Identification of Friction / Vibration Interaction between Solids

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 microand macroscale that yield negative slope in friction force/sliding speed relation, a well known source of dynamic instability.


Introduction
Contact phenomena involving deformable bodies thrive in industry and everyday life.Many authors have considered the mechanical system vibration due to surface interaction and examine the surface theories with the presence of friction (Brockley, et al. 1970;Soom and Kim, 1983;Ibrahim, and Rivin, 1994;Ibrahim, 1994;Aronov et  « «M MÉ Éà àØ ØŸ ŸG G ä äG GO Oô ôØ ØŸ ŸG G .á°ùeÓàŸG ΩÉ°ùL'G ÚH »µ«eÉaej~dG πYÉØàdG -RGõàg'G h ∑ÉµàM'G ÚH πYÉØàdG :á (Panovko, et al. 1965;Nayfeh and Mook, 1979;Mitropolskii and Nguyen, 1997;Popp, 1991;Tondl, 1991;McMillan, 1997 andThomsen andFidlin, 2002), and it will be in this present study.The physical contact model and governing equations with the inclusion of the viscoelastic properties are presented in this paper to study the dynamic effects of the surface irregularities as a result of mechanical interactions.
Several analytical or computational methods have been applied to solve the dry friction problems to increase the understanding of friction and vibration phenomenon in mechanical systems.Some of the elastic, elastic-plastic, and plastic contact models are reformulated to estimate a normal and tangential contact force, contact area and contact stiffness.The elastic and the elastic-plastic models developed by Abdo and Farhang (2005) and by Abdo and Shamseldin (2005) in addition to the research results pertinent to the estimation of contact stiffness, Abdo (2006) and Abdo (2005) are used in this study along with the addition of the viscoelastic effects of contact.In this work, 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.There is no assumption of friction/velocity relation made in the formulation of the governing differential equations.The elastic and ratedependent contact force components between rough surfaces are presented to obtain the equations governing the vibration response of mass-spring-damper system.

The Dynamic Model
The basic assumptions of the elastic model along with the shoulder-to-shoulder contact model (Abdo and Farhang 2005), and Abdo 2005 and(Abdo andShamseldin, 2005) are used in this study to account for the elastic contribution of rough surfaces.Elastic contact model has viewed the surface irregularities as identical asperities with spherical summits differing only in their heights that can be measured from the mean plane separation of two rough surfaces.The misaligned asperities permit shoulder-to-shoulder contact.When two solid bodies are brought into contact, the real contact area will only be a fraction of the apparent macroscopic contact area.The introduction of contact slope allowed the consideration of normal and tangential components of the force.The governing equation of the tangential motion can be expressed as

Referring to the free body diagram shown in
no where F na is the normal force applied to mass M. F nc is the normal contact force between the two contacting rough surfaces.It is the scalar sum of the resisting and assisting viscoelastic contact components (3) In this section, we are interested in developing a mathematical f ormulation relating the tribological events at microscopic and the macroscopic scales vibration response of a mass on a moving belt.An example of the physical system is shown in Fig. 1 which consists of a mass m on a belt that moves at constant speed, y .The mass is a rigid body, at time t positioned at x(t) in a fixed frame of coordinates.It is subjected to a normal applied static load F na , linear spring-loading force Kx, plus damping force Cdx / dt.The h 0 indicates the ini tial mean plane separation of the two contacting rough surfaces corresponding to the normal applied static load, F na .The model doesn't assume a friction coefficient; therefore, friction is instead obtained as a result of system's dynamic response that in cludes time -dependent forces at the contact.The tangential -to-normal contact load ratio, the classical definition of the friction coefficient, is shown to depend on system response that in turn is dependent on both the structural and contact characteristics.
The terms ne F − and ne F + denote, respectively, the resisting and assisting elastic normal contacts at the asperity interference slopes (Bengisu et al. 1997;Bengisu et al. 1999), as shown in Fig. 2. It remains a challenge to determine whether there is a net elastic influence on slidi ng surfaces.As shown in Fig. 2 , when in static equilibrium, without the presence of an applied tangential force, the contact force due to negative (4)

Approximation of Elastic Contact Forces
The elastic normal contact force per unit area is the sum of the resisting and assisting elastic normal contacts at the asperity interference slopes (5) The analytical solution of the force in Eq. ( 5) may be approximated as the product of the two terms as follows: where D ne and f ne , are a function of surface properties and a function of dimensionless mean plan separation between two rough surfaces, respectively.Two integrations are involved in the contact force calculations.The function D ne is defined in Abdo (2006) as where h is the normalized mean plane separation.The f ne function consists of double integral function over r and s domains.The integration of the contact function occurs contact slope will be equal to that due to the positive slope; therefore, the net tangential force on a surface is zero.In the presence of an applied tangential force, the equilibrium condition dictates that the net tangential force to be the equilibrating force.Bengisu and Akay (Bengisu et al. 1997) for example assume that the net elastic tangential force, ie.those due to positive slopes minus those due to negative slopes, is negligible.We have chosen instead to include the factor of the friction due to the positive or resisting slopes.In the presence of an applied tangential load that is within the bounds of the static friction force, the statistical summation of asperity contacts corresponding to positive slopes is no longer equal to that for contacts with negative slopes.
According to (Bengisu and Akay 1997) the assisting and resisting contacts balance in the case of sliding, with the expected conclusion that friction force is a result of mainly adhesive forces.It is expected that as tangential applied load is increased an d the tangential contact loads due to positive -slope contacts continue to rise while those due to negative -slope contacts decrease.In the limiting case, or the so -called maximum static friction, it is postulated that the load is completely supported by p ositive-slope contacts.The views forwarded by Bengisu and Akay and postulated herein perhaps represent two extremes of what may actually take place in a frictional contact.In the former, the friction is assumed to be mainly due to the adhesive forces whereas in the latter the friction force is assumed to be borne solely by mechanical interaction of roughness.In this way we can obtain a generic formulation that can be easily adapted to any assumption.The scenario for contact of asperities when a maximum force occurs is shown in Fig. 3 .
Likewise, nv F − and nv F + are those fo rces due to viscous effects, ie. the rate dependent effect of interference at positive and negative asperity slopes.The elastic and viscous terms ar e a consequence of a viscoelastic contact, but they are defined separately here for the purpose of implementing the approximate analytical description of each term derived later.The tangential force, F t , in Eq. ( 2) exerted through the frictional contact on mass M. It is the scalar sum of the elastic interactions of the collection through both resisting and assisting viscoelastic contact components .
over the range of asperity-offset misalignment, referred to as r-integration.This integration is to account for the total expected interference.The height sum of the rough surface, s, is represented by a Gaussian probability density function, φ(s).Hence, in relating micron-scale events to the resulting macro-scale expectations, two integrations must be performed: firstly, the integration over possible asperity offset and secondly the statistical integration over all possible asperity height sums, s-integration, by implementing a Gaussian probability function.The resulting integral forms describing an elastic contact of rough surfaces are analytically non-integrable.Hence, their solution must be obtained numerically.This fact imposes serious limitations on their use in a dynamic analysis of a mechanical system.To circumvent this shortcoming, we developed approximate analytical functions for viscoelastic contact of rough surfaces.Using an interactive search involving dynamic plotting of the function, f ne is approximated using a considerably simpler function.The result is an approximate function that depends on two independent variables, β, h.Using the dynamic plotting technique, f ne can be approximated by a product of exponential function of h and a polynomial function of β.The piecewise analytical solution of f ne is obtained at three different intervals as shown in the following equations: The analytical solution of elastic tangential contact per unit area is developed.The analytical approximation can be described as the product of two functions: a function of surface properties D te and a function of normalized mean plane separation, h, between two rough surfaces, f te , as follows: Note that the constants D ne and D te are identical.f te is expressed as (13) Using dynamic fitting technique, the piecewise analytical solution of the function f te is approximated in closedequations for two intervals of h as (14) (15)

Approximation of Viscoelastic Contact Forces
Therefore, a modified form of Hertz contact results in which the modulus operator is used instead of the usual elastic modulus (16) where ω the interference between the two asperities and is derived in (Abdo 2005;and Abdo et al. 2005) as ( 17) Substitution of modulus operator in the Hertz equation of contact and subsequent expansion revealed two forces.One is an elastic force having the form encountered previously, and a second is viscous force defined by ( 18) 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.To develop an expression for the normal and tangential components of the viscous forces, the formulations of low to moderate contact force between interacting asperities developed in Abdo (2005) and verified in Abdo (2006) are considered.The interacting asperities are not assumed to occur only at the asperity peaks, thus allowing the possibility of oblique contacts wherein the local contact surfaces are no longer parallel to the mean planes of the mating surfaces.The normal component of the viscous force can be written so as to include the oblique contact ( ) ( ) 114 96 320 01 3 0 4 5 ( ) 5 3001 0 2396 0 000598 1 694 0 0847 0 00029 0 27 192 79 91601 0 02498 2 7 4 5 The viscoelastic model is developed using the Kelvin-Voigt's constitutive equation relating stress to strain.In Kelvin -Voigt model, the composite modulus of the material ′ E and the loss modulus E v are used to introduce the viscoelastic modulus differential operator Since the interference, ω, depends on the normalized mean plane separation, h, and the tangential offset between two asperities, r, its derivative with respect to time may be written as (21) Utilizing Eq. ( 18) then ( 22) The total expected contact force per unit area introduced in the (Greenwood and Trip 1971) is employed to develop the total expected viscous normal contact force per unit area as follows (23) Utilizing Eq. ( 19) through Eq. ( 22), then Eq. ( 23) may be written as The total expected viscous tangential contact force may be written as (30) Utilizing Eqs. ( 20) through ( 22) and Eq. ( 29), then Eq. ( 30) may be written as (31) where α s is the interference slope defined by two points of interactions with respect to the mean plane.To simplify Eq .( 19), we co nsider first the derivation of the rate of change of the interference function, ω is the loss factor for the friction film or material and σ is the standard deviation of the surface heights.The solution for the viscous contact normal force consists of two parts, involving the rate change of the normalized mean plane separation, h , and the rate of change of the normalized tangential misalignment, ie. the sliding speed, = − r x y.The total expected viscous contact nor mal force, due to the interaction on both shoulders, is ( ) where nv D is a constant term associated with h and r that depends on the surface parameters and material properties.It is define as follows As before, the solution for the viscous contact tangential force per unit area consists of two parts involving h and r .Therefore, Eq .( 31) may be written as (32) where D tv is identical to D nv .f tvh and f tvr are the analytical functions for the normalized viscous tangential force.Using the dynamic plotting technique, the piecewise analytical solutions for f tvh and f tvr can be approximated by a product of exponential function of h and a polynomial function of β as follows ( 33) and ( 34)

The Model Analysis and Simulation
According to the dynamic model described in  37) and ( 38) govern the motion of mass M in the normal and tangential directions.The equations tie the vibration behavior of the system to both structural and contact properties of the friction interface.
The dynamic model was developed based on several assumptions to simplify the study.The elastic normal and tangential contact components are due to both positive and negative interference slopes of an asperity as shown in Fig. 2 and are added algebraically to provide their combined effect.
There may or may not be a net resisting elastic tangential force when the two surfaces in relative sliding contact.When this force exists, it is described as a fraction of the maximum static friction force that is due to only positive slopes.
The viscoelastic tangential forces are only produced by the resisting contacts.

(
) where o h is the initial mean plane separation of the contacting rough surfaces due to an applied normal force and ( ) ĥ h σ is the dimensionless displacement in the normal direction.

(
) When h is negative, all viscoelastic components exist and are additive for normal motion.
The responses of the dynamic model are studied at different belt speeds and a dimensionless average radius of a summit curvature, , β to investigate the roughness effect.The equation of motion was solved using a Matlab Simulink program.Unlike in other publications the friction force is accounted for as a result of interaction of the rough surfaces and response of the system.Therefore, the tangential -to-normal contact ratio is a time -dependent variable based on structure and contact characteristics of the system.Table 1 summarizes the parameters used.The simulations include moderate applied load condition corresponding to a high initial separation, h 0 = 3.5.The total range of dimensionless mean plane separation is from 0 to 4.5 for contact force functions.The β values considered in the simulatio n are 100 and 140 to ensure low plasticity index so that the asperity deformation will remain predominantly elastic.For cases performed but in order to reduce the length of the article only the results of case 1 will be presented.

Table 1. Parameters from a steel sample
The analytical solutions of normal to tangential contact forces presented in this work are applicable to dimensionless mean plane separation, h = 0 to 4.6 and the dimensionless average radius of asperity summit curvature, β = 10 to 140.T he approximate analytical solutions developed included the normal and tangential contact forces due to viscoelastic effects.The purely elastic term or rate dependent term is presented as a product of two functions.The first corresponds to a constant te rm for a given surface and the second depends on the separation of mean surfaces as well as tangential asperity offset.It suffices to evaluate the approximate equations with respect to these functions.Therefore the evaluations will only involve The results for case (1) are depicted in Fig. 7 through Fig. 10. Figure 7a shows the trajectory of tangential motion, tangential speed versus tangential displacement of the mass m.The zoom-in view (Fig. 7b) shows decreasing spiral, which represents a mass response of decaying oscillation corresponding to a stable behavior.The trajectories in Fig. 7 and the time history in Fig. 8 indicate an initial decay followed by a growth in vibration up to a level at which the vibration is sustained.
In case of the normal motion of the mass the structural decay is absent consistent with the system's depiction in Fig. 1 in which no structural stiffness or damping exists for the vibration of mass M along the normal to the platform.As shown in Fig. 9(a,b) the vibration is grown and sustained at a value of 0.005 mm.The friction coefficient defined as the tangential to normal friction shown in Fig. 10 depicts a steady state value of 0.026 for case 1.

Conclusions
This paper has addressed the dynamic effects of the surface irregularities (roughness) as a result of mechanical interactions.We have exhibited the cross-influence of tribological interactions that occur at the micron-scale and the dynamics of a mechanical system involving macroscale events.The utility of the contact theories in this work was demonstrated through the dynamic analysis of a simple mechanical system.We have considered a mechanical system comprising a spring-damper-mass and a platform in relative tangential and normal movements while maintaining frictional contact between mass and platform rough surfaces.
The results pertaining to the prediction of contact forces were used along with the addition of the viscoelastic effects of contact.The approximate analytical solutions were obtained for normal and tangential contact components in viscoelastic contact.The term viscoelastic was separated into components of elastic and rate-dependent interactions.It was shown that the formulation of contact forces allow the inclusion of frictional contact without the necessity of including the friction/velocity relation phenomenologically.No presumption of a friction coefficient has been made.The friction force is accounted for as a result of interaction of the rough surfaces.Rather the tangential (friction) force has been a result of the consideration of contact at the micron-scale level between two asperities on the surfaces and the statistical summation of this interaction to obtain the macrolevel expectation functions.This led to a set of nonlinear ordinary differential show good agreements between the original and the approximation functions throughout the ranges of .β The relative percentage errors are between 9% and 10%.The comparisons of the original and viscous functions for the same values of β are also performed.The estimation of viscous the functions shows a relative percentage error between the original and the approximation functions is less than 20%.This is below the inherent uncertainty involved in surface measurements that can climb to as much as 50%.Comparisons between the original integral function of the elastic normal contact force, ne f , as depicted in Eq. 8 and its piecewise analytical solutions, and ne f 3 for the ranges of h = 0 to 1.5, h = 1.5 to 3.0 and h = 3.0 to 4.5, respectively, at β = 100 and 140 are shown in Fig. 4.
As seen in the figures, the comparisons show good agreement between the original and the approximate functions for β = 100 and 140.The relative percentage error between ne f and ne ne f , f 1 2 and ne f 3 is 9% at β = 100 and 10% at β =140.The comparison between the approximate functions te f and its analytical solutions and tvh f and its analytical solutions for the ranges of h = 0 to 1.5, h = 1.5 to 3.0 and h = 3.0 to 4.5, respectively, at β = 100 and 140 are shown in figures 5 and 6.The relative percentage error between the original and the approximation functions for both functions are 17% and 19%, respectively.equations that directly relate the vibration of the system to the surface parameters, mechanical system parameters and physical parameters.
The study has shown the following: • The tangential trajectories of the four cases showed initial decay dominated by structural damping followed by a growth in vibration up to a level at which vibration is sustained that is due to the interaction of surface asperities of the moving platform and those of the structure mass.• The vibration of the normal direction of the four cases grow to a level at which it is sustained that is due to frictional interaction between the platform surface roughness and that of mass M and the absence of the structural decay.• The low tangential to normal contact load ratio of the four cases is due in part to the relatively low surface roughness in the present study in order to ensure the predominance of elastic interaction at the contact.

Figure 1 .
Figure 1.Schematics of block on a moving platform and its free-body diagram h o Figure 2. Asperity shoulder contact at positive and negative interference slopes and f nvr are the analytical functions for the normalized viscous normal force.Using the dynamic plotting technique, the piecewise analytical solutions for f nvh and f nvr can be approximated by a product of exponential function of h and a polynomial function of β as follows

Fig
ne te nvh tvh nvr f , f , f , f , f and tvr f , not the constant terms D ne , D te , D nv , and D tv .The comparison between the original integral function of the elastic normal and tangential contact forces, ne f , te f as depicte d in Eqs. 8 and 13, respectively, and their piecewise analytical solutions ne ne f of h = 0 to 1.5, h = 1.5 to 3.0 and h = 3.0 to 4.5 at diff erent values of β are performed.The comparisons at β = 10, 40, 75, 100, 125, and 140 Figure 4. Elastic normal force at β = 100 & 140