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The Coulomb coefficients of friction between balls, lanes, and pins are mysterious and only loosely regulated. This chapter presents two methods for determining these coefficients and describes how the necessary moments of inertia are found. The ball is in contact with the lane as it travels, and subsequently it collides with the pins. The pins collide with themselves and with the lane. This yields 4 coefficients of interest.


Friction measurement via IMU

One way to measure the friction between a ball and lane is to instrument the ball with an inertial measurement unit (a combination of three mutually perpendicular accelerometers and three mutually perpendicular gyroscopes) and monitor its kinematics as it traverses the lane. This experiment determines if a Coulomb friction model is appropriate and determines the coefficient of friction assuming it is.  The 2D planar equations of motion for a ball rolling on a plane with slipping are as follows:

$$m \ddot x = -F$$

$$I \ddot \theta = F r$$

where \(\ddot{x}\) is the scalar linear acceleration, \(\ddot{\theta}\) is the scalar angular acceleration, \(F\) is the friction force, and \(r\) is the radius of the ball. These equations are only for planar motion; the angular velocity cannot be in an arbitrary direction, it must be perpendicular to the plane of motion.


The IMU is placed in a recess in the side of the ball approximately along the angular velocity vector emanating from the center of the ball. Data is stored on flash memory inside the IMU and wirelessly transferred to a computer after every few trials.  The ball is thrown with minimal spin to clearly see how it transitions from sliding to rolling.  The coefficient of friction is

$$\mu = -\frac{I \ddot \theta}{m g r}$$


\(\mu\) is computed from two representative data points in a time history of the angular speed, as shown in the following figure.

IMU data

For the particular ball, lane, and oil pattern used, the overall coefficient is 0.0516.  Split into two piecewise chunks, the coefficients are 0.0633 and 0.0375.


Moment of inertia via trifilar pendulum

The moment of inertia of the ball is essential to compute the coefficient of friction in the above IMU experiment.  The fixed plate used is a board with hooks spaced along a circle of equal diameter to the ball. Strings are cut to equal lengths and tied to the hooks. The axis of interest is marked on the ball, and a tape measure (and other strings) are used to marked 3 equally spaced points along the corresponding diameter. The strings are tied together to form a cradle for the ball, and taped to the ball on bottom and at the marked points along the ball's diameter. The ball is twisted roughly 20 degrees, and the time to complete 50 oscillations is measured. This time yields the natural frequency (\(\omega_{n} = 2 \pi / T\)), and in turn the moment of inertia and radius of gyration come from the following equations:

$$I = \frac{m g r^2}{l \omega_n ^2}$$

$$k = \sqrt{\frac{I}{m}}$$

The setup is shown in the figure below.

Trifilar pendulum

For the ball used in the IMU experiment, all radii of gyration are essentially equal at 2.781".


Friction measurement via inclined plane

A low-tech way to compute a Coulomb coefficient of friction is to place one material atop the other, and tilt the lower body until slip occurs.   \(\mu\) is simply \(\tan \theta\).  The experimental setups are shown below.

Ball-lane friction
Ball-pin friction
Lane-pin friction



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