Course code: wb1412

Course name: Linear and nonlinear vibrations in mechanical systems

This concerns a Course

ECTS credit points: 3

Faculty of Mechanical Engineering and Marine Technology

Section of Engineering Mechanics

Lecturer(s): Woerkom, dr. ir. P.T.L.M. van

Tel.:  015 - 27 82792 /      

Catalog data:

- Introduction and review of linear vibration theory.

- Occurrence and types of linear and nonlinear mechanical vibrations.

- Analysis of linear and nonlinear vibrations in discrete mechanical systems.

- Suppression of vibrations.

- Introduction of nonlinear vibrations in continuum systems.

Course year:

MSc 1^st year

Course language:

English

In case of Dutch: Please contact the lecturer about an English alternative, whenever needed.

Semester:

2A / 2B

Hours per week:

2

Other hours:

Assessment:

Written report

Assessment period:

 /  /

(see academic calendar)

Prerequisites (course codes):

wb1216, wi2051wb, wi3097wb

Follow up (course codes):

Detailed description of topics:

- Introduction: review of linear vibration theory, sources of excitation, nonlinear vibrations in mechanical systems.

- Occurrence and types of mechanical vibrations: forced vibrations, self-excited vibrations, stick-slip vibrations, limit cycles, jump resonance, transient response due to impulse excitation, effect of impact, effect of vibrations on humans (hearing, comfort), machine vibrations, machine-tool chatter, vibration of structures to due fluid-structure interaction, intended vibrations in micro-electro-mechanical systems (MEMS), dynamics of buckling.

- Analysis of linear and nonlinear vibrations in discrete systems: phase plane analysis, stability of equilibrium, stability of motion, stability criteria (Routh-Hurwitz, Sylvester, Lyapunov, Mathieu), Duffing's method, method of averaging (Krylov-Bogoliubov, Van der Pol), Poincaré perturbation method, Poincaré-Lindstedt perturbation method, two-time-variable perturbation method, bifurcations.

- Suppression of vibrations: isolation, damping, properties of metal and rubber springs, and composites, passive dynamic damping, passive configuration damping, active damping.

- Introduction of nonlinear vibrations in continuum systems: nonlinear sound wave propagation, nonlinear vibration of a string.

Course material:

• Course notes, on Blackboard (in preparation).

References from literature:

- Dimarogonas, A. Vibration for Engineers. Second edition. Prentice-Hall, 1996.

- Harris, C.M. and Piersol, A.G. Harris's Shock and Vibration Handbook. Fifth edition. McGraw-Hill, 2002.

- Inman, D.J. Engineering Vibration. Prentice-Hall, 1996. See especially chapter 10 on nonlinear vibrations (only in this first edition!)

- Jordan, D.W. and Smith, P. Nonlinear Ordinary Differential Equations - an Introduction to Dynamical Systems. Third edition. Oxford University Press, 1999.

- Kelly, S.G. Fundamentals of Mechanical Vibrations. Second edition. McGraw-Hill International Editions, 2000.

- Rao, S.S. Mechanical Vibrations. Fourth edition. Prentice-Hall, 2004.

- Thomson, J.J. Vibrations and Stability - Order and Chaos. McGraw-Hill, London, 1997.    

Remarks assessment, entry requirements, etc.:

The course consists of two parts:

- presentation of a number of topics selected from the above outline, by the lecturer;

- investigation of a specific topic, by the participant. The topic for the assignment will be selected in consultation between participant and lecturer. The participant will carry out an exploratory study and document his findings in the form of a written progress report and a written final report.

The assessment (grading) will be based on the quality of the investigation as documented in the report.

Learning goals:

The student must be able to:

1. demonstrate understanding of the essentials of linear vibration theory, for single degree-of-freedom (“dof”) systems and for multi dof systems

2. model and analyse four classes of response suppression techniques applicable to multi dof linear systems, namely passive isolation, passive damping, active isolation, active damping

3. model and analyse suppression of response in continuum linear systems (specifically a clamped-free beam), using piezo patch and piezo film as sensor/actuator

4. identify physical sources of nonlinear dynamic behaviour of multi dof and of continuum systems, occurring in a wide field of engineering endeavour

5. analyse system stability under small perturbations (linearisation; Routh-Hurwitz, Sylvester, first method of Lyapunov) and under large perturbations (global stability, using second method of Lyapunov)

6. describe global nonlinear dynamic behaviour single dof systems, using the phase plane

7. analyse weakly nonlinear dynamic behaviour (“perturbed motion”) of single and multi dof systems using general perturbation theory (Poincaré expansion, Krylov-Bogoliubov method, averaging methods, two-variable method), to justify equivalent linearization, and to apply these techniques in the analysis of the dynamics of various physical systems

8. analyse periodic behaviour of single dof nonlinear systems (Lindstedt method, Duffing method, averaging methods) and to analyse stability of periodic behaviour (Floquet analysis, Mathieu analysis)

9. discuss physical sources of parametric excitation in linear systems, to analyse resulting periodic motion including presence of viscous damping (generalised Mathieu equation)

10. model and analyse the dynamics of nonlinear vibrations in distributed systems - specifically sound propagation, string vibration, and dynamic buckling of beams

11. carry out an independent investigation of a mutually agreed research topic in linear or nonlinear system dynamics

Computer use:

Matlab, if desired as part of take-home assignment.

Laboratory project(s):

Take-home assignment (see above).

Design content:

Percentage of design:     %