Group - E

Group Members

Burdukov, Ilya IgorevichLewczyk, Tomasz

Sithambaram, SasitharanPenmatcha, Bharath

Steszewski, Andrew Joseph

MAE412

Introduction

Objective of this project is to design and optimize a four bar mechanism for a vehicle to climb over obstacle of a given height.

Use different approaches to find

1. Shortest Link2. Reducing Peak Torque.3. Reducing The Fluctuation of the

torque.

Matlab code for solving the 12 equationsMatlab code for solving the 12 equations P1 = x1 +i*y1;P1 = x1 +i*y1; P2 = x2 +i*y2;P2 = x2 +i*y2; P3 = x3 +i*y3;P3 = x3 +i*y3;

delta2 = P2-P1;delta2 = P2-P1; delta3 = P3-P1;delta3 = P3-P1; alpha2 = Alpha2*cv;alpha2 = Alpha2*cv; alpha3 = Alpha3*cv;alpha3 = Alpha3*cv;

Xp = [Mx Nx];Xp = [Mx Nx]; Yp = [My Ny];Yp = [My Ny]; for n = 1:2for n = 1:2 R1 = (x1-Xp(n)) + i*(y1-Yp(n));R1 = (x1-Xp(n)) + i*(y1-Yp(n)); R2 = (x2-Xp(n)) + i*(y2-Yp(n));R2 = (x2-Xp(n)) + i*(y2-Yp(n)); R3 = (x3-Xp(n)) + i*(y3-Yp(n));R3 = (x3-Xp(n)) + i*(y3-Yp(n)); R1mag = abs(R1);R1mag = abs(R1); R2mag = abs(R2);R2mag = abs(R2); R3mag = abs(R3);R3mag = abs(R3); zeta1 = atan2(imag(R1),real(R1));zeta1 = atan2(imag(R1),real(R1)); zeta2 = atan2(imag(R2),real(R2));zeta2 = atan2(imag(R2),real(R2)); zeta3 = atan2(imag(R3),real(R3));zeta3 = atan2(imag(R3),real(R3));

C1 = R3mag*cos(alpha2+zeta3)-C1 = R3mag*cos(alpha2+zeta3)-R2mag*cos(alpha3+zeta2);R2mag*cos(alpha3+zeta2);

C2 = R3mag*sin(alpha2+zeta3)-C2 = R3mag*sin(alpha2+zeta3)-R2mag*sin(alpha3+zeta2);R2mag*sin(alpha3+zeta2);

C3 = R1mag*cos(alpha3+zeta1)-R3mag*cos(zeta3);C3 = R1mag*cos(alpha3+zeta1)-R3mag*cos(zeta3); C4 =-R1mag*sin(alpha3+zeta1)+R3mag*sin(zeta3);C4 =-R1mag*sin(alpha3+zeta1)+R3mag*sin(zeta3); C5 = R1mag*cos(alpha2+zeta1)-R2mag*cos(zeta2);C5 = R1mag*cos(alpha2+zeta1)-R2mag*cos(zeta2); C6 =-R1mag*sin(alpha2+zeta1)+R2mag*sin(zeta2);C6 =-R1mag*sin(alpha2+zeta1)+R2mag*sin(zeta2);

A1 = -C3^2 - C4^2;A1 = -C3^2 - C4^2; A2 = C3*C6 - C4*C5;A2 = C3*C6 - C4*C5; A3 = -C4*C6 - C3*C5;A3 = -C4*C6 - C3*C5; A4 = C2*C3 + C1*C4;A4 = C2*C3 + C1*C4; A5 = C4*C5 - C3*C6;A5 = C4*C5 - C3*C6; A6 = C1*C3 - C2*C4;A6 = C1*C3 - C2*C4;

K1 = A2*A4 + A3*A6;K1 = A2*A4 + A3*A6; K2 = A3*A4 + A5*A6;K2 = A3*A4 + A5*A6; K3 = (A1^2-A2^2-A3^2-A4^2-A6^2)/2;K3 = (A1^2-A2^2-A3^2-A4^2-A6^2)/2;

if n == 1if n == 1 beta3_1 = 2*atan2(K2 + sqrt(K1^2+K2^2-K3^2),K1+K3);beta3_1 = 2*atan2(K2 + sqrt(K1^2+K2^2-K3^2),K1+K3); beta3_2 = 2*atan2(K2 - sqrt(K1^2+K2^2-K3^2),K1+K3);beta3_2 = 2*atan2(K2 - sqrt(K1^2+K2^2-K3^2),K1+K3); beta2_1 = atan2(-(A3*sin(beta3_1)+A2*cos(beta3_1)+A4), ...beta2_1 = atan2(-(A3*sin(beta3_1)+A2*cos(beta3_1)+A4), ... -(A5*sin(beta3_1)+A3*cos(beta3_1)+A6));-(A5*sin(beta3_1)+A3*cos(beta3_1)+A6)); beta2_2 = atan2(-(A3*sin(beta3_2)+A2*cos(beta3_2)+A4), ...beta2_2 = atan2(-(A3*sin(beta3_2)+A2*cos(beta3_2)+A4), ... -(A5*sin(beta3_2)+A3*cos(beta3_2)+A6));-(A5*sin(beta3_2)+A3*cos(beta3_2)+A6)); if (abs(beta2_1-alpha2) < 10^(-6) && abs(beta3_1-alpha3) < 10^(-6))if (abs(beta2_1-alpha2) < 10^(-6) && abs(beta3_1-alpha3) < 10^(-6)) beta2 = beta2_2;beta2 = beta2_2; beta3 = beta3_2;beta3 = beta3_2; elseelse beta2 = beta2_1;beta2 = beta2_1; beta3 = beta3_1;beta3 = beta3_1; endend elseelse sigma3_1 = 2*atan2(K2 + sqrt(K1^2+K2^2-K3^2),K1+K3);sigma3_1 = 2*atan2(K2 + sqrt(K1^2+K2^2-K3^2),K1+K3); sigma3_2 = 2*atan2(K2 - sqrt(K1^2+K2^2-K3^2),K1+K3);sigma3_2 = 2*atan2(K2 - sqrt(K1^2+K2^2-K3^2),K1+K3); sigma2_1 = atan2(-(A3*sin(sigma3_1)+A2*cos(sigma3_1)+A4), ...sigma2_1 = atan2(-(A3*sin(sigma3_1)+A2*cos(sigma3_1)+A4), ... -(A5*sin(sigma3_1)+A3*cos(sigma3_1)+A6));-(A5*sin(sigma3_1)+A3*cos(sigma3_1)+A6)); sigma2_2 = atan2(-(A3*sin(sigma3_2)+A2*cos(sigma3_2)+A4), ...sigma2_2 = atan2(-(A3*sin(sigma3_2)+A2*cos(sigma3_2)+A4), ... -(A5*sin(sigma3_2)+A3*cos(sigma3_2)+A6));-(A5*sin(sigma3_2)+A3*cos(sigma3_2)+A6)); if (abs(sigma2_1-alpha2) < 10^(-6) && abs(sigma3_1-alpha3) < 10^(-6))if (abs(sigma2_1-alpha2) < 10^(-6) && abs(sigma3_1-alpha3) < 10^(-6)) sigma2 = sigma2_2;sigma2 = sigma2_2; sigma3 = sigma3_2;sigma3 = sigma3_2; elseelse sigma2 = sigma2_1;sigma2 = sigma2_1; sigma3 = sigma3_1;sigma3 = sigma3_1; endend endend endend

Matlab Code For SynthesisMatlab Code For Synthesis

Amat = [(exp(i*beta2) -1), (exp(i*alpha2) -1);(exp(i*beta3) -1), Amat = [(exp(i*beta2) -1), (exp(i*alpha2) -1);(exp(i*beta3) -1), (exp(i*alpha3) -1)];(exp(i*alpha3) -1)];

Bvec = [delta2;delta3];Bvec = [delta2;delta3]; X_L = inv(Amat)*Bvec;X_L = inv(Amat)*Bvec; W1 = X_L(1,1); Z1 = X_L(2,1); A = P1-Z1-W1;W1 = X_L(1,1); Z1 = X_L(2,1); A = P1-Z1-W1; Amat = [(exp(i*sigma2) -1), (exp(i*alpha2) -1);(exp(i*sigma3) -1), Amat = [(exp(i*sigma2) -1), (exp(i*alpha2) -1);(exp(i*sigma3) -1),

(exp(i*alpha3) -1)];(exp(i*alpha3) -1)]; Bvec = [delta2;delta3];Bvec = [delta2;delta3]; X_R = inv(Amat)*Bvec;X_R = inv(Amat)*Bvec; U1 = X_R(1,1); S1 = X_R(2,1); A_p = P1-S1-U1;U1 = X_R(1,1); S1 = X_R(2,1); A_p = P1-S1-U1;

Compact Design

How did we find the shortest linklengths?

Solving the vector equations Confining the four bar path to go

through three points We obtained 12 equations and 18

unknowns We were left with 6 free choices

Results:

Sum of Four-links = 0.8685m

Animation for the lowest link lengths

Lowest peak Torque

How did we find the lowest torque? Using Energy Method The main equation used for torque was

Results:Lowest Peak Torque = 0.2665Nm

Mat lab Code For lowest Peak TorqueMat lab Code For lowest Peak Torque

r1 = abs(A_p-A); r2 = abs(W1); r1 = abs(A_p-A); r2 = abs(W1); r3 = abs(A_p+U1-A-W1); r4 = abs(U1);r3 = abs(A_p+U1-A-W1); r4 = abs(U1); Links =[r1,r2,r3,r4];Links =[r1,r2,r3,r4];

suml = sum(Links)suml = sum(Links) if (max(Links)+min(Links) <= suml/2) if (max(Links)+min(Links) <= suml/2)

&& (suml < 89)&& (suml < 89) cond = 1 % Grashofcond = 1 % Grashof Mlink(c1,c2) = suml; Mlink(c1,c2) = suml; else cond = 0 % Non Grashofelse cond = 0 % Non Grashof Mlink(c1,c2) = 9;Mlink(c1,c2) = 9; endend if cond == 1;if cond == 1; y = 1;y = 1; for th2 =0:.1:(2*pi)for th2 =0:.1:(2*pi) %th2 =atan2(imag(W1),real(W1));%th2 =atan2(imag(W1),real(W1)); th3 =atan2(imag(A_p+U1-A-W1),real(A_p+U1-th3 =atan2(imag(A_p+U1-A-W1),real(A_p+U1-

A-W1));A-W1)); th4 =atan2(imag(U1),real(U1));th4 =atan2(imag(U1),real(U1)); V = inv([r3*sin(th3), -r4*sin(th4); -V = inv([r3*sin(th3), -r4*sin(th4); -

r3*cos(th3), r4*cos(th4)]) * [-r3*cos(th3), r4*cos(th4)]) * [-r2*sin(th2)*(1); r2*cos(th2)*(1)];r2*sin(th2)*(1); r2*cos(th2)*(1)];

th2dot = 1;th2dot = 1; th3dot = V(1,1);th3dot = V(1,1); th4dot = V(2,1);th4dot = V(2,1); Acc = inv([r3*sin(th3), -r4*sin(th4); -Acc = inv([r3*sin(th3), -r4*sin(th4); -

r3*cos(th3), r4*cos(th4)]) * r3*cos(th3), r4*cos(th4)]) * [r4*cos(th4)*(th4dot)^2, -r2*cos(th2), -[r4*cos(th4)*(th4dot)^2, -r2*cos(th2), -r3*cos(th3)*(th3dot)^2; r3*cos(th3)*(th3dot)^2; r4*sin(th4)*(th4dot)^2, -r2*sin(th2), -r4*sin(th4)*(th4dot)^2, -r2*sin(th2), -r3*sin(th3)*(th3dot)^2];r3*sin(th3)*(th3dot)^2];

th3ddot = Acc(1,1);th3ddot = Acc(1,1); th4ddot = Acc(2,1);th4ddot = Acc(2,1); m2 = (2690.66) * (.0001) * r2;m2 = (2690.66) * (.0001) * r2; m3 = (2690.66) * (.0001) * r3;m3 = (2690.66) * (.0001) * r3; m4 = (2690.66) * (.0001) * r4;m4 = (2690.66) * (.0001) * r4; I3 = (1/12) * (.0001)*(r3)^3;I3 = (1/12) * (.0001)*(r3)^3; I4 = (1/12) * (.0001)*(r4)^3;I4 = (1/12) * (.0001)*(r4)^3; VxR2 = .5*r2*sin(th2)*th2dot;VxR2 = .5*r2*sin(th2)*th2dot; VxR3 = -r2*sin(th2)*th2dot VxR3 = -r2*sin(th2)*th2dot

- .5*r3*sin(th3)*th3dot;- .5*r3*sin(th3)*th3dot; VxR4 = -.5*r4*sin(th4)*th4dot;VxR4 = -.5*r4*sin(th4)*th4dot; VyR2 = .5*r2*cos(th2)*th2dot;VyR2 = .5*r2*cos(th2)*th2dot; VyR3 = r2*cos(th2)*th2dot VyR3 = r2*cos(th2)*th2dot

+ .5*r3*cos(th3)*th3dot;+ .5*r3*cos(th3)*th3dot; VyR4 = .5*r4*cos(th4)*th4dot;VyR4 = .5*r4*cos(th4)*th4dot; AxR2 = -.5*r2*cos(th2)*(th2dot)^2;AxR2 = -.5*r2*cos(th2)*(th2dot)^2; AxR3 = -(r2*cos(th2)*(th2dot)^2) - AxR3 = -(r2*cos(th2)*(th2dot)^2) -

(.5*r3*cos(th3)*(th3dot)^2 (.5*r3*cos(th3)*(th3dot)^2 + .5*r3*sin(th3)*(th3ddot));+ .5*r3*sin(th3)*(th3ddot));

AxR4 = -(.5*r4*cos(th4)*th4dot^2 + .5*r4*sin(th4)*th4ddot);AxR4 = -(.5*r4*cos(th4)*th4dot^2 + .5*r4*sin(th4)*th4ddot);

AyR2 = -.5*r2*sin(th2)*(th2dot)^2;AyR2 = -.5*r2*sin(th2)*(th2dot)^2; AyR3 = -r2*sin(th2)*(th2dot)^2 - .5*r3*sin(th3)*(th3dot)^2 AyR3 = -r2*sin(th2)*(th2dot)^2 - .5*r3*sin(th3)*(th3dot)^2

+ .5*r3*cos(th3)*(th3ddot);+ .5*r3*cos(th3)*(th3ddot); AyR4 = -.5*r4*sin(th4)*th4dot^2 + .5*r4*cos(th4)*th4ddot;AyR4 = -.5*r4*sin(th4)*th4dot^2 + .5*r4*cos(th4)*th4ddot;

Torque2 = abs((m2*AxR2*VxR2 + m2*VyR2*(AyR2+9.81) + Torque2 = abs((m2*AxR2*VxR2 + m2*VyR2*(AyR2+9.81) + m3*AxR3*VxR3 + m3*VyR3*(AyR3+9.81) + I3*th3ddot*th3dot + m3*AxR3*VxR3 + m3*VyR3*(AyR3+9.81) + I3*th3ddot*th3dot + m4*AxR4*VxR4 + m4*VyR4*(AyR4+9.81) + I4*th4ddot*th4dot));m4*AxR4*VxR4 + m4*VyR4*(AyR4+9.81) + I4*th4ddot*th4dot));

a(y) = Torque2;a(y) = Torque2; y = y+1;y = y+1; endend aa max(a)max(a) end end

Animation For The Lowest Peak TorqueAnimation For The Lowest Peak Torque

Lowest Torque Fluctuation

How did we get the lowest Torque Fluctuation?

Results:Lowest torque fluctuation =

Matlab Code For The Lowest Torque FluctuationMatlab Code For The Lowest Torque Fluctuation

Animation For The Lowest Torque Fluctuation

Solid Edge For Compact Design

Solid Edge For Lowest Torque

Solid Edge For Lowest Torque Fluctuation

Questions?Questions?

The EndThe End