Course Code: wb1412
Course name: Non-linear Vibrations

DUT credits: 2
ECTS credits: 3

Subfaculty of Mechanical Engineering and Marine Technology

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

Tel.: 015 - 27 82792

Catalog data:

• Elementary examples of system nonlinearities, amongst which: spring, pendulum, dry friction, aerodynamic damping, speaker, beam.
• Describing functions (equivalent linearisation). Premature linearisation. Phase plane analysis.
• Regular perturbations. Asymptotic series. Poincaré method. Poincaré-Lindstedt method. Two-variable method. Periodic solutions. Duffing method. Van der Pol method (method of slowly varying parameters). Limit cycles.
• Stability criteria: Routh-Hurwitz, Lyapunov. Stability under parametric excitation (Mathieu equation). Time delays. Bifurcations. Chaos.

Course year: 4
Period: 0/0/2/2
Hours p/w: 2
Other hours: -
Assessment: take-home
Assessment period: 4
(see academic calendar)

Prerequisites:

Follow-up:

Detailed description of topics:

Most engineers will sooner or later be confronted with issues involving dynamics of mechanical systems. It is therefore prudent and in fact essential to also acquire some elementary knowledge of nonlinear aspects of the dynamics of mechanisms and structures during their formative years at the university. Consider some issues:

• in real life mechanical systems display nonlinear behavior (heavily deformed spring, dry friction, buckling, tight as well as loose cables, periodic external excitation, sensor quantisation, and so on). Linearisation may lead to misleading information;

• non-linearities may be responsible for hard-to-explain differences between predicted measurements and real measurements;

• interpretation and validation of results from numerical simulations of nonlinear dynamics may be difficult or even meaningless in the absence of insight of nonlinear behavior;

• non-linearities may change the dynamic characteristics of a mechanical system considerably. This may be quite desirable and to be exploited; it may also be most undesirable.

The course addresses issues in modelling and analysis of the dynamics of non-linear systems found in daily life. In particular vibrations around one or more system equilibria will be investigated. More specifically:

Oscillations in a “pin-on-disk” tribometer. Application of “dither” to achieve equivalent linearisation of mechanisms with non-linear friction. Angular motion of rigid bodies. Automatic nutation damping. Deliberately non-linear vibration damping. Galopping cables (Erasmus bridge; power lines). Wheel blockage during braking. Wheel shimmy. Ship roll in sea waves. Surge of a ship moored to an articulating tower. Dynamics of a loudspeaker. Stability of systems with parametric excitation (pendulum with vibrating support, cable and beam with periodic axial load). Chaos in een simple control system. Chaos with inkjet printing. Instability and chaos during buckling. Stability and chaos during metal cutting.

Course material:

Hand-outs may be distributed during the course.

At the same time, the following two paperback books are recommended for study:

• Jordan, D.W and Smith, P. Nonlinear Ordinary Differential equations, second edition. Clarendon Press, Oxford, 1995.

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

 References from literature:

• Moon, F.C. Chaotic Vibrations – an Introduction for Applied Scientists and Engineers. J. Wiley and Sons, N.Y., 1987.

• Moon, F.C. Dynamics and Chaos in Manufacturing Processes. J. Wiley and Sons, Inc., N.Y., 1998.

• Verhulst, F. Nietlineaire Differentiaalvergelijkingen en Dynamische Systemen. Epsilon Uitgaven, Utrecht, 1985. (An English version is also on the market.)

Remarks (specific information about assessment, entry requirements, etc.):
At the end of the course a take-home assignment will be issued. This assignment may be purely theoretical in nature. Alternatively it may contain a strong numerical component, involving the simulation of system dynamics using e.g. MATLAB. The assignment will be drafted in consultation with the participating students.

Goals:
Provide insight in the fundamental aspects of nonlinearities in mechanical systems and their analysis. Review of some mathematical tools for analysis. Review of non-linear behavior of systems as occurring in daily life within a variety of engineering fields.

Computer use:
Possibly MATLAB to solve problems defined in the take-home assignment.

Laboratory projects:

Design content:
Examples are taken from applications. The course therefore assists the designer to obtain insight into dynamics of mechanical systems such as the ones mentioned above.

Percentage of design: