A Floquet-Based Analysis of Parametric Excitation Through the Damping Coefficient

The Floquet theory has been classically used to study the stability characteristics of linear dynamic systems with periodic coefficients and is commonly applied to Mathieu’s equation, which has parametric stiffness. The focus of this article is to study the response characteristics of a linear oscillator for which the damping coefficient varies periodically in time. The Floquet theory is used to determine the effects of mean plus cyclic damping on the Floquet multipliers. An approximate Floquet solution, which includes an exponential part and a periodic part that is represented by a truncated Fourier series, is then applied to the oscillator. Based on the periodic part, the harmonic balance method is used to obtain the Fourier coefficients and Floquet exponents, which are then used to generate the response to the initial conditions, the boundaries of instability, and the characteristics of the free response solution of the system. The coexistence phenomenon, in which the instability wedges disappear and the transition curves overlap, is recovered by this approach, and its features and robustness are examined.

Floquet-Based Analysis of General Responses of the Mathieu Equation

Solutions to the linear unforced Mathieu equation, and their stabilities, are investigated. Floquet theory shows that the solution can be written as a product between an exponential part and a periodic part at the same frequency or half the frequency of excitation. In the current work, an approach combining Floquet theory with the harmonic balance method is investigated. A Floquet solution having an exponential part with an unknown exponential argument and a periodic part consisting of a series of harmonics is assumed. Then, performing harmonic balance, frequencies of the response are found and stability of the solution is examined over a parameter set. The truncated solution is consistent with an existing infinite series solution for the undamped case. The truncated solution is then applied to the damped Mathieu equation and parametric excitation with two harmonics.

A New Analytical Approach for the Determination of the Transient Response in Elastic Mechanisms

Recent work on multi-regime per cycle machinery has shown that for such systems the transient response of the link deflections, which is governed by the homogeneous solution to a Hill-type differential equation, cannot be neglected in a meaningful simulation. Based on mathematical work by E. T. Whittaker, a new analytical approach for obtaining such transient responses is given. It is achieved by the determination of the characteristic exponent in the stable regions of the Floquet solution to the Hill’s equation involved. This technique is applied to a specific elastic mechanism. Several sample computations show that the transient response may vary from near-periodic to totally aperiodic.

A New Analytical Approach for the Determination of the Transient Response in Elastic Mechanisms

Recent work on multi-regime per cycle machinery has shown that for such systems the transient response of the link deflections, which is governed by the homogeneous solution to a Hill-type differential equation, cannot be neglected in a meaningful simulation. Based on mathematical work by E.T. Whittaker, a new analytical approach for obtaining such transient responses is given. It is achieved by the determination of the characteristic exponent in the stable regions of the Floquet solution to the Hill’s equation involved. This technique is applied to a specific elastic mechanism. Several sample computations show that the transient response may vary from near-periodic to totally aperiodic.