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TECHNICAL PAPERS

Natural Frequencies of Rotating Uniform Beams with Coriolis Effects

[+] Author and Article Information
S. M. Hashemi, M. J. Richard

Department of Mechanical Engineering, Laval University, Québec, Canada G1K 7P4

J. Vib. Acoust 123(4), 444-455 (Apr 01, 2001) (12 pages) doi:10.1115/1.1383969 History: Received March 01, 2000; Revised April 01, 2001
Copyright © 2001 by ASME
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References

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Simpson,  A., 1973, “Calculation of Natural Frequencies and Modes of Steadily Rotating Systems: A Teaching Note,” Aeronaut. Q., 24, pp. 139–146.
Meirovitch,  L., 1974, “A New Method of Solution of the Eigenvalue Problem for Gyroscopic Systems,” AIAA J., 12, pp. 1337–1342.
Laurenson,  R. M., 1976, “Modal Analysis of Rotating Flexible Structures,” AIAA J., 14, No. 10, pp. 1444–1450.
Hoa,  S. V., 1979, “Vibration of a Rotating Beam With Tip Mass,” J. Sound Vib., 67, No. 3, pp. 369–381.
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Hashemi, S. M., Richard, M. J., and Dhatt, G., 1997, “A Dynamic Finite Element (DFE) Formulation for Free Vibration Analysis of Centrifugally Stiffened Uniform Beams,” Proceedings of 16th Canadian Congress of Applied Mechanics (CANCAM 1997), June 1–6, Québec, Québec, pp. 443–444.
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Hashemi, S. M., Richard, M. J., and Dhatt, G., 1997, “A Bernoulli-Euler Stiffness Matrix Approach for Vibrational Analysis of Spinning Linearly Tapered Beams,” 42nd ASME Gas Turbine Paper No. 97-GT-500.
Williams,  F. W., and Wittrick,  W. H., 1970, “An Automatic Computational Procedure for Calculating Natural Frequencies of Skeletal Structures,” Int. J. Mech. Sci., 12, pp. 12781–12791.
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Banerjee,  J. R., 1989, “Coupled Bending-Torsional Dynamic Stiffness Matrix for Beam Elements,” Int. J. Numer. Methods Eng., 28, pp. 1283–1298.
Hashemi, S. M., Richard, M. J., and Dhatt, G., 1996, “A Bernoulli-Euler Stiffness Matrix Approach for Vibrational Analysis of Linearly Tapered Beams,” Proceedings of Acoustics Week in Canada 1996, October 7–11, Calgary, Alberta, p. 87.
Hashemi,  S. M., Richard,  M. J., and Dhatt,  G., 1999, “A New Dynamic Finite Element (DFE) Formulation for Lateral Free Vibrations of Euler-Bernoulli Spinning Beams Using Trigonometric Shape Functions,” J. Sound Vib., 220, No. 4, pp. 602–623.
Hashemi, S. M., Richard, M. J., and Dhatt, G., 1998, “A Dynamic Finite Element (DFE) Approach for Coupled Bending-Torsional Vibrations of Beams,” Proceedings of the 39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Long Beach, CA, pp. 2614–2624, Paper No. 98-AIAA-2020.
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Figures

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An example of a spinning two-dimensional frame
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An individual member of a frame
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The domain, discretized by a number of 2-node elements
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Rotating uniform beam geometry
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Comparison of spin-induced effects for α=90 deg; the FREQUENCY RATIO- (first lead-lag mode (ωx)/ωx0 vs SPIN RATIO-Ω/ωz0;—, preload plus centripetal and Coriolis Accelerations included; — * —, only the centripetal Acceleration included: —+—, only the Coriolis Acceleration included; —□—, only the preload included
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Comparison of spin-induced effects for α=90 deg; the FREQUENCY RATIO- (first flapping mode (ωz)/ωz0 vs SPIN RATIO- Ω/ωz0; —, preload plus centripetal and Coriolis Acceleration included; — * —, only the centripetal Acceleration included; —+—, only the Coriolis Acceleration included; —□—, only the preload included
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A cantilever uniform radial beam
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The basis functions (Pifz,i=1,2,3,4) for in-plane (lead-lag) flexural vibration of a uniform beam rotating in horizontal plane; E=1 GPa, A=1 m2 , L=1 m, ρ=1 kg/m3 , and l=1 m4 . (a) Standard (static) cubic basis functions; (b) dynamic basis functions for a nonrotating beam; (c) dynamic basis function for a rotating beam and λ=Ω.l2.m/E.I=12; —, P1; —○—, P2; —+—, P3; —⋆–, P4.  
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The variation of the dynamic in-plane (lead-lag) flexural shape functions Ni vs frequency changes for a uniform beam rotating in horizontal plane; E=1 GPa, A=1 m2 , L=1 m, ρ=1 kg/m3 , l=1 m4 , and λ=Ω.l2.m/E.I=12. (a) N1; (b) N2; (c) N3; (d) N4; –×, ω1, 1st natural frequency;  −Δ, ω2, 2nd natural frequency; –⋆, ω3, 3rd natural frequency; −⋄, ω4, 4th natural frequency.  
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The change of the fourth dynamic shape function, N4, at the third natural frequency, ω3, vs spinning speed for the same beam as in Fig. 9. –×, Ω=4 rad/s; –Δ, Ω=8 rad/s, –⋆, Ω=12 rad/s.

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