Euler Equations. 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.

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Presentation transcript:

Euler Equations

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. x1x1 x2x2 x3x3

Angular Velocity Matrix  The velocity vector can be found from the rotation.  The matrix  is related to the time derivative of the rotation. Antisymmetric matrixAntisymmetric matrix Equivalent to angular velocity vectorEquivalent to angular velocity vector

Matching Terms  The terms in the  matrix correspond to the components of the angular velocity vector.  The angular velocity is related to the S matrix.

Body Frame  The angular velocity can also be expressed in the body frame. Body version of matrixBody version of matrix x1x1 x2x2 x3x3

Body System Tensor  The moment of inertia tensor may change in time. Components are not constant in space systemComponents are not constant in space system Use body system with constant componentsUse body system with constant components

Angular Momentum Change  Differentiate the angular momentum to get the dynamic behavior. Use the change in the inertia tensor J r p

Euler Equations  Choose the body system with principal axes.  Choose the space system to instantaneously coincide with the body system.  The set of three equations are the Euler equations. next