Temple H. Fay

The pendulum equation

We investigate the pendulum equation θ + ⋋ 2 sin θ = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin θ 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.

Energy and the nonsymmetric nonlinear spring

We give aderivation of the nonlinear spring equation mx - f(x) = 0 where f(x) represents the restoring force and m is the mass of a weight attached to the spring. This derivation shows that even powers may appear in the Maclaurin expansion of the restoring force function. We demonstrate how the ‘energy approach’ facilitates the investigation of the resultant motions and makes phase plane analysis easy by reducing the characterization of critical points to determining the sign of asecond-order partial derivative of the energy function. This circumvents the sometimes difficult eigenvalue determination of the linearization procedure commonly taught. This approach is no more difficult than the approach usually takes in textbooks for much simpler equations and it leads to interesting harmonic oscillator examples and problems suitable for undergraduate research.

Rotating structures and Bryan’s effect

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.

Second-order van der Pol plane analysis

A technique is developed, called second-order van der Pol plane analysis, for finding and classifying approximate initial conditions that lead to harmonic solutions to differential equations of the form where is periodic in the variable t. The classical first-order van der Pol plane analysis may incorrectly classify critical points as centres, suggesting a harmonic solution for these initial conditions. This new technique correctly classifies all critical values. An approximate boundary curve is found in the x -phase plane that separates bounded solutions from unbounded ones for the forced soft spring equation All of this suggests computer laboratory problems and student research problems, as the technique is general and can be applied to many classical nonlinear differential equations.

A storm in a wineglass

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.

Using the homotopy method to find periodic solutions of forced nonlinear differential equations

This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the differential equation x + k ẋ + g(x,t) = ε(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; ε(t) is continuous and periodic of period P, and when ∂g/∂x satisfies some additional boundedness conditions. This means that there exist initial values x(0) = α* and ẋ (0) = β* 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) = α and ẋ(0) = β, a path in the phase plane can be produced, starting at (α, β) and terminating at (α*, β*). 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 Shens hypotheses. The ideas used reinforce concepts from beginning courses in ordinary differential equations, linear algebra, and numerical analysis.

The Forced van der Pol Equation.

We report on a study of the forced van der Pol equation by solving numerically the differential equation for a variety of values of the parameters ϵ,F and ω. 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.

Convergence for Fourier series solutions of the forced harmonic oscillator II

Harmonic oscillator equations of the form ÿ + ⋋2y = h(t) where ⋋ is a real constant and h(t) is a continuous, piecewise smooth, periodic ‘forcing’ function are considered. The exact solution, obtained through the Laplace transform is cumbersome to handle over long t intervals, and thus solving ‘term-by-term’ by replacing h(t) by its Fourier series is an attractive and accurate alternative. But this solution is an infinite series involving sums of sine and cosine terms, and thus one should worry about convergence of a solution in this form. In the article, it is shown that such a series solution indeed converges uniformly over the entire real line and is twice continuously differentiable, the derivatives being calculated ‘term-by-term’. Only results commonly available in the undergraduate literature are used to verify this and in so doing, a non-trivial application of these results is given. Also included are some interesting problems suitable for undergraduate research.

The Forced Hard Spring Equation.

Through numerical investigations, various examples of the Duffing type forced spring equation with ϵ positive, are studied. Since ϵ 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 2π/ω and coincide with the harmonic trajectory of the undamped equation. This is a case of a limit cycle for a nonautonomous equation. Student computer laboratory investigations and research problems arise naturally and several are suggested.

Nonlinear resonance and Duffing's spring equation II

The paper discusses the boundary in the frequency–amplitude plane for boundedness of solutions to the forced spring Duffing type equation For fixed initial conditions and for representative fixed values of the parameter ϵ, the results are reported of a systematic numerical investigation into the global stability of solutions to the initial value problem as the parameters F and ω 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 indicate that the low resonance frequency (to two decimal places) is independent of the value of ϵ and that near a higher jump frequency phenomena the behaviour of solutions is very unstable. Computer laboratory problems suitable for student research and small group projects are included.