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tumble

# tumble - Tumble The wobble of a freely spinning body...

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Tumble The wobble of a freely spinning body becomes unstable under a very special condition. The condition is called the tumble condition . The tumble condition is developed below. Begin with Eq. (11.4 – 5). Let Σ M CX = M CY = M CZ = 0 and assume that the body is initially spinning about the Z axis, almost, so the angular velocity components of the body about the X and Y axes are initially small. In Eq. (11.4 – 5), the term ϖ X Y is “small squared” since the terms X and Y are each small. Neglecting the X Y term in Eq. (11.4 – 5), it reduces to ( a – c ) Z X Z Y Z X Y Z Y X Y Z X I I I I I I = - + = - + = 0 ) ( 0 ) ( 0 From ( c ), Z is a constant. Equations ( a ) and ( b ) are now linear, simultaneous equations in terms of X and Y . To decouple the equations, differentiate ( a ) with respect to time and substitute ( b ) into the result eliminating Y . Similarly, differentiate ( b ) with respect to time and substitute ( a ) into the result eliminating X . This yields

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tumble - Tumble The wobble of a freely spinning body...

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