Qualitative analysis of classes of motion for multiresonant systems I. An algebraic method

Summary.

A general multiresonant system is considered, in which the linear frequency and, possibly, a forcing frequency are involved in a set of linear conditions. The nature of the resonances is first discussed, by distinguishing independent and dependent equations, and both the analysis and design problems of the system are addressed. Rules are then given to construct the qualitative form of the AMEs to any desired order. Two families of terms are identified: improper resonant terms (not associated with any resonance conditions) and proper resonant terms (depending on the specific conditions), sub-divided into primary (of lower order) and secondary (of higher order). Theorems are proved to show that both improper and secondary resonant terms have no qualitative but only quantitative effects on classes of motion; reference is therefore made to reduced equations. An algebraic algorithm is illustrated to determine classes of motion, using only the integer resonance coefficients. The concept of degree of constraint of a given resonance condition is introduced, entailing a hierarchic order among the resonance conditions, the implications of which are discussed. Finally, some numerical simulations are shown to test the robustness of classes of motion to higher-order terms not accounted for in the asymptotic analysis.