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me304_hw7_sol

# me304_hw7_sol - MIDDLE EAST TECHNICAL UNIVERSITY MECHANICAL...

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ME304/Spring 2005/Sections 02+03+04/HW7Solutions Page 1 / 11 SOLUTIONS TO HOMEWORK 7 Prepared by Atilla BAYRAM (A-148), Gökhan BAYAR (C-204), Erdinç İ Y İ AY (C-209) Problem 1-b i) The unit step response of a system is given as ) 2 cos( ) 2 ( sin 4 1 ) ( t e t e t y t t us = In case of zero initial conditions, the unit impulse response can be determined as ( ) ) 2 sin( 2 ) 2 cos( )) 2 ( cos 2 ( 4 ) 2 ( sin 4 ) ( ) ( t e t e t e t e t y t y t t t t us ui + = = & ( ) ) 2 sin( 2 ) 2 cos( ) 2 ( cos 8 ) 2 ( sin 4 ) ( t t t t e t y t ui + + = ( ) ) 2 ( cos 7 ) 2 ( sin 6 ) ( t t e t y t ui = ii) The unit ramp response can be determined as ( ) τ τ τ τ τ τ τ d e e d y t y t t us ur = = 0 0 ) 2 cos( ) 2 sin( 4 1 ) ( ) ( = t t t ur d e d e d t y 0 0 0 ) 2 cos( ) 2 sin( 4 ) ( τ τ τ τ τ τ τ Note that + + + + = C bx e b a a bx e b a b dx bx e ax ax ax ) ( cos ) sin( ) cos( 2 2 2 2 and + + + + = C bx e b a a bx e b a b dx bx e ax ax ax ) ( sin ) cos( ) sin( 2 2 2 2 Using the above formulas for a = -1 and b = 2, one obtains t t t ur e e e e t y 0 0 0 ) 2 cos( 5 1 ) 2 ( sin 5 2 ) 2 sin( 5 1 ) 2 cos( 5 2 4 ) ( + + = τ τ τ τ τ τ τ τ τ MIDDLE EAST TECHNICAL UNIVERSITY MECHANICAL ENGINEERING DEPARTMENT ME 304 CONTROL SYSTEMS SPRING 2005

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ME304/Spring 2005/Sections 02+03+04/HW7Solutions Page 2 / 11 Rearranging the above expression + = 5 1 ) 2 cos( 5 1 ) 2 ( sin 5 2 5 2 ) 2 sin( 5 1 ) 2 cos( 5 2 4 ) ( t e t e t e t e t t y t t t t ur 5 9 ) 2 sin( 5 2 ) 2 cos( 5 9 ) ( + + = t e t e t t y t t ur The MATLAB plots of y ui (t) and y ur (t) are given below with the M-file. %PLOTTING OF THE UNIT IMPULSE RESPONSE AND UNIT RAMP RESPONSE OF THE SYSTEM FOR PART (b) OF ME 304 – HW 7 - PROBLEM 1. % DEFINE THE DISCRETE TIMES AT WHICH THE PLOT DATA WILL BE OBTAINED % AS FROM t=0 TO t=10 WITH 0.01 INCREMENTS. t=0:0.01:10; % GENERATE yui(t) AND yur(t) VALUES AT THESE DISCRETE TIMES for j=1:1001; yui(j)= exp(-t(j))*(6*sin(2*t(j))-7*cos(2*t(j))); yur(j)=t(j)+(9/5)* exp(-t(j))*cos(2*t(j))+ (2/5)* exp(-t(j))*sin(2*t(j))- (9/5) ; end % PLOT yui(t) VERSUS t figure (1) plot(t,yui,'-') % PUT THE LABEL OF HORIZONTAL AND VERTICAL AXES xlabel('t (s)') ylabel('yui(t)') % PUT THE TITLE OF THE PLOT title('Unit Impulse Response y(t) of the System given in part (b) of Problem 1') % PLOT yur(t) VERSUS t figure (2) plot(t,yur,'-') % PUT THE LABEL OF HORIZONTAL AND VERTICAL AXES xlabel('t (s)') ylabel('yur(t)') % PUT THE TITLE OF THE PLOT title('Unit Ramp Response y(t) of the System given in part (b) of Problem 1')
ME304/Spring 2005/Sections 02+03+04/HW7Solutions Page 3 / 11

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