An analytical solution to non-similar normal modes in a strongly non-linear discrete system
✍ Scribed by G.S. Happawana; A.K. Bajaj; M. Azene
- Publisher
- Elsevier Science
- Year
- 1995
- Tongue
- English
- Weight
- 384 KB
- Volume
- 183
- Category
- Article
- ISSN
- 0022-460X
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✦ Synopsis
In non-linear vibration theory, the concept of a ''non-linear normal mode'', similar to the ''normal mode'' in linear vibration theory, was introduced by Rosenberg [1-3]. It is used for studying the free oscillations of discrete as well as continuous undamped strongly non-linear systems. According to Rosenberg, ''non-linear normal modes'' are defined as free motions where all co-ordinates vary equi-periodically, reaching their extremum values at the same instants of time. The motion in a normal mode of vibration for a non-linear system, in contrast to the linear case, could be a straight line or a curved line in the configuration space. These are termed ''similar'' and ''non-similar'' non-linear normal modes, respectively, in the literature [3]. These curved modal lines in the configuration space become singular at the maximum equipotential surface [3]. Also, for non-linear systems, unlike the case of a linear system, the number of existing normal modes can exceed the number of degrees of freedom of the system. A more complete discussion of the concept of non-linear normal modes can be found in references [1,5,6,7]. Techniques for developing asymptotic solutions that approximate the non-similar normal modes can be found in references [7][8][9][10].
In a recent work by Vakakis [10], the free oscillations of a two-degree-of-freedom strongly non-linear system of discrete oscillators have been examined by computing its ''non-similar non-linear normal modes''. The Mikhlin-Manevich asymptotic methodology introduced in reference [7] has been implemented for solving the singular functional equations that describe the ''non-similar modes'' of the strongly non-linear conservative oscillators, shown in Figure 1. The asymptotic techniques developed in reference [10], for approximating the modal lines in the configuration space, are valid only for sufficiently small values of the functional variables. The fact that the functional equations are singular at the maximum equipotential surface further complicates the problem, limiting the range of independent parameter values to small open intervals not containing the singular points at the equipotential surface.
The objective of this note is to establish a closed form asymptotically valid solution for the non-similar normal modes for the strongly non-linear discrete system studied in reference [10]. Since a relatively complete and detailed analysis of the problem of the non-linear normal modes is given in reference [10], the discussion here is mostly restricted to the equations describing the asymptotic approximation to the non-similar normal modes. The equation governing the existence of the non-linear normal mode for the system is singular at the boundaries (or at the equipotential surface) and depends on a small mistuning parameter. The singularity in the sequence of linear problems obtained for an