Solved Problems in Nonlinear Oscillations

Solved Problems in Nonlinear Oscillations is Zeng He, Wen Jiang and Lin Wang's companion to Nayfeh and Mook's Nonlinear Oscillations. The 2025 volume organizes worked solutions from conservative and forced oscillators through coupled systems, continuous media and traveling waves. This guide covers the problem index and seven solution chapters with method-selection tasks and a verification plan.

How to study this book

Use this sourcebook alongside the relevant theory and original exercise statements in Nayfeh and Mook. Review ordinary differential equations, linear vibrations and perturbation notation. Before viewing a solution, derive the model, identify small parameters and select a method. Afterward, explain each solvability condition and approximation order in your own words. Numerical agreement is most useful when parameter choices and tolerances are documented.

Chapter-wise explanations and study tasks

Chapter 1: Problem List

The opening chapter is a navigation tool rather than a new theory chapter. It organizes the exercises and gives short descriptions, with solutions aligned to the chapter names and problem numbering of Nayfeh and Mook's Nonlinear Oscillations. Use it to select problems by physical setting and analytical method. The sourcebook is intended as a supplement, so consult the original problem statement and relevant theory before reading a worked solution. Record what you attempted independently and where the derivation first became unclear. This turns the solution into feedback on reasoning rather than a sequence to copy.

Study task: Select three problems at increasing difficulty and classify the system, excitation and proposed method. Attempt the first setup before consulting its solution.

Chapter 2: Conservative Single-Degree-of-Freedom Systems

Conservative oscillators provide a setting in which energy and phase-plane reasoning can organize nonlinear behavior. The chapter's problems include potential-energy interpretation and a range of approximation methods, such as multiple scales, averaging, harmonic balance and equivalent linearization. Frequency can depend on amplitude, and trajectories need not resemble the circular picture of a linear harmonic oscillator. Derive the equation and identify the conserved quantity before applying an approximation. Compare the assumptions and retained terms of different methods instead of expecting their formulas to be identical at every order. Numerical checks can reveal the range in which an approximation remains informative.

Study task: For an illustrative conservative oscillator, derive the energy integral and sketch possible trajectories. Explain how you would compare an approximate period with numerical results at several amplitudes.

Chapter 3: Nonconservative Single-Degree-of-Freedom Systems

Nonconservative systems allow energy exchange, making equilibria, stability and limit cycles central concerns. The chapter's problem list includes singularity analysis and comparisons involving Rayleigh and van der Pol equations. Linearization describes local behavior near an equilibrium but does not automatically determine the global phase portrait. A closed orbit seen in a short numerical plot also needs a stability and convergence interpretation. Relate the sign and structure of energy input or dissipation to the observed response, and distinguish an equilibrium's stability from that of a periodic motion. Check that numerical transients are not mistaken for long-term behavior.

Study task: Compare trajectories starting near and away from an equilibrium. Identify what local linearization can establish and what additional evidence is needed to discuss a limit cycle.

Chapter 4: Forced Oscillations of Systems Having a Single Degree of Freedom

External forcing introduces the relationship between response, excitation frequency, damping and nonlinearity. The worked material includes multiple-scales analysis of resonance with explicit scaling and detuning choices. Those choices determine which terms balance at a given approximation order. Frequency-response equations can have multiple branches, but finding algebraic roots is not the same as establishing their stability or accessibility. Follow the derivation from nondimensionalization through the slow amplitude and phase equations, then compare steady solutions with direct integration. Distinguish transient motion from a settled response and report the assumptions behind each plotted curve.

Study task: Trace a resonance calculation by listing the small parameters and their orders. Explain why excitation, damping and nonlinear terms must be balanced consistently.

Chapter 5: Parametrically Excited Systems

Parametric excitation changes a coefficient of the governing equation, rather than simply adding an external force. The chapter includes transition-curve calculations for Hill-type equations, using Floquet reasoning and perturbation methods. Stability depends on the parameter region, and boundaries require an argument about the periodic or growth behavior of solutions. A forcing-frequency analogy alone can hide the distinctive mechanism of parametric resonance. Read each derivation by identifying the periodic coefficient, the assumed scaling and the condition used to locate a transition. Numerical integration should test nearby parameter values as well as the boundary itself.

Study task: Compare additive forcing with a periodically varying stiffness term. Describe a numerical check on both sides of a proposed stability boundary.

Chapter 6: Systems Having Finite Degrees of Freedom

Coupled finite-dimensional systems require keeping track of several coordinates, modal behavior and interactions that do not appear in a one-coordinate model. Read this problem section with the governing equations and coupling assumptions visible. A coordinate transformation may simplify the linear part while leaving nonlinear interactions between modes. Separate the procedure for constructing an approximation from the interpretation of its physical response. Check initial conditions and parameter assumptions before comparing solutions. The study task below provides a conceptual route through this section without reproducing its lengthy worked derivations.

Study task: For a two-coordinate model, identify the coupling terms and describe how linear modal coordinates would be found. Explain why the nonlinear terms may still couple the transformed equations.

Chapter 7: Continuous Systems

Continuous systems describe fields depending on position and time, introducing boundary conditions alongside evolution equations. The chapter includes a worked derivation for a nonuniform nonlinear elastic rod, connecting a local force balance with a distributed model. Keep material properties, geometry and loading distinct when constructing the equation. A later reduction to a finite number of modes is an approximation whose basis and boundary conditions matter. Check whether the reduced model preserves the physical features relevant to the question, and compare results as the retained representation is refined.

Study task: Outline a force balance for a small rod element and identify the required boundary conditions. Explain one way a modal truncation could affect an oscillation prediction.

Chapter 8: Traveling Waves

Traveling-wave problems connect spatial and temporal dependence and introduce questions about propagation and dispersion. The chapter begins by using a wave-form substitution to examine a dispersion relation and phase speed. A constant speed for one model does not establish nondispersive behavior for every nonlinear or modified equation. State the governing assumptions before interpreting a wave ansatz, and distinguish a periodic-wave calculation from a localized traveling structure. Substitution, dimensional checks and numerical comparison help evaluate whether a proposed solution actually satisfies the equation and the intended conditions.

Study task: Substitute a simple harmonic wave form into a linear wave model and derive its dispersion relation. Explain what dependence of phase speed on wavenumber would imply.

Worked-solution verification

For one problem, keep four separate records: your independent setup, a reconstruction of the published derivation, a numerical comparison and a statement of validity limits. Check dimensions and initial or boundary conditions. Change the small parameter or amplitude to identify where agreement deteriorates, and explain whether the discrepancy is expected from the approximation. The study tasks here are original companions, not replacements for the source's detailed solutions.

Suggested study plan

1. Use Chapter 1 to select a problem sequence matched to your theory course rather than reading solutions indiscriminately.
2. Work through representative problems from Chapters 2–3, combining energy and phase-plane interpretation with approximation methods.
3. Study Chapters 4–5 by documenting detuning, excitation order and the distinction between forced and parametric effects.
4. Use Chapters 6–7 to track coupling, mode selection and boundary conditions.
5. Finish Chapter 8 by deriving a wave relation and checking the conditions behind the proposed solution.

Questions for deeper understanding

• Which neglected terms limit the range of an approximation?
• What makes a resonance scaling internally consistent?
• How do local stability, a phase portrait and long-time numerical behavior differ as evidence?
• When does a coordinate or modal transformation simplify only part of a nonlinear model?
• How can a numerical check fail to reveal an analytical assumption that has been violated?

Edition and source coverage

Source edition: Zeng He, Wen Jiang and Lin Wang, Solved Problems in Nonlinear Oscillations: A Sourcebook for Scientists and Engineers, Springer, 2025, DOI 10.1007/978-981-97-6113-5. These original explanations follow the verified chapter sequence and published chapter descriptions, using the official book record and linked chapter material. They provide a conceptual study companion rather than reproducing the book's proofs, worked solutions, code or illustrations. Consult the original chapters for complete assumptions, derivations, examples and references. The verified problem index and available chapter material establish the full sequence. The guide to the finite-degrees-of-freedom chapter is a conceptual study orientation based on the verified topic and standard modelling distinctions; access to that chapter's detailed worked text was unavailable during preparation. Its individual derivations and numerical examples are therefore not presented as independently checked.