Determining Angle Solutions for a Spherical Quadrilateral - Prof. Carl D,Iii Crane, Assignments of Mechanical Engineering

Download Determining Angle Solutions for a Spherical Quadrilateral - Prof. Carl D,Iii Crane and more Assignments Mechanical Engineering in PDF only on Docsity! Problem 6.3 A spherical quadrilateral is to be formed from the following four links: α12 = 75°, α23 = 110°, α34 = 60°, α41 = 80° . The value of θ1, the input angle for this case, is 120°. Determine the two sets of solutions for the remaining joint angles of the quadrilateral. The angle θ2 will be solved for first (this was an arbitrary selection) from the spherical cosine law Z12 = c34 . Expanding the definition of Z12 gives s23 (X1s2 + Y1c2) + c23 Z1 – c34 = 0 . This is grouped as A c2 + B s2 + D = 0 where A = s23 Y1 , B = s23 X1 , D = c23 Z1 – c34 Numerical values for the coefficients A, B, and D can be obtained by expanding and evaluating the definitions X1 = s41 s1 = 0.8529 Y1 = -(s12c41 + c12s41c1) = -0.0403 Z1 = c12c41 – s12s41c1 = 0.5206 . This gives A = -0.0379 B = 0.8014 D = -0.6781 Solving the equation Ac2+Bs2+D=0 for θ2 gives θ2A = 125.0° θ2B = 60.4° Corresponding values for θ3 can be obtained from the following fundamental sine and sine- cosine laws X12 = s34 s3 Y12 = s34 c3