On the Stability of the Uniformly Accelerated Motion of an Elastic Cylinder on an Inclined Flat Surface

Abstract

Rolling with the slipping of an infinite elastic cylinder with a horizontal axis in the gravity field on a flat surface made of the same material is considered. The angle between the nondeformed boundary of the surface and the horizon is nonzero. Based on solution of the corresponding contact problem in Carter’s quasi-static formulation, the reaction of the flat surface in dynamic equations is determined. It is shown that if the tangent of the inclination angle does not exceed the product of the friction coefficient and the coefficient that depends on the elastic properties of the material and the mass distribution, motion with constant relative slip δ and a constant acceleration of the center of mass is possible; there is a region in the contact area where the materials stick together. This motion is found to be asymptotically stable with respect to δ. The dependence of the cylinder-axis acceleration on the mechanical properties of the material and the friction coefficient is examined. The results obtained are compared with those of the classical problem of rigid-disk motion along an inclined straight line.

APPENDIX 1

Stability of uniformly accelerated motion. We examine stability of solution (5.2) with respect to variable ${{\delta }}$ [19]. We introduce the variables:

$$x = {{\delta }} - {{\delta }}\text{*},\quad y = \tilde {V}.$$

The equations of perturbed motion have then the form:

$$x' = \frac{{G(x,{\text{sign}}(y))}}{y},\quad y' = F(x,{\text{sign}}(y)),$$

were $F\left( {x,\operatorname{sign} \left( y \right)} \right) = \tilde {Q}\left( {x + \delta \text{*},\operatorname{sign} \left( y \right)} \right) + \tan \varphi$,

$$G\left( {x,\operatorname{sign} \left( y \right)} \right) = - (1 + {{j}^{{ - 1}}})\tilde {Q}\left( {x + \delta \text{*},\operatorname{sign} \left( y \right)} \right) - \tan \varphi $$

$$ - \;(\tilde {Q}\left( {x + \delta \text{*},\operatorname{sign} \left( y \right)} \right) + \tan \varphi )\left( {x + \delta \text{*}} \right).$$

Since for values close to unperturbed motion $\tilde {V}\left( 0 \right) > 0$, $y ' > 0$, we have $y\left( {{\tau }} \right) > 0$ for all ${{\tau }} > 0$. Consequently,

$$y ' = F\left( {x,1} \right) > 0,\quad x ' = \frac{{G\left( {x,1} \right)}}{y},\quad G\left( {0,1} \right) = 0,\quad F\left( {0,1} \right) = w\left( {\delta \text{*}} \right).$$

(A.1)

We consider the auxiliary equation

$$\bar {x} ' = \frac{{G\left( {\bar {x},1} \right)}}{{F\left( {0,1} \right)\tau + {{y}_{0}}}}.$$

(A.2)

This equation has the solution $\bar {x} = 0$. We have in the vicinity of this solution

$$G\left( {\bar {x},1} \right) = {{k}_{1}}\bar {x} + O({{\bar {x}}^{2}}),\quad {{k}_{1}} = \frac{{\partial G}}{{\partial \bar {x}}}\left( {0,1} \right).$$

We now prove that the coefficient ${{k}_{1}} < 0$. Indeed,

$${{k}_{1}} = \frac{{\partial G}}{{\partial x}}\left( {0,1} \right) = - (1 + {{j}^{{ - 1}}} + \delta \text{*})\frac{{\partial \tilde {Q}}}{{\partial \delta }}\left( {\delta \text{*},1} \right) - w\left( {\delta \text{*}} \right).$$

We note that since $- \mu {\text{/}}\left( {2\kappa } \right) < {{\delta }}\text{*} < 0$, it follows from Eq. (4.3) that

$$\frac{{\partial \tilde {Q}}}{{\partial \delta }}\left( {\delta \text{*},1} \right) = 4\kappa \left( {1 + \frac{{2\kappa \delta \text{*}}}{\mu }} \right) > 0.$$

In addition, $w\left( {{{\delta }} \text{*}} \right) > 0$, consequently, ${{k}_{1}} < 0$.

Disregarding the terms on the order of $O({{\bar {x}}^{2}})$, we obtain from Eq. (8.2) an equation with separable variables:

$$\bar {x} ' = \frac{{{{k}_{1}}\bar {x}}}{{{{k}_{2}}\tau + {{y}_{0}}}},\quad {{k}_{1}} < 0,\quad {{k}_{2}} = w\left( {\delta \text{*}} \right) > 0.$$

We arrive then at:

$$\ln (\bar {x}) - \ln ({{\bar {x}}_{0}}) = \frac{{{{k}_{1}}}}{{{{k}_{2}}}}\left( {\ln (\tau + {{y}_{0}}{\text{/}}{{k}_{2}}) - \ln {{y}_{0}}{\text{/}}{{k}_{2}}} \right),$$

whence

$$\bar {x} = {{\bar {x}}_{0}}{{\left( {\frac{{{{k}_{2}}{{\tau }} + {{y}_{0}}}}{{{{y}_{0}}}}} \right)}^{{{{k}_{1}}/{{k}_{2}}}}}.$$

The index of power is negative implying that the $\bar {x}$ function defined by auxiliary equation (8.2) decreases as a power function of time. Since majorizing equations can be derived for initial system (8.1) based on the auxiliary equation, solution (5.2) of the initial system is asymptotically stable with respect to variable ${{\delta }}$.