Abstract
In this work (also, preprint ANL/MCS-P3020-0812, Argonne National Laboratory) we introduce a complementarity-based rolling friction model to characterize dissipative phenomena at the interface between moving parts. Since the formulation is based on differential inclusions, the model fits well in the context of nonsmooth dynamics, and it does not require short integration timesteps. The method encompasses a rolling resistance limit for static cases, similar to what happens for sliding friction; this is a simple yet efficient approach to problems involving transitions from rolling to resting, and vice-versa. We propose a convex relaxation of the formulation in order to achieve algorithmic robustness and stability; moreover, we show the side effects of the convexification. A natural application of the model is the dynamics of granular materials, because of the high computational efficiency and the need for only a small set of parameters. In particular, when used as a micromechanical model for rolling resistance between granular particles, the model can provide an alternative way to capture the effect of irregular shapes. Other applications can be related to real-time simulations of rolling parts in bearing and guideways, as shown in examples.
Similar content being viewed by others
References
Anitescu M, Hart GD (2004) A fixed-point iteration approach for multibody dynamics with contact and friction. Math Program, Ser B 101(1):3–32 (ANL/MCS P985-0802)
Anitescu M, Tasora A (2010) An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput Optim Appl 47(2):207–235
Anitescu M, Potra FA, Stewart D (1999) Time-stepping for three-dimensional rigid-body dynamics. Comput Methods Appl Mech Eng 177:183–197
Arora S, Barak B (2009) Computational complexity: a modern approach. Cambridge University Press, Cambridge
Bardet J (1994) Observations on the effects of particle rotations on the failure of idealized granular materials. Mech Mater 18(2):159–182. Special Issue on Microstructure and Strain Localization in Geomaterials
Calvetti F, Nova R (2004) Micromechanical approach to slope stability analysis. Degradations and instabilities in geomaterials. Springer, Berlin
Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Geotechnique 29(1):47–65
de Coulomb CA (1821) Théorie des machines simples en ayant égard au frottement de leurs parties et à la roideur des cordages. Bachelier, Paris
Estrada N, Azema É, Radjaï F, Taboada A (2011) Identification of rolling resistance as a shape parameter in sheared granular media. 1. Phys Rev E, Stat Nonlinear Soft Matter Phys 84(1):011306
Facchinei F, Pang J (2003) Finite-dimensional variational inequalities and complementarity problems, vol 1. Springer, Berlin
Flores P, Leine R, Glocker C (2012) Application of the nonsmooth dynamics approach to model and analysis of the contact-impact events in cam-follower systems. Nonlinear Dyn 69:2117–2133. doi:10.1007/s11071-012-0413-3
Hairer E, Nørsett SP, Wanner G (2010) Solving ordinary differential equations. Springer, Berlin
Haug EJ (1989) Computer-aided kinematics and dynamics of mechanical systems. Prentice-Hall, Englewood Cliffs
Iwashita K, Oda M (1998) Rolling resistance at contacts in simulation of shear band development by DEM. J Eng Mech 124(3):285–292
Jiang M, Yu H-S, Harris D (2005) A novel discrete model for granular material incorporating rolling resistance. Comput Geotech 32(5):340–357
Jourdan F, Alart P, Jean M (1998) A Gauss Seidel like algorithm to solve frictional contract problems. Comput Methods Appl Mech Eng 155:31–47
Ketterhagen WR, am Ende MT, Hancock BC (2009) Process modeling in the pharmaceutical industry using the discrete element method. J Pharm Sci 2(98):442–470
Kinderleher D, Stampacchia G (1980) An introduction to variational inequalities and their application. Academic Press, New York
Kruggel-Emden H, Rickelt S, Wirtz S, Scherer V (2008) A study on the validity of the multi-sphere discrete element method. Powder Technol 188(2):153–165
Leine RI, Glocker C (2003) A set-valued force law for spatial Coulomb-Contensou friction. Eur J Mech A, Solids 22(2):193–216
Negrut D, Tasora A, Mazhar H, Heyn T, Hahn P (2012) Leveraging parallel computing in multibody dynamics. Multibody Syst Dyn 27:95–117. doi:10.1007/s11044-011-9262-y
Nocedal J, Wright SJ (1999) Numerical optimization, vol. 39. Springer, Berlin
Oda M, Konishi J, Nemat-Nasser S (1982) Experimental micromechanical evaluation of strength of granular materials: effects of particle rolling. Mech Mater 1(4):269–283
Pacejka HB (2005) Tire and vehicle dynamics, 2nd edn. SAE International, Warrendale
Pfeiffer F, Glocker C (1996) Multibody dynamics with unilateral contacts. Wiley, New York
Rankine WJM (1868) Manual of applied mechanics. Charless Griffin, London
Shabana AA (2005) Dynamics of multibody systems, 3rd edn. Cambridge University Press, Cambridge
Stewart D, Pang J-S (2008) Differential variational inequalities. Math Program 113(2):345–424
Stewart DE (2001) Reformulations of measure differential inclusions and their closed graph property. J Differ Equ 175:108–129
Stewart DE, Trinkle JC (1996) An implicit time-stepping scheme for rigid-body dynamics with inelastic collisions and Coulomb friction. Int J Numer Methods Eng 39:2673–2691
Studer C, Glocker C (2007) Solving normal cone inclusion problems in contact mechanics by iterative methods. J Syst Des Dyn 1(3):458–467
Tasora A, Anitescu M (2010) A convex complementarity approach for simulating large granular flows. J Comput Nonlinear Dyn 5(3):1–10
Tasora A, Anitescu M (2011) A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics. Comput Methods Appl Mech Eng 200(5–8):439–453
Tasora A, Negrut D, Anitescu M (2008) Large-scale parallel multi-body dynamics with frictional contact on the graphical processing unit. J Multi-Body Dyn 222(4):315–326
Terzaghi K, Peck RB, Mesri G (1996) Soil mechanics in engineering practice. Wiley-Interscience, New York
Tordesillas A, Walsh D (2002) Incorporating rolling resistance and contact anisotropy in micromechanical models of granular media. Powder Technol 124(1–2):106–111
Weisbach JL (1870) A manual of the mechanics of engineering and of the construction of machines, vol 3. Van Nostrand, New York
Acknowledgements
A. Tasora thanks Ferrari Automotive and TP Engineering for financial support. Mihai Anitescu was supported by the U.S. Department of Energy, under Contract No. DE-AC02-06CH11357.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix A: Kinematics of rolling in three-dimensional space
Let \(\mathbf {\omega}_{A,W}^{(W)}\) and \(\mathbf {\omega}_{B,W}^{(W)}\) denote the angular velocities of two bodies A and B, relative to the absolute reference frame W and expressed in the basis of the frame (W). We assume A and B to be rigid or with negligible deformations. Introducing the rotation matrix R A,W ∈SO(ℝ,3) that represents the rotation of A respect to W, we have \(\mathbf {\omega}_{A,W}^{(W)} = {R}_{A,W} \mathbf {\omega}_{A,W}^{(A)}\) and \(\mathbf {\omega}_{B,W}^{(W)} = {R}_{B,W} \mathbf {\omega}_{B,W}^{(A)}\).
Let the unimodular quaternion \(\mathbf {\varepsilon }_{A,W} \in \mathbb{H}_{1}\) represent the rotation of the frame A respect to absolute frame W. We recall that, for unimodular quaternions, the inverse \(\mathbf {\varepsilon }^{-1}\) is also the conjugate \(\mathbf {\varepsilon }^{*}\). We also recall that it is possible to compute R from \(\mathbf {\varepsilon }\) and vice versa.
Thanks to a property of quaternion algebra [13] the relative rotation of two references is
By performing differentiation respect to time, we get
From the result in [27], the quaternion derivative can be transformed in angular velocity, using pure quaternions:
Since \(( \mathbf {\varepsilon }_{1}^{*} \mathbf {\varepsilon }_{2})^{*} = \mathbf {\varepsilon }_{2}^{*} \mathbf {\varepsilon }_{1}\), and remembering that \(\mathbf {\varepsilon } \mathbf {\varepsilon }^{*} =1\), we can develop Eq. (48) into
The product \(\dot{ \mathbf {\varepsilon }}_{B,W} \mathbf {\varepsilon }_{B,W}^{*}\) in the second term of the summation can be replaced with the pure quaternion \(\frac{1}{2}[0, \mathbf {\omega}_{B,W}^{(W)}]\) using Eq. (47). Also, the first term can be premultiplied by \(\mathbf {\varepsilon }_{A,W}^{*} \mathbf {\varepsilon }_{A,W} =1\), becoming \(2 \mathbf {\varepsilon }_{A,W}^{*} \mathbf {\varepsilon }_{A,W} \dot{ \mathbf {\varepsilon }}_{A,W}^{*} \mathbf {\varepsilon }_{A,W}\); here the product between the second and third quaternion can be replaced with the pure quaternion \(\frac{1}{2}[0, \mathbf {\omega}_{A,W}^{(W)}]^{*}\), again using Eq. (47). Thus we have
A rotation in 3D space of the vector part of a pure quaternion can be obtained with unitary quaternions, that is, \([0,\mathbf {v}^{(W)}]= \mathbf {\varepsilon }_{A,W} [0,\mathbf {v}^{(A)}] \mathbf {\varepsilon }_{A,W}^{*}\).
Hence, recalling that \([0, \mathbf {\omega}_{A,W}^{(W)}]^{*}= -[0, \mathbf {\omega}_{A,W}^{(W)}]\) by the property of conjugate quaternions, we can rewrite Eq. (50) and obtain the expected result for relative angular velocity \(\mathbf {\omega}_{r}\):
Appendix B: Formulation of D vectors
We assume that the vector of generalized velocities v contains the speeds of the centers of mass of the bodies \(\dot{\mathbf {x}}^{(W)}\), expressed in absolute coordinates (W) and the angular velocities \(\mathbf {\omega}^{(i)}\) expressed in the local coordinates of the ith body, as \(\mathbf {v}= [ \dot{\mathbf {x}}^{(W)}_{1}, \mathbf {\omega}^{(1)}_{1}, \dot{\mathbf {x}}^{(W)}_{2}, \mathbf {\omega}^{(2)}_{2}, \ldots ]^{T}\).
Given a contact between a pair of two rigid bodies A and B, we define the positions of the two contact points with respect to the centers of mass, expressed in the coordinate systems of the two bodies, as \(\mathbf {s}_{A}^{(A)}\) and \(\mathbf {s}_{B}^{(B)}\). The absolute rotations of the coordinate systems of the bodies are \({R}_{A}^{(W)}, {R}_{B}^{(W)} \in \mathsf{SO}(\mathbb{R},3)\) and the absolute rotation of the contact plane is \({R}_{P}^{(W)} \in \mathsf{SO}(\mathbb{R},3) = [ \mathbf {n}, \mathbf {u}, \mathbf {w} ]\). Thus, the vectors \(\mathbf {D}_{\gamma_{n}}\), \(\mathbf {D}_{\gamma_{u}}\), \(\mathbf {D}_{\gamma_{w}}\) can be computed as \({D}_{\gamma}= [ \mathbf {D}_{\gamma_{n}}, \mathbf {D}_{\gamma_{u}}, \mathbf {D}_{\gamma_{w}} ] \in \mathbb{R}^{3 \times m_{v}}\),
where \({ \tilde {s}}\) is the skew symmetric matrix such that \({ \tilde {s}}\mathbf {x} = \mathbf {s} \wedge \mathbf {x}\).
Similarly, recalling the result in Eq. (51), one can compute the vectors \(\mathbf {D}_{\tau_{n}}\), \(\mathbf {D}_{\tau_{u}}\), \(\mathbf {D}_{\tau_{w}}\) as \({D}_{\tau}= [ \mathbf {D}_{\tau_{n}}, \mathbf {D}_{\tau_{u}}, \mathbf {D}_{\tau_{w}} ] \in \mathbb{R}^{3 \times m_{v}}\):
We remark that, because of the extreme sparsity of (52) and (53), only the following four 3×6 matrices need to be stored per each contact
Here we considered B as the reference body: otherwise, if A were the reference for contact coordinates, signs should be swapped in all terms in Eqs. (52)–(57).
Appendix C: Computing projections on intersections of cones
We describe the procedure to compute the Euclidean projection of a point x on an intersection of circular cones that have one common component (in the case studied here, that component is the normal force). We assume that a generic point x is structured as follows:
and that the m circular cones are second-order cones defined by
where μ i >0, i=1,2,…,m. We are interested in computing the projection of a vector x on ⋂K i , that is,
For example, in the case treated in this work, we are interested in simultaneous modeling of sliding, rolling, and spinning friction in three dimensional configurations. That is, we have three cones, m=3 and x is a six-dimensional vector, x=(γ n ,γ u ,γ w ,τ u ,τ w ,τ n ). The mapping (58) is the following: x 0=γ n , l 1=(γ u ,γ w ), l 2=(τ u ,τ w ), l 3=τ n . The friction coefficients are μ 1=μ, μ 2=ρ, μ 3=σ.
The crucial observation that simplifies the computation of the projection is that the component \(\tilde{l}_{i}\) of the projection \(\tilde{x}\) must be collinear with l i . Indeed, if this is not the case, then rotating \(\tilde{l}_{i}\) over l i will preserve feasibility but will necessarily reduce \(\| x-\tilde{x} \|\), a contradiction. Therefore, there exists t i such that \(\tilde{l}_{i}=t_{i} l_{i}\). The optimization that defines the projection then becomes
We have normalized the component of y in terms of y 0 to allow for the range of t i to be the same. For a given y 0, the optimal t i , which we denote by t i (y 0), is easy to compute. Indeed we obtain the following
Substituting t i for the optimal values t i (y 0) in the optimization problem, we obtain that the problem is equivalent to
Here I is the indicator function of a set. It is immediately apparent that this function is piecewise quadratic and that it is convex. Indeed, convexity follows from the fact that each term function is convex, the first term as a quadratic, and the other terms as their graphs are the union of a parabola with a flat line.
To find its optimum, we can do the following.
-
1.
Define and order the breakpoints 0, and \(\frac{\| l_{i} \|}{\mu_{i}}\), with i=1,2,…,m. Successive breakpoints define a piece.
-
2.
On each piece find the minimum of the quadratic function.
-
3.
Compute the overall minimum, which is the lowest value of all such minima.
Once \(\tilde{x}_{0}=y_{0}\) is determined, t i (y 0) is computed, and the other components of the projection are computed as \(\tilde{l}_{i}=t_{i}(\tilde{x}_{0})\frac{\mu_{i} \tilde{x}_{0}}{\| l_{i} \|}\).
For a large number of breakpoints we can exploit convexity of ψ, by noting that we can evaluate the function at the breakpoints, and find the minimum value. Then, by convexity, the overall minimum must occur in a segment that neighbors the breakpoint with the minimum value. Hence, one minimizes the quadratic only in those intervals.
To summarize:
-
1.
Define and order the breakpoints 0, and \(\frac{\| l_{i} \|}{\mu_{i}}\), with i=1,2,…,m. Successive breakpoints define a piece. We assume without loss of generality that the labels have been permuted so that the natural order has the breakpoints in increasing order, that is, i<j⇒ \(\frac{\| l_{i} \|}{\mu_{i}} < \frac{\| l_{j} \|}{\mu_{j}}\). If two breakpoints have the same value, we delete their index.
-
2.
Enumerate the objective function ψ at the breakpoints, and find the i for which \(\psi(\frac{\| l_{i} \|}{\mu_{i}}) \leq \psi(\frac{\| l_{j} \|}{\mu_{j}})\), ∀j. If there is one such i, the overall minimum is on a neighboring segment; if there are two, it is on the segment in between (there cannot be three different indices, since the function is not piecewise constant).
-
3.
Minimize the piecewise quadratic on either the one or two segments identified, and report the result.
For a small number of breakpoints (i.e., the number of cones m is small), it is not likely that this reduced method would practically be much faster than comprehensive enumeration.
Rights and permissions
About this article
Cite this article
Tasora, A., Anitescu, M. A complementarity-based rolling friction model for rigid contacts. Meccanica 48, 1643–1659 (2013). https://doi.org/10.1007/s11012-013-9694-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-013-9694-y