Analysis of Crank-Slider Angular Velocity, omega3

The angular and translational velocities are the rates at which the positions of the links change with respect to time.
Beginning with the complex loop equation, the velocities are derived by first taking the derivative with respect to time.

d/dt(r2*exp(i*theta2) + r3*exp(i*theta3)) = d/dt(r1*exp(i*theta1) + r4*exp(i*theta4))

d/dt(r2)*exp(i*theta2) + i*d/dt(theta2)*r2*exp(i*theta2) + d/dt(r3)*exp(i*theta3) + i*d/dt(theta3)*r3*exp(I*theta3) =

d/dt(r1)*exp(i*theta1) + i*d/dt(theta1)*r1*exp(i*theta1) + d/dt(r43)*exp(i*theta4) + i*d/dt(theta4)*r4*exp(i*theta4)

For the crank-slider the lengths r2, r3, r4, theta1, and theta4 are constant, thus

d/dt(r2) = d/dt(r3) = d/dt(r4) = 0 and d/dt(theta1) = d/dt(theta4) = 0

By definition:

d/dt(theta2) is omega2, d/dt(theta3) is omega3 and d/dt(r1) is v1

Thus upon substitution, we obtain

V1 = v1*exp(i*theta1) = i*omega2*r2*exp(i*theta2) + i*omega3*r3*exp(i*theta3)

This is the vector describing the velocity of link 1, the slider. In order to find the scalar values omega3 and v1 first multiply both sides by exp(-i*theta1) to produce

v1 =i*omega2*r2*exp[i(theta2-theta1)] + i*omega3*r3*exp[i(theta3-theta1)] =0

Equating the imaginary parts of both sides and solving for omega3 gives

omega3 =[-omega2*r2*cos(theta2-theta1)]/[r3*cos(theta3-theta1)]

Please enter the data to find the angular velocity.

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

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