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Research Papers

On Some Problems With Modeling of Coulomb Friction in Self-Locking Mechanisms

[+] Author and Article Information
Marek Wojtyra

Institute of Aeronautics and Applied Mechanics,
Warsaw University of Technology,
Nowowiejska 24,
Warsaw 00-665, Poland
e-mail: mwojtyra@meil.pw.edu.pl

1Corresponding author.

Manuscript received October 31, 2014; final manuscript received April 11, 2015; published online June 30, 2015. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 11(1), 011008 (Jan 01, 2016) (10 pages) Paper No: CND-14-1276; doi: 10.1115/1.4030386 History: Received October 31, 2014; Revised April 11, 2015; Online June 30, 2015

Friction significantly influences the mechanical system dynamics, especially when self-locking property is observed. The Coulomb model is frequently adopted to represent friction in multibody analysis and simulation. It can be shown that in some extreme cases of joint friction modeling, problems with solution uniqueness and existence are encountered, even when only bilateral constraints and kinetic regime of friction are considered. These problems are studied in detail in the paper. To approach the investigated subject, a wedge mechanism, viewed as a simplified model of a speed reducer, is studied. Two different mathematical models of joint friction are used, both based on the Coulomb friction law. The first version of the model is purely rigid, i.e., no deflections of the mechanism bodies are allowed. Constraints are imposed to maintain the ratio between input and output velocity. The second version of the model allows deflection of the frictional contact surface, and forces proportional to this deflection are applied to contacting bodies (no constraints to maintain the input–output velocity ratio). Using the rigid body model, it is shown that when friction is above the self-locking limit, paradoxical situations may be observed when kinetic friction is investigated. For some sets of parameters of the mechanism (gearing ratio, friction coefficient, and inertial parameters), two distinct solutions of normal and friction forces can be found. Moreover, for some combinations of external loads, a solution that satisfies equations of motion, constraints, and the Coulomb friction law does not exist. Furthermore, for appropriately chosen loads and parameters of the mechanism, infinitely many feasible sets of normal and friction forces can be found. Investigation of the flexible body model reveals that in nonparadoxical situations the obtained results are closely similar to those predicted by the rigid body model. In previous paradoxical situations, no multiple solutions are found; however, problems with stability of solutions emerge. It is shown that in regions for which the paradoxes were observed only unstable solutions are available. The origins of paradoxical behavior are identified and discussed. The key factors determining the model performance are pointed out. Examples of all indicated problematic situations are provided and analyzed. Finally, the investigated problems are commented from more general perspectives of multibody system dynamics.

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Figures

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Fig. 1

Distribution of forces in a wedge mechanism

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Fig. 2

Normal force N (top) and acceleration of body I (bottom) versus driving force F. Nonparadoxical case.

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Fig. 3

Normal force N (top) and acceleration of body I (bottom) versus driving force F. Paradoxical case.

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Fig. 4

Normal force N (top) and acceleration of body I (bottom) versus driving force F. Case of critical mass ratio.

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Fig. 5

Acceleration of body I versus normal force N. Case of critical mass ratio and critical force F.

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Fig. 6

Displacements of bodies I and O

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Fig. 7

Normal force N (top) and acceleration of body I (bottom) versus time t. Mass ratio greater than critical.

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Fig. 8

Normal force N (top) and acceleration of body I (bottom) versus time t. Mass ratio less than critical, stable solutions.

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Fig. 9

Normal force N (top) and acceleration of body I (bottom) versus time t. Mass ratio less than critical, unstable solutions.

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Fig. 10

Normal force N (top) and acceleration of body I (bottom) versus time t. Mass ratio and force F less than critical.

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Fig. 11

Normal force N (top) and acceleration of body I (bottom) versus time t. Results for various stiffness.

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