MD final problem 2-6 - Machine Dynamics Test Problems...

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Machine Dynamics Test Problems Problem Statement : Given a fourbar linkage with the link lengths L 1 =d=150 mm, L 2 =a=10 mm, L 3 =b=50 mm, L 4 =c=120 mm. For θ 2 =60° find all possible values of θ 3 and θ 4 . Solution : K 1 = d a = 150 10 = 15 K 2 = d c = 150 120 = 1.25 K 3 = a 2 b 2 + c 2 + d 2 2 ac = 10 2 50 2 + 120 2 + 150 2 2 ( 10 )( 120 ) = 14.375 A= cos θ 2 - K 1 -K 2 cos θ 2 + K 3 = cos(60) – 15 – 1.25cos(60) + 14.375 = -0.75 B= -2sin θ 2 = -2sin(60) = -1.732 C= K 1 – (K 2 + 1)cos θ 2 + K 3 = 15 – (1.25 +1) cos(60) + 14.375 = 28.25 θ 4open = 2 arctan B B 2 4 AC 2 A = 2 arctan [ 1.732 1.732 2 4 (− 0.75 )( 28.25 ) 2 (− 0.75 ) ] = 157.77° θ 4crossed = 2 arctan B + B 2 4 AC 2 A = 2 arctan [ 1.732 + 1.732 2 4 (− 0.75 )( 28.25 ) 2 (− 0.75 ) ] = -164.607° K 4 = d b = 150 50 = 3
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K 5 = c 2 d 2 a 2 b 2 2 ab = 120 2 150 2 10 2 50 2 2 ( 10 )( 120 ) = -10.7 D = cos θ 2 - K 1 + K 4 cos θ 2 + K 5 = cos(60) – 15+ 3cos(60) = -13.0 E= -2 sin θ 2 = -2 sin (60) = -1.732 F= K 1 + (K 4 - 1)cos θ 2 + K 5 = 15 + (3 – 1)cos(60)= 5.3 θ 3open = 2 arctan ¿ E E 2 4 DF 2 D = 2 arctan [ 1.732 1.732 2 4 (− 13 )( 5.3 ) 2 (− 13 ) ] = 59.83° θ 3crossed = 2 arctan ¿ E + E 2 4 DF 2
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