Kinematic Analysis is required to review points of interest in the revolution of our triple crank Watt's linkage mechanism. Our goal with this analysis is to achieve a "flicking" motion while striking a ball.

Figure 1. Triple Crank Watt’s Linkage Mechanism (Source)

Determining Link Lengths

Figure 2. Determining Link Lengths in CAD

A snapshot of Figure 2 was imported into CAD, and the lengths of the links were recorded. These lengths were over 27", so the team "normalized" them by dividing by the longest link. Doing so yielded the proportions. We then multiplied these proportions by a scale factor of 4" in order to obtain more feasible link lengths. See table below.

The 4" scaled lengths were used in the the following calculation procedures as well as in manufacturing.

Mobility Calculations

The following schematic was used to model the proposed mechanism:

Figure 3. Mobility Calculations Joint and Linkage References

Gruebler's Equation

Grashof Condition

MotionGen.io Analysis

Using motiongen.io, the team created a preliminary analysis to visualize the path of the links. The path of interest is in joint 6 - the connection between links 5 and 6. The team will attach an end effector here to "kick" the ball. As seen below, there is a peak in the velocity and acceleration at t = 0.2. This signals the "flicking" motion in the rotation which provides a basis for our Python analysis.

Figure 4. MotionGen.io Position Profile

Figure 5. Position Profile of Joint 6

Figure 6. Velocity Analysis of Joint 6

Figure 7. Acceleration Analysis of Joint 6

Python Analysis

Files utilized for this analysis can be found in the team's GitHub repository.

Figure 8. Vector Loop Graphical Representation

Position Profile of Links

Conducting kinematic analysis in Python, the team generated a position profile of the six-bar triple crank Watt's linkage mechanism. 

Figure 9. Position Profile of Links

We noticed that in our best attempt at animating the linkage, the coupler links seem to change in length slightly as they complete a revolution. We attempted to debug this issue but were not successful in doing so. 

Velocity Analysis of Output

Figure 10. Angular Velocity of Joint 6 as a Function of Theta2

Force Analysis

To conduct the force analysis, the team calculated the mechanical advantage of the mechanism as a function of theta2 throughout one revolution.

Mechanical Advantage

As shown in the graph below, the mechanical advantage dips at ~80 degrees. This corresponds to the moment when the end effector has the most velocity (as shown in Figure 13). This dip in mechanical velocity makes sense due to the inverse relationship between force and velocity.

Figure 11. Mechanical Advantage of Link 6 as a Function of Theta2

Velocity Ratio

We were interested in understanding how our output link's velocity (link 6) would change with respect to our input link.

The input to output velocity ratio of a system is inversely related to its mechanical advantage, which measures the force or distance amplification of a machine. When mechanical advantage is high, output force or distance is greater than input, sacrificing speed for increased force. Conversely, when mechanical advantage is low, the system operates faster but with less force. Thus, if a system exhibits a minimum mechanical advantage when velocity is at a minimum, it indicates optimization for speed rather than force at that point.

Figure 12. Theta2 vs Angular Velocity Ratio

The velocity and mechanical advantage were used to understand how we could achieve our objective of imparting the velocity on the ball. Our goal could be achieved by imparting a momentum proportional to the velocity at which the link would strike the ball. However, this means that the highest velocity ratio, is where the lowest mechanical advantage happens, meaning to achieve our goal we would need to reduce the impact forces to make it feasible for our design to impart the velocity on the ball. This led us to select a ping pong ball to be used with our design, given the negligible force it would have on the end effector. 

Acceleration Analysis 

Figure 13. Theta2 vs Angular Acceleration of Link 6

Analysis of Point P

We decided to mount our end effector on Link 5, allowing for a non-circular end-effector curve. The position vector for point P was found by the addition of vectors O₂A and PA. We define PA using the internal link offset angle 𝝳  and the position angle of the coupler link, θ₃.

Notes for Future Iterations

1. The team would like to play around with a more complex calibration procedure, capable of hitting different targets at different distances.