Mechanical systems, equivalent in Lyapunov's sense to systems not containing non-conservative positional forces
✍ Scribed by V.N. Koshlyakov; V.L. Makarov
- Publisher
- Elsevier Science
- Year
- 2007
- Tongue
- English
- Weight
- 328 KB
- Volume
- 71
- Category
- Article
- ISSN
- 0021-8928
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✦ Synopsis
Developing results obtained previously (Refs. Koshlyakov VN. Structural transformations of the equations of perturbed motion of a certain class of dynamical systems. Ukr Mat Zh 1997; 49 (4): 535-539; Koshlyakov VN. Structural transformations of dynamical systems with gyroscopic forces. Prikl Mat Mekh 1997; 61 (5): 774-780; Koshlyakov VN, Makarov VL. The theory of gyroscopic systems with non-conservative forces. Prikl Mat Mekh 2001; 65 (4): 698-704; Koshlyakov VN, Makarov VL. The stability of non-conservative systems with degenerate matrices of dissipative forces. Prikl Mat Mekh 2004; 68 ( 6): 906-913), the general problem of eliminating non-conservative positional structures from the second-order differential equation with constant matrix coefficients, obtained when modelling many mechanical systems, is considered. It is assumed that the matrices of the dissipative and non-conservative positional structures may, in particular, be degenerate. Under fairly general assumptions, theorems containing the necessary and sufficient conditions for a Lyapunov transformation to exist are proved. This converts the initial matrix equation to an equivalent autonomous form (in Lyapunov's sense) with a symmetrical matrix of the positional forces. An illustrative example is considered.