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
$$ f_S = f_1 – f_2 $$
| Electromagnetic Oscillation | Mechanics and Oscillation |