Set6_2DConservativeDynamics

Set6_2DConservativeDynamics - 2-Dimensional Systems...

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1 2-Dimensional Systems (autonomous) Every system is of the form Ex: The damped pendulum   x f(x,x) g(x) 0 g l θ m O R e e R Kinematics: r e R r e e    2 R r e e e e
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2 Applying Newton‟s second law to the particle Let us define two nondimensional variables:   mr F e : m mgsin c  2 r e : T mgcos m 2 nn 2 sin 0 2 n g c g 2m ( mgsin c ) e R F (mgcos T) e g θ m R e e mg T c l
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3 Write the equations in first-order form: (1) We first study the Undamped problem in detail. 1. Equilibria of the system (i.e., when 2. Now, linearize (1) about an equilibrium: 12 xx 2 2 n 2 n 1 x 2 x sin(x ) ( 0)  x x 0) 21 x 0 and x 0, n ,n 1, 2, 3
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4 Let are small perturbations Substituting into equation (1) and 11 x n y 22 x 0 y n 0,1, 2,3 12 n y 0 y (with 0) 2 2 n 1 0 y sin( n y ) Here y and y yy 2 2 n 1 1 2n n1 y (sin( n )cosy cos( n )siny ) ( 1) siny
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5 Assume small variations from equilibrium point: (Taylor Series expansion for small y 1 and y 2 , and retain only linear terms ) Case 1 : n = even 11 2 n 1 22 n 01 yy ( 1) 0  2 1 2 2 n 1 y y y y 12 2 2 n1 We can integrate equations in phase plane dy y first write as dy y 2 n 1 1 2 2 y dy y dy 0 2 2 2 n 1 2 Integrating, this gives ( y y )/ 2 C (cons defines an ellipse around the equilib u t) () ri m
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6 Phase portrait around a center > with(DEtools): phaseportrait([D(x1)(t)=x2(t),D(x2)(t)=-x1(t)],\[x1(t),x2(t)],t=- 7...7,[[x1(0)=0,x2(0)=1],[x1(0)=0,x2(0)=0.5],[x1(0)=0,x2(0)=0.2]],stepsize =0.02,title=`Phase portrait around a center`,colour=magenta,linecolor=[gold,blue,red]);
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7 Case 2 : n = odd Note that there are two special curves: These curves (straight lines for C=0) pass through the equilibrium with specified slope 12 yy 2 2 n 1 2 2 n1 Proceeding along thesame lines, we can integrate dy y as follows : first write dy y 2 n 1 1 2 2 y dy y dy 2 2 2 n 1 2 defines a hyperbola around the equilibrium, C de In pe tegrating ( y y )/ nds on initial co 2 C (c ndit o i nst) ( ons ) 2 n 1 n
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8 Phase portrait around a Saddle > with(DEtools): phaseportrait([D(x1)(t)=x2(t),D(x2)(t)=x1(t)],\[x1(t),x2(t)],t=- 1.5. ..1.5,[[x1(0)=0,x2(0)=0.5],[x1(0)=0.5,x2(0)=0],[x1(0)=0,x2(0)=- 0.5],[x1(0)=-0.5,x2(0)=0]],stepsize=0.02,title=`Phase portrait around a saddle`,colour=magenta,linecolor=[green,gold,blue,red]);
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9 From last time: The equations for pendulum motion are ( undamped pendulum ) The equilibria are at Linearized EOM about each equilibrium: n = even n = odd 12 1 2 2 nn 2 2 1 xx x x x 2 x sinx 0 2 1 x0 x n ,n 0, 1, 2  2 1 2 2 n 1 y y y y 2 1 2 2 n 1 y y y y
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10 Phase portrait for the undamped pendulum > with(DEtools):phaseportrait([D(x1)(t)=x2(t),D(x2)(t)=-sin(x1(t))],\[x1(t),x2(t)],t=- 7...7,[[x1(0)=0,x2(0)=1],[x1(0)=0,x2(0)=2],[x1(0)=0,x2(0)=2.1],[x1(0)=0,x2(0)=- 2],[x1(0)=0,x2(0)=-2.1],[x1(0)=6.28,x2(0)=1],[x1(0)=- 6.28,x2(0)=1],[x1(0)=9.424777962,x2(0)=0],[x1(0)=-2*Pi,x2(0)=2],[x1(0)=-2*Pi,x2(0)=- 2],[x1(0)=2*Pi,x2(0)=2],[x1(0)=2*Pi,x2(0)=-2]],stepsize=0.02, title=`Phase portrait for the undamped Pendulum`,colour=magenta,linecolor=[blue,blue,blue,blue,blue,blue,blue,blue,blue,blue,bl ue,blue]); - 0 -2 2 separatrix saddle center or libration rotation
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This note was uploaded on 12/29/2011 for the course ME 680 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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Set6_2DConservativeDynamics - 2-Dimensional Systems...

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