0
TECHNICAL PAPERS

Combined Torsional-Bending-Axial Dynamics of a Twisted Rotating Cantilever Timoshenko Beam With Contact-Impact Loads at the Free End

[+] Author and Article Information
Sunil K. Sinha

GE Aircraft Engines, M.D.: P-30, General Electric Company, 1 Neumann Way, Cincinnati, OH 45215sunil.sinha@ae.ge.com

J. Appl. Mech 74(3), 505-522 (May 31, 2006) (18 pages) doi:10.1115/1.2423035 History: Received February 15, 2005; Revised May 31, 2006

In this paper, consideration is given to the dynamic response of a rotating cantilever twisted and inclined airfoil blade subjected to contact loads at the free end. Starting with the basic geometrical relations and energy formulation for a rotating Timoshenko beam constrained at the hub in a centrifugal force field, a system of coupled partial differential equations are derived for the combined axial, lateral and twisting motions which includes the transverse shear, rotary inertia, and Coriolis effects, as well. In the mathematical formulation, the torsion of the thin airfoil also considers a very general case of shear center not being coincident with the CG (center of gravity) of the cross section, which allows the equations to be used also for analyzing eccentric tip-rub loading of the blade. Equations are presented in terms of axial load along the longitudinal direction of the beam which enables us to solve the dynamic pulse buckling due to the tip being loaded in the longitudinal as well as transverse directions of the beam column. The Rayleigh–Ritz method is used to convert the set of four coupled-partial differential equations into equivalent classical mass, stiffness, damping, and gyroscopic matrices. Natural frequencies are computed for beams with varying “slenderness ratio” and “aspect ratio” as well as “twist angles.” Dynamical equations account for the full coupling effect of the transverse flexural motion of the beam with the torsional and axial motions due to pretwist in the airfoil. Some transient dynamic responses of a rotating beam repeatedly rubbing against the outer casing is shown for a typical airfoil with and without a pretwist.

FIGURES IN THIS ARTICLE
<>
Copyright © 2007 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

A pretwisted Timoshenko beam and the local coordinate system

Grahic Jump Location
Figure 2

(a) Schematic representation of an inclined rotating beam with respect to the fixed global frame of reference as viewed along the spin axis. (b) Airfoil cross section and its equivalent Timoshenko beam representation as viewed from the free end of the blade.

Grahic Jump Location
Figure 3

Finite-element model of the blade (L=15.8cm) with a total twist of (βR−βr)=−25deg

Grahic Jump Location
Figure 4

Change in cantilever beam frequencies with no-twist as a function of aspect ratio (slenderness ratio=0.01)

Grahic Jump Location
Figure 5

Change in cantilever beam frequencies twisted at 45deg as a function of aspect ratio (slenderness ratio=0.01)

Grahic Jump Location
Figure 6

Change in the twisted cantilever beam frequencies with aspect ratio (chord/span)=0.125 as a function of the total twist angle (slenderness ratio=0.01, angular velocity Ω=0.0 (stationary)—, angular velocity Ω=300rad∕s (rotating)- - -)

Grahic Jump Location
Figure 7

Change in the twisted cantilever beam frequencies with aspect ratio (chord/span)=0.25 as a function of the total twist angle (slenderness ratio=0.01, angular velocity Ω=0.0 (stationary)—, angular velocity Ω=300rad∕s (rotating)- - -)

Grahic Jump Location
Figure 8

Change in the twisted cantilever beam frequencies with aspect ratio (chord/span)=0.5 as a function of the total twist angle (slenderness ratio=0.01, angular velocity Ω=0.0 (stationary)—, angular velocity Ω=300rad∕s (rotating)- - -)

Grahic Jump Location
Figure 9

Change in the twisted cantilever beam frequencies with aspect ratio (chord/span)=0.667 as a function of the total twist angle (slenderness ratio=0.01, angular velocity Ω=0.0 (stationary)——, angular velocity Ω=300rad∕s (rotating)– – –)

Grahic Jump Location
Figure 10

Comparison of measured airfoil root strain gage data (……) versus results from the present analytical model (—) during repeated radial incursion of 0.1mm at the blade tip with 72deg circumferential rub zone for the first four rubs

Grahic Jump Location
Figure 11

Analytically computed Coriolis forces at the blade root during repeated radial incursion of 0.1mm at the blade tip with 72deg circumferential rub zone for the first four rubs

Grahic Jump Location
Figure 12

Comparison of analytically computed transient lateral tip displacements for an untwisted beam (—) versus a 45deg twisted beam (.....) during repeated rubs (one pulse per revolution) at the blade tip with a periodic contact force of magnitude Fmax=0.1× (Euler critical buckling load)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In