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ME429 Fall 2009 HW4 Solution

# ME429 Fall 2009 HW4 Solution - the relevant MATLAB code...

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Assigned on 26.10.2009 Monday Due date 02.11.2009 Monday, beginning of the lecture ME429 Fall 2009 HW4 - SOLUTION 1) Effective stiffness coefficient of the system can be found from the static deflection relation as follows: st st F k ρ = where 5000 N st F = 0.05 m st ρ = 5 5000 10 N/m 0.05 k = = Steady state amplitude of a system with hysteretic damping is given as follows: 2 2 2 2 1 o n F k X ω η ω = - + a) When the force is applied at resonant frequency, that is n ω ω = , the equation becomes; o F X k η = Then the loss factor can be found as follows: ( )( ) 5 350 0.03182 10 0.11 o F kX η = = = b) 0.5 n ω ω = ( ) 5 3 2 2 2 350 10 4.662 10 m 1 0.5 0.03182 X - = = × - + c) 2 n ω ω = ( ) 5 3 2 2 2 350 10 1.167 10 m 1 2 0.03182 X - = = × - +

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Assigned on 26.10.2009 Monday Due date 02.11.2009 Monday, beginning of the lecture 2) ( ) ( ) 2 * 2 2 2 2 1 1 1 o o F F r i k k X r i r η η η - - = = - + - + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 5 * 2 2 2 2 2 2 2 2 2 2 3 2 5 * 2 2 2 2 2 2 2 2 350 0.03182 1.1137 10 10 Im 1 1 0.03182 1 0.001013 350 1 1 3.5 10 1 10 Re 1 1 0.03182 1 0.001013 o o F k X r r r F r r r k X r r r η η η - - - - × = = = - + - + - + - - × - = = = - + - + - + The graph showing the complex amplitude of the system response is below, along with
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Unformatted text preview: the relevant MATLAB code. clear,clc format long Fs = 5000; d = 0.05; Fo = 350; X = 0.11; k = Fs/d; eta = Fo/(k*X); r1 = 0.5; r2 = 2; X1 = abs((Fo/k)/(1-r1*r1+i*eta)); X2 = abs((Fo/k)/(1-r2*r2+i*eta)); r=0:0.001:2; for j = 1:numel(r) X(j) = (Fo/k)/(1-r(j)*r(j)+i*eta); end plot(real(X),imag(X)) xlabel( 'real part of the complex amplitude [m]' ) ylabel( 'imaginary part of the complex amplitude [m]' ) grid on n = find(r==1) text(real(X(n)),imag(X(n)), '\leftarrow \omega = \omega_n' , 'FontSize' ,18) Assigned on 26.10.2009 Monday Due date 02.11.2009 Monday, beginning of the lecture -0.06-0.04-0.02 0.02 0.04 0.06-0.12-0.1-0.08-0.06-0.04-0.02 real part of the complex amplitude [m] imaginary part of the complex amplitude [m] ← ω = ω n...
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ME429 Fall 2009 HW4 Solution - the relevant MATLAB code...

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