Perturbation Analysis of Parametric Resonance

Mathematical Models for Suspension Bridges

“…From the Mathieu diagram and from the asymptotic behavior of the separating lines (see again [267]) we learn that the instability regions become more narrow as a D n 2 increases. Let us recall that for common bridges one has i > , see (1.4): since the parametric lines (3.55) take their origin when a D 2 i = 2 > 1 (see the right picture in Fig.…”

“…We analysed in particular the two first instability regions. From [267] we know that .q; a/ lies in the first or second instability region for small enough q if, respectively, 1 q CO.q 2 /<a < 1Cq CO.q 2 / or 4 1 12 q 2 CO.q 4 /<a < 4C 5 12 q 2 CO.q 4 /:…”

“…The Mathieu equations are (3.70) with a.t/ D a C 2q cos.2t/, see (3.53) and precise stability criteria are known according to the values of a and q, see e.g. [267]. However, also the Mathieu functions are difficult to employ, mainly because of the impossibility of analytically representing them in a simple and handy way, see e.g.…”

Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features

“…This remarkable example was evidently …rst considered by A. Stephenson in 1908 [17] and [18]. Additional examples that have the same governing equations are, for instance: a frequency-modulated tuned circuit, the Paul trap for charged particles, stability of a ‡oating body, the mirror trap for neutral particles [7], certain autoparametric vibration absorbers [19], stability of elastic systems (bars, for example) under certain time-varying loading [20], asymmetric shaft and bearings in rotor dynamics [21], torsional motions of a rotor in contact with a stator [22], etc. Additional examples from other …elds include those from aerospace engineering: for example, helicopter rotor blades in forward ‡ight, attitude stability of satellites in elliptic orbits) and biology (for instance, heart rhythms, membrane vibrations in the inner ear).…”

Renormalization Group as a Probe for Dynamical Systems

“…We have already dealt with this system in the previous chapter. Here, we consider a nonlinear parametric equation which is far more general and complex having the form [379],…”