A mass $m$ attached to a spring with spring constant $k$ and damping coefficient $c$ is modeled by Newton's Second Law. An external driving force $F(t)$ can be applied to the mass.

Dividing the general equation by $m$ gives the standard form, which uses the natural frequency $\omega_n$ and the damping ratio $\zeta$.

• Free Undamped Motion ($c=0, F(t)=0$): Simple Harmonic Motion ($x'' + \omega_n^2 x = 0$). The solution is $x(t) = C_1\cos(\omega_nt) + C_2\sin(\omega_nt)$.

• Free Damped Motion ($c > 0, F(t)=0$): The auxiliary equation is $m r^2 + cr + k = 0$. The roots determine the type of damping based on the discriminant $D = c^2 - 4mk$.

  • Overdamped ($D > 0$): Two real distinct roots. System slowly returns to equilibrium without oscillating. $x(t) = c_1e^{r_1t} + c_2e^{r_2t}$.
  • Critically Damped ($D = 0$): One repeated real root. System returns to equilibrium as fast as possible without oscillating. $x(t) = c_1e^{rt} + c_2te^{rt}$.
  • Underdamped ($D < 0$): Complex conjugate roots ($\alpha \pm \beta i$). System oscillates with exponentially decreasing amplitude. $x(t) = e^{\alpha t}(c_1\cos(\beta t) + c_2\sin(\beta t))$.
• Forced Motion ($F(t) \neq 0$): Add the particular solution $x_p(t)$. Resonance occurs when the driving frequency matches the natural frequency ($\omega = \omega_n$) in an undamped system, causing amplitude to grow infinitely over time ($x_p(t) \propto t\sin(\omega_nt)$).

A simple pendulum of mass $m$ attached to a string of length $L$ swinging under gravity $g$ is governed by a non-linear differential equation.

For small angles ($\theta \approx 0$), we can use the Taylor series approximation $\sin(\theta) \approx \theta$. The linearized equation becomes a simple harmonic oscillator where the natural frequency is $\omega_n = \sqrt{g/L}$, and the period is $T = 2\pi\sqrt{L/g}$.

An electrical circuit with a resistor ($R$), inductor ($L$), and capacitor ($C$) connected in series is modeled by Kirchhoff's Voltage Law. The charge on the capacitor $q(t)$ and the current $i(t)$ exhibit behavior directly analogous to a damped mechanical system.

• Mass $m \leftrightarrow$ Inductance $L$
• Damping $c \leftrightarrow$ Resistance $R$
• Spring Constant $k \leftrightarrow$ Inverse Capacitance $1/C$
• Displacement $x(t) \leftrightarrow$ Charge $q(t)$
• Velocity $v(t) \leftrightarrow$ Current $i(t)$

The vertical deflection $y(x)$ of an elastic beam under a distributed transverse load $w(x)$ is governed by a fourth-order linear differential equation. This is solved by integrating four times and applying boundary conditions at the supports (e.g., fixed ends have $y=0, y'=0$; pinned ends have $y=0, y''=0$).