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Homework2bsol.fall11

# Homework2bsol.fall11 - ME 352 Machine Design I Fall...

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ME 352 - Machine Design I Name of Student ______________________________ Fall Semester 2011 Lab Section Number ___________________________ Homework No. 2. Parts (i)-(iv) due at the beginning of lecture, Wednesday, September 7th. (20 points). Part (v) due at the beginning of lab, Monday, September 12th, or Tuesday, September 13th. (20 points). 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 planar four-bar linkage. This homework focuses on these techniques for the position analysis of a planar four-bar linkage with the following link dimensions: Ground Link 1: 20.0 cm Input Link 2: 9.0 cm Coupler Link 3: 15.0 cm Output Link 4: 17.0 cm 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= ° counterclockwise; i.e., oriented above the X-axis. (i) For the input angle 2 30 θ= D counterclockwise from the X-axis, choose a suitable scale and accurately draw the four-bar linkage in its two possible configurations. Measure the values of the joint variables 3 θ and 4 θ for each configuration. (ii) Use the law of sines and the law of cosines to determine the joint variables 3 θ and 4 θ for the open configuration. (iii) Use the closed-form solution to solve for the joint variables 3 θ and 4 θ . (Recall that the closed- form solution for a planar four-bar linkage is commonly referred to as Freudenstein's equation, see Uicker, et al., Section 10.13, Equations (10.23)-(10.26), pages 451 and 452. Note, however, that these equations are for o 1 0 ). Compare your answers for the two joint variables 3 θ and 4 θ with the answers from Part (i). (iv) Set up and carry out the Newton-Raphson iteration procedure (see Uicker, et al., Section 2.8, page 65) to solve for the joint variables 3 θ and 4 θ (for the open configuration) when the input angle 2 30 . D Use the initial estimates for 3 θ and 4 θ given by Part (i) above. Continue to iterate until 3 θ and 4 θ converge to within 0.01 . D Please show all the steps for each iteration. (v) Write a computer program using Matlab which uses the Newton-Raphson iteration procedure to solve for 3 θ and 4 θ (for the open configuration) given initial estimates for these joint variables (and the input angle 2 θ and the link lengths). To check your program, use the initial estimates given by Part (i) above and print out the joint variables 3 θ and 4 θ when the input angle 2 30 . D

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2 Solution to Homework Set 2. (i) The link dimensions and the joint angles that are specified in the problem statement are: Ground Link: R 1 = 20 cm θ 1 = 15 ˚ Input Link: R 2 = 9 cm θ 2 = 30 ˚ Coupler Link: R 3 = 15 cm θ 3 = ? ˚ Output Link: R 4 = 17 cm θ 4 = ? ˚ For the specified input angle 2 30 θ= D
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Homework2bsol.fall11 - ME 352 Machine Design I Fall...

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