An introduction to self-excited oscillations

This paper describes an approach used to introduce a type of nonlinear problems in an undergraduate class on mechanical vibrations. Self-excited oscillations are encountered in a number of practical applications including brakes, clutches, belts, tires, and violins. To go beyond the derivation of the equations of motion for simplified models and examine the effect of various parameters requires the ability to find numerical solutions. It was found that developing numerical solutions using a simple integration technique such as Euler's method with a spreadsheet program like Excel was most effective because: (1) Euler's method is easy to implement; (2) Excel is widely available; (3) students are able to develop the solution themselves; (4) it can be done quickly. In this case students were able to explore problems with one or more degrees of freedom and compare their results with those found in recent publications which presents several advantages: students develop confidence in their ability to explore different models and examine the effects of different complicating factors, they develop their own solutions and are able to focus on understanding the physics of the problem, and they develop a sense that they are working on problems of current interest instead of some overly simplified textbook problem. Examples dealing with brake squeal problem were used and the effects of mass, stiffness, damping and friction were studied. Many different friction models are available and several of them were used to determine the effect of friction on the appearance of self-excited vibrations. The appearance of a limit cycle in the phase portrait is discussed along with the dynamics of the system. It is also shown that a short high frequency excitation can be used to squelch those self-excited oscillations.