9 - 4 Instructions: Show procedure (if applicable). An electro-mechanical vibrat

Question

9 - 4 Instructions: Show procedure (if applicable).

An electro-mechanical vibrator is schematically shown below. The electromagnetic force induced in the mass m is F_m = alpha i and the induced voltage is e_34 = alpha v Write your own matlab program to generate the Bode plots (ie A ~ omega on log-log scale, omega on semi-log scale). From the plots, determine the frequency of the voltage source beyond which the amplitude of the vibrator displacement is less than 10% of that when co is near zero. Also, determine the phase angle between the input and output at this driving frequency. Make sure you use ATAN2 function to calculate (omega) and the plot should be a smooth curve. If you see a jump in 2ti in your plot, program in a correction to bring it to a smooth curve. Turn in your plots and the matlab program.

Explanation / Answer

you need to mathmatical expression of above circuit for that

Apply KVL to electrical loop

e=Ri+Ld(i)/dt+e34

e=Ri+Ld(i)/dt+v

taking laplace of above equation

e(s)=Ri(s)+sLi(s)+v(s) .......eq1

and

v=d(x)/dt

taking laplace v(s)=sx(s).......eq2

diffrential equation for machanical system is

F=md(v)/dt+bv+kx

F=md2(x)/dt2+bd(x)/dt+kx

taking laplace of above equation

F(s)=s2mx(s)+sbx(s)+kx(s)

i(s)=s2mx(s)+sbx(s)+kx(s) .........eq3 as F=i

substuting i(s) value of eq3 and v(s) of eq2 in eq1 we get

e(s)=R(s2mx(s)+sbx(s)+kx(s))+sL(s2mx(s)+sbx(s)+kx(s))+sx(s)

e(s)=(R(s2m+sb+k)+L(s3m+s2b+ks)+s)x(s)

so x(s)/e(s)=1/((R(s2m+sb+k)+L(s3m+s2b+ks)+s))

x(s)/e(s)=1/((Rs2m+Rsb+Rk+Ls3m+Ls2b+Lks+s))

x(s)/e(s)=1/((s3Lm+s2(Lb+Rm)+s(Lk++Rb)+Rk)) ........eq4 is the transferfunction for above network

this equation should be used to plot bode diagram

ploting of bode in matlab

i. input all parameter values ie., values of L,m,b,R,k, in command window

ii.create a transfer function by using below matlab syntax

  A=tf([1],[Lm (Lb+Rm) (Lk++Rb) Rk])

iii. plot bode by using following matlab syntax

bode(A)

iv. from bode plot by using data curser we can find frequency for 10% less gain of zero frequency.

v.plot of phase curve by atan2 function

select f=0:0.1:1000

ef=sin(2*pi*f)

xf=ef/(((2*pi*f)3Lm+(2*pi*f)2(Lb+Rm)+2*pi*f(Lk++Rb)+Rk))

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