VIBRATIONS OF ROTATING CIRCUMFERENTIALLY PERIODIC STRUCTURES

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Abstract
The paper considers the free, undamped vibrations of rotating, circumferentially periodic structures. Due to the rotation Coriolis effects are present as well as centrifugal and geometric stiffness effects. It proves to be the presence of Coriolis effects, represented by skew-symmetry in the equations of motion, that the natural modes of the system consist of backward and forward rotating mode shapes. The backward and forward rotating mode shapes with the same wave number inherently vibrate with different frequencies. Therefore, standing natural mode shapes cannot be found in rotating, circumferentially periodic structures (except in those cases where the deflection is the same at corresponding points on every substructure in which case the mode shape may be interpreted as standing). A method of complex constraints is applied to the substructures which reduces the equations of motion for the complete structure to a set of equations of motion pertaining to a single substructure. The number of equations is reduced in the same proportions and the computational benefits are dramatic. As an application the in-plane vibrations of the rotating regular polygon consisting of point masses and massless springs are studied. For this example some interesting features are found. Under certain circumstances double latent roots are found, but there is only one associated latent vector. Under certain other circumstances the possible natural frequencies of vibration are dependent on the angular velocity of rotation. In particular, when the rotational velocity is increased from zero and reaches a threshold value an additional natural mode of vibration appears. In fact, the frequency of vibration of this mode is zero, but as soon as the rotational velocity has increased beyond the threshold value, the natural frequency of vibration of the newly-born mode shape is greater than zero. The phenomenon may be regarded as the birth of a natural mode shape.

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