QUICK-RETURN MECHANISM SOLUTION BY THE NEWTON-RAPHSON METHOD

Transcription

1 QUICK-RETURN MECHANISM SOLUTION BY THE NEWTON-RAPHSON METHOD

2 Loop Equations 1st loop O K KO O O 0 i i 2 f se a e a e i f s cos a cos a f ssin a sin 0 2

3 Loop Equations 2nd loop O L LM MN NO 0 i i i i f a e a e e a a e 0 f a cos a cos a a f a sin a sin 0 5 5

4 Newton Raphson Method Position Equations f s cos a cos a f ssin a sin 0 2 f a cos a cos a a f a sin a sin J f 0 J f f1 f1 f1 f1 k k k k 5 s k f 2 f2 f2 f 2 f1 k k k k k 5 s f 5 2 f3 f3 f3 f 3 k f s 3 k k k k 5 s f k f f f f k k k k 5 s k k k k k s sin 0 cos 0 k s cos a2 cos 2 a 1 k k k k k s cos 0 sin 0 k s sin 5 a2 sin 2 a sin a sin 0 0 a cos a cos a a k k k k k a cos a5 cos k a sin a5 sin 5 k k k k k 5 5 s

5 Velocity Equations f1 v1 Vs cos ssin 2a2 sin 2 0 t f2 v2 Vs sin s cos 2a2 cos 2 0 t f3 v3 a sin 5a5 sin 5 0 t f v a cos 5a5 cos 5 V 0 t a ssin 0 cos 0 2a2 sin 2 s cos 0 sin 0 5 2a2 cos 2 a 0 sin 5 sin 5 0 0Vs V 0 a cos a5 cos 5 0 1

6 Acceleration Equations scos sin 2 ssin s cos 2a2 sin 2 2 a2 cos 2 0 ssin cos 2 scos s sin 2a2 cos 2 2 a2 sin 2 0 a cos A a s V 1 A a s V 2 A a sin a sin a cos a cos 0 A a 5a5 5 5 a5 5 a a a ssin 0 cos 0 s cos 0 sin 0 5 a sin 5 sin 5 0 0as a a cos 5 cos sin cos sin 0 a cos 5 a5 cos5 a sin a sin 2V s sin s cos 2a2 sin a2 cos 2 2V s cos ssin 2a2 cos a2 sin

7 MATLAB PROGRAM %CLEAR: Variables and command window in MATLAB clc,clear % INPUT: PHYSICAL PARAMETERS of MECHANISM (meter) a1=0;a2=15;a=70;a5=30;a6=22; % INPUT: Maximum Iteration Number Nmax Nmax=100; % INPUT: INITIAL GUESS VALUES for th, th5, s and to respectively x=[30*pi/180,60*pi/180,0,20]; % INPUT: ERROR TOLERANCE xe=0.001*abs(x); % INPUT: SYSTEM INPUTS (th2,w2,al2) dth=5*pi/180; th2=0*pi/180:dth:180*pi/180; w2=-20.9*ones(1,length(th2)); al2=0*ones(1,length(th2)); % xe=transpose(abs(xe)); kerr=1; %If kerr=1, results are not converged

8 for k=1:1:length(th2); for n=1:nmax % %Assign initial guess to unknowns th(k)=x(1);th5(k)=x(2); s(k)=x(3);to(k)=x(); % INPUT: JACOBIAN Matrix J=zeros(,); J(1,1)=-s(k)*sin(th(k));J(1,3)=cos(th(k)); J(2,1)=s(k)*cos(th(k));J(2,3)=sin(th(k)); J(3,1)=-a*sin(th(k));J(3,2)=a5*sin(th5(k)); J(,1)=a*cos(th(k));J(,2)=-a5*cos(th5(k));J(,)=1; % INPUT: Function f f=zeros(,1); f(1,1)=-(s(k)*cos(th(k))-a2*cos(th2(k))-a1); f(2,1)=-(s(k)*sin(th(k))-a2*sin(th2(k))); f(3,1)=-(a*cos(th(k))-a5*cos(th5(k))-a1-a6); f(,1)=-(a*sin(th(k))-a5*sin(th5(k))+to(k)); % eps=inv(j)*f;x=x+transpose(eps);

9 if abs(eps)<xe kerr=0;break end end if kerr==1 'Error nr' end th(k)=x(1);th5(k)=x(2); s(k)=x(3);to(k)=x(); %---velocity fv(1,1)=-w2(k)*a2.*sin(th2(k)); fv(2,1)=w2(k)*a2.*cos(th2(k)); fv(3,1)=0; fv(,1)=0; vel=inv(j)*fv; w(k)=vel(1);w5(k)=vel(2); Vs(k)=vel(3);Vt(k)=vel();

10 %---acceleration fa(1,1)=-al2(k)*a2*sin(th2(k))- w2(k)^2*a2*cos(th2(k))+2*w(k)*vs(k)*sin(th(k))+w(k)^2*s(k)*cos(th(k)); fa(2,1)=al2(k)*a2*cos(th2(k))-w2(k)^2*a2*sin(th2(k))- 2*w(k)*Vs(k)*cos(th(k))+w(k)^2*s(k)*sin(th(k)); fa(3,1)=w(k)^2*a*cos(th(k))-a5*w5(k)^2*cos(th5(k)); fa(,1)=w(k)^2*a*sin(th(k))-a5*w5(k)^2*sin(th5(k)); acc=inv(j)*fa; al(k)=acc(1);al5(k)=acc(2); as(k)=acc(3);at(k)=acc(); End % Angle: radian --> degree th2d=th2*180/pi; thd=th*180/pi; th5d=th5*180/pi; % Plots figure(1), subplot(,3,1),plot(th2d,thd,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('\theta_ (^o)'),xlim([0 180]) subplot(,3,2),plot(th2d,w,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('\omega_ (r/s)'),xlim([0 180]) subplot(,3,3),plot(th2d,al,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('\alpha_ (r/s^2)'),xlim([0 180])

11 subplot(,3,),plot(th2d,th5d,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('\theta_5 (^o)'),xlim([0 180]) subplot(,3,5),plot(th2d,w5,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('\omega_5 (r/s)'),xlim([0 180]) subplot(,3,6),plot(th2d,al5,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('\alpha_5 (r/s^2)'),xlim([0 180]) subplot(,3,7),plot(th2d,s,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('s (cm)'),xlim([0 180]) subplot(,3,8),plot(th2d,vs,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('v_s (cm/s)'),xlim([0 180]) subplot(,3,9),plot(th2d,as,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('a_s (cm/s^2)'),xlim([0 180]) subplot(,3,10),plot(th2d,to,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('t (cm)'),xlim([0 180]) subplot(,3,11),plot(th2d,vt,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('v_t (cm/s)'),xlim([0 180]) subplot(,3,12),plot(th2d,at,'r','linewidth',2),xlabel('\theta_2 (^o)'),ylabel('a_t (cm/s^2)'),xlim([0 180])

12 %% Writing position values to file knm=[transpose(th2d) transpose(thd) transpose(th5d) transpose(s) transpose(to)]; str = 'th2 th th5 s t '; %# A string fname = 'konum.txt'; %# A file name fid = fopen(fname,'w'); %# Open the file if fid ~= -1 fprintf(fid,'%s\r\n',str); %# Print the string fclose(fid); %# Close the file end dlmwrite(fname,knm,'-append',... %# Print the matrix 'delimiter','\t', 'newline','pc'); %% Writing velocity values to file hz=[transpose(th2d) transpose(w) transpose(w5) transpose(vs) transpose(vt)]; str = 'th2 w w5 Vs Vt '; %# A string fname = 'hız.txt'; %# A file name fid = fopen(fname,'w'); %# Open the file if fid ~= -1 fprintf(fid,'%s\r\n',str); %# Print the string fclose(fid); %# Close the file end dlmwrite(fname,hz,'-append',... %# Print the matrix 'delimiter','\t', 'newline','pc');

13 %% Writing acceleration values to file ivme=[transpose(th2d) transpose(al) transpose(al5) transpose(as) transpose(at)]; str = 'th2 al al5 as at '; %# A string fname = 'ivme.txt'; fid = fopen(fname,'w'); if fid ~= -1 fprintf(fid,'%s\r\n',str); fclose(fid); end dlmwrite(fname,hz,'-append',... 'delimiter','\t',... 'newline','pc'); %# A file name %# Open the file %# Print the string %# Close the file %# Print the matrix

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