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
$$ f_S = f_1 – f_2 $$