Pulsation and precession of the resonant swinging spring
Item Type:Journal Article
Citation:Peter Lynch and Conor Houghton 'Pulsation and precession of the resonant swinging spring' in Physica D: Nonlinear Phenomena, 190, (1-2), 2004, pp 38 - 62.
Pulsation and Precession of the Resonant.pdf (published (arXive copy) peer reviewed) 487.9Kb
When the frequencies of the elastic and pendular oscillations of an elastic pendulum or swinging spring are in the ratio 2:1, there is a regular exchange of energy between the two modes of oscillation. We refer to this phenomenon as pulsation. Between the horizontal excursions, or pulses, the spring undergoes a change of azimuth which we call the precession angle. The pulsation and stepwise precession are the characteristic features of the dynamics of the swinging spring. The modulation equations for the small-amplitude resonant motion of the system are the well-known three-wave equations. We use Hamiltonian reduction to determine a complete analytical solution. The amplitudes and phases are expressed in terms of both Weierstrass and Jacobi elliptic functions. The strength of the pulsation may be computed from the invariants of the equations. Several analytical formulas are found for the precession angle. We deduce simplified approximate expressions, in terms of elementary functions, for the pulsation amplitude and precession angle and demonstrate their high accuracy by numerical experiments. Thus, for given initial conditions, we can describe the envelope dynamics without solving the equations. Conversely, given the parameters which determine the envelope, we can specify initial conditions which, to a high level of accuracy, yield this envelope.
Author: HOUGHTON, CONOR JAMES
Type of material:Journal Article
Physica D: Nonlinear Phenomena
Availability:Full text available
Keywords:Pure & Applied Mathematics