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Unformatted text preview: ME 5659 Lecture Notes Dr. N. Jalili 1 Department of Mechanical and Industrial Engineering ME 5659: Control and Mechatronics MATHEMATICAL MODELS: Mathematical modeling of a dynamic system refers to the process of describing the system in terms of governing (differential) equations. These equations are typically obtained from either a direct approach or numerical methods (e.g., finite element method). Concerning the direct approaches, there are two different modeling strategies; i ) Newtonian approach, and ii ) Analytical method. The former method is based on deriving the equations of motion using the free-body-diagram of the system and taking into account the effects of external forces applied on the boundary of the system. This typically requires a system decomposition exercise where the dynamic system is considered to be built-up based on its components. The second modeling approach, i.e., the analytical approach, is an energy-based modeling framework in which interactions between different fields (e.g., electrical, mechanical, magnetic) can be conveniently established and presented. Linear vs. Nonlinear Models: It is clear that most natural and practical systems are nonlinear in nature. Examples include gearboxes with inherent backlash, machine components with dry frictions and linear systems possessing dead-zones (due to manufacturing deficiencies) or undergoing large-amplitude vibrations. Figure 1 depicts some demonstrable examples of these naturally nonlinear systems. g m x m k 2 k 1 m k V x Figure 1. Schematic representation of nonlinear systems; a) nonlinear pendulum due to large-amplitude vibration, b) linear mass-spring with dead-zone representing backlash in geared systems, and c) simple model of friction- limited mass-spring system with inherent dry friction. a b c ME 5659 Lecture Notes Dr. N. Jalili 2 The linearized models developed for these naturally nonlinear systems are our own idealization which may not be justifiable. However, for the small-amplitude vibrations considered in this book, linear assumptions can be made. Lumped-parameters vs. Distributed-parameters Models: Similar to linear and nonlinear modeling viewpoints, physical systems can be mathematically modeled as either discrete or continuous systems. All real systems are made of physical parameters that cannot be assumed isolated, and hence, are continuous by their nature. An idealization of these naturally continuous systems is the discretization of these systems into many isolated components that can be described by independent degrees-of-freedom (DOFs). Figure 2 demonstrates this idealization process on a flexible beam where only one mode (fundamental) of vibration is considered when discretizing this continuous system....
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- Spring '10