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Unformatted text preview: 1 & 1. INTRODUCTION Applications of nonlinear dynamics Structural dynamics (inertial nonlinearities, material or geometric nonlinearities) Automotive systems (dry friction, nonlinear suspensions, engine mounts and isolators, vibrations with impact or clearance) Satellite dynamics, celestial mechanics Chemical reactions Fluid dynamics Stock market modeling Population/Ecosystem dynamics Climate modeling Classification: We deal with systems which can be modeled or put into the following form of a system of ordinary differential equations: & Dots: & & N phase space variables & P system parameters = & 2 2 1 2 N 1 2 P x f (x ,x x , , ,t) = & 1 1 1 2 N 1 2 P x f (x ,x x , , ,t) = & N N 1 2 N 1 2 P x f (x ,x x , , ,t) d dt { } 1 2 N x ,(t),x (t) x (t) : { } 1 2 P , , , : 2 & & If all are independent of time, then such systems are called autonomous . & Systems are Nonautonomous when there is explicit dependence on time in the functions & For autonomous systems, N is the dimension of phase space or of the system, i.e., it is the number of independent variables needed to describe the dynamics of the system i f 's. 1 2 f (....), f (....),.... 1 N (x ,......,x ) & Whenever the right hand side, or functions, are nonlinear in the phase space variables, the system is nonlinear ; e.g., if we have terms like etc., the system is nonlinear (these terms are all nonlinear). & Note that terms like are not nonlinear. & A linear system is a special case of nonlinear systems, and can be written as: 2 1 2 1 x x ,or x & & i 1 1 f (x , , , ,t) & & & = + 1 1 1 2 2 2 N N N x g (t) x x x g (t) A( s,t) x x g (t) 1 x t ,or cost 3 & EXAMPLES: 1. Free planar pendulum position vector is: acceleration is then = = = & & & R R r e , r e e =  && & && 2 R centripetal...
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This note was uploaded on 12/28/2011 for the course ME 580 taught by Professor Na during the Fall '10 term at Purdue UniversityWest Lafayette.
 Fall '10
 NA

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