It can be predicted by Rayleigh's interlacing eigenvalue theorem that structural modifications of a spinning disk can shift the first critical speed of the system up to a known limit. In order to corroborate this theorem, it is shown numerically, and verified experimentally that laterally constraining of a free spinning flexible thin disk with tilting and translational degree-of-freedom (DOF), the first critical speed of the disk cannot increase more than the second critical speed of the original system (the disk with no constraint). The governing linear equations of transverse motion of a spinning disk with rigid body translational and tilting DOFs are used in the analysis of eigenvalues of the disk. The numerical solution of these equations is used to investigate the effect of the constraints on the critical speeds of the spinning disk. Experimental tests were conducted to verify the results.

References

References
1.
Rayleigh
,
J. W. S.
,
1877
,
Theory of Sound
, Vol. 1,
Macmillan and Co
,
London
, Chap. 4.
2.
Mohammadpanah
,
A.
,
2012
, “Idling and Cutting Vibration Characteristics of Guided Circular Saws,”
Master's thesis
, University of British Columbia, Vancouver, BC, Canada.
3.
Mohammadpanah
,
A.
,
2015
, “Flutter Instability Speed of Guided Splined Disks, With Applications to Sawing,”
Ph.D thesis
, University of British Columbia, Vancouver, BC, Canada.
4.
Mohammadpanah
,
A.
, and
Hutton
,
S. G.
,
2015
, “
Flutter Instability Speeds of Guided Splined Disks: An Experimental and Analytical Investigation
,”
J. Shock Vib.
,
2015
, p. 942141.
5.
Mohammadpanah
,
A.
, and
Hutton
,
S. G.
,
2015
, “
Maximum Operation Speed of Splined Saws
,”
J. Wood Mater. Sci. Eng.
,
1
(3), pp. 142–146.
6.
Mohammadpanah
,
A.
, and
Hutton
,
S. G.
,
2016
, “
Dynamics Behavior of a Guided Spline Spinning Disk, Subjected to Conservative in-Plane Edge Loads, Analytical and Experimental Investigation
,”
ASME J. Vib. Acoust.
,
138
(4), p. 041005.
7.
Mohammadpanah
,
A.
,
Hutton
,
S. G.
, and
Khorasany
,
R. M. H.
,
2011
, “
Critical Speeds of Guided Circular Saws, a Sensitivity Analysis to Design Variables
,”
23rd Canadian Congress of Applied Mechanics
(
CANCAM
), Vancouver, BC, Canada, June 5–9, pp. 344–347.
8.
Mohammadpanah
,
A.
, and
Hutton
,
S. G.
,
2017
, “
Theoretical and Experimental Verification of Dynamic Behaviour of a Guided Spline Arbor Circular Saw
,”
J. Shock Vib.
,
2017
, p. 6213791.
9.
Khorasany
,
R. M. H.
,
Mohammadpanah
,
A.
, and
Hutton
,
S. G.
,
2012
, “
Vibration Characteristics of Guided Circular Saws: Experimental and Numerical Analyses
,”
ASME J. Vib. Acoust.
,
134
(
6
), p.
061004
.
10.
Kaczmarek
,
A.
,
Orlowski
,
K.
, and
Javorek
,
L.
,
2016
, “
Comparison of Natural Frequencies of a Circular Saw Blade Obtained Empirically and With FEM
,”
Ann. Warsaw Univ. Life Sci. (SGGW), For. Wood Technol.
,
95
, pp.
46
50
.
11.
Tobias
,
S. A.
, and
Arnold
,
R. N.
,
1957
, “
The Influence of Dynamical Imperfections on the Vibration of Rotating Disks
,”
Inst. Mech. Eng., Proc.
,
171
(
1
), pp.
669
690
.
12.
Mote
,
C. D.
,
1977
, “
Moving Load Stability of a Circular Plate on a Floating Central Collar
,”
J. Acoust. Soc. Am.
,
61
(
2
), pp.
439
447
.
13.
Hutton
,
S. G.
,
Chonan
,
S.
, and
Lehmann
,
B. F.
,
1987
, “
Dynamic Response of a Guided Circular Saw
,”
J. Sound Vib.
,
112
(
3
), pp.
527
539
.
14.
Chen
,
J. S.
, and
Hsu
,
C. M.
,
1997
, “
Forced Response of a Spinning Disk Under Space-Fixed Couples
,”
J. Sound Vib.
,
206
(
5
), pp.
627
639
.
15.
Chen
,
J. S.
, and
Wong
,
C. C.
,
1995
, “
Divergence Instability of a Spinning Disk With Axial Spindle Displacement in Contact With Evenly Spaced Stationary Springs
,”
ASME J. Appl. Mech.
,
62
(
2
), pp.
544
547
.
16.
Yang
,
S. M.
,
1993
, “
Vibration of a Spinning Annular Disk With Coupled Rigid-Body Motion
,”
ASME J. Vib. Acoust.
,
115
(
2
), pp.
159
164
.
17.
Chen
,
J. S.
, and
Bogy
,
D. B.
,
1993
, “
Natural Frequencies and Stability of a Flexible Spinning Disk-Stationary Load System With Rigid Body Tilting
,”
ASME J. Appl. Mech.
,
60
(
2
), pp.
470
477
.
18.
Price
,
K. B.
,
1987
, “Analysis of the Dynamics of Guided Rotating Free Centre Plates,” Ph.D. dissertation, University of California, Berkeley, CA.
19.
Raman
,
A.
, and
Mote
,
C. D.
,
2001
, “
Experimental Studies on the Non-Linear Oscillations of Imperfect Circular Disks Spinning Near Critical Speed
,”
Int. J. Non-Linear Mech.
,
36
(
2
), pp.
291
305
.
20.
Kang
,
N.
, and
Raman
,
A.
,
2006
, “
Vibrations and Stability of a Flexible Disk Rotating in a Gas-Filled Enclosure—Part 2: Experimental Study
,”
J. Sound Vib.
,
296
(
4–5
), pp.
676
689
.
21.
D'Angelo
,
C.
, and
Mote
,
C. D.
,
1993
, “
Aerodynamically Excited Vibration and Flutter of a Thin Disk Rotating at Supercritical Speed
,”
J. Sound Vib.
,
168
(1), pp.
15
30
.
22.
Chen
,
J. S.
, and
Wong
,
C. C.
,
1996
, “
Modal Interaction in a Spinning Disk on a Floating Central Collar and Restrained by Multiple Springs
,”
J. Chin. Soc. Mech. Eng.
,
17
(
6
), pp.
251
259
.
23.
Chen
,
J. S.
, and
Bogy
,
D. B.
,
1992
, “
Mathematical Structure of Modal Interactions in a Spinning Disk-Stationary Load System
,”
Am. Soc. Mech. Eng. J. Appl. Mech.
,
59
(
2
), pp.
390
397
.
24.
Adams
,
G. G.
,
1980
, “
Procedures for the Study of the Flexible-Disk to Head Interface
,”
IBM J. Res. Develop.
,
24
(
4
), pp.
512
517
.
25.
Weisensel
,
G. N.
, and
Schlack
,
A. L.
,
1993
, “
Response of Annular Plates to Circumferentially and Radially Moving Loads
,”
ASME. J. Appl. Mech.
,
60
(
3
), pp.
649
661
.
26.
Chen
,
J.
, and
Hua
,
C.
,
2004
, “
On the Secondary Resonance of a Spinning Disk Under Space-Fixed Excitations
,”
ASME J. Vib. Acoust.
,
126
(
3
), pp.
422
429
.
27.
Young
,
T. H.
, and
Lin
,
C. Y.
,
2006
, “
Stability of a Spinning Disk Under a Stationary Oscillating Unit
,”
J. Sound Vib.
,
298
(
1–2
), pp.
307
318
.
28.
Deqiang
,
M.
, and
Suhuan
,
C.
,
2001
, “
Effect of the Guides on the Lowest Critical Rotational Frequencies of Circular Saw
,”
Chin. J. Mech. Eng.
,
14
(2), pp.
166
170
.
29.
Khorasany
,
R. M. H.
, and
Hutton
,
S. G.
,
2011
, “
On the Equilibrium Configurations of an Elastically Constrained Rotating Disk: An Analytical Approach
,”
Mech. Res. Commun.
,
38
(
2011
), pp.
288
293
.
30.
Courant
,
R.
,
1943
, “
Variational Methods for the Solution of Problems of Equilibrium and Vibration
,”
Bull. Am. Math. Soc.
,
49
, pp.
1
23
.
31.
Satt
,
I.
,
1992
, “
Aeroelastic Divergence of Lifting Surface
,”
Mechanical Engineering Conference, Haifa, Israel
, May 21–22, pp.
1
3
.
32.
Timoshenko
,
S.
, and
Goodier
,
J. N.
,
1951
,
Theory of Elasticity
,
McGraw-Hill
,
New York
.
33.
Gladwell
,
G. M. L.
,
2004
,
Inverse Problems in Vibration
,
2nd ed.
, Springer-Verlag, New York.
34.
Chen
,
J. S.
, and
Bogy
,
D. B.
,
1992
, “
Effects of Load Parameters on the Natural Frequencies and Stability of a Spinning Disk With a Stationary Load System
,”
ASME J. Appl. Mech.
,
59
(
2S
), pp.
S230
S235
.
35.
Meirovitch
,
L.
,
1997
,
Principles and Techniques of Vibrations
, Prentice Hall, Upper Saddle River, NJ.
36.
Mohammadpanah
,
A.
,
Lehmann
,
B.
, and
White
,
J.
,
2017
, “
Development of a Monitoring System for Guided Circular Saws: An Experimental Investigation
,”
J. Wood Mater. Sci. Eng.
, epub.
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