Multi-Multiplier Ambient-Space Formulations of Constrained Dynamical Systems, with an Application to Elastodynamics

Various formulations of the equations of motion for both finite- and infinite-dimensional constrained Lagrangian dynamical systems are studied. The different formulations correspond to different ways of enforcing constraints through multiplier fields. All the formulations considered are posed on ambient spaces whose members are unrestricted by the need to satisfy constraint equations, but each formulation is shown to possess an invariant set on which the constraint equations and physical balance laws are satisfied. The stability properties of the invariant set within its ambient space are shown to be different in each case. We use the specific model problem of linearized incompressible elastodynamics to compare properties of three different ambient-space formulations. We establish the well-posedness of one formulation in the particular case of a homogeneous, isotropic body subject to specified tractions on its boundary. Accepted October 11, 2000?Published online April 23, 2001     

M-P inverse method of topological variation of frame structures

In this paper a topological modification method for structural variations of space frame structures is studied by using the Moore-Penrose inverse theory and a new kind of factorization of the stiffness matrix. A set of explicit formulations of variations are obtained. A numerical example is given to illustrate the valid of the present method.     

Nonlocal and gradient rate plasticity

This paper deals with a formulation of nonlocal and gradient plasticity with internal variables. The constitutive model complies with local internal variables which govern kinematic hardening and isotropic softening and with a nonlocal corrective internal variable defined either as the sum between a new internal variable and its spatial weighted average or as the gradient of a measure of plastic strain. The rate constitutive problem is cast in the framework provided by the convex analysis and the potential theory for monotone multivalued operators which provide the suitable tools to perform a theoretical analysis of such nonlocal and gradient problems. The validity of the maximum dissipation theorem is assessed and constitutive variational formulations of the rate model are provided. The structural rate problem for an assigned load rate is then formulated. The related variational formulation in the complete set of state variable is contributed and the methodology to derive variational formulations, with different combinations of the state variables, is explicitly provided. In particular the generalization to the present nonlocal and gradient model of the principles of Prager–Hodge, Greenberg and Capurso–Maier is presented. Finally nonlocal variational formulations provided in the literature are derived as special cases of the proposed model.     

Fractional-order relaxation laws in non-linear viscoelasticity

Viscoelastic constitutive equations are constructed by assuming that the stress is a nonlinear function of the current strain and of a set of internal variables satisfying relaxation equations of fractional order. The dependence of the relaxation equations on the strain can also be nonlinear. The resulting constitutive equations are examined as mapping between appropriate Sobolev spaces. The proposed formulation is easier to implement numerically than history-based formulations.      

A velocity transformation method for the nonlinear dynamic simulation of railroad vehicle systems

The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton–Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.     

Formulating Three-Dimensional Contact Dynamics Problems

ABSTRACT

Complementarity formulations are a promising approach for solving dynamic multi-rigid-body contact problems. Two aspects of simulating contact in a complementarity setting are addressed here. First, an explicit formulation of the differential equations governing contact points for bodies of general surface geometry is developed. These equations may be used to integrate the contact position and to set up the basic dynamics equations. Second, an efficient method for handling frictionless planar contacts of arbitrary boundary shape is presented. Throughout, the problem is set up as explicitly as possible, with special attention being given to the way that the contact geometry is related to the dynamics.     

Plasticity size effects in voided crystals

The shear and equi-biaxial straining responses of periodic voided single crystals are analysed using discrete dislocation plasticity and a continuum strain gradient crystal plasticity theory. In the discrete dislocation formulation, the dislocations are all of edge character and are modelled as line singularities in an elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and annihilation are incorporated through a set of constitutive rules. Over the range of length scales investigated, both the discrete dislocation and strain gradient plasticity formulations predict a negligible size effect under shear loading. By contrast, under equi-biaxial loading both plasticity formulations predict a strong size dependence with the flow strength approximately scaling inversely with the void spacing. Excellent agreement is obtained between predictions of the two formulations for all crystal types and void volume fractions considered when the material length scale in the non-local plasticity model is chosen to be (about 10 times the slip plane spacing in the discrete dislocation models).     

On the Computer Formulations of the Wheel/Rail Contact Problem

In this investigation, four nonlinear dynamic formulations that can be used in the analysis of the wheel/rail contact are presented, compared and their performance is evaluated. Two of these formulations employ nonlinear algebraic kinematic constraint equations to describe the contact between the wheel and the rail (constraint approach), while in the other two formulations the contact force is modeled using a compliant force element (elastic approach). The goal of the four formulations is to provide accurate nonlinear modeling of the contact between the wheel and the rail, which is crucial to the success of any computational algorithm used in the dynamic analysis of railroad vehicle systems. In the formulations based on the elastic approach, the wheel has six degrees of freedom with respect to the rail, and the normal contact forces are defined as function of the penetration using Hertzs contact theory or using assumed stiffness and damping coefficients. The first elastic method is based on a search for the contact locations using discrete nodal points. As previously presented in the literature, this method can lead to impulsive forces due to the abrupt change in the location of the contact point from one time step to the next. This difficulty is avoided in the second elastic approach in which the contact points are determined by solving a set of algebraic equations. In the formulations based on the constraint approach, on the other hand, the case of a non-conformal contact is assumed, and nonlinear kinematic contact constraint equations are used to impose the contact conditions at the position, velocity and acceleration levels. This approach leads to a model, in which the wheel has five degrees of freedom with respect to the rail. In the constraint approach, the wheel penetration and lift are not permitted, and the normal contact forces are calculated using the technique of Lagrange multipliers and the augmented form of the system dynamic equations. Two equivalent constraint formulations that employ two different solution procedures are discussed in this investigation. The first method leads to a larger system of equations by augmenting all the contact constraint equations to the dynamic equations of motion, while in the second method an embedding procedure is used to obtain a reduced system of equations from which the surface parameter accelerations are systematically eliminated. Numerical results are presented in order to examine the performance of various methods discussed in this study.     

导弹适配器受压及弹射时摩擦力研究

针对导弹适配器的结构特征,分别建立了可压缩橡胶泡沫和不可压橡胶圆筒轴对称平面应变问题有限变形的平衡方程,基于Blatz-Ko应变能函数和三次缩减多项式应变能函数,得到了相应的位移和应力模式;在此基础上求解了适配器受压问题的非线性方程组和导弹发射时适配器所受到的摩擦力.算例分析与有限元数值模拟比较表明:解析解与数值解非常吻合,径向应力在发射筒内外表面误差最大为0.558%,周向应力在粘合面误差最大为0.246%,导弹发射时的最大量纲为一的摩擦力为1.0228.适配器径向应力在材料粘合交界面上最小,在适配器外表面最大,均为压应力;橡胶泡沫和不可压橡胶的周向应力均为压应力,橡胶泡沫的周向应力由内向外变大,不可压橡胶的周向应力由内向外变小.橡胶泡沫的径向受压大于周向受压,不可压橡胶的周向受压大于径向受压.研究不同过盈量对应力和摩擦力的影响表明:过盈量每增加0.0013,橡胶泡沫 层和不可压橡胶层的径向应力约增加0.13,不可压橡胶层的周向应力约增加2.14,而摩擦力约增加0.22.过盈量对不可压橡胶层的周向应力和导弹所受到的摩擦力影响非常大,对橡胶泡沫的径向应力有一定的影响,周向应力变化很小.     

Nonlocal damage theory in hybrid-displacement formulations

This paper discusses three hybrid-displacement finite element formulations for the simulation of strain localization based on nonlocal damage theory. An isotropic integral nonlocal damage model is chosen. The hybrid finite element formulations adopted in this work are developed from first principles of Mechanics. The first one defines the domain approximations using the Trefftz functions derived for the linear elastic regime. When damage appears the hybrid-Trefftz displacement formulation degenerates into an hybrid-displacement formulation. The second formulation uses an enriched Trefftz basis with the consideration of local Heaviside functions. The third formulation uses orthonormal Legendre polynomials for the domain approximations. A set of benchmark tests is presented and discussed in order to compare the performance and accuracy of the different models. It is shown that the proposed hybrid-Trefftz formulation allows the reproduction of the general behavior of the structure but does not lead to a correct simulation of the strain tensor evolution. The hybrid-displacement formulation that uses orthonormal Legendre polynomials gives coherent results, so it appears to be a promising field of investigation.     

Variational principles of fluid full-filled elastic solids

IntroductionManypracticalproblemsinengineeringinvolveanalysisoffluidfijll-filledelasticsolids.Energyexplorationand"utilizationaretwoexamples.ThefieldequationsofBlot'sstaticalanddynamicaltheoryoffluidfijll-filledelasticsolidswereestablishedinRefs.[1,Zj.BecausetheitisdifficulttogetexactanswersInumericalmethodsareadopted,especiallythet'initCelementmethod.Theelementmethodbasedonvariationalprinciplesisappliedextensively.GhaboussiandWilsonderivedvariationalprinciplesonthebasisofBlot'sequationsan…     

On the use of linear graph theory in multibody system dynamics

Multibody dynamics involves the generation and solution of the equations of motion for a system of connected material bodies. The subject of this paper is the use of graph-theoretical methods to represent multibody system topologies and to formulate the desired set of motion equations; a discussion of the methods available for solving these differential-algebraic equations is beyond the scope of this work. After a brief introduction to the topic, a review of linear graphs and their associated topological arrays is presented, followed in turn by the use of these matrices in generating various graph-theoretic equations. The appearance of linear graph theory in a number of existing multibody formulations is then discussed, distinguishing between approaches that use absolute (Cartesian) coordinates and those that employ relative (joint) coordinates. These formulations are then contrasted with formal graph-theoretic approaches, in which both the kinematic and dynamic equations are automatically generated from a single linear graph representation of the system. The paper concludes with a summary of results and suggestions for further research on the graph-theoretical modelling of mechanical systems.     

On unilateral contact of two plates aligned at an angle to each other

The contact problem for two elastic plates aligned at a prescribed angle to each other is considered. The set of contact points is assumed to be unknown in advance and to be determined only after the problem is solved. Various formulations of the problem are given, and their equivalence is proved. A complete set of boundary conditions fulfilled on the contact domain is found, and the character of satisfaction of these conditions is described. The asymptotic properties of solutions are studied for rigidity parameters of the contacting plates tending to infinity. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 42–58, July–August, 2008.     

广义特征值问题中重特征值的特征向量导数

本文把重特征值的特征向量导数的计算方法，推广到非亏损矩阵的广义特征值问题，并给出了特征值导数也有重根时特征向量导数的计算式。本方法的优点是只需已知所考虑的重特征值的特征向量，因而计算量小，对于大型复杂结构更为适用。     

广义特征值问题中重特征值的特征向量导数

本文把重特征值的特征向量导数的计算方法，推广到非亏损矩阵的广义特征值问题，并给出了特征值导数也有重根时特征向量导数的计算式。本方法的优点是只需已知所考虑的重特征值的特征向量，因而计算量小，对于大型复杂结构更为适用。     

New spatial curved beam and cylindrical shell elements of gradient-deficient Absolute Nodal Coordinate Formulation

Based on previous studies, a new spatial curved slender-beam finite element and a new cylindrical shell finite element are proposed in the frame of gradient-deficient Absolute Nodal Coordinate Formulation (ANCF). The strain energy of the beam element is derived by using the definition of the Green?CLagrange strain tensor in continuum mechanics so that the assumption on small strain can be relaxed. By using the differential geometry and the continuum mechanics, the angle between two base vectors of a defined local coordinate frame of the cylindrical shell element is introduced into the strain energy formulations. Therefore, the new shell element can be used to model parallelogram shells. The analytical formulations of elastic forces and their Jacobian for the above two finite elements of gradient-deficient ANCF are also derived via the skills of tensor analysis. The generalized-alpha method is used to solve the huge set of system equations. Finally, four case studies including both static and dynamic problems are given to validate the proposed beam and cylindrical shell elements of gradient-deficient ANCF.     

On the Interior Transmission Problem in Nondissipative, Inhomogeneous, Anisotropic Elasticity

In this paper, the interior transmission problem for the non absorbing, anisotropic and inhomogeneous elasticity is investigated. The direct scattering problem for the penetrable inhomogeneous, anisotropic and nondissipative scatterer is first studied and the existence and uniqueness of its solution are established. In the sequel, the interior transmission problem in its classical and weak form is presented and suitable variational formulations of it are settled. Finally, it is proved that the interior transmission eigenvalues constitute a discrete set. This revised version was published online in June 2006 with corrections to the Cover Date.     

Min-max criterion to the optimal design of vibration absorber in a system with Coulomb friction and viscous damping

In this paper, an optimal design of damped dynamic vibration absorber using minimax method for a primary system with Coulomb friction and viscous damping is presented. Meanwhile, to reduce sensitivity of the primary system response to the variations of the forcing frequency, a set of constraint equations that include the distribution characteristic of the resonant frequencies of the primary system is established for the optimal design. Effectiveness of the proposed design is illustrated through numerical simulations. The formulations and results in this paper extend previous results given by Den Hartog and Pennestrì.