2.4 - Kinematic Analysis

Kinematic Analysis for Final Prototype Design:

For the basic kinematic analysis, we applied the Gruebler Equation to determine the mobility of the system in units of degrees of freedom. This process is documented below:

Gruebler Equation

M = 3(L-1) - 2(J₁) - J₂

M = Mobility in Degrees of Freedom

L = Number of Links

J₁ = Number of Full Joints

J₂ = Number of Half Joints

L = 4

J₁ = 4

J₂ = 0

M = 3(4-1) - 2*4 - 0

M = 9 - 8

M = 1 DOF (Rotation about O1)

We then determined which class of motion that our mechanism is classified under by calculating the Grashof Condition of the system. This calculation is documented below:

Grashof Condition

S + L ? P + Q

S = Shortest Link = 2.125in

L = Longest Link = 4 in

P = Other Link = 4 in

Q = Last Link = 2.625 in

Result:

6.125in < 6.625in

It was determined that the ternary-link model was a member of the Class I Grashof Classification which allows for at least one link in the mechanism to complete a full revolution. This resulted in a crank-rocker mechanism because the link adjacent to S (shortest link of length 2.125in) was grounded.

After calculating the Gruebler and Grashof Condition, we moved to kinematic analysis with equations of motion in order to calculate the position, velocity, acceleration, velocity ratio, and mechanical advantage of points of interest. Specifically, we analyzed the spoon, as it is the point that we want to make a figure-8 path.

Below is the full position, velocity, and acceleration analysis set-up:

Figure 1: Equations of Motion Used for Animation and Analysis

Once we derived the equations of motion that we needed for our specific mechanism, we implemented these equations in a python script to create animations and calculate values of interest to plot. Our animations show the planned motion of the mechanism, as well as current values of angular position, angular velocity, angular acceleration, angular velocity ratio, and mechanical advantage.

Below are the path and mechanism animations with calculation readouts:

Figure 2: Path Profile Animation with Position, Velocity, and Acceleration Calculations of Points A and B

Figure 3: Path Profile Animation with Velocity, Acceleration, Angular Velocity Ratio, and Mechanical Advantage Calculations of Spoon

After creating the animations, we created plots for the position, velocity, acceleration, angular velocity ratio, and mechanical advantage of the spoon as a function of the input angle during one full rotation. As seen in Figure 5, the velocity of the spoon never reaches 0 in/sec, where in the initial inverted parallelogram design, the velocity of the spoon did reach 0 in/sec when it reach the toggle points, which caused all of our issues earlier.

Below are the kinematic analysis plots of the spoon vs. input angle:

Figure 4: Position of Spoon vs. Input Angle

Figure 5: Velocity of Spoon vs. Input Angle

Figure 6: Acceleration of Spoon vs. Input Angle

Figure 7: Angular Velocity Ratio of Spoon vs. Input Angle

Figure 8: Mechanical Advantage of Spoon vs. Input Angle

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