A two-parameter friction model

Rolling resistance modelling in the Celtic stone dynamics

“…Rolling friction is there modelled as a resistance against motion of the area of deformation over the body [11] and, ignoring the problem of rotation of the elliptical region, is consistent with the results obtained under the assumption of the linear hysteresis [12][13][14][15]. Karapetyan [16] proposed a two-parametric model of friction forces for spherical contact between a ball and planar base being a result of deformations of the contacting bodies. This model joins three different kinds of friction (Coulomb, Contensou and rolling friction) and can be seen as a development of Zhuravlev model.…”

Dynamics of a rolling and sliding disk in a plane. Asymptotic solutions, stability and numerical simulations

“…The possibility of detachment of the body from the plane and aspects of the friction law were discussed in [3,11]. There are also models of friction when the contact between the rigid body and the plane is no longer a point but a small region and the friction force is obtained by integration over the whole contact region, see, e.g., [6,14,26].…”

The interconnection of sliding and rolling in the problem of the motion of a homogeneous sphere on a rough horizontal plane

“…Unlike Zhuravlev's assumptions, 1,2 we shall assume that the interaction of the sphere with the plane is described by the friction model proposed by Karapetyan. 3,4 , We will assume that e 1 , e 2 , e 3 is a right-handed, orthonormalized moving frame of reference such that the unit vector e 1 is directed along the sliding velocity, u = ue 1 , of the sphere (that is, along the velocity of the point of contact of the sphere with the support plane), the unit vector e 2 is orthogonal to the sliding velocity u and, like e 1 , lies in the horizontal plane, and the unit vector e 3 is directed along the ascending vertical, u s = u + [, ae 3 ] is the velocity of the centre of mass of the sphere and = 1 e 1 + 2 e 2 + 3 e 3 is the angular velocity of the sphere.…”

“…The investigation of the dynamics of the sphere for the motions (3) reduces to an analysis of the system of equations (4) where 6 (5) k > 0 is the coefficient of friction and ∈ [0,1] and ∈ [0,1] are the parameters of the friction model ( ≤ ). Theorem.…”

“…Using formula (5), we will prove that the sliding velocity u and the angular rolling velocity of the sphere 2 simultaneously vanish. To do this, we write the system of equations (4) in the form of the equation and change from the variables u and w to the variables z = u/w and =−ln[w/w(0)]. As a result, we obtain (6) We now consider the function (7) It is well known that a third order polynomial has three real roots if and only if its discriminant is positive.…”