4,366 research outputs found
Pulsation and Precession of the Resonant Swinging Spring
When the frequencies of the elastic and pendular oscillations of an elastic
pendulum or swinging spring are in the ratio two-to-one, 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.Comment: 33 pages, 9 eps figure
Precession and Recession of the Rock'n'roller
We study the dynamics of a spherical rigid body that rocks and rolls on a
plane under the effect of gravity. The distribution of mass is non-uniform and
the centre of mass does not coincide with the geometric centre.
The symmetric case, with moments of inertia I_1=I_2, is integrable and the
motion is completely regular. Three known conservation laws are the total
energy E, Jellett's quantity Q_J and Routh's quantity Q_R.
When the inertial symmetry I_1=I_2 is broken, even slightly, the character of
the solutions is profoundly changed and new types of motion become possible. We
derive the equations governing the general motion and present analytical and
numerical evidence of the recession, or reversal of precession, that has been
observed in physical experiments.
We present an analysis of recession in terms of critical lines dividing the
(Q_R,Q_J) plane into four dynamically disjoint zones. We prove that recession
implies the lack of conservation of Jellett's and Routh's quantities, by
identifying individual reversals as crossings of the orbit (Q_R(t),Q_J(t))
through the critical lines. Consequently, a method is found to produce a large
number of initial conditions so that the system will exhibit recession
Stepwise Precession of the Resonant Swinging Spring
The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at
cubic order in its approximate Lagrangian. The corresponding modulation
equations are the well-known three-wave equations that also apply, for example,
in laser-matter interaction in a cavity. We use Hamiltonian reduction and
pattern evocation techniques to derive a formula that describes the
characteristic feature of this system's dynamics, namely, the stepwise
precession of its azimuthal angle.Comment: 28 pages, 10 figure
Counting of discrete Rossby/drift wave resonant triads (again)
The purpose of our earlier note (arXiv:1309.0405 [physics.flu-dyn]) was to
remove the confusion over counting of resonant wave triads for Rossby and drift
waves in the context of the Charney-Hasegawa-Mima equation. A comment by
Kartashov and Kartashova (arXiv:1309.0992v1 [physics.flu-dyn]) on that note has
further confused the situation. The present note aims to remove this
obfuscation
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