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Forced or Driven Harmonic Oscillator

 

When an external periodic force is applied on a system, the force imports a periodic pulse to the system so that the loss in energy in doing work against the dissipative forces is recovered

 

As a result, the system is continuously oscillates. In the initial stages, the system tends to execute oscillations with the natural frequency (or frequency of free vibrations), while the impressed periodic force tries to impose its own frequency on it. Therefore, the free vibrations of the body soon die out and ultimately, the system starts oscillating with a constant amplitude and with the frequency equal to that of the impressed force. This is called the steady state of an oscillator. These vibrations are called the forced vibrations. The force impressed on the system is called the driver and the system which executes forced vibrations, is called the forced or driven harmonic oscillator.

 

Thus a particle executing the forced harmonic oscillations is acted upon by the following three forces:

 

(i) A linear restoring force(= – Kx), which is directly proportional to the displacement from a fixed point and is in a direction opposite to the displacement.

 

(ii) A damping force ( =- r dx/dt),which is directly proportional to the velocity and is in

direction opposite to the motion.

 

An external periodic force(F=F0 sin ωt)where F0 is the amplitude of the impressed force and w  is the angular frequency of the impressed force.

 

If m be the mass of particle executing forced oscillations and d2x /dt2: is its acceleration at dt any instant, then by Newton’s law.

 

Total force acting on the particle = m d 2x /dt2

 

The equation of motion of forced harmonic oscillations will be

where, r/m = 2b, K/m= n2 and F0/m = f

 

Obviously from equation (1), the differential equation for the free oscillations of the system will be d2x + n2x = 0, where n = √K/m =angular frequency of free oscillations.

 

Transient and Steady States

 

We have read that on applying an external periodic force, initially the system tends to oscillate with an angular frequency w1 = √(n2-b2) due to damping, but the driving force of angular frequency w acting on the system forces it to oscillate with its own frequency. In this way, the actual motion of system is obtained by the superposition of two oscillations, whose angular frequencies are w1 = √(n2-b2) and w2 = w respectively. Hence if w≠ n, the solution of the equation (1) can be written as follows

x = x1 + x2

 

where, x1 is the solution of equation (1), when the external force is zero. Then from equation (1)

 

and

 

Remember that in the complete solution x = x1 + x2, x1 is known as the complementary

function and x2 is known as the particular integral.

 

The complementary function represented by equation (2) decreases exponentially with time and after some time, this term vanishes, hence it is also known as the transient solution of the forced harmonic oscillator. In this way, the system in transient state, oscillates with a frequency different from the natural frequency or the frequency of the driving force.

 

After a long time, when t >> t, the natural oscillations of the system vanishes due to damping and then the system oscillates with the frequency of the driving force. This state of the system is known as the steady state.

 

Let the solution of equation (3) in the steady state be x2 = A sin (wt-8). [Since in the steady state, the amplitude of forced oscillation is constant l (=A, say) and the frequency is equal to the frequency of impressed force i.e., ω / 2π

 

Here θ is the phase difference between the displacement and the impressed force.

Then

dx2 / dt = A ω cos (ω t- θ) and   d2x2 / dt2 = -A ω 2 sin (ω t- θ) dt

 

Substituting these values in equation (3)

-A ω2 - sin (ω t- θ) + 2bAw cos (ω t- θ) + n2A sin (ω t- θ) = f sin (ω t- θ + θ)

or                     A(n2-w2) sin (ω t- θ) + 2bAw cos (ω t- θ) =/[sin (ω t- θ) cos θ + cos (ω t- θ) sin θ]

 

 

Since this equation is valid for all value of t, therefore by equating the coefficients of sin(ω t- θ) and cos(ω t- θ) separately, we get

 

A (n2-w2) = f cos θ and 2bAw = f sin θ

 

Squaring and adding the above equations

 

A2[(n2-w2)2 + 4b2 ω2 ] = f2[cos2 θ+ sin2 θ]Putting these values of A and 8 in x2 =A

sin (rot-8), the solution of equation (3) is

Substituting the values of x1 and x2; the complete solution of differential equation (1) of forced oscillator is given as

 

In the above equation, the first term on the RHS which is transient part, decreases with time and finally its role vanishes. The time upto which it plays its role, depends on the amount of damping. More damping, more rapidly this term decreases to zero.

 

When the damping is zero (i.e., when b = 0), in steady state x =  f /(n2-w2) sin wt

and θ = 0° (i.e., the driving force and displacement will be in same phase). It is concluded that the phase difference between the displacement and driving force of a forced oscillator is due to damping. It is also clear that when w = n, the amplitude of oscillations become infinite. This condition is known as Resonance.

 

Figure 1.14 represents the variation of the transient displacement xl steady displacement x2 and their sum, i.e., total displacement x with time t of a forced oscillator for low damping.

(a) Variation of transient displacement x1 with time

(b) Variation of steady displacement x2 with time

(c) Variation of total displacement

(c) Variation of total displacement x with time

FIGURE 1.14 Variation of displacement of a forced oscillator with time

 

Above equation (7) gives the displacement of the forced oscillator at any instant t. It is clear that the amplitude of the driven oscillator in steady state does not depend on time t (i.e., it remains constant with time), but it depends on the frequency roof the external periodic force.

 

Now we will study the following three cases:

(i)                 When ω<< n, i.e., the frequency of the driving force is much less.

(ii)               When ω = n, i.e., the state of resonance.

(iii)             When ω > n, i.e., the frequency of the driving force is much high.

 

Case (i) When ω << n, i.e., the driving frequency is less than the natural frequency of the driven. For low damping (when b –7 0), from equation (4) and equation (5)

and tan θ = 0 or e = ∞

 

Thus in this case, the displacement is in phase with the driving force and the amplitude of oscillations does not depend on the mass and damping, but only depends on the force constant.

 

Case (ii). When ω = n, i.e., the frequency of driver is equal to the natural frequency of the driven. Then from equation (4)

Since t = 1/2b and ω = n ] .

 

This is called the state of resonance. Thus at resonance, the amplitude of oscillation depends on the damping coefficient r. Low the damping, more is the amplitude. If damping is zero (i.e., r = 0 or ‘t = ∞), Amax = ∞(infinite)

 

From, equation (5) tan θ = ∞ or θ= π/ 2 … (10)

 

i.e., at resonance, the displacement of the oscillator lags behind the driving force in phase by π/ 2. Remember that the amplitude represented by equation (9) is not maximum. The reason behind it is as follows:

 

From, A =

it is clear that for A to be maximum, the value of tterm

must be minimum. i.e.,

8b2ω + 2(n22)2 (- 2 ω) = 0

ω2 = n2 – 2b2 = ω2r(Say)

Thus, at a particular frequency of the driver, the amplitude of oscillator becomes maximum. This phenomenon is called the amplitude resonance and this particular frequency is called the resonance frequency. The resonant frequency of forced harmonic oscillator

If damping is zero (i.e., b = 0), than ω, = n (i.e., the resonant angular frequency of the oscillator is equal to the natural angular frequency of the driven) and maximum amplitude Amax =infinite. Figure 1.15 shows the resonant amplitude of different case.

FIGURE 1.16 Variation of phase difference with frequency in steady state

 

1.8.1 Power Absorption by a Forced Oscillator

 

We know that there is some loss in energy in doing work against damping by each oscillating system but in a forced oscillator, the oscillations are maintained by providing energy by external impressed force. Hence it becomes essential to know that at what rate of the energy should be supplied to the system to maintain the oscillations in the steady state, i.e., what is the absorption of average energy by the oscillating system? By definition at any instant t, power P = force F- x velocity v.

 

But the solution in steady state

where φ= θ – π/2

where φ= θ – π/2

where φ= θ – π/2and

Then at any instant t, power P = F0 sin rot x v0 sin (rot-<j)) or average power absorbed

=1/2 F0 v0

In the condition of resonance

Thus, in the condition of resonance, the average power absorbed by the forced oscillator is maximum because in the condition of resonance, velocity and impressed force true in the same phase.

 

The value of maximum absorbed average power depends on the amplitude of the impressed force.

 

Figure 1.17 represents the variation of absorbed average power with frequency.

FIGURE 1.17 Variation of average power with frequency

 

From equation (4), the amplitude of a forced oscillator is

At resonance [From education(11)]

or         (n2-2b2-wh2)2 + 4b2n2-4b4 = 2 (4b2n2-4b4)

But      w r2 =n2 -2b : (wr2-wh2)2=4b2n2 -4b2

or (wr2 -wh2)2 = (4b2wr2 + 4b4)

 

If damping is low, we can neglect the term 4b4. Then

Hence, half width of resonance curve

∆w =|wh-wr |= b 1 /2t

where ‘t is the relaxation time

 

Now, quality factor Q =2π x Average energy stored / Energy loss in one period

From equation (17)

 

Quality factor, Q =w/ 2∆w

 

= frequency when Average power is maximum /  total width of half power point

 

Thus, for the sharp resonance, the value of Q must be high, i.e., the half with ∆w of the resonance curve must be small.

1.8.5 Velocity Resonance

 

From equation of the displacement of a forced oscillator is

and velocity of the oscillator at any instant

Where φ =θ –π/2

 

Hence amplitude of velocity

It is clear from the above expression (20) that -as the frequency w  of the driving force changes, the amplitude of velocity also changes. Figure 1.19 shows the variation of velocity amplitude v0 with the frequency vo of the driving force.

When w = 0, vo =0

 

When w = n2 /w    or    w = n  ,   vo =f /2b maximum

 

This is called the condition of velocity resonance. When the value o f w exceeds n , the value of v0 decreases. Thus, when the frequency of the driving force is equal to the natural frequency of the oscillator, the velocity of oscillator is maximum. Hence the kinetic energy of the oscillator is also maximum. At resonance, the amplitude of velocity depends on the damping (lower the damping, high is the velocity amplitude).

 

At velocity resonance, θ= tan-1 (∞) = π/2

. . Phase difference between the velocity and the driving force is.

φ = θ –π/2 = π/2 =π/2=0

i.e., at resonance, the velocity of. oscillator is in phase with the driving force.