Effect of contacting bodies’ mechanical properties on the dynamics of a rolling cylinder

Abstract

This paper investigates the influence of the material properties on the deceleration dynamics of a deformable cylinder rolling with slipping on a half-space of the same material. The interaction of the cylinder and the half-space is described by the 2D quasistatic contact problem of viscoelasticity (Goryacheva: J Appl Math Mech 37(5):877–885, 1973; Contact mechanics in tribology. Kluwer, Dordrecht 1998) which includes as limiting cases the absolutely rigid and elastic materials. Full dynamical analysis of the problem including the phase portrait, the dependence of the deceleration distance on the mechanical properties of the contacting bodies and on the friction coefficient is provided. The qualitative features of deceleration are justified by asymptotic analysis.

Appendix

Statement 1

(full dissipativity). The kinetic energy of the cylinder

$$\begin{aligned} T(\tilde{V},\tilde{\omega }) = \frac{\tilde{V}^2}{2} + j\frac{\tilde{\omega }^2}{2} \end{aligned}$$

(14)

decreases in time for all the motions except for full rest ($\tilde{V}= 0$ and $\tilde{\omega }= 0$).

Proof

Using Eq. (5), the time derivative of the kinetic energy on the solutions $\tilde{V}(\tilde{t})$, $\tilde{\omega }(\tilde{t})$ can be expressed in two equivalent forms:

$$\begin{aligned} \frac{\mathrm{d}T}{\mathrm{d}\tilde{t}}= & {} \gamma \left( \tilde{Q}- \frac{\tilde{M}}{2}\right) \tilde{V}- \gamma \left( \tilde{Q}+ \frac{\tilde{M}}{2}\right) \tilde{\omega }= -\gamma \left( \Big (\tilde{Q}+\frac{\tilde{M}}{2}\Big )\tilde{V}\delta +\tilde{M}\tilde{V}\right) , \end{aligned}$$

(15)

$$\begin{aligned} \frac{\mathrm{d}T}{\mathrm{d}\tilde{t}}= & {} -\gamma \left( \tilde{Q}\tilde{V}\delta +\frac{\tilde{M}}{2}\tilde{V}(2 + \delta )\right) . \end{aligned}$$

(16)

Based on the calculation results shown in Fig. 2, in the full slipping regime $|\delta |\ge \delta _\alpha ^*(\tilde{V})$, we consider the term $\tilde{Q}= \mu \,\mathrm {sign}(\tilde{\omega }-\tilde{V})$ as the leading one in Eq. (5). Hence, $|\tilde{Q}|>|\tilde{M}|$ and $\mathrm {sign}\Big (\tilde{Q}+\tilde{M}/2\Big ) =\mathrm {sign}\tilde{Q}$. Besides, we have (see Sect. 3)

$$\begin{aligned} \tilde{Q}\tilde{V}\delta = \mu |\tilde{\omega }-\tilde{V}|>0\quad \text {and} \quad \tilde{M}\tilde{V}>0\ \text {if}\ \tilde{V}\ne 0. \end{aligned}$$

Therefore, from (15) we get $\dfrac{\mathrm{d}T}{\mathrm{d}\tilde{t}}<0$ for $\tilde{V}\ne 0$ (since $|\delta |\ge \delta _\alpha ^*(\tilde{V})$, then $\tilde{\omega }\ne \tilde{V}$).

If $|\delta |<\delta _\alpha ^*(\tilde{V})$, then $2+\delta >0$ and from (16), we get again $\dfrac{\mathrm{d}T}{\mathrm{d}\tilde{t}}<0$ for $\tilde{V}\ne 0, \tilde{\omega }\ne 0$. Besides, we notice that $\frac{\mathrm{d}T}{\mathrm{d}\tilde{t}} = 0$ if and only if $\tilde{\omega }= 0$ and $\tilde{V}= 0$ simultaneously. $\square$

Statement 2

(Global asymptotical stability of the equilibrium). There exists a unique asymptotically stable fixed point $\tilde{V}= 0$, $\tilde{\omega }= 0$ of Eq. (5); all solutions of these equations tend to this point: $\tilde{V}(\tilde{t})\rightarrow 0,\tilde{\omega }(\tilde{t})\rightarrow 0$.

Proof

follows from Barbashin–Krasovskii–LaSalle principle [31, 32] with a full energy of the system as Lyapunov function. $\square$

Statement 3

(slow manifold). For any initial condition, there exists an instant $\tilde{t}^*$ such that for any $\tilde{t}>\tilde{t}^*$, the phase curve goes in ${\mathcal {O}}(\kappa ^{-1})$—neighborhood of the slow manifold that is defined by the equation

$$\begin{aligned} \tilde{Q}(\tilde{V},\tilde{\omega }) =-\frac{1-j}{1+j}\cdot \frac{\tilde{M}(\tilde{V})}{2} . \end{aligned}$$

(17)

Thus, the estimate

$$\begin{aligned} \tilde{Q}(\tilde{V}(\tilde{t}),\tilde{\omega }(\tilde{t})) =-\frac{\tilde{M}(\tilde{V}(\tilde{t}))}{2}\frac{1-j}{1+j}(1 + {\mathcal {O}}(\kappa ^{-1})) \end{aligned}$$

(18)

holds true on the solution $\tilde{V}(\tilde{t})$, $\tilde{\omega }(\tilde{t})$ of Eq. (5) for $\tilde{t}>\tilde{t}^*$.

Proof

Let us consider dynamics of the system in the variables

$$\begin{aligned} C = \tilde{V}+j\tilde{\omega },\quad V_s = \tilde{\omega }-\tilde{V}, \end{aligned}$$

(19)

where C is the value of the first integral for rigid and elastic case defined in Eq. (11) and $V_s = \delta \tilde{V}$ is the slippage. Then, the governing equations equivalent to Eq. (5) have the form

$$\begin{aligned} \frac{\mathrm{d} C}{\mathrm{d} \tilde{t}}= & {} -{\gamma } {\tilde{M}}, \end{aligned}$$

(20)

$$\begin{aligned} \frac{\mathrm{d} V_s}{\mathrm{d} \tilde{t}}= & {} -\gamma \tilde{Q}\frac{1+j}{j} -\gamma \frac{\tilde{M}}{2}\frac{1-j}{j}. \end{aligned}$$

(21)

Since the leading term in the right-hand side of Eq. (21) is $\tilde{Q}$ and $\tilde{M}= {\mathcal {O}}({\kappa ^{-1}})$, then $C(\tilde{t})$ is a slow variable and $V_s(\tilde{t})$ is a fast variable. For any fixed value of C, there exists a unique asymptotically stable critical point $V_s^* = V_s^*(C)$ of the fast equation (Eq. (21)). The critical points are defined in the following equation where the inverse to Eq. (19) coordinate change is used:

$$\begin{aligned} \tilde{Q}(\tilde{V}(C,V_s^*),\tilde{\omega }(C,V_s^*)) =-\frac{\tilde{M}(\tilde{V}(C,V_s^*))}{2}\frac{1-j}{1+j}. \end{aligned}$$

(22)

Thus, the critical points of Eq. (21) in initial variables have the form $\tilde{V}^* = \tilde{V}^*(V_s^*(C),C)$ and $\tilde{\omega }^* = \tilde{\omega }^*(V_s^*(C),C)$.

These critical points form a slow manifold that includes the origin of the phase space $\tilde{V}= \tilde{\omega }= 0$. After a finite time $\tilde{t}^*$, the phase point moves in its ${\mathcal {O}}(\kappa ^{-1})$ – neighborhood slowly [33]; thus,

$$\begin{aligned} \tilde{V}(\tilde{t}) = \tilde{V}^*(V_s^*(C),C)(1+{\mathcal {O}}(\kappa ^{-1})), \ \tilde{\omega }(\tilde{t}) = \tilde{\omega }^*(V_s^*(C),C)(1+{\mathcal {O}}(\kappa ^{-1})). \end{aligned}$$

Substituting it into Eq. (17), we get Eq. (18). $\square$

Lemma

$$\begin{aligned} \left. \frac{\mathrm{d}\tilde{M}}{\mathrm{d}\tilde{V}}\right| _{\tilde{V}= 0} = \frac{\alpha -1}{\alpha } \end{aligned}$$

Proof

In [34, Appendix] using the asymptotics of Bessel functions, it has been shown that $\lambda = 1$, $\varepsilon = 0$ and $1/\zeta = 0$ for $\tilde{V}= 0$. Hence, from Eq. (7) we get

$$\begin{aligned} \left. \frac{\mathrm{d}\tilde{M}}{\mathrm{d}\tilde{V}}\right| _{\tilde{V}= 0} = \left. \kappa ^{-1} \frac{\mathrm{d}}{\mathrm{d}\tilde{V}}\left[ \frac{\kappa \tilde{V}}{\lambda }\left( \lambda ^2 -\frac{1}{\alpha }\right) \right] \right| _{\tilde{V}= 0} = \frac{\alpha -1}{\alpha }. \end{aligned}$$

$\square$

Statement 4

(asymptotically exponential deceleration). For any initial conditions and for any constant $k>0$, there exists a time instant $\tilde{t}^{**}=\tilde{t}^{**}(k)<+\infty$ and a constant $K=K(k)$ such that the velocity $\tilde{V}$ decreases faster than exponentially:

$$\begin{aligned} |\tilde{V}(\tilde{t})|\le K\exp \left( {-\gamma \frac{\alpha -1}{(1+j)(1+k)\alpha }\tilde{t}}\right) \quad \text {for }\tilde{t}>\tilde{t}^{**}. \end{aligned}$$

Note that the constant K depends also on initial conditions.

Proof

We calculate the time derivative with respect to Eq. (5):

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\tilde{t}}\frac{\tilde{V}^2}{2} = \gamma \tilde{V}\left( \tilde{Q}- \frac{\tilde{M}}{2}\right) . \end{aligned}$$

Substituting here $\tilde{Q}$ from Eq. (18), we get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\tilde{t}}\frac{\tilde{V}^2}{2} = -\gamma \frac{\tilde{M}}{1+j}\tilde{V}\left( 1+O(\kappa ^{-1})\right) \quad \text {for }\tilde{t}>\tilde{t}^{*}. \end{aligned}$$

Having $\tilde{V}(\tilde{t})\rightarrow 0$ (Statement 2) and based on the lemma, we conclude that for any $k>0$, there exists a finite time instant $\tilde{t}^{**}\ge \tilde{t}^*$ such that

$$\begin{aligned} |\tilde{M}(\tilde{V})|\ge \frac{\alpha -1}{\alpha }\frac{|\tilde{V}|}{1+k}, \end{aligned}$$

and therefore,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\tilde{t}}\frac{\tilde{V}^2}{2} \le -\gamma \frac{\alpha -1}{(1+j)\alpha }\frac{2}{1+k}\frac{\tilde{V}^2}{2}\left( 1+O(\kappa ^{-1})\right) . \end{aligned}$$

Integrating through time and neglecting the terms of order $O(\kappa ^{-1})$, we get

$$\begin{aligned} \frac{\tilde{V}^2}{2}\le & {} K_0\exp \left( -\gamma \frac{\alpha -1}{(1+j)\alpha }\frac{2}{1+k}\tilde{t}\right) \Rightarrow \\ |\tilde{V}|\le & {} K\exp \left( -\gamma \frac{\alpha -1}{(1+j)\alpha }\frac{1}{1+k}\tilde{t}\right) \text { for any } k> 0 \text { and }\tilde{t}>\tilde{t}^{**} \end{aligned}$$

$\square$

Theorem

(finiteness of the distance till stop). For viscoelastic material with parameters $\kappa \gg 1$, $\alpha >1$ and friction coefficient $\mu >0$ for any initial conditions $\tilde{V}_0$ and $\tilde{\omega }_0$, the axis of the cylinder moves on a finite distance.

Proof

The distance till stop is

$$\begin{aligned} |\tilde{\xi }(+\infty )| = \left| \int \limits _0^{+\infty } \tilde{V}(\tilde{t})\mathrm{d}\tilde{t}\right| \le \int \limits _0^{\tilde{t}^{**}} |\tilde{V}(\tilde{t})|\mathrm{d}\tilde{t}+ \int \limits _{\tilde{t}^{**}}^{+\infty } |\tilde{V}(\tilde{t})|\mathrm{d}\tilde{t}. \end{aligned}$$

Since the kinetic energy (14) does not increase, we have

$$\begin{aligned} |\tilde{V}(\tilde{t})|\le \sqrt{2T(\tilde{V}_0,\tilde{\omega }_0)} = \sqrt{\tilde{V}_0^2 + j\tilde{\omega }_0^2} \end{aligned}$$

and using Statement 4, we have

$$\begin{aligned} |\tilde{\xi }(+\infty )|\le \tilde{t}^{**}\sqrt{\tilde{V}_0^2 + j\tilde{\omega }_0^2} + K\frac{(1+j)\alpha (1+k)}{\gamma (\alpha -1)} \end{aligned}$$

$\square$

Remark

The theorem is not true for elastic materials $(\alpha = 1)$. In this case, the distance till full stop is infinite for almost all initial conditions.