The Control Of Breach In Numerical Analysis Of Multi-body Systems

The mathematical model of Multi-body system can usually be described as a differential form of kinetic equations,which are a set of differential-algebraic equations(DAES).During the process of solving these differential-algebraic equations,one of the major problems encountered is the breach of contract of the integration process.It can be said that the breach of numerical integration process is an important factor for numerical stability.For the rigid Multi-body system problems,slight breadth and high frequency oscillation exists in the body of the system.As the integral formats of the equation solver in common use are mostly conditional stability.When solving such a highly nonlinear and rigid equation of the multi-body dynamics system,it is always integral with a very unreasonable small step.There will be a bad numerical form or an unstable numerical solution sometimes.In view of this, some scholar have attempted to adopt several absolute and steady numerical methods whose stability are not restricted within the integral step.However,the multi-body system dynamic equations are nonlinear,these absolute and steady numerical methods may not be able bo guarantee the numerical stability.Some of the literature proposed security algorithm,which limit the integral format by controlling the energy balance.The results obtained in this way could ensure numerical solution stability,but the accuracy of the numerical solution is not satisfactory.The current numerical methods for solving initial value problem of differential equations are mostly based on the basic mathematical theory,and are carried out through the given numerical integration formats,which are restricted with principles of mathematics such as the difference equality and taylor expansion.These traditional methods neglect the implicit mechanics meaning of the equations.Explained from the mechanics,the traditional numerical methods assumes the campaign mode in the process of integration for the system,and force the system move in accordance with the campaign mode.These modes added to the system with a set of constraints.If this set of constraints continuously does positive works to the system, adding energy,it will lead to divergence.In fact,this constraint does not exist,so it's called pseudo-bound.As the restriction always reflected by constrained force,the constrained force corresponding to the pseudo-bound is known as the pseudo-bound constrained force.In this paper,the concept of the pseudo-bound constrained force is imported in the mechanical category.With the kinematics recurrence relations between the adjacent objects in multi-body systems,the recursive formula of the pseudo-bound constrained force is given. With the use of Cotes formula and trapezoidal integration formula,also with the Simpson formula and central difference scheme,the assumed movement pattern of the system in a integral step were derived,which is called pseudo-bound.On this basis,and the multi-body dynamic equations is successfully solved by controlling the pseudo-bound constrained force converging to zero at the integral point.