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Damped Simple Harmonic Motion

When the motion of an oscillator reduces due to an external force, the oscillator and its motion are damped. These periodic motions of gradually decreasing amplitude are damped simple harmonic motion. An example of damped simple harmonic motion is a simple pendulum.

In the damped simple harmonic motion, the energy of the oscillator dissipates continuously. But for small damping, the oscillations remain approximately periodic. The forces which dissipate the energy are generally frictional forces.

Expression of damped simple harmonic motion

Let’s take an example to understand what damped simple harmonic motion is. Consider a block of mass m connected to an elastic string of spring constant k. In an ideal situation, if we push the block down a little and then release it, its angular frequency of oscillation is ω = √k/ m.

However, in practice, an external force (air in this case) will exert a damping force on the motion of the block and the mechanical energy of the block-string system will decrease. This energy that is lost will appear as the heat of the surrounding medium.

The damping force depends on the nature of the surrounding medium. When we immerse the block in a liquid, the magnitude of damping will be much greater and the dissipation energy is much faster. Thus, the damping force is proportional to the velocity of the bob and acts opposite to the direction of the velocity. If the damping force is F_d, we have,

F_d = -bυ                              (I)

where the constant b depends on the properties of the medium(viscosity, for example) and size and shape of the block. Let’s say O is the equilibrium position where the block settles after releasing it. Now, if we pull down or push the block a little, the restoring force on the block due to spring is F_s = -kx, where x is the displacement of the mass from its equilibrium position. Therefore, the total force acting on the mass at any time t is, F = -kx -bυ.

Now, if a(t) is the acceleration of mass m at time t, then by Newton’s Law of Motion along the direction of motion, we have

ma(t) = -kx(t) – bυ(t)                        (II)

here, we are not considering vector notation because we are only considering the one-dimensional motion. Therefore, using first and second derivatives of s(t), v(t) and a(t), we have,

m(d²x/dt²) + b(dx/dt) + kx =0                      (III)

This equation describes the motion of the block under the influence of a damping force which is proportional to velocity. Therefore, this is the expression of damped simple harmonic motion. The solution of this expression is of the form

x(t) = Ae^-bt/2m cos(ω′t + ø)                        (IV)

where A is the amplitude and ω′ is the angular frequency of damped simple harmonic motion given by,

ω′ = √(k/m – b²/4m² )                             (V)

The function x(t) is not strictly periodic because of the factor e^-bt/2m which decreases continuously with time. However, if the decrease is small in one-time period T, the motion is then approximately periodic. In a damped oscillator, the amplitude is not constant but depends on time. But for small damping, we may use the same expression but take amplitude as Ae^-bt/2m

∴ E(t) =1/2 kAe^-bt/2m                                    (VI)

This expression shows that the damping decreases exponentially with time. For a small damping, the dimensionless ratio (b/√km) is much less than 1. Obviously, if we put b = 0, all equations of damped simple harmonic motion will turn into the corresponding equations of undamped motion.

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