Approximate Methods for Analysing Nonlinear Vibrations

Most nonlinear systems have no easily obtainable exact analytical solution. As a result, one has to use either graphical methods (via numerical solution) or approximate analytical methods. In this chapter a range of the most common approximation methods will be discussed. In particular, the effect of nonlinearity on resonance peaks is often analysed using approximate methods. For example, vibrating systems where the natural frequency changes as the amplitude of oscillation increases will be examined in detail. The discussion starts with the methods of harmonic balance and averaging. Then perturbation approximation methods are considered, the most important of which is multiple scales method. Finally the method of normal forms is discussed when applied to vibration problems.