A weak energy–momentum method for stochastic instability induced by dissipation and random excitations

• A weak energy–momentum method is presented for stochastic instability.
• The stochastic instability criteria of gyroscopic systems are obtained.
• A Lagrange top subjected to random excitations is employed to expound the method.

This paper focuses on the problem of stochastic instability resulting from the action of dissipation and random excitations. The energy–momentum theorem is extended from deterministic Hamiltonian systems to stochastic Hamiltonian systems, and then a weak energy–momentum method is presented for stochastic instability analysis of random systems suffering destabilizing effects of dissipation and random excitations. The presented method combines the stochastic averaging procedure to formulate the equivalent systems of random systems for obtaining the stochastic instability criteria in probability, and can be applied to a class of systems including random gyroscopic systems with positive or negative definite potential energy. As an example, the stochastic instability conditions of a Lagrange top subjected to random vertical support excitations are formulated to express the stochastic instability induced by dissipation and random excitations.