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

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