1,721,306 research outputs found
A transfer matrix perturbation approach to the buckling analysis of nonlinear periodic structures
No full textCenter Manifold & Normal Forms vs Multiple Scale: two alternative approaches to bifurcation analysis of dynamical systems
No full textFree-vibrations and sensitivity analysis of a defective 2-degree-of-freedom system
No full textFree oscillations of a two degree-of-freedom system with nonproportional damping are analyzed. By a suitable choice of parameters, a family of defective systems having a noncomplete set of eigenvectors is selected. Free motions of underdamped and overdamped defective systems are studied in the four-dimensional state space, and their main characteristics are discussed. In particular, the rate at which the trajectories are attracted by the eigenvectors is determined. Small perturbations of order epsilon of the parameters are then considered, and asymptotic expressions for the modified system eigensolutions are obtained. These allow qualitative discussion of the effects of modifications on the mechanical behavior of nearly defective systems. Marked sensitivities of order epsilon(1/2) or epsilon(1/4) are found. These depend strongly on the damping magnitude. An extensive numerical analysis is performed
Perturbation methods for nonlinear autonomous discrete-time dynamical systems
No full textTwo perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincare and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincare method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions
Eigensolutions of perturbed nearly defective matrices
No full textPerturbation methods for approximating eigenvalues and eigenvectors of perturbed nearly defective (mistuned) systems have been discussed. It is shown that the mistuning causes Taylor expansions to be not uniformly valid even in small intervals of the perturbation parameter, subsequently rendering them useless for practical purposes. The problem is overcome by starting the perturbation expansion from an exactly defective (tuned) system ''close'' to the mistuned one. Asymptotic expansions are then obtained in terms of fractional powers of two parameters: the modification and the mistuning parameter. However, as the tuned system is unknown, an inverse problem has to be solved in order to determine it. An algorithm valid for the particular but frequent case of several couples of nearly coincident eigenvalues has been detailed for this problem; an outline of a more general case is also given. Illustrative examples are presented. (C) 1995 Academic Press Limite
On the use of the multiple scale method in solving 'difficult' bifurcation problems
No full textSeveral algorithms consisting in 'non-standard' versions of the Multiple Scale Method are illustrated for 'difficult' bifurcation problems. Preliminary, the 'easy' case of bifurcation from a cluster of distinct eigenvalues is addressed, which requires using integer power expansions, and it leads to bifurcation equations all of the same order. Then, more complex problems are studied. The first class concerns bifurcation from a defective eigenvalue, which calls for using fractional power expansions and fractional time-scales, as well as Jordan or Keldysh chains. The second class regards the interaction between defective and non-defective eigenvalues. This problem also requires fractional powers, but it leads to differential equations which are of a different order for the involved amplitudes. Both autonomous and parametrically excited non-autonomous systems are studied. Moreover, the transition from a codimension-3 to a codimension-2 bifurcation is explained. As a third class of problems, singular systems possessing an evanescent mass, as Nonlinear Energy Sinks, are considered, and both autonomous systems undergoing Hopf bifurcation and non-autonomous systems under external resonant excitation, are studied. The algorithm calls for a suitable combination of the Multiple Scale Method and the Harmonic Balance Method, the latter is applied exclusively to the singular equations. Several applications are shown, to test the effectiveness of the proposed methods. They include discrete and continuous systems, autonomous, parametrically and externally excited systems
Mode localization in dynamics and buckling of linear imperfect continuous structures
No full textLocalization phenomena in one-dimensional imperfect continuous structures are analyzed, both in dynamics and buckling. By using simple models, fundamental concepts about localization are introduced and similarities between dynamics and buckling localization are highlighted. In particular, it is shown that strong localization of the normal modes is due to turning points in which purely imaginary characteristic exponents assume a non zero real part; in contrast, if turning points do not occur, only weak localization can exist. The possibility of a disturbance propagating along the structure is also discussed. A perturbation method is then illustrated, which generalizes the classical WKB method; this allows the differential problem to be transformed into a sequence of algebraic problems in which the spatial variable appears as a parameter. Applications of the method are worked out for beams and strings on elastic soil. All these structures are found to have nearly-defective system matrices, so their characteristic exponents are highly sensitive to imperfections
Eigensolutions sensitivity for nonsymmetric matrices with repeated eigenvalues
No full textA perturbation algorithm is developed for evaluating eigenvalue and eigenvector directional derivatives of nonsymmetric defective matrices. Some properties of these matrices are recalled; in particular, chains of generalized right and left eigenvectors and their orthogonal properties are defined. Small perturbations of the matrix are then considered. An asymptotic expansion of the eigensolutions of the perturbed problem is obtained in terms of noninteger powers of the perturbation parameter. Marked sensitivity of the eigensolutions is highlighted. Particular attention is devoted to the eigenvectors of the perturbed system and to the strong coupling that occurs between the chains. An example is developed to illustrate the algorithm and compare perturbative and numerical solutions
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