Vibrations of Pre-Twisted Cantilever Blading

Abstract

This paper reports a study of the vibrations of pre-wisted cantilever blading. The blading is pre-twisted linearly about the centroid of its cross-section up to an angle of π/2 rad and is considered to be mounted encastré at the root. Accurate prediction of the frequencies of vibration of such blading is of considerable practical importance. Two theoretical approaches to the problem are discussed. The first one, involving a direct solution of the equations of motion, is not made use of, although the characteristic equations are derived in an appendix for reference. The second approach is based on the application of Rayleigh's principle. For blades of uniform rectangular cross-section, frequency equations are derived for the fundamental mode of lateral motion and all the modes of torsional motion. Similar types of equation are shown to apply to aerofoil cross- section blading since direct coupling effects between bending and torsion, being extremely small, can be neglected. Vibration tests were conducted on sets of blades of both uniform rectangular and uniform aerofoil cross-section pre-twisted between 0 and π/2 rad, the objects being to determine the frequencies and corresponding nodal patterns contained within a specified frequency band. The effect of pre-twist on the measured frequencies of the lateral, torsional, and, for the aerofoil cross-section, coupled modes of vibration is shown, with certain exceptions, to be considerable. For the modes of motion where a theoretical solution is provided, reasonable agreement is shown to exist between the calculated and corresponding measured frequencies. Hence, the use of Rayleigh's method is justified, giving, as it does, relatively simple expressions for the natural frequencies of vibration. The torsional frequencies of pre-twisted blades are dependent on the torsional stiffness values. Hence, a theoretical study of torsional stiffness variation with pre-twist is presented, the results obtained being confirmed experimentally by a series of simple torsion tests.