On the stability of nonconservative systems with small dissipation (original) (raw)

Effect of small dissipative and gyroscopic forces on the stability of nonconservative systems

Doklady Physics, 2003

We analyze the effect of small forces proportional to the generalized velocity vector on the stability of a linear autonomous mechanical system with nonconservative positional forces. It is known that arbitrarily small dissipation generally destabilizes a nonconservative system . Necessary and sufficient conditions on the matrix of dissipative and gyroscopic forces under which the system is asymptotically stable are obtained. The two-dimensional system is studied in detail. The problem of the stability of the Ziegler-Herrmann-Jong pendulum is considered as a mechanical example.

Analysis of Stability and Bifurcation in Nonlinear Mechanics with Dissipation

Entropy, 2011

The analysis of stability and bifurcation is studied in nonlinear mechanics with dissipative mechanisms: plasticity, damage, fracture. The description is based on introduction of a set of internal variables. This framework allows a systematic description of the material behaviour via two potentials: the free energy and the potential of dissipation. In the framework of standard generalized materials the internal state evolution is governed by a variational inequality which depends on the mechanism of dissipation. This inequality is obtained through energetic considerations in an unified description based upon energy and driving forces associated to the dissipative process. This formulation provides criterion for existence and uniqueness of the system evolution. Examples are presented for plasticity, fracture and for damaged materials.

Stability and Bifurcation in non linear mechanics

HAL (Le Centre pour la Communication Scientifique Directe), 2023

Analysis of stability and bifurcation is studied in non linear mechanics with mechanisms of dissipation : plasticity, damage, fracture. With introduction of a set of internal variables, this framework allows a systematic description of the material behaviour via two potentials : the free energy and the potential of dissipation. For standard generalized materials internal state evolution is governed by a variational inequality depending on the mechanism of dissipation. This inequality is obtained through energetic considerations in an unified description based upon energy and driving forces associated to internal variable evolution. This formulation provides criterion for existence and uniqueness of the system evolution. Examples are presented for plasticity, fracture and for damaged materials

Destabilization of Disc Brake Mechanical System Due to Nonproportional Damping

2020

The paper deals with the study of a possible dissipation induced instability in a disc brake system, which leads to a self-excited vibration and thus to a brake squeal. The work describes an experimental estimation of the damping properties of a disc component of the simplified disc brake system. The stability of the system is analyzed numerically using the Finite Element approach and the Complex eigenvalue analysis. From the evolution of real parts of the eigenvalues with friction, the friction threshold value destabilizing the system is defined for the undamped and non-proportionally damped system. From the results, it can be seen how the damping non-proportionality can lead to the dissipation-induced instability of the disc brake system.

Phenomena of Instability in Non-Conservative Dynamical Systems

САВРЕМЕНА ТЕОРИЈА И ПРАКСА У ГРАДИТЕЉСТВУ, 2018

If the loading is non-conservative, the loss of stability may not manifest itself as the system going into another equilibrium state, but as exhibiting oscillations of increasing amplitude. To take account of this possibility, we must consider the dynamic behavior of the system, because stability is essentially a dynamic concept. In the paper the author’s theory, named the rheological-dynamical analogy (RDA), is used to examine the phenomena of instability in linear internally damped inelastic (LIDI) dynamical systems. Apart from quantitative research, qualitative research is presented to demonstrate the influence of inelasticity and internal friction on dynamic response.

Article Analysis of Stability and Bifurcation in Nonlinear Mechanics with Dissipation

2011

Static Bifurcation in Mechanical Control Systems

Lecture Notes in Control and Information Sciences, 2004

Feedback regulation of nonlinear dynamical systems inevitably leads to issues concerning static bifurcation. Static bifurcation in feedback systems is linked to degeneracies in the system zero dynamics. Accordingly, the obvious remedy is to change the system input-output structure, but there are other possibilities as well. In this paper we summarize the main results connecting bifurcation behavior and zero dynamics and illustrate a variety of ways in which zero structure degeneracy can underly bifurcation behavior. We use several practical examples to illustrate our points and give detailed computational results for an automobile that undergoes loss of directional and cornering stability.

On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints

Mathematical and Computer Modelling, 2010

The first Lyapunov method, extended by V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the equilibrium position of a mechanical system moving in the field of potential and dissipative forces. The motion of the system is subject to the action of the ideal linear nonholonomic nonhomogeneous constraints. Five theorems on the instability of the equilibrium position of the above mentioned system are formulated. The theorem formulated in [V.V. Kozlov, On the asymptotic motions of systems with dissipation, J. Appl. Math. Mech. 58 (5) (1994) 787-792], which refers to the instability of the equilibrium position of the holonomic scleronomic mechanical system in the field of potential and dissipative forces, is generalized to the case of nonholonomic systems with linear nonhomogeneous constraints. In other theorems the algebraic criteria of the Kozlov type are transformed into a group of equations required only to have real solutions. The existence of such solutions enables the fulfillment of all conditions related to the initial algebraic criteria. Lastly, a theorem on instability has also been formulated in the case where the matrix of the dissipative function coefficients is singular in the equilibrium position. The results are illustrated by an example.

Analysis of discontinuous bifurcations in nonsmooth dynamical systems

Regular and Chaotic Dynamics, 2012

Dynamical systems with discontinuous right-hand sides are considered. It is well known that the trajectories of such systems are nonsmooth and the fundamental solution matrix is discontinuous. This implies the presence of the so-called discontinuous bifurcations, resulting in a discontinuous change in the multipliers. A method of stepwise smoothing is proposed allowing the reduction of discontinuous bifurcations to a sequence of typical bifurcations: saddlenode, period doubling and Hopf bifurcations. The results obtained are applied to the analysis of the well-known dry friction oscillator, which serves as a popular model for the description of self-excited frictional oscillations of a braking system. Numerical techniques used in previous investigations of this model did not allow general conclusions to be drawn as to the presence of self-excited oscillations. The new method makes it possible to carry out a complete qualitative investigation of possible types of discontinuous bifurcations in this system and to point out the regions of parameters which correspond to stable periodic regimes.

On the stability of nonconservative systems with small dissipation (original) (raw)

Related papers