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### Understanding Rotational Dynamics in Inertial and Non-Inertial Coordinate Systems ###

This text delves into the principles of rotational dynamics as they pertain to inertial and non-inertial systems. It discusses the relationship between a rigid body's movement and the linear transformation connecting different coordinate systems. Key concepts include orthogonal matrices, eigenvalues, and eigenvectors, which describe rotation and angular velocity. The preservation of distance and the properties of matrices involved in rotation are highlighted, with emphasis on the significance of orthogonality and symmetry in analyzing the behavior of rotating bodies. ###

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### Understanding Rotational Dynamics in Inertial and Non-Inertial Coordinate Systems ###

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  1. Rotations

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. The angular velocity can also be expressed in the body frame. Body version of matrix Body Rotation x3 x2 x1

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