Linear Gyroscopic Systems Research Articles

Perturbation bounds for stable gyroscopic systems in motion about an unstable equilibrium position

Relative perturbation bounds for stable linear gyroscopic systems Mẍ(t)+Gẋ(t)+Kx(t)=0 are considered, where M,K∈Rn×n are symmetric, M is positive definite, K negative definite and G∈Rn×n is skew-symmetric such that |G|>kM−k−1K for some k>0. This means that all of the eigenvalues of the system are purely imaginary and semi-simple. For this kind of systems, we present an upper bound for the relative change in eigenvalues as well as the sinΘ type bound for the corresponding eigenvectors under the perturbation of the system matrices. The performance of obtained bounds is illustrated in numerical experiments.

On the stability of linear systems with a quadratic integral

The problem of stability of non-degenerate linear systems admitting a first integral in the form of a non-degenerate quadratic form is considered. New algebraic criteria for stability, as well as complete instability of such systems, have been established in the form of equality to zero of traces of products of matrices, which include an additional symmetric matrix. These conditions are closely related to the symplectic geometry of the phase space, which is determined by the matrix of the original linear system and the symmetric matrix defining the first integral. General results are applied to finding conditions for complete instability of linear gyroscopic systems.

Existence/nonexistence of instability regions in a parametrically excited linear gyroscopic system

The present work investigates the existence/nonexistence of instability regions for a parametrically excited linear gyroscopic system. To achieve this goal, a new approach is proposed to determine the boundaries of parametric instability regions. As long as the gyroscopic system's undamped equations of motion are derived, one can easily use this approach to determine its instability regions. For convenience, a rotor-bearing system with periodic axial loaded is used as a parametrically excited representative gyroscopic system. The approach rewrites the second order differential equations in the state space form. Then the generalized eigenvalue problem is solved to derive the left and right eigenvectors. They are used to decouple the governing equations and reduce the order. In the subsequent theoretical derivation, the multiple scale method is applied to obtain the analytical solutions of the boundaries of instability regions. The numerical simulation is also carried out to validate the analytical boundaries. Wherein the numerical instability regions are obtained by applying the discrete state transition matrix method. From the theoretical and numerical analysis, we find out: (1) the analytical boundaries match well with the numerical results; (2) only the sum type instability regions can be observed; (3) the primary difference type instability regions do not exist; (4) the secondary or higher order difference type instability regions also may not exist.

A note on the Lyapunov stability of linear conservative gyroscopic systems

A note on the Lyapunov stability of linear conservative gyroscopic systems

Multiple Time Analysis of Weakly Coupled Non- linear Gyroscopic Systems

Multiple Time Analysis of Weakly Coupled Non- linear Gyroscopic Systems

Methods for obtaining sufficient conditions for the stability of autonomous conservative systems

A computational method for obtaining sufficient conditions for the stability of the stationary solution of autonomous conservative systems is proposed in the paper. This method is adapted to linear autonomous gyroscopic systems with three degrees of freedom. It is based on the positive definiteness of a parametric quadratic form composed of the gyroscopic force matrices and the potential function. The control parameters for the stability of the zero solution of the gyroscopic system are the entries of the gyroscopic force matrix. The algorithm of the computational method includes estimating one gyroscopic force parameter in the equation constructed from a necessary stability condition.

Eigensensitivity analysis of rotating clamped uniform beams with the asymptotic numerical method

In this paper, free vibrations of a rotating clamped Euler–Bernoulli beams with uniform cross section are studied using continuation method, namely asymptotic numerical method. The governing equations of motion are derived using Lagrange׳s method. The kinetic and strain energy expression are derived from Rayleigh–Ritz method using a set of hybrid variables and based on a linear deflection assumption. The derived equations are transformed in two eigenvalue problems, where the first is a linear gyroscopic eigenvalue problem and presents the coupled lagging and stretch motions through gyroscopic terms. While the second is standard eigenvalue problem and corresponds to the flapping motion. Those two eigenvalue problems are transformed into two functionals treated by continuation method, the Asymptotic Numerical Method. New method proposed for the solution of the linear gyroscopic system based on an augmented system, which transforms the original problem to a standard form with real symmetric matrices. By using some techniques to resolve these singular problems by the continuation method, evolution curves of the natural frequencies against dimensionless angular velocity are determined. At high angular velocity, some singular points, due to the linear elastic assumption, are computed. Numerical tests of convergence are conducted and the obtained results are compared to the exact values. Results obtained by continuation are compared to those computed with discrete eigenvalue problem.

Simultaneous diagonalization of three real symmetric matrices

We formulate and prove necessary and sufficient conditions of simultaneous diagonalization of three real symmetric matrices of regular pencil. The conditions are algebraic and consist, in particular, of two spectral requirements and one matrix equality. For the degenerate matrix pencil we suggest an approach that allows reducing of the analysis to a regular pencil. With the use of obtained theorems we investigate a decomposition of linear gyroscopic system into subsystems of an order not higher than two and the stability of trivial solution to a system.

Lagrangians for Damped Linear Multi-Degree-of-Freedom Systems

This paper deals with finding Lagrangians for damped, linear multi-degree-of-freedom systems. New results for such systems are obtained using extensions of the results for single and two degree-of-freedom systems. The solution to the inverse problem for an n-degree-of-freedom linear gyroscopic system is obtained as a special case. Multi-degree-of-freedom systems that commonly arise in linear vibration theory with symmetric mass, damping, and stiffness matrices are similarly handled in a simple manner. Conservation laws for these damped multi-degree-of-freedom systems are found using the Lagrangians obtained and several examples are provided.

Spectral properties of linear gyroscopic systems

We study how the natural frequencies of a linear gyroscopic system behave under variations in the kinetic and potential energy matrices as well as in the gyroscopic force matrix.

Instabilities in a MEMS Gyroscope Subjected to Angular Rate Fluctuations

Instabilities in a vibrating MEMS gyroscope that is subject to periodic fluctuations in input angular rates are investigated. For the purpose of acquiring stability conditions, when the angular rate input is subject to small intensity periodic fluctuations, dynamic behavior of periodically perturbed linear gyroscopic systems is studied in detail. An asymptotic approach based on the method of averaging has been employed for this purpose, and closed-form conditions for the onset of instability due to parametric resonances have been obtained. A numerical approach based on the Floquet-Lyapunov theory is employed for validating the analytical stability predictions. Furthermore, for characterizing the effect due to change in angular rate input, an in-depth natural frequency analysis has been performed. Stability predictions have been illustrated via stability diagrams in the excitation amplitude-frequency space. Based on these results, the dangerous critical frequencies can be avoided during the design process.

Solvability condition in multi-scale analysis of gyroscopic continua

The solvability condition is investigated for the method of multiple scales applied to gyroscopic continua. The general framework of the multi-scale analysis is proposed for a linear gyroscopic continuous system under small nonlinear time-dependent disturbances. The solvability condition is derived from the properties of the systems. The condition holds only for appropriate boundary conditions. The appropriateness of the boundary conditions can be examined for unperturbed linear systems. An example is presented to highlight the requirements on the boundary conditions.

Dynamic stability of ring-based angular rate sensors

Dynamic stability of a rotating ring subjected to harmonic perturbations in input angular rate is examined using an asymptotic approach. The governing equations that represent the transverse and tangential in-plane motion of the ring are derived via Hamilton's principle. The equations of motion, after discretization and suitable linearization, represent a two-degree-of-freedom time-varying linear gyroscopic system. Such a system can exhibit instability behaviour characterized by exponential growth in response amplitudes. Employing the method of averaging, conditions for instability are obtained in closed-form. Instability boundaries for the ring in the excitation intensity-frequency space are then established for small excitation amplitudes. In addition, effects of damping, input angular rate variations, and imperfection due to the ring asymmetry are discussed.

MEMS vibratory angular rate sensors: stability considerations for design

Stability of a class of vibratory angular rate sensors that are subject to periodic fluctuations in input angular rates is investigated. Two typical configurations are considered in the present paper: a single-axis ring-type and a dual-axis disk-type. For the purpose of acquiring stability conditions, when the angular rate input is subject to small intensity periodic fluctuations, dynamic stability behaviour of periodically perturbed linear gyroscopic systems are studied in detail. An asymptotic approach based on the method of averaging has been employed for this purpose, and closed-form conditions for the onset of instability due to parametric resonances have been obtained for these two types of vibrating gyroscopes. Stability predictions have been illustrated via plots in the excitation amplitude–frequency space. Damping is considered and in general it has been demonstrated that the introduction of damping tends to stabilize the sensor systems. Furthermore, for characterizing the effect of input angular rate on the stability, an in-depth natural frequency analysis has been performed for the disk-type angular rate sensor. In addition, the effect of mass mismatch on the stability region has been presented for the ring-type angular rate sensor. Copyright © 2005 John Wiley & Sons, Ltd.

On stability boundary of linear multi-parameter Hamiltonian systems

In this paper an approximate equation is derived to describe smooth parts of the stability boundary for linear Hamiltonian systems, depending on arbitrary number of parameters. With this equation, we can obtain parameters corresponding to the stability boundary, as well as to the stability and instability domains, provided that one point on the stability boundary is known. Then differential equations describing the evolution of eigenvalues and eigenvectors along a curve on the stability boundary surface are derived. These equations also allow us to obtain curves belonging to the stability boundary. Applications to linear gyroscopic systems are considered and studied with examples.

Bifurcations of Eigenvalues of Gyroscopic Systems With Parameters Near Stability Boundaries

This paper deals with stability problems of linear gyroscopic systems Mx¨+Gx˙+Kx=0 with finite or infinite degrees-of-freedom, where the system matrices or operators depend smoothly on several real parameters. Explicit formulas for the behavior of eigenvalues under a change of parameters are obtained. It is shown that the bifurcation (splitting) of double eigenvalues is closely related to the stability, flutter, and divergence boundaries in the parameter space. Normal vectors to these boundaries are derived using only information at a boundary point: eigenvalues, eigenvectors, and generalized eigenvectors, as well as first derivatives of the system matrices (or operators) with respect to parameters. These results provide simple and constructive stability and instability criteria. The presented theory is exemplified by two mechanical problems: a rotating elastic shaft carrying a disk, and an axially moving tensioned beam.

Exact stationary solutions of stochastically excited and dissipated gyroscopic systems

The Hamilton's equations of gyroscopic systems are derived. It is pointed out that gyroscopic forces are derived from kinetic energy and/or potential energy, so they should be regarded as a part of Hamiltonian systems rather than externally generalized forces. A symplectic transformation matrix is proposed to decouple the Hamilton's equations of linear gyroscopic systems. It is shown that the exact stationary solution of a stochastically excited and dissipated gyroscopic system is not affected by the gyroscopic forces when the gyroscopic system as a Hamiltonian system is non-integrable. When the gyroscopic system is integrable, the stationary probability density and the statistics for various generalized coordinates vary with the gyroscopic forces while the mean total energy of the system is unchanged. A stochastically excited and dissipated gyropendulum is studied in detail to demonstrate the inferences.

Delay-induced Instabilities in Gyroscopic Systems

It is shown that stable linear gyroscopic systems of the form Mx''+ Tx'+ Kx= 0 (M > 0) always become unstable when an arbitrarily small delay is introduced in the gyroscopic term. In the case where K is negative definite, then the system will be unstable for all positive delays. On the other hand, examples are given showing that some of these systems may actually become asymptotically stable for larger values of the delay parameter.

ALMOST-SURE STABILITY OF LINEAR GYROSCOPIC SYSTEMS

This paper studies the stability behaviour of a linear gyroscopic system parametrically perturbed by a (multiplicative) real noise of small intensity. To this end, its maximal Lyapunov exponent is calculated using the method of Sri Namachchivaya et al.[1]. The results derived are suitable for cases where the response frequenciesω1 , ω2are non-commensurable and the infinitesimal generator associated with the noise process, ξ (t) has a simple zero eigenvalue. These results are then employed to determine the almost-sure stability boundaries of a rotating shaft subjected to random axial loading.

A stability theorem for gyroscopic systems

This paper deals with the problem, already considered by Lord Kelvin and Tait, of the stability of linear conservative gyroscopic systems. A theorem which provides a necessary and sufficient conditions for a certain class of the systems is given. These conditions are directly in terms of the coefficient matrices. The proof is based on an orthogonal transformation converting the original system into uncoupled two-dimensional subsystems, and applying spectral analysis to the latter. An example is given.