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. The work of 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.

Chapters 4 and 5 are relatively independent of each other, and some instructors prefer to discuss linearization near an equilibrium point for a nonlinear system (Section 5.1) immediately after Chapter 3. We do not adopt that approach mainly because our engineering students take electric circuit theory concurrently, and therefore it is important that we study forced, second-order linear equations as soon as possible.

DETools

The main tool for this chapter is ForcedMassSpring. However, we also use the Pendulums tool with nonzero F to illustrate the concept of external forcing in our introduction to the material.

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 1-19 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 sinusoidal 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.

DETools

ForcedMassSpring and HPGSystemSolver are the two main programs for this section.

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.

Exercise 16 illustrates the linearity of the equation.

Exercise 17 involve matching the graph of a solution with its corresponding differential equation. You may want to try this exercise before you assign it.

Exercises 18-20 involve the theory and practice for solving equations where the forcing function includes both exponential and trigonometric functions.

Exercises 21-23 make the connection between the phasor approach that engineering students often see in their circuit theory courses and the approach discussed in this section.

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.

DETools

ForcedMassSpring, BeatsAndResonance, and HPGSystemSolver are the three main tools for this section.

Comments on selected exercises

Exercises 1-14 provide practice with the Method of Undetermined Coefficients.

Exercises 15-18 give practice determining the frequency of beats and the frequency of their rapid oscillations.

Exercise 21 is a matching problem similar to the one in Section 4.2.

Exercises 20 and 22 are essay questions on resonance. We believe that the story in Exercise 22 is true, but we cannot remember the name of the university.

4.4 Amplitude and Phase of the Steady State

This brief section is 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.

DETools

AmplitudeAndPhase illustrates the results of this section.

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 the work of A.C. Lazer and P.J. McKenna (see The American Mathematical Monthly, Vol. 106, No. 1, 1999, pp. 1-18, 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 oscillates 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 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.

Review Exercises

Exercises 1-10 are "short answer" exercises. The answers are (usually) one or two sentences. Most (but not all) are relatively straightforward.

Exercises 11-14 are true/false problems. We always expect our students to justify their answers.

Exercises 15-22 are routine exercises in finding the general solution of a forced second-order equation.

Exercise 23 is a matching exercise that includes both homogeneous and forced equations.

In Exercises 24--27, the student must find the general solution of sinusoidally forced equations as well as compute the amplitude and phase of the steady-state solution. Exercise 27 is somewhat tricky because the forcing is expressed in terms of sine rather than cosine.

Comments on the Labs

Lab 4.1. Two Magnets and a Spring Revisited

In this lab, the students continue their study of the system involving two magnets and a spring from Lab 2.1. In this case, the system is subject to periodic forcing.

Lab 4.2. 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.

Lab 4.3. The Tacoma Narrows Bridge

This lab illustrates the ideas discussed in Section 4.5. It is designed to follow Lab 2.4. The forced harmonic oscillator parts of the lab are relatively straightforward. However, the students will probably need help with the computation and meaning of the large-amplitude periodic solution. (See the article by Lisa Humphreys and Ray Shammas, College Mathematics Journal, Vol. 31, No. 5, 2000, pp. 338-346.)