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## Rotations

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**Space coordinates are an inertial system.**Fixed in space Body coordinates are a non-inertial system. Move with rigid body Space and Body x3 x3 x2 x2 x1 x1**A linear transformation connects the two coordinate systems.**The rotation can be expressed as a matrix. Use matrix operations Distance must be preserved. Matrix is orthogonal Product is symmetric Must have three free parameters Matrix Form x3 x2 x1**Axis of Rotation**• An orthogonal 3 x 3 matrix will have one real eigenvalue. • Real parameters • Cubic equation in s • The eigenvalue is unity. • Matrix leaves length unchanged • The eigenvector is the axis of rotation. x3 x2 x1 +1 for right handedness**The eigenvector equation gives the axis of rotation.**Eigenvalue = 1 The trace of the rotation matrix is related to the angle. Angle of rotation c Trace independent of coordinate system Single Rotation**Rotating Vector**• A fixed point on a rotating body is associated with a fixed vector. • Vector z is a displacement • Fixed in the body system • Differentiate to find the rotated vector. x3 x2 x1**The velocity vector can be found from the rotation.**The matrix W is related to the time derivative of the rotation. Antisymmetric matrix Equivalent to angular velocity vector Angular Velocity Matrix**The terms in the W matrix correspond to the components of**the angular velocity vector. The angular velocity is related to the S matrix. Matching Terms**The angular velocity can also be expressed in the body**frame. Body version of matrix Body Rotation x3 x2 x1