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Viscous damping is commonly discussed in beginning differential equations and physics texts but dry friction or Coulomb friction is not despite dry friction being encountered in many physical applications. One reason for avoiding this topic is that the equations involve a jump discontinuity in the damping term. In this article, we adopt an energy approach which permits a general discussion on how to investigate trajectories for second-order differential equations representing mechanical vibration models having dry friction. This approach is suitable for classroom discussion and computer laboratory investigation in beginning courses, hence introduction of dry friction need not be delayed for more advanced courses in mechanics or modelling. Our method is applied to a harmonic oscillator example and a pendulum model. One advantage of this method is that the values of the maximum deflections of a solution can be calculated without solving the differential equation either analytically or numerically, a technique that depends on only the initial conditions. (Contains 3 tables and 6 figures.)

Quadratic friction involves a discontinuous damping term in equations of motion in order that the frictional force always opposes the direction of the motion. Perhaps for this reason this topic is usually omitted from beginning texts in differential equations and physics. However, quadratic damping is more realistic than viscous damping in many models and is applicable to problems concerning aerodynamic and hydrological drag. In this article, we discuss how to use a generalized energy approach that permits us to discuss trajectories for initial value problems of dissipative oscillators like the harmonic oscillator and the pendulum having quadratic damping. This technique permits the computing of maximum deflections of a solution without having to compute the solution either analytically or numerically. This discussion is suitable for classroom presentation and for computer laboratory investigation. (Contains 6 figures and 1 note.)

In this article we examine 2 x 2 first-order systems of ordinary differential equations and show how to identify separatrices for phase plane portraits when the system has a saddle point critical value. We describe how to use a computer algebra system to generate trajectories from contour plots, when possible, and determine the equation of the separatrix. When this approach is not possible, we describe how to use numerical investigations to determine the separatrix. Generating a phase plane portrait is useful, for at a glance one can observe what initial values give rise to bounded solutions, periodic solutions and other important features. It also permits the instructor to concentrate on the qualitative aspects of the model under investigation rather than the calculational difficulties associated with finding solutions. A list of examples suitable for student study is given in an appendix. (Contains 3 figures.)

Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega] "F"-plane for fixed [epsilon], examine harmonic and subharmonic trajectories and van der Pol plane analysis. All these invite further numerical and analytic investigations suitable for student or small group computer laboratory research problems. (Contains 11 figures.)

We report on a study of the forced van der Pol equation x + [epsilon](x[superscript 2] - 1)x + x = F cos[omega]t, by solving numerically the differential equation for a variety of values of the parameters [epsilon], F and [omega]. In doing so, many striking and interesting trajectories can be discovered and phenomena such as frequency entrainment, almost periodic solutions, space filling trajectories and seemingly chaotic behaviour are explored. These examples naturally give rise to computer laboratory problems suitable for student research and small group projects. (Contains 7 figures.)

We discuss the boundary in the Poincare phase plane for boundedness of solutions to spring model equations of the form [second derivative of]x + x + epsilonx[superscript 2] = Fcoswt and the [second derivative of]x + x + epsilonx[superscript 3] = Fcoswt and report the results of a systematic numerical investigation on the global stability of solutions to initial value problems as the initial conditions are allowed to vary. We demonstrate that the unforced spring models explain some of the unbounded behaviour observed with the forced models and thus give at least an idea of where to expect bounded behaviour to occur. We also discuss the behaviour when viscous damping is added to the unforced models. (Contains 17 figures.)

This note discusses the boundary in the frequency--amplitude plane for boundedness of solutions to the forced spring Duffing type equation. For fixed initial conditions and fixed parameter [epsilon] results are reported of a systematic numerical investigation on the global stability of solutions to the initial value problem as the parameters F and [omega] are allowed to vary. This can be interpreted as varying the forcing amplitude and forcing frequency to a nonlinear spring problem and asking for the threshold between bounded oscillatory responses and unbounded unstable responses. These preliminary results show interesting behaviour, illustrate a low resonance frequency and a higher jump frequency phenomena, and give rise to computer laboratory problems suitable for student research and small group projects. (Contains 3 figures.)

Through numerical investigations, various examples of the Duffing type forced spring equation with epsilon positive, are studied. Since [epsilon] is positive, all solutions to the associated homogeneous equation are periodic and the same is true with the forcing applied. The damped equation exhibits steady state trajectories with the interesting property that these steady states are periodic of period 2[pi]/[omega] and coincide with the harmonic trajectory of the undamped equation. This is a case of a limit cycle for a non-autonomous equation. Student computer laboratory investigations and research problems arise naturally and several are suggested.

Finding a periodic solution to a nonlinear ordinary differential equation is in general a difficult task. Only in a very few cases can direct methods be applied to an equation to find initial values leading to a solution of the corresponding initial value problem that is periodic. Oscillatory periodic solutions have such practical importance that great effort has gone into developing techniques to find such initial conditions. Recently a new technique was presented in this journal for finding periodic solutions having the same period as the forcing part of the equation, the so called harmonic solution. In this article, the authors give a related technique for discovering initial values leading to solutions having period a multiple of the forcing, so called subharmonic solutions. They also give some problems suitable for undergraduate research that are designed to show the broad applicability of the technique. This article concentrates on Duffing's equation as representative, but the techniques developed can certainly be applied to other equations.

The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto an unbounded one. This is a common occurrence when numerically solving differential equations with initial values very close to a separatrix that distinguishes between stable (bounded) solutions and unstable (unbounded) solutions. This numerical phenomenon is not discussed in most texts and it is the purpose of this article to describe the effect is such a way as to make it suitable for beginning students to understand why things happen the way they do. Given the modern trend for computer laboratory projects in beginning differential equations courses, it is important for students to be aware of one of the common failings of numerical solutions. (Contains 4 figures.)

Non-linear second-order differential equations whose solutions are the elliptic functions "sn"("t, k"), "cn"("t, k") and "dn"("t, k") are investigated. Using "Mathematica", high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are correct to at least 11 decimal places. These formulas have the advantage over numerically generated data that they are computationally efficient over the entire real line. This approach is seen as further justification for the early introduction of Fourier series in the undergraduate curriculum, for by doing so, models previously considered hard or advanced, whose solution involves elliptic functions, can be solved and plotted as easily as those models whose solutions involve merely trigonometric or other elementary functions. (Contains 1 footnote.)

This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the differential equation x[dots above] + kx[dot above] + g(x,t) = [epsilon](t) where k is a constant; g is continuous, continuously differentiable with respect to x , and is periodic of period P in the variable t; [epsilon](t) is continuous and periodic of period P, and when [partial derivative of g with respect to x] satisfies some additional boundedness conditions. This means that there exist initial values x(0) = [alpha]* and x[dot above](0) = [beta]* so that the solution to the corresponding initial value problem is periodic of period P and is unique (up to a translation of the time variable) with this property. The proof of this result is constructive, so that starting with any initial conditions x(0) = [alpha] and x[dot above](0) = [beta], a path in the phase plane can be produced, starting at ([alpha], [beta]) and terminating at ([alpha]*, [beta]*). Both the theoretical proof and a constructive proof are discussed and a "Mathematica" implementation developed which yields an algorithm in the form of a Mathematica notebook (which is posted on the webpage http://pax.st.usm.edu/downloads). The algorithm is robust and can be used on differential equations whose terms do not satisfy Li and Shen's hypotheses. The ideas used reinforce concepts from beginning courses in ordinary differential equations, linear algebra, and numerical analysis. (Contains 7 figures.)

We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are appropriately chosen very small and the time interval is short. On the other, we suggest that computationally, there is no advantage taking these approximations. We further justify this by employing an approach to deriving Fourier series approximations to the pendulum equation accurate to at least eleven decimal places. Students can generate highly accurate Fourier series solutions to nonlinear equations and thus concentrate on the qualitative aspects of the model rather than the computational difficulties. (Contains 13 figures.)

This paper compliments two recent articles by the author in this journal concerning solving the forced harmonic oscillator equation when the forcing is periodic. The idea is to replace the forcing function by its Fourier series and solve the differential equation term-by-term. Herein the convergence of such series solutions is investigated when the forcing function is bounded, piecewise continuous, and piecewise smooth. The series solution and its term-by-term derivative converge uniformly over the entire real line. The term-by-term differentiation produces a series for the second derivative that converges pointwise and uniformly over any interval not containing a jump discontinuity of the forcing function. (Contains 1 figure.)

Given three points in the plane, interest is in the locus of all points for which the sum of the distances to the given points is a prescribed constant. These curves turn out to be sixth degree polynominals in x and y , and thus are complicated. However, it turns out that often there is a point, within the triangle formed by the three given points, for which the constant is a minimum. This point is determined for isosceles triangles, the general case is discussed, although all details are not revealed. Student problems, such as supplying the missing details mentioned and further generalizations, are suggested. (Contains 10 figures.)

Presents an exercise suitable for beginning calculus students that may give insight into series representations and allow students to see some elementary application of these representations. The Fourier series is used to approximate by taking sums of trigonometric functions of the form sin(ns) and cos(nx) for n is greater than or = zero. (PVD)

Described is an approach to the derivation of numerical integration formulas. Students develop their own formulas using polynomial interpolation and determine error estimates. The Newton-Cotes formulas and error analysis are reviewed. (KR)

Discusses a numerical technique called the method of adjoints, turning a linear two-point boundary value problem into an initial value problem. Described are steps for using the method in linear or nonlinear systems. Applies the technique to solve a simple pendulum problem. Lists 15 references. (YP)

In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates at a rate proportional to the rate of rotation. During investigations of the effect in various solid and fluid-filled objects of various shapes, an interesting commonality was found in connection with the gyroscopic effects of the rotating object. The effect has also been discussed in connection with a rotating fluid-filled wineglass. A linear theory is developed, assuming that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. The associated physics and mathematics are easy enough for undergraduate students to understand. [DOI: 10.1119/1.3088877]

The qualitative effect of a major disturbance such as an earthquake or a hurricane or, on a lesser scale, a powerboat moving along the edge of a bay, harbor, or lake, can be observed in a partially filled wineglass. We simulate this small-scale disturbance and do a quantitative analysis to explain how a resonance can occur on the liquid surface in the wineglass. An explanation is also given why such circumstances can occur in scaled-up situations such as bays, rivers, and harbors. [DOI: 10.1119/1.2742397]

Describes the use of a bell in the classroom to demonstrate the phenomena of beats and resonance. Explains properties of the bell, and its use in place of a demonstration with a spring is discussed. (PVD)

The authors draw together a variety of facts concerning a nonlinear differential equation and compare the exact solution with approximate solutions. Then they provide an expository introduction to the elliptic sine function suitable for presentation in undergraduate courses on differential equations. (MNS)

An old way to determine asymptotes for curves described in polar coordinates is presented. Practice in solving trigonometric equations, in differentiation, and in calculating limits is involved. (MNS)

The behavior of certain functions in advanced calculus is discussed, with the mathematics explained. (MNS)

Provides examples to show that parallel coverage of convergence theorems for both series and improper integrals will tend to strengthen each other. Indicates that such coverage should also help students to better understand the concept of asymptote. (JN)

Results are presented of an impromptu exploration of polar formulas for volumes of revolution for certain plane regions. The material is thought to be unique, and to offer room for student exploration. It is felt pupil investigation can lead to increased pupil interest in both polar coordinates and calculus. (MP)

The Perron-Frobenius Theorem states that let B = [b(sub ij)] be a positive n*n matrix and accordingly the following statements hold like the multiplicity of the eigenvalue r is one, there are a right- and left-eigenvectors associated with r that have positive entries among others which are discussed. The Perron-Frobenius theorem makes possible the determination of the limit (sub n tends to infinity) A(sub n) = 1/6 A1 + 1/3 A2 + 1/2 A3.

An extension of the integration by parts formula, useful in the classroom for products of three functions, is illustrated with several examples. (MNS)