Energy Consistent Numerical Integration Method For Planar Greatly Deformed Euler-bernoulli Beam Structure

Numerical integration method is one of the most important methods for solving structure dynamic problems,and stability is one of the major properties of numerical integration method.The stability of conventional numerical integration method like central differential method and Newmark-? method in linear system has been studied thoroughly.But the stability property in linear system is not suitable in nonlinear system,methods like spectral radius method and pole method are impossible to identify the stability of an algorithm in nonlinear system as well.On the other hand,the energy principle always works,so there are energy method,including energy conserving methods and energy consistent methods,adding energy constrain equation to the algorithm to remain unconditionally stable even in nonlinear system.Most of the energy methods are specific to a particular object to simplify the process of formulae derivation.There also exist some energy methods that theoretically adaptable to any circumstances,but its performance needs to be proved by simulations.The Euler-Bernoulli beam is a very regular element in structure analysis,but there is no research about the application of energy method in planar Euler-Bernoulli beam under large deformation so far.So it's necessary to do the research about the application of energy method in planar EulerBernoulli beam.This paper focuses on the energy consistent numerical integration method for planar greatly deformed Euler-Bernoulli beam structure and the contents of the paper is as follow:(1)The establishment of the model of planar Euler-Bernoulli beam under large deformation.By independent interpolation to its horizontal and vertical displacement,taking the influence of larger rotation into consideration and adapting the engineering strain to the description of its axial strain,the model is able to describe the Euler-Bernoulli beam under large deformation.A comparison of the theoretical solution is performed and the result shows that the numerical solution approach to the theoretical solution well and thus the reliability of the model is conformed.(2)Theoretical analysis to the conventional energy method is performed,the result shows that the nonlinear correction method fits the model well as the numerical integration method.The method is perfected theoretically by adjusting the expression of the potential energy,exhibiting the expression of the method in damped system,proving the energy consistent property of the method in energy decaying system and proposing the iteration method for the algebraic equations according to the mathematical property of the method with a iteration graph.A muti-coefficients correction method is proposed according to the development of the nonlinear correction method,which is believed to have a better accuracy.(3)The verification of the stability of the energy method and reliability of the iteration method is performed.The energy method and the average acceleration method are applied to several simulations.The properties of the algorithm are discussed in accuracy,stability and computation cost.The result shows that the energy method has almost the same accuracy with the average acceleration method when having the same time interval,and the stability of the energy method is better than the average acceleration method.In the simulations,the average acceleration method,which is believed to be unconditionally stable in linear system,becomes unstable when the time interval is larger but the energy method stays stable.When having the same time interval,the energy method spend more time than acceleration method but the cost can be decreased by increasing the time interval.And the proposed iteration method works fine in the calculation.