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1

Using symbolic computation in the characterization of frictional instabilities involving orthotropic materials

Agwa M. A., Pinto da Costa A.

International Journal of Applied Mathematics and Computer Science

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2015

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Vol. 25, no. 2

259--267

EN

The present work addresses the problem of determining under what conditions the impending slip state or the steady sliding of a linear elastic orthotropic layer or half space with respect to a rigid flat obstacle is dynamically unstable. In other words, we search the conditions for the occurrence of smooth exponentially growing dynamic solutions with perturbed initial conditions arbitrarily close to the steady sliding state, taking the system away from the equilibrium state or the steady sliding state. Previously authors have shown that a linear elastic isotropic half space compressed against and sliding with respect to a rigid flat surface may get unstable by flutter when the coefficient of friction μ and Poisson's ratio ν are sufficiently large. In the isotropic case they have been able to derive closed form analytic expressions for the exponentially growing unstable solutions as well as for the borders of the stability regions in the space of parameters, because in the isotropic case there are only two dimensionless parameters (μ and ν). Already for the simplest version of orthotropy (an orthotropic transversally isotropic material) there are seven governing parameters (μ, five independent material constants and the orientation of the principal directions of orthotropy) and the expressions become very lengthy and literally impossible to manipulate manually. The orthotropic case addressed here is impossible to solve with simple closed form expressions, and therefore the use of computer algebra software is required, the main commands being indicated in the text.

2

Dynamic analysis of mechanisms with joint friction in the presence of redundant constraints

Frączek J., Wojtyra M.

International Journal of Applied Mechanics and Engineering

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2010

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Vol. 15, no 3

685-697

EN

This paper discusses the problem of calculating the joint friction forces in redundantly constrained rigid body mechanisms, described in absolute Cartesian coordinates. At the beginning the systems without joint friction are considered. It is shown that, under certain conditions, selected joint reaction forces can be uniquely determined despite the redundant constraints existence. A numerical method of such joints detection is proposed. Afterwards the equations of motion for mechanisms with redundant constraints and Coulomb friction in joints are formulated. In general, motion of such system is not defined uniquely. If, however, friction forces appear only in joints with uniquely determined reactions, motion can be determined uniquely. An example of dynamic analysis of a mechanism with redundant constraints and joint friction is presented.

3

Modelowanie tarcia w układach mechanicznych

Brodny J.

Górnictwo i Geologia

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2010

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T. 5, z. 2

7-17

PL

Tarcie jest zjawiskiem fizycznym powszechnie występującym w przyrodzie i mającym istotny wpływ na charakter pracy układów mechanicznych. W artykule omówiono podstawowe rodzaje tarcia i przedstawiono jego podział. Zaprezentowany został najczęściej wykorzystywany przy modelowaniu tarcia w układach mechanicznych model tarcia Coulomba. Przeanalizowano wpływ prędkości na wartość siły tarcia oraz przejście z tarcia statycznego w kinematyczne. Dla układu o jednym stopniu swobody wyznaczony został wpływ współczynnika tarcia wewnętrznego na charakterystykę jego ruchu.

EN

Friction is a physical phenomenon widespread in nature and having major influence on the character of mechanical systems' work. The article describes the basic types of friction and their classification. The Coulomb friction model most commonly used for friction modelling in mechanical systems has been presented. The effect of speed on the friction force values and transition from static to kinematic friction have been investigated. The effect of internal friction coefficient on its motion's characteristics has been identified for a single degree-of-freedom system.

4

The penalty and lagrange multiplier methods in the frictional 3d beam-to-beam contact problem

Litewka P.

Civil and Environmental Engineering Reports

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2005

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No. 1

189-207

EN

This paper is concerned with the frictional contact between beams. The purpose is to compare the results obtained by the penalty and Lagrange multiplier methods. The advantages and disadvantages of both methods are generally known and widely discussed but not in the quantitative manner and not specifically for the problem of beam-to-beam contact. The paper sketches briefly both formulations pointing out the main differences and features of both methods. The section with examples presents several cases of analysed beam-to-beam contact scenarios. The accuracy of results is taken into account, also the comparison to the full 3D analysis performed using the program ABAQUS is made. The computation times and the length of the codes for both methods are compared as well. These criteria allow one to conclude that for the beam-to-beam contact problem the Lagrange multiplier method is more attractive than the penalty method.

PL

Artykuł dotyczy kontaktu z tarciem między belkami. Jego celem jest porównanie wyników obliczeń uzyskanych metodami współczynnika kary i mnożników Lagrange'a. Zalety i wady obu tych metod są ogólnie znane i dyskutowane w literaturze, brakuje jednak ich porównania ilościowego dla przypadku kontaktu między belkami. W artykule krótko przedstawiono oba sformułowania i wskazano na podstawowe różnice między nimi. Przedstawiono wyniki obliczeń dla kilku przykładów kontaktujących się belek. Wzięto pod uwagę dokładność obliczeń, wyniki porównano z pełną analizą trójwymiarową wykonaną za pomocą programu ABAQUS. Porównano także długość kodu i czas obliczeń. Te kryteria pozwoliły stwierdzić, że w przypadku kontaktu między belkami metoda mnożników Lagrange’a jest bardziej atrakcyjna niż metoda współczynnika kary.

5

Modelowanie kontaktu w układzie ślizgowym obciążonym dynamiczną siłą normalną

Grudziński K., Kostek R.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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2004

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z. 23

143-148

PL

W pracy rozważano dwa modele kontaktu ciernego: tradycyjny model tarcia Coulomba o jednym stopniu swobody (ID) oraz model o dwóch stopniach swobody (2D), uwzględniający nieliniową podatność kontaktową w kierunku normalnym. Dla tych modeli, metodą symulacji komputerowej, wyznaczono charakterystyki obrazujące zależność reakcji normalnej i siły tarcia w kontakcie od amplitudy i częstotliwości obciążenia zewnętrznego. Wykazano, że sił dynamicznych występujących w kontakcie ciernym nie można należycie opisać za pomocą modelu ID.

EN

In this paper two models of frictional contact have been investigated: the classical Coulomb's friction model with one degree of freedom (ID) and a model with two degrees of freedom (2D) which takes into account the normal contact flexibility of the interface. For these two models characteristics have been computed which show the dependencies of the normal reaction and the friction force in the contact from the amplitude and frequencies of the dynamical force. It has been proved that the dynamical forces at the friction contact can not be described relevant using ID model of that contact.

6

On thermoviscoelastic wheel-rail contact problems

Chudzikiewicz A., Myśliński A.

Vibrations in Physical Systems

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2004

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Vol. 21

119--122

EN

This paper is concerned with the numerical solution of wheel - rail rolling contact problems. The unilateral dynamic contact problem between a viscoelastic body and a rigid foundation is considered. The contact with Coulomb friction law occurs at a portion of the boundary of the body. The contact condition is described in velocities. The friction coefficient is assumed to be bounded and suitable small. A frictional heat generation and heat transfer across the contact surface as well as Archard's law of wear in contact zone are assumed. The equlibrium state of this contact problem is described by the coupled hyperbolic variational inequality of the second order and a parabolic equation. To solve numerically this contact problem we will decouple it into mechanical and thermal parts. Finite difference and finite element methods are used to discretize the contact problem. The Augmented Lagrangian technique combined with the active set method are employed to solve the discretized contact problem. Numerical examples are provided.

7

A Pivoting Solver for Dynamic Planar Multi-Contact Friction Problems

Funk K., Pfeiffer F.

Machine Dynamics Problems

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2000

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Vol. 24, No. 1

87-99

EN

An important application of complementarity theory is the determination of the contact forces in multi-body systems with rigid contacts. As long as the contacts are friction-free the formulation of the multi-contact problem results in a well posed linear complementarity problem (LPC). But once a friction force-depending on the normal contact force - is introduced the problem gets a lot more complicated, and the resulting LPC gets ill posed for many solving algorithms. The paper addresses a linear complementarity formulation in a generalised form, which is appropriate to handle planar friction problems. In addition, a direct solving method based on Lemke's method is presented, that can handle the generalised linear complementarity problem.