Homework2sol.fall10

Homework2sol.fall10 - ME 560 Kinematics Fall Semester 2010...

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ME 560 Kinematics Fall Semester 2010 Homework No. 2 Wednesday, September 1st Section 2.12, see pages 77-79, presents five different approaches for the position analysis of planar single degree of freedom linkages. The approaches are illustrated by Example 2.5 which is a sliding- block linkage and Example 2.6 which is a four-bar linkage. This homework focuses on two of these techniques for the position analysis of a planar four-bar linkage with the following link dimensions: Ground Link 1: 16.0 inches Input Link 2: 8.0 inches Coupler Link 3: 14.0 inches Output Link 4: 12.0 inches The fixed X and Y-axes are specified as horizontal and vertical, respectively, and the origin is coincident with the ground pivot of the input link 2. The orientation of the ground link relative to the X- axis is 1 15 θ= D counterclockwise; i.e., oriented above the X-axis. For the input angle 2 30 D counterclockwise from the X-axis: (i) Use Freudenstein's equation to solve for the joint variables 3 θ and 4 θ for the open configuration. A reference is Chapter 10, see Section 10.11, Equations (10.23) - (10.26), pages 451 and 452. Note, however, that these equations are only valid for the X-axis coincident with the ground link (that is, the angle o 1 0 ). (ii) Set up and carry out the Newton-Raphson iteration procedure by hand (see Section 2.8, page 65) to solve for the joint variables 3 θ and 4 θ . For the initial estimates of the coupler angle and the output angle use a scaled drawing of the four-bar linkage in the open configuration. Continue to iterate until 3 θ and 4 θ converge to within 0.01 . D Please show all the steps for each iteration. Three iterations should be sufficient. (iii) Write a general purpose computer program (Matlab is recommended) which uses the Newton- Raphson iteration procedure to solve for the joint variables 3 θ and 4 θ .
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2 Solution to Homework Set 2. Graphical Approach. The link dimensions and the joint angles that are specified in the problem statement are: Ground Link: R 1 = 16 inches θ 1 = 15 ˚ Input Link: R 2 = 8 inches θ 2 = 30 ˚ Coupler Link: R 3 = 14 inches θ 3 = ? ˚ Output Link: R 4 = 12 inches θ 4 = ? ˚ For the specified input angle 2 30 , θ= D the two possible configurations of the four-bar linkage are as shown in Figure 1. The loop 21 4 O AB O is the open configuration of the four-bar linkage and the loop 22 4 O AB O is the closed (or crossed-over) configuration of the four-bar linkage. Figure 1. The graphical solution for the four-bar linkage. Scale: 1 inch = 8 inches. From measurements of the scaled drawing, the answers for the coupler angle and the output angle in the two configurations are: The Open Configuration 4 OABO : θ 3 = 60° and θ 4 = 97° The Closed Configuration 4 OABO: θ 3 = – 57° and θ 4 = – 95°
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3 Trigonometry. The notation for the open and the closed (or the crossed-over) configurations of the four-bar linkage are as shown in Figure 2.
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Homework2sol.fall10 - ME 560 Kinematics Fall Semester 2010...

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