Vibration Analysis Of a Self–Excited Elastic Beam

 

M. A. Barrón–Meza

 

Departamento de Materiales Universidad Autónoma Metropolitana Azcapotzalco Av. San Pablo 180, Col. Reynosa–Tamaulipas C.P. 02200, México, D.F., MÉXICO. E–mail: bmma@correo.azc.uam.mx

 

ABSTRACT

The vibration behavior and the energy exchange among the normal modes of a clamped–free self–excited elastic beam are analyzed in this work. To model this kind of beam, the damping term of a van der Pol oscillator is directly added to the equation of a linear elastic beam, yielding a single nonlinear partial differential equation. To solve this equation, a spectral method is employed. Three vibration modes are considered in the analysis, and the values of the self–exciting constant are varied in order to cover from linear to nonlinear vibration behavior. Multiple frequencies of the nonlinear beam are determined through the power spectral density of the beam free–end time series. Given that this relatively simple model mimics at least in a qualitative way some key issues of the fluid–structure problem, it could be potentially useful for fatigue studies and vibration analysis of rotating blades in turbomachinery.

Keywords: Beam vibration, fluid–structure problem, modal interaction, self–excited beam, spectral method, turbine blade vibration, van der Pol oscillator.

 

RESUMEN

En este trabajo se analizan el comportamiento bajo vibración y el intercambio de energía entre los modos normales de una barra elástica autoexcitada con un extremo fijo y el otro libre. Para modelar esta clase de barra se le agrega directamente el término de amortiguamiento de un oscilador van der Pol a la ecuación de una barra elástica lineal, obteniéndose una sola ecuación diferencial parcial. Para resolver esta ecuación se usa el método espectral. En el análisis se consideran tres modos de vibración, y los valores de la constante de auto–excitación se varían a modo de cubrir un comportamiento a la vibración desde lineal hasta no lineal. Las múltiples frecuencias de la barra no lineal se determinan mediante el espectro de potencias de las series de tiempo del extremo libre. Dado que este modelo relativamente simple reproduce, al menos cualitativamente, algunos aspectos clave del problema fluido–estructura, puede ser potencialmente útil para estudios de fatiga y análisis de la vibración de álabes rotatorios en turbomaquinaria.

 

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