Mechanical Oscillation
| $F_I$ | $\mathrm{N}$ | internal forces |
| $F_R$ | $\mathrm{N}$ | repelling force |
| $F_G$ | $\mathrm{N}$ | weight force |
| $m$ | $\mathrm{kg}$ | mass |
| $a$ | $\mathrm{m/s^2}$ | acceleration |
| $k$ | $\mathrm{N/m}$ | spring constant |
| $x$ | $\mathrm{m}$ | deflection, elongation |
| $s_h$ | $\mathrm{m}$ | horizontal deflection |
| $\varphi$ | $\mathrm{rad}, \deg$ | excursion angle |
| $\omega_0$ | $\mathrm{s^{-1}}$ | angular eigenfrequency |
| $f$ | $\mathrm{Hz}$ | eigenfrequency |
| $T$ | $\mathrm{s}$ | period duration |
| $l$ | $\mathrm{m}$ | thread length |
| $g$ | $\mathrm{m/s^2}$ | gravitational acceleration |
| $M_R$ | $\mathrm{Nm}$ | repelling torque |
| $J_A$ | $\mathrm{kg\,m^2}$ | moment of inertia referred to center of rotation |
| $J_S$ | $\mathrm{kg\,m^2}$ | moment of inertia referred to center of gravity |
| $r$ | $\mathrm{m}$ | distance between center of rotation and center of gravity |
| $\alpha$ | $\mathrm{s^{-2}}$ | angular acceleration |
| $D$ | $\mathrm{Nm/rad}$ | torsional coefficient, torsional spring constant |
| $t$ | $\mathrm{s}$ | time |
Spring-Mass System
1-D free, straight, and undamped oscillation.
| Condition: | $\sum$ external forces $= \sum F_A = 0$ |
$$ \sum F_I = m \, a $$
$$ F_R = – k \, x \qquad
F_R = m \, a $$
$$ m \, a + k \, x = 0 \qquad
m \, \ddot{x} + k \, x = 0 $$
Differential equation:
$$ \ddot{x} + \frac{k}{m} \, x = 0 $$
$$ \omega_0 = \sqrt{\frac{k}{m}} \qquad
\ddot{\omega} + \omega_0^2 \, x = 0 $$
$$ T = 2\,\pi\, \sqrt{\frac{m}{k}} \qquad
f = \frac{1}{2\,\pi} \, \sqrt{\frac{k}{m}} $$
Phase-shifted oscillation
$$ x(t) = \hat{x} \, \sin \left( \omega_0 \, t + \varphi_0 \right) $$
Mathematical Pendulum
Horizontal deflection
$$ s_h = l \, \sin\varphi \qquad
F_R = F_G \, \sin\varphi $$
Differential equation:
$$ \ddot{\varphi} + \frac{g}{l} \, \sin\varphi = 0 $$
| Condition: | $\sin\varphi \approx \varphi $ for $ \varphi < 5\deg$ |
Simplified:
$$ \ddot{\varphi} + \frac{g}{l} \, \varphi = 0 $$
$$ \varphi(t) = \hat{\varphi} \, \cos\left( \omega_0 \, t \right) \qquad
\varphi(t) = \hat{\varphi} \, \cos\left( \omega_0 \, t + \varphi_0 \right) $$
$$ \omega_0 = \sqrt{\frac{k}{m}} \qquad
T = 2 \, \pi \, \sqrt{\frac{m}{k}} $$
$$ k = \frac{F_R}{s_h} \qquad
k = \frac{m \, g}{l} $$
Eigenfrequency and period duration are independent on mass:
$$ \omega_0 = \sqrt{\frac{g}{l}} \qquad
T = 2 \, \pi \, \sqrt{\frac{l}{g}} $$
Physical Pendulum
$$ M_R = \vec{r} \times \vec{F}_G \qquad
M_R = r \, m \, g \, \sin\varphi $$
$$ \sum M = J \, \alpha \qquad
r \, m \, g \, \sin\varphi + J_A \, \alpha = 0 $$
Huygens–Steiner theorem:
$$ J_A = J_S + m \, r^2 $$
Differential equation:
$$ \ddot{\varphi} + \frac{m\, g\, r}{J_A} \, \sin\varphi = 0 $$
| Condition: | $\sin\varphi \approx \varphi $ for $ \varphi < 5\deg$ |
Simplified:
$$ \ddot{\varphi} + \frac{m\, g\, r}{J_A} \, \varphi = 0 $$
$$ \varphi(t) = \hat{\varphi} \, \cos\left( \omega_0 \, t \right) \qquad
\varphi(t) = \hat{\varphi} \, \cos\left( \omega_0 \, t + \varphi_0 \right) $$
$$ \omega_0 = \sqrt{\frac{m \, g \, r}{J_A}} \qquad
\omega_0 = \sqrt{\frac{g}{r}} $$
$$ T = 2 \, \pi \, \sqrt{\frac{J_A}{m\, g\, r}} \qquad
\alpha = \frac{m\, g\, r}{J_A} \, \varphi $$
$$ J_S = m \, r \, \left( \frac{g\, T^2}{4\, \pi^2} – r \right) $$
Torsional Pendulum
$$ M_R = \vec{r} \times \vec{F}_R \qquad
M_R = J \, \alpha \qquad
M_R = – D \, \varphi $$
$$ \sum M = J \, \alpha \qquad
\alpha = \ddot{\varphi} $$
$$ J = – D \, \frac{\varphi}{\alpha} \qquad
J = \frac{T^2}{4 \, \pi^2} \, D $$
Differential equation:
$$ \ddot{\varphi} + \frac{D}{J} \, \sin\varphi = 0 $$
$$ \omega_0 = \sqrt{\frac{D}{J}} \qquad
T = 2 \, \pi \, \sqrt{\frac{J}{D}} $$
| Deformable Body Mechanics | Mechanics and Oscillation | Electromagnetic Oscillation |