III. Kinematic Analysis and Synthesis - JV
The first step of analyzing the kinematics of this mechanism was to determine its mobility using Grueblers equation. There are 6 linkages in this mechanism and 7 lower pairs, giving one Degree of Freedom for this system.
The knowns of my mechanism would be Theta 2, Omega 2, and Alpha 2, for this analysis I am assuming that the motor is running at a constant speed and that there is no angular acceleration. The origin (0,0) of this assembly will be placed at O4.
To determine the positions (theta and lengths), velocities (angular and sliding), and accelerations (angular, sliding, and Coriolis) I split the mechanism into two parts. The first part analyzed was just Links 2,3, and 4. A vector loop equation was used to solve for the length of sliding link 3 from O4 as well as the theta of links 3 and 4.
Once these positions were determined the next step was to find the sliding velocity of link 3 and the angular velocity of link 4.
Finally we can now solve for the accelerations of links 3 and 4 which will allow us to move onto the second part of the mechanism to observe the output sliding link 6.
The remaining part of this mechanism will include links 4,5, and 6, this can conveniently be modeled as a Four bar crank slider as shown below, the equations used are those provided in the textbook also from vector loop analysis.
With all of this done the mechanism could finally be analyzed all together, this was done using MATLAB. A for loop was used to iterate through theta 2 from 0 to 360 degrees, at each iteration the states for each linkage would be calculated using functions created to solve for the relevant positions, velocities and accelerations. These functions were included at the end of the script file to make sharing easier, the script file will be linked in the appendix section of this page. This script file was used in conjunction with the design process to ensure this mechanism would function properly, imaginary numbers were being produced with the textbooks given values so these were tweaked until proper performance was achieved. Two figures were created from these calculations, the first is an animation of the mechanism throughout its full input rotation there is also an arrow to indicate the direction of the Coriolis acceleration for Link 3. The second figure shows the variations of Link 6's (g) position, velocity, and acceleration. The return direction is the positive direction, we can see the vast increase in the velocity as link 6 returns.