Linear-rotational Analogs - Mechanical Oscillators

Mechanical Oscillators

SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.

Equations of motion
Physical situation Nomenclature Translational equations Angular equations
SHM
  • x = Transverse displacement
  • θ = Angular displacement
  • A = Transverse amplitude
  • Θ = Angular amplitude

Solution:

Solution:
Unforced DHM
  • b = damping constant
  • κ = torsion constant

Solution (see below for ω'):

Resonant frequency:

Damping rate:

Expected lifetime of excitation:

Solution:

Resonant frequency:

Damping rate:

Expected lifetime of excitation:
Angular frequencies
Physical situation Nomenclature Equations
Linear undamped unforced SHO
  • k = spring constant
  • m = mass of oscillating bob
Linear unforced DHO
  • k = spring constant
  • b = Damping coefficient
Low amplitude angular SHO
  • I = Moment of inertia about oscillating axis
  • κ = torsion constant
Low amplitude simple pendulum
  • L = Length of pendulum
  • g = Gravitational acceleration
  • Θ = Angular amplitude
 Approximate value

Exact value can be shown to be:
Energy in mechanical oscillations
Physical situation Nomenclature Equations
SHM energy
  • T = kinetic energy
  • U = potenial energy
  • E = total energy
 Potential energy

Maximum value at x = A:

Kinetic energy

Total energy
DHM energy

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