A New Spin On Teaching 3 D Kinematics And Gyroscopic Motion

Students typically struggle with sophomore level dynamics – this difficulty is compounded when the material is extended to three dimensions. Coriolis acceleration can be daunting in planar kinetics – adding additional rotating frames can totally confuse students. Because most courses only offer a rudimentary coverage of three-dimensional motion, textbook problems are typically posed “at an instant in time.” For convenience, the rotating axes are usually chosen to be coincident with the non-rotating axes, and there are seldom rotations about multiple axes. Because of this, students rarely understand the time dependent nature of three-dimensional kinematics and how body-fixed rotating axes systems change over time. To help address this problem, a project was created that requires students to create a model of a motion-based flight simulator. In addition, hands-on mini-labs demonstrating gyroscopic motion were developed to provide a physical feel for three-dimensional kinematics. In the 3D Matlab simulation project, students were first provided a description of how the simulator moves. The team of 3-4 students created a physical model of the simulator with a representation of the different axis systems. This model was used to help the teams develop coordinate transformation matrices between the different axis systems. The angular velocities for each of the different motors (planetary, pitch, and roll) were provided to the student teams and they were asked to determine (a) the angular velocity and acceleration that a pilot in the gondola would experience and (b) the linear accelerations at the pilot’s head (i.e., the vestibular system). After calculating the inertial tensors for the gondola and the planetary arm, students determined the kinetic energy of the system and the moments applied at different bearings in the system as functions of time. We have found that students gain a good appreciation for the time varying nature of 3D kinematics, and understand how to do problems where the rotating and non-rotating axes are not coincident. An animation using the Matlab Virtual Reality Toolbox allowed the students to visualize the kinematic solution and also provided a visual check that the solution is valid. To help students gain a physical appreciation for gyroscopic motion, a set of mini-labs was created. Although most dynamics instructors routinely use a spinning bicycle wheel as a demo, few students get to experience the motion first hand. Similarly, toy gyroscopes can be used to help teach students about precession and demonstrate how gyroscopic navigational devices operate. These hands-on laboratories can be much more powerful than demonstrations and lecturing – the students can actually feel the gyroscopic moments generated. These demonstrations were assessed through two problems on the final examination. The first asked what happens to the motion of a gyroscope when you push gently on the outer gimble. The second involved the action-reaction moments involved with gyroscopic motion (e.g., if you are riding your bike and lean to the left, which way to do you have to push on your handlebars). Scores on these different problems along with subjective survey results were used to assess the effectiveness of the mini-labs.