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TECHNICAL PAPERS

Spatial Modulation of Repeated Vibration Modes in Rotationally Periodic Structures

[+] Author and Article Information
M. Kim, J. Moon, J. A. Wickert

Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213-3890

J. Vib. Acoust 122(1), 62-68 (Jul 01, 1997) (7 pages) doi:10.1115/1.568443 History: Received August 01, 1996; Revised July 01, 1997
Copyright © 2000 by ASME
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References

Zenneck,  J., 1899, “Ueber die freiea Schwingungen nur annähernd vollkommener kreisformiger Platten,” Ann. Phys. (Leipzig), 67, pp. 165–184.
Tobias,  S. A., 1951, “A Theory of Imperfection for the Vibrations of Elastic Bodies of Revolution,” Engineering,172, pp. 409–411.
Tobias,  S. A., 1958, “Non-linear Forced Vibrations of Circular Disks,” Engineering, 186, pp. 51–56.
Thomas,  D. L., 1974, “Standing Waves in Rotationally Periodic Structures,” J. Sound Vib., 37, pp. 288–290.
Thomas,  D. L., 1979, “Dynamics of Rotationally Periodic Structures,” Int. J. Numer. Methods Eng., 14, pp. 81–102.
Ewins,  D. J., 1969, “The Effects of Detuning Upon the Forced Vibrations of Bladed Disks,” J. Sound Vib., 9, pp. 65–79.
Stange,  W. A., and MacBain,  J. C., 1983, “An Investigation of Dual Mode Phenomena in a Mistuned Bladed Disk,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, 105, pp. 402–407.
Charnley,  T., and Perrin,  R., 1978, “Studies With an Eccentric Bell,” J. Sound Vib., 58, pp. 517–525.
Nelson,  R. L., and Thomas,  D. L., 1978, “Free Vibration Analysis of Cooling Towers With Column Supports,” J. Sound Vib., 57, pp. 149–153.
Allaei,  D., Soedel,  W., and Yang,  T. Y., 1986, “Natural Frequencies and Modes of Rings that Deviate from Perfect Axisymmetry,” J. Sound Vib., 111, pp. 9–27.
Fox,  C. H. J., 1990, “A Simple Theory for the Analysis and Correction of Frequency Splitting in Slightly Imperfect Rings,” J. Sound Vib., 142, pp. 227–243.
Allaei,  D., Soedel,  W., and Yang,  T. Y., 1987, “Eigenvalues of Rings with Radial Spring Attachments,” J. Sound Vib., 121, pp. 547–561.
Yu,  R. C., and Mote,  C. D., 1987, “Vibration and Parametric Excitation in Asymmetric Circular Plates Under Moving Loads,” J. Sound Vib., 119, pp. 409–427.
Shen,  I. Y., and Mote,  C. D., 1992, “Dynamic Analysis of Finite Linear Elastic Solids Containing Small Elastic Imperfections: Theory With Application to Asymmetric Circular Plates,” J. Sound Vib., 155, pp. 443–465.
Tseng,  J.-G., and Wickert,  J. A., 1994a, “Vibration of an Eccentrically Clamped Annular Plate,” ASME J. Vibr. Acoust., 116, pp. 155–160.
Parker,  R. G. and Mote,  C. D., 1996a, “Exact Boundary Condition Perturbation Solutions in Eigenvalue Problems,” ASME J. Appl. Mech., 63, pp. 128–135.
Parker,  R. G., and Mote,  C. D., 1996b, “Exact Perturbation for the Vibration of Almost Annular or Circular Plates,” ASME J. Vibr. Acoust., 118, pp. 436–445.
Tseng,  J.-G., and Wickert,  J. A., 1994b, “On the Vibration of Bolted Plate and Flange Assemblies,” ASME J. Vibr. Acoust., 116, pp. 468–473.
Rayleigh, 1887, The Theory of Sound, New York: Dover, second edition (1945 reissue).
Mallik,  A. K., and Mead,  D. J., 1977, “Free Vibration of Thin Circular Rings on Periodic Radial Supports,” J. Sound Vib., 54, pp. 13–27.
Mead,  D. J., 1973, “A General Theory of Harmonic Wave Propagation in Linear Periodic Systems with Multiple Coupling,” J. Sound Vib., 27, pp. 235–260.
Orris,  R. M., and Petyt,  M., 1974, “A Finite Element Study of Harmonic Wave Propagation in Periodic Structures,” J. Sound Vib., 33, pp. 223–236.
Courant, R., and Hilbert, D., 1953, Methods of Mathematical Physics, Wiley-Interscience reprint (1989).
Morse, P. M., and Feshbach, H., 1953, Methods of Theoretical Physics, McGraw-Hill.
Wagner,  L. F., and Griffin,  J. H., 1993, “A Continuous Analog Model for Grouped-Blade Vibration,” J. Sound Vib., 165, pp. 421–438.

Figures

Grahic Jump Location
Schematic of the test apparatus. Aluminum disks were preloaded by a hydraulic press against spacers to obtain rotationally periodic contact, or against a solid axisymmetric flange.
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Measured collocated transfer functions of the disk with nominally clamped-free boundary conditions (upper), and with six equally-spaced displacement and slope constraints around the inner edge (lower). Values of ND, and the sine or cosine orientation of the split modes, are indicated.
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Measured sine and cosine members of the split three nodal diameter doublet, having distinct frequencies 196 and 219 Hz; N=6 stiffness features with locations indicated at top. The shapes are sectioned along r=c to highlight variation in θ.
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Measured mode shape of the disk which illustrates spatial modulation; N=6 stiffness features. This mode has repeated natural frequency, and is asymptotic to (0,4). As indicated by the Fourier decomposition, significant contamination occurs at k=2.
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Doublet modes companion to Fig. 4, as predicted by a finite element model; N=6 stiffness features. The S (——) and C (– – –) members have repeated frequency, but each is spatially modulated, an is any linear combination of them. Contamination occurs for each mode at k=2, and to lesser degrees at k=8 and 10.
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Isometric and section views of a measured mode shape of the disk; N=6 stiffness features. This mode has repeated natural frequency, and is asymptotic to (0,5).
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Predicted antisymmetric mode shape which is asymptotic to (0,5) S;N=6 stiffness features. The symmetric companion has repeated frequency, but it is not shown here for clarity. Contamination occurs at k=1, and to a lesser degree at k=7.
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Checkerboard diagram depicting the contamination wavenumbers for each base mode in the presence of six features
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Measured mode shape of the disk which illustrates spatial modulation; N=5 inertia features. This mode has repeated natural frequency, and is asymptotic to (0,4). As indicated by the Fourier decomposition, significant contamination occurs at k=1.
Grahic Jump Location
Checkerboard diagram depicting the contamination wavenumbers for each base mode in the presence of five features
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Measured mode shape of the disk which illustrates spatial modulation; N=5 inertia features. This mode has repeated natural frequency, and is asymptotic to (0,6). As indicated by the Fourier decomposition, significant contamination occurs at k=1 and 4.
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Photograph of a typical automotive disk brake rotor
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Predicted circumferential sections of the rotor’s ND=3 base mode taken around the (a) outer and (b) inner edges of the rotor’s cheeks. N=4 stiffness features model the constraint of the mounting studs. A radial gradient of modulation at k=1 exists, being most acute around the inner periphery.

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