Research Papers

Analysis and Computation of Two Body Impact in Three Dimensions

[+] Author and Article Information
Yan-Bin Jia

Department of Computer Science,
Iowa State University,
Ames, IA 50011
e-mail: jia@iastate.edu

Feifei Wang

Department of Computer Science,
Iowa State University,
Ames, IA 50011
e-mail: wangff@iastate.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 22, 2016; final manuscript received November 23, 2016; published online January 25, 2017. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 12(4), 041012 (Jan 25, 2017) (16 pages) Paper No: CND-16-1398; doi: 10.1115/1.4035411 History: Received August 22, 2016; Revised November 23, 2016

A formal impulse-based analysis is presented for the collision of two rigid bodies at single contact point under Coulomb's friction in three dimensions (3D). The tangential impulse at the contact is known to be linear in the sliding velocity whose trajectory, parametrized with the normal impulse and referred to as the hodograph, is governed by a generally nonintegrable ordinary differential equation (ODE). Evolution of the hodograph is bounded by rays in several invariant directions of sliding in the contact plane. Exact lower and upper bounds are derived for the number of such invariant directions, utilizing the established positive definiteness of the matrix defining the governing ODE. If the hodograph reaches the origin, it either terminates (i.e., the contact sticks) or continues in a new direction (i.e., the contact resumes sliding) whose existence and uniqueness, only assumed in the literature, are proven. Closed-form integration of the ODE becomes possible as soon as the sliding velocity turns zero or takes on an invariant direction. Assuming Stronge's energy-based restitution, a complete algorithm is described to combine fast numerical integration (NI) with a case-by-case closed-form analysis. A number of solved collision instances are presented. It remains open whether the modeled impact process will always terminate under Coulomb's friction and Stronge's (or Poisson's) restitution hypothesis.

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Fig. 1

Impact between two bodies. The impulse I consists of a component Iz along their contact normal (aligned with the z-axis), and a tangential component I⊥ (in the xy plane) due to contact friction. The line of impact ℓ through c is normal to the two contacting surfaces.

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Fig. 2

Impacts (a) between two balls and (b) between a ball and a half-space with infinite mass

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Fig. 3

Configuration of impact (μ=0.8 and e = 0.95) between an icosahedron B1 and a tetrahedron B2, both with uniform mass density, respectively. The x–y plane coincides with a pentagon face of B1, whose center coincides with a vertex of B2 at c. All pentagons and hexagons on B1 have side length 0.1. The vector from c to the vertex of the contacting pentagon with the largest y-coordinate rotates from the y-axis about the z-axis (through B1 's center of geometry) through π/10. The remaining three verticesof the tetrahedron are located at (0,0,−0.75)T, (0.5,0,−0.5)T, and (0,0.5,−0.5)T. Other geometric and physicalparameters include the followings: o1=(0,0,0.232744)T, o2=(0.125,0.125,−0.4375)T, m1=3.0, m2=1.0, Q1=0.0671673 U3, and Q2=diag(0.017239,0.022813,0.027135).

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Fig. 4

Hodographs of four instances of the icosahedron–tetrahedron impact in Fig. 3 generated by changing the values of some physical parameters listed in its caption and Eq. (13): (a) no change; (b) μ=0.4 and V1z−=0.2; (c) μ=0.25 and V1z−=0.2; and (d) μ=3.0 and V1z−=0.5. Here, V1z− is the z-component of the pre-impact velocity V1− of the icosahedron. The direction out of the origin along a dashed (blue) line represents a centripetal invariant direction. The line is labeled with an arrow pointing toward the origin to indicate the magnitude change of the sliding velocity in this direction. The direction out of the origin along a dashed (red) line represents a centrifugal invariant direction. This line is labeled with an arrow pointing away from the origin.

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Fig. 5

Impulse curve for the icosahedron–tetrahedron impact. Also shown are the plane Pc of compression and the line Ls of sticking.

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Fig. 6

Bundle of 15 hodographs resulting from unit γ− valuesfor the icosahedron–tetrahedron collision (μ=0.8 and vz=−0.2)

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Fig. 7

Hodograph of the impact instance in Fig. 3 with the following changed parameter values: μ=0.25 and V1−=(−0.7,−0.35,−0.5)T

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Fig. 8

Bundle of 13 hodographs resulting from unit γ− valuesfor the icosahedron–tetrahedron collision (μ=0.25 and vz=−0.8)

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Fig. 9

Eight possible event sequences. A dashed line segment connecting two events represents a period over which the impulse growth has to be computed via numerical integration, while a solid line segment represents a period during which the impulse growth has a closed form.

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Fig. 10

Hodographs of the icosahedron–tetrahedron impact in Fig. 3 with the following changed parameter values: (a) μ=0.35 and V1z−=0.2; (b) μ=0.25 and V1z−=−1; and (c) μ=0.55 and V1z−=0.9

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Fig. 11

Configuration of impact (μ=0.7 and e = 0.7) between abowling pin (B1) and a ball (B2), both with uniform massdistribution. Here, o1=(0.068436,−0.00000242,0.073735)T, o2=(−0.000061,−0.000890,−0.108458)T, m1=1.63293, m2=6.35029, Q1=diag (0.0021186,0.012092,0.014029)T, and Q2=diag(0.0295723, 0.029669,0.030068)T. The shape of the pin is specified in9. The pin is originally symmetric about the z-axis with its center of mass at the origin. To engage in contact, it rotates about the y-axis through  tan−1(3.15435) and then translates by (0.068436,0,0.0737351)T. The ball has radius 0.108458. Its three holes are each created from subtracting a cylinder parallel to the y-axis. Holes 1 and 2 each has radius 0.0127, while hole 3 has radius 0.0142875. The bottom faces of the three cylinders are centered, respectively, at (0.03302,0.067845,−0.126238)T, (0.03302,0.067845,−0.090678)T, and (−0.0381,0.0676971,−0.108458)T.

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Fig. 12

Proofs of part (iv) of Theorem 8: (a) ‖Σ−1p‖<1, (b) ‖Σ−1p‖=1, (c) ‖Σ−1p‖>1, and (d) of part (v) of the theorem

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Fig. 13

Unit circle C maps to an ellipse E2 centered at p under β°α

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Fig. 14

Inverse mapping sequence α−1°β−1

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Fig. 15

The unit vectors ŝl and ŝr in the directions from o to the points of tangency sl and sr on E2 delimits an arc A on the unit circle that contains the point û with û∼(β°α)û, if it exists

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Fig. 16

Four possible cases (a)–(d) of overlapping between the elliptic segments E2+ between sl and sr, and A between ql and qr

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Fig. 17

The arc A is contained in either (a) two adjacent quadrants or (b) just one quadrant




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