Free and Forced Oscillations Cheat Sheet
Understanding the behavior of oscillating systems requires distinguishing between undamped, damped, and forced oscillations, each governed by specific differential equations and characterized by unique parameters.
Core Principles
- Undamped Free Oscillation: System oscillates indefinitely with constant amplitude and frequency when no dissipative forces are present.
- Damped Free Oscillation: Amplitude of oscillation decays over time due to dissipative forces like friction or air resistance.
- Forced Oscillation: External periodic force drives the system, leading to oscillations at the driving frequency, potentially resulting in resonance.
Formulas
- Undamped Free Oscillation: $m\frac{d^2x}{dt^2} + kx = 0$
- Damped Free Oscillation: $m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$
- Forced Oscillation (with damping): $m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F_0 \cos(\omega_d t)$
- Natural Frequency (Undamped): $\\omega_0 = \sqrt{\frac{k}{m}}$
- Damping Ratio: $\\zeta = \frac{c}{2\sqrt{mk}}$
- Resonant Frequency (for underdamped systems): $\\omega_r = \sqrt{\\omega_0^2 - \frac{c^2}{2m^2}} = \omega_0 \sqrt{1 - 2\\zeta^2}$
- Amplitude of Forced Oscillation: $A = \frac{F_0/m}{\sqrt{(\\omega_0^2 - \omega_d^2)^2 + (c\omega_d/m)^2}}$
Key Terms
- Mass (m): The inertia of the oscillating object, measured in kilograms.
- Spring Constant (k): A measure of the stiffness of the spring or restoring force, measured in Newtons per meter.
- Damping Coefficient (c): Represents the strength of the dissipative force, measured in Newton-seconds per meter.
- Driving Force Amplitude (F₀): The maximum magnitude of the external periodic force applied to the system, measured in Newtons.
- Driving Frequency (ωd): The angular frequency of the external periodic force, measured in radians per second.
- Natural Frequency (ω₀): The frequency at which a system would oscillate if there were no damping or external forces, measured in radians per second.
- Resonance: The phenomenon where the amplitude of oscillation becomes very large when the driving frequency is close to the natural frequency of the system.
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