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ME 320 HW#4 Lec 11 through 14 (Spring2022) 1. The following diagram illustrates a Stephenson III mechanism used to guide a digging tool. The dimensions for the illustrated mechanism is shown in the table below. In the diagram only vectors, W1 and U1 are given. Add remaining vectors of V1, G1, L1, V1*, G1*, U1* in the mechanism diagram below. 2. Repeat Problem 1 for the following Stephenson III mechanism applied for the gripper. W1 U 1 W 1 a 0 b 0 b 0 * U 1 3. For the Stephenson III mechanism used to guide a digging tool, sketch the configuration of mechanism after the crank rotates 60 degrees CCW. The following is the link rotation values solved by Matlab when the crank rotates 60 degrees CCW. The digging arm and bucket positions are provided and you need to sketch corresponding link configuration in the space provided below using hinge positions, a1, b1, b1* after crank rotation from the data provided below. Be sure to add link names of W1, V1, etc. in your sketch. a0 b 0 b 0 * 1 V1 V1* L1 a 1 b 1 b 1 * 60o ao b o b o* X Y ~1 .0 4. For the Watt II mechanism configuration given below, corresponding link parameters are given for =0o. When the crank W1 is rotated 120 o CCW, the following rotation angles for the couplers and followers are given below. These were obtained by solving two loop equations for left and right four bar mechanisms. Sketch the Watt II mechanism for =+120o using the output provided above using hinge positions, a1, a1*, b1, b1* after crank rotation. Be sure to add link names of W1, V1, etc. in your sketch. W1 = 1; th = 90o; V1 = 1.5; rho = 19.3737 o; U1 = 1.5; sigma = 93.2461 o; G1x=1.5; G1y=0; W1*=2; th*=25o; V1*=2; rho*=-59.4144 o; U1*=2; sigma*=-10.8507 o; G1x*=0.866;G1y*=-0.5; W1 U 1 V 1 W 1 * V 1 * U 1 * 120o a1 a 1 * b 1 b 1 * 5. For the following Stephenson III mechanism velocity equations are given below: * * * *( ) ( ) ( ) ( )* * * * 1 1 1 1 0 (1) j j j ji i i i j j j ji W e i L e i V e i U e + + + + + + − = Stephenson III Mechanism : Derive the following acceleration equation for the Stephenson III mechanism from (1) noting that • four bar motion of ( )j j and ( )j j are given • velocity of ( )* *j j are solved by velocity equation * * * * * * * * 1 * * * 1 1 1 1 1 1 2 2 * * * 1 1 1 11 1 Stephenson III: j j j jj j j j j j j j j j j j j j V S U S W C W S L C L S W S W C L S L CV C U C − + + + ++ + + + + ++ + − − − = − + − + − ( ) ( ) * * * * * * * * * * 1 12 2 * * * * 1 1 j j j j j j V C U C V S U S + + + + − 6. Answer the following for synthesizing a four-bar linkage for the following three latch positions using arbitrary angles of 2 3 2 320 , 50 , 20 , 40 = − = = = − when the associated coupler link attached to the hatch has rotations of 2 330 , 50 = − = − . Modify the input for the m-file Synthesis4BarLatch_Lec12.m posted under lecture 12. (1) Based on three precision points of p1, p2, and p3 given above, calculate the vectors 1 1 1 1 1 1 1 1 and T T x y x y x y x yW W Z Z U U S S . (2) Determine two ground hinge positions a0 and b0 for the 4 bar mechanism. (3) Sketch link configuration for latching position 2 and 3, i.e. 2 and 2 for second configuration and 3 and 3 for the third configuration. 7. Synthesize a planar four bar mechanism for the three positions of the bucket shown below. As shown below, the diagram shows only the configuration for the position 1. Complete link configuration of the position 2 and 3 for 2 3 2 330 , 40 , 20 , 40 = − = − = = when the associated coupler link attached to the bucket has rotations of 2 345 , 90 = − = − . You can either sketch it by hand using calculated data or plot them by Matlab. - 45o -90 o Position 1 Position 2 Position 3 a1 b 1 Configuration of 4 bar at Position 1 a1 b 1 8. Derive the following equation in Lecture 13 Stephenson III mechanism synthesis by repeating the derivation for P31. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) * * * * 2 2 2 2 2 * * * * 2 2 2 2 2 3* * * * 3 3 3 3 3 * * * * 3 3 3 3 1 * 21 1 * 211 * 1 31 * 1 31 1 1 1 1 1 1 1 1 x y x y C S C S P CU S C S C P SU V P CC S C S V P S S C S C − − − − − − − = − − − − − − 9. Synthesize a planar four-bar mechanism for a 90 deg CW total rotation range of the follower link and a 60 deg CW crank rotation as shown below. Validate your answer by sketching three precision positions with each link configurations. Use Gj=[1 0]. You can modify any of Matlab programs posted on Webcampus under Lecture 14. Precision Point j (deg) j (deg) j (deg) 1 0 0 0 2 -20 -45 5 3 -40 -90 -5 Position 3 Position 1 Position 2