Nonlinear formulation for flexible multibody system with large deformation

In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of the geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation.

An exact nonlinear hybrid-coordinate formulation for flexible multibody systems

The previous low-order approximate nonlinear formulations succeeded in capturing the stiffening terms, but failed in simulation of mechanical systems with large deformation due to the neglect of the high-order deformation terms. In this paper, a new hybrid-coordinate formulation is proposed, which is suitable for flexible multibody systems with large deformation. On the basis of exact strain- displacement relation, equations of motion for flexible multi-body system are derived by using virtual work principle. A matrix separation method is put forward to improve the efficiency of the calculation. Agreement of the present results with those obtained by absolute nodal coordinate formulation （ANCF） verifies the correctness of the proposed formulation. Furthermore, the present results are compared with those obtained by use of the linear model and the low-order approximate nonlinear model to show the suitability of the proposed models.

Partition method for impact dynamics of flexible multibody systems based on contact constraint

The impact dynamics of a flexible multibody system is investigated. By using a partition method, the system is divided into two parts, the local impact region and the region away from the impact. The two parts are connected by specific boundary conditions, and the system after partition is equivalent to the original system. According to the rigid-flexible coupling dynamic theory of multibody system, system＇s rigid-flexible coupling dynamic equations without impact are derived. A local impulse method for establishing the initial impact conditions is proposed. It satisfies the compatibility con- ditions for contact constraints and the actual physical situation of the impact process of flexible bodies. Based on the contact constraint method, system＇s impact dynamic equa- tions are derived in a differential-algebraic form. The contact/separation criterion and the algorithm are given. An impact dynamic simulation is given. The results show that system＇s dynamic behaviors including the energy, the deformations, the displacements, and the impact force during the impact process change dramatically. The impact makes great effects on the global dynamics of the system during and after impact.

The bond graph model of planar flexible multibody mechanical systems and its dynamic principle

In order to increase the efficiency and reliability of the dynamic analysis for flexible planar linkage containing the coupling of multi-energy domains, a method based on bond graph is introduced. From the viewpoint of power conservation, the peculiar property of bond graph multiport element MTF is discussed. The procedure of modeling planar flexible muhibody mechanical systems by bond graphs and its dynamic principle are deseribed. To overcome the algebraic difficulty brought by differential causality anti nonlinear junction structure, the constraint forces at joints can be considered as unknown effort sources and added to the corresponding O-junctions of system bond graph model. As a result, the automatic modeling on a computer is realized. The validity of the procedure is illustrated by a practical example.

Numerical and experimental studies on impact dynamics of a planar flexible multibody system

In this paper a computational methodology on impact dynamics of the flexible multibody system is presented. First, the floating frame of reference approach and nodal coordinates on the basis of finite element formulation are used to describe the kinematics of planar deformable bodies. According to the kinematic description of contact conditions, the contact constraint equations of planar flexible bodies are derived. Based on the varying topology technique the impact dynamic equations for a planar multibody system are established. Then the initial conditions of the equations in each contact stage are determined according to the discontinuity theory in continuum mechanics. The experiments between the aluminum rods are performed to check the correctness of the proposed method. Through the comparison between the numerical and experimental results the proposed method is validated. Experimental results also show that the impulse momentum method cannot accurately predict the complex impact dynamic phenomena and the continuous model may lead to a serious error when used to simulate the impact problems with significant wave propagation effects.

THE COUPLING DYNAMICAL MODELING THEORY OF FLEXIBLE MULTIBODY SYSTEM

Based on the deformation theory of elastic beams, the coupling effect between the coupling displacements of a point on the middle line of beam and large overall motion is presented. The 'coupling matrix library' and Jourdain's variation principle and single direction recursive formulation method are used to establish the general coupling dynamical equations of flexible multibody system. Two typical examples show the coupling effect between coupling displacements and large overall motion on the dynamics of flexible multibody system consisting of beams.

Numerical Solution for Stiff Dynamic Equations of Flexible Multibody System

A nonlinear numerical integration method, based on forward and backward Euler integration methods, is proposed for solving the stiff dynamic equations of a flexible multibody system, which are transformed from the second order to the first order by adopting state variables. This method is of A0 stability and infinity stability. The numerical solutions violating the constraint equations are corrected by Blajer＇s modification approach. Simulation results of a slider-crank mechanism by the proposed method are in good agreement with ones from other literature.

FLEXIBLE MULTIBODY SYSTEM DYNAMICS-FINITE SEGMENT METHOD

The principle and method of flexible multibody system dynamics is presented. The dynamic equation have been developed by means of Huston's method based on Kane's equation. In which the flexible members with general cross-section characters were divided into finite segment models under the assumption of small strain. In order to decrease the degrees of freedom of the system and increase the efficiency of numerical calculation. the mode transformation has been introduced. A typical example is presented. and the foregoing method has been perfectly verified.

Multibody dynamic analysis using a rotation-free shell element with corotational frame

Rotation-free shell formulation is a simple and effective method to model a shell with large deformation. Moreover, it can be compatible with the existing theories of finite element method. However, a rotation-free shell is seldom employed in multibody systems. Using a derivative of rigid body motion, an efficient nonlinear shell model is proposed based on the rotation-free shell element and corotational frame. The bending and membrane strains of the shell have been simplified by isolating deformational displacements from the detailed description of rigid body motion. The consistent stiffness matrix can be obtained easily in this form of shell model. To model the multibody system consisting of the presented shells, joint kinematic constraints including translational and rotational constraints are deduced in the context of geometric nonlinear rotation-free element. A simple node-to-surface contact discretization and penalty method are adopted for contacts between shells. A series of analyses for multibody system dynamics are presented to validate the proposed formulation. Furthermore,the deployment of a large scaled solar array is presented to verify the comprehensive performance of the nonlinear shell model.

Multi-variable approach of contact-impact issue in variable topology system

Contact-impact processes occur at most cases in multibody systems. Sub-periods and sub-regional methods are frequently used recently, and different coordinates are introduced in both of the approaches. However, the sub-regional method seems to be more effective. Floating frame of reference formulation is widely used for contact treatment, which describes displacements by the rigid body motion and a small superposed deformation, and the coordinates depicting the deformation include finite element nodal coordinates and modal coordinates, the former deals with the contact/impact region, and the later describes the non-contact region. In this paper, free interface substructure method is used in modeling, and the dynamic equation of a single body is derived. Then, using the Lagrange equation of the first kind, the dynamic equations of multibody systems are established. Furthermore, contact-impact areas are treated through additional constraint equations and Lagrange multipliers. Using such approach, the number of system coordinates and the dimensions of mass matrix are significantly reduced with the modal truncation, therefore both of the efficiency and accuracy are guaranteed. Finite element method in the local contact region can deal with contact/impact between arbitrarily complex interfaces, whereas, additional contact constraints used in the nodal description region can avoid the customized parameters that are used in the continuous force model. C 2013 The Chinese Society of Theoretical and Applied Mechanics.[doi： 10.1063/2.1301307]

A GENERAL PROCEDURE TO CAPTURE THE "DYNAMIC STIFFNESS

A general procedure to capture the 'dynanmic Stiffness' is presented in this paper. The governing equations of motion are formulated for an arbitrary flexible body in large overall motion based on Kane's equations . The linearization is performed peroperly by means of geometrically nonlinear straindisplacement relations and the nonlinear expression of angular velocity so that the 'dynamical stiffness' terms can be captured naturally in a general tcase. The concept and formulations of partial velocity and angular velocity arrays of Huston's method are extended to the flexible body and form the basis of the analysis. The validity and generality of the procedure presented in the paper are verified by numerical results of its application in both the beam and plate models.

MODAL TEST AND ANALYSIS OF CANTILEVER BEAM WITH TIP MASS

The phenomenon of dynamic stiffening is a research field of general interest for flexible multi-body systems.In fact,there are not only dynamic stiffening but also dynamic softening phenomenon in the flexible multi-body systems.In this paper,a non-linear dynamic model and its linearization characteristic equations of a cantilever beam with tip mass in the centrifugal field are established by adopting the general Hamilton Variational Principle.Then,the problems of the dynamic stiffening and the dynamic softening are studied by using numerical simulations.Meanwhile, the modal test is carried out on our centrifuge.The numerical results show that the system stiffness will be strengthened when the centrifugal tension force acts on the beam (i.e.the dynamic stiffening).However,the system stiffness will be weakened when the centrifugal compression force acts on the beam (i.e.the dynamic softening). Furthermore,the equilibrium position of the system will lose its stability when the inertial force reaches a critical value.Through theoretical analysis,we find that this phenomenon comes from the effect of dynamic softening resulting from the centrifugal compression force.Our test results verify the above conclusions and confirm that both dynamic stiffening and softening phenomena exist in flexible multi-body systems.

Development of flexible telescopic boom model using ANCF sliding joint constraints with LuGre friction

In this investigation, a modeling procedure of a telescopic boom of cranes is developed using the absolute nodal coordinate formulation together with the sliding joint constraints. Since telescopic booms are extracted and retracted under various operating conditions, the overall length of the boom changes dynamically, leading to the time-variant vibration characteristics. For modeling the telescopic structure of booms, a special care needs to be exercised since the location of the sliding contact point moves Mong the deformable axis of the flexible boom and the solution to a moving boundary problem is required. This issue indeed makes the modeling of the telescopic boom difficult, despite the significant needs for the analysis. It is, therefore, the objective of this investigation to develop a modeling procedure for the flexible telescopic boom by considering the sliding contact condition with the dynamic frictional effect. To this end, the sliding joint constraint developed for the absolute nodal coordinate formulation is employed for describing relative sliding motion between flexible booms, while flexible booms are modeled using the beam element of the absolute nodal coordinate formulation, which allows for modeling the large rotation and deformation of the structure.

COMPUTER AIDED ANALYSIS OF FLEXI－BLE MEMBER IN CONSIDERATION OF THE EFFECTS OF DYNAMIC STI－FFEN－ING

A recently developed procedure to capture the dynamic stiffening of an arbitrary flexiblemember in large overall motion accompanied by small elastic vibrations is presented. A mechanicalsystem that consists of one or more flexible members is called a flexible mechanical system. If thesystem is considered as a multibody system, the flexiblemember can be considered as a flexible bodyin a flexible multibody system. Having retained the nonlinearitites up to an appropriate point in theanalysis, the linearization is then performed properiy so that the dynamic stiffening terms can befound naturally, while the explicit formulation of the governing equations for the deformation mo-tion is ultimately linear. Based on the procedure, the effects of dynamic stiffening are investigatedqualitatively and quantitatively with analytical and numerical examples. The results are useful incomputer aid analysis of the dynamic behavior of flexible mechanical systems.

Deployment dynamics and topology optimization of a spinning inflatable structure

Inflatable space structures may undergo the vibration of a long duration because of their features of dynamic deployment,high flexibility,and low-frequency modes.In this paper,a topology optimization methodology is proposed to reduce the vibration of a spinning inflatable structure.As the first step,a variable-length shell element is developed in the framework of arbitrary Lagrange-Euler(ALE)and absolute nodal coordinate formulation(ANCF)to accurately model the deployment dynamics of the inflatable structure.With the help of two additional material coordinates,the shell element of ALE-ANCF has the ability to describe the large deformation,large overall motion,and variable length of an inflatable structure.The nonlinear elastic forces and additional inertial forces induced by the variable length are analytically derived.In the second step,a topology optimization procedure is presented for the dynamic response of an inflatable structure through the integration of the equivalent static loads(ESL)method and the density method.The ESL sets of the variable-length inflatable structure are defined to simplify the dynamic topology optimization into a static one,while the density-based topology optimization method is used to describe the topology of the inflatable structure made of two materials and solve the static optimization problem.In order to obtain more robust optimization results,sensitivity analysis,density filter,and projection techniques are also utilized.Afterwards,a benchmark example is presented to validate the ALE-ANCF modeling scheme.The deployment dynamics and corresponding topology optimization of a spinning inflatable structure are studied to show the effectiveness of the proposed topology optimization methodology.