Question

1. The orientation of a reference frame F2 is obtained from the reference frame F by a 2 - 3 - 1 rotation sequence, with angles Oy, 0, and ex: • A rotation dy about the y-axis of frame F1, • A rotation 0, about the z-axis of the intermediate frame Fi, • A rotation 0x about the z-axis of the transformed frame Ft. (note that we used rotation sequence 3 – 2 – 1 in class to find the Eulerian angles.) (a) Obtain the rotation matrix C21. (b) Using the result in (a), obtain expressions for computing 02, 0y and 0, from the rotation matrix. (c) Where is the singularity of the 2 – 3 – 1 rotation sequence? What does this physically mean? (d) Given that frame F2 rotates with angular velocity Wa1 = FI w21 relative to frame Fi, obtain the kinematical relationship between w21 and (e) Using the result in (d), what happens to the kinematical relationship if the angles 0x, Dy, 02 and rates èx, dy, 7, are small?

Answer 1

Now putting Equation this substitution , we get, in the tvcotx = cosecx . dx Now, integrating factor of EQ @ is. =) Ref & & Scotx dx. & & locsinx). u sinx. Now, multiplying f= sinx both sides of a we get oor du at V cotx sinx à Cosecx sinx. da = sinx du txcosx=1. -) de(2x crsina) = 1 & alreina)=dx. Integrating on both sides, we get. - Usinxe sdx. I u X&C, where c is - mga sinx=xte. - sink sy = xtc. constant,