The eigenvalue problems of grouped turbo blades were numerically formulated by using the generalized differential quadrature method (GDQM). Different boundary approaches accompanying the GDQM to transform the partial differential equations of grouped turbo blades into a discrete eigenvalue problem are discussed. Effects of the number of sample points and the different boundary approaches on the accuracy of the calculated natural frequencies are also studied. Numerical results demonstrated the validity and the efficiency of the GDQM in treating this type of problem.

1.
Bellman
,
R.
, and
Casti
,
J.
,
1971
, “
Differential Quadrature and Long-Term Integration
,”
J. Math. Anal. Appl.
,
34
, pp.
235
238
.
2.
Bellman
,
R. E.
,
Kashef
,
B. G.
, and
Casti
,
J.
,
1972
, “
Differential Quadrature: A Technique for Rapid Solution of Nonlinear Partial Differential Equations
,”
J. Comput. Phys.
,
10
, pp.
40
52
.
3.
Chen
,
W.
, and
Zhong
,
T.
,
1997
, “
The Study on the Nonlinear Computations of the DQ and DC Methods
,”
Numer. Methods Part. Differ. Eqs.
,
13
, pp.
57
75
.
4.
Quan
,
J. R.
, and
Chang
,
C. T.
,
1989
, “
New Insights in Solving Distributed System Equations by the Quadrature Method—I. Analysis
,”
Comput. Chem. Eng.
,
13
, pp.
779
788
.
5.
Quan
,
J. R.
, and
Chang
,
C. T.
,
1989b
, “
New Insights in Solving Distributed System Equations by the Quadrature Method—II. Numerical Experiments
,”
Comput. Chem. Eng.
,
13
, pp.
1017
1024
.
6.
Shu
,
C.
, and
Richards
,
B. E.
,
1992
, “
Application of Generalized Differential Quadrature to Solve Two-Dimensional Incompressible Navier-Stokes Equations
,”
Int. J. Numer. Methods Fluids
,
15
, pp.
791
798
.
7.
Li
,
H.
, and
Lam
,
K. Y.
,
1998
, “
Frequency Characteristics of a Thin Rotating Cylindrical Shell Using the Generalized Differential Quadrature Method
,”
Int. J. Mech. Sci.
,
40
(
5
), pp.
443
459
.
8.
Bert
,
C. W.
,
Jang
,
S. K.
, and
Striz
,
A. G.
,
1988
, “
Two New Approximate Methods for Analyzing Free Vibration of Structural Components
,”
Int. J. Numer. Methods Eng.
,
28
, pp.
561
577
.
9.
Bert
,
C. W.
, and
Malik
,
M.
,
1996
, “
Free Vibration Analysis of Tapered Rectangular Plates by Differential Quadrature Method: A Semi-analytical Approach
,”
J. Sound Vib.
,
190
(
1
), pp.
41
63
.
10.
Bert
,
C. W.
,
Wang
,
X.
, and
Striz
,
A. G.
,
1993
, “
Differential Quadrature for Static and Free Vibration Analysis of Anisotropic Plates
,”
Int. J. Solids Struct.
,
30
, pp.
1737
1744
.
11.
Bert
,
C. W.
,
Wang
,
X.
, and
Striz
,
A. G.
,
1994
, “
Static and Free Vibration Analysis of Beams and Plates by Differential Quadrature Method
,”
Acta Mech.
,
102
, pp.
11
24
.
12.
Bert
,
C. W.
,
Wang
,
X.
, and
Striz
,
A. G.
,
1994
, “
Convergence of the DQ Method in the Analysis of Anisotropic Plates
,”
J. Sound Vib.
,
170
, pp.
140
144
.
13.
Han
,
J. B.
, and
Liew
,
K. M.
,
1999
, “
Axisymmetric Free Vibration of Thick Annular Plates
,”
Int. J. Mech. Sci.
,
41
, pp.
1089
1109
.
14.
Malik
,
M.
, and
Bert
,
C. W.
,
1996
, “
Implementing Multiple Boundary Conditions in the DQ Solution of Higher-Order PDE Application to Free Vibration of Plates
,”
Int. J. Numer. Methods Eng.
,
39
, pp.
1237
1258
.
15.
Striz
,
A. G.
,
Chen
,
W.
, and
Bert
,
C. W.
,
1994
, “
Static Analysis of Structures by the Quadrature Element Method (QEM)
,”
Int. J. Solids Struct.
,
31
, pp.
2807
2818
.
16.
Du
,
H.
,
Liew
,
K. M.
, and
Lim
,
M. K.
,
1996
, “
Generalized Differential Quadrature Method for Buckling Analysis
,”
J. Eng. Mech.
,
122
(
2
), pp.
95
100
.
17.
Rao
,
J. S.
,
1972
, “
Flexural Vibration of Pretwisted Tapered Cantilever Blades
,”
J. Eng. Ind.
,
94
, No.
1
, pp.
343
346
.
18.
Rao
,
J. S.
,
1977
, “
Vibration of Rotating, Pretwisted and Tapered Blades
,”
Mech. Mach. Theory
,
12
, pp.
331
337
.
19.
Hodges
,
D. H.
,
Chung
,
Y. Y.
, and
Shang
,
X. Y.
,
1994
, “
Discrete Transfer Matrix Method for Non-uniform rotating beams
,”
J. Sound Vib.
,
169
, pp.
276
283
.
20.
Abrate
,
S.
,
1995
, “
Vibrations of Non-uniform Rods and Beams
,”
J. Sound Vib.
,
185
(
4
), pp.
703
716
.
21.
Dawson
,
B.
,
1968
, “
Coupled Bending-Bending Vibrations of Pre-twisted Cantilever Blading Treated by Rayleigh-Ritz Energy Method
,”
J. Mech. Eng. Sci.
,
10
, pp.
381
386
.
22.
Dawson
,
B.
, and
Carneige
,
W.
,
1969
, “
Model Curves of Pretwisted Beams of Rectangular Cross-Section
,”
J. Mech. Eng. Sci.
,
11
, pp.
1
13
.
23.
Gupa
,
R. S.
, and
Rao
,
S. S.
,
1978
, “
Finite Element Eigenvalue Analysis of Tapered and Twisted Timoshenko Beams
,”
J. Sound Vib.
,
56
(
2
), pp.
187
200
.
24.
Swaminathan
,
M.
, and
Rao
,
J. S.
,
1977
, “
Vibrations of Rotating, Pretwisted and Tapered Blades
,”
Mech. Mach. Theory
,
12
, pp.
331
337
.
25.
Subrahmanyam
,
K. B.
,
Kulkarni
,
S. V.
, and
Rao
,
J. S.
,
1981
, “
Coupled Bending-Bending Vibrations of Pre-twisted Cantilever Blading Allowing for Shear Deflection and Rotary Inertia by the Reissner Method
,”
Int. J. Mech. Sci.
,
23
(
9
), pp.
517
530
.
26.
Subrahmanyam
,
K. B.
, and
Rao
,
J. S.
,
1982
, “
Coupled Bending-Bending Vibrations of Pretwisted Tapered Cantilever Beams Treated by the Reissner Method
,”
J. Sound Vib.
,
82
(
4
), pp.
577
592
.
27.
Chen
,
W. R.
, and
Keer
,
L. M.
,
1993
, “
Transverse Vibrations of a Rotating Twisted Timshenko Beam Under Axial Loading
,”
ASME J. Vibr. Acoust.
,
115
, pp.
285
294
.
28.
Storiti
,
D.
, and
Aboelnaga
,
Y.
,
1987
, “
Bending Vibrations of a Class of Rotating Beams With Hypergeometric Solutions
,”
ASME J. Appl. Mech.
,
54
, pp.
311
314
.
29.
Wagner
,
J. T.
,
1967
, “
Coupling of Turbomachine Blade Vibrations Through the Rotor
,”
ASME J. Eng. Gas Turbines Power
,
89
, pp.
502
512
.
30.
Griffin
,
J. H.
,
1980
, “
Friction Damping of Resonant Stresses in Gas Turbine Engine Airfoils
,”
ASME J. Eng. Gas Turbines Power
,
102
, pp.
329
333
.
31.
Griffin
,
J. H.
, and
Sinha
,
A.
,
1985
, “
The Interaction Between Mistuning and Friction in the Forced Response of Bladed Disk Assemblies
,”
ASME J. Eng. Gas Turbines Power
,
107
, pp.
107
205
.
32.
Sinha
,
A.
, and
Griffin
,
J. H.
,
1984
, “
Effects of Static Friction on the Forced Response of Frictionally Damped Turbine Blades
,”
ASME J. Eng. Gas Turbines Power
,
106
, pp.
65
69
.
33.
Wagner
,
L. F.
, and
Griffin
,
J. H.
,
1996
, “
Forced Harmonic Response of Grouped Blade Systems: Part I—Discrete Theory
,”
ASME J. Eng. Gas Turbines Power
,
118
, pp.
130
136
.
34.
Wagner
,
L. F.
, and
Griffin
,
J. H.
,
1996
, “
Forced Harmonic Response of Grouped Blade Systems: Part II—Application
,”
ASME J. Eng. Gas Turbines Power
,
118
, pp.
137
145
.
35.
Anderson
,
G. L.
,
1975
, “
On the Extensional and Flexural Vibration of Rotating Bars
,”
Int. J. Non-Linear Mech.
,
10
, pp.
223
236
.
36.
Young
,
T. H.
,
1991
, “
Dynamic Response of a Pretwisted, Tapered Beam With Non-constant Rotating Speed
,”
J. Sound Vib.
,
167
(
3
), pp.
529
539
.
37.
Sherbourne
,
A. N.
, and
Pandey
,
M. D.
,
1991
, “
Differential Quadrature Method in the Buckling Analysis of Beams and Composite Plates
,”
Comput. Struct.
,
40
, pp.
903
913
.
38.
Shu
,
C.
, and
Du
,
H.
,
1997
, “
A Generalized Approach for Implementing General Boundary Conditions in the GDQ Free Vibration Analysis of Plates
,”
Int. J. Solids Struct.
,
34
(
7
), pp.
837
846
.
39.
Jang
,
S. K.
,
Bert
,
C. W.
, and
Striz
,
A. G.
,
1989
, “
Application of Differential Quadrature to Static Analysis of Structural Components
,”
Int. J. Numer. Methods Eng.
,
28
, pp.
561
577
.
40.
Kuang, J. H., and Huang, B. W., 2000, “Mode Localization in a Grouped Bladed Disk,” ASME Paper No. 00-GT-369.
You do not currently have access to this content.