Abstract:

This paper showcases the application of diverse mathematical techniques in the simulation of planar cam mechanisms. The developed model serves as an applied component within the elective course "Mathematical Methods in Engineering Practice", tailored for Bachelor's students in the fields of Mechanical Engineering and Mechatronics. Each of the mathematical topics covered in this course, including fundamental algebra, iterative algorithms for nonlinear systems of equations, coordinate transformations, interpolation techniques, numerical integration and differentiation methods, and the numerical solution of initial-value problems, is explored through the lens of the developed model of a planar cam mechanism. Students engage in a holistic learning experience, where these individual topics are interlinked and collectively employed in the analysis and simulation of the intricacies presented by the model.

In the initial stages of this study, focus is placed on the development of the kinematic and kinetic model of a rotating rod situated on an eccentrically mounted circular disk. The necessary constraint equations for this system can be readily formulated based on geometric considerations. As a preparatory step for modeling cam disks with arbitrary geometries, contour variables are introduced. Both the rod and the circular disk are then defined by discrete points, enabling a more generalized approach. Subsequently, progression occurs to the next phase of investigation through the development of a model featuring an elliptically shaped cam disk. This advancement allows for the exploration of more intricate geometric configurations, laying the groundwork for the comprehensive modeling of cam mechanisms with diverse shapes.

Upon completion of the course, students acquire a comprehensive understanding of various mathematical methods, applying them directly within the realm of computer-aided analysis. Through hands-on experiences, they implement these methods in the computational exploration of a concrete problem. Additionally, students gain insights into challenges inherent in multibody simulation, such as index reduction of differential-algebraic systems of equations and singular mass matrices arising from contour variables.

Keywords:

Mathematical modeling, instructional design, cam mechanisms, multibody simulation.