The Parabolic-Hyperbolic System Governing the Spatial Motion of Nonlinearly Viscoelastic Rods

This paper treats initial-boundary-value problems governing the motion in space of nonlinearly viscoelastic rods of strain-rate type. It introduces and exploits a set of physically natural constitutive hypotheses to prove that solutions exist for all time and depend continuously on the data. The equations are those for a very general properly invariant theory of rods that can suffer flexure, torsion, extension, and shear. In this theory, the contact forces and couples depend on strains measuring these effects and on the time derivatives of these strains.The governing equations form an eighteenth-order quasilinear parabolic- hyperbolic system of partial differential equations in one space variable (the system consisting of two vectorial equations in Euclidean 3-space corresponding to the linear and angular momentum principles, each equation involving third-order derivatives). The existence theory for this system or even for its restrictced version governing planar motions has never been studied. Our work represents a major generalization of the treatment of purely longitudinal motions of [12], governed by a scalar quasilinear third-order parabolic-hyperbolic equation. The paper [12] in turn generalizes an extensive body of work, which it cites. Our system has a strong mechanism of internal friction embodied in the requirement that the consitutive function taking the strain rates to the contact forces and couples be uniformly monotone. As in [12], our system is singular in the sense that certain constitutive functions appearing in the principal part of the differential operator blow up as the strain variables approach a surface corresponding to a total compression. We devote special attention to those inherent technical difficulties that follow from the underlying geometrical significance of the governing equations, from the requirement that the material properties be invariant under rigid motions, and from the consequent dependence on space and time of the natural vectorial basis for all geometrical and mechanical vector-valued functions. (None of these difficulties arises in [12].) In particular, for our model, the variables defining a configuration lie on a manifold, rather than merely in a vector space. These kinematical difficulties and the singular nature of the equations prevent our analysis from being a routine application of available techniques.The foundation of our paper is the introduction of reasonable consitutive hypotheses that produce an a priori pointwise bound preventing a total compression and a priori pointwise bounds on the strains and strain rates. These bounds on the arguments of our consitutive functions allow us to use recent results on the extension of monotone operators to replace the original singular problem with an equivalent regular problem. This we analyze by using a modification of the Faedo-Galerkin method, suitably adapted to the peculiarities of our parabolic-hyperbolic system, which stem from the underlying mechanics. Our consitutive hypotheses support bounds and consequent compactness properties for the Galerkin approximations so strong that these approximations are shown to converge to the solution of the initial-boundary-value problem without appeal to the theory of monotone operators to handle the weak convergence of composite functions.Acknowledgement The work of Antman reported here was supported in part by grants from the NSF and the ARO.     

Invariant Measures for Dissipative Systems and Generalised Banach Limits

Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier–Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521–540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on metric spaces that have compact attracting sets, taking limits evaluated along individual trajectories. We also show that if the space is a reflexive separable Banach space, or if the dynamical system has a compact absorbing set, then rather than taking limits evaluated along individual trajectories, we can take an ensemble of initial conditions: the generalised Banach limit can be used to construct an invariant measure based on an arbitrary initial probability measure, and any invariant measure can be obtained in this way. We thus propose an alternative to the classical Krylov–Bogoliubov construction, which we show is also applicable in this situation.     

Spatial Analyticity of the Solutions to the Subcritical Dissipative Quasi-geostrophic Equations

We employ a new bilinear estimate to show that solutions to the subcritical dissipative quasi-geostrophic equations with initial data in the scaling-invariant Lebesgue space are analytic in space variables. Some decay in time estimates for space–time derivatives are also obtained.     

Invariant Solutions of the Thermal-Diffusion Equations for a Binary Mixture in the Case of Plane Motion

The group properties of the thermal-diffusion equations for a binary mixture in plane flow are studied. Optimal systems of first-and second-order subalgebras are constructed for the admissible Lie operator algebra, which is infinite-dimensional. Examples of the exact invariant solutions are given, which are found by solving ordinary differential equations. Exact solutions are found that describe thermal diffusion in an inclined layer with a free boundary and in a vertical layer in the presence of longitudinal temperature and concentration gradients. The effect of the thermal-diffusion parameter on the flow regime is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 95–108, January–February, 2006.     

Spatial Independence of Invariant Sets of Parabolic Systems

In this paper we study the -limit sets of semiflows generated by systems of autonomous parabolic systems under Neumann boundary conditions. Under weaker assumptions than previous works, we show the -limit sets consist of space independent solutions. We also prove similar results for shadow systems and some systems with periodic time dependence and calculate the Conley indices of invariant sets.     

Motion of Planar Link Mechanisms Revisited

The problem of finding the positions of a four-bar linkage at which the coupler link and rocker have extreme angular velocities is solved. A method is proposed for kinematic analysis of class III mechanisms. The method is based on joining a fictitious link to the original mechanism. The kinematic analysis of a class III mechanism is reduced to successive kinematic analyses of two four-bar linkages     

Dynamics of Geometrically Nonlinear Rods: I. Mechanical Models and Equations of Motion

Slender thread like bodies (like cables, ropes, textilethreads or belts) are often used in technical applications. Becauseof their dimensions the one-dimensional continuum is the appropriatemechanical model for bodies of this type. Making use of the basicrelations of three-dimensional continua as a starting point the paperdevelops the general kinematic and kinetic relations of one-dimensionalcontinua for the case that the cross-sections will remain plane (Bernoullihypothesis), that large deflections are possible but the strains remainsmall and that the material is homogeneous and isotropic and behaveslinearly elastic. This results in the equations of motion of shearableand extensible rods (Timoshenko-beams). By neglection of shear deformationand of the rotational inertia of the cross-sections (assumptions thatcan be done in most technical applications) the equations of motionof Euler–Bernoulli-beams are derived in standard and concentratedform. The Euler–Bernoulli-beam equations contain the equations ofmotion of threads with zero bending and torsional stiffness. It isshown that the neglection of bending and torsional stiffness is onlyvalid if the tension is always positive. The second part of this paper[1] selects and develops appropriate numerical solution methods.The derived algorithms are used to solve problems from space and marineengineering.     

A General Method for Estimating Dynamic Parameters of Spatial Mechanisms

Dynamic equations of motion require a large number of parameters for each element of the system. These can include for each part their mass, location of center of mass, moment of inertia, spring stiffnesses and damping coefficients. This paper presents a technique for estimating these parameters in spatial mechanisms using any joint type, based on measurements of displacements, velocities and accelerations and of external forces and torques, for the purpose of building accurate multibody models of mechanical systems. A form of the equations of spatial motion is derived, which is linear in the dynamic parameters and based on multibody simulation code methodologies. Singular value decomposition is used to find the essential parameter set, and minimum parameter set. It is shown that a simulation of a four-bar mechanism (with spherical, universal, and revolute joints) and based on the estimated parameters gives accurate response.     

Invariant and Partially Invariant Solutions of the Green-Naghdi Equations

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 26–35, November–December, 2005.     

Spatial Motion of a Rod in a Viscous Fluid Flow

Equations of spatial motion of a curved finitelength rod in a viscous fluid flow are derived. Analytical solutions of problems on the motion of a straight rod under conditions of pure shear, simple shear, and uniaxial extension of the fluid are obtained. Longitudinal stability of the straight rod during its spatial motion is considered. Effective viscosity of a suspension filled by rigid straight rods is evaluated.     

Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear Rate Dependent Viscosity

In the present paper we prove the existence of weak solutions to the equations of non-stationary motion of an incompressible fluid with shear rate dependent viscosity in a cylinder Q = Ω × (0,T), where denotes an open set. For the power-low model with we are able to construct a weak solution with ∇ · u = 0.     

Invariant helical subspaces for the Navier-Stokes equations

Three-dimensional solutions with helical symmetry are shown to form an invariant subspace for the Navier-Stokes equations. Uniqueness of weak helical solutions in the sense of Leray is proved, and these weak solutions are shown to be regular (strong) solutions existing for arbitrary time t. The global universal attractor for the infinite-dimensional dynamical system generated by the corresponding semi-group of helical flows is shown to be compact and finite-dimensional. The Hausdorff and fractal dimensions of the global attractors are estimated in terms of the governing physical parameters and in terms of the helical parameters for several problems in the class, with the most detailed results obtained for rotating Hagen-Poiseuille (pipe) flow. In this case, the dimension, either Hausdorff or fractal, up to an absolute constant is bounded from above by , where is the axial wavenumber, n is the azimuthal wavenumber and Re is the Reynolds number based on the radius of the pipe. These upper bounds are independent of the rotation rate.     

Modeling and Designing the Barreling Machine Drive with Complex Spatial Motion of the Container

International Applied Mechanics - The motion of the processing medium inside the container performing complex spatial motion in a Turbula-type barreling machine was studied experimentally. It was...     

On the Material Symmetry of Elastic Rods

This paper presents a treatment of material symmetry for hyperelastic rods. The rod theory of interest is based on a Cosserat (or directed) curve with two director fields, and was developed in a series of works by Green, Naghdi and several of their co-workers. The treatment is based on Murdoch and Cohen's work on material symmetry of Cosserat surfaces. Two material symmetry groups are discussed: one pertains to the strain-energy function, while the other pertains to the response functions. The paper closes by showing how the treatment relates to the form-invariant approach used in Green and Naghdi's papers and a treatment proposed recently by Cohen. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Sharp Upper Bound for the¶Torsional Rigidity of Rods¶by Means of Web Functions

Using web functions, we approximate the Dirichlet integral which represents the torsional rigidity of a cylindrical rod with planar convex cross-section Ω. To this end, we use a suitably defined piercing function, which enables us to obtain bounds for both the approximate and the exact value of the torsional rigidity. When Ω varies, we show that the ratio between these two values is always larger than ¾; we prove that this is a sharp lower bound and that it is not attained. Several extremal cases are also analyzed and studied by numerical methods.     

Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods

Reduced order models for the dynamics of geometrically exact planar rods are derived by projecting the nonlinear equations of motion onto a subspace spanned by a set of proper orthogonal modes. These optimal modes are identified by a proper orthogonal decomposition processing of high-resolution finite element dynamics. A three-degree-of-freedom reduced system is derived to study distinct categories of motions dominated by a single POD mode. The modal analysis of the reduced system characterizes in a unique fashion for these motions, since its linear natural frequencies are near to the natural frequencies of the full-order system. For free motions characterized by a single POD mode, the eigen-vector matrix of the derived reduced system coincides with the principal POD-directions. This property reflects the existence of a normal mode of vibration, which appears to be close to a slow invariant manifold. Its shape is captured by that of the dominant POD mode. The modal analysis of the POD-based reduced order system provides a potentially valuable tool to characterize the spatio-temporal complexity of the dynamics in order to elucidate connections between proper orthogonal modes and nonlinear normal modes of vibration.     

边坡预应力锚杆蠕变的数值分析

运用非线性蠕变分析理论模型,采用MSC.MARC软件重点研究了锚杆的应力指数、蠕变系数和几何参数的变化对于锚杆应力松弛的影响.通过对预应力锚杆支护的边坡进行数值分析,探讨了锚杆的蠕变规律以及锚杆预应力松弛现象.锚杆的参数取不同数值时,比较数值模拟结果发现:锚杆的轴力和剪力主要分布在锚固段的前端;随着锚杆应力指数及蠕变系数的增加,蠕变应变增大,应力松弛加快.     

Elliptic Two-Dimensional Invariant Tori for the Planetary Three-Body Problem

The spatial planetary three-body problem (i.e., one star and two planets, modelled by three massive points, interacting through gravity in a three dimensional space) is considered. It is proved that, near the limiting stable solutions given by the two planets revolving around the star on Keplerian ellipses with small eccentricity and small non-zero mutual inclination, the system affords two-dimensional, elliptic, quasi-periodic solutions, provided the masses of the planets are small enough compared to the mass of the star and provided the osculating Keplerian major semi-axes belong to a two-dimensional set of density close to one.     

Invariant manifolds for products of random diffeomorphisms

This paper is concerned with the construction of invariant families of submanifolds for products of random diffeomorphisms on a compact Riemannian manifold. These submanifolds can be obtained for almost arbitrary parameters disjoint from the Lyapunov spectrum of the resulting cocycle. Local measurable families are constructed and the globalization problem is discussed. We present a globalization result for generalized stable and unstable manifolds.