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Introduction of Nonlinear Dynamics into a Undergraduate Intermediate Dynamics Course Bongsu Kang

Department of Mechanical Engineering Indiana University – Purdue University Fort Wayne

Abstract

This paper presents a way to introduce nonlinear dynamics and numerical analysis tools to junior and senior mechanical engineering students through an intermediate dynamics course. The main purpose of introducing nonlinear dynamics into a undergraduate dynamics course is to increase the student awareness of the rich dynamic behavior of physical systems that mechanical engineers often encounter in real world applications, while a more rigorous and quantitative treatment of this subject is left to graduate-level dynamics and vibration courses. Employing relatively simple mechanical systems, well known nonlinear dynamics phenomena such as the jump phenomenon due to stiffness hardening/softening, self excitation/limit cycle, parametric resonance, irregular motion, and the basic concepts of stability are introduced. In addition, the students are introduced to essential analysis techniques such as phase diagrams, Poincaré maps, and nondimensionalization. Equations of motion governing such nonlinear systems are numerically integrated with the Matlab ODE solvers rather than analytically to ensure that the student’s comprehension of the subject is not hindered by mathematics which are beyond the typical undergraduate engineering curricula. Discussions of the subject in the course underline the qualitative behavior of nonlinear dynamic systems rather than quantitative analysis.

Introduction

Typical undergraduate mechanical engineering curricula layout a sequence of dynamics and vibration courses, e.g., dynamics (first dynamics course), kinematics and kinetics of machinery or similar courses, system dynamics, and relevant technical electives. The Bachelor of Science in Mechanical Engineering program at Indiana University – Purdue University Fort Wayne (IPFW) requires three core courses and offers two technical elective courses in the areas of dynamics and vibration. A technical elective course titled “Intermediate Dynamics with Computer Applications” has been recently developed and taught by the author to meet a strong demand from mechanical engineering students for a dynamics course that is oriented to real world engineering applications. This course covers three major topics: three dimensional rigid body dynamics, time-response analysis of dynamic systems, and dynamics of one dimensional distributed parameter systems. Introduction to nonlinear dynamics is treated under the scope of the time-response analysis of dynamic systems. In the first dynamics course, students learn how to apply the Newtonian mechanics to solve dynamics problems of particles and rigid bodies. However, more specifically, what they learn is a static version of dynamics. That is they solve a particular state, “a snap shot of the motion”, of a dynamic system for certain unknown kinematic quantities with the application of relatively simple algebra and trigonometry. From the author’s perspective it is interesting to observe that

Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright © 2005, American Society for Engineering Education

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Kang, B. (2005, June), Introduction Of Nonlinear Dynamics Into An Undergraduate Intermediate Dynamics Course Paper presented at 2005 Annual Conference, Portland, Oregon. https://peer.asee.org/14153