A pool game is modeled to show the transmission of momentum between one impacting
ball and 15 impacted balls.
The purpose of this example is to investigate the transmission of momentum between
several balls. Contact with the various interfaces using the Penalty and Lagrange
Multipliers' method is analyzed.
An initial velocity of 1.5 ms-1 in X direction is applied to all nodes of
the white (cue) ball. Figure 1. Initial Translational Velocities of the Impacting Ball
All nodes of the lower face of the table are completely fixed (translations and
rotations).
Gravity is considered for all the balls nodes. A function defines the gravity
acceleration in the Z direction compared with time. Gravity is activated using
/GRAV. Figure 2. Gravity Function (-0.00981 mm.ms-2 )
Pool is a game consisting of 16 balls, each 50.8 mm in diameter. It is played on a small billiard
table measuring 1800 mm x 900 mm. Fifteen (15) balls are placed in a triangle to
enable their tight grouping. The initial velocity of the shooting (cue) ball is
presumed equal to 1.5 ms-1. Elastic rebounds are observed. Figure 3. Pool Game
Units: mm, g, N, MPa.
The material is subjected to a linear elastic law (/MAT/LAW1) with the
following properties:
Material Properties: Balls (pheonolic resin)
Initial density
0.00137 g.mm-3
Young's modulus
10500
Poisson ratio
0.3
Material Properties: Frame (polymer)
Initial density
0.001 g.mm-3
Young's modulus
1000
Poisson ratio
0.49
Material Properties: Plate (slate)
Initial density
0.0028 g.mm-3
Young's modulus
62000
Poisson ratio
0
Model Method
The balls are meshed with 16–node solid shells (quadratic elements) in order to
improve the conditions of contact by taking into account the curvatures. The frame
of the table is made of 16-node solid shells to comply with the interface used. The
plate is modeled using only one solid element. The 16-node thick shells are
considered as solid elements. They are defined by a thick TYPE20 shell property
(number 16 solid formulation for quadratic 16-node thick shells, fully-integrated
with 2x2x2 integration points). Figure 4. Pool Game Mesh Figure 5. Mesh for Balls Figure 6. 16-node Thick Shell Element
The TYPE16 interface with the Lagrange Multipliers method is used to model the
ball/ball and balls/table contacts. An interface must be defined for each ball (that
is: 16 interfaces in total). An additional interface is used to define the contacts
between the balls and the table (plate and frame). Figure 7. TYPE16 Interface: Secondary SHEL16 for Balls and Main SHEL16 for the
Table Figure 8. Example of the TYPE16 Interface Defined for the Contact Between
Balls
Secondary nodes (red) are extracted from the external surfaces of the parts.
Results
Curves and Animations
Due to the faceting of the ball, contact between the impacting ball and the impacted
balls is not perfectly symmetrical and momentum is not homogeneously transmitted
among the balls. An apparent physical strike thus results. Figure 9. Figure 10. Collision of the Balls
Figure 11. History of the Balls' Motions (contact control: TYPE16 interface)
