Kinematic Analysis and Synthesis - JJ

Mobility Analysis: 

After determining the different materials and dimensions of the six-bar window mechanism, the next step was to evaluate the kinematics of my design. The first step of the kinematic analysis involved determining the mobility of the entire assembly. This was possible by applying Gruebler’s equation. 

The mobility of the entire assembly was determined to be 1. 

Kinematic Analysis: 

The next step of the Kinematic Analysis was to determine the different positions, velocities, and accelerations of each link. I was able to determine these variables by simplifying the vector loop equation. Essentially, two different vector loop equations were used to evaluate the positions, velocities, and accelerations of each different link. 

Loop 1 

Figure 3

Figure 3 above defines the different directions and notations for the vector loop equation. The vector loop equation is defined as below: 

The position equations are defined as below: 

The velocity equations are defined as below: 

As the slider link travels, the length of vector c will change over time. This makes the vector c time dependent. 

The acceleration equations are defined as below: 

Loop 2 

Figure 4

Figure 4 above defines the different directions and notations for the vector loop equation. The vector loop equation is defined as below: 

The position equations are defined as below: 

I utilized a local reference plane to simplify the calculations for the position analysis. An angular offset of 90 degrees was used. 

The velocity equations are defined as below: 

As the slider link travels, the length of vector c will change over time. This makes the vector c time dependent. 

The acceleration equations are defined as below: 

MATLAB 

The equations for position analysis, velocity analysis, and acceleration analysis were all imported into Matlab. With input at Link 2, the input crank rotated at 1 rad/sec in the counter-clockwise direction at a constant angular velocity (no angular acceleration). Although I was able to plot the position, velocity, and acceleration analysis of all of the links (excluding Link 1), I encountered some errors. For Link 5, there was an error for the position plot. I believe the results were inaccurate due to the change in the geometric space from local to global. In addition, I had difficulties with the second vector loop equation as I was not too sure about the configuration. I had difficulty determining whether the vector loop was in a closed-loop or open-loop configuration. The plots for these links are available in the appendix; the Matlab code will be provided there.