Forced Oscillations We have seen how the amplitude of a damped oscillator decreases in time due to the presence of resistive forces. As a result, the energy and the frequency of oscillation of the system, decreases and eventually the oscillations die down. To maintain oscillations and to compensate for this energy decrease, we have to feed energy to the system by applying an external periodic force that does positive work on the system. In general, the frequency of the external applied force, and that of the damping system, may not match. However, as we shall see , irrespective of its natural frequency, the system oscillates with the frequency of the applied force in the steady state. Such oscillations are called forced oscillations. If , however, the frequency of the applied force exactly matches the natural frequency of the vibrating system, we would observe the phenomenon of resonance-- a spectacular effect in which, as we shall find in the discussion given below, the amplitude of forced oscillations becomes very large.

We start our discussion by writing the equation of motion of a system driven by an externally applied harmonic force and attempt to find the solution of this equation.

Equation of Motion for a Weakly Damped Forced Oscillator

We consider again the spring mass system as an example of a forced weakly damped harmonic oscillator (Fig.6.5 ). As the mass is displaced from its

equilibrium position and then released, it is subjected, at any instant, to

(i) a restoring force –kx ,

(ii) a damping force and

(iii) a driving force, Fo cos ω t.

The equation of motion for a forced oscillator is, therefore,

Dividing by m, and defining the coefficients:

,

we rewrite eq. (6.24) as

This is now an inhomogeneous second order differential equation with constant coefficients. This equation is of a general nature describing the motion of any oscillator (not only of a spring mass system) whose natural frequency is and which is subjected to a harmonic driving force fo.

General solution for the displacement of a Forced oscillator A general procedure to solve Eqn.(6.25) is to express the general solution, for the displacement x(t) , as a sum of two components, i.e.,

x(t) = x1 (t) + x2 (t),

where x1(t) is a solution of the equation for which the right hand side of Eqn. (6.25) is zero

and x2(t)is a solution that satisfies the equation

In mathematical language, x1 is called the complementary function and x2 is called the particular integral, with (x1 + x2) representing the complete solution of Eqn. (6.25). Recall that in the absence of the external driving force,

the displacement,

As we noticed earlier, this (complementary function) part of the (complete) solution decays exponentially and will disappear after sometime. During this transient state, the system oscillates with a frequency ωd which is different from its natural frequency or the frequency of the driving force. After a sufficiently long time (t>> ), the damped oscillations of the system will disappear. However, as the general solution of Eqn. (6.25) does not decay, the system will oscillate with the frequency of the driving force. The system is then said to be in the steady state.

To obtain the steady state solutions of Eq. (6.25), let the displacement of the forced oscillator be given by

where A and δ are the constants to be determined. Comparing this equation with the driving force F = Fo cos t, we note that the force leads the displacement, in phase, by the angle δ. To determine the constants, A and δ, we differentiate Eqn.(6.27) twice with respect to time :

and substitute these back in Eqn.(6.26(a)). This gives

Writing explicitly (for the two terms of the form cos(α – β) and sin(α – β)), we re-express the above equation as

Now since sint and cost are orthogonal functions, we can compare their coefficients on both sides and get

Now since sint and cosωt are orthogonal functions, we can compare their coefficients on both sides and get

Using the relation (6.31(a)) we write

On substituting the expressions for cosδ and sinδ in Eqn.(6.30(a)) , we find the expression for A, i.e.

This gives us the steady state solution of Eqn.(6.25) as

Where the phase difference δ is given by Eqn.(6.31(b).

Using equations (6.31(b), (6.31(c) and (6.32) we summarize below the three pieces of information regarding the steady state behavior of a force driven oscillator:

The phase difference δ between the displacement and the driving force depends upon the angular frequency of the driving force as well as the (two) constants b and the natural frequency, of the oscillating system.

The amplitude A of the forced oscillations depends on the constants Fo and the damping constant, b, of the oscillator.

The motion of the oscillator is found to be completely independent of the initial conditions, i.e. the way the oscillator is set into motion. No matter how we start the oscillator, its motion will ultimately settle down to the form determined by Eqn.(6.32) (forgetting about how it originally started).

It is clear from Eqn.(6.32) that the steady solution has the same frequency as that of the driving force. It has a constant amplitude and a phase,defined completely, relative to the driving force. Indeed, the complete solution would be a superposition of the transient part given by and the steady state given by Eqn. (6.32). The resultant sum is

The plot of the individual terms ,and their resultant, showing the variation of net displacement with time is depicted in Fig.6.6.

Fig. 6.6 Time variation of (a) the transient solution; (b) the steady state solutions and ( c) the general solution given by Eqn.(6.33).

Behavior of the amplitude and phase of a forced oscillator We now proceed to analyze the behavior of the amplitude of the steady state solution (given by Eqn.(6.31 (c))) , with respect to the variation of the frequency of the driving force. We have here three cases:

(a) Case I: When ω<< ω (low driving frequency)

At low driving frequency, Eqn.(6.31(c) can be rewritten as

If ω<<ω0 , the ratio << 1 and therefore the squared, and higher order terms ,can be neglected. This reduces the amplitude to

This shows that , at sufficiently low frequencies, the steady state amplitude of the oscillation is essentially determined by the stiffness constant k and the magnitude of the driving force Fo. Under this condition, the phase difference, given by Eqn.(6.31 (a)), reduces to

That is the phase difference, between the steady state displacement x2 and the driving force is zero.

(b) Case II. When = (Resonance Frequency)

The expression (6.31(c)), for the amplitude at = , gets simplified as

This shows that ,at resonance, the amplitude depends upon the damping coefficient. It is inversely proportional to the damping coefficient b; smaller the value of b, the greater is the amplitude. Since

damping is always present and it never becomes zero, the amplitude can never be infinite in a practical situation.

For the phase difference δ, when we put ω=ω0 in Eqn.(6.31(a), we find

Thus the driving force and the displacement have a phase difference of ∕2 at = ,.

Is the value of A(ω) given by Eqn.(6.36) maximum at ω=ω0 = ,? To answer this question, we can maximize expression (6.31(c)) for A(), i.e. differentiate it with respect to , and find at what value of , the first derivative becomes zero and the second derivative is negative. We have

The derivative will be zero if the numerator in the above expression is zero; i.e.

This equation gives the acceptable root

It is easy to check, by taking the second derivative, that at is indeed negative.

Thus we conclude that the peak value of A(ω) does not occur precisely at = but

at . This value is less than by a factor depending on the damping of the oscillating system. On substituting for , from Eqn.(6.38) in Eqn.(6.31 (c)), we get

When at a particular frequency = , the amplitude becomes maximum, we say resonance has occurred at the resonance frequency .

(c) Case III. When >> (i.e, the driving frequency is large)

Again, for ω>>ω0, we rewrite Eqn.(6.31(c) as

In the iimit of high driving frequency , we can neglect terms containing and

also and get

This shows that at high frequencies the amplitude falls as , ultimately reducing to zero as ω→∞.

Similarly, using Eqn.(6.31(a)), we find that the phase δ becomes

which gives δ = (6.41)

This shows that, at high frequencies, the driving force leads the displacement by .

Based on the above discussion, we can now pictorially summarize the behaviour of the amplitude as well as of the phase difference, between the driving force and the displacement, by depicting a plot of A (ω) and δ (ω), with respect to over a wide range of frequencies. This is shown in Figs. 6.7((a) and 6.7(b)).

Quality Factor, Power Dissipation and Sharpness of Resonance of a Forced Oscillator Quality Factor of an oscillator, damped or driven, can, in general, be defined as

(6.42)

To obtain the expression for quality factor of a forced oscillator, we need the expressions for (i) the average energy stored in the system and (ii) the average power dissipated.

Let us first recall that the instantaneous energy of a driven oscillator consists of two parts, (i) the kinetic energy and (ii) the potential energy.

The expression ,for the instantaneous kinetic energy,is

where

and ,

The time average, of the kinetic energy, over one complete cycle ,of period T, is defined as

(6.43)

On substituting the above expression for the K.E, we get

(6.44)

To obtain this expression, we use the relations

The instantaneous power dissipated, i.e, the rate at which work is done is

P(t)= F(t) x velocity,

Taking the time average over one complete cycle of the power, i.e, F(t) v, given above, and

using the relations

we get

In the above equation, the expression for

obtained in the preceding section has also been used.

Using the expressions, for <E> and <P> ,we can now obtain, by using Eqn.(6.42), the expression for the quality factor, i.e.

A more useful definition of the quality factor can be given in terms of the amplitudes. The quality factor can be defined as the ratio of the amplitude, at resonance, to that at low frequencies ( i.e. as → 0).

We have seen above that the amplitude at resonance [see Eqn.(6.39],

and the amplitude at low frequencies has been obtained as (See Eqn.(6.34))

Thus, the quality factor can be expressed as

, (6.48)

which can be further simplified for the case when damping is small when . In such a case, neglecting b2 in Eqn.(6.48), we get

(6.49)

Using the above relations, we can re-express Amax in terms of the quality factor. Thus

Substituting ,from Eqn.(6.49) for , the amplitude at resonance can be expressed in terms of the quality factor as

This shows that Amax increases as the quality factor Q increases. Starting from Eqn.(6.31(c)) for A as a function of , i.e,

Let us rewrite this expression as

Now use the expression, Eqn.(6.49), for the quality factor, i.e to get

The plot depicting the behaviour of the amplitude A ,as a function of driving frequency ,for different values of Q is as shown in Fig.6.7.

Fig. 6.7

There is yet another way to study the influence of quality factor, on the behaviour of average power absorbed, with respect to the driving frequency, near resonance condition.

In the above, we obtained the expression for the average power absorbed by a forced oscillator, as

On comparing the right hand sides of Eqns.(6.53) and (6.54), we get the condition

Or

In other words,

Considering first the (-) sign, we have , which gives

Here the root, with negative sign, cannot be accepted as it would give a negative frequency(which is physically unrealistic). Thus the acceptable root is

(6.55)

Similarly, considering the (+) sign in the quadratic equation, we get

which has the roots

Note that these roots of , both of which represent just the two half power points, are on the opposite sides of the resonance position. And the frequency interval between the two half power points is given by

Fig.6.8

Clearly this interval goes on decreasing as the value of Q becomes larger and larger, resulting in an increase in the sharpness of the resonance. Quality factor of a resonating system can thus also be regarded as measure of its sharpness of resonance. This has been depicted in the figure, where the plot of vs the frequency of a forced oscillator shows that the resonance becomes sharp for high Q (Fig.6.8). A low Q-system, on the other hand, has a flat resonance with a large band-width, (ω1-ω2).Sharpness of resonance, as the name suggests, implies a rapid rate of fall of power,(from its( resonant frequency) maximum), with frequency, on either side of the resonance frequency. These results help us to understand the importance of Q-factor, in electrical LCR circuits ,using an A.C. source of variable frequency.