0
Research Papers

Limitation on Increasing the Critical Speed of a Spinning Disk Using Transverse Rigid Constraints, An Application of Rayleigh's Interlacing Eigenvalues Theorem

[+] Author and Article Information
Ahmad Mohammadpanah

Department of Mechanical Engineering,
The University of British Columbia,
Vancouver, BC V6T 1Z4, Canada;
FPInnovations,
Vancouver, BC V6T 1Z4, Canada
e-mail: ahmadpa20@gmail.com

Stanley G. Hutton

FPInnovations,
Vancouver, BC V6T 1Z4, Canada

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 15, 2017; final manuscript received February 12, 2018; published online March 30, 2018. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 140(4), 041017 (Mar 30, 2018) (10 pages) Paper No: VIB-17-1322; doi: 10.1115/1.4039421 History: Received July 15, 2017; Revised February 12, 2018

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.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Rayleigh, J. W. S. , 1877, Theory of Sound, Vol. 1, Macmillan and Co, London, Chap. 4.
Mohammadpanah, A. , 2012, “Idling and Cutting Vibration Characteristics of Guided Circular Saws,” Master's thesis, University of British Columbia, Vancouver, BC, Canada.
Mohammadpanah, A. , 2015, “Flutter Instability Speed of Guided Splined Disks, With Applications to Sawing,” Ph.D thesis, University of British Columbia, Vancouver, BC, Canada.
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.
Mohammadpanah, A. , and Hutton, S. G. , 2015, “ Maximum Operation Speed of Splined Saws,” J. Wood Mater. Sci. Eng., 1(3), pp. 142–146.
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.
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.
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.
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. [CrossRef]
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.
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Yang, S. M. , 1993, “ Vibration of a Spinning Annular Disk With Coupled Rigid-Body Motion,” ASME J. Vib. Acoust., 115(2), pp. 159–164. [CrossRef]
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. [CrossRef]
Price, K. B. , 1987, “Analysis of the Dynamics of Guided Rotating Free Centre Plates,” Ph.D. dissertation, University of California, Berkeley, CA.
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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.
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. [CrossRef]
Adams, G. G. , 1980, “ Procedures for the Study of the Flexible-Disk to Head Interface,” IBM J. Res. Develop., 24(4), pp. 512–517. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Courant, R. , 1943, “ Variational Methods for the Solution of Problems of Equilibrium and Vibration,” Bull. Am. Math. Soc., 49, pp. 1–23. [CrossRef]
Satt, I. , 1992, “ Aeroelastic Divergence of Lifting Surface,” Mechanical Engineering Conference, Haifa, Israel, May 21–22, pp. 1–3.
Timoshenko, S. , and Goodier, J. N. , 1951, Theory of Elasticity, McGraw-Hill, New York.
Gladwell, G. M. L. , 2004, Inverse Problems in Vibration, 2nd ed., Springer-Verlag, New York.
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. [CrossRef]
Meirovitch, L. , 1997, Principles and Techniques of Vibrations, Prentice Hall, Upper Saddle River, NJ.
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.

Figures

Grahic Jump Location
Fig. 2

Variation of natural frequencies as a function of rotation speed for a free disk

Grahic Jump Location
Fig. 1

Schematic of a disk, with inner and outer radius of a, and b, and thickness of h

Grahic Jump Location
Fig. 9

Variation of natural frequencies as a function of rotation speed for a disk with an area constraint over one third of the surface of the disk, - solid line: constrained disk, - - dash line: free spinning disk

Grahic Jump Location
Fig. 3

Variation of natural frequencies as a function of rotation speed for a disk with one point constraint at inner radius, - solid line: constrained disk, - - dash line: free spinning disk

Grahic Jump Location
Fig. 4

Variation of natural frequencies as a function of rotation speed for a disk with one point constraint at outer radius, - solid line: constrained disk, - - dash line: free spinning disk

Grahic Jump Location
Fig. 5

Variation of natural frequencies as a function of rotation speed for a disk with a diametric line constraint from inner to outer radius, - solid line: constrained disk, - - dash line: free spinning disk

Grahic Jump Location
Fig. 6

Variation of natural frequencies as a function of rotation speed for a disk with two diametric line constraint from inner to outer radius, with angular separation of 45 deg, - solid line: constrained disk, - - dash line: free spinning disk

Grahic Jump Location
Fig. 7

Variation of natural frequencies as a function of rotation speed for a disk with clamped at inner radius (representation of a clamped disk), - solid line: - solid line: constrained disk, - - dash line: free spinning disk

Grahic Jump Location
Fig. 8

Variation of natural frequencies as a function of rotation speed for a disk with an area constraint over one quarter of the surface of the disk, - solid line: constrained disk, - - dash line: free spinning disk

Grahic Jump Location
Fig. 12

Variation of excited frequencies with rotation speed for a disk with no constraint

Grahic Jump Location
Fig. 13

Variation of excited frequencies with rotation speed for a disk with one pin constraint at the rim

Grahic Jump Location
Fig. 14

Variation of excited frequencies with rotation speed for a disk with one pin constrain at the eye

Grahic Jump Location
Fig. 10

Experimental setup

Grahic Jump Location
Fig. 11

Deflection of a spline guided disk, during idling run-up from 0 to 3600 rpm, measured by displacement probe (left), and variation of excited frequencies of a guided splined disk as a function of rotation speed (right)

Grahic Jump Location
Fig. 15

Variation of excited frequencies with rotation speed for a disk with one radius line constraint

Grahic Jump Location
Fig. 16

Variation of excited frequencies with rotation speed for a disk with two radius line constraints

Grahic Jump Location
Fig. 17

Variation of excited frequencies with rotation speed for a disk with area constraint over 1/4 surface of the disk

Grahic Jump Location
Fig. 18

Variation of excited frequencies with rotation speed for a clamped disk

Tables

Errata

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