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Example 9 - Billiards (pool)

Summary

The impact and rebound between balls on a small billiard table is studied. This example deals with the problem of defining interfaces and transmitting momentum between the balls. The study is divided into three parts:

At first, a general study is used to see the results of a cue ball when coming into contact with the 15 other balls arranged in a triangle. The balls are meshed for the purpose using 16-node shell elements (for the curvature) and a type 16 interface between each ball as well as between the balls and the table. The results show that the momentum is not homogenously transmitted: the balls on the table are not being evenly spread out.

Secondly, the collision between two balls is studied. All parameters are the same as in the first part. The reaction of those two balls is then compared to the analytical results.

Finally, six different interfaces are compared: types 16 and 17 tied or sliding interfaces using the Lagrange Multipliers method and a type 7 tied or sliding interface using the Lagrange Multipliers or the Penalty method. The study is also initiated using a quasi-static gravity application prior to dynamic behavior. When comparing the kinetic energy transmission, the results show that interfaces without the tied option provide better results than the others, and that the type 16 interface seems to be the best.

The following studies are depicted:

Billiards, Pool

Collision between two balls

Study on interfaces

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Billiards (Pool)

Title

Billiards (Pool)

Number

9.1

Brief Description

A pool game is modeled to show the transmission of momentum between one impacting ball and 15 impacted balls.

Keywords

• 16-node thick shell, sphere mesh

• Type 7 interface using the Lagrange Multipliers method and the Penalty method

• Type 16 sliding and tied interface, type 17 sliding and tied interface, quadratic surface contact

• Elastic shock

• Momentum transmission, shock wave

RADIOSS Options

• Interfaces 7 (/INTER/TYPE7), 16 (/INTER/LAGMUL/TYPE16) and 17 (/INTER/LAGMUL/TYPE17)

• Initial velocities (/INIVEL)

• 16-node thick shell property type 20 (/PROP/TSHELL)

Input File

Billiard_game / Interface_16: <install_directory>/demos/hwsolvers/radioss/09_Billiards/Billiards_model/BILLA

RD*

<install_directory>/demos/hwsolvers/radioss/09_Billiards/Billiards_model/Suppl

ement_Interface7Lag/BILLARD*

RADIOSS Version

51e

Technical / Theoretical Level

Advanced

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Overview

Aim of the Problem

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.

Physical Problem Description

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 ball is presumed equal to 1.5 ms-1. Elastic rebounds are observed.

Fig 1: Pool game.

Units: mm, g, N, MPa.

The material is subjected to a linear elastic law (/MAT/LAW1) with the following properties:

Balls: phenolic resin

Frame: polymer Plate: slate

Initial density 0.00137 g.mm-3 0.001 g.mm-3 0.0028 g.mm-3

Young's modulus

10500 MPa 1000 MPa 62000 MPa

Poisson ratio 0.3 0.49 0

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Analysis, Assumptions and Modeling Description

Modeling Methodology

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 type 20 shell property (number 16 solid formulation for quadratic 16-node thick shells, fully-integrated with 2x2x2 integration points).

Fig 2: Pool game mesh.

Fig 3: Mesh for balls.

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Fig 4: 16-node thick shell element.

The type 16 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).

Fig 5: Type 16 interface: slave SHEL16 for balls and master SHEL16 for the table.

Fig 6: Example of the type 16 interface defined for the contact between balls.

Slave nodes (red) are extracted from the external surfaces of the parts.

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RADIOSS Options Used

An initial velocity of 1.5 ms-1 in the X direction is applied to all nodes of the white ball.

Fig 7: 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.

Fig 8: Gravity function (-0.00981 mm.ms

-2 )

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Simulation Results and Conclusions

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.

Fig 9: Collision of the balls

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Fig 10: History of the balls’ motions (contact control: type 16 interface).

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Collision between Two Balls

Study on Trajectories

Title

Collision between two balls

Number

9.2

Input File

Collision study: <install_directory>/demos/hwsolvers/radioss/09_Billiards/Collision_simulation/

COLLISION*

Two balls are now considered in order to study the behavior of impacting spherical balls.

The balls’ behavior is described using the parameters (angles and velocities) shown in Fig 11. The numerical results are compared with the analytical solution, assuming a perfect elastic rebound (coefficient of restitution is equal to 1).

Fig 11: Problem data.

Initial values: V1 = 0.7m.s-1 ; V2 = 1m.s-1 ; 1 = 40°; 2 = 30; massball = 44.514g.

Modeling Methodology

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The balls and the table have the same properties, previously defined for a pool game. The dimensions of the table are 900 mm x 450 mm x 25 mm and the balls’ diameter is 50.8 mm. The balls and the table are meshed with 16-node thick shell elements for using the type 16 Lagrangian interface.

Fig 12: Mesh of the problem (16-node thick shells).

The initial translational velocities are applied to the balls in the /INIV Engine option. Velocities are projected on the X and Y axes.

Fig 13: Initial velocities applied on the balls (initial position).

Gravity is considered for the balls (0.00981 mm.ms-2 ).

The ball/ball and balls/table contact is modeled using the type 16 interface (slave nodes / master 16-node thick shells contact). The interface defining the ball/ball contact is shown in Fig 14.

Fig 14: Master and slave sides for the type 16 Lagrangian interface.

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Analytical Solution

Take two balls, 1 and 2 from masses m1 and m2, moving in the same plane and approaching each other on a collision course using velocities V1 and V2, as shown in Fig 15.

Fig 15: General problem of collision between two balls.

Velocities are projected onto the local axes n and t. To obtain the velocities and their direction after impact, the momentum conservation law is recorded for the two balls:

(1)

or

(2)

The shock is presumed elastic and without friction. Maintaining the translational kinetic energy is respected as there is no rotational energy:

(3)

Such equality implies that the recovering capacity of the two balls corresponds to their tendency to deform.

This condition equals one of the elastic impacts, with no energy loss. Maintaining the system’s energy gives:

(4)

This relation means that the normal component of the relative velocity changes into its opposite during the elastic shock (coefficient of restitution value e is equal to the unit).

The following equations must be checked for normal components:

The equations system using V’1 and V’2 as unknowns is easily solved:

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It should be noted that these relations depend upon the masses ratio.

As the balls do not suffer from velocity change in the t-direction, maintaining the tangential component of each sphere’s velocity provides:

The norms of velocities after shock result from the following relations.

and

In this example, balls have the same mass: m1 = m2.

Therefore:

and (7)

The norms of the velocities are given using the following relations, depending on the initial velocities and angles:

(8)

By recording the projection of the velocities, directions after shock can be evaluated using relation (9):

(9)

Equations (8) and (9) are used for determining the analytical solutions (angles and velocities after collision).

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Simulation Results: Comparison of Numerical Results with the Analytical Solution

The following diagram shows the trajectories of the balls’ center point obtained using numerical simulation before and after collision.

Fig 16: Trajectories of balls (center of gravity).

Fig 17: Variation of velocities (collision at 40 ms).

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Fig 18: Energy assessment.

For given initial values of V1, V2, 1 and 2, simulation results are reported in Table 1.

Table 1: Comparison of results for after collision

Numerical Results Analytical Solution

1’ 42.27° 1’ 44.72°

2’ 26.75° 2’ 26.48°

V1’ 0.731 m/s V1’ 0.731 m/s

V2’ 0.969 m/s V2’ 0.977 m/s

Conclusion

The simulation corroborates with the analytical solution. The 16-node thick shells are fully-integrated elements without hourglass energy. This modeling provides a good transmission of momentum. However, the type 16 interface does not take into account the quadratic surface on the slave side (ball 2), due to the node to thick shell contact. Accurate results are obtained for a collision without penetrating the quadratic surface of the slave side in order to confirm impact between the spherical bodies.

A fine mesh could improve the results.

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Study on Interfaces

Comparison of Results Obtained using Different Interfaces

Title

Study on interfaces

Number

9.3

Input File

Inter_7_Penalty: <install_directory>/demos/hwsolvers/radioss/09_Billiards/Contact_modelling/Int

er_7_Penalty/TEST7P*

Inter_7_Lagrangian: <install_directory>/demos/hwsolvers/radioss/09_Billiards/Contact_modelling/Int

er_7_Lagrangian/TEST7L*

Inter_16_tied: <install_directory>/demos/hwsolvers/radioss/09_Billiards/Contact_modelling/Int

er_16_tied/TEST16T*

Inter_16_sliding: <install_directory>/demos/hwsolvers/radioss/09_Billiards/Contact_modelling/Int

er_16_sliding/TEST16S*

Inter_17_tied: <install_directory>/demos/hwsolvers/radioss/09_Billiards/Contact_modelling/Int

er_17_tied/TEST17ST*

Inter_17_sliding: <install_directory>/demos/hwsolvers/radioss/09_Billiards/Contact_modelling/Int

er_17_sliding/TEST17S*

The balls and the table have the same properties as previously defined. The dimensions of the table are 900 mm x 450 mm x 25 mm and the balls’ diameter is 50.8 mm.

Six interfaces are used to model the contacts (ball/ball and balls/table):

Table 2: Interfaces used in the problems.

Type 16 (Lagrange Multipliers) tied or sliding: slave nodes / master solids contact

Type 17 (Lagrange Multipliers) tied or sliding: slave 16-node shells / master 16-node shells contact

Type 7 (Lagrange Multipliers): slave nodes / master surface contact

Type 7 (Penalty) sliding: slave nodes / master surface contact

The type 16 interface defines contact between a group of nodes (slaves) and a curved surface of quadratic elements (master part). The type 17 interface is used for modeling a surface-to-surface contact.

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For both interfaces, the Lagrange Multipliers method is used to apply the contact conditions; gaps are not required. Contact between the balls and the table is set as tied or sliding. Contact between the balls themselves is always considered as sliding. The type 7 interface enables the simulation of the most general contact types occurring between a master surface and a set of slave nodes. The Coulomb friction between surfaces is not modeled here (sliding contact) and the gap is fixed at 0.1 mm. The other parameters are set to default values.

The type 7 interface with the Penalty method is not available with 16-node thick shell elements. Thus, brick elements replace the 16-nodes shells in this case (check in the input file).

Contact modeling between balls (always sliding).

Fig 19: Definition of slave and master sides for contact.

The symmetrical interface definition is not recommended when using the Lagrange Multipliers method (types 16, 17 and 7-Lag). The problem using the interface with the Penalty method uses two interfaces to model the symmetrical impact.

Fig 20: Symmetrical configuration of the type 7 interface using the Penalty method

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Interface Slave (red) and Master (blue) Objects

Type 16 – tied Slave: nodes / Master: solids (16-node shell)

Type 16 – sliding Slave: nodes / Master: solids (16-node shell)

Type 17 – tied Slave: 16-node shell / Master: 16-node shell

Type 17 – sliding Slave: 16-node shell / Master: 16-node shell

Type 7 – Lagrange Multipliers

Slave: nodes / Master: surface (segments)

Type 7 – Penalty method Slave: nodes / Master: surface (segments)

Contact between the balls and the table (sliding or tied depending on the problem):

Fig 21: Definition of slave and master objects for balls/table contacts.

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Interface Slave (red) and Master (blue) Objects

Type 16 – tied Slave: nodes / Master: solids (16-node shell)

Type 16 – sliding Slave: nodes / Master: solids (16-node shell)

Type 17 – tied Slave: 16-node shell / Master: 16-node shell

Type 17 – sliding Slave: 16-node shell / Master: 16-node shell

Type 7 – Lagrange Multipliers

Slave: nodes / Master: surface (segments)

Type 7 – Penalty method Slave: nodes / Master: surface (segments)

Pre-loading: quasi-static gravity loading to reach static equilibrium.

The explicit time integration scheme starts with nodal acceleration computation. It is efficient for the simulation of dynamic loadings. However, a quasi-static simulation via a dynamic resolution method needs to minimize the dynamic effects for converging towards static equilibrium and describes the pre-loading case before the dynamic analysis. Thus, the quasi-static solution of gravity loading on the model shows a steady state in the transient response.

To reduce the dynamic effect, dynamic relaxation can be used (/DYREL in the D01 Engine file). A

diagonal damping matrix proportional to the mass matrix is introduced into the dynamic equation:

with:

being the relaxation value by default, equal to 1.

T being the period to be damped (less than or equal to the largest period of the system).

Thus, a viscous stress tensor is added to the stress tensor:

In an explicit code, the application of the dashpot force modifies the velocity equation:

without relaxation

with relaxation

with:

This option is activated in the D01 file using /DYREL (inputs: = 1 and T = 0.2).

The dynamic problem (impact between balls) is considered in a second run managed by the D02 Engine

file with a time running from 30 ms to 130 ms.

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Simulation Results: Kinetic Energy Transmission between Balls during Collision

Type 17 Interface

Contact between quadratic surfaces

Balls/table

contact: tied

Ball/ball contact: sliding

Type 17 Interface

Contact between quadratic surfaces

Balls/table

contact: sliding

Ball/ball contact: sliding

Type 16 Interface

Contact nodes /

quadratic surface

Balls/table

contact: tied

Ball/ball contact: sliding

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Type 16 Interface

Contact nodes /

quadratic surface

Balls/table

contact: sliding

Ball/ball contact: sliding

Type 7 Interface

Lagrange Multipliers

method

Contact nodes / linear

surface (sliding contact)

Type 7 Interface

Penalty method

Contact nodes / linear

surface Balls/table

contact: sliding

Ball/ball contact: sliding

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Conclusion

Interface 16 Tied

Interface 16 Sliding

Interface 17

Tied

Interface 17 Sliding

Interface 7 Lagrange Multipliers

Interface 7

Penalty

Cycles 241392 241385 241387 241385 241385 773099

Error on Energy

-30.8% -1.4% -55.5% -10.8% -1.2% -46.1%

Rolling yes no yes no no no

Momentum Transmission

partial quasi-perfect

partial good good partial

Quadratic surface

master side

master side

master and slave

sides

master and slave

sides no no

A non-elastic collision appears using the type 7 interface Penalty method. After impact, each ball has about half of the initial velocity. The momentum transmission is partial and can be improved by increasing the stiffness of the interface despite the hourglass energy and degradation of the energy assessment.

Error on energy is more noticeable for interfaces using the Tied option, due to taking into account the rolling simulation.

This study shows the high sensitivity of the numerical algorithms for the modeling impact on elastic balls. Regarding the interface type, the kinematics of the problem and the transmission of momentum are more or less satisfactory. Type 16 interface allows good results to be obtained.