No 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.

No abstract

No abstract

The motion of the elastic pendulum is ruled by coupling between the pendulum mode and the harmonic oscillator mode. The bob of the pendulum executes trajectories in a domain of the vertical plane, the shape and size of which are set by the initial conditions and the total energy of the pendulum. The limits of this libration, in exact resonance, are studied using parabolic coordinates. Other Hamiltonians, with similar interaction terms, exhibit similar libration limits.