Approximate closed-form equations are derived for normal and tangential contact forces of rough surfaces in dry friction. Using an extension of the Greenwood and Tripp model, in which the derivations permit asperity shoulder-to-shoulder contact and viscoelastic asperity behavior, mathematical formulae are derived for normal and tangential components of the contact force that depend not only on the proximity of the two surfaces but also the rate of approach and relative sliding. A statistical approach is forwarded in which dependence of the asperity tangential contact force on relative tangential velocity of two asperities can be cast as corrective factors in the mathematical description of tangential force. In this regard two corrective coefficients are derived: force directionality corrective coefficient and force-velocity directionality corrective coefficient. The results show that for a moderate to high load ranges the contact force can be analytically described to within 20 percent accuracy of that from a numerical integration of the contact equations, well below the uncertainties due to surface profile measurement.
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