Damped Harmonic Oscillation

Consider the mass-spring system discussed in Section 2.1. Suppose that, as it slides over the horizontal surface, the mass is subject to a frictional damping force that opposes its motion, and is directly proportional to its instantaneous velocity. It follows that the net force acting on the mass when its instantaneous displacement is takes the form

where is the mass, the spring force constant, and a constant (with the dimensions of angular frequency) that parameterizes the strength of the damping. The time evolution equation of the system thus becomes [cf., Equation (2)]

where is the undamped oscillation frequency [cf., Equation (6)]. We shall refer to the preceding equation as the

Let us search for a solution to Equation (63) of the form

where , , , and are all constants. By analogy with the discussion in Section 2.1, we can interpret the preceding solution as a periodic oscillation, of fixed angular frequency , and phase angle , whose amplitude decays exponentially in time as . It can be demonstrated that

(65) | ||

(66) |

Hence, Equation (63) becomes

0 | ||

(67) |

The only way that the preceding equation can be satisfied at all times is if the (constant) coefficients of and separately equate to zero, so that

(68) | ||

(69) |

These equations can be solved to give

(70) |

and

(71) |

Thus, the solution to the damped harmonic oscillator equation is written

assuming that (because cannot be negative). We conclude that the effect of a relatively small amount of damping, parameterized by the

Although the angular frequency, , and decay rate, , of the damped harmonic oscillation specified in Equation (72) are determined by the constants appearing in the damped harmonic oscillator equation, (63), the initial amplitude, , and the phase angle, , of the oscillation are determined by the initial conditions. In fact, if and then it follows from Equation (72) that

(73) | ||

(74) |

giving

(75) | ||

(76) |

The damped harmonic oscillator equation is a linear differential equation. In other words, if is a solution then so is , where is an arbitrary constant. It follows that the solutions of this equation are superposable, so that if and are two solutions corresponding to different initial conditions then is a third solution, where and are arbitrary constants.

Multiplying the damped harmonic oscillator equation, (63), by , we obtain

(77) |

which can be rearranged to give

where

(79) |

is the total energy of the system: that is, the sum of the kinetic and potential energies. Because the right-hand side of (78) cannot be positive, and is only zero when the system is stationary, the total energy is not a conserved quantity, but instead decays monotonically in time due to the action of the damping. The net rate at which the force (62) does work on the mass is

(80) |

The spring force (i.e., the first term on the right-hand side) does negative work on the mass (i.e., it reduces the system kinetic energy) when and are of the same sign, and does positive work when they are of the opposite sign. It can be demonstrated that, on average, the spring force does no net work on the mass during an oscillation cycle. The damping force, on the other hand, (i.e., the second term on the right-hand side) always does negative work on the mass, and, therefore, always acts to reduce the system kinetic energy.