4. Determine the values of the constants needed for finding 0, from equations 4.11b and 4.12....

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4. Determine the values of the constants needed for finding 0, from equations 4.11b and 4.12. |: - 8 - 2 6 K4 = 1.6296 (2 a.b) D(02):=cos (82)-K +K4-cos (82)+K5 1 K5 := E(82) :=-2.sin(82) 5. Use equation 4.13 to find values of 3 for the crossed circuit. F (e 2) :=K1 + (K4-1) cos(82)+K5 8 3 (82) :=2- (atan2 (2-D (8). -E (8) + E (3 ) - 4D(0 2). F (0 2))) 6. Determine 0; for the position shown and after the crank has moved to the horizontal position. 831 03 (21) 8 31 = 275.1 -deg 32 = 3 (821 +482) 7. Use equations 4.28 to find the transmission angles. 01-031-841 | 2nd Edition, 1999 are 1 *2=32-842 * = 46.1 deg The transmission angle is smaller when the crank is in the horizontal position. 8. Check the Grashof condition of the linkage. Condition (S,L,P,Q) := |SL-S+L PQ+P+Q arg 2 8 32 = 255.9.deg * = 87.5 deg Condition (b, d,a,c) = "non-Grashof" 9. Using equations 4.33, determine the crank angles (relative to the line AD) at which links 3 and 4 are in toggle. return "Grashof" if SL PROBLEM 4-18a Statement: Given: Scale Figure P4-5a to obtain dimensions for the problem. Find the angular displacement of link CD when link AB rotates clockwise from the position shown to horizontal. What is the smallest transmission angle between those two positions? Find the toggle positions of this linkage in terms of the angle of link AB. Measured link lengths: Crank (AB) Rocker (CD) Crank angle for position shown (relative to AD): 2nd Edition, 1999 a : 29 mm c=28mm Crank rotation angle from position shown to horizontal: 821=62-deg 40 2:=-37.deg Two argument inverse tangent 8 atan2 (x, y) return 0.5 if x=0 return atan atan + otherwise Solution: See Figure P4-5a and Mathcad file P0418a. 1. Use equations 4.8a and 4.10 to calculate 84 as a function of 2 (for the crossed circuit). K3 K :== d K :== K2 K -1.5172 K = 1.5714 A(02) := cos(82)-K-K2-cos (82) + K3 B(02):=-2-sin(82) Coupler (BC) Ground link (AD) if x>0 8 = 41 8 4(82) := 2. atan2 (2-A(82). -B(82) + B (0) - 4.4(0) C (82)), 2. Determine 84 for the position shown and after the crank has moved to the horizontal position. 0 41 = 4(021) 182.6+deg 8 21=0 4(0 21 +48 ) 42 42 b=27-mm d=44-mm c(02):=K - (K 2+1) cos(e 2) + K3 = 209.8-deg _a-b+c +d (2.a.c) K3 = 1.7438 4 3. Subtract the two values of 04 to find the angular displacement of link CD when link AB rotates clockwise from the position shown to the horizontal. 48 40 42-841 404 = 27-deg 4. Determine the values of the constants needed for finding 0, from equations 4.11b and 4.12. |: - 8 - 2 6 K4 = 1.6296 (2 a.b) D(02):=cos (82)-K +K4-cos (82)+K5 1 K5 := E(82) :=-2.sin(82) 5. Use equation 4.13 to find values of 3 for the crossed circuit. F (e 2) :=K1 + (K4-1) cos(82)+K5 8 3 (82) :=2- (atan2 (2-D (8). -E (8) + E (3 ) - 4D(0 2). F (0 2))) 6. Determine 0; for the position shown and after the crank has moved to the horizontal position. 831 03 (21) 8 31 = 275.1 -deg 32 = 3 (821 +482) 7. Use equations 4.28 to find the transmission angles. 01-031-841 | 2nd Edition, 1999 are 1 *2=32-842 * = 46.1 deg The transmission angle is smaller when the crank is in the horizontal position. 8. Check the Grashof condition of the linkage. Condition (S,L,P,Q) := |SL-S+L PQ+P+Q arg 2 8 32 = 255.9.deg * = 87.5 deg Condition (b, d,a,c) = "non-Grashof" 9. Using equations 4.33, determine the crank angles (relative to the line AD) at which links 3 and 4 are in toggle. return "Grashof" if SL PROBLEM 4-18a Statement: Given: Scale Figure P4-5a to obtain dimensions for the problem. Find the angular displacement of link CD when link AB rotates clockwise from the position shown to horizontal. What is the smallest transmission angle between those two positions? Find the toggle positions of this linkage in terms of the angle of link AB. Measured link lengths: Crank (AB) Rocker (CD) Crank angle for position shown (relative to AD): 2nd Edition, 1999 a : 29 mm c=28mm Crank rotation angle from position shown to horizontal: 821=62-deg 40 2:=-37.deg Two argument inverse tangent 8 atan2 (x, y) return 0.5 if x=0 return atan atan + otherwise Solution: See Figure P4-5a and Mathcad file P0418a. 1. Use equations 4.8a and 4.10 to calculate 84 as a function of 2 (for the crossed circuit). K3 K :== d K :== K2 K -1.5172 K = 1.5714 A(02) := cos(82)-K-K2-cos (82) + K3 B(02):=-2-sin(82) Coupler (BC) Ground link (AD) if x>0 8 = 41 8 4(82) := 2. atan2 (2-A(82). -B(82) + B (0) - 4.4(0) C (82)), 2. Determine 84 for the position shown and after the crank has moved to the horizontal position. 0 41 = 4(021) 182.6+deg 8 21=0 4(0 21 +48 ) 42 42 b=27-mm d=44-mm c(02):=K - (K 2+1) cos(e 2) + K3 = 209.8-deg _a-b+c +d (2.a.c) K3 = 1.7438 4 3. Subtract the two values of 04 to find the angular displacement of link CD when link AB rotates clockwise from the position shown to the horizontal. 48 40 42-841 404 = 27-deg