On the linear theory of the elastic pendulum

Abstract

A harmonic oscillator in a gravitational field, such as a weight on a spring, sometimes exhibits loss of lateral stability due to parametric resonance. The resulting horizontal, or pendulum oscillation is analysed by means of a locus line drawn in the Ince-Strutt stability chart of the respective Mathieu equation, to determine graphically the range of mass leading to instability for a particular amplitude of the initially vertical oscillation and the corresponding growth coefficient.

Steady states of the parametric rotator and pendulum

“…We notice also that the steady-state phase Θ 0 is completely determined by the non-linear dynamics and the damping, without reference to the initial conditions which, for times τ ≫ 1/ λ, are completely erased by the damping interactions. We conclude that, in physical units of time, at small ∆ and λ satisfying (11), steady-state rotations occur only with the same angular frequency γ as the parametric excitation, θ(t) = ±(γt + Θ 0 ) + O(∆), either clockwise or counter-clockwise.…”

“…which generalises condition (11) to elliptic excitations of the rotator. We see also that contrarian motions are allowed by (28), for small enough λ, except for ǫ = −π/2.…”

“…We must therefore assume that λ = Ø(∆), which expresses the physically reasonable condition that, for small parametric excitations to cause steadystate rotations, friction must be correspondingly small. (The relation between λ and ∆ is made precise below in equation (11).) Thus, substituting (7) in (6) and neglecting terms of Ø(∆ 2 ) yields…”

Synchronous motion of two vertically excited planar elastic pendula

“…The equations of motion are easy to write but, in general, impossible to solve analytically, even in the Hamiltonian case. The elastic pendulum exhibits a wide and surprising range of highly complex dynamic phenomena [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16].…”

“…They considered small oscillations of the planar pendulum and identified the linear normal modes of two distinct types, vertical or springing oscillations in which the elasticity is the restoring force and quasi-horizontal swinging oscillations in which the system acts like a pendulum. When the frequency of the springing and swinging modes are in the ratio 2 : 1, an interesting non-linear phenomenon occurs, in which the energy is transferred periodically back and forth between the springing and swinging motions [1,2,3,4,5,6]. The most detailed treatment of small amplitude oscillations of both plane and spherical elastic pendula is presented in the works of Lynch and his collaborators [7,10,11,12].…”