Dynamic vibrations of a damageable viscoelastic beam in contact with two stops

A model for the material damage, due to dynamic vibrations of a Kelvin‐Voigt viscoelastic beam whose tip is constrained to move between two stops, is presented and numerically analyzed. The contact of the free tip with the stops is described by the normal compliance condition. The evolution of damage of the beam's material, which measures the reduction of its load carrying capacity, is modeled with a parabolic inclusion. The existence of the unique local solution is stated. A numerical algorithm is presented, in which spatially it is approximated by finite elements, and the time derivatives are discretized with the Euler scheme. Error estimates are derived for sufficiently regular solutions, and four numerical simulations are shown. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Dynamic frictionless contact of a nonlinear beam with two stops

In this work, we consider mathematical and numerical approaches to a dynamic contact problem with a highly nonlinear beam, the so-called Gao beam. Its left end is rigidly attached to a supporting device, whereas the other end is constrained to move between two perfectly rigid stops. Thus, the Signorini contact conditions are imposed to its right end and are interpreted as a pair of complementarity conditions. We formulate a time discretization based on a truncated variational formulation. We prove the convergence of numerical trajectories and also derive a new form of energy balance. A fully discrete numerical scheme is implemented to present numerical results.     

Vibrations of a beam between two rigid stops: strong solutions in the framework of vector-valued measures

Abstract

We study a problem that models the dynamics of an elastic beam vibrating between two rigid stops, so we use the Signorini non-penetration condition to describe the contact process. This allows for impacts and velocity jumps. Motivated by the need to better understand this kind of dynamics, we introduce a new formulation of the problem in the framework of vector-valued measures, somewhat similar to the case of a discrete mechanical system. We prove the existence of a strong solution and establish the main properties of the reaction shear stress that acts on the system at impacts, which is a measure with a singular part.     

The study of the dynamic contact in ultrasonic motor

In this paper, we consider a mathematical model which describes the dynamic contact between an elastic body and an obstacle. The process is assumed to be dynamic and the contact is modeled with the normal compliance. We present a variational formulation of the problem and prove the existence and the uniqueness of a weak solution. An efficient numerical method is presented to analyze this dynamic contact. This approach exploits the augmented Lagrangian concept and a special time integration algorithm. This is exploited to study the dynamic contact between the rotor and the stator inside an ultrasonic motor SHINSEI USR 60. Numerical results are presented and show the interest of this method to forecast the origin of the principal failure mode of this motor.     

Dynamic contact of a beam against rigid obstacles: Convergence of a velocity-based approximation and numerical results

Motivated by the study of vibrations due to looseness of joints, we consider the motion of a beam between rigid obstacles. Due to the non-penetrability condition, the dynamics is described by a hyperbolic fourth order variational inequality. We build a family of fully discretized approximations of this problem by combining some classical space discretizations with velocity based time-stepping algorithms for discrete mechanical systems subjected to unilateral constraints. We prove the stability and the convergence of these numerical methods. Finally we propose some examples of implementation using either Hermite or B-spline finite element approximations.     

Qualitative behaviour of stochastic delay equations with a bounded memory

A dichotomy is proved concerning recurrence properties of the solution of certain stochastic delay equations. If the solution process is recurrent, there exists an invariant measure π on the state space C which is unique (up to a multiplicative constant) and the tail-field is trivial. If π happens to be a probability measure, then for every initial condition, the distribution of the process converges to it as t→∞. We will formulate a sufficient condition for the existence of an invariant probability measure (ipm) in icrnia of Lyapunov junctionals and give two examples, one Heing the stochastic-delay version of the famous logistic equation of population growth. Finally we study approximations of delay equations by Markov chains.     

Non-probabilistic convex models and interval analysis method for dynamic response of a beam with bounded uncertainty

This paper is concerned with the comparison of two non-probabilistic set-theoretical models for dynamic response measures of an infinitely long beam. The beam is on an uncertain foundation and subjected to a moving force with constant speed. The steady state vibration is analyzed with finite element method. The dynamic responses of the beam are approximated to the first-order respect of the uncertainty variables. As a rule, in convex models and interval analysis, the uncertainties are considered to be unknown, but they give out their allowable vector space. Comparing the convex models with interval analysis in mathematical proofs and numerical calculations, it’s shows that under the condition of transform an interval vector to an outer enclosed ellipsoid, the dynamic response of the infinitely long beam predicted by interval analysis is smaller than that by convex models; under the condition of transform a hyperellipsoid to an outer enclosed interval vector, the dynamic response of the infinitely long beam calculated by convex models is smaller than that by interval analysis method.     

Dynamic model of a flying manipulator with two highly flexible links

Nonlinear dynamic model of a flying manipulator with two revolute joints and two highly flexible links is obtained using Hamilton’s principle. Flying base of the manipulator is a rigid body. Stress is treated three dimensionally in the isotropic linearly-elastic links, but the in-plane and out-of-plane warpings of the links’ cross-sections are neglected. Although the links’ cross-sections undergo negligible elastic orientation, their models are more accurate than a nonlinear 3D Euler–Bernoulli beam. Tension, compression, twisting and spatial deflections of each link are coupled to each other by some nonlinear terms including two new ones. In the issue of flying flexible-link manipulators new terminologies, namely forward/inverse kinetics instead of forward/inverse kinematics are suggested, since determination of position and orientation of the end-effector is coupled to the partial differential motion equations.     

Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique

The paper deals with the numerical solution of the quasi-variational inequality describing the equilibrium of an elastic body in contact with a rigid foundation under Coulomb friction. After a discretization of the problem by mixed finite elements, the duality approach is exploited to reduce the problem to a sequence of quadratic programming problems with box constraints, so that efficient recently proposed algorithms may be applied. A new variant of this method is presented. It combines fixed point with block Gauss–Seidel iterations. The method may be also considered as a new implementation of fixed point iterations for a sequence of problems with given friction. Results of numerical experiments are given showing that the resulting algorithm may be much faster than the original fixed point method and its efficiency is comparable with the solution of frictionless contact problems.     

On a solvability of hydro-mechanical problem based on contact problem with visco-plastic friction in Bingham rheology

This paper deals with the solvability and numerical solution of contact problem with a local visco-plastic friction in the visco-plastic Bingham rheology. The model problem discussed represents a simple hydro-mechanical model of the global project on a security of regions endangered by great hurricanes and deluges. The main goal of the idea of this project is to connect the climatic observations and the corresponding climatic models with the thermo-hydro-dynamic and the thermo-hydro-mechanic models, with the possibility to estimate future destructions of such endangered regions with landslides of unstable slopes. The investigated mathematical model is based on the visco-plastic Bingham rheology. The numerical approach is based on the semi-implicit scheme in time and the FE approximation in space. The algorithm is shortly discussed.     

Zeros of a random algebraic polynomial with coefficient means in geometric progression

This paper provides the mathematical expectation for the number of real zeros of an algebraic polynomial with non-identical random coefficients. We assume that the coefficients {a_j}ⁿ⁻¹_j=0 of the polynomial T(x)=a₀+a₁x+a₂x²+?+a_n−1xⁿ⁻¹ are normally distributed, with mean E(a_j)=μ^j+1, where μ≠0, and constant non-zero variance. It is shown that the behaviour of the random polynomial is independent of the variance on the interval (−1,1); it differs, however, for the cases of |μ|<1 and |μ|>1. On the intervals (−∞,−1) and (1,∞) we find the expected number of real zeros is governed by an interesting relationship between the means of the coefficients and their common variance. Our result is consistent with those of previous works for identically distributed coefficients, in that the expected number of real zeros for μ≠0 is half of that for μ=0.     

Elastic solutions of a functionally graded cantilever beam with different modulus in tension and compression under bending loads

This paper considers a functionally graded cantilever beam with different modulus in tension and compression. The beam is subjected to bending loads, including pure bending, shear force at the free end and uniform pressure on the upper lateral, respectively. Its modulus values in tension and compression both change with the thickness coordinate as arbitrary functions, which could bring the beam a broader range of applications in engineering. The problem is treated as a plane stress case and described by Airy stress function. By using semi-inverse method, the elastic solutions for the beam are obtained, which can be easily degenerated into the ones for homogeneous beams. An example is finally presented to show the effect of nonhomogeneous materials with different modulus on the elastic field in a cantilever beam.     

Simulation of the transverse vibrations of a cantilever beam with an eccentric tip mass in the axial direction using integral transforms

The main purpose of the current work is to employ an integral transform approach based on eigenfunction expansion and on an implicit filter scheme in order to solve the governing equations for the transverse vibrations of a cantilever beam clamped at one end and with an eccentric tip mass in the axial direction at the other end. Numerical results are obtained for both the undamped and damped natural frequencies of the system, as well as for its transverse displacement due to arbitrarily time-varying load and imposed displacement at the clamped end. The numerical results reported in the current work are highly accurate and new in the literature. New exact results are also provided for the transient displacement and its higher-order spatial derivatives to allow computation of bending stresses and strains. The relative merits of the proposed approach are finally pointed out.     

Smoothness of certain functions in two kinds of risk models with a barrier dividend strategy

In this paper,we study the smoothness of certain functions in two kinds of risk models with a barrier dividend strategy.Mainly using technique from the piecewise deterministic Markov processes theory,we prove that the function is continuously differentiable in the first risk model.Using the weak infinitesimal generator method of Markov processes,we prove that the function is twice continuously differentiable in the second risk model.Intego-differential equations satisfied by them are derived.     

Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths

In this note we consider a chain of N$N$ oscillators, whose ends are in contact with two heat baths at different temperatures. Our main result is the exponential convergence to the unique invariant probability measure (the stationary state). We use the Lyapunov’s function technique of Rey-Bellet and coauthors [Luc Rey-Bellet, Statistical mechanics of anharmonic lattices, in: Advances in Differential Equations and Mathematical Physics (Birmingham, AL, 2002), in: Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 283–298. MR MR1991548 (2005a:82068) [11]; Luc Rey-Bellet, Lawrence E. Thomas, Fluctuations of the entropy production in anharmonic chains, Ann. Henri Poincaré 3 (3) (2002) 483–502. MR MR1915300 (2003g:82060); Luc Rey-Bellet, Lawrence E. Thomas, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Comm. Math. Phys. 225 (2) (2002) 305–329. MR MR1889227 (2003f:82052); Luc Rey-Bellet, Lawrence E. Thomas, Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators, Comm. Math. Phys. 215 (1) (2000) 1–24. MR MR1799873 (2001k:82061) [12]; Jean-Pierre Eckmann, Claude-Alain Pillet, Luc Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Comm. Math. Phys. 201 (3) (1999) 657–697. MR MR1685893 (2000d:82025); Jean-Pierre Eckmann, Claude-Alain Pillet, Luc Rey-Bellet, Entropy production in nonlinear, thermally driven Hamiltonian systems, J. Statist. Phys. 95 (1–2) (1999) 305–331. MR MR1705589 (2000h:82075)], with different model of heat baths, and adapt these techniques to two new case recently considered in the literature by Bernardin and Olla [Cédric Bernardin, Stefano Olla, Fourier’s law for a microscopic model of heat conduction, J. Statist. Phys. 121 (3–4) (2005) 271–289. MR MR2185330] and Lefevere and Schenkel [R. Lefevere, A. Schenkel, Normal heat conductivity in a strongly pinned chain of anharmonic oscillators, J. Stat. Mech. Theory Exp. 2006 (02) (2006) L02001].     

Inverse prediction of frictional heat flux and temperature in sliding contact with a protective strip by iterative regularization method

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent frictional heat flux at the interface of two semi-spaces, one of them is covered by a strip of coating, during a sliding-contact process from the knowledge of temperature measurements taken within one of the semi-space. It is assumed that no prior information is available on the functional form of the unknown heat generation; hence the procedure is classified as the function estimation in inverse calculation. Results show that the relative position between the measured and the estimated quantities is of crucial importance to the accuracy of the inverse algorithm. The current methodology can be applied to the prediction of heat generation in engineering problems involving sliding-contact elements.     

The asymptotic behaviour of solutions for a class of quasilinear wave equations with cubic nonlinearity in two space dimensions

For a class of quasilinear wave equations with small initial data, first we give the lower bound of lifespan of classical solutions, then we discuss the long time asymptotic behaviour of solutions away from the blowup time. This project is supported by the Tianyuan Foundation of China and Laburay of Mathematics for Nonlinear Problems, Fudan University.