Dynamical Modeling with Second-Order Systems: Oscillations, Resonance & Rocket Dynamics

1. Newton's Second Law for the Spring-Mass-Damper System

Consider a mass $m$ attached to a linear spring of stiffness constant $k$ and a viscous damper with damping coefficient $c$. By Newton's second law, $m x'' = F_{\text{spring}} + F_{\text{damping}} + F_{\text{ext}}(t) = -kx - cx' + F(t)$:

Dividing by $m$, we write the canonical form: $x'' + 2\zeta \omega_0 x' + \omega_0^2 x = \frac{F(t)}{m}$, where $\omega_0 = \sqrt{\frac{k}{m}}$ is the undamped natural angular frequency and $\zeta = \frac{c}{2\sqrt{km}}$ is the dimensionless damping ratio.

2. Free Motion Classification ($F(t) \equiv 0$)

• Undamped Motion ($\zeta = 0, c = 0$): Simple Harmonic Motion (SHM): $$x(t) = c_1 \cos(\omega_0 t) + c_2 \sin(\omega_0 t) = A \cos(\omega_0 t - \phi)$$ where amplitude $A = \sqrt{c_1^2 + c_2^2}$ and period $T = \frac{2\pi}{\omega_0}$.
• Overdamped Motion ($\zeta > 1, c^2 > 4km$): Characteristic roots are real, negative, and distinct: $r_{1,2} = -\zeta\omega_0 \pm \omega_0\sqrt{\zeta^2 - 1}$. The system returns to equilibrium non-oscillatingly.
• Critically Damped Motion ($\zeta = 1, c = 2\sqrt{km}$): Repeated real root $r = -\omega_0$. Solution: $$x(t) = (c_1 + c_2 t) e^{-\omega_0 t}$$ This provides the fastest possible return to equilibrium without oscillation (crucial in automobile shock absorbers and gun recoil mechanisms).
• Underdamped Motion ($\zeta < 1, c^2 < 4km$): Complex roots $r = -\alpha \pm i \omega_d$, where $\alpha = \zeta \omega_0 = \frac{c}{2m}$ and $\omega_d = \omega_0 \sqrt{1 - \zeta^2}$ is the quasi-frequency. $$x(t) = A e^{-\alpha t} \cos(\omega_d t - \phi)$$ The amplitude decays exponentially. The rate of decay is quantified by the logarithmic decrement $\delta$: $$\delta = \ln\left(\frac{x(t)}{x(t + T_d)}\right) = \alpha T_d = \frac{2\pi \zeta}{\sqrt{1 - \zeta^2}}$$

1. Forced Oscillations with Harmonic Excitation

When subjected to a periodic external force $F(t) = F_0 \cos(\omega t)$:

The general solution consists of a transient complementary solution $x_c(t)$ (which dies out exponentially as $t \to \infty$) and a persistent steady-state particular solution $x_p(t) = x_{ss}(t)$:

Using the method of undetermined coefficients, the steady-state amplitude is derived exactly as:

and the phase lag is $\phi = \arctan\left(\frac{c\omega/m}{\omega_0^2 - \omega^2}\right) = \arctan\left(\frac{2\zeta (\omega/\omega_0)}{1 - (\omega/\omega_0)^2}\right)$.

2. Resonance Phenomena

• Pure Secular Resonance (No Damping, $\zeta = 0$): When $\omega = \omega_0$, the denominator vanishes. The particular solution undergoes linear secular growth: $$x_p(t) = \frac{F_0}{2 m \omega_0} t \sin(\omega_0 t)$$ The amplitude grows without bound as $t \to \infty$, leading to structural destruction.
• Resonance with Damping ($\zeta < 1/\sqrt{2}$): Maximizing $X(\omega)$ by minimizing the radicand yields the resonant frequency $\omega_r$: $$\omega_r = \omega_0 \sqrt{1 - 2\zeta^2}$$ At resonance, the peak amplitude is $X_{\text{max}} = \frac{F_0/k}{2\zeta\sqrt{1 - \zeta^2}}$. The sharpness of the resonance peak is measured by the Quality Factor $Q$: $$Q = \frac{1}{2\zeta} = \frac{\sqrt{km}}{c}$$

1. Series RLC Electrical Network Modeling

By Kirchhoff's voltage law, the sum of voltage drops across an inductor $L$, resistor $R$, and capacitor $C$ in a closed loop equals the applied electromotive force $E(t)$:

Since electric current is the time derivative of charge, $I(t) = \frac{dq}{dt}$, substituting yields the second-order linear ODE for charge $q(t)$:

Differentiating with respect to $t$ yields the dual equation for the electric current $I(t)$:

Mechanical-Electrical Analogy: Inductance $L \leftrightarrow$ Mass $m$; Resistance $R \leftrightarrow$ Damping $c$; Inverse capacitance $1/C \leftrightarrow$ Spring stiffness $k$; Charge $q \leftrightarrow$ Position $x$; Current $I \leftrightarrow$ Velocity $v$.

2. Rocket Motion Dynamics (Variable Mass Systems)

A rocket burning propellant is an open variable-mass mechanical system. Let $M(t) = M_0 - \alpha t$ denote the total mass of the rocket at time $t$, where $\alpha = -\frac{dM}{dt}$ is the constant mass burn rate, and let $u$ denote the exhaust velocity of gases relative to the rocket.

By conservation of momentum in a vertical gravitational field $g$ with linear atmospheric drag $-k v$:

Dividing by $M(t)$ yields the first-order linear differential equation for velocity $v(t)$:

In the vacuum limit ($k = 0$), integrating directly yields the famous Tsiolkovsky rocket equation: