Damped Oscillator | Oscillations, Waves & Optics - Physics PDF Download

Damped Harmonic Oscillation

The majority of the oscillatory systems in everyday life suffer some sort of irreversible energy loss due to frictional or viscous heat generation while they are oscillating. Their

amplitude of oscillation dies away with time. Such oscillation is called damped harmonic

oscillation.

Consider the mass-spring system, body of mass m attached to spring with spring

constant k is released from position x₀ (measured from equilibrium position) with

velocity v ; the mass is subject to a frictional damping force which opposes its motion,

and is directly proportional to its instantaneous velocity F_res = −bv

The quantity v is a positive constant, whose value depends on the properties of the

material providing the resistance. The minus sign indicates that the force resists the

motion, so it is directed opposite to the velocity.

The total force on the body is the sum of the restoring force F = −kx , and the resistive

force

F_res: F_net = F + F_res = −kx − bv

Using Newton second law = 0 ⇒ = 0

Here, we define the constant γ such that, 2γ = b/m called damping constant and

ω₀ = √k /m is the natural frequency of undamped oscillator.

Assuming a solution of the form x(t) = Ce^αt

On differentiation, we get  = 0

The solution (forω ) from the quadratic formula is: α =

⇒ α =

Thus the solution is x(t) =  =  where ω ' =

If we consider the quantity under the square root sign, we see that there are three

possibilities.

Case 1: Overdamped Case: If the damping coefficient is large, then γ >ω₀ and ω′

will be imaginary. Hence x (t )will be a negative exponential function. It is shown in

figure (a).

Case 2: Critically Damped Case: If ω₀ =γ , then the square root vanishes. In this case,

the solution is again a negative exponential function which goes to zero quicker than the

overdamped case, as shown in figure (b).

Case 3: Underdamped Case: If ω₀  >γ then the quantity under the square root is

positive and we have a real number for ω ′ . The solution for ‘ x ’ is then

This is the solution of the damped harmonic oscillator. The oscillatory motion is shown

in figure (c).

Time Period: of the damped harmonic oscillator is: T' =

This shows that due to damping the time period slightly increased.

Logarithmic Decrement: This measure the rate at which the amplitude decay

Mean Life time(τ_m)  : It is the time taken for the amplitude to decay to 1/e of the initial

value. When t = , Amplitude = A/e

Energy of the Damped Oscillator

Kinetic energy (K)

The displacement of a damped harmonic oscillator is x (t ) =

The instantaneous velocity is

u =

The approximation is done as  γ <<ω₀

Thus, kinetic energy is K = 1/2mu² =

Potential energy (U)

U =

Total energy (E)

E = K + U =

⇒E =  Since  γ <<ω₀ thus ω′ =

This shows that the energy of the oscillator decreases with time, the exponential decay of

energy is show below.

Power Dissipation: It is the rate at which the energy is lost

P =

Relaxation Time: It is the time taken for the total energy to decay to 1/e of its initial value E₀. If τ is the relaxation time, then at t =τ , we shall have E = .

Thus =  This gives relaxation time τ =

Thus we can also write power dissipated as P = E/τ

and energy can be expressed as E(t) = .

Quality Factor (Q): It is defined as the 2π times the ratio of the energy stored in the

system to the energy lost per period.

Q =

Energy stored in the system is E while the energy loss per period is PT , thus 

This shows that, lower the damping, higher the value of Q.

Now energy can be written in term of Q as E(t) =  

It means that Q is related to the number of oscillation over which the energy fall to 1/e of

its original value E₀ , which is also called the relaxation time. This happens in time, t =τ ,

where τ is given by

In one period (T ) number of oscillation is =1

In time τ the number of oscillation = n , then n =

Thus the energy falls to 1/e of its original value after n=  Q/2π = cycle of free oscillation.

Relation between relaxation time, mean time and Quality factor:

Let N is the number of oscillation in time

While n is the number of oscillation in time τ =

Thus, relation between mean and relaxation time is  τ_m = 2t

In one period (T ) number of oscillation is =1

In time τ_m the number of oscillation = N

 or