## Main Article Content

## Abstract

Dry-friction forces have been shown to depend not only on the characteristics of the surface in contact but also on the dynamic interaction of the contacting bodies. A viscoelastic mathematical model that accounts for the interaction at micro-scale of rough surfaces is developed. The mathematical formulation relates the tribological events at microscopic and macroscopic scales vibration response of a "mass on moving belt". The viscoelastic properties are presented by combining loss modulus with Young's modulus to obtain a differential operator on the interference, reminiscent of the Kelvin-Voigt model. The analysis of the system establishes the relation between friction force and speed and supports observed behavior of many systems with friction. The derivations do not rely on a phenomenological account of friction, which requires a presumed friction coefficient. Instead the friction force is accounted for as a result of interaction of the rough surfaces. This has led to a set of nonlinear ordinary differential equations that directly relate the vibration of the system to the surface parameters. It is shown that, as a result of coupling of the macrosystem's dynamics and contact, there are combinations of parameters at micro- and macroscale that yield negative slope in friction force/sliding speed relation, a well known source of dynamic instability.

## Keywords

## Article Details

*How to Cite*

*The Journal of Engineering Research [TJER]*,

*5*(1), 62–70. https://doi.org/10.24200/tjer.vol5iss1pp62-70

* * References

- Abdo, J. A. and Farhang, K., 2005, "Elastic-Plastic Contact Model for Rough Surfaces based on Plastic Asperity Concept," Int. J. of Non-Linear Mechanics, Vol. 40, pp 495-506.
- Abdo, J. A. and Shamseldin, E., 2005, "Modeling of Contact Area, Contact Force, and Contact Stiffness of Mechanical Systems with Friction," ASME Mechanical Engineering Congress Paper no. IMECE2005-82980, Orlando, Florida.
- Abdo, J. A., 2005, "Experimental Technique to Study Tangential-to-normal Contact Load Ratio," Tribology Transactions, Vol. 48, pp 389- 403.
- Abdo, J. A., 2006, "Modeling of Normal and Tangential Contact Stiffness of Rough Surfaces," Int. J. of Modeling and Simulation, Vol. 26(4), pp 295-302.
- Aronov, V., D'souza, A. F., Kalpakjian, S. and Sharper, I., 1984a, "Interaction among Friction, Wear and System Stiffness-Part 1: Effect of Normal Load on System Stiffness," J. Tribol., Vol. 106, pp. 54-58.
- Aronov, V., D'souza, A. F., Kalpakjian, S. and Sharper, I., 1984b "Interaction among Friction, Wear and System Stiffness-Part 2: Vibrations Induced by dry Friction," J. Tribology, V. 106, pp. 59-64.
- Aronov, V., D'souza, A. F., Kalpakjian, S. and Sharper, I., 1984c "Interaction among Friction, Wear and System Stiffness-Part 3: Wear Model," J. Tribology, Vol. 106, pp. 59-64.
- Bengisu, M. T. and Akay, A., 1997, "Relation of Dry Friction to Surface Roughness," ASME J. of Trib, Vol. 119, pp. 18-25.
- Bengisu, M. T. and Akay, A., 1999, "Stick-Slip Oscillations: Dynamics of Friction and Surface Roughness," J. of Acoust. Soc. Am., Vol. 105, pp. 194-205.
- Brockley, C. A. and Ko, P. L., 1970, "The Measurement of Friction and Friction-Induced Vibration - Trans. ASME - pp. 543-549.
- Greenwood, J. A. and Tripp, J. H., 1971, "The Contact of Two Rough Nominally Flat Rough Surfaces," Proc. Instn. Mech. Engrs., Vol. 185, pp. 625-633.
- Ibrahim, R. A., 1994, "Friction Induced Vibration, Chatter, Squeal and Chaos, Part II: Dynamic an Modeling - Applied Mechanics Reviews," Vol. 47, pp. 227-253.
- Ibrahim, R. A. and Rivin, E., 1994, "Friction-Induced Vibration, Part I: Mechanics of Contact and Friction - Applied Mechanics Reviews," Vol. 47, pp. 209-226.
- McMillan, J., 1997, "A Non-linear Model for Self-excited Vibration," J. Sound Vibration, Vol. 205(3), pp 323- 335.
- Mitropolskii, Y.A. and Nguyen, V. D., 1997, "Applied Sympotic Methods in Nonlinear Oscillation," Kluwer, Dordrecht.
- Nayfeh, A. H. and Mook, D. T., 1979, "Nonlinear Oscillations," Wiley, New York.
- Panovko, Y. G. and Gubanova, I. I., 1965, "Stability and Oscillation of Elastic Systems; Paradoxes, Fallacie and New Concepts," Consultants Burea, New York.
- Popp, K., 1992, "Some Model Problems Showing Stickslip Motion and Chaos, Friction-Induced Vibration, Chatter, Squeal, and Chaos," ASME DE, Vol. 49, ASME Design Engineering Division, New York, pp. 1-12.
- Soom, A. and Kim, C., 1983, "Interactions between Dynamic Normal and Frictional Forces during Unlubricated sliding," Transactions of the ASME, J. of Lubrication Technology, Vol. 105, pp. 221-229.
- Tan, X. and Roger, R., 1998, "Simulation of Friction in Multi-Degree-of-Freedom Vibration System," ASME J. Dyn. Syst., Control, Vol. 120, pp. 144-146.
- Thomsen, J. J. and Fidlin, A., 2002, "Analytical Approximations for Stick-slip Vibration Amplitudes," Int. J. of Non-Linear Mechanics, Vol. 38, pp 389-403.
- Tondl, A., 1991, "Quenching of Self-Excited Vibration," Elsevier, Amsterdam.
- Tworzydlo, W., Becker, E. and Oden, J., 1992, "Numerical Modeling of Friction-Induced Vibration and Dynamic Instabilities," App.l Mech. Rev. Vol. 45(7), pp. 255- 274.
- Tzou, K., Wickert, J. and Akay, A., 1998, "In-plane Vibration Modes of Arbitrarily Thick Disks," ASME J. Vibr. Acout., Vol. 120, pp. 384-391.

#### References

Abdo, J. A. and Farhang, K., 2005, "Elastic-Plastic Contact Model for Rough Surfaces based on Plastic Asperity Concept," Int. J. of Non-Linear Mechanics, Vol. 40, pp 495-506.

Abdo, J. A. and Shamseldin, E., 2005, "Modeling of Contact Area, Contact Force, and Contact Stiffness of Mechanical Systems with Friction," ASME Mechanical Engineering Congress Paper no. IMECE2005-82980, Orlando, Florida.

Abdo, J. A., 2005, "Experimental Technique to Study Tangential-to-normal Contact Load Ratio," Tribology Transactions, Vol. 48, pp 389- 403.

Abdo, J. A., 2006, "Modeling of Normal and Tangential Contact Stiffness of Rough Surfaces," Int. J. of Modeling and Simulation, Vol. 26(4), pp 295-302.

Aronov, V., D'souza, A. F., Kalpakjian, S. and Sharper, I., 1984a, "Interaction among Friction, Wear and System Stiffness-Part 1: Effect of Normal Load on System Stiffness," J. Tribol., Vol. 106, pp. 54-58.

Aronov, V., D'souza, A. F., Kalpakjian, S. and Sharper, I., 1984b "Interaction among Friction, Wear and System Stiffness-Part 2: Vibrations Induced by dry Friction," J. Tribology, V. 106, pp. 59-64.

Aronov, V., D'souza, A. F., Kalpakjian, S. and Sharper, I., 1984c "Interaction among Friction, Wear and System Stiffness-Part 3: Wear Model," J. Tribology, Vol. 106, pp. 59-64.

Bengisu, M. T. and Akay, A., 1997, "Relation of Dry Friction to Surface Roughness," ASME J. of Trib, Vol. 119, pp. 18-25.

Bengisu, M. T. and Akay, A., 1999, "Stick-Slip Oscillations: Dynamics of Friction and Surface Roughness," J. of Acoust. Soc. Am., Vol. 105, pp. 194-205.

Brockley, C. A. and Ko, P. L., 1970, "The Measurement of Friction and Friction-Induced Vibration - Trans. ASME - pp. 543-549.

Greenwood, J. A. and Tripp, J. H., 1971, "The Contact of Two Rough Nominally Flat Rough Surfaces," Proc. Instn. Mech. Engrs., Vol. 185, pp. 625-633.

Ibrahim, R. A., 1994, "Friction Induced Vibration, Chatter, Squeal and Chaos, Part II: Dynamic an Modeling - Applied Mechanics Reviews," Vol. 47, pp. 227-253.

Ibrahim, R. A. and Rivin, E., 1994, "Friction-Induced Vibration, Part I: Mechanics of Contact and Friction - Applied Mechanics Reviews," Vol. 47, pp. 209-226.

McMillan, J., 1997, "A Non-linear Model for Self-excited Vibration," J. Sound Vibration, Vol. 205(3), pp 323- 335.

Mitropolskii, Y.A. and Nguyen, V. D., 1997, "Applied Sympotic Methods in Nonlinear Oscillation," Kluwer, Dordrecht.

Nayfeh, A. H. and Mook, D. T., 1979, "Nonlinear Oscillations," Wiley, New York.

Panovko, Y. G. and Gubanova, I. I., 1965, "Stability and Oscillation of Elastic Systems; Paradoxes, Fallacie and New Concepts," Consultants Burea, New York.

Popp, K., 1992, "Some Model Problems Showing Stickslip Motion and Chaos, Friction-Induced Vibration, Chatter, Squeal, and Chaos," ASME DE, Vol. 49, ASME Design Engineering Division, New York, pp. 1-12.

Soom, A. and Kim, C., 1983, "Interactions between Dynamic Normal and Frictional Forces during Unlubricated sliding," Transactions of the ASME, J. of Lubrication Technology, Vol. 105, pp. 221-229.

Tan, X. and Roger, R., 1998, "Simulation of Friction in Multi-Degree-of-Freedom Vibration System," ASME J. Dyn. Syst., Control, Vol. 120, pp. 144-146.

Thomsen, J. J. and Fidlin, A., 2002, "Analytical Approximations for Stick-slip Vibration Amplitudes," Int. J. of Non-Linear Mechanics, Vol. 38, pp 389-403.

Tondl, A., 1991, "Quenching of Self-Excited Vibration," Elsevier, Amsterdam.

Tworzydlo, W., Becker, E. and Oden, J., 1992, "Numerical Modeling of Friction-Induced Vibration and Dynamic Instabilities," App.l Mech. Rev. Vol. 45(7), pp. 255- 274.

Tzou, K., Wickert, J. and Akay, A., 1998, "In-plane Vibration Modes of Arbitrarily Thick Disks," ASME J. Vibr. Acout., Vol. 120, pp. 384-391.