Project 2 Report - Analysis

Project 2 Report - Analysis - Analysis A complete kinematic...

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Analysis A complete kinematic analysis of the geared five-bar linkage has been done. The analysis section of this report is broken into four sections. The first section analyzes the position of each link for a given input angle. Also, the singular configurations of the geared five-bar linkage are identified. The second section solves for the first and second order kinematic coefficients of links 3, 4, and 5. Included in this section is the method used to calculate the angular velocity and angular acceleration of links 3, 4, and 5. Section three shows the position analysis of the coupler point P. The approach used to compute first- and second-order kinematic coefficients for the coupler point P are also included in section 3. The equations used to determine the velocity and acceleration of coupler point P are also shown in section three. Section four of the analysis shows how the unit tangent and unit normal vector to the path of coupler point P were calculated. The radius of curvature and the center of curvature of coupler point P is also determined in section four. Section 1 : Position Analysis Figure 3: Vector Loops Used For a given set of link dimensions the objective is to determine the position of each link as the input is rotated. The Newton-Raphson approach was used to solve the position analysis problem of the linkage. Equation 1 below is the vector loop equation that represents the linkage of the geared five bar mechanism.
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(1 ) Where, in Eq. (1) is the error vector due to the initial guesses of the two unknown variables. Table 1 depicts the known and unknown quantities for each vector. Table : Eq. 1 Known / Unknown Quantities VECTOR Length Angle KNOWN INPUT KNOWN ? KNOWN ? KNOWN CONSTRAIN T KNOWN KNOWN It is important to discuss the constraint for gear 3. The change in the angle of gear 3 is related to the gear ratio, radius of gear 2 to gear 3, and the change in the angle of link 2 by Eq. (2) below. Eq. (2) is commonly referred to the rolling contact equation, and will be referred in that manner for the continuation of this paper. (2) Where and represent the radius of gear 2 and gear 3. Notice the negative sign used in equation 2. This is used because of the external contact of gears 2 and 3. Using the known and unknown values along with the rolling contact equation, the joint variables for and can be found using the Newton-Raphson method as follows. The X and Y components of Eq. (1) are (3a) and (3b) To linearize Eq. (3), the first-order Taylor’s Series can be written as
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and (4a) (4b) The terms and in the equations are referred to as the corrections on the joint angles of links 3 and 4. These are the variables that must be solved in order to complete the position analysis using the Newton-Raphson method. The coefficients on the left hand side of Eq. (4) are shown below.
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This note was uploaded on 02/03/2012 for the course ME 352 taught by Professor Staff during the Fall '08 term at Purdue.

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Project 2 Report - Analysis - Analysis A complete kinematic...

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