Some aspects of destabilization in reversible dynamical systems with application to follower forces

This paper examines the destabilization of the equilibria of reversible dynamical systems which is induced by the addition of irreversible perturbations. Attention is restricted to reversible dynamical systems which have frequently appeared in the literature on elastic stability. There they are often referred to as follower force problems. The destabilization phenomenon is linear in nature and explicit criteria are established to determine the particular eigenvalue splittings. The post-destabilization dynamics are also examined using the appropriate normal forms for two specific cases, one where the eigenvalues are non-resonant and the other where the eigenvalues are in a strong one-to-one resonance. Finally, the destabilization criteria and certain features of the post-destabilization dynamics are illustrated using two examples of follower force systems.     

Stochastic averaging using elliptic functions to study nonlinear stochastic systems

In this paper, a new scheme of stochastic averaging using elliptic functions is presented that approximates nonlinear dynamical systems with strong cubic nonlinearities in the presence of noise by a set of Itô differential equations. This is an extension of some recent results presented in deterministic dynamical systems. The second order nonlinear differential equation that is examined in this work can be expressed as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qeguuDJXwAKbacfiGaf8hEaGNbamaacqGHRaWkcaWGJbadcaaIXaGc% cqWF4baEcqGHRaWkcaWGJbadcaaIZaGccqWF4baEdaahaaWcbeqaai% aaiodaaaGccqGHRaWkcqaH1oqzcaWGMbGaaiikaiab-Hha4jaacYca% cqWFGaaicuWF4baEgaGaaiaacMcacqGHRaWkcqaH1oqzdaahaaWcbe% qaaiaaigdacaGGVaGaaGOmaaaaruWrL9MCNLwyaGGbcOGaa43zaiaa% cIcacqWF4baEcaGGSaGae8hiaaIaf8hEaGNbaiaacaGGSaGae8hiaa% IaeqOVdGNaaeikaiaadshacaqGPaGaaiykaiabg2da9iaaicdaaaa!645D!\[\ddot x + c1x + c3x^3 + \varepsilon f(x, \dot x) + \varepsilon ^{1/2} g(x, \dot x, \xi {\text{(}}t{\text{)}}) = 0\] where c ₁ and c ₃ are given constants, (t) is stationary stochastic process with zero mean and 1 is a small parameter. This method involves the laborious manipulation of Jacobian elliptic functions such as cn, dn and sn rather than the usual trigonometric functions. The use of a symbolic language such as Mathematica reduces the computational effort and allows us to express the results in a convenient form. The resulting equations are Markov approximations of amplitude and phase involving integrals of elliptic functions. Finally, this method was applied to study some standard second order systems.     

Multipulse Orbits in the Motion of Flexible Spinning Discs

Summary. The global dynamics of flexible spinning discs are studied. The discs studied are parametrically excited in their spin rate, and have imperfections that cause symmetry-breaking. After determining the equations of motion in a suitable form, the energy-phase method is employed to show the existence of chaotic dynamics by identifying multipulse jumping orbits in the perturbed phase space. We provide restrictions on the damping, forcing, and symmetry-breaking parameters in order for these complicated dynamics to occur. The dissipative version of the energy-phase method predicts a wider range of values for which chaotic dynamics occurs than the traditional Melnikov method. The results are then discussed in terms of the physical motion of the spinning disc system. The multipulse orbits are manifested in the physical system as a shifting between two different nodal configurations of the disc. When the motion is chaotic, an observer will see a random jumping between the two nodal configurations of the disc. Received February 7, 2000; accepted November 18, 2001     

Rigorous Stochastic Averaging at a Center with Additive Noise

We consider a random perturbation of a two-dimensional Hamiltonian system with an isolated elliptic fixed point; that is, a center. Under an appropriate change of time, we identify a reduced stochastically-averaged model. We give a rigorous proof of averaging at the center. Our main technique is to use the martingale problem. Our formulation of the result is in a sufficiently abstract setting that it agrees with more complicated averaging results.     

Reduced normal forms for nonlinear control of underactuated hoisting systems

Nearly every cargo that is transported by ship is boxed in standardized containers. The land to ship and ship to land handling of containers is performed by container cranes. These cranes dominate the throughput of the containers in ports. The operators need to be well-trained and skilled because of the requirements of the task to load or unload the ships. During container handling, the crane-load system can be compared to a pendulum of rope length L and deflection f{\phi} . It is referred to as an underactuated system. Whenever the fixed point of the pendulum is displaced, the load starts oscillating. In the nonlinear dynamics literature, one finds the field of resonant coupling to transfer energy from actuated controllable modes to underactuated uncontrollable modes. This work presents a novel normal form approach in order to identify the mode coupling. The proposed method decomposes a nonlinear control system into controllable and uncontrollable subsystem. Only the normal form terms in the controllable subsystem are utilized to design a general controller. Simulation results are given to highlight the load oscillation suppression.     

Particle filtering for chaotic dynamical systems using future right-singular vectors

Nonlinear Dynamics - In this paper, we combine tools from the study of chaotic dynamical systems with nonlinear non-Gaussian data assimilation algorithms to produce novel particle filtering...     

Reversible dynamical systems: Dissipation-induced destabilization and follower forces

This paper uses results from the theory of reversible dynamical systems to examine the nonlinear stability of undamped follower force systems. The well-known dissipation-induced destabilization found in these systems is also examined and it is indicated how the postdestabilization dynamics can be determined using normal forms and an eigenvalue splitting criterion.     

Spindle Speed Variation for the Suppression of Regenerative Chatter

Summary. A phenomenon commonly encountered during machining operations is chatter. It manifests itself as a vibration between workpiece and cutting tool, leading to poor dimensional accuracy and surface finish of the workpiece and to premature failure of the cutting tool. A chatter suppression method that has received attention in recent years is the spindle speed variation method, whereby greater widths of cut are achieved by modulating the spindle speed continuously. By adapting existing mathematical techniques, a perturbative method is developed in this paper to obtain finite-dimensional equations in order to systematically study the mechanism of spindle speed variation for chatter suppression. The results indicate both modest increase of stability and complex nonlinear dynamics close to the new stability boundary. The method developed in this paper can readily be applied to any other system with time-delay characteristics.