Analysis of Crank-Slider Angular Position, theta3

The angular position, theta3, can be solved by using a loop equation.

R2 + R3 = R1 + R4

A general vector r can be represented in complex polar with two parameters; a length r and an angle theta. Thus each vector can be represented as

R = r*exp(i*theta)

In complex polar form the loop equation becomes

r2*exp(i*theta2) - r3*exp(i*theta3) = r1*exp(i*theta1) + r4*exp(i*theta4)

Rearranging the equation to solve for r1 and theta3 gives

r1*exp(i*theta1) - r3*exp(i*theta3) = r2*exp(i*theta2) - r4*exp(i*theta4)

Converting the right side into Cartesian form where

a=r2*cos(theta2) - r4*cos(theta4)
and
b=r2*sin(theta2) - r4*sin(theta4)
gives
r1*exp(i*theta1) - r3*exp(i*theta3) = a + i*b

Multiplying by exp(-i*theta1) gives

r1 - r3*exp[i(theta3-theta1)] = exp(-i*theta1)*(a + i*b)

and equating the imaginary parts of both sides eliminates r1 and produces

-r3*sin(theta3-theta1) = a*sin(-theta1) + b*cos(-theta1)

Solving for theta3 yields two solutions

theta3 = theta1 + [sin{(a*sin(theta1)-b*cos(theta1))/r3}]^-1

theta3 = theta1 + PI - [sin{(a*sin(theta1)-b*cos(theta1))/r3}]^-1

Please enter the data to calculate the angular position of link 3.

Unit Type:
Link lengths (m or ft): r2: r3: r4: rp:
Angles: theta1: theta2: beta:

Output option: