Difference between harmonic motion and simple harmonic motion?

The name simple harmonic motion suggests that its the simple version of "harmonic motion".

Does harmonic motion exist and if so is there a difference between these 2 terms?

The OP should perhaps have asked "Does simple harmonic motion exist?" given that it is harder to find examples of simple harmonic motion than harmonic motion.

Any motion which repeats itself can be called harmonic motion and so simple harmonic motion is in the set of harmonic motions.

One way of defining simple harmonic motion is via the differential equation

$\dfrac{d^2X}{dt^2} = - \omega^2\,X$ where $X$ is some parameter which could be a displacement, an angle etc.

In mechanics it can be interpreted as the motion of a mass acted on by a restoring force which is proportional to the displacement of the mass from a fixed point.

Simple harmonic motion has important properties, for example, the period of oscillation does not depend on the amplitude of the motion and lots of systems do undergo simple harmonic motion even if sometimes it is an approximation.

A good example of the difference between harmonic motion and simple harmonic motion is the simple pendulum.

The simple pendulum undergoes harmonic motion, however for small angles of oscillation it does approximately undergo simple harmonic motion.

• On the contrary, I found numerous resources on SHM and not a single on harmonic motion. Nov 16, 2021 at 13:23
• One reason is that a lot of systems exhibit approximate simple harmonic motion and the more general harmonic motion is often much more difficult to analyse. Nov 16, 2021 at 18:48

Harmonic motion would just be any motion that is periodic. For example, motion following a kind of square wave would be harmonic.

Simple harmonic motion is motion that is specifically sinusoidal; that is, it can be described by a sine wave. The reason simple harmonic motion is "simple" is because any kind of harmonic motion can be constructed as superpositions of SHMs of different frequencies, via the Fourier transform.

Simple Harmonic motion is defined as motion where the restoring force is proportional to the displacement, and pointed towards equilibrium.

The example of the simple pendulum is a good one, though a pendulum is not actually simple harmonic motion. You can treat it as such by substituting theta for sin(theta) when calculating the restoring force. When the displacement is small enough this works, and if you do a Taylor expansion of sin(theta) you can see why. For small displacements, the higher-order terms go to zero.

Simple Harmonic Motion is the basis of all harmonic motion. Plotting the back-and-forth oscillatory motion over time gives a sinusoidal wave and at some point any sine or cosine wave has that sort of motion as its components.

Since any wave can be analyzed as a combination of sine and cosine waves, they can all be thought of as combinations of particles in simple harmonic motion at different frequencies. I think the simple part is just because it is the most basic component, and concretely defined. It is easy to spot all around us, although it is usually damped because of friction, it is still simple harmonic when it is a single oscillator where the restoring force is proportional to the displacement, and it is moving periodically about an equilibrium position. shock absorbers, instruments, you find it in electrical circuits... and the equations can work for almost any periodic oscillatory motion for small displacements.

• I found this and I thought you may be interested. For the technicalities of the differences between harmonic motion and non-harmonic periodic motion see the information starting on page 17. Not all periodic wave motion is harmonic. google.com/books/edition/Mechanical_Vibrations/… Jun 2, 2023 at 2:26

Harmonic motion refers to repetitive oscillations around an equilibrium position. Simple harmonic motion (SHM) is a specific type of harmonic motion where the restoring force is directly proportional to the displacement from the equilibrium position and is directed towards that position. This results in a sinusoidal pattern of motion, commonly seen in phenomena like a swinging pendulum or a mass-spring system.