Parametric and autoparametric resonance

Parametric and autoparametric resonance play an important part in many applications while posing interesting mathematical challenges. The linear dynamics is already nontrivial whereas the nonlinear dynamics of such systems is extremely rich and largely unexplored. The role of symmetries is essential, both in the linear and in the nonlinear analysis

An application of the ince algebraization to the stability of non-linear normal vibration modes

International audienceA normal vibration mode stability in conservative non-linear systems is investigated. The algebraization by !nee (transition from linear equations with periodic coefficients to equations with singular points) is used. The normal mode stability in homogeneous systems, whose potential is an even homogeneous function of the variables and systems close to the homogeneous one, is investigated. Eigenvalues and eigenfunctions are obtained. Conditions when a number of instability zones in a non-linear system parameters space are finite (finite zoning or finite-gap conditions) are also obtained

Stability of parametrically forced linear systems

The stability analysis of constant-coefficient linear systems is extended to systems with periodically-varying coefficients. Although this theory is mathematically well-understood, little work has been done regarding its application to physical problems. All previous results are based on asymptotic analysis. A review of the theory of parametrically-forced linear systems will be presented, followed by a detailed stability analysis of a pendulum with a harmonically moving base

The influence of gyroscopic forces on the dynamic behavior and flutter of rotating blades

The structural dynamics of a cantilever turbomachine blade mounted on a spinning and precessing rotor are investigated. Both stability and forced vibration are considered with a blade model that increases in complexity (and verisimilitude) from a spring-restrained point mass, to a uniform cantilever, to a twisted uniform cantilever turbomachine blade mounted on a spinning and precessing rotor are investigated. Both stability and forced vibration are considered with a blade model that increases in complexity (and verisimilitude) from a spring-restrained point mass, to a uniform cantilever, to a twisted uniform cantilever, to a tapered twisted cantilever of arbitrary cross-section. In every instance the formulation is from first principles using a finite element based on beam theory. Both ramp-type and periodic-type precessional angular displacements are considered. In concluding, forced vibrating and flutter are studied using the final and most sophisticated structural model. The analysis of stability is presented and a number of numerical examples are worked out

A Method for Analysing Parametrically Excited System by Matrix Function

This paper describes a method for analysing parametrically excited system of higher order. The method is based on the theory of matrix function and the discrete Fourier transform. As a numerical example, we deal with a kind of Hill's equation derived from the synchronous generator circuit with unbalanced capacitive load and give its stability charts