Reduction methods play a fundamental role in nonlinear dynamics. They allow the essential dynamics of the original system to be captured using models with a very low number of degrees of freedoms, thus avoidinng the use of brutal numerical modelling. Reduction methods have been thoroughly discussed by Sheindl and Troger [1], who compared linear and nonlinear Galerkin methods, the center manifold and the approximate inertial manifold theories. Nayfeh [2] also developed a modified version of the Galerkin method, able to overcome the shortcomings of the classical procedure. Nayfeh and co-workers [3-4] have also widely applied the Multiple Scale Method in the so called direct form, i.e. by dealing with the partial differential equations and boundary conditions rather than with algebraic systems obtained by means of a priori discretization. Within the framework of bifurcation theory, the authors have systematically applied the Multiple Scale Method to finite dimensional systems to derive the relevant bifurcation equations [5-7]. The method make it possible to avoid both the search for the center manifold and the use of the normal form theory, since the algorithm furnishes bifurcation equations directly in normal form. In this paper an attempt is made to extend the method to infinite-dimensional systems. A key point of the procedure lies in defining a scalar product and using the bilinear identity to get the linear adjoint problem. The solution of the relevant homogeneous problem supplies the tool to enforce the solvability conditions at any order of the perturbation procedure. After reconstitution [8], these equations furnish the desired bifurcation equations. The method is directly applied, without discretization, to a one-dimensional continuous model of an inextensible and shear-undeformable planar beam, equipped with a lumped visco-elastic device and loaded by an axial follower force. The nonlinear equations of motion, expanded up-to cubic terms, are derived via the generalized Hamilton principle. After integration, the longitudinal displacement and the axial reactive force are condensed, and a single integro-differential equation in the transversal displacement component is finally drawn. The spectral properties of the linear operator are then studied. First an adjoint operator is built-up, which differs from the original one for the presence of the non-conservative follower force in the boundary conditions. Then the linear stability of the trivial equilibrium is analyzed. It is found that the beam loses stability for (a) divergence, (b) Hopf and (c) double-zero bifurcations. The stability regions are plotted on the parameter plane, consisting of the load and of a stiffness ratio. Both right and left eigenvectors are evaluated at the critical conditions. At the double-zero bifurcation point the system is defective, i.e. it possess an incomplete set of eigenvectors; this entails the need to find an index-two generalized eigenvector. The Multiple Scale Method is then applied as a reduction method to obtain the bifurcation equations capturing the asymptotic nonlinear dynamics of the infinite-dimensional system around the divergence, the Hopf and the double-zero bifurcations. A set of linear perturbation equation, with singular operator, is obtained, and solved in chain. From the solvability conditions, requiring the known terms are orthogonal to the left eigenvectors of the singular operator, amplitude-equations on the different time-scales are obtained. When these are recombined on the true time-scale, the bifurcation equations are drawn. They are found to be already in normal form. These equations are then numerically integrated to describe the whole scenario around the bifurcation points. References [1] A. Steindl and H. Troger, 2001. Methods for Dimension Reduction and their Application in Nonlinear Dynamics, Int. J. of Solids and Structures, 38, 3131-2147. [2] A. H. Nayfeh, 1998. Reducted-Order Models of Weakly Nonlinear Spatially Continuous Systems, Nonlinear Dynamics, 16, 105-125. [3] A. H. Nayfeh and W. Lacarbonara, 1998. On the Discretization of Spatially Continuous Systems with Quadratic and Cubic Nonlinearities, JSME International Journal, 41, 510-531. [4] G. Rega, W. Lacarbonara, A. H. Nayfeh, and C. Chin, 1999. Multiple Resonances in Suspended Cables: Direct versus Reduced-Order Models, International Journal of Non-Linear Mechanics, 34, 901-924. [5] A. Luongo, A. Paolone and A. Di Egidio, 2000. Sensitivity and Linear Stability Analysis Around a Double Zero Eigenvalues, AIAA Journal, 38/4, 702-710. [6] A. Luongo, A. Di Egidio, and A. Paolone, 2003. Multiple Time Scale Analysis for Bifurcation from a Multiple-Zero Eigenvalue, AIAA Journal, 41/6, 1143-1150. [7] A. Luongo, A. Di Egidio, and A. Paolone, 2002. Multiple Scale Bifurcation Analysis for Finite-Dimensional Autonomous Systems, Recent Research Developments in Sound & Vibration, Transworld Research Network, Kerala, India, 1, 161-201. [8] Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1991.

### Divergence, Hopf and Double-Zero Bifurcations of a nonlinear Planar Beam

#####
*LUONGO, Angelo;DI EGIDIO, ANGELO*

##### 2004-01-01

#### Abstract

Reduction methods play a fundamental role in nonlinear dynamics. They allow the essential dynamics of the original system to be captured using models with a very low number of degrees of freedoms, thus avoidinng the use of brutal numerical modelling. Reduction methods have been thoroughly discussed by Sheindl and Troger [1], who compared linear and nonlinear Galerkin methods, the center manifold and the approximate inertial manifold theories. Nayfeh [2] also developed a modified version of the Galerkin method, able to overcome the shortcomings of the classical procedure. Nayfeh and co-workers [3-4] have also widely applied the Multiple Scale Method in the so called direct form, i.e. by dealing with the partial differential equations and boundary conditions rather than with algebraic systems obtained by means of a priori discretization. Within the framework of bifurcation theory, the authors have systematically applied the Multiple Scale Method to finite dimensional systems to derive the relevant bifurcation equations [5-7]. The method make it possible to avoid both the search for the center manifold and the use of the normal form theory, since the algorithm furnishes bifurcation equations directly in normal form. In this paper an attempt is made to extend the method to infinite-dimensional systems. A key point of the procedure lies in defining a scalar product and using the bilinear identity to get the linear adjoint problem. The solution of the relevant homogeneous problem supplies the tool to enforce the solvability conditions at any order of the perturbation procedure. After reconstitution [8], these equations furnish the desired bifurcation equations. The method is directly applied, without discretization, to a one-dimensional continuous model of an inextensible and shear-undeformable planar beam, equipped with a lumped visco-elastic device and loaded by an axial follower force. The nonlinear equations of motion, expanded up-to cubic terms, are derived via the generalized Hamilton principle. After integration, the longitudinal displacement and the axial reactive force are condensed, and a single integro-differential equation in the transversal displacement component is finally drawn. The spectral properties of the linear operator are then studied. First an adjoint operator is built-up, which differs from the original one for the presence of the non-conservative follower force in the boundary conditions. Then the linear stability of the trivial equilibrium is analyzed. It is found that the beam loses stability for (a) divergence, (b) Hopf and (c) double-zero bifurcations. The stability regions are plotted on the parameter plane, consisting of the load and of a stiffness ratio. Both right and left eigenvectors are evaluated at the critical conditions. At the double-zero bifurcation point the system is defective, i.e. it possess an incomplete set of eigenvectors; this entails the need to find an index-two generalized eigenvector. The Multiple Scale Method is then applied as a reduction method to obtain the bifurcation equations capturing the asymptotic nonlinear dynamics of the infinite-dimensional system around the divergence, the Hopf and the double-zero bifurcations. A set of linear perturbation equation, with singular operator, is obtained, and solved in chain. From the solvability conditions, requiring the known terms are orthogonal to the left eigenvectors of the singular operator, amplitude-equations on the different time-scales are obtained. When these are recombined on the true time-scale, the bifurcation equations are drawn. They are found to be already in normal form. These equations are then numerically integrated to describe the whole scenario around the bifurcation points. References [1] A. Steindl and H. Troger, 2001. Methods for Dimension Reduction and their Application in Nonlinear Dynamics, Int. J. of Solids and Structures, 38, 3131-2147. [2] A. H. Nayfeh, 1998. Reducted-Order Models of Weakly Nonlinear Spatially Continuous Systems, Nonlinear Dynamics, 16, 105-125. [3] A. H. Nayfeh and W. Lacarbonara, 1998. On the Discretization of Spatially Continuous Systems with Quadratic and Cubic Nonlinearities, JSME International Journal, 41, 510-531. [4] G. Rega, W. Lacarbonara, A. H. Nayfeh, and C. Chin, 1999. Multiple Resonances in Suspended Cables: Direct versus Reduced-Order Models, International Journal of Non-Linear Mechanics, 34, 901-924. [5] A. Luongo, A. Paolone and A. Di Egidio, 2000. Sensitivity and Linear Stability Analysis Around a Double Zero Eigenvalues, AIAA Journal, 38/4, 702-710. [6] A. Luongo, A. Di Egidio, and A. Paolone, 2003. Multiple Time Scale Analysis for Bifurcation from a Multiple-Zero Eigenvalue, AIAA Journal, 41/6, 1143-1150. [7] A. Luongo, A. Di Egidio, and A. Paolone, 2002. Multiple Scale Bifurcation Analysis for Finite-Dimensional Autonomous Systems, Recent Research Developments in Sound & Vibration, Transworld Research Network, Kerala, India, 1, 161-201. [8] Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1991.##### Pubblicazioni consigliate

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