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# Ball dynamics

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### Friction model

Several previous investigators have modeled the friction force acting on the ball as Coulomb friction with more or less arbitrary friction coeﬃcients (Hopkins & Patterson, 1977; Frohlich, 2004). The force is modeled as:

$$\mathbf{F}=-\mu m g \frac{\mathbf{v_{con}}}{|\mathbf{v_{con}}|}$$

where vcon is the vector velocity of the contact point.  Alternatively, a viscous friction model could be used:

$$\mathbf{F}=-\mu \frac{A}{h} \mathbf{v_{con}}$$

where $$\mu$$ is the viscosity of the oil, $$A$$ is the contact area, $$h$$ is the depth of the oil layer, and $$v$$ is the velocity of the contact area. The viscosity of the oil $$\mu$$ is required to be between 12 and 81 centiPoise by the USBC. An average of all oils made by Kegel (a major bowling oil manufacturer) was taken to give $$\mu$$ = 34.68cP = 0.03468 kgms$$^2$$ . The contact area $$A$$ can be calculated by Hertz Contact Theory.  It was determined experimentally that Coulomb friction is a more appropriate model (see Experiments).

### Equations of motion with Autolev and Matlab

The equations of motion are complicated greatly by the presence of mass center offset, general inertia tensor (instead of a scalar inertia), and the distinction between slipping and rolling.  In the case of slipping, the 2D position of the ball and four Euler parameters are used as generalized coordinates.  The 2D velocity and 3D angular velocity are the generalized speeds.  For rolling, the same generalized coordinates are used, but only three generalized speeds are needed to describe the motion of the ball.  The three components of angular velocity are chosen.  For either slipping or rolling, using the Newton-Euler method to generate equations of motion would be quite difficult.  And using Kane's method by hand would be quite tedious.  Instead, a symbolic manipulation software, Autolev, is used to facilitate equation generation.  The equations are output in a Matlab-compatible format, but a framework must be coded to integrate them, implement a variable coefficient of friction across and along the lane, detect if the ball has entered the gutter, and to halt the simulation once the ball reaches the end of the lane.  The equations themselves occupy hundreds of lines, but a full trajectory can be computed in just a few seconds.

The output measures include ﬁnal state such as position, speed, and angle, as well as time histories of velocities and angular velocities (see video below).  To facilitate simulation a GUI has been built on top of this to allow input of initial conditions and parameters, oil pattern, and friction model. It also has preset initial conditions used by several previous articles.

### Ball dynamics results

Conclusions from many simulations are briefly presented in the form of bowling myths:

• Balls with higher radii of gyration slide further and achieve greater entry angle.  Mostly true.
• Balls with mass center shifted toward the fingers slide further and achieve greater entry angle.  Mostly true.
• Mass center location is a crock and has negligible effect on ball kinematics.  False.
• Placing the pin (minimum principal axis) near the positive axis point (PAP) makes the axis precess sooner and achieves smaller entry angle.  Placing the pin 6 3/4'' (one quarter of the ball's circumference) from the PAP makes the ball slide further and achieve greater entry angle.  Placing the pin 3 3/8'' from the PAP achieves maximal hook and entry angle.  True.
• A ball with greater difference between its radii of gyration achieves greater entry angle.  True.
• Greater axis rotation gives more slide and entry angle.  There is an intermediate level of axis rotation that maximizes hook.  True.