Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams

TLDR: In this article, the critical and post-critical behavior of a non-conservative nonlinear structure undergoing statical and dynamical bifurcations is analyzed, with particular emphasis on the role of damping on the critical scenario.

Abstract: The critical and post-critical behavior of a non-conservative non-linear structure, undergoing statical and dynamical bifurcations, is analyzed. The system consists of a purely flexible planar beam, equipped with a lumped visco-elastic device, loaded by a follower force. A unique integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the parameter space. Particular emphasis is given to the role of the damping on the critical scenario. The occurrence of different mechanisms of instability is highlighted, namely, of divergence, Hopf, double zero, resonant and non-resonant double Hopf, and divergence–Hopf bifurcation. Attention is then focused on the two latter (codimension-two) bifurcations. A multiple scale analysis is carried-out directly on the continuous model, and the relevant non-linear bifurcation equations in the amplitudes of the two interactive modes are derived. The fixed-points of these equations are numerically evaluated as functions of two bifurcation parameters and some equilibrium paths illustrated. Finally, the bifurcation diagrams, illustrating the system behavior around the critical points of the parameter space, are obtained.

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Linear and nonlinear interactions between static and
dynamic bifurcations of damped planar beams
Angelo Di Egidio, Angelo Luongo, Achille Paolone
To cite this version:
Angelo Di Egidio, Angelo Luongo, Achille Paolone. Linear and nonlinear interactions between static
and dynamic bifurcations of damped planar beams. International Journal of Non-Linear Mechanics,
Elsevier, 2007, 42 (1), pp.88. �10.1016/j.ijnonlinmec.2006.12.010�. �hal-00501740�

www.elsevier.com/locate/nlm
Author’s Accepted Manuscript
Linear and nonlinear interactions between static and
dynamic bifurcations of damped planar beams
Angelo Di Egidio,Angelo Luongo,Achille Paolone
PII: S0020-7462(07)00026-1
DOI: doi:10.1016/j.ijnonlinmec.2006.12.010
Reference: NLM 1311
To appear in: International Journal of Non-
Linear Mechanics
Received date: 6 October 2006
Revised date: 27 November 2006
Accepted date: 6 December 2006
Cite this article as: Angelo Di Egidio, Angelo Luongo and Achille Paolone, Linear and non-
linear interactions between static and dynamic bifurcations of damped planar beams, Inter-
national Journal of Non-Linear Mechanics (2007), doi:10.1016/j.ijnonlinmec.2006.12.010
This is a PDF file of an unedited manuscript that has been accepted for publication. As
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Accepted manuscript
Linear and Nonlinear Interactions Between Static and
Dynamic Bifurcations of Damped Planar Beams
Angelo Di Egidio, Angelo Luongo
1
and Achille Paolone
Total number of pages: 40
Total number of tables: 4
Total number of figures: 12
1
Author to whom all correspondence should be addressed
1

Accepted manuscript
Contents
1 Introduction 4
2 Model 5
2.1 The equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The linear direct and adjoint problems . . . . . . . . . . . . . . . . . . . . . . 7
3 Linear Stability Analysis 8
4 Post-Critical Analysis 11
4.1 Bifurcation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 Conclusions 16
A Equations of Motion 17
B Eigenvalue Transversality Condition 20
C Coefficients of the bifurcation equations 20
2

Accepted manuscript
Linear and Nonlinear Interactions Between Static and
Dynamic Bifurcations of Damped Planar Beams
Angelo Di Egidio
†
, Angelo Luongo
†
and Achille Paolone
‡
†
DISAT, University of L’Aquila, Monteluco di Roio, 67040 L’Aquila, Italy
‡
DISG, University of Rome “La Sapienza”, via Eudossiana, 18, 00184 Rome, Italy
Abstract
The critical and post-critical behavior of a nonconservative nonlinear structure, un-
dergoing statical and dynamical bifurcations, is analyzed. The system consists of a purely
flexible planar beam, equipped with a lumped visco-elastic device, loaded by a follower force.
A unique integro-differential equation of motion in the transversal displacement, with relevant
boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear
configuration is built-up in the parameter space. Particular emphasis is given to the role
of the damping on the critical scenario. The occurrence of different mechanisms of insta-
bility is highlighted, namely, of divergence, Hopf, double zero, resonant and non-resonant
double Hopf, and divergence-Hopf bifurcation. Attentions is then focused on the two latter
(codimension-two) bifurcations. A Multiple Scale analysis is carried-out directly on the con-
tinuous model, and the relevant nonlinear bifurcation equations in the amplitudes of the two
interactive modes are derived. The fixed-points of these equations are numerical evaluated
as functions of two bifurcation parameters and some equilibrium paths illustrated. Finally,
the bifurcation diagrams, illustrating the system behavior around the critical points of the
parameter space, are obtained.
Keywords: Stability analysis, damping effects on stability, non-conservative loads,
bifurcation, Multiple Scales Method, direct perturbation approach, divergence and Hopf bi-
furcations, beam continuous model.
3