Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle

The dynamics of non-smooth oscillators has not yet sufficiently been investigated, when damping is simultaneously due to friction and impact. Because of the theoretical and practical interest of this type of systems, an effort is made in this paper to lighten the behaviour of a single-degree-of-freedom oscillator colliding with an obstacle and excited by a moving base, which transfers energy to the system via friction. The different nature of discontinuities arising in the combined problem of friction and impact has been recognized and discussed. Closed-form solutions are presented for both transient and steady-state response, assuming Coulomb's friction law and a rigid stop-limiting motion. Furthermore, a deformable (hysteretic) obstacle has been considered, and its influence on the response has been investigated.     

Impact dynamics of multibody systems with frictional contact using joint coordinates and canonical equations of motion

This paper presents a methodology in computational dynamics for the analysis of mechanical systems that undergo intermittent motion. A canonical form of the equations of motion is derived with a minimal set of coordinates. These equations are used in a procedure for balancing the momenta of the system over the period of impact, calculating the jump in the body momentum, velocity discontinuities and rebounds. The effect of dry friction is discussed and a contact law is proposed. The present formulation is extended to open and closed-loop mechanical systems where the jumps in the constraints' momenta are also solved. The application of this methodology is illustrated with the study of impact of open-loop and closed-loop examples.     

Modelling the dynamics of large block structures

This paper summarizes the main critical points that arise when the problem of modelling the dynamics of block structures is tackled. In the first sections, a rigorous formulation of dynamics and impact problem is presented for a single rigid block freely supported on rigid ground, in order to illustrate the basic difficulties concerning the modelling of more complicated structures. Then, a critical review is presented on the numerous researches performed on this subject and the results achieved, and the problems still open, are put in evidence.

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Sommario In questo lavoro, si illustrano i punti salienti e critici che devono essere affrontati nella modellazione del comportamento dinamico di structture costituite da grandi blocchi assemblati a secco. Nei primi paragrafi, viene presentato e discusso il problema generale della dinamica e dell' urto del blocco singolo semplicemente appoggiato su suolo rigido: è questa la base necessaria per affrontare in modo rigoroso la modellazione di strutture più complesse. Viene quindi presentata una rassegna critica di vari modelli proposti in letteratura, evidenziando i problemi risolti e quelli ancora aperti.

     

Nonlinear dynamics of a cable–pulley system using the absolute nodal coordinate formulation

It is reasonable to develop models and to investigate the dynamic behaviour of systems composed of cables since cable vibration can have an important effect on the motion of these mechanical systems. This paper deals with the application of the nonlinear formulation for flexible body dynamics called the absolute nodal coordinate formulation (ANCF). It is used for modelling the systems composed of cables, pulleys, other rigid bodies and a motor with prescribed motion. The ANCF was chosen as a suitable approach, which that can allow to consider a detailed interaction of the cable and the pulley with its nonlinear dynamical behaviour. The ANCF uses absolute positions of nodes (reference vectors) and slopes (reference vector derivations) as a set of nodal coordinates. An in-house modelling tool in the MATLAB system was created based on the proposed modelling methodology and two case studies were performed. A simple system containing a pulley and a cable with two attached bodies was used in order to test the simulation tool based on the proposed modelling methodology with respect to different parameters. A more complex mechanical system composed of a driven weight joined with a motor by a cable led over a pulley was numerically and also experimentally investigated. The comparison of obtained numerical and experimental results shows sufficient agreement and proves that the proposed modelling approach can be used for dynamic analyses of such systems.     

Time- and frequency-domain analysis using lumped mass global collocation

Quite recently, a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures was proposed (Archive of Applied Mechanics: DOI: ). This paper extends the latter methodology on several levels, in both the time and frequency domain. Firstly the formulation is updated so that it can also deal with rods of variable cross section. Then, the fully populated mass matrices of the previous formulation are properly replaced by lumped masses, thus saving still more computer effort. Subsequently, a new general formulation for the transient response analysis is proposed. Finally, a novel procedure for the coupling of two neighboring collinear rods is presented. The theory is supported by six test cases concerning elastic rods of constant and variable cross sections. Among these, transient analysis refers to the response of a single rod due to a Heaviside-type loading as well as to the impact between two collinear rods of different cross sections.     

On penalty function methods in the finite-element analysis of flow problems

In this paper the penalty function method is reviewed in the general context of solving constrained minimization problems. Mathematical properties, such as the existence of a solution to the penalty problem and convergence of the solution of a penalty problem to the solution of the original problem, are studied for the general case. Then the results are extended to a penalty function formulation of the Stokes and Navier-Stokes equations. Conditions for the equivalence of two penalty-finite element models of fluid flow are established, and the theoretical error estimates are verified in the case of Stokes's problem.     

Hybrid dynamics of two coupled oscillators that can impact a fixed stop

We consider two linearly coupled masses, where one mass can have inelastic impacts with a fixed, rigid stop. This leads to the study of a two degree of freedom, piecewise linear, frictionless, unforced, constrained mechanical system. The system is governed by three types of dynamics: coupled harmonic oscillation, simple harmonic motion and discrete rebounds. Energy is dissipated discontinuously in discrete amounts, through impacts with the stop. We prove the existence of a non-zero measure set of orbits that lead to infinite impacts with the stop in a finite time. We show how to modify the mathematical model so that forward existence and uniqueness of solutions for all time is guaranteed. Existence of hybrid periodic orbits is shown. A geometrical interpretation of the dynamics based on action coordinates is used to visualize numerical simulation results for the asymptotic dynamics.     

柔性多体系统的递推组集建模与仿真软件的实现

简要地阐述多体系统动力学单向递推组集建模方法，介绍根据这种方法开发而成的仿真软件系统ＤＡＦＭＢ及它的功能与特点。通过双摆算例指出本文提出的模型在计算效率与计算精度方面优于笛卡尔坐标的微分－代数方程。     

Geometry of mesoscopic dynamics and thermodynamics

Contact geometry provides a natural setting for classical thermo-dynamics. In this paper we use it to derive the structure of mesoscopic dynamics (GENERIC) expressing its compatibility with thermodynamics. In the second part we derive kinematics (Poisson brackets) of a large family of mesoscopic state variables.     

On the geometrically exact formulation of structural mechanics and its applications to dynamics,control and optimization

In this survey paper we re-examine the theoretical formulation of structural mechanics, introducing no restrictions with respect to the size of displacements, rotations or deformations, which is commonly referred to as geometrically exact. A special attention is given to clarifying the computational aspects of finite rotations as the key ingredient of any such formulation. We briefly discuss several novel applications of the geometrically exact formulation to dynamics, control and optimization. To cite this article: A. Ibrahimbegovic, C. R. Mecanique 331 (2003).     

Off-center impact of an elastic column by a rigid mass

Based on the Timoshenko beam model the equations of motion are obtained for large deflection of off-center impact of a column by a rigid mass via Hamilton's principle. These are a set of coupled nonlinear partial differential equations. The Newmark time integration scheme and differential quadrature method are employed to convert the equations into a set of nonlinear algebraic equations for displacement components. The equations are solved numerically and the effects of weight and velocity of the rigid mass and also off-center distance on deformation of the column are studied.     

Effect of permanent indentation on the delamination threshold for small mass impact on plates

Small mass impacts on composite structures are common cases caused by hailstones and runway debris. Small mass impactors usually result in a wave controlled local response, which is independent of boundary conditions. This response occurs before the reflection of waves from the boundaries and cannot be modeled by large mass drop weight tests. An elasto-plastic contact law, which accounts for permanent indentation and damage effects, was used here to study small mass impact on laminated composite plates. By comparing with results from the Hertzian contact law, it was found that damage can change the dynamic response of the structure significantly with increasing impact velocity. Due to smaller contact force generated for the case of using elasto-plastic contact, the central displacement of the plate is also less than the one using Hertzian contact law. The linearized version of the contact law was then used to derive the closed-form approximations of the contact force, indentation and plate central displacement for the impact loading of composite laminates. The threshold velocity for delamination onset under small mass impact was predicted analytically based on the obtained peak contact forces by combining with an existing quasi-static delamination threshold load criterion. A good agreement was found between the predicted threshold values and published experimental results.     

A finite element approach to the chaotic motion of geometrically exact rods undergoing in-plane deformations

The paper is concerned with a hybrid finite element formulation for the geometrically exact dynamics of rods with applications to chaotic motion. The rod theory is developed for in-plane motions using the direct approach where the rod is treated as a one-dimensional Cosserat line. Shear deformation is included in the formulation. Within the elements, a linear distribution of the kinematical fields is combined with a constant distribution of the normal and shear forces. For time integration, the mid-point rule is employed. Various numerical examples of chaotic motion of straight and initially curved rods are presented proving the powerfulness and applicability of the finite element formulation.     

An investigation of an Emden-Fowler equation from thin film flow

A third-order ordinary differential equation (ODE) for thin film flow with both Neumann and Dirichlet boundary conditions is transformed into a second-order nonlinear ODE with Dirichlet boundary conditions.Numerical solutions of the nonlinear second-order ODE are investigated using finite difference schemes.A finite difference formulation to an Emden-Fowler representation of the second-order nonlinear ODE is shown to converge faster than a finite difference formulation of the standard form of the second-order nonlinear ODE.Both finite difference schemes satisfy the von Neumann stability criteria.When mapping the numerical solution of the second-order ODE back to the variables of the original third-order ODE we recover the position of the contact line.A nonlinear relationship between the position of the contact line and physical parameters is obtained.     

Basic features of the time finite element approach for dynamics

Very general weak forms may be developed for dynamic systems, the most general being analogous to a Hu-Washizu three-field formulation, thus paralleling well-established weak methods of solid mechanics. In this work two different formulations are developed: a pure displacement formulation and a two-field mixed formulation. With the objective of developing a thorough understanding of the peculiar features of finite elements in time, the relevant methodologies associated with this approach for dynamics are extensively discussed. After having laid the theoretical bases, the finite element approximation and the linearization of the resulting forms are developed, together with a method for the treatment of holonomic and nonholonomic constraints, thus widening the horizons of applicability over the vast world of multibody system dynamics. With the purpose of enlightening on the peculiar numerical behavior of the different approaches, simple but meaningful examples are illustrated. To this aim, significant parallels with elastostatics are emphasized. Paper presented at the ‘International Technical Specialists' Meeting on Rotorcraft Basic Research’, March 25–27, 1991, Georgia Institute of Technology, Atlanta, Georgia, USA.     

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Continuum mechanics with dislocations, with the Cattaneo-type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov-type system of the first-order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).     

About the formulation,verification and validation of the hypersonic flow solver Eilmer

We describe the formulation of the gas dynamics and high‐temperature thermochemical modules of the Eilmer code, an open‐source Navier–Stokes solver for transient compressible flow in two and three dimensions. The core gas dynamics formulation is based on finite‐volume cells, and the thermochemical effects are handled with specialised updating schemes that are coupled into the overall time‐stepping scheme. Verification of the code is explored via a number of case studies that use analytic and semi‐analytic solutions as comparison. These include both smooth and shocked flows and are used to demonstrate the order of spatial accuracy of the code. Cases include manufactured solutions for rather abstract inviscid and viscous flow, an idealised detonation wave supported by a curved body, and the transient flow of an idealised but high‐performance shock tube. Validation of the inviscid gas dynamics and thermochemical models is then explored using data from a selection of experimental studies. These studies include ballistic range experiments with chemically‐inert noble gases and high‐temperature chemically‐reacting air. These comparisons show that the code performs well and they provide a lesson in considering a range of experimental data rather than relying upon isolated data points for validation. These verification and validation cases are described in full detail and will be useful for other code developers of high‐temperature compressible flow solvers. Copyright © 2013 John Wiley & Sons, Ltd.