t)$$

$$\ddot{x} = 2 \, \delta \, \dot{x} + \omega_0^2 \, x \qquad \ddot{x} = \frac{\hat{F_E}}{m} \, \cos( \omega_E \, t )$$

$$\alpha = \arctan \frac{\omega_E \, b}{m \, (\omega_0^2 – \omega_E^2) } \qquad \alpha = \arctan \frac{ 2\, \omega_E \, \delta}{\omega_0^2 – \omega_E^2}$$

$$x(t) = \hat{x} \, \omega_E \, \sin ( \omega_E \, t + \varphi_0 \, \omega_E ) \qquad \hat{x} = \frac{ \hat{F_E} }{\sqrt{ (m \, \omega_0^2 – k^2) + b^2\, \omega_E^2 }}$$

$$\varphi(\omega_E) = \arctan \frac{b \, \omega_E}{m \, \omega_E^2 – k}$$

Resonance

$$\omega_{RES} = \sqrt{ \omega_0^2 – \frac{b^2}{2 \, m^2} } \qquad \omega_{RES} = \sqrt{ \omega_0^2 – 2 \, \delta^2 }$$

$$\hat{x}_{RES} = \frac{\hat{F}_E}{ b \, \sqrt{\omega_0^2 – \delta^2} } \qquad \hat{x}_{RES} = \frac{\hat{F}_E}{b \, \omega_d^2} \qquad \hat{x}_{RES} = \frac{\hat{F}_E}{2 \, \delta \, m \, \omega_d}$$

$$G = \frac{\pi \, \omega_0^2}{\Lambda \, \omega_d^2} \qquad G \approx \frac{\pi}{\Lambda}$$

$$\frac{\hat{x}}{\hat{x}_{STAT}} = \frac{\omega_0^2}{\sqrt{ (\omega_0^2 – \omega_E^2)^2 + (2\,\delta\,\omega_E)^2 }}$$

Superposition

The same orientation and the same frequency

$$\hat{x}_R = \sqrt{ \hat{x}_1^2 + \hat{x}_2^2 + 2 \, \hat{x}_1 \, \hat{x}_2 \, \cos( \varphi_{01} – \varphi_{02}) }$$

$$\varphi_{0R} = \arctan \frac{\hat{x}_1 \,\sin \varphi_{01} + \hat{x}_2 \,\sin \varphi_{02} }{\hat{x}_1 \,\cos \varphi_{01} + \hat{x}_2 \,\cos \varphi_{02}}$$

 Condition: $\hat{x}_1 = \hat{x}_2$

$$\hat{x}_R = 2 \, \hat{x}_1 \, \cos\frac{\varphi_{01} – \varphi_{02}}{2}$$

 Condition: $\Delta\varphi = \pi$

Cancelling, subtraction of amplitudes

The same orientation but different frequencies:
beat, low differences of frequencies

$$x_R(t) = 2 \, \hat{x} \, \cos\left( \frac{\omega_1 – \omega_2}{2}\, t \right) \, \sin\left( \frac{\omega_1 + \omega_2}{2}\, t \right)$$

beat frequency
$$f_S = f_1 – f_2$$