Kinematic Analysis:
The figure below shows our drawing used for the kinematic analysis. The figure contains the variables we used along with their definitions. Θ2Θ2, theta3 Θ3, theta4Θ4, c, r, and d all change with respect to time. The mechanism’s degrees of freedom was calculated with the equation:
M = 3*L - 3 - 2*J_1 - J_2 (1)
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To perform the analysis we used two separate vector loops to solve for all of the unknown variables. Loop one contains the points O2, A, B, and O1. Vector loop two contains the points C, D, E, O3, and O1. Using these two vector loops, shown in figures 3.2 and 3.3, we were able to solve for all the unknowns for position, velocity, and acceleration. We performed hand calculations to set up our two equations and two unknowns in each case. From there we turned to MATLAB to symbolically solve those equations for our desired unknowns. Then we turned those symbolic equations into MATLAB functions we could call in our simulation. After that we created our point position vectors for A, B, C, D, and E with O2 being at the origin. By taking the time derivatives of the position vectors we also calculated the magnitude of velocity and acceleration over time for each point. Then to end our hand calculations we grouped together the terms for the present coriolis accelerationCoriolis acceleration.
Figure 3.2. Vector Loop 1
Figure 3.3. Vector Loop 2
For our simulation we used a for loop to go through all of the possible values of theta2Θ2, from 0 to two pi, and calculate calculating the resulting Θ_3, c, w3ω3, c_dot, a3α3, c_Ddot, theta4Θ4, r, d, w4ω4, d_dot, a4α4, and d_Ddot. With those resulting values we could then calculate the position, magnitude of velocity, and magnitude of acceleration for each point of interest. All of the hand calculations, MATLAB functions, MATLAB code, and the resulting figures are located in the appendix. The figures below were created in MATLAB and are some of the important figures showing we met our goals.
Figure 3.4 Path of Each Point of Interest
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One challenging part of the analysis was the variable r(t). One reason was because of its dependence on c(t) and the length from point B to O1. Another reason was because depending on theta2, c(t) and r(t) could overlap each other. While coding the analysis in matlab MATLAB we needed to make sure r(t) was correctly handled because without it our mechanism would not simulate properly. A second challenge we encountered was when solving for theta3Θ3, c(t), theta4Θ4, and d(t) since we had two possible solutions to choose from. This meant we had to decide which one was the correct solution when simulating our mechanism. We used trial and error to determine which solutions were the correct ones.