Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is
executed by any quantity obeying the Differential Equation

(1) |

(2) | |||

(3) |

where the two constants and (or and ) are determined from the initial conditions.

Many physical systems undergoing small displacements, including any objects obeying Hooke's law, exhibit simple harmonic motion. This equation arises, for example, in the analysis of the flow of current in an electronic CL circuit (which contains a capacitor and an inductor ). If a damping force such as Friction is present, an additional term must be added to the Differential Equation and motion dies out over time.

Adding a damping force proportional to , the first derivative of with
respect to time, the equation of motion for *damped* simple harmonic motion is

(4) |

(5) |

(6) |

(7) |

(8) |

- 1. is Positive: overdamped,
- 2. is Zero: critically damped,
- 3. is Negative: underdamped.

If a periodic (sinusoidal) forcing term is added at angular frequency , the same three solution regimes are again obtained. Surprisingly, the resulting motion is still periodic (after an initial transient response, corresponding to the solution to the unforced case, has died out), but it has an amplitude different from the forcing amplitude.

The ``particular'' solution to the forced second-order nonhomogeneous Ordinary Differential Equation

(9) |

(10) |

(11) |

Critical damping is a special case of damped simple harmonic motion in which

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

(18) |

(19) | |||

(20) |

so

(21) | |||

(22) |

For sinusoidally forced simple harmonic motion with critical damping, the equation of motion is

(23) |

(24) |

Plugging this into the equation for the particular solution gives

(25) |

In order to put this in the desired form, note that we want to equate

(26) |

This means

(27) | |||

(28) |

so

(29) | |||

(30) |

Plugging in,

(31) | |||

(32) |

The solution in the requested form is therefore

(33) |

where is defined by (32).

Overdamped simple harmonic motion occurs when

(34) |

(35) |

(36) | |||

(37) |

where

(38) |

(39) |

(40) | |||

(41) |

so

(42) | |||

(43) |

For a cosinusoidally forced overdamped oscillator with forcing function
, the particular solutions
are

(44) | |||

(45) |

where

(46) | |||

(47) |

These give the identities

(48) | |||

(49) |

and

(50) |

The Wronskian is

(51) |

The particular solution is

(52) |

(53) | |||

(54) |

Therefore,

(55) |

where

(56) |

Underdamped simple harmonic motion occurs when

(57) |

(58) |

(59) |

(60) |

(61) |

(62) |

(63) |

(64) |

(65) | |||

(66) |

so the general solution is

(67) |

(68) | |||

(69) |

so and can be expressed in terms of the initial conditions by

(70) | |||

(71) |

For a cosinusoidally forced underdamped oscillator with forcing function
, use

(72) | |||

(73) |

to obtain

(74) | |||

(75) | |||

(76) |

The particular solutions are

(77) | |||

(78) |

The Wronskian is

(79) |

The particular solution is given by

(80) |

(81) | |||

(82) |

Using computer algebra to perform the algebra, the particular solution is

(83) |

where

(84) |

(85) |

(86) |

© 1996-9

1999-05-26