# Nonlinear Vibration

By Prof. S. K Dwivedy   |   IIT Guwahati
Learners enrolled: 318
Most of the vibrating structure are nonlinear in nature. But for simplification of the analysis they have been considered to be linear. Hence, to actually know the response of the system one should study the nonlinear behavior of the system. Here one may encounter multiple equilibrium points or solutions which may be stable or unstable. The response may be periodic, quasiperiodic or chaotic. The present course is a simulation based course where one can visualize the response of different mechanical systems for different resonance conditions. Out of 9 modules, first 8 modules are on developing the equations of motion, solution procedure of these equations and application of them to general single and multi-degree of freedom systems. The last modules which will be covered in 3 weeks taking 3 different applications of current interest are project based and it will give a very good practical exposure. The course will be very useful for undergraduate, post graduate and PhD students in Academic institutions and also practicing engineers in Industry.

INTENDED AUDIENCE
Senior under graduate or post graduate and PhD students in Mechanical Engineering can take this course under Advanced Dynamics domain
PREREQUISITES : None
INDUSTRIES  SUPPORT     : All the industry dealing with manufacturing, automobile, aerospace etc. will require nonlinear vibration analysis to improve their productivity
Summary
 Course Status : Completed Course Type : Elective Duration : 12 weeks Category : Mechanical Engineering Advanced Dynamics and Vibration Credit Points : 3 Level : Undergraduate/Postgraduate Start Date : 24 Jan 2022 End Date : 15 Apr 2022 Enrollment Ends : 07 Feb 2022 Exam Date : 23 Apr 2022 IST

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

Week 1:Module 1: Introduction to Nonlinear Mechanical Systems
Introduction to mechanical systems, Superposition rule, familiar nonlinear equations: Duffing equation, van der Pol’s equation, Mathieu-Hill’s equation, Lorentz system, Equilibrium points: potential function
Week 2:Module 2: Development of Nonlinear Equation of Motion using Symbolic Software
Force and moment based Approach, Lagrange Principle, Extended Hamilton’s principle, use of scaling and book-keeping parameter for ordering
Week 3: Module 3: Solution of Nonlinear Equation of Motion
Numerical solution, Analytical solutions: Harmonic Balance method, Straight forward expansion and Lindstd-Poincare’ method
Week 4:Method of Averaging, Method of multiple scales, Method of 3 generalized Harmonic Balance method
Week 5:Module 4: Analysis of Nonlinear SDOF system with weak excitation
Free vibration of undamped and damped SDOF systems with quadratic and cubic nonlinearity, and forced vibration with simple resonance
Week 6:Analysis of Nonlinear SDOF system with hard excitation
Nonlinear system with hard excitations, super and sub harmonic resonance conditions, Bifurcation analysis of fixed-point response
Week 7: Vibration Analysis of Parametrically Excited system
Principal and combination parametric resonance conditions, Floquet theory, frequency and forced response of nonlinear parametrically excited system.
Week 8:Analysis of Periodic, quasiperiodic and Chaotic System
Stability and bifurcation analysis of periodic response, analysis of quasi-periodic system, analysis of chaotic System
Week 9:Numerical Methods for Nonlinear system Analysis
Solutions of a set of nonlinear equations, Numerical Solution of ODE and DDE equations, Time response, phase portraits, frequency response, Poincare section, FFT, Lyapunov exponent
Week 10:Practical Application 1: Nonlinear Vibration Absorber
Equation of motion, Solution of EOM: Use of Harmonic Balance method, Program to obtain time and frequency response
Week 11: Practical Application 2: Nonlinear Energy Harvester
Development of Equation of motion: symbolic software, Solution of EOM: Use of method of Multiple Scales, Program to obtain time and frequency response
Week 12:Practical Application 3: Analysis of electro-mechanical system
Development of Equation of motion and its solution, Use of Floquet theory, Parametric instability regions, Study of periodic, quasiperiodic and chaotic response

### Books and references

1. A. H. Nayfeh, and D. T. Mook, Nonlinear Oscillations, Wiley-Interscience,1995
2. M. P. Cartmell, Introduction to linear, parametric, and nonlinear vibrations, Chapman and Hall, 1990.
3 A. H. Nayfeh, and B. Balachandran, Applied Nonlinear Dynamics, Wiley,1995.
4. R. Seydel, From Equilibrium to Chaos: Practical Bifurcation and StabilityAnalysis, Elsevier, 1988.
5.Moon, F. C., Chaotic & Fractal Dynamics: An Introduction for AppliedScientists and Engineers, Wiley, 1992.
6. R. H. Rand, 2005.Lecture Notes on Nonlinear Vibrations, version 52. Available online athttp://audiophile.tam.cornell.edu/randdocs/nlvibe52.pdf
7. J. Guckenheimer, and P. Holmes, 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag.
8. S. H. Strogatz, 1994. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology,Chemistry, and Engineering. Addison-Wesley.
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### Prof. S. K Dwivedy

IIT Guwahati
Prof. S. K. Dwivedy has started working in the field of nonlinear Vibration during his PhD program at IIT Kharagur. He is teaching the Nonlinear Vibration Course for the last 20 years at IIT Guwahati. He has developed the web and video courses on Nonlinear Vibration in NPTEL. He has guided 10 PhD students and more than 40 M. Tech students and published more than 150 papers in the International Journals and Conferences in the field of nonlinear vibration. These works have been in diverse areas such as structural applications, energy harvester, robotic manipulators, metallic ion polymer composites, pneumatic artificial muscles, vibration absorbers in biomedical applications.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 23 April 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Guwahati .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team