# What Is Damped Simple Harmonic Motion?

Simple harmonic motion is a type of motion which repeats itself after a specific interval of time. It is a repetitive motion in which the body keeps following a specific pattern of motion, in simple harmonic motion the object covers some distance about a central position called the mean position. The point at which the object reaches after covering the maximum distance is called the extreme position. The distance between the maximum position and the mean position is called amplitude of the motion.

However once this to and fro motion is started the body will always not touch the maximum or extreme position in reality its distance will start decreasing gradually. This simple harmonic motion is found in several objects, some very fine examples are the motion of pendulum and movements of an object tied to a spring. It is assumed that when any object with its one end tied to a fixed point is displaced from its position it can execute simple harmonic motion, and there is always some work done to displace the object initially which is then used to keep the object moving.

Damped motion on the other hand is a motion in which an attempt is made to reduce the amplitude of the vibrating body. So a damped simple harmonic motion is the motion in an object tied to a fixed point is displaced from its position, and it will start executing simpler harmonic motion. Once the motion is started through some external source or through the inheritance of the system the amplitude of the motion is damped.
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Simple harmonic motion is periodic motion (repeats in specific manner at standard intervals i.e. Sinusoidal) with constant amplitude. Example is motion of simple harmonic oscillator. Damping is effect used to reduce the amplitude of oscillation of any oscillatory system. So damped simple harmonic motion is simple harmonic motion with some effects to reduce amplitude of its waves.    This type of motion satisfies the second order differential equation.In real physical environment, an oscillation is not as perfect. Forces like friction and air resistance acts on system and decrease the speed and amplitude of oscillation until system achieve rest or equilibrium. Damping force is main dissipative force which always acts opposite to the direction of velocity, result in decrease of amplitude. Solving the equation of damped harmonic motion we have three damping conditions i.e. Over damping, critical damping and under damping.    Different results are drawn by solving their linear equations.  Example of damped harmonic oscillator is mass attached with spring. The undamped mass attached to spring moves to and fro with constant amplitude and frequency while in real world damped harmonic oscillation of mass attached with spring exists i.e. It shows oscillation but its oscillation decays with passage of time due to damping forces like air resistance, surface friction; these forces tend to reduce the velocity of mass hence decrease in its amplitude.
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The Information and Communication Revolution has applied the spheres of math and engineering to the fullest. The applications take care of the complex operations within the computing and other industries. The study of the dynamics applicable to a dedicated sphere of application involves the coding and decoding of a lot of direct formulae and equations and variations of the formulae, according to the specific industry. Damped simple harmonic motion is an engineering specific application and the calculation is derived from a specially constructed equation with three variables: w, x and beta.

When you add a damping force proportional to ' ' to the existing equation for simple harmonic motion and apply the first derivative of ' ' with respect to the time involved, the result is the damped simple harmonic motion equation:
[ x+ ßx+w2x=0]

Here, ' ß' is the damping constant. This equation helps to calculate the analysis of the flow of current in an electronic CLR circuit. A CLR circuit is one that contains a capacitor, an inductor, and a resistor. The curve that is produced as a result of the application of the two damped harmonic oscillators at right angles to each other is referred to as a 'harmonograph' in engineering and it helps to simplify the operation further to a 'Lissajous' curve, if ß1=ß2=0.
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The motion in which acceleration in the body is directly proportional to the displacement and its direction is towards mean position is called simple harmonic motion. If a force of friction is present, then the motion of a harmonic oscillator is damped by the friction and is called damped harmonic motion. The resistive or frictional force in a damped oscillator is called damping force. The net force on the oscillating body is the sum of restoring force '-kx' and the damping force '-bv'.

Net force = restoring force - damping force

ma= -kx - bv

ma + bv + kx = 0

a + bv/ m + kx/ m = 0

as, v= dx/ dt and a= dv/dt => a= d2x/ dt2

d2x/dt2 + b/m dx/dt + k/m x =0

putting b/m = 2B and k/m = w2

d2x/dt2 + 2B dx/dt + w2x = 0 is the required equation for damped simple harmonic motion.

Moreover, B is a constant and known as beta and w is also a constant and known as omega.
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When a spring is stretched a distance x from its equilibrium position, according to
Hooke's law it exerts a restoring force F = - kx where the constant k is called the spring
constant. If the spring is attached to a mass m, then by Newton's second law, −kx = m x .. ,
where x .. Is the second derivative of x with respect to time. This differential equation has
the familiar solution for oscillatory (simple harmonic) motion:

X = Acos( ωt + φ), (1)
where A and φ are constants determined by the initial conditions and ω= k / m is the
angular frequency. The period is T = 2 π m k . By differentiating Eq.(1) we determine the
velocity
v = −A ωsin(ωt + φ),
which can be rewritten as
v = Aωbecause(ωt + π 2
+ φ). (2)
By differentiating again we obtain the acceleration
a = −Aω2 because( ωt + φ),
which can be rewritten as
a = Aω2 because( ωt + π+ φ). (3)
From the acceleration we find the force,
F = mAω2 because( ωt + π+ φ). (4)
From these four equations we see that the velocity leads the displacement in phase by
π/2 while the force and acceleration lead by π.
For actual oscillating masses the motion is frequently not quite this simple since
frictional forces act to retard the motion.
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Anonymous commented
To my understanding of the question, i think the question is yet to be answered. What is damped simple harmonic motion not what is the equation of damped s.h.m nonetheless i stand to be corrected if am wrong.