Crank-Slider Slider Acceleration Analysis

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))

The real part of

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)

gives the x component of the slider acceleration and the imaginary part gives the y component.

ax = alpha2*r2*sin(theta2) -omega2^2*r2*cos(theta2)
+alpha3*r3*sin(theta3) -omega3^2*r3*cos(theta3)

ay = alpha2*r2*cos(theta2) -omega2^2*r2*sin(theta2)
+alpha3*r3*cos(theta3) -omega3^2*r3*sin(theta3)

Please enter the data to find the accleration of the slider.

Unit Type:
Link lengths (m or ft): r2: r3: r4: rp:
Angles: theta1: theta2: beta:
Angular Velocity: omega2:
Angular Acceleration: alpha2: