fa_09lecture_set1_[Compatibility_Mode]

fa_09lecture_set1_[Compatibility_Mode] - 1 & 1....

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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 Non-autonomous 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 University-West Lafayette.

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fa_09lecture_set1_[Compatibility_Mode] - 1 & 1....

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