lab-inf-4_tasks / 2000_521

Jianqun Gao

Si C. Lee

Department of Mechanical Engineering,

The Ohio State University,

Columbus, OH 43210

Xiaolan Ai

Harvey Nixon

The Timken Company,

Canton, OH 44706

An FFT-Based Transient Flash Temperature Model for General Three-Dimensional Rough Surface Contacts

A transient ¯ash temperature model was developed based on a Fast Fourier Transform method. An analytical expression for the heat partition function was obtained. Together, these substantially increase the speed of ¯ash temperature calculations. The effect of surface topography on the ¯ash temperature was examined. According to the simulation results, the surface with a longitudinal roughness produced a noticeably higher ¯ash temperature than the surface with a transverse roughness. The simulation results also indicate that there is a signi®cant cross-heating activity between the asperities; the temperature pro®les appeared surprisingly gradual although their contact pressures had extremely sharp peaks. @S0742-4787~00!04002-9#

1 Introduction

The friction at sliding contact interfaces where two surfaces come together generates heat. Most of this heat is conducted away through local rubbing asperities. The study of asperity ¯ash temperature is important in boundary lubricated contacts where the hydrodynamic lifting action of the lubricant is negligible. This temperature may be responsible for scuf®ng ~Blok @1#!. In partialehl contacts, the ¯ash temperature reduces the concentration of the lubricant adsorbate that protects the surfaces from scuf®ng while the lubrication pressure increases the adsorbate concentration ~Lee and Cheng @2#!.

The total asperity temperature consists of the bulk temperature and the ¯ash temperature. The bulk temperature is easily measured while the ¯ash temperature is typically calculated since it does not readily lend itself to measurement. Much work has been done in the past for the prediction of the ¯ash temperature. Blok @3# ®rst proposed the concept of ¯ash temperature and derived simpli®ed formulas for the maximum temperature rise on moving surfaces. Jaeger @4# formalized the mathematical models for the ¯ash temperature on a semi-in®nite medium for moving uniform rectangular heat sources. Many other ¯ash temperature models have appeared in the literature since. These studies extended Jaeger's theory to various heat source shapes and to multiple asperity contacts based on steady-state conditions ~Archard @5#, Francis @6#, Kuhlmann-Wilsdorf @7#; Lee and Cheng @2#, Tian and Kennedy, @8#, Boes and Moes @9#!. The transient ¯ash temperature models include the works by Ling and Pu @10# who simulated the dry contact of a square protrusion sliding against a smooth plane and by Lai and Cheng @11# who derived a numerical solution for the lubricated rough surface sliding on a ¯at. Qiu and Cheng @12# improved the numerical ef®ciency of Lai and Cheng's model, resulting in a noticeable gain in the calculation speed.

The simulation of ¯ash temperature in real surfaces requires an even greater ef®ciency. This is especially true for when the asperity contact footprint changes with time, as in real surface contacts. The required numerical integration for temperature in the space domain can be performed more ef®ciently in the frequency domain. By taking the Fast Fourier Transform ~FFT! and its inverse ~IFFT!, its computing time can be signi®cantly reduced. Ju and Farris @13# and Stanley and Kato @14# demonstrated this ef®ciency

Contributed by the Tribology Division for publication in the JOURNAL OF TRI- BOLOGY. Manuscript received by the Tribology Division February 5, 1999; revised manuscript received September 21, 1999. Associate Technical Editor: T. Lubrecht.

in dry contact simulations of rough surfaces. Ju and Farris @15# also derived analytical expressions for the thermoelastic solutions for an arbitrary moving heat source in the frequency domain.

In this paper, a general ¯ash temperature model is developed for the three-dimensional rough surface contacts based on the FFT method. The time and spatial dependent partition function for the heat ¯ux was solved using a transform function. The present method signi®cantly reduces the computational time required in simulations of ¯ash temperature of real surface contacts.

2 Model

The friction generates heat at the asperity contact interface. The rate of frictional heat generation is given by

where m is the coef®cient of friction, pa , the asperity contact pressure, and Vs , the sliding speed.

Figure 1 is a schematic representation of a rough surface contact. The highest peaks of the asperities within the nominal contact area support the load. Since the heat ¯ux is proportional to the contact pressure, the pressure must be known at every instance of time. In Fig. 1, ~x,y,z! are ®xed Cartesian coordinate axes whose origin is at the center of the nominal area. The two interacting bodies slide along the x-axis with velocities V1 and V2 . A single heat source qdx8d y 8dt8 emitted at (x8, y 8,0,t8) with no heat loss at the boundary divides itself between the two surfaces according to the heat partition function @12 f (x8, y 8,t8) #q(x8, y 8,0,t8) for surface 1 and f (x8, y 8,t8)q(x8, y 8,0,t8) for surface 2.

The temperature rise of the body 1 due to the moving point heat

The corresponding temperature rise of the body 2 is: