III - Kinematic Analyses and Synthesis

First, we determined that the three bar mechanism the links composed of had 1 degree of freedom. The mechanism consisted of three links, two pin joints, and one half joint: the slider joint. Using Grubler’s equation below:

M = 3L - 2J1 - J2 - 3

We found that M = 3(3) - 2(2) - 1 - 3 = 1.

After determining the mechanisms’ mobility, we also conducted position, velocity, acceleration, angular velocity and angular acceleration analysis on the links. We calculated the position of point C versus angle 2, the velocity and acceleration of C versus angle 2, along with the angular velocities and angular accelerations of link 3. The hand calculations and plots are attached below. 

All hand calculations were put into MatLab to generate the plots below. The plots show the movements across one revolution of our links, which was 0 to 90 degrees and back. 

Figure 1: Position of Point C versus Angle 2. 

This plot shows point C as it moves from an input angle of 0 to 90 degrees. At the starting point, point C is at the horizontal distance of 150 mm, which is the length of the two links laying flat. As the servo turns towards 90 degrees, the position of C gets closer to the servo, until it reaches 90 and C is directly above. 

Figure 2: Sliding velocity of C versus Angle 2. 

This plot shows the relationship of the sliding velocity of point C vs input angle. Initially, the velocity is 0 mm/s as the point and links are at rest. But starting the servo, the point moves in the negative x-direction, hence the negative velocity for 0 to 90 degrees. As the links return back to their original position (0 degrees), the velocity of C becomes positive as it goes towards the positive x-direction. 

Figure 3: Angular Velocity of Link 3 versus Angle 2. 

This plot shows that the magnitude of angular velocity of link 3 is always constant. This is because the angular velocity at which the servo turns is constant, making the links have the same constant velocity as well. Going towards the servo, link 3 rotates in the clockwise direction, which is why its angular velocity is negative from 0 to 90 degrees. Going back to its original position, its angular velocity is positive since it's rotating counterclockwise. 

Figure 4: Acceleration of Link 1 versus Angle 2. 

This plot shows the acceleration of link 1, aka point C, as it moves towards and away the servo. The acceleration is negative going towards the servo since C moves in the negative x-direction, and positive going away from the servo since C moves in the positive x-direction. 

Figure 5: Angular Acceleration of Link 3 versus Angle 2. 

Our final plot shows the angular acceleration of link 3 versus angle 2. Theoretically, since the angular velocity of link 3 is always constant, there shouldn't be any angular acceleration. This is why our angular acceleration values are very small, raised to the power of 10^5.