Interactive Four-Bar Linkage Position Analysis Position analysis begins with formulating the loop-closure equation for the fourbar mechanism shown below. R2+ r3= r1+ r4 (1) Incorporating complex numbers, r2*exp(i*theta2)+r3*exp(i*theta3)=r1*exp(i*theta1)+r4*exp(i*theta4) (2) Note link lengths r1,r2,r3, and r4 along with theta1 are constants.

Let theta2 be the independent variable, and theta3 and theta4 be the dependent variables. Rearanging the equation, we have r3*exp(i*theta3)-r4*exp(i*theta4)=r1*exp(i*theta1)-r2*exp(i*theta2). (3) Let R1=r3, phi1=theta3, R2=-r4, phi2=theta4, and z=(x3,y3)=r1*exp(i*theta1)-r2*exp(i*theta2). We now have a general complex equation R1*exp(i*phi1)+R2*exp(i*phi2)=z. (4) Angular positions theta3 and theta4 can now be solved for given parameters r1,r2,r3,r4,theta1, and theta2. From equation (4) we obtain cos(phi1)=(x3-R2cos(phi2))/R1 (5) sin(phi1)=(y3-R2sin(phi2))/R1.

(6) Substituting these results into the trig identity sin^2(phi1)+cos^2(phi1)=1 and simplifying we obtain y3*sin(phi2)+x3*cos(phi2)=(x3^2+y3^2+R2^2-R1^2)/2*R2. (7) From this equation we can obtain formulas for phi1 and phi2 phi2=atan2(y3,x3)± acos((x3^2+y3^2+R2^2-R1^2)/2*R2*sqrt(x3^2+y3^2)) (8) phi1=atan2(sin(phi1),cos(phi1)) =atan2((y3-R2sin(phi2))/R1, (x3-R2cos(phi2))/R1). (9) Similar equations can be derived assuming either theta3 or theta4 is known with the other two angles as parameters. Please enter link lengths, theta1 and one other known angle to find the other two angles.

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