Resonances, waves and fields: Summing impulses with calculus

Monday, June 11, 2012

3.9 Summing impulses with calculus

Next we will consider a whole series of impulses designed to mimic a sine wave as shown in Fig. 18a. Since our resonator is a linear system, the response will simply be the sum of the responses as shown in Fig. 17, each delayed to start at the time of the particular impulse.

Fig. 18a. A series of force pulses, i.e. impulses, arranged in magnitude and sign to simulate a sinusoidal forcing function. The impulses are off at times before zero.

Fig. 18b. This figure shows a few of the impulses (a, b, c, and d) similar to those in Fig. 18a along with the decaying sinusoid that results from each impulse (each on its own axis). We will next sum these sinusoids using integration in the case of an infinite number of infinitesimally short impulses.

In Eqns. (91) and(92) we have the response from a single pulse at time equal to t'. We need to sum an infinite number of such responses, each in reponse to an impulse of pulse height equal to f = f₀sin ωt'. We'll do this summation via the process of integration (2^nd reference).

We start by repeating Equation (91):

. (91)

We next substitute f₀sin ωt' in for force f and replace dt with dt' (a dummy integration variable) and integrate in order to have an expression for the displacement x as a function of time:

. (95)

Note that at each time t we need to sum (i.e. integrate) the impulse function from 0 (the start of the impulses) to our present time t to sum up the effects of all impulses that have occurred up to the present time t. The calculational method we are using here is basically the method of Laplace transforms, a widely used mathematical method of expressing a function in terms of exponentially decaying (or increasing) sinusoids.

You can integrate (95) by substituting sin θ = (1/2i)(e^iθ − e^−iθ) and cranking through the integral. An easier option is to use Wolfram online integrator to do the indefinite part of the integral to get:

(96)

where we have done the last step manually. Alternatively, a person could use the free software Maxima to do the integration in one step, however there is a reasonably steep learning curve to be able to use this very powerful algebra solving software.

Incidentally, if we had used an exciting force proportional to cos ωt' instead of sin ωt' (in (95) above), we would have the result of:

. (96a)

The real part of (96) is plotted in Fig. 19 below for a simulated sinusoidal driving force that is turned off at time t = t_off .

Fig. 19. Graph showing the build up of the resonance with time for three different driver frequencies. The resonant frequency of the resonator is 2.5Hz. The top series of curves represent the displacement of the mass versus time for the three driver frequencies. Note that when the driver frequency matches the resonant frequency (the green line) the greatest amplitude is eventually obtained. The dashed green line is the envelope for this build up. After the driver is turned off, all three cases decay with an exponential envelope, all at the same decay frequency, independent of the frequency used to excite the resonance.

The middle series of curves show the displacement of the driver for the three frequencies. You can see the phase of the driver versus that of the mass displacement for the three frequencies. In the case of the green curve, the driver lags the mass by 90 degrees (π/2).

The phase difference between driver and mass is also plotted in the bottom series of curves. We see that the green curve settles to the optimal relative phase of −π/2, while a driver frequency that is too low or too high results in relative phases that are greater or less than this optimal value. After the driver is turned off, the relative phase between mass and driver is undefined.

Phase of the oscillations relative to that of the driver

One interesting aspect of the above graph is the comparison of the drive phase compared with the phase of the displacement of the mass. Calculation of these phase curves required some extra care. Equation (96) (reproduced below) is a mix of the real and imaginary representations required when a calculation involves the product of two phasors.

. (96)

There are three terms in (96). The oscillating part of the first term is a complex phasor (e^st), whereas the oscillating parts of the last two terms are real sinusoids, a sine and a cosine. We take the real part of (96) as we must in order to find the actual displacement:

. (97)

We see that the complex multipliers of e^st affect the phase of the first term, i.e. the −i, s, and s² + ω² affect the phase of the first term. When we take the real parts of the second and third terms the sine and cosine functions are real, so we just multiply these by the real part of their complex multipliers. In this form, these complex multipliers affect the amplitude of the sine and cosine functions, but not their phases. The multipliers become simple real constants.

The most straightforward way to calculate the phase is to put all the terms of (97) in a complex form. To this end, we substitute −i e^iωt for sin ωt and e^iωt for cos ωt in (97) to have all phase information in a complex format. This substitution (and dropping the Re( ) operation where possible) yields:

. (98)

The advantage of the purely complex representation is that the phase is readily accessible, in contrast to the real representation of a complicated equation. The graphs were made using gnuplot, a free professional grade graphing program. This program has an "arg" function which was used to calculate the phase angle of a complex expression and produce the phase curves at the bottom of Fig. 19.

You may notice in Fig. 19 that the red and blue phase lines are not simple. These represent the phases for the system driven below and above resonance. They all start at 0 then move close to π/2, then close to either 0 (for the blue line) or π (for the red line) then finally seem to drift a little towards π/2 . We can qualitatively explain this behavior as follows:

At the start, the bob is just being pushed by the external driver and is therefore in phase with it.
Shortly after the start, the phase difference between driver and the built up oscillations hasn't had time to develop, so the behavior is similar to the case of being driven at resonance, i.e. with a π/2 phase difference. The amplitude grows as though the system is being driven at resonance.
Later, the frequency difference between the driver and the built up oscillations finally has had time to cause a considerable phase difference from the "on-resonance" case.
Finally, the considerable phase difference causes some destructive interference between the built up oscillations and the driver, and reduces the size of the built up oscillations somewhat. The destructive interference operates mostly on the oldest component of the oscillations, effectively reducing the decay time some.
After the amplitude is reduced a little, we move into a stable steady state phase and amplitude.
We see this overshoot and settling behavior in the earlier animations: Fig. 15a, Fig. 15e, Fig. 16a, and Fig. 16d when the drive frequency did not match the resonant frequency. At the risk of getting ahead of ourselves, we shall see it again in Fig. 21 of the next posting.