Team 24 - Kinematic Analysis & Synthesis

Mobility calculation

Our mechanism has six links (including ground), seven full revolute joints, and no half joints.

M = 3(L-1) - 2J₁-J₂

M = 3(6-1) - 2(7) - 0

M = 1

Therefore, our mechanism has one degree of freedom, as expected and desired.

Position analysis

To conduct a position analysis of our mechanism, we used the vector loop process to solve for all the angles as a function of the input angle. Due to the complex nature of a six-bar mechanism, we needed to use two vector loops and two angle relations based on ternary link geometry to end up with six equations to find the six unknown angles. Our positional vector loops are shown below.

We used the fsolve function in MATLAB to solve these nonlinear equations, resulting in the angles shown in the figure below. Unfortunately, because the functions are not smooth around 90° and 270°, the solver did not adequately converge at those points, leading to some inaccuracy in the calculated angles. Our MATLAB code can be found in the Appendix.

The functioning of our mechanism was mostly dependent on velocities rather than exact position, so we did not solve for the position of any specific points during the operation, especially since we knew the shape of the mechanism’s motion from the simulation tool (see Design Process).

Velocity analysis

After solving for the mechanism’s angles as a function of the input link’s angle, we conducted another vector loop analysis to find the angular velocities of the links.

Once again, we ended up with six equations and six unknowns, but since these equations were linear, MATLAB was able to converge on solutions at every angle. Unfortunately, since these equations depend on the angles we solved for previously, there are resulting inaccuracies in the calculated velocities as well. See the figure below.

Finally, we used the angular velocity of the output ternary link to solve for the linear velocity of the launch point for the dart. Hand calculations and MATLAB plot are shown below. Our MATLAB code can be found in the Appendix.

Animation