Project Equations (2)

# Project Equations (2) - Deliverable 2 Baker TASK 1 The...

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Deliverable 2 ME 352 – Division 2 Garrett Baker TASK 1 The vector loop equation (VLE) can be written as VECTOR Length Angle KNOWN INPUT KNOWN ? KNOWN ? KNOWN CONSTRAIN T KNOWN KNOWN The constraint for link 3 is the rolling contac t equation. Rolling Contact Equation: In the vector loop equation, is the error vector due to the initial guesses of the two unknown variables. The X and Y components are: and To linearize these two equations, the first-order Taylor’s Series can be written as and where and are referred to as the corrections on the joint angles of links 4 and 5. The coefficients on the left-hand side ; i.e., the partial derivatives, can be written as In matrix form, the two linear equations can be written as Using Cramer’s rule, the two corrections can be written as

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Deliverable 2 ME 352 – Division 2 Garrett Baker and where the determinate is Task 2 : Identify Singular Configurations The singular configurations occur when the determinate of the coefficient matrix is zero i.e., Therefore, the conditions are When the above it true, a special configuration of the linkage is obtained. This is referred to as the toggle positions. For the toggle position, links 4 and 5 are either aligned or are folded on top of each other. In the toggle position, the first- and second-order kinematic coefficients as well as the mechanical advantage will go towards infinity. In the toggle position, only a small change is required to obtain a significant change in the angular displacements of and . [1] Task 3: Solve for the first-order and second-order kinematic coefficients of links 3,4, and 5. VLE: X – Component of VLE: Y – Component VLE: Differentiating Equations with respect to input Equations above have 3 unknowns. The other equation comes from differentiating the rolling contact equation. By definition, is equal to 1 (input link). This gives as
Deliverable 2 ME 352 – Division 2 Garrett Baker Putting equations in Matrix Form: Using Cramer’s Rule (*note – The determinate is the same as in Task 1) Finding second order Kinematic Coefficients Differentiating again with respect to input Equations above have 3 unknowns. The constraint is again, the rolling contact equation. Differentiating the rolling contact equation again gives the second-order kinematic coefficient for That is Putting equations in Matrix Form: Using Cramer’s Rule: and Task 5: Angular Velocity and Angular Acceleration of links 3, 4, and 5. Using first-order kinematic coefficients the equations used to determine the angular velocity of links 3, 4, and 5 are: The equations used to determine the angular acceleration of links 3,4, and 5 are shown below In the equations above, it is important to note that is zero. This is because the angular velocity of the input link, , if said to be constant.

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Deliverable 2 ME 352 – Division 2 Garrett Baker Task 6 : Position of the coupler point P and the first-order and second-order kinematic coefficients of this coupler point. VLE:
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Project Equations (2) - Deliverable 2 Baker TASK 1 The...

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