harmonic-oscillatornewtonian-mechanicsoscillatorsterminology
I have gone through the Phys.SE question Can a simple pendulum be considered a simple harmonic oscillator?. From the 1st answer of this Question it seems to me that another type of "Harmonic motion" is "Damped Harmonic Motion".
But can you please explain what the other type of Pendulum is? Is this Compound pendulum or Damped Pendulum or Something else?
Why would you consider the amplitude to be equal to the horizontal displacement? There's also a vertical displacement, which is arguably more important, without vertical displacement there would be no oscillation, since the potential energy would remain the same. Or take for example a torsion pendulum:
You wouldn't describe the amplitude as $r\sin{\theta}$, because that would give an amplitude of zero when the pendulum swings 360 degrees. It makes more sense to say that the amplitude is equal to the length of the path the pendulum travels, i.e. the length of the arc.
The basic equation for the simple harmonic oscillator is $\frac{d^2x}{dt^2} =- \omega^2 x$. One solution that works here is $x=x_0 e^{i\omega t}$. Notice how the complex conjugate, $x^* = x_0^* e^{-i\omega t}$ is also a valid solution. The reason for this is because, while the body may be oscillating in one dimensional motion, it can be thought of as going either clockwise or counterclockwise in circle in the complex plane, where its angular velocity is either $i\omega$ for counterclockwise or $-i\omega$ for clockwise.
Best Answer
In the case of a simple pendulum (also called a mathematical pendulum of simple gravity pendulum), one assumes that all of the mass is the bob and the rest of the pendulum is massless. An example of the simple pendulum is given in the image below.
The simple pendulum (see wikipedia or hyperphysics) leads to a simple differential equation by using Newton's second law: $$\ddot{\theta}+\frac{g}{l}\sin(\theta)=0.$$
This pendulum gives the easiest way te look at harmonic motion. The above case is what they call the simple pendulum.
You could add an extrernal source so it would be a driven simple pendulum, of friction so that it becomes a simple pendulum with friction.
If you want to further complexify it, you could drop the assumption that all of the mass is in the bob and add inertia to the picture (so a real-life pendulum). This kind of pendulum is called the physical pendulum (or compound pendulum), the swinging body is no longer considered a point mass, but a mass with finite measurements. An example (compared to the simple pendulum) is given on the figure below.
This also leads to equations using Newton's second law applied on angular momentum (just as the ones which were used to derive the equation for the simple pendulum).