Mechanical Vibrations

Definition

Mechanical vibrations are modeled by second-order linear ODEs describing the displacement $y(t)$ of a mass $m$ attached to a spring, subject to damping $\delta$ and external forcing $F(t)$: $my'' + \delta y' + ky = F(t)$

• How to read: “The mass m times the second derivative of y, plus the damping constant delta times the first derivative of y, plus the spring constant k times y, is equal to the forcing function F of t.”
• Meaning: Mass times acceleration plus damping force plus spring force equals external driving force—the full vibration equation.

Why It Matters

Resonance can destroy a bridge or a rocket; understanding mechanical vibrations is essential for designing systems that can withstand the cyclic stresses of their environment without shaking themselves to pieces.

Core Concepts

• Free Vibrations ($F(t)=0$):

  • Undamped ($\delta=0$): Simple Harmonic Motion. $y(t) = A\cos(\omega t - \phi)$ where $\omega = \sqrt{k/m}$.

    • How to read: “The position y of t is equal to the amplitude A times the cosine of the quantity omega t minus phi; where the angular frequency omega is the square root of k divided by m.”
    • Meaning: No damping gives perpetual oscillation at natural frequency $\omega = \sqrt{k/m}$.

  • Damped ($\delta > 0$): Behavior depends on the damping ratio $\zeta = \delta / (2\sqrt{mk})$:

    • How to read: “The damping ratio zeta is equal to delta divided by two times the square root of m times k.”

    • Meaning: $\zeta$ compares damping to critical damping; determines under/critical/over-damped behavior.

      • Underdamped ($\zeta < 1$): Oscillations with decaying amplitude $e^{-(\delta/2m)t}$.
      • Critically Damped ($\zeta = 1$): Fastest return to equilibrium without oscillation.
      • Overdamped ($\zeta > 1$): Sluggish return to equilibrium with no oscillation.

• Forced Vibrations ($F(t) \neq 0$): External energy input driving the system.

• Resonance: Occurs when the forcing frequency matches the natural frequency $\omega$ in an undamped system, leading to unbounded amplitude.

Connected Concepts

• Physics
• Second-Order Linear Homogeneous Equations
• Thermodynamics
• Conformal Maps