Whitworth quick return mechanism - Displacement, velocity and acceleration analysis

 Whitworth quick return mechanism - Displacement, velocity and acceleration analysis

In the below picture shown is the whitworth quick return mechanism. Where crank length is L2, fixed link length is L1, oscillating link length is L4 for any given crank angle theta2, angle between vertical and crank is theta2 and angle between vertical and oscillating slotted link is theta4. Analysis of Slider and coupler links are ignored in this analysis.

In the below pictures shown the procedure to find angles theta4, oscillating slotted lever length L4 is described. Clock wise angles considered negative and counter clock wise angles considered positive. Angular velocity of the crank omega2  taken counter clockwise positive and angular acceleration of the crank alpha2  counter clockwise taken as positive.

In the below pictures described the procedure to find angular velocity of the slotted link and angular acceleration of the slotted link. Also described the procedure to find velocity L4 dot. and acceleration L4 double dot. along the slotted link.

After having derived all the necessary equations, a problem is solved using Microsoft excel and the results are plotted in the below pictures. In this problem crank length L2 = 240 mm, fixed link length L1 = 150 mm, angular velocity of the crank omega2 is 0.8 rad/sec. and angular acceleration alpha is 0 rad/sec^2.

Crank and slotted lever quick return mechanism - Displacement, Velocity and acceleration analysis

 Crank and slotted lever quick return mechanism - Displacement, Velocity and acceleration analysis

In the below picture shown is a crank and slotted lever quick return mechanism. Where crank length is r, fixed link length is 3r, oscillating link length is x for any given crank angle theta, angle between horizontal and oscillating slotted link is phy. Slider and coupler link are ignored in this analysis.

In the below pictures shown the procedure to find angles phy, oscillating slotted lever length x is described. Clock wise angles considered negative and counter clock wise angles considered positive. Angular velocity of the crank omega  taken counter clockwise positive and angular acceleration of the crank alpha  counter clockwise taken as positive.

In the below pictures described the procedure to find angular velocity of the slotted link and angular acceleration of the slotted link. Also described the procedure to find velocity x dot. and acceleration x double dot. along the slotted link.

After having derived all the necessary equations, a problem is solved using Microsoft excel and the results are plotted in the below pictures. In this problem crank length r = 200 mm, fixed link length 3r = 600 mm, angular velocity of the crank omega is 10 rad/sec. and angular acceleration alpha is 0 rad/sec^2. In the below pictures read L4 as x, L4 dot. as x dot and L4 double dot as x double dot.

Crank and slotted lever quick return mechanism - Torque and Force analysis

 Crank and slotted lever quick return mechanism - Torque and Force analysis

In the below picture shown is the Crank and slotted lever quick return mechanism (Slider and coupler link is not shown), length of the fixed link is L1, length of crank is L2, angle between vertical axis and crank is theta 2, angle between slotted link and vertical axis is theta 3, torque on the crank is T2 and reaction torque on the slotted link is T3. Length of slotted link at any given crank angle theta 2 is L3, the procedure to calculate the same is discussed below. (Note: In the below pictures read theta 1 as theta 2)

In the below picture described the procedure to calculate slotted link length L3 at any given crank angle theta 2 and angle between vertical axis and slotted link Theta 3 for any given crank angle theta 2.

In the below picture shown, torque on the crank is T2 and reaction torque on the slotted link is T3.

In this analysis friction is neglected along slotted link, so force along the slotted link is zero.

In the below picture described the procedure to calculate reaction torque on the slotted link. 

In the above pictures derived an equation to calculate reaction torque on the slotted link due to applied torque T2 on the crank. In the below picture a problem is solved using Microsoft excel. In this problem, crank length L2 = 62.5 mm, fixed link length L = 150 mm and torque on the crank T = 106.38 Nm

In the below picture Theta 2 along horizontal axis and Theta 3 along vertical axis is plotted for crank angle from zero degrees to 360 degrees.

In the below picture Theta 2 along horizontal axis and Torque T3 along vertical axis is plotted for crank angle from zero degrees to 360 degrees.

Whitworth quick return mechanism - Torque and Force analysis

 Whitworth quick return mechanism - Torque and Force analysis

In the below picture shown is the whitworth quick return mechanism (Slider and coupler link is not shown), length of the fixed link is L1, length of crank is L2, angle between vertical axis and crank is theta 2, angle between slotted link and vertical axis is theta 3, torque on the crank is T2 and reaction torque on the slotted link is T3. Length of slotted link at any given crank angle theta 2 is L3, the procedure to calculate the same is discussed below.

In the below picture described the procedure to calculate slotted link length L3 at any given crank angle theta 2 and angle between vertical axis and slotted link for any given crank angle theta 2.

In the below picture shown, torque on the crank is T2 and reaction torque on the slotted link is T3, angle between crank and slotted link for any given crank angle theta 2 is Phy.

In this analysis friction is neglected along slotted link, so force along the slotted link is zero.

In the below picture described the procedure to calculate reaction torque on the slotted link. 

In the above pictures derived an equation to calculate reaction torque on the slotted link due to applied torque T2 on the crank. In the below picture a problem is solved using Microsoft excel. In this problem, crank length L2 = 250 mm, fixed link length L = 150 mm and torque on the crank T = 2.5 Nm

In the below picture Theta 2 along horizontal axis and Theta 3 along vertical axis is plotted for crank angle from zero degrees to 360 degrees.

In the below picture Theta 2 along horizontal axis and Torque T3 along vertical axis is plotted for crank angle from zero degrees to 360 degrees.

Stamping mechanism kinematic analysis - Applications of four bar mechanism

 Stamping mechanism kinematic analysis - Applications of four bar mechanism

In the below picture shown is a Stamping mechanism, an example of four bar linkage / mechanism. When the handle moved to right or left, the punch at E moves up and down. In this article explained the procedure to derive equations to find position of punch with respect to the handle position varying from 45 degrees to 135 degrees and crank angle versus punch height H is plotted.

 

Length of the fixed link is L1, length of the crank is L2, length of the coupler is L3, length of the follower link is L4 and length of the link DE is L5. Angle between crank / handle and fixed link is theta 2 and height of the punch from fixed link is H.

In the above pictures explained the procedure to derive equations, now let us solve a problem using Microsoft excel. In this problem length of the link L1 is 220 mm, L2 is 240 mm, L3 is 250 mm, L4 is 240 mm, L5 is 210 mm and theta 5 is 50 degrees.

In the below picture crank angle along horizontal axis and punch height along vertical axis is plotted.