Synthesis
The original mechanism design was made using the Planar Mechanism Kinematic Simulator (PMKS) web app.
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Initial design of mechanism in PMKS
Mobility
The first step in the kinematic analysis was to use Gruebler's equation to assess the mobility of the device. In this equation, L is the number of links, J1 is the number of full joints, and J2 is the number of half joints. The joint between the first coupler and ground is a half joint because it allows for sliding and rotating motion. M is equal to one, so there is one degree of freedom.
Analysis of First Loop
Position Analysis
The 5-Bar mechanism can be broken into two vector loops as shown. The three vector loop was evaluated first, which was then used to complete the analysis of the four vector loop. Position analysis of the first loop yields the following equations:
Schematic of mechanism showing vector loops
Vector loop equation
Position equations of link 3 angle and length of b
Velocity Analysis
The time derivative of the position vector loop was taken to yield the velocity equations.
Velocity equations of link 3 angle and b
Acceleration Analysis
The second time derivative of the vector loop was taken to yield the acceleration equations.
Acceleration equations of link 3 angle and b
Analysis of Second Loop
Position Analysis
In the second vector loop, the vector from B to D is broken into components. This reduces the number of unknown variables in the equations to two, and makes them solvable.
Vector loop equation
Position equations of link 4 angle and length of e
Additional geometry is needed to solve for the length
Velocity Analysis
Acceleration Analysis
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c. It can be calculated using the law of cosines.
Length of c
Velocity Analysis
The time derivative of the second vector loop was taken to yield the velocity equations.
Velocity equations of link 4 angle and e
Acceleration Analysis
The second time derivative of the second vector loop was taken to yield the acceleration equations.
Acceleration equations of link 4 angle and e
MATLAB Code
Position analysis was conducted in Matlab, however the results are inaccurate, as can be seen from the figure. The problem likely comes from the angle induced by the first coupler link. The slider is offset at an angle, and I had difficulty trying to model that in my analysis.
Matlab figure of position analysis