EXACT SOLUTION FOR FINITE DEFORMATION PROBLEMS OF CANTILEVER BEAM WITH VARIABLE SECTION UNDER THE ACTION OF ARBITRARY TRANSVERSE LOADS

This paper deals with finite deformation problems of cantilever beam with variable sec- tion under the action of arbitrary transverse loads.By the use of a method of variable replacement, the nonlinear differential equation with varied coefficient for the problem can be transformed into an equation with variable separable.The exact solution can be obtained by the integration method. Some examples are given in the paper,and the results of these examples show that this exact solution includes the existing solutions in references as special cases.

Some new criteria for lag synchronization of chaotic Lur'e systems by replacing variables control

Some new criteria for the chaotic lag synchronization are proposed. At first,lag synchronization scheme for identical master-slave Lur‘ e systems by replacing variables control and the relevant error system are given, and the relations between absolute stability of the error system and the chaotic lag synchronization are described. Then, based on a quadratic Lyapunov function, two new Lur‘ e criteria for the above chaotic lag synchronization are proved. Four corresponding frequency domain criteria are further derived by means of Meyer-Kalman-Yacubovia Lemma. These frequency domain criteria are applied to analyze the lag synchronization of general master-slave Chua's circuits so that some ranges of the parameters in which the master-slave Chua's circuits achieve chaotic lag synchronization by replacing single-variable control are attained. Finally, some examples are given to verify the theoretical results.

Reduction of Linear Systems of ODEs with Optimal Replacement Variables

In this exploratory study,we present a new method of approximating a large system of ODEs by one with fewer equations,while attempting to preserve the essential dynamics of a reduced set of variables of interest.The method has the following key elements:(i)put a(simple,ad-hoc)probability distribution on the phase space of the ODE;(ii)assert that a small set of replacement variables are to be unknown linear combinations of the not-of-interest variables,and let the variables of the reduced system consist of the variables-of-interest together with the replacement variables;(iii)find the linear combinations that minimize the difference between the dynamics of the original system and the reduced system.We describe this approach in detail for linear systems of ODEs.Numerical techniques and issues for carrying out the required minimization are presented.Examples of systems of linear ODEs and variable-coefficient linear PDEs are used to demonstrate the method.We show that the resulting approximate reduced system of ODEs gives good approximations to the original system.Finally,some directions for further work are outlined.

System Reduction Using an LQR-Inspired Version of Optimal Replacement Variables

Optimal ReplacementVariables(ORV)is amethod for approximating a large system of ODEs by one with fewer equations,while attempting to preserve the essential dynamics of a reduced set of variables of interest.An earlier version of ORV[1]had some issues,including limited accuracy and in some rare cases,instability.Here we present a newversion of ORV,inspired by the linear quadratic regulator problemof control theory,which provides better accuracy,a guarantee of stability and is in some ways easier to use.