Interactive Simulation Modules (ISMs) in Vibrations

This is a work-in-progress. A typical course in Vibrations primarily involves the study of the motion of single (SDOF) and multiple degree-of-freedom (MDOF) systems subjected to nonzero initial displacements or velocities (free response) and non-zero forces (forced response). The time-dependent nature of the motion, along with the sheer number of cases that are typically considered in an undergraduate-level class (SDOF: free response-undamped, underdamped, critically damped, overdamped; harmonic force response, periodic force response, general force response; MDOF: free response, forced harmonic response, natural frequencies and mode shapes) leads to many different solutions and approaches that the students are introduced to throughout the term without enabling them to see the physics of each response. To address this deficiency, we have developed eleven interactive simulation modules (ISMs) in MATLAB which depict the motion of the system under the aforementioned responses and allow the student to control many of the parameters of vibration to see the effect of each of them on the response. We are currently implementing these ISMs in a Vibrations course taught by one of the authors. A preliminary assessment of five of the ISMs has indicated that most students have found the ISMs to be very helpful in visualizing and understanding the topics that have been discussed in class. We intend to use the assessment of the students in the class to determine the effectiveness of providing these ISMs and improvements that may be implemented in the future. Introduction and Motivation A course in Vibrations is a mainstay of virtually every undergraduate Mechanical Engineering curriculum, offered either as a mandatory course or a technical elective. At a minimum, an undergraduate course in Vibrations involves the study of the motion of single (SDOF) and multiple degree-of-freedom (MDOF) systems subjected to non-zero initial displacements or velocities (free response) and/or non-zero forces (forced response). As shown in Figure 1, each of these categories contains subcategories that relate to a different physical response. In a typical Vibrations course, the topics shown in Figure 1 are usually supplemented by a discussion on vibration control (which is an application of the SDOF and MDOF forced response), continuous systems and basic finite element analysis (through which continuous systems are approximated as MDOF systems). The breadth of topics that are covered in a typical Vibrations course and the complexities associated with the mathematical solution of each subcategory can often be frustrating and confusing for students. One manner in which we have attempted to simplify the sheer number of mathematical approaches that the students need to consider in our Vibrations course is to consider any linear SDOF and MDOF problem using a state-space approach, in which only four matrices, , , , A B C D , need to be obtained from the system in order to fully evaluate the