Closed form solutions for the dynamic response of Euler–Bernoulli beams with step changes in cross section

This paper focuses on the nonlinear oscillations and the steady-state responses of a thin-walled compressor blade of gas turbine engines with varying rotating speed under high-temperature supersonic gas flow. The rotating compressor blade is modeled as a pre-twisted, presetting, thin-walled rotating cantilever beam. The model involves the geometric nonlinearity, the centrifugal force, the aerodynamic load and the perturbed angular speed due to periodically varying air velocity. Using Hamilton’s principle, the nonlinear partial differential governing equation of motion is derived for the pre-twisted, presetting, thin-walled rotating beam. The Galerkin’s approach is utilized to discretize the partial differential governing equation of motion to a two-degree-of-freedom nonlinear system. The method of multiple scales is applied to obtain the four-dimensional nonlinear averaged equation for the resonant case of 2:1 internal resonance and primary resonance. Numerical simulations are presented to investigate nonlinear oscillations and the steady-state responses of the rotating blade under combined parametric and forcing excitations. The results of numerical simulation, which include the phase portrait, waveform and power spectrum, illustrate that there exist both periodic and chaotic motions of the rotating blade. In addition, the frequency response curves are also presented. Based on these curves, we give a detailed discussion on the contributions of some factors, including the nonlinearity, damping and rotating speed, to the steady-state nonlinear responses of the rotating blade.

Analysis on nonlinear oscillations and resonant responses of a compressor blade

A new dynamic finite element (dfe) formulation for lateral free vibrations of euler–bernoulli spinning beams using trigonometric shape functions

Fundamental vibration of rotating cantilever blades with pre-twist

The governing differential equations and the boundary conditions for the coupled bending-bending-extensional vibration of a rotating nonuniform beam with arbitrary pretwist and an elastically restrained root are derived by Hamilton’s principle. The semianalytical solution procedure for an inextensional beam without taking account of the coriolis forces is derived. The coupled governing differential equations are transformed to be a vector characteristic governing equation. The frequency equation of the system is derived and expressed in terms of the transition matrix of the vector governing equation. A simple and efficient algorithm for determining the transition matrix of the general system with arbitrary pretwist is derived. The divergence in the Frobenius method does not exist in the proposed method. The frequency relations between different systems are revealed. The mechanism of instability is discovered. The influence of the rotatory inertia, the coupling effect of the rotating speed and the mass moment of inertia, the setting angle, the rotating speed and the spring constants on the natural frequencies, and the phenomenon of divergence instability are investigated. @DOI: 10.1115/1.1408615#

/pdf/the-instability-and-vibration-of-rotating-beams-with-10vbl4zutm.pdf

The Instability and Vibration of Rotating Beams With Arbitrary Pretwist and an Elastically Restrained Root

Coupled bending–bending–torsion vibration analysis of rotating pretwisted blades: an integral formulation and numerical examples

Various methods have been developed for the determination of the dynamic characteristics of rotating beams or blades [1]. This note follows the steps presented in [2] and a finite element discretisation of the obtained equations of motion is performed. The main features of this approach are the determination of the mass matrix, Coriolis matrix and the stiffening effect of the rotation in the stiffness matrix in the manner of [3] and using the symbolic code Mathematica.

V. Anghel

Papers

Coupled bending–bending–torsion vibration analysis of rotating pretwisted blades: an integral formulation and numerical examples

A Finite Element Formulation for Modal Analysis of Twisted Rotating Elastic Beams