1.6 Driven Harmonic Oscillator

The mass, spring and damping force combination described in the previous section is referred to as a damped harmonic oscillator. If a time varying force of the form F_o cos w t , where F_o and w are constants, is applied to a damped harmonic oscillator the system is then referred to as a driven harmonic oscillator and this system also serves as a useful model for many real systems.  The equation of motion of such an oscillator is

(1.17)             m d²x/dt² + R dx/dt + K x = F_o cos w t

This type of differential equation is said to be non-homogeneous and it is shown in courses in differential equations that the general solution is made up of the sum of two parts: the general solution of the homogeneous part, m d²x/dt² + R dx/dt + Kx=0 , and any particular solution of the entire equation. The solution of the homogeneous part is exactly that of the damped oscillator which was given in the previous section. Observations made on real systems suggest that a particular solution might be of the form

(1.18)             x = C sin(w t - q )

where C and q are constants. Differentiating this expression to obtain dx/dt and d2x/dt2 and substituting in (1.17) yields

                    -Cw² m sin(w t - q ) + RCw cos(w t - q ) + KC sin(w t - q ) = F_o cos w t

Substituting (sin w t cos q -cos w t sin q ) for sin(w t - q ), (cos w t cos q + sin w t sin q ) for cos(w t - q ) and rearranging, the equation above becomes

(1.19)             {C [w² m sin(w t - q ) + R w cos q ] -Fo} cos w t + C sin w t [(K-w²m) cos q +Rw sin q ]

This expression must be identically zero for all possible times if (1.18) is to be a solution of (1.17). It is apparent that if one can make

                    C [w² m sin(w t - q ) + R w cos q ] -Fo = 0
and

                    (K-w²m) cos q +Rw sin q = 0

by a proper choice of C and q then (1.19) would be zero for all times. The second of these conditions is satisfied if tan q = -(K-w²m) /Rw . This makes

(1.20)             sin q = (w m - K/w ) / Ö (R² +(w m - K/w )² )
and
(1.21)             cos q = R/Ö (R² +(w m - K/w )² )

By substituting for sin q and cos q in the first of the conditions, one solves for C and finds

                    C = [Fo/w ]/ Ö (R² +(w m - K/w )² )

A particular solution of (1.17) is thus

                    x = [Fo/w ] sin(w t - q ) / Ö (R² +(w m - K/w )² )

and the general solution is

(1.22)             A exp(-a t) cos(w_d t + f ) + [Fo/w ] sin(w t - q ) / Ö (R² +(w m - K/w )² )

where

            a = R/(2m)            w_d = Ö (K/m - (R/(2m))²)             tan q = (w m - K/w )/R

and A and f are arbitrary constants. The first term of the solution is called the transient part since after a sufficient time has elapsed this term is negligibly small. The system is then said to be in the steady state and its motion is described only by the second term, called the steady state solution. Note that the second term is of the same form as equation (1.5) which describes the motion of a simple harmonic oscillator. The steady state motion of the driven oscillator is thus simple harmonic motion and has the same frequency as that of the driving force. If, for convenience, one lets

(1.23)             Z = Ö (R² +(w m - K/w )² )

then one may write

(1.24)                  x       = F_o/(w Z) sin (w t - q )
(1.25)               dx/dt   = F_o/Z cos (w t - q )
(1.26)             d²x/dt² = -F_ow/Z sin(w t - q )

Note that dx/dt, d²x/dt² and the driving force F_o cos w t all have the same frequency and period but no two of these quantities are in phase with each other. It should be apparent that x and d2x/dt2 differ in phase by 180 degrees and that the driving force and dx/dt differ in phase by q . A more complete discussion of the phase relations will be deferred until later, when the steady state solution of (1.17) is written in complex form.

1.7 Mechanical Resonance

Any driven oscillator whose steady state motion is described by Equation (1.24) is executing harmonic motion with a constant amplitude F_o/(w Z) . A constant amplitude implies that the driving force must be supplying enough energy to the system to make up for the amount dissipated by the damping force. One interprets the rate at which the driving force does work as the rate at which it is supplying energy to the system. Now, if a force F is acting on a particle which undergoes a displacement ds, the work dW done by the force in this displacement is, by definition F · ds. If dt is the time during which the displacement takes place, then the rate, dW/dt at which the force is doing work is F · v, where v is the velocity of the particle. Thus, in the steady state, the rate at which the driving force is supplying energy at time t is

If one substitutes for cos q from (1.21) and for Z from (1.23) the average rate at which the driving force supplies energy may be written

                    P_av = ½ F_o² R/[R²+(w m – K/w )²]

This average power is a constant for any given oscillator and driving force. If the frequency w changes or the driving force is changed, the average input power and the peak values of x and dx/dt changes, since both are functions of w . If one calculates P_av for a number of different values of w and plots the average input power as a function of w , a curve similar to that of Figure 1.7 is obtained. For a given damped harmonic oscillator, i. e., for fixed values of R, m, and K, and for a constant Fo, the average input power P_av is largest for w =Ö (K/m, a result that should be evident from an examination of (1.24). This value of w corresponds to an actual frequency

(1.28)            f_r = 1/(2p ) Ö (K/m)

which is called the resonant frequency of the system. (In practice w_r = 2p f_r is also known as the resonant frequency of the system.) The resonant frequency of a driven oscillator is exactly the same as the the frequency of oscillation of an undamped simple harmonic oscillator with the same mass and spring constant as the driven system. If the damping constant R of the system is small, the resonant frequency will be nearly the same as the natural frequency of the system (i. e. the frequency of oscillation when there is no driving force).

The resonant frequency and the shape of the P_av versus frequency curve are two important characteristics of an oscillating system. As a quantitative measure of the shape of the curve, one uses a dimensionless quantity called the Q (or Quality Factor) of the system which is defined by

(1.29)             Q = w_r / (w₂ - w₁)

where w₁ and w₂ are the two values of w for which the input power P_av is one half of the input power at resonance. These two frequencies are indicated in Figure 1.7. If they lie close to each other and hence Q is large, then P_av decreases rapidly on either side of the resonant frequency, and the resonance is said to be sharp. If w₁ and w₂ are widely spaced and consequently Q is small, then P_av is approximately constant over a range of frequencies in the neighborhood of the resonant frequency. When this is true, the resonance is said to be broad. One can determine which parameters of the oscillating system determine its Q by calculating w₂ - w₁ as follows. If w_x is one of the angular frequencies for which P_av is one half of its maximum value then

(1.30)             ½ F_o² R/[R²+ (w_xm – K/w_x)² ] = ½ F_o² /R

Rearranging and simplifying one obtains

                    w_xm – K/w_x = ± R

This equation gives rise to two quadratic equations, one for +R and onefor -R. The solutions for +R and -R are respectively

                    w_x = R/(2m) ± Ö [(R/(2m))² + K/m]
and
                    w_x = -R/(2m) ± Ö [(R/(2m))² + K/m]

There are thus four values of w_x which satisfy (1.30). However, one notes that two of these values are negative and have no physical meaning. Setting the larger of the positive values equal to w₂ and the smaller of the positive values to w₁, and substituting into (1.29) gives

(1.31)             Q = w_r m/R

Since the resonance frequency w_r = Ö (K/m) , it follows that

(1.32)             Q = Ö (Km) / R

Thus, the Q of a mechanical oscillator may be increased by making the spring stiffer, the mass larger, or the damping smaller. The Q of a system is of interest because it strongly influences the response of the system to a change in the excitation of the system. For example, if the driving force is removed from a system which is oscillating at its resonant frequency, the amplitude of the oscillation will decay at a rate determined by a = R/(2m). Solving for R in (1.31) we find that a =(w_r m/Q)/(2m) = ½ w_r/Q. Thus, the higher the Q of the system, the more slowly the oscillation will die out.

Example 1.3

When the power to the machine of Example 1.1 is shut off it is observed that as the rotational speed of the machine decreases there is a point where the machine and platform are set into oscillation. This oscillation gradually dies out. During the oscillation, the peak excursion of the platform in the upward direction, measured from its equilibrium position, is observed to be 4 cm at a time to and 20 cycles later is observed to be 2 cm.

(a) What is the value of the damping constant, R, for this system?

(b) What is the Q of this system?

(c) If this system were driven at its resonant frequency by a harmonic driving force with a peak value of 20 N, what would be the maximum displacement of the platform from its equilibrium position, after a steady state has been established?

Solution

(a) Assume the damping is sufficiently small so that the frequency of the damped system can be taken equal to that of the undamped system, namely, 4.4 Hz. Using Equation (1.16) with A_n=4 cm, A_m=2 cm, t_n=to, t_m=to + 20T, where T=1/(4.4), and solving for a yields
            a = 4/4 ln(2) / 20 = 0.152 s^-1

            R = 2 m a = (2)(525)(0.152) = 160 kg/s

b) Q = w r m/R = (2p )(4.4)(525)(.152) = 90.7

(c) At resonance Z=R and the peak value of x [Equation (1.21)] becomes

            x_peak = F_o/(w_r R) = [20]/[(2p )(4.4)(160)] = 4.5 x 10^-3 m = 0.45 cm
 

1.8 Stiffness, Resistance, Mass Controlled Oscillators

For a given driven harmonic oscillator, it may happen that over a certain range of frequencies, one of the three terms R, w m, or K/w is much larger than the other two. At frequencies considerably below resonance, Kw is usually much larger than R and w m, so that Z» K/w . For the range of frequencies for which this is a good approximation, the oscillator is said to be stiffness controlled and the steady state solution reduces to

1.9 Solution of Driven Oscillator in Complex Form

In Section 1.6 the equation of motion

(1.17) m d²x/dt² + R dx/dt + K x = F_o cos w t

of a driven harmonic oscillator was found to have a steady state solution in the form of

(1.24)             x = F_o/(w Z) sin (w t - q )
(1.25)             dx/dt = F_o/Z cos (w t - q )
(1.26)             d²x/dt² = -F_ow/Z sin(w t - q )

where

                    tan q = (w m - K/w )/R                     Z = Ö (R² +(w m - K/w )² )
 

If one is interested only in the steady state solution, as is often the case, it turns out one can obtain such a solution with less algebra by the following technique. Suppose that a force F_o sin w t rather than F_o cos w t is applied to the oscillator and that y rather than x is used to measure the displacement. The equation of motion in this case is

(1.34)             m d²ydt² + R dy/dt + Ky = F_o sin w t

If one multiplies (1.34) by i and adds it to (1.17) one obtains

                    m d²(x + iy)/dt² + R d(x+iy)/dt + K(x+iy) = F_o (cos w t + i sin w t )

By setting x = x+iy and expressing the term on the right in exponential form this becomes

(1.35)             m d²x/dt² + R dx/dt + Kx = F_o exp(iw t)

Remembering that the equality of any two complex expressions requires equality of both the real parts and the imaginary parts, it should be evident that if one can find a solution of (1.35) of the form x(t) = x₁(t) + i x₂(t) then x₁(t) will be a solution of (1.17) and x₂(t) will be a solution of (1.34). It is readily verified that there exists a solution of 1.35) in the form

(1.36)            x = A exp(iw t)

where A is a (complex) constant. Substituting into (1.35)

                    [-mw²A + R iwA + K A ] exp(iw t) = Fo exp(iw t)

and solving for A, yields

(1.37)         A = i F_o/w /[ R + i (w m - K/w )]

Thus,

(1.38)         x = i(Fo/w ) exp(iw t) / [R + i (w m - K/w )]

is a solution of (1.35). The denominator of this expression is a complex number of the form a+ib. A complex number can also be expressed as m exp(iq ), with magnitude m, and phase q . The magnitude of the denominator is Ö [R² + (w m - K/w )²], and tan q = R/(w m - K/w ) . Now writing x in exponential form we have

                    x = -i(Fo/w ) exp( i(w t - q )) / Ö [R² + (w m - K/w )²]

The real part of this expression is exactly the same as the steady state solution of the driven oscillator obtained by solving the real differential equation (1.17). For reasons mentioned earlier, one prefers to regard (1.38) as the steady state solution and to regard F_oexp(iw t) as the driving force. If A has the value given in (1.37), then

(1.39)            x = A exp(iw t)

(1.40)             dx/dt = i wA exp(iw t) = i x

(1.41)             d²x/dt² = -Aw² exp(iw t) = - w²x

The real parts of these three equations correspond exactly to equations (1.24), (1.25) and (1.26) respectively. If, at an arbitrarily chosen instant of time, one represents x, dx/dt and d²x/dt² in the complex plane, one obtains a figure similar to that of Figure 1.8.

Figure 1.8

Although the position of x is arbitrary, since it depends on the particular instant of time chosen, the positions of dx/dt and d²x/dt² are fixed relative to x from the relations dx/dt = iwx. and d²x/dt² = -w²x . Note again that the angle between any two complex quantities is exactly the phase difference between the corresponding real quantities.

If, in the equation (1.40) one substitutes the value of A from (1.37), one obtains after rearranging

(1.42)             F_o exp(iw t) = R dx/dt + i(mw - K/w ) dx/dt

which relates the driving force and the velocity. Representing dx/dt in the complex plane in an arbitrarily chosen position, the positions of R dx/dt and i(w m-K/w ) dx/dt and their sum, F_oexp(iw t) are determined as indicated in Figure 1.9.
 

Figure 1.9

From this figure it is seen that the angle between the vector representing F_o exp(iw t) and that representing dx/dt is the angle whose tangent is (w m - K/w )/R. This is the angle q of equations (1.24), (1.25) and (1.26), and is the phase difference between the driving force and the velocity. In drawing the figure, it was assumed that w m > K/w . When this is true, the driving force ``leads'' the velocity by the angle q .

Because of the ease of manipulation and the fact that the phase relations are more readily apparent, one usually prefers to do algebraic manipulations with the quantities x , dx/dt, d²x/dt² and F_oexp(iw t), remembering that by taking the real parts of these quantities, one can obtain x, dx/dt, d²x/dt² and the real driving force F_o cos w t. It should be pointed out that in dealing with work and power one must use real quantities. In calculating the average work done by the driving force, as was done in Section 1.7, it was necessary to use the real expressions for the force and velocity.

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