Linear vibrations are studied for a straight uniform finite beam element of general orientation spinning at a constant angular speed about a fixed axis in the inertial space. The gyroscopic and circulatory matrices and also the geometric stiffness matrix of the beam element are presented. The effect of the centrifugal static axial load on the bending and torsional dynamic stiffnesses is thereby accounted for. The Rayleigh/Timoshenko/Saint-Venant theory is applied, and polynomial shape functions are used in the construction of the deformation fields. Nonzero off-diagonal elements in the gyroscopic and circulatory matrices indicate coupled bending/shearing/torsional/tensional free and forced modes of a generally oriented spinning beam. Two numerical examples demonstrate the use and performance of the beam element.

Issue Section:
Gas Turbines: Combustion and Fuels
Topics:
Spin (Aerodynamics), Spinning (Textile), Construction, Deformation, Die cutting, Linear vibration, Polynomials, Shapes, Shearing (Deformation), Stiffness, Stress
1.
Ruhl, R. L., “Dynamics of Distributed Parameter Rotor Systems: Transfer Matrix and Finite Element Techniques,” PhD Dissertation, Cornell University, 1970.
2.
Nelson
H. D.
, “
A Finite Rotating Shaft Element Using Timoshenko Beam Theory
,”
ASME Journal of Mechanical Design
, Vol.
102
,
1980
, pp.
793
–
803
.
3.
Kang
Y.
,
Shih
Y.-P.
, and
Lee
A.-C.
, “
Investigation on the Steady-State Responses of Asymmetric Rotors
,”
ASME Journal of Vibration and Acoustics
, Vol.
144
(
2
),
1992
, pp.
194
–
208
.
4.
Likins
P. W.
, “
Finite Element Appendage Equations for Hybrid Coordinate Dynamic Analysis
,”
International Journal of Solids and Structures
, Vol.
8
(
5
),
1972
, pp.
709
–
731
.
5.
Wittrick
W. H.
, and
Williams
F. W.
, “
On the Free Vibration Analysis of Spinning Structures by Using Discrete or Distributed Mass Models
,”
Journal of Sound and Vibration
, Vol.
82
(
1
),
1982
, pp.
1
–
15
.
6.
Leung
A. Y. T.
, and
Fung
T. C.
, “
Spinning Finite Elements
,”
Journal of Sound and Vibration
, Vol.
125
(
3
),
1988
, pp.
523
–
537
.
7.
Lundblad
H. M.
, “
Forced Harmonic Vibration of Rotating Beam Systems in Space Analysed by Use of Exact Finite Elements
,”
International Journal for Numerical Methods in Engineering
, Vol.
32
(
3
),
1991
, pp.
571
–
594
.
8.
Archer
J. S.
, “
Consistent Matrix Formulations for Structural Analysis Using Finite-Element Techniques
,”
AIAA Journal
, Vol.
3
(
10
),
1965
, pp.
1910
–
1918
.
9.
Przemieniecki, J. S., Theory of Matrix Structural Analysis, McGraw-Hill, New York, 1968.
10.
Sa¨llstro¨m
J. H.
, “
Fluid-Conveying Damped Rayleigh-Timoshenko Beams in Transverse Vibration Analyzed by Use of an Exact Finite Element, Part II: Applications
,”
Journal of Fluids and Structures
, Vol.
4
(
6
),
1990
, pp.
573
–
582
.
11.
Cook, R. D., Malkus, D. S., and Plesha, M. E., Concepts and Applications of Finite Element Analysis, 3rd ed., Wiley, New York, 1989.
12.
Timoshenko, S. P., and Gere, J. M., Theory of elastic Stability, 2nd ed., McGraw-Hill, Singapore, 1963.
13.
Gu¨rgo¨ze
M.
, “
On the Dynamical Behaviour of a Rotating Beam
,”
Journal of Sound and Vibration
, Vol.
143
(
2
),
1990
, pp.
356
–
363
.
14.
Bauer
H. F.
, and
Eidel
W.
, “
Vibration of a Rotating Uniform Beam, Part II: Orientation Perpendicular to the Axis of Rotation
,”
Journal of Sound and Vibration
, Vol.
112
(
2
),
1988
, pp.
357
–
375
.
This content is only available via PDF.
Copyright © 1996
 by The American Society of Mechanical Engineers
You do not currently have access to this content.