Module 1: Introduction and Fundamentals of Nonlinear Vibrations (6 Hours)
Introduction to Nonlinear Vibrations
Difference between linear and nonlinear systems
Sources of nonlinearity in mechanical systems
Linear Vibration Review
Free and forced vibrations of single-degree-of-freedom (SDOF) systems
Damping and resonance

Module 2: Free Vibrations of Undamped and Damped Nonlinear Systems (8 Hours)
Free Vibrations with Nonlinear Restoring Forces
Hardening and softening springs
Approximate analytical methods (e.g., perturbation methods, Lindstedt-Poincaré method)
Damped Free Oscillations and Geometry of Integral Curves
Study of singular points
Applications using the notion of singularities

Module 3: Forced Oscillations in Nonlinear Systems (7 Hours)
Forced Vibrations with Nonlinear Restoring Forces
Response to periodic excitation
Jump phenomenon and bifurcations
Resonance in Nonlinear Systems
Subharmonic and superharmonic resonance

Module 4: Self-Sustained Oscillations (8 Hours)
Free Oscillations in Self-Sustained Systems
Examples of self-excited oscillations (Van der Pol oscillator)
Forced Oscillations in Self-Sustained Systems
Nonlinear limit cycles and bifurcations
Applications in engineering and physics

Module 5: Stability and Hill’s Equation (7 Hours)
Hill’s Equation and Stability Analysis
Floquet theory and parametric excitation
Stability of periodic solutions
Applications to Nonlinear Oscillations
Stability in engineering and physical systems
Practical case studies

To introduce the fundamental principles of nonlinear vibrations and their distinctions from linear vibration systems.

To develop analytical and numerical methods for studying free and forced oscillations in nonlinear systems.

To provide insights into self-sustained oscillations, singularities, and stability analysis in nonlinear dynamic systems.

To equip students with the ability to apply mathematical models and stability criteria to solve real-world nonlinear oscillation problems in engineering and applied sciences.

(CO1) Explain the fundamental differences between linear and nonlinear vibration systems and their significance in real-world applications.
(CO2) Analyze the behavior of undamped and damped nonlinear oscillators using mathematical and graphical techniques.
(CO3) Evaluate the response of nonlinear systems under forced oscillations and resonance conditions.
(CO4) Investigate self-sustained oscillations and limit cycles in nonlinear dynamic systems.
(CO5) Apply stability criteria and Hill’s equation to assess the behavior of nonlinear oscillations in engineering problems.

vana Kovacic and Michael J. Brennan, Nonlinear Oscillations: Exact Solutions and Their Approximations, Springer, 1st edition , 2020

David N. Cheban, Nonautonomous Dynamics: Nonlinear Oscillations and Global Attractors, Springer, 1st edition , 2020

Ali H. Nayfeh, Dean T. Mook, Solved Problems in Nonlinear Oscillations, Springer, 1st edition , 2020

Steven H. Strogatz, Nonlinear Dynamics and Chaos, CRC Press , 2020

Communications in Nonlinear Science and Numerical Simulation, Elsevier

Nonlinear Dynamics, Springer