Previous:  Damped Oscillations

The previous section looked at the problem of damped harmonic motion, in which both a linear restoring force and a velocity dependent force act on our block of mass m.  We will now extend this further by considering an additional force, an external force that varies sinusoidally with time.  The following diagram illustrates this.

The equation of motion can then be written in the form

where we define two new constants:

Note that we choose to write the sinusoidal driving force as a complex exponential.  This is for algebraic convenience.  What it does mean is that the displacement x is now strictly a complex number.  However, it is only the real part of x that we are interested in, and as long as we bear this in mind, there is no mathematical or physical difficulty with writing the driving force as a complex exponential.

The solution of this equation consists of two components:

1.  A solution of the homogeneous equation, that is the equation of motion with the right-hand side set to zero;

2.  A particular solution to the full equation, which we must seek by finding a trial solution.

The homogeneous equation was solved in the previous article, and as we saw there the solution to the homogeneous equation shows exponential decay in all cases.  This “transient” solution is not of interest here, and we will not consider it further.

To find a particular solution of the differential equation, we try a (complex) solution of the form

Substituting this trial solution into the differential equation and equating real and imaginary components on the left and right-hand sides yields the following, after some fairly simple algebra:

The most interesting aspect of this result is that the amplitude of the motion A is now a function of the driving frequency ω.  In particular, it takes on a maximum value that is give by

This frequency is known as the “amplitude resonant frequency”.  The maximum value for the amplitude is then found by substituting this frequency into the expression for A.

The occurrence of a maximum amplitude of oscillation is referred to as “amplitude resonance”.  It should be noted that if the damping is sufficiently strong, then resonance does not occur, because the square root in the expression for the resonant frequency must be a real number.  The condition for resonance is therefore

The following graph illustrates the variation of the amplitude A with frequency ω, for varying degrees of damping.  In all cases F = 10 N, m = 1 kg, ω₀ = 5 s^-1. 

It can be seen that as the degree of damping increases, so the width of the amplitude resonance curve increases and the peak decreases.  That is, as the damping increases, so the sharpness of the resonance decreases.  As noted above, if the damping increases sufficiently, then the resonance peak vanishes (the black curve in the figure above).

The theory of resonance can be extended by considering quantities such as the quality factor Q of the resonance, defined by

            ,

and various limiting cases.  However, we will not pursue this rather “dry” (in my opinion) aspect of the theory of resonance here.

Next:  The Simple Pendulum