Kinematic Analysis - Catapult

Garrett’s Matlab

Initially, the group encountered difficulties in simulating the Coriolis effect and the motion of a ball being thrown from the mechanism utilizing that movement. As a result, a kinematic analysis was conducted, avoiding the use of traditional Coriolis equations. The basis of this analysis was a comparison of the current radial acceleration on the ball due to gravity with the radial acceleration required to maintain a circular path. The group reasoned that if the arm speed was forcing the ball's angular velocity to be such that the radial gravitational acceleration was less than the radial acceleration required to keep the ball on a circular path, the ball would begin to move radially outwards, accelerating at a rate equal to the difference in accelerations. The equations explaining this are presented below:

acm= -r2 (where acmis radial acceleration for circular motion, r is the current distance from the ball to the pivot, and is the angular velocity of the throwing link.)

ag = -sin()*g (where ag is the radial acceleration from gravity, is the angle of the throwing link, and g is the acceleration from gravity.)

aball  = r2 -sin()*g (where aball  is the radial acceleration of the ball, with positive being outwards)

Using the equations described above, along with the PMKS simulation values, the team developed a MATLAB script to simulate the motion of a ball being thrown from the mechanism for a given linkage and basket length. With this simulation method, the team was able to generate a prediction for the ball's flight path based on the measurements of the final design which is shown below:

Noah’s Matlab

The analysis began with the PKMS model shown below. The kinematics data was exported from it with L2 being yellow and the throwing arm being blue.

The second set of calculations was done using the equations for coriolis effect from the textbook in conjunction with ODE45 in MATLAB. The values for theta, omega, and alpha were assumed to be known as provided from the PKMS export with the MATLAB script only calculating Rp, Vp, and Ap via the equations below with gravity being added to the Y component of Ap as -mg.

The ODE45 portion of the script ran from the greatest theta from the horizon to the moment the ball exceeded an Rp value of 21.5 meaning that it had traveled the 3.5in length of the basket. After ODE45, a simple model of free flight was used assuming constant x velocity and y velocity being affected by gravity. The script ended when the balls Y coordinates reached below 0 indicating that it would have hit the ground.

Originally we knew we could operate at around 10 RPM with our motor but we quickly found that this would not work for our design as we could only throw an estimated 40in. We looked to speed up ]link two to 30 RPM and achieved estimated distances of 200in. This was due to the ball leaving the basket earlier causing a max acceleration vector 30 degrees from the horizon as opposed to parallel to the horizon for the 10rpm case. This would allow the ball to continue to gain altitude after leaving the basket. The simulated data points for the 30rpm case produced an exit velocity of 282.4 in/s or 16.05 in mph and a max acceleration of around 14g. Too late it was learned that the motor could not supply enough torque or did not have enough speed to reach this RPM.

The blue portion of this chart represents the ball being in the basket still and red is free trajectory.

Jason’s Solidworks

We modeled all of the linkages in Solidworks and created an assembly. Using the motion analysis tool, we tested different torque values for the input link and launched the ball until we found one that worked (80 N-mm or 0.7 lb-in). This released the ball at a 55 degree angle from the horizontal. The ball traveled approximately 10 feet. According to the motor spec sheet, this simulation validated that our motor was able to launch the ball. However, this simulation did not account for the friction or inertia of the gearbox and planetary gears. This and the incorrect spec sheet led to the final prototype not performing as expected.