Rolling Resistance

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Two and three-dimensional boundary element formulations of compressible isotropic, transversely isotropic and orthotropic viscoelastic layers of arbitrary thickness, applied to the rolling resistance of rigid cylinders and spheres (new)

Gérard-Philippe Zéhil and Henri P. Gavin

Article featured in Advances in Engineering: "By modeling the resistance to motion on orthotropic viscoelastic layers, Dr. Gérard-Philippe Zéhil and his co-author have discovered important behavioral features that the scientific and engineering communities were previously unaware of. In a key scientific article contributing to excellence in engineering research, the authors show that the level of dissipation varies with the direction of motion, following a trend depending on speed. The moving object is furthermore subjected to a lateral force which tends to deviate it from its initial course to follow a path governed by velocity. The authors discuss potential impacts of broad significance of their findings, such as on the design of metamaterials and the engineering of systems to achieve motion control and optimize directional energy transfers in various applications."

European Journal of Mechanics - A/Solids

Journal article, 2014

Submitted manuscript, 2013

Abstract

New two-dimensional and three-dimensional boundary element formulations of compressible viscoelastic layers of arbitrary thickness are presented in this work. The formulations are derived in increasing order of complexity for: (i) compressible isotropic layers, (ii) transversely isotropic layers, and (iii) fully orthotropic layers. It is further shown that existing 2D and 3D models for incompressible isotropic layers may be regarded as particular instances of case (i). The proposed formulations are based on Fourier series and support any linear viscoelastic material model characterized by general frequency-domain master-curves. These approaches result in a compliance matrix for the layer's upper boundary, which includes the effects of steady-state motion. This characterization may be used as a component in various problem settings to generate sequences of high fidelity solutions for varying parameters. The proposed modeling techniques are applied, in combination with appropriate contact solvers, to the rolling resistance of rigid cylinders and spheres on compressible isotropic, transversely isotropic and orthotropic layers. The latter case reveals that the dissipated power varies with the direction of motion, which suggests new ways of optimizing the level of damping in various engineering applications of very high impact. Interesting lateral viscoelastic effects resulting from material asymmetry are unveiled. These phenomena could be harnessed to achieve smooth and `invisible' guides across three-dimensional viscoelastic surfaces, and hence suggest new ways of controlling trajectories, with a broad range of potential applications.

Rolling resistance of a rigid sphere with viscoelastic coatings (new)

Gérard-Philippe Zéhil and Henri P. Gavin

International Journal of Solids and Structures

Journal article, 2014

Submitted manuscript, 2013

Abstract

We present a novel three-dimensional boundary-element formulation that fully characterizes the mechanical behavior of the external boundary of a multi-layered viscoelastic coating attached to a hard rotating spherical core. The proposed formulation incorporates both, the viscoelastic, and the inertial effects of the steady-state rolling motion of the sphere, including the Coriolis effect. The proposed formulation is based on Fourier-domain expressions of all mechanical governing equations. It relates two-dimensional Fourier series expansions of surface displacements and stresses, which results in the formation of a compliance matrix for the outer boundary of the deformable coating, discretized into nodes. The computational cost of building such a compliance matrix is optimized, based on configurational similarities and symmetry. The proposed formulation is applied, in combination with a rolling contact solving strategy, to evaluate the viscoelastic rolling friction of a coated sphere on a rigid plane. Steady-state results generated by the proposed model are verified by comparison to those obtained from running dynamic simulations on a three-dimensional finite element model, beyond the transient. A detailed application example includes a verification of convergence and illustrates the dependence of rolling resistance on the applied load, the thickness of the coating, and the rolling velocity.

Three-dimensional boundary element formulation of an incompressible viscoelastic layer of finite thickness applied to the rolling resistance of a rigid sphere

Gérard-Philippe Zéhil and Henri P. Gavin

International Journal of Solids and Structures

Journal article, 2013

Submitted manuscript, 2012

Abstract

A three-dimensional boundary element formulation of an incompressible viscoelastic layer of finite thickness is proposed, in a moving frame of reference. The formulation is based on two-dimensional Fourier series expansions of relevant mechanical fields in the continuum of the layer. The linear viscoelastic material is characterized, in the most general way, by its frequency-domain master curves. The presented methodology results in a compliance matrix for the layer's upper boundary, which includes the effects of steady-state motion and can be used in any contact problem-solving strategy. The proposed formulation is used, in combination with a contact solver, to build a full three-dimensional model for the steady-state rolling/sliding resistance incurred by a rigid sphere on the layer. Energy losses include viscoelastic damping and surface friction. The model is tested and its results are found to be consistent with existing solutions in limiting cases. An example is explored and the corresponding results are used to illustrate the influence of different parameters on the rolling resistance. General aspects of previously-described dependences are confirmed.

Simple algorithms for solving steady-state frictional rolling contact problems in two and three dimensions

Gérard-Philippe Zéhil and Henri P. Gavin

International Journal of Solids and Structures

Journal article, 2013

Submitted manuscript, 2012

Abstract

This paper presents simple, yet robust and efficient algorithms for solving steady-state, frictional, rolling/sliding contact problems, in two and three dimensions. These are alternatives to powerful, well established, but in particular instances, possibly ‘cumbersome’ general-purpose numerical techniques, such as finite-element approaches based on constrained optimization. The cores of the solvers rely on very general principles: (i) resolving motional conflicts, and (ii) eliminating unacceptable surface tractions. The proposed algorithms are formulated in the context of small deformations and applied to the cases of a rigid cylinder and a rigid sphere rolling on a linear viscoelastic layer of finite thickness, in two and three dimensions, respectively. The underlying principles are elucidated, relevant mathematical expressions derived and details given about corresponding implementation techniques. The proposed contact algorithms can be extended to more general settings involving a deformable indenter, material nonlinearities and large deformations.

Simplified approaches to viscoelastic rolling resistance

Gérard-Philippe Zéhil and Henri P. Gavin

International Journal of Solids and Structures

Journal article, 2013

Submitted manuscript, 2012

Abstract

Modeling approaches yielding rolling resistance estimates for rigid spheres (and cylinders) on viscoelastic layers of finite thicknesses are introduced as lower-cost alternatives to more comprehensive solution-finding strategies. Detailed examples are provided to illustrate the capabilities of the different approaches over their respective ranges of validity.