Before your read this, I suggest you read post 24.5.

In post 24.5, we saw that the displacement, x, of a damped simple harmonic oscillator is given by equation 1.

Here t represents time, dx/dt represents speed and d²x/dt² represents acceleration. As in post 24.5 we are considering an object that oscillates along a line, so that we can use scalar notation. Then k is the stiffness of the system, M(dx/dt) is the drag force acting on the object and m is its mass. Post 24.5 also gives the analogues of equation 1 for rotational motion and for motion of an electric charge as well as a general expression for all three. But here I am going to concentrate on a simple mechanical system – translational oscillation along a line that I can represent using scalar notation.

For an oscillator that has its maximum displacement, x₀, when t = 0 the solution of equation 1 is given by equation 2, where ω is given by equation 3.

When M = 0, equation 2 and 3 become identical to the corresponding equations for the undamped harmonic oscillator (see post 18.11), as expected, because the drag force is then negligible.

Equations 2 and 3 follow from equation 1 provided that M² < 4km. I derive them in the appendix.

Before we get obsessed by mathematics, let’s think about the physical system that these equations represent. M depends on the dimensions and shape of the oscillating object and is proportional to the viscosity of the medium in which it oscillates, as explained in post 17.17; the greater the value of M, the greater the drag force.

The picture at the beginning of this post compares the time-dependent displacement of a simple harmonic oscillator (M = 0) with an example of the displacement of a damped simple harmonic oscillator (M = 4) as predicted by equation 2. The picture shows that the effect of damping is to progressively decrease the amplitude of the oscillations until they are imperceptible. The picture below shows that the greater the value of M (provided M² < 4km), the greater the effect.

We can also use the concept of energy as an alternative way of thinking about what is happening. A simple harmonic oscillator (M = 0) oscillates indefinitely because it is constantly exchanging kinetic and potential energy. When M ≠ 0, the system is dissipating energy so that, in each cycle, the oscillator has less kinetic energy, so the oscillations decrease until, eventually, they are imperceptible.

Related posts

24.5 Linear second-order systems
22.18 Coupled oscillators
21.1 An electrical simple harmonic oscillator
19.17 Computer modelling – the simple harmonic oscillator
18.17 Euler’s relation, oscillations and waves
18.11 Motion in a circle
18.7 The simple pendulum
18.6 The pendulum

Appendix

To solve equation 1, note that the first second and third terms must have a similar form if they add up to zero. Also note that the terms contain, in order, no derivative, a first-order derivative (dx/dt) and a second order derivative (d²x/dt²). What sort of function has a similar form when it isn’t differentiated, is differentiated once and is differentiated twice? An exponential because e^x, d(e^x)/dt and d²(e^x)/dt² are all the same (see post 18.15).

So, let’s see if we can find a function of the form

where C and p are constants, that is a solution to equation 1. Differentiating equation 4 gives

Substituting equations 4 and 5 into equation 1 gives

This equation has two solutions. One is that   Ce^pt = 0. Since   Ce^pt = x,   this solution means that there is no displacement, so no oscillation – this solution is of no interest here because it describes a system that is not moving. The other is that

which is a quadratic equation with two solutions that can be expressed (post 18.15) as

Equation 7 has three types of solution.

Type 1 

so that its square root is a real number.

Type 2  

so that p = –M/2m which means that the two solutions of equation 6 are identical.

Type 3  

which means that

where i is the square root of -1 (see post 18.16).

I am going to pursue the type 3 solution because I note that when   M = 0 

which describes the displacement of a simple harmonic oscillator when substituted into equation 4 (see post 18.17).

So when   M > 0,   the type 3 solution looks like a plausible solution for a damped harmonic oscillator.

Then, according to equation 4,

If the final step isn’t clear, see post 18.2. Noting that the term in red is the angular frequency of a simple harmonic oscillator, modified by a term that vanished when M = 0, which would be the case for a simple harmonic oscillator, I am going to define ω by equation 3, in the main text. Then

Let’s consider that the oscillator has its maximum displacement, x₀, at time t = 0, so that the cosine term now provides the mathematical solution of the physical system (see post 18.17). Then, x is given by equation 2 in the main text.