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Analysis for the vibration characteristic of a rotating beam with varying section
Last modified: 2017-05-19
Abstract
A flexible beam attached to a rotating hub is a good representative of many engineering machines, such as helicopter blades, solar panels of space vehicles, robotic arms, etc. The vibration characteristics of such a simple system are relevant to the analysis of the dynamical response of a flexible multibody system. In existing literature, researchers adopted the conventional methods in structure dynamics to analyze the vibration characteristics of the flexible beam, in which the couplings between the hub’s dynamics and the beam’s dynamics are completely ignorable. Nevertheless, a rotating beam essentially suffers from a dynamical boundary markedly different from the static boundary conditions assumed in structure dynamics. How to quantify the vibration characteristics of the rotating beam is still a tough issue in the field of flexible multibody system and vibration analysis.
In this work, we study a flexible beam with varying section where the height and width of the beam change linearly with the length (see Fig.1). We first employ the Hamilton principle to establish a set of nonlinear partial differential equations (PDE) governing the motion of the hub-beam system, then reduce the PDE into a set of nonlinear ordinary differential equations (ODE) in a spatial domain via a Fourier series expansion within a finite time. By solving the ODEs either numerically or theoretically, we finally obtain the vibration frequencies and associated mode shapes for the vibration of the rotating beam in a steady motion. We validate our results by comparing with the experimental data and the past numerical results reported in existing literature. Comprehensive investigations show that the coupling between the hub’s dynamics and the beam’s dynamics, under certain configurations of the system, is much stronger to change significantly the vibration characteristics of the elastic response of the rotating beam. We show via numerical investigations that the coupling mainly depends on the ratio of the hub’s inertia and beam’s inertia, the shape of the varying cross section, as well as the rotary speed of the hub. This supplies a guideline of optimizing the flexible structures when the elastic vibration needs to be controlled.
In this work, we study a flexible beam with varying section where the height and width of the beam change linearly with the length (see Fig.1). We first employ the Hamilton principle to establish a set of nonlinear partial differential equations (PDE) governing the motion of the hub-beam system, then reduce the PDE into a set of nonlinear ordinary differential equations (ODE) in a spatial domain via a Fourier series expansion within a finite time. By solving the ODEs either numerically or theoretically, we finally obtain the vibration frequencies and associated mode shapes for the vibration of the rotating beam in a steady motion. We validate our results by comparing with the experimental data and the past numerical results reported in existing literature. Comprehensive investigations show that the coupling between the hub’s dynamics and the beam’s dynamics, under certain configurations of the system, is much stronger to change significantly the vibration characteristics of the elastic response of the rotating beam. We show via numerical investigations that the coupling mainly depends on the ratio of the hub’s inertia and beam’s inertia, the shape of the varying cross section, as well as the rotary speed of the hub. This supplies a guideline of optimizing the flexible structures when the elastic vibration needs to be controlled.