Browsing by Author "Marsden, Jerrold E."
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Item Dissipation Induced Instabilities(1992) Bloch, Anthony M.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; Ratiu, Tudor S.; ISRThe main goal of this paper is to prove that if the energy- momentum (or energy-Casimir) method predicts formal instability of a relative equilibrium in a Hamiltonian system with symmetry, then with the addition of dissipation, the relative equilibrium becomes spectrally and hence linearly and nonlinearly unstable. The energy-momentum method assumes that one is in the context of a mechanical system with a given symmetry group. Our result assumes that the dissipation chosen does not destroy the conservation law associated with the given symmetry group -- thus, we consider internal dissipation. Our result also includes the special case of systems with no symmetry and ordinary equilibria. Our result is proved by combining the techniques of Chetaev, who proved instability theorems using a special Chetaev- Lyapunov function, those of Hahn, which enable one to strengthen the Chetaev results from Lyapunov instability to spectral instability. Our main achievement is to strengthen these results to the context of the block diagonalization version of the energy momentum method given by Lewis. Marsden, Posbergh, and Simo. However, we also give the eigenvalue movement formulae of Krein, MacKay and others both in general and adapted to the context of the normal form of the linearized equations given by the block diagoanl form as provided by the energy-momentum method. A number of specific examples, such as the rigid body with internal rotors, are provided to illustrate the results.Item The Dynamics of Coupled Planar Rigid Bodies, Part II: Bifurcations, Periodic Solutions, and Chaos.(1988) Oh, Y.G.; Sreenath, N.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; ISRPart I of this paper, namely Sreenath, Oh, Krishnaprasad, and Marsden [1987], hereafter denoted [I], studied the Hamiltonian structure and equilibria for interconnected planar rigid bodies, with the primary focus being on the case of three bodies coupled with hinge joints. The Hamiltonian structure was obtained by the reduction technique, starting with the canonical Hamiltonian structure in material representation and then quotienting by the group of Euclidean motions.Item The Dynamics of Coupled Planar Rigid Bodies.(1986) Sreenath, N.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; ISRThis paper studies the dynamics of coupled planar rigid bodies, concentrating on the case of two bodies coupled with a hinge joint. The Hamiltonian structure is non-canonical and is obtained using the methods of reduction, starting from canonical brackets on the cotangent bundle of the configuration space in material representation. The dynamics on the reduced space occurs on cylinders in R^3; stability of the equilibria is studied using the Energy - Casimir method and is confirmed numerically. The phase space contains a homoclinic orbit which will produce chaotic solutions when the system is perturbed.Item The Dynamics of Two Coupled Rigid Bodies.(1987) Grossman, R.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; ISRIn this paper we derive a Poisson bracket on the phase space so(3)*x so(3)*x S0(3) such that the dynamics of two three- dimensional rigid bodies coupled by a ball and socket joint can be written as a Hamiltonian system.Item The Hamiltonian Structure of Nonlinear Elasticity: The Convective Representation of Solids, Rods, and Plates.(1987) Simo, J.C.; Marsden, Jerrold E.; Krishnaprasad, Perinkulam S.; ISRABSTRACT NOT AVAILABLE.Item Hamiltonian Structures and Stability for Rigid Bodies with Flexible Attachments.(1987) Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; ISRThe dynamics of a rigid body with flexible attachments is studied. A general framework for problems of this type is established in the context of Poisson manifolds and reduction. A simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration. Second, the Energy- Casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out. The Poisson structure and stability results take into account vibrationa of the atring, rotations of the rigid body, their coupling at the point of attachment, and centrifugal and Coriolis forces.Item Stability Analysis of a Rigid Body with a Flexible Attachment Using the Energy-Casimir Method.(1987) Posbergh, T.A.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; ISRWe consider a system consisting of a rigid body to which a linear extensible shear beam is attached. For such a system the Energy- Casimir method can be used to investigate the stability of the equilibria. In the case we consider, it can be shown that a test for (formal) stability reduces to checking the positive definiteness of two matrices which depend on the parameters of the system and the particular equilibrium about which the stability is to be ascertained.Item Stabilization of Rigid Body Dynamics by Internal and External Torques(1990) Bloch, Anthony M.; Krishnaprasad, Perinkulam S.; Marsden, Jerrold E.; Sanchez de Alvarez, G.; ISRIn this paper we discuss the stabilization of the rigid body dynamics by external torques (gas jets) and internal torques (momentum wheels). We compare the stabilizing quadratic quadratic feedback law for a single external torque recently analyzed in Bloch and Marsden [1989b,c] with quadratic feedback torques for internal rotors. We show that with such torques, the equations for the rigid body with momentum wheels are Hamiltonian with respect to a Lie-Poisson bracket structure. Further, these equations are shown to generalize the dual-spin equations analyzed by Krishnaprasad [1985] and Sanchez de Alvarez [1986]. We establish stabilization with a single rotor by using the energy-Casimir method. We also show how to realize the external torque feedback equations using internal torques. Finally, extending some work of Montgomery [1990], we derive a formula for the attitude drift for the rigid body-rotor system when it is perturbed away from stable equilibrium and we indicate how to compensate for this.