Analysis of Crank-Slider Angular Acceleration, alpha3

The accelerations of the links is the rate at which their velocities change with respect with time. The accelerations are derived by taking the derivative with respect to time of the velocity vector V1,

d/dt(V1) =d/dt(v1*exp(i*theta1) =d/dt[i*omega2*r2*exp(i*theta2) +i*omega3*exp(i*theta3)]

d/dt(v1)*exp(i*theta1) +i*d/dt(theta1)*v1*exp(i*theta1)=

i*d/dt(omega2)*r2*exp(i*theta2) +i*omega2*d/dt(r2)*exp(i*theta2) +i*i*d/dt(theta2)*omega2*r2*exp((i*theta2) +

i*d/dt(omega3)*r3*exp(i*theta3) +i*omega3*d/dt(r3)*exp(i*theta3) +i*i*d/dt(omega3)*r3*exp(i*theta3)

By definition

d/dt(v1) is a1, d/dt(omega2) is alpha2, d/dt(omega3) is alpha3, and i*i is -1

Substituting these and the previously defined values gives

A1 = a1*exp(i*theta1) = i*alpha2*r2*exp(i*theta2) - omega2*omega2*r2*exp(i*theta2) + i*alpha3*r3*exp(i*theta3) - omega3*omega3*r3*exp(i*theta3)

Multiplying by exp(-i*theta1) yields

a1 =i*alpha2*r2*exp[i(theta2-theta1)] -omega2*omega2*r2*exp[i(theta2-theta1)] +
i*alpha3*r3*exp[i(theta3-theta1)] -omega3*omega3*r3*exp[i(theta3-theta1)]

Equating the imaginary parts and solving for alpha3 gives

alpha3 =[omega2^2*r2*sin(theta2-theta1) +omega3^2*r3*sin(theta3-theta1) -alpha2*r2*cos(theta2-theta1)]/(r3*cos(theta3-theta1))

Please enter the data to calculate the angular acceleration, alpha3.

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

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