Abstract
In this paper is Analyzed the local dynamical behavior of a slewing flexible structure considering nonlinear curvature. The dynamics of the original (nonlinear) governing equations of motion are reduced to the center manifold in the neighborhood of an equilibrium solution with the purpose of locally study the stability of the system. In this critical point, a Hopf bifurcation occurs. In this region, one can find values for the control parameter (structural damping coefficient) where the system is unstable and values where the system stability is assured (periodic motion). This local analysis of the system reduced to the center manifold assures the stable / unstable behavior of the original system around a known solution.
Center manifold; equilibrium solution; Nonideal dynamical systems
Application of the center manifold theory to the study of slewing flexible Nonideal structures with nonlinear curvature: a case study
A. Fenili^{I}; J. M. Balthazar^{II}; D. T. Mook^{III}; H. I. Weber^{IV}
^{I}Instituto Nacional de Pesquisas Espaciais Centro Técnico Aeroespacial (ITA) INPE Av. dos Astronautas, 1758 12227010 São José dos Campos, SP. Brazil. Email: andre_fenili@yahoo.com ^{II}Universidade Estadual Paulista Campus Rio Claro Cx. P. 178 13500230 Rio Claro, SP. Brazil. Email: jmbaltha@rc.unesp.br ^{III}Virginia Polytechnic Institute and State University 200610219  Blacksburg, VA USA. Email: dtmook@vt.edu ^{IV}PUC RJ Departamento de Engenharia Mecânica Rua Marquês de São Vicente, 225, Gávea 22453900 Rio de Janeiro, RJ. Brazil. Email: Hans@ mec.pucrio.br
ABSTRACT
In this paper is Analyzed the local dynamical behavior of a slewing flexible structure considering nonlinear curvature. The dynamics of the original (nonlinear) governing equations of motion are reduced to the center manifold in the neighborhood of an equilibrium solution with the purpose of locally study the stability of the system. In this critical point, a Hopf bifurcation occurs. In this region, one can find values for the control parameter (structural damping coefficient) where the system is unstable and values where the system stability is assured (periodic motion). This local analysis of the system reduced to the center manifold assures the stable / unstable behavior of the original system around a known solution.
Keywords: Center manifold, equilibrium solution, and Nonideal dynamical systems
Introduction
The study of the dynamic behavior of slewing flexible structures, complicated by its nonlinear nature and of continuing interest to researchers and scientists, has in view the study of lightweight and faster structures for space and robotics applications (basically).
The slewing structures were first considered in literature by (Book et al., 1975) that applied a truncated modal model on a two beams with two joints system and included a discussion of a control design with a torque source. Considerable efforts are being performed since then by other authors. Hardware set up to experimentally study slewing control for flexible structure was developed by (Juang and Horta, 1987). This paper was the motivation of the present work.
Low inherent damping, small natural frequencies, and extreme light weights are among common characteristics of these systems which make them vulnerable to any external / internal disturbances such as slewing maneuvers, impacts, etc. Robot arms with the characteristics quoted above are easy to carry out, they need smaller actuators and can reach objectives in a greater workspace since they are thinner and longer than the rigid ones usually used for the same task. DC motors used in these applications are popular actuators not only because they can generate a wide range of torque's and angular velocities, but also because they are quiet, clean and efficient. Nowadays, the research for these kind of mechanical systems is an increasing preoccupation in a competitive world.
Many of the papers found in this area of research are concerning the dynamics or (and) control of the flexible structure itself. Little effort has been focused on actuatorstructure interaction that obviously affects the dynamic properties of the slewing system. This work deals with this kind of interaction. A first announcement of this work was done in (Fenili et.al, 1999).
Finally, it is important to mention that there are have few works in the current literature concerning this mutual interaction in any system. (Kononenko, 1969) was the first one to systematize the study of this kind of systems using asymptotic methods taking into account the energy of the driving source.
(De Mattos et al., 1997) investigated the problem of an unbalanced mass attached to the shaft of a DC motor that was fixed to the end of a cantilever beam. It was supposed that the power supply to the motor was limited. Increasing the unbalanced mass could increase the extension of the associated jump phenomena. It was also observed that in some cases the amplitude and frequency of the motion were modulated and that simple harmonic motion does not exist. Several harmonics were plainly evident in the FFT analysis of the accelerometer output and all attempts of activating nonlinear and parametric resonances were successful. Recently (Balthazar et.al, 1999) presented a complete review of current literature on this kind of problems from 1902 to 1999. Finally we remarked that some experimental results are presented in (Fenili et al, 2001). The authors obtained good agreement between the theory and experiment.
The goal of this present work is the study of the problem of slewing maneuvers considering the existence of a mutual interaction between the energy source and the structure dynamics. This fact turns the problem a Nonideal one (Kononenko, 1969). The present problem is a kind of this new class of problems discussed in (Balthazar et al, 2001).
Here, the dynamics of the nonlinear and Nonideal system is analyzed in the neighborhood of an equilibrium solution (fixed point). To this end, the theory of reduction to the center manifold is utilized and the nonlinear governing equations of motion are simplified (Carr, 1981).
The dynamical system studied here is similar to the one presented in Fig. 1.
Nomenclature
Mathematical Modeling
Let the kinetic and strain energies for the flexible beam be given by (Fenili, 2001):
Introducing both expression in the Extended Hamilton Principle:
One obtain the governing equations of motion for the variables (t) and n(x,t) that after the discretization of the variable n(x,t) (assumed modes method), nondimensionalizing and scaling (for the inclusion of the small parameter ) results :
where:
The governing equations of motion for the slewing flexible structure (nonlinear curvature and Nonideal) in state space form are presented in Eqs. (1). In these equations µ_{1} represents the structural damping.
The parameter represents a small parameter multiplying all the nonlinearities in the system to make them perturbations around a known linear system.
Rewriting Eq.(1) in the form = F(x), expanding it on Taylor series around one of the equilibrium solutions (the fixed point (,,,) = (x_{1},x_{2},x_{3},x_{4}) = (0, 0, 0,0)) and eliminating the equation for the variable x_{1} (decoupled), one can find:
The jacobian matrix for the system (2) is given by:
In the cases where J have eigenvalues equal to zero or eigenvalues with zero real parts, the linear analysis (analysis of the matrix (3)) is not conclusive about the stability of the equilibrium solution. In these cases, some terms of superior order must be included in the analysis and the theory of center manifold reduction is utilized (Carr, 1981).
The conditions for the occurrence of a Hopf bifurcation in this kind of system are studied in this work. In the sequence, the system is reduced to the center manifold in the neighborhood of this point.
The characteristic equation associated to (3) is given by:
where:
The Hurwitz criterion for which the jacobian matrix (3) may have a pair of purely imaginary eigenvalues, condition for the existence of a Hopf bifurcation, is stated as:
In (4), will be developed a study of the values of that satisfy the Hurwitz criterion. Analyzing the roots of the equation in obtained from the relation , one concludes that the set of condition (4) will be satisfied only in the cases where the dimensionless length of the structure assumed values minor than 3.517332965. In these calculations, a_{1} = 1.571401043, a7 = and b_{1} = 1.
Hopf Bifurcation
The following parameters are defined: beam cross section: 0.010 X 0.009; Ng = 20 (gear box reduction); U_{max} = 1.5 V (in the DC motor); L = 3 (beam length); = 7.4999 10^{7} and µ_{1} is the control parameter.
The values of a_{1 }(i = 1,2,3) in (4) will be: a_{1}= 13.440613874; a_{2} = 0.798526437 and a_{3} = 10.732685509. These values satisfy conditions (4) and the critical parameter found (in dimensionless form) and the associated eigenvalues are, respectively: µ_{1c}= 5.009291768 10^{11}; _{1} = 0.893603065 i ; _{2}=  0.893603065 i ; _{3}= 13.440613874.
A simple test is done in order to verify the changing in the system behavior one obtains by crossing the µ_{1C} value as shown in Table 1.
For the results in Table 1, one can verify that for values of structural damping below than 5.009291768 10^{11} the system is unstable while for values above this one the system is stable. It may be stated that the value 5.009291768 10^{11} represents a bifurcation value for µ_{1}.
The order of magnitude of
^{2} ( 10^{13}) is justified by the fact that the dimensional variables of the system are rescaled by orders of 1/. The equations of motion presented here are dimensionless.For one to study the behavior of the system governed by the set of Eqs. (2) around the critical value, one may put µ_{1}= 5.009291768 10^{11} + µ_{2} in these equations. The system of Eqs. (2) is then written as:
The system of equation (5) is integrated by a fourth order RungeKutta.
In the numerical simulations, the value of µ_{2} is changed and the behavior of the states q_{1} (x2; amplitude of deflection of the structure) and its time derivative (x4) are plotted in the phase plane for each case. A general view of this evolution with increasing µ_{2} is showed in the Fig. 2.
The maximum deflections of the structure obtained by the variation of the damping parameter (µ_{2}) are plotted against this parameter in the Fig. 3.
The circles in the phase plane plots of the Figures 4 and 5 indicate the final state of the system in that condition of damping after a specified time is reached.
A Case Study: Center Manifold Reduction
In Eqs. (5), making the coordinate transformation
where the matrix P represents the matrix of the linear system eigenvectors (associated to
_{1} = 0.893603065 i , _{2}=  0.893603065 i and _{3}= 13.440613874 ), one have:
Utilizing the notation and the theory of reduction of the system dynamics to the center manifold in the way it is proposed in (Wiggins, 1990), one have:
The expression for h_{1} will be given by:
The unknown parameters a e b in Eq. (15) may be found by the expression:
Substituting Eq. (7)Eq. (14) in Eq. (16) and equating the terms in equal to zero yields:
And results:
Equating the terms in equal to zero yields:
And results:
Substituting Eq. (17) and Eq. (18) in Eq. (15), one obtain:
The graphic of the center manifold expressed by Eq. (19) is plotted in Fig. 9.
Substituting Eq. (19) in the first and second of Eqs. (6) and expanding the resulting system of equations around µ_{2}=0, one obtains the reduction of the system expressed by Eq. (1) to the center manifold (Eqs. (20)).
The suspension trick (additional equation for the control parameter (Eq. (20b))) assures that the influence of others values of this parameter on the dynamics of the system may be verified (bearing in mind that the center manifold is only valid for a specified value of µ_{2}).
There are stability theorems that proof that if an equilibrium solution of the reduced equations (Eqs. (20)) is stable (unstable), the same equilibrium solution for the complete equations (Eq. (1) ) is also stable ( unstable ) ( Nayfeh,1994).
The integration of Eq. (19) in time was made beneath a fourth order RungeKutta. Three situations were considered (one for each value of µ_{2} around the critical value of µ_{1} (named µ_{1C})), according to the specifications in Fig. 6 Fig. 8. The variables u_{2} and u_{3} and the time in these figures are adimensional and local quantities.
The analysis of these figures yields information about the system stability in the neighborhood of the fixed point (or equilibrium solution) (0,0,0,0).
In the cases where µ_{2} < 0, no equilibrium solution is reached. The same don't occurs in the cases where µ_{2}=0 and in the cases where µ_{2} > 0, according to the Fig. 7 and Fig. 8, respectively.
The initial conditions for the following simulations are: u2 = 0.30/ and u3 = 0.30/ .
Conclusions
The Hopf bifurcation was verified. When the parameter µ_{1} crosses its critical value, the stability of the system represented by the set of Eqs. (2) changes from unstable to stable.
It must be recalled that when one put the terms of perturbation in Eqs. (2) equal to zero ( = 0) , the stability of the linear system associated with Eqs. (2) don't depend on the parameter µ_{1}. In this case, the Hopf bifurcation is impossible to occur.
One can conclude that this kind of bifurcation, in the system under analysis, is associated with the perturbation (nonlinear terms) imposed to the system.
The local analysis in the neighborhood of an equilibrium solution for the complete nonlinear and Nonideal dynamical system studied gives information's about the stability of the system behavior when one varies the structural damping coefficient.
The stability of the system changes of unstable to stable when this control parameter is altered around a critical value in a considered fixed point solution (or equilibrium solution). The expression for the center manifold is not unique. New considerations for Eq. (15) can be proposed and the results compared with the ones obtained here.
Acknowledgements
The authors thank Fundação de Amparo à Pesquisa do Estado de São Paulo, Brazil and Conselho Nacional de Pesquisas, Brazil for the financial supports, in several grants, in order to, develop the scientific researches in Nonideal problems and their applications. The authors also thank Department of Mechanical Deign of State University of Campinas, SP.
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Publication Dates

Publication in this collection
27 Aug 2003 
Date of issue
July 2002