A mechanical system sliding on a moving surface with Coulomb friction is a rich area for study. Despite much past work, there is still something to be gleaned by closed-form expressions for the system behavior. Consider a spring-mass-damper system (K, M, C) with deflection x, base moving in the +x direction at velocity V, sliding friction F, and sticking friction Fs. An initial condition of x0 at rest can be considered general because all possible motions will follow. Two dimensionless schemes are used. For the abstract, we focus on the scheme normalized by x0 with variable z = x/x0, τ = (ωnt, ωn = [K/M]1/2, ζ = c/[2(KM)1/2], ν̄ = V / (ωnx0), f = F/(Kx0), and fs = Fs/(Kx0). Since the solution is piecewise linear, this allows closed-form results.
For this abstract, we consider C = 0, Fs = F. (Other cases are in the paper.) There are three critical ground speeds. The first, ν̄d, is when sticking first occurs (at z = f). At the second speed, ν̄c, sticking has moved to z = −f. Thereafter, the sticking point again increases, reaching z = f at the third speed, ν̄b. For higher ν̄, there is no sticking.
In this paper, closed form expressions are presented for the three critical speeds:(1)ν¯d=[(1+3f)(1−5f)]12,ν¯c=[(1+f)(1−3f)]12,ν¯b=1−f
These formulas are verified by numerical simulation. The insight is that there is a limited range of f for which certain critical points can be reached. Thus, 0 < f < 1/5 has different dynamics than 1/5 < f < 1/3.
Formulas are also derived for the second maximum of z, which gives an indication of decay or growth of the system. For example, with f = fs and C = 0, the second maximum z with f < 1/5 is:(2)zmax=f+((1−f)2−ν¯2−4f)2+ν¯2ν¯d<ν¯<ν¯czmax=ν¯+fν¯c<ν¯<ν¯bzmax=1ν¯>ν¯c
Formulas will also be given for the times at which the maximum occurs and the times at which a transition occurs from static to sliding for all cases.
Ternary Logic of Motion to Resolve Kinematic Frictional Paradoxes
