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hw6_sol - MEEN 364 Fall 2004 Homework Set 6 Due 5:00 PM 1(a...

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MEEN 364 Homework Set 6 Fall 2004 October 8, 2004 Homework Set 6 – Due October 15, 2004 @ 5:00 PM 1. (a) We have (1.1) 0 sin ) ( 2 = + + + θ θ θ θ & & & R g R l To obtain the equilibrium point, the derivative terms are equated to zero. ( ) 0 , 0 = = θ θ & & & 0 ) sin( 0 = θ g π θ n = 0 , n = 0, ± 1, ± 2 The only meaningful value, however is 0 0 = θ (b) Expanding each term in (1.1) by Taylor’s series about the equilibrium point and neglecting the higher order terms, we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 ) 0 ( sin sin ) 0 ( ) 0 ( ) 0 ( 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = + + + + + + + + = = = = = = = = = = = = = = = = = = θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ & & & & & & & & & & & & & & & & & & & & & & R R g g R R R l l 0 0 0 0 0 0 0 = + + + + + + + θ θ g l & & 0 l g θ θ + && = 0 g l θ θ + && = (c) The simulation results for the linear and non-linear cases are given below: 0 1 2 3 4 5 6 7 8 9 10 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 time ( s ) θ ( t ) ( rad ) θ vs time Linear Non-linear 0 1 2 3 4 5 6 7 8 9 10 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 time ( s ) ω ( t ) ( rad ) ω vs time Linear Non-linear 1
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MEEN 364 Homework Set 6 Fall 2004 October 8, 2004 1) M-file ---------------------------------------------------------------------------------------- % single-pendulum function xp=sing(t,x) % linear state-equations for a single-pendulum. % x=[theta thetadot] % xp is the time derivative of the state-vector, x. R=0.2;l=1;g=9.8; x1=x(1); xp(1,1)=x(2); xp(2,1)=-g/l*x1; ---------------------------------------------------------------------------------------- function xp=sing(t,x) % nonlinear state-equations for a single-pendulum. R=0.2;l=1;g=9.8; x1=x(1); % the angle xp(1,1)=x(2); % angular derivative is the angular velocity xp(2,1)=-(g*sin(x1)+R*x(2)*x(2))/(l+R*x1); % Newton's 2nd law ---------------------------------------------------------------------------------------- % plot [t1,x1]=ode45(@Lsing,[0 10],[pi/6 0]'); %[0 10]: time span, [pi/6 0]: initial conditions [t2,x2]=ode45(@sing,[0 10],[pi/6 0]');
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