Parametric resonance and nonlinear string vibrations

Abstract

Periodic changes in the tension of a taut string parametrically excite transverse motion in the string when the driving frequency is close to twice the natural frequency of any transverse normal mode of the string. The literature on this phenomenon is synthesized and extended to include the effects of damping as well as nonlinearity. It is shown that it is nonlinearity rather than damping that limits the growth of a resonantly excited mode, although damping is needed for steady-state oscillations to occur. The validity of the usual approximation that the string tension depends only on time and not on space is checked by modeling a string as point masses joined by massless linear springs. It is found that although this approximation is likely to be violated in practice, the violation does not have a significant effect on the results. The source of the disagreement in the literature for the speed of longitudinal waves in a stretched string is identified.

Small amplitude transverse waves on taut strings: exploring the significant effects of longitudinal motion on wave energy location and propagation

“…The algorithm for solving (23) and (24) for a string driven at X = 0 is therefore as follows. If the waves in the string are fully caused by the boundary conditions η(0, t ) and ξ (0, t ), then 1. η(X, t ) = η(0, t − X/c T ) 2. ξ T (0, t ) can be determined by solving (27) and using ξ T (0, t 0 ) = 0, where t 0 is the time when the boundary first starts to be displaced 3.…”

“…which implies that if ξ + 1 2 η 2 is independent of position, then the tension, and the potential energy density through (B.2), is always uniform across the entire string and is determined by the length of the string at every instant (e.g. [26,27], section 5 of [30]). (Note the difference with the previous subsection: in the c L → ∞ limit, the tension is exactly uniform across the entire string with τ > τ 0 , while in the general case considered in subsection 3.2, the tension is only exactly uniform over a pure T-wave with the tension in that case given by τ 0 .…”

Microwave absorption by an array of carbon nanotubes: A phenomenological model

“…1. The boundaries between stable and unstable regions near the values ␣ =1/4, ␤ = 0 are given by ␣ Ϸ 1/4±␤ /2 − ␤ 2 /8 20,27,28 It is seen that the point ␣ =1/4, ␤ =1/2 lies in an unstable region and the general solution has the form…”

“…The displacements are solutions of a nonlinear Mathieu equation derived from the third-order wave equations constrained to two-dimensional wave motion. Initial conditions of the nonlinear Mathieu's equation are considered given ͑referred to as a "seed" by Rowland 20 ͒ and initiated by thermal excitations. ͑Experiments 6-8 were done at room temperature in the work cited in Ref.…”

Hyperbolic chaos of standing wave patterns generated parametrically by a modulated pump source

“…In this note we propose a way to organize an analogous kind of chaotic dynamics in parametric excitation of standing wave patterns by modulated pumping in a spatially extended system with nonlinear dissipation. Particularly, the model we consider may be associated with mechanical vibrations of a string and regarded as a modification of classic Melde experiment [12][13][14].…”