The differential equations of motion for the lateral and torsional deformations of twisted rotating beams are developed for application to helicopter rotor and propeller blades. No assumption is made regarding the coincidence of the neutral, elastic, and mass axes, and the generality is such that previous theories involving various simplifications are contained as subcases to the theory presented in this paper. Special attention is given the terms which are not included in previous theories. These terms are largely coupling-type terms associated with the centrifugal forces. Methods of solution of the equations of motion are indicated by selected examples.

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Differential equations of motion for combined flapwise bending, chordwise bending, and torsion of twisted nonuniform rotor blades

Finite element vibration analysis of rotating timoshenko beams

A finite element technique is used to determine the natural frequencies and the mode shapes of a cantilever plate mounted on the periphery of a rotating disc. The plane of the plate is assumed to make any arbitrary angle with the plane of rotation of the disc. The distributed centrifugal force is resolved into two components—one acting in the plane of the plate and the other normal to the plate. The stresses produced in the middle surface of the plate due to the in-plane forces are first determined. The increase in the bending stiffness of the plate elements due to these in-plane stresses is obtained in a manner similar to that used in the stability analysis of plates. The component of the distributed centrifugal force normal to the plate surface is added to the inertia force.

From the results of computations carried out for various values of the aspect ratio, the speed of rotation, the disc radius and the setting angle, empirical formulae are derived giving the effect of these parameters on the natural frequencies. These empirical formulae are observed to be in agreement with the corresponding known formulae for rotating cantilever beams, when the aspect ratio is high.

Vibration analysis of rotating cantilever plates

Modal analysis and shape optimization of rotating cantilever beams

Initial stresses due to centrifugal forces or other causes change the stiffness characteristics of a solid in a manner which can be simply formulated.The effect is incorporated in a programme using isoparametric solid elements and it is shown that excellent results can be obtained for a diverse range of problems. Indeed the use of ‘reduced integration’ technique allows the extreme problems of thin plates and shells to be treated by the same element.The proposed formulation is particularly useful for turbine and compressor blades where the geometry changes from thick solid section to thin plate or slab.

Free vibration of initially stressed solids, with particular reference to centrifugal-force effects in rotating machinery

Note presenting a Rayleigh energy approach utilizing the bending mode of the nonrotating beam in the determination of the bending frequency of the rotating beam, which is evaluated and is found to give good practical results for helicopter blades. Charts are presented for the rapid estimation of the first three bending frequencies for rotating and nonrotating cantilever and hinged beams.

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Robert T Yntema

Papers

Simplified Procedures and Charts for the Rapid Estimation of Bending Frequencies of Rotating Beams