Project 37: Resonance - Maximal Response to Forcing

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Resonance is a peak in vibration as you vary the frequency at which you "shake" a system. For example, many old cars have speeds where they hum loudly, say at 47 mph. Slower, it's quieter and faster it's also quieter. We will see this phenomenon in the solution of the linear system


where we vary the frequency w and observe the amplitude of the response.

By our study of "transients" in Project 36, we know that the initial conditions do not matter for long-term behavior of the solutions. We are interested only in the amplitude of the steady-state solution.

The values of h and k in the steady-state solution,


were found in Hint 36.5.1. You should verify the solutions:

37.1 Some Useful Trig

Given h and k, we want to compute a and so that for all ,


Conversely, given a and , we want to compute the values of h and k that make the equation hold for all . The number a is called the amplitude and the "angle" is called the "phase." (This is not the phase variable trick, but a different "phase.")

It is remarkable that any linear combination of sine and cosine is actually another sine. And conversely, that any phase shift may be written as a combination of only plain sine and cosine. The usefulness of this form comes up in the study of resonance below.

HINTS IN THE CASE WHERE h AND k ARE GIVEN: Approach 1) We could use calculus to maximize for all values of . The max of is clearly a at the angle where the sine peaks.

Approach 2) For any pair of numbers satisfying there is an angle , so that

Why is this so? (Think of an appropriate parametric curve. What is the definition of radian measure? What does this problem have to do with the unit circle? Draw the vector . What is its length?)

If we know how to find so a given , let , so that


satisfies and produces an angle as above.

Use this angle to write


Finish proving that these values of a and make

by using the addition formula for sine.

We won't need the converse part, but you can probably see why it holds once you find the formulas for a and in terms of h and k.

37.2 Resonance in Forced Linear Oscillators

  • RESONANCE QUESTION 1
    How much is when we write the solution

    in the form

    Express your answer in terms of the physical parameters m, c, and s as well as the variable forcing frequency w.

  • RESONANCE QUESTION 2
    Simplify the expression for and show that is maximized when the square of its reciprocal is minimized. What frequency minimizes


    Figure 37.1: vs


    This is called the resonant frequency, and now you should analyze it in various cases of the physical constants. Use the computer to plot versus for various choices of the parameters m, c, and s. First observe that we have resonance only when 2c2<4ms. Next observe that if 2c2<4ms, then we must have c2-4ms<0. The autonomous system oscillates with "natural frequency"


    provided that c2-4ms<0. Explain in your project what this autonomous oscillation means. Why is this called the "natural frequency?"

    What happens when , but c2<4ms? What happens if c2>4ms? Verify this mathematical split with some computer plots or experiments.

    37.3 An Electrical Circuit Experiment

    This is an optional suggestion for an experiment to verify your theoretical calculations. The series R-L-C electrical experiments work very well, because the components are very nearly linear at low current. If you can get access to a variable resistor box, a variable capacitor box, an inductor (the bigger the inductance, the better), a wave generator, and an oscilloscope, all of which are in most engineering labs, you can easily build a lab apparatus that illustrates ALL of the things in the above project. An R-L-C circuit satisfies a linear second-order equation and the wave generator provides the forcing term. By varying the resistor and capacitor, you can see all the things we have learned - the graphs appear on the oscilloscope! (It is much harder to perform the mechanical oscillator experiments, because it is difficult to vary spring constants and damping friction.)

    37.4 Nonlinear Damping

    The most suspect term in our mechanical oscillator is the damping force


    For damping caused by air resistance, we have investigated nonlinear laws like

    (in the bungee project.) The nonlinear 7/5 power makes the damping force grow faster as the speed increases. This optional section asks how we might detect nonlinearity from the behavior of solutions of the system.

    The frequency of oscillation of a linear system


    is always the same (provided it oscillates). What is this frequency?

    Run the computer program ComplexRoots and observe that initial values that begin on a line in the phase plane remain in line. Why is this just the geometric representation of the fact that the frequency is independent of the initial condition?

    Conduct some computer experiments with the system


    beginning with the ComplexRoots program, by modifying it to solve the nonlinear equation. With suitable choices of the parameters and suitable scales for your flow, you will be able to see nonlinearity arising as frequencies that depend on the initial conditions. Explain what you observe. If the nonlinearity of a shock absorber is small, say

    with , will this be easy to detect in a physical system? What if p is large, say p=3?


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