In physics, complex harmonic motion is a complicated realm based on the simple harmonic motion. The word "complex" refers to different situations. Unlike simple harmonic motion, which is regardless of air resistance, friction, etc., complex harmonic motion often has additional forces to dissipate the initial energy and lessen the speed and amplitude of an oscillation until the energy of the system is totally drained and the system comes to rest at its equilibrium point.
Damped harmonic motion is a real oscillation, in which an object is hanging on a spring. Because of the existence of internal friction and air resistance, the system will over time experience a decrease in amplitude. The decrease of amplitude is due to the fact that the energy goes into thermal energy.
Damped harmonic motion happens because the spring is not very efficient at storing and releasing energy so that the energy dies out. The damping force is proportional to the velocity of the object and is at the opposite direction of the motion so that the object slows down quickly. Specifically, When an object is damping, the damping force will be related to velocity by a coefficient :
The diagram shown on the right indicates three types of damped harmonic motion.
An object or a system is oscillating in its own natural frequency without the interference of an external periodic force or initial motion. Damped oscillation is similar to forced oscillation except that it has continuous and repeated force. Hence, these are two motions that have opposite results.
Resonance occurs when the frequency of the external force (applied) is the same as the natural frequency (resonant frequency) of the system. When such a situation occurs, the external force always acts in the same direction as the motion of the oscillating object, with the result that the amplitude of the oscillation increases indefinitely, as it's shown in the adjacent diagram. Away from the value of resonant frequency, either greater or lesser, the amplitude of the corresponding frequency is smaller.
In a set of driving pendulums with different length of strings hanging objects, the one pendulum with the same length of string as the driver gets the biggest amplitude of swinging.
See video: https://www.youtube.com/watch?v=aCocQa2Bcuc
A double pendulum is a simple pendulum hanging under another one, which is an epitome of the compound pendulum system. It shows an abundant dynamic behavior. The motion of a double pendulum seems chaotic. We can hardly see a regulated routine that it is going, which makes it complicated. Also, whether the lengths and masses of the two arms are equal to each other, makes it hard to identify the centers of the two rods. Moreover, a double pendulum may exert motion without the restriction of only two-dimension (usually vertical) plane. In other words, the complex pendulum can move to anywhere within the sphere, which has the radius of the total length of the two pendulums. However, for a small angle, the double pendulum can act similar to the simple pendulum because the motion is determined by sine and cosine functions as well.
The image shows a marine clock with motor springs and double pendulum sheel.