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:
$$ \ddot{\varphi} + \frac{D}{J} \, \sin\varphi = 0 $$

$$ \omega_0 = \sqrt{\frac{D}{J}} \qquad
T = 2 \, \pi \, \sqrt{\frac{J}{D}} $$