This paper considers the free vibrations of the non-uniform axially functionally graded cantilever beam with a tip body. It is assumed that the mass center of the tip body is eccentrically displaced in both axial and transverse direction relative to the beam tip. All considerations are carried out within the Euler–Bernoulli beam theory. The in-plane transverse and axial deformations of the beam are considered. It is shown that there is a coupling between the axial and transverse deformations of the cantilever beam due to the tip body mass center eccentricity in the transverse direction. An universal, numerically efficient rigid element method which is able to analyze the cantilever beam with any law of changes of the geometric parameters of the cross section or the characteristics of the material along the beam was formed. Theoretical considerations are accompanied by numerical examples. There is a good agreement of the results obtained with the results available in the literature.

Free vibration analysis of a non-uniform axially functionally graded cantilever beam with a tip body

The determination of model frequencies and mode shapes of rotating blades, which are either straight or tapered, is usually accomplished via application of conventional assumed displacement finite element or assumed modes methodologies. To ensure accuracy of modal information at low mode num bers, a large number of elements or admissible functions must be utilized in the model. Modal frequencies converge to the exact solutions from above with the increase of number of elements or assumed admissible functions. To apply the resulting large degree-of-freedom model in a dynamic simulation (forced or transient), or to embed such a model in a real-time model-based control problem, can be impractical. As a result, the model order must be reduced via static or dynamic condensation procedures to a practical number of degrees-of-freedom, which is constrained by simulation time, or the control interval in a digital control system. Therefore, a method is proposed, based on a spectral finite element technique, to develop such a low degree-of-freedom dynamic model directly and without resorting to a condensation procedure. The method exploits semi-analytical progressive wave solutions of the governing partial differential equations. We have calculated such results for a number of examples such as a straight beam and beams with uniform taper or compound tapers. Only one single spectral finite element is needed to obtain any modal frequency or mode shape, which is as accurate or better than other approaches reported in the literature for a straight or uniformly tapered beams. The minimum number of such spectral finite element correspond to the number of substructures (i.e., beam sections with different uniform tapers) in a rotating beam in order to capture the complete system dynamic characteristics. The element assembly procedure is accomplished in the same fashion as the conventional finite element approach. Overall, for a rotating blade system, our spectral finite element method provides highly accurate predictions for any modal frequency using a single element or very few elements corresponding to the number of uniform taper changes in the blade system.Copyright © 2003 by ASME

Free Vibration Analysis of Rotating Blades With Uniform Tapers

A Bayesian methodology for detection of railway ballast damage using the modified Ludwik nonlinear model

Effect of pre-load on wave propagation characteristics of hexagonal lattices

This paper presents a new approach to the problem of determining the frequencies and mode shapes of Euler–Bernoulli tapered cantilever beams with a tip mass and a spring at the free end. The approach is based on the replacement of the flexible beam by a rigid multibody system. Beams with constant thickness and exponentially and linearly tapered width, as well as double-tapered cantilever beams are considered. The influence of the tip mass, stiffness of the spring, and taper on the frequencies of the free transverse vibrations of tapered cantilever beams are examined. Numerical examples with results confirming the convergence and accuracy of the approach are given.

A rigid multibody method for free vibration analysis of beams with variable axial parameters

The traditional beam element was improved to consider the variable axial parameters. The variable axial parameters were formulated in terms of a power series, and the general forms of elementary mass and stiffness matrices which depend on the power order were derived. The mass and stiffness matrices of the improved beam element were obtained in terms of an elementary matrix series. The beam elements for various tapered beams and a beam under linearly axial temperature distribution were derived. The vibrations of the beams with various taper shapes were studied and the variations of natural frequencies and modal shapes were investigated. A uniform beam under linearly axial temperature distribution was modeled and studied. The influences of axial temperature difference on the natural frequencies and modal shapes were investigated. Results show that the improved beam element could consider the variable axial parameters of beam conveniently.

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Chun Li

Papers

An Improved Beam Element for Beams with Variable Axial Parameters