In this chapter we study a few important second-order equations with nonautonomous terms due to external forcing. Sections 4.1--4.4 cover nonhomogeneous linear equations, and our treatment is relatively traditional. For the most part we rely on the Method of Undetermined Coefficients to obtain particular solutions. We discuss steady-state solutions and resonance. We augment the usual discussion with a little more qualitative analysis than is often the case, but on the whole, the methods discussed in these sections are versions of those used in most texts. Sections 4.1--4.3 include the standard material, and Section 4.4 is a brief and more technical section designed for courses that are oriented toward engineering applications.
In Section 4.5 we discuss the collapse of the Tacoma Narrows Bridge. Recent work by A. C. Lazer and P. J. McKenna indicates that resonance due to the forcing caused by vortex shedding was not the cause of the collapse. (It is extremely unlikely that vortex shedding could have occurred at sufficiently regular intervals for traditional resonance to have been the cause.) Their work indicates that a more likely explanation is that the collapse was due to nonlinear effects occurring during large oscillations. Like the Lorenz equations, we include this section to give our students a glimpse of current mathematical research. The results discussed there do not reappear in subsequent sections and can safely be skipped if necessary.
4.1 Forced Harmonic Oscillators
In this section we introduce the nonhomogeneous linear equation for a forced harmonic oscillator, and we discuss the Extended Linearity Principle and the Method of Undetermined Coefficients. Only exponential forcing terms are used in the section, but other forcing functions are covered in the exercises. Trigonometric forcing is covered in Section 4.2. Our approach here is traditional except that we are honest that the "method" is really just a a guess-and-test technique.
The section ends with an RLC circuit example. The coefficients of this example are realistic values, so the time scales in Figures 4.7 and 4.8 are typical.
Comments on selected exercises
Exercises 1--17 involve the use of the Method of Undetermined Coefficients for standard exponential forcing functions.
The remaining exercises involve other forms of forcing---both for damped and undamped systems.
4.2 Sinusoidal Forcing
Harmonic oscillators with periodic (i.e., trigonometric) forcing is covered in this section. The approach is again fairly standard except for the fact that complex exponentials are used in the guess-and-test step of the Method of Undetermined Coefficients. The use of complex exponentials simplifies the guessing procedure and reinforces the use of complex numbers begun in Chapter 3. This technique is also consistent with the notion of phasors, which are standard descriptors used in the theory of electric circuits.
Unlike many discussions of this technique, we continue to employ the use of the phase plane, which provides a nice geometric representation for the steady-state solution. In this section, we concentrate on the damped case leaving the undamped phenomena for Section 4.3.
Comments on selected exercises
Exercises 1--14 are typical exercises that use the techniques described in this section.
Exercise 15 presents an alternate approach that does not use complex numbers.
Exercises 16--19 involve matching, and many of the rest of the exercises involve the theory and practice for solving equations where the forcing function is a sum of exponential and trigonometric functions.
4.3 Undamped Forcing and Resonance
In this section we continue the discussion initiated in Section 4.2, and we study the effects of periodic forcing in the undamped case. In particular we study beats and resonance using the techniques introduced in Section 4.2.
Comments on selected exercises
Exercises 1--10 provide practice with the Method of Undetermined Coefficients.
Exercises 11--14 give practice determining the frequency of beats and the frequency of their rapid oscillations.
Exercises 15--18 are matching problems.
Exercises 20 and 21 are essay problems on resonance. We believe the story in Exercise 21 is true, but we cannot remember the name of the university. Do you know?
4.4 Amplitude and Phase of the Steady State
This is a brief section intended for courses that are significantly oriented toward engineering applications. It uses the complex arithmetic developed in Section 4.2 to discuss the amplitude and phase of the steady state solutions derived in Section 4.2. The exercises have a distinct engineering orientation.
4.5 The Tacoma Narrows Bridge
The official investigation into the collapse of the Tacoma Narrows bridge concluded that resonant forcing was not the cause of the collapse. Periodic forcing caused by the shedding of vortices during high winds could not reasonably be expected to maintain a sufficiently precise period for a sufficient time to cause the collapse.
In this section, we discuss recent work of A.C. Lazer and P.J. McKenna (see SIAM Reviews, Vol. 32, No. 4, 1990, pp. 537--578 and the references cited there). They model the motion of light flexible suspension bridges and observe that there is a significant nonlinearity that occurs when the bridge oscillators with moderately high amplitude. When the bridge is below equilibrium, the stretched cables act as springs pulling the bridge up and a linear harmonic oscillator model is reasonable. On the other hand, when the bridge is significantly above its equilibrium position, the cables are slack and do not push down. Only the constant force of gravity pulls down on the bridge. They show that a simple nonlinear system with this sort of (discontinuous) nonlinearity can exhibit stable, large amplitude oscillations in the presence periodic forcing over a range of frequencies. Based on this idea, they also develop more accurate models for the full motion of the bridge. These models employ partial differential equations rather than ordinary differential equations.
This section develops the Lazer-McKenna system and reports on the results of their numerical and analytical investigations. While the numerical work is fairly delicate and the theorems moderately difficult, the results are within the reach of the typical student.
There are many good examples where resonance leads to dramatic behavior in physical systems. However, this is not the case for the Tacoma Narrows Bridge. The real culprit was the use of a linear model when a nonlinear model was required. This is an important lesson that we certainly want our students to understand and appreciate.
Comments on selected exercises
Exercises 1--4 are modifications of the Lazer-McKenna system for suspension bridges. Exercises 5--8 discuss another application given by Lazer and McKenna to objects bobbing up and down in water but which can also lift out of the water.
Comments on the Lab
Lab 4.1. A Periodically Forced RLC Circuit
The equations modeling RLC circuits, which are frequently given as a main motivation why second-order equations must be studied instead of first-order systems, are actually first developed as a first-order system. The dependent variables are voltage over the capacitor and the current. This system is then converted into a second-order equation. This lab is meant to emphasize this point. It can be assigned after Section 4.2 and goes nicely with Lab 3.2.