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A linear vibrational system with multiple degrees of freedom subjected to parametric excitation is considered. It is assumed that the system is statically unstable but close to a critical point, the excitation amplitude and damping are small, and the excitation frequency is arbitrary. A new stabilization condition is derived in terms of integrals depending on eigenfrequencies and modes of the undisturbed conservative system and the symmetric excitation matrix. As a special case, an approximation for high-frequency excitation is deduced from this condition. Influence of damping on stabilization region is shown to be very small. Two examples for systems with one and two degrees of freedom are presented. It is shown that stabilization of statically unstable systems is possible for low, medium and high excitation frequencies. [Copyright &y& Elsevier]

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Linear multi-input dynamical systems smoothly depending on parameters are considered. A set of parameter values corresponding to uncontrollable systems (an uncontrollability set) is studied. The typical (generic) structure of the uncontrollability set is described. A constructive method of perturbation analysis of the uncontrollability set is developed. Formulae of first-order approximations for the uncontrollability set and generalized eigenvalues (uncontrollable modes) are derived and used for numerical construction of the uncontrollability set. The method is based on the versal deformation theory for matrix pairs under feedback equivalence. As an example, the uncontrollability set is found for a three-parameter two-degree-of-freedom mechanical system. [ABSTRACT FROM AUTHOR]

Families of matrices smoothly depending on a vector of parameters are considered. Arnold [Russian Math. Surveys, 26 (1971), pp. 29–43] and Galin [Uspekhi Mat. Nauk, 27 (1972),pp. 241–242] have found and listed normal forms of families of complex and real matrices (miniversal deformations), to which any family of matrices can be transformed in the vicinity of a point in the parameter space by a change of basis, smoothly dependent on a vector of parameters, and by a smooth change of parameters. In this paper a constructive method of determining functions describing a change of basis and a change of parameters, transforming an arbitrary family to the miniversal deformation, is suggested. Derivatives of these functions with respect to parameters are determined from a recurrent procedure using derivatives of the functions of lower orders and derivatives of the family of matrices. Then the functions are found as Taylor series. Examples are given. The suggested method allows using efficiently miniversal deformations for investigation of different properties of matrix families. This is shown in the paper where tangent cones (linear approximations) to the stability domain at the singular boundary points are found. [ABSTRACT FROM AUTHOR]

This paper deals with the study of generic singularities of a boundary of the stability domain in a parameter space for systems governed by autonomous linear differential equations y = Ay or x( m) + a1x( m-1) + ··· + amx = 0. It is assumed that elements of the matrix A and coefficients of the differential equation of mth order smoothly depend on one, two, or three real parameters. A constructive approach allowing the geometry of singularities (orientation in space, magnitudes of angles, etc.) to be determined with the use of tangent cones to the stability domain is suggested. The approach allows the geometry of singularities to be described using only first derivatives of the coefficients ai of the differential equation or first derivatives of the elements of the matrix A with respect to problem parameters with its eigenvectors and associated vectors calculated at the singular points of the boundary. Two methods of study of singularities are suggested. It is shown that they are constructive and can be applied to investigate more complicated singularities for multiparameter families of matrices or polynomials. Two physical examples are presented and discussed in detail. [ABSTRACT FROM AUTHOR]