Homework4bsol.spring11

Homework4bsol.spring11 - ME 352 Machine Design I Spring...

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ME 352 - Machine Design I Name of Student __________________________ Spring Semester 2011 Lab Section Number ______________________ Homework No. 4 (30 points). Due at the beginning of lecture on Friday, February 11th. Consider Problem 3.29, see Figure P3.29, page 162, of the text book. In addition to determining the magnitude and direction of the angular velocity of the oscillating follower (that is, link 4) also determine the magnitude and direction of the angular velocity of the roller (that is, link 3). To solve this problem use the following two methods: (i) The method of kinematic coefficients. Clearly document your solution to the position analysis problem. Then clearly show all the steps for the method of kinematic coefficients beginning with the vector loop equation. List the unknown variables, and any constraint equations. In particular, write the rolling constraint equation between the cam and the roller. Use Cramer’s rule to determine the first-order kinematic coefficients of the mechanism. Specify the determinant of the coefficient matrix. Also, determine when the mechanism is in a singular configuration. (ii) The method of instantaneous centers of velocity. Draw the mechanism to a good scale and accurately measure the locations of the instantaneous centers of velocity. Then use the appropriate measurements to determine the first-order kinematic coefficients of the mechanism. (iii) Compare the answers for the angular velocity of the roller (link 3) and the angular velocity of the oscillating follower (link 4) that you obtained from Part (i) with the answers that you obtained from Part (ii).
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2 Solution to Homework 4 (30 Points). Position Analysis. The following data is given in the problem statement, see Figure 1: 2 2 R O B 1.25 inches, = = 7 R BD 2.5 inches, = = 4 4 R O D 3.5 inches, = = 11 2 R O K 3 inches, = = 12 4 R KO 3 inches, = = 2 = 135 , θ ° 11 = 0 , θ ° and 12 = 90 . θ ° There are several methods that could be used for the position analysis of the mechanism, the method that is chosen here is trigonometry. Figure 1. Dimensions of the cam-follower mechanism. The triangle O 2 KO 4 is a right-angled triangle, therefore, the distance from pin 2 O (or A) to pin 4 O (or E) can be written as 2 2 2 4 O O AK KE = + (1a) Substituting the given data into this equation gives 2 2 2 4 O O 3 3 3 2 inches 4.243inches = + = = (1b)
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3 Note that the angles KO 2 O 4 and KO 4 O 2 are both 45°. Also, note that the triangle O 2 BO 4 is a right-angled triangle (this is clear by starting with the angle of link 2 and subtracting the 45° of angle KO 2 O 4 ). Therefore, the distance from point B to the ground pin E (or O 4 ) is 2 2 4 BO AB AE = + (2a) Substituting the given data into this equation gives ( ) 2 2 4 BO 1.25 3 2 4.423inches = + = (2b) For the triangle O 2 BO 4 , the law of sines gives inches rads inches 3 2 sin90 sin O BO 0.959 2 4 4.423 ° = = (3a) Therefore, the angle is 1 O BO sin 0.959 73.58 2 4 = = ° (3b) Since the sum of the interior angles of the triangle O 2 BO 4 is 180 ° then the angle BO O 180 90 73.58 16.42 4 2 = °− °− ° = ° (4) For the triangle BDO 4 , the law of cosines gives 2 2 2 3.5 2.5 4.423 (2)(2.5)(4.423)cos O BD 4 = + (5a) Rearranging this equation gives 2 2 2 2.5 4.423 3.5 cos O BD 0.6133
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