scholarly journals Asymptotic stability and instability criteria for some elastic systems by Liapunov’s direct method

1972 ◽  
Vol 29 (4) ◽  
pp. 535-540 ◽  
Author(s):  
R. H. Plaut
Keyword(s):  
Asymptotic Stability ◽  
Direct Method ◽  
Instability Criteria ◽  
Stability And Instability ◽  
Elastic Systems ◽  
Liapunov's Direct Method

Stability Conditions for a Class of Distributed-Parameter Systems

1966 ◽  
Vol 88 (2) ◽  
pp. 475-479 ◽  
Author(s):  
R. E. Blodgett
Keyword(s):  
Equilibrium State ◽  
Local Stability ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Chemical Reactor ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Stability Conditions ◽  
Stability And Instability ◽  
Liapunov's Direct Method

The purpose of this paper is to obtain stability conditions for a class of nonlinear distributed-parameter systems by using a generalization of Liapunov’s direct method. Sufficient conditions for local stability and instability of the equilibrium state are derived. An application is given in which conditions are obtained for stability of a chemical-reactor process.

Dynamic Stability of Nonlinear Antisymmetrically-Laminated Cross-Ply Rectangular Plates

1989 ◽  
Vol 56 (2) ◽  
pp. 375-381 ◽  
Author(s):  
Andrzej Tylikowski
Keyword(s):  
Asymptotic Stability ◽  
Dynamic Stability ◽  
Material Properties ◽  
Stability Problem ◽  
Large Deflection ◽  
Direct Method ◽  
Rectangular Plates ◽  
Stability Criteria ◽  
The Stability ◽  
Liapunov's Direct Method

The dynamic stability problem is solved for rectangular plates that are laminated antisymmetrically about their middle plane and compressed by time-dependent deterministic or stochastic membrane forces. Moderately large deflection equations taking into account a coupling of in-plane and transverse motions are used. The asymptotic stability and almost-sure asymptotic stability criteria involving a damping coefficient and loading parameters are derived using Liapunov’s direct method. A relation between the stability of nonlinear equations and linearized ones is analyzed. An influence on the number of orthotropic layers, material properties for different classes of parametric excitation on stability domains is shown.

Stability of a Pin-Ended Bar in Torsion and Compression

1973 ◽  
Vol 40 (2) ◽  
pp. 405-410 ◽  
Author(s):  
J. A. Walker
Keyword(s):  
Direct Method ◽  
System Parameters ◽  
Torque Vector ◽  
Instability Criteria ◽  
Stability And Instability ◽  
Liapunov Functionals ◽  
The Stability

The stability of a uniform pin-ended bar having equal principal stiffnesses, subjected to combined torsion and compression, is studied via the direct method of Liapunov for cases in which the torque vector is circulatory; i.e., the torque vector does not bisect the angle between the undeformed and deformed axes of the bar. Liapunov functionals are presented and applied to both the case of a viscous environment and the case of a viscoelastic environment. Several simple stability and instability criteria are obtained in terms of the system parameters.

On the Application of Liapunov’s Direct Method to Linear Lumped-Parameter Elastic Systems

1974 ◽  
Vol 41 (1) ◽  
pp. 278-284 ◽  
Author(s):  
J. A. Walker
Keyword(s):  
Direct Method ◽  
Liapunov Functional ◽  
Lumped Parameter ◽  
Elastic Systems ◽  
Large Velocity ◽  
Sufficient Stability ◽  
Necessary And Sufficient ◽  
Sufficient Stability Conditions ◽  
Size And Structure ◽  
Liapunov's Direct Method

A specialized Liapunov functional is presented for stability analysis of linear lumped-parameter elastic systems involving completely general types of forces not explicitly dependent on time. This functional appears to have the simplest possible form capable of providing necessary and sufficient stability conditions for virtually all such systems. Complete analyses are carried out for systems with and without velocity-dependent forces. Simple approaches are given for investigating systems involving “arbitrarily small” and “arbitrarily large” velocity-dependent forces, as well as for determining whether and how the “size” and “structure” of the velocity-dependent forces affect stability. Applications are made to several examples.

scholarly journals A stability theory for perturbed differential equations

1979 ◽  
Vol 2 (2) ◽  
pp. 283-297
Author(s):  
Sheldon P. Gordon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Direct Method ◽  
Mathematical Technique ◽  
General Differential Equation ◽  
Stability And Instability ◽  
The Difference ◽  
Perturbed Differential Equation ◽  
Eventual Stability ◽  
Liapunov's Direct Method

The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered isX′=f(t,X)and the associated perturbed differential equation isY′=f(t,Y)+g(t,Y).The approach used is to examine the difference between the respective solutionsF(t,t0,x0)andG(t,t0,y0)of these two differential equations. Definitions paralleling the usual concepts of stability, asymptotic stability, eventual stability, exponential stability and instability are introduced for the differenceG(t,t0,y0)−F(t,t0,x0)in the case where the initial valuesy0andx0are sufficiently close. The principal mathematical technique employed is a new modification of Liapunov's Direct Method which is applied to the difference of the two solutions. Each of the various stabillty-type properties considered is then shown to be guaranteed by the existence of a Liapunov-type function with appropriate properties.

scholarly journals Almost sure asymptotic stability analysis of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations

2012 ◽  
Vol 15 ◽  
pp. 71-83 ◽  
Author(s):  
Gregory Berkolaiko ◽  
Evelyn Buckwar ◽  
Cónall Kelly ◽  
Alexandra Rodkina
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Equilibrium Solution ◽  
Test System ◽  
Dimensional System ◽  
Step Size ◽  
Stochastic Perturbations ◽  
Stability And Instability

AbstractWe perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our test equation a two-dimensional system of Itô differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure asymptotic stability of the equilibrium solution. For small values of the constant step-size parameter, we derive close-to-sharp conditions for the almost sure asymptotic stability and instability of the equilibrium solution of the discretisation that match those of the original test system. Our investigation demonstrates the use of a discrete form of the Itô formula in the context of an almost sure linear stability analysis.

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