A Variational Approach to Dynamics of Flexible Multibody Systems

Abstract

This paper presents a variational formulation of constrained dynamics of flexible multibody systems, using a vector-variational calculus approach. Body reference frames are used to define global position and orientation of individual bodies in the system, located and oriented by position of its origin and Euler parameters, respectively. Small strain linear elastic deformation of individual components, relative to their body reference frames, is defined by linear combinations of deformation modes that are induced by constraint reaction forces and normal modes of vibration. A library of kinematic couplings between flexible and/or rigid bodies is defined and analyzed. Variational equations of motion for multibody systems are obtained and reduced to mixed differential-algebraic equations of motion. A space structure that must deform during deployment is analyzed, to illustrate use of the methods developed     

Equilibrium of Phases with Interfacial Energy: A Variational Approach

The paper proves the existence of equilibrium two phase states with elastic solid bulk phases and deformation dependent interfacial energy. The states are pairs (y,E) consisting of the deformation y on the body and the region E occupied by one of the phases in the reference configuration. The bulk energies of the two phases are polyconvex functions representing two wells of the substance. The interfacial energy is interface polyconvex. The last notion is introduced and discussed below, together with the interface quasiconvexity and interface null Lagrangians. The constitutive theory and equilibrium theory of the interface are discussed in detail under appropriate smoothness hypotheses. Various forms of the interfacial stress relations for the standard and configurational (Eshelby) interfacial stresses are established. The equilibrium equations are derived by a variational argument emphasizing the roles of outer and inner variations.     

Parabolic Systems with Nowhere Smooth Solutions

We construct smooth 2×2 parabolic systems with smooth initial data and C^α right-hand side which admit solutions that are nowhere C¹. The elliptic part is in variational form and the corresponding energy ϕ is strongly quasiconvex and in particular satisfies a uniform Legendre-Hadamard (or strong ellipticity) condition.     

A Variational Approach to Particles in Lipid Membranes

A variety of models for the membrane-mediated interaction of particles in lipid membranes, mostly well-established in theoretical physics, is reviewed from a mathematical perspective. We provide mathematically consistent formulations in a variational framework, relate apparently different modelling approaches in terms of successive approximation, and investigate existence and uniqueness. Numerical computations illustrate that the new variational formulations are directly accessible to effective numerical methods.     

Identification of Mechanical Properties by Displacement Field Measurement: A Variational Approach

We study a parameter identification problem associated with a two-dimensional mechanical problem. In the first part, the experimental technique of determining the displacement field is briefly presented. The variational method proposed herein is based on the minimization of either a separately convex functional or a convex functional that leads to the reconstruction of the elastic tensor and the stress field. These two reconstructed fields are continuous and piecewise linear on a triangulation of the two-dimensional domain. Some numerical and experimental examples are presented to test the performance of the algorithms.     

第二类抛物型变分不等式中的MRM方法

本文讨论了第二类抛物型变分不等式中的MRM(多重互易方法)方法。首先采用时间项半离散和隐格式方法将抛物型变分不等式化解为一个椭圆变分不等式，然后利用MRM-边界积分方程，将其化解为MRM-边界混合变分不等式，并给出了MRM-边界混合变分不等式解的存在唯一性。说明了该MRM-边界混合变分不等式与常规边界积分方程得到的边界混合变分不等式是一致的，并且具有更容易编程实现。这为使用MRM边界元方法数值求解抛物型变分不等式提供了方法和理论依据。文末给出了数值算例。     

Entire Solutions to Equivariant Elliptic Systems with Variational Structure

We consider the system Δu ? W _u (u) = 0, where \({u : \mathbb{R}^n \to \mathbb{R}^n}\) , for a class of potentials \({W : \mathbb{R}^n \to \mathbb{R}}\) that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial G-equivariant entire solutions connecting the global minima of W along certain directions at infinity.     

A Variational Approach to Obstacle Problems for Shearable Nonlinearly Elastic Rods

We use variational methods to study obstacle problems for geometrically exact (Cosserat) theories for the planar deformation of nonlinearly elastic rods. These rods can suffer flexure, extension, and shear. There is a marked difference between the behavior of a shearable and an unshearable rod. The set of admissible deformations is not convex, because of the exact geometry used. We first investigate the fundamental question of describing contact forces, which we necessarily treat as vector‐valued Borel measures. Moreover, we introduce techniques for describing point obstacles. Then we prove existence for a very large class of problems. Finally, using nonsmooth analysis for handling the obstacle, we show that the Euler‐Lagrange equations are satisfied almost everywhere. These equations provide very detailed structural information about the contact forces. Accepted June 3, 1996     

A Variational Approach to the Macroscopic Electrodynamics of Anisotropic Hard Superconductors

We consider Bean’s critical state model for anisotropic superconductors. A variational problem solved by the quasi-static evolution of the internal magnetic field is obtained as the Γ-limit of functionals arising from Maxwell’s equations combined with a power law for the dissipation. Moreover, the quasi-static approximation of the internal electric field is recovered, using a first order necessary condition. If the sample is a long cylinder subjected to an axial uniform external field, the macroscopic electrodynamics is explicitly determined.     

A Variational Approach for the 2-Dimensional Semi-Geostrophic Shallow Water Equations

Existence of weak solutions to the 3-D semi-geostrophic equations with rigid boundaries was proved by Benamou and Brenier [3], using Monge transport theory. This paper extends the results to a free surface boundary condition, which is more physically appropriate. This extension is at present for the 2-D shallow water case only. In addition, we establish stronger time regularity than was possible in [3]. Accepted October 9, 2000?Published online February 14, 2001     

A Variational Approach to the Isoperimetric Inequality for the Robin Eigenvalue Problem

The isoperimetric inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions was recently proved by Daners in the context of Lipschitz sets. This paper introduces a new approach to the isoperimetric inequality, based on the theory of special functions of bounded variation (SBV). We extend the notion of the first eigenvalue λ₁ for general domains with finite volume (possibly unbounded and with irregular boundary), and we prove that the balls are the unique minimizers of λ₁ among domains with prescribed volume.     

Complexity Versus Complex Systems: A New Approach to Scientific Discovery

Extraction of quantitative features from observations via suitable measuring devices M means that the words of science are coded as numbers, and the syntaxis is a set of mathematical rules. Once general premises are available all consequences can be worked in a purely deductive way. This characteristic of science displays two orders of drawbacks, namely, undecidability of deductive procedures, and intractability of computer modelings of complex situations. The way out of such a crisis consists in an adaptive strategy, that is, in a frequent readjustment of M suggested by the observed events. As a consequence, M provides different data streams (words) for the same observed events, as it is tuned to different resolutions. The adaptive strategy here introduced should by no means be confused with the adaptivity of a learning machine, which—inputted by a data stream—readjusts itself over a class of theoretical explanations in order to select the optimal one, thus providing knowledge conditional on the assigned input. On the contrary, physics aims at extracting regular patterns out of things, by a trial and error procedure which includes not only modifications of the explanations for fixed data sets, but also exploring different data sets via modified M's. This M-adjustment is a pre-linguistic endeavour, not expressible by a formal language. Such an essential characteristic of the physical program means that physics can not be performed by a machine.     

Variational Principles for Entropies of Nonautonomous Dynamical Systems

In this paper, we first introduce the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we show that there is certain variational relation for nonautonomous dynamical systems.     

A Dynamic Systems Approach to Understanding Reaching Movements with a Prosthetic Arm

A case study was conducted on an experienced upper extremity prosthetic user that required him to perform a reaching and grasping task with both his prosthetic and normal anatomical hand. We used a scanning task (Wallace, Stevenson, Spear, & Weeks, 1994; Button, Bennett, & Davids, 1998) that required the participant to perform a wide range of coordination patterns defined by the relative phasing between the aperture of the fingers (or artificial prehensor) and the arm. Visual templates of the required finger trajectories in the various required phase conditions served as environmental information for the subject to follow. Based on previous work, we hypothesized that the participant would exhibit at least one stable reaching and grasping pattern in both his anatomical and prosthetic arm. In support of this hypothesis, the results showed a negative sloping relationship between the required relative phase and the mean delta relative phase (required relative phase minus the actual relative phase). The smallest delta relative phase occurred at approximately 80° and 115° relative phase for the anatomical and prosthetic arm, respectively during the scanning task. These results confirm our previous work of the presence of only one attractor in reaching and grasping movements using either the anatomical or prosthetic arm.     

A Causality Approach to Particle Dynamics for Systems

Causality in physics is an old idea which emphasizes the notion that there are ‘causes’ and ‘responses’ to those causes; this relationship is expressed through a fundamental equation which is supposed to describe the evolution of a dynamical system of particles. In a most elementary example, one may identify the power input to a system as the ‘cause’ and the rate of change of the internal plus kinetic energy of the system as representing the ‘response’. In this case, the balance of one against the other is a statement of balance of energy for the dynamical system; such balance is often postulated as fundamental in mechanics. It is the purpose of this paper to investigate a more fundamental causality based approach for describing the dynamics of a system of particles. Forces intrinsic and extrinsic to the system are distinguished and an evolutionary causality law, which is form-invariant under a change of frame, is proposed. We assume, as is common, that intrinsic forces are objective under a general change of frame. Extrinsic forces are assumed to be objective only under a Galilean (that is, inertial) change of frame. This limited objectivity will play a major role in reducing the fundamental proposed ‘law of causality’ to what is recognized as the classical balance of energy for the system. The concept of mass is not introduced as primitive in this work; the existence of ‘inertial constants’, each of which associates with a specific particle of the system, is a result of the theory. Double and triple binding forces between pairs and triples of particles are admitted as fundamental to the intrinsic force structure of this theory. The balance of linear momentum for the system is derived and a generalized intrinsic force action-reaction principle is obtained. The emergence from the causality theory of pair- and triple-particle potential functions is discussed and these potential functions are shown to be intimately related to the internal binding force structure. Moreover, in Theorem 6.1 the underlying invariance structure of the theory shows that the double and triple binding forces must be separately ‘moment-balanced’ and that this gives rise to the balance of moment-of-momentum for the system. Finally, a constitutive theory for the binding forces is posited and it is shown that the double binding forces are determined by the pair-particle potential function and that the triple binding forces are determined by the triple-particle potential function.     

The Interaction between a Uniformly Loaded Circular Plate and an Isotropic Elastic Halfspace: A Variational Approach

ABSTRACT

The axially symmetric flexural interaction of a uniformly loaded circular plate resting in smooth contact with an isotropic elastic halfspace is examined by using an energy method. In this development the deflected shape of the plate is represented in the form of a power series expansion which satisfies the kinematic constraints of the plate deformation. The flexural behavior of the plate is described by the classical Poisson-Kirchhoff thin plate theory. Using the energy formulation, analytical solutions are obtained for the maximum deflection, the relative deflection, and the maximum flexural moment in the circular plate. The results derived from the energy method are compared with equivalent results derived from numerical techniques. The solution based on the energy method yields accurate results for a wide range of relative rigidities of practical interest.     

Persistence in Forward Nonautonomous Competitive Systems of Parabolic Equations

In the present paper forward nonautonomous competitive systems of two parabolic second order partial differential equations are studied. The concept of forward uniform persistence for such systems is introduced. Sufficient conditions, expressed in terms of the principal spectrum, are given for those systems to be forward uniformly persistent.     

A Variational Approach to the Fredholm Alternative for the p-Laplacian near the First Eigenvalue

We are concerned with the existence of a weak solution to the degenerate quasi-linear Dirichlet boundary value problem