Stabilization via parametric excitation of multi-dof statically unstable systems

• Statically unstable multi-dof Hamiltonian systems are stabilized by parametric excitation.
• A general system is considered and several resonances are studied by the Multiple Scale Method.
• Integer and fractional series expansions are used.
• Restabilization is possible at any excitation frequency, except in the resonance zones.
• Numerical results are displayed for a triple-pendulum.

The problem of re-stabilization via parametric excitation of statically unstable linear Hamiltonian systems is addressed. An n-degree-of-freedom dynamical system is considered, at rest in a critical equilibrium position, possessing a pair of zero-eigenvalues and n − 1 pairs of distinct purely imaginary conjugate eigenvalues. The response of the system to a small static load, making the zero eigenvalues real and opposite, simultaneous to a harmonic parametric excitation of small amplitude, is studied by the Multiple Scale perturbation method, and the stability of the equilibrium position is investigated. Several cases of resonance between the excitation frequency and the natural non-zero frequencies are studied, calling for standard and non-standard applications of the method. It is found that the parametric excitation is able to re-stabilize the equilibrium for any value of the excitation frequencies, except for frequencies close to resonant values, provided a sufficiently large excitation amplitude is enforced. Results are compared with those provided by a purely numerical approach grounded on the Floquet theory.