1) Consider any two body-fixed coordinate systems, F_(BI) and F_(B2) where F_(B2) is related to F_(01) by a positive rotation about the y_(11) axis through an angle (C) (a) Determine the nine angles between the axes of F_(a1) and those of F_(B2) Use this information to write down T_(B2,81) using the direction cosine matrix (b) For a 321 Euler angle sequence from F_(B1) to F_(B2) determine the three angles Theta _(x),Theta _(y) and Theta , Use this Information to write down T_(82,01) using the Euler angle representation of the direction cosine matrix. (c) Determine the eigenaxis and angle that relates F_(B2) to F_(B1) Use this Information to write down T_(B2,B1) using the Euler parameter representation of the direction cosine matrix.

Question

Answer 1

(a) To determine the nine angles between the axes of

and

, we need to consider the rotation that relates

to

. Since

is related to

by a positive rotation about the

axis through an angle (C), we can say that the rotation is of the form:

where

,

, and

are the rotation matrices about the

,

, and

axes, respectively, and

and

are the angles between the

and

axes of

and

, respectively.The nine angles between the axes of

and

are given by:

where

denotes the dot product.Using this information, we can write down the direction cosine matrix

as:

(b) For a 321 Euler angle sequence from

to

, we can determine the three angles

,

, and

as follows:

Using this information, we can write down the direction cosine matrix

using the Euler angle representation as:

(c) To determine the eigenaxis and angle that relates

to

, we need to find the eigenvector and eigenvalue of the transformation matrix

. The eigenaxis is given by the eigenvector, and the angle is given by the argument of the eigenvalue.The transformation matrix

can be written using the Euler parameter representation as:

where

,

, and

are the Euler parameters.Using this information, we can write down the direction cosine matrix

using the Euler parameter representation as:

Soru