0
Research Papers

# Semi-Exact Natural Frequencies for Kirchhoff–Love Plates Using Wave-Based Phase Closure

[+] Author and Article Information
Michael J. Leamy

George W. Woodruff School
of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0405
e-mail: michael.leamy@me.gatech.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 27, 2015; final manuscript received November 30, 2015; published online January 20, 2016. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 138(2), 021008 (Jan 20, 2016) (10 pages) Paper No: VIB-15-1286; doi: 10.1115/1.4032183 History: Received July 27, 2015; Revised November 30, 2015

## Abstract

This paper presents semi-exact, closed-form algebraic expressions for the natural frequencies of Kirchhoff–Love plates by analyzing plane waves, their edge reflections, and their phase closure. The semi-exact nature is such that the analysis exactly satisfies plate boundary conditions along each edge when taken in isolation, but not fully when combined, and thus is approximate near a corner. As frequency increases, the expressions become increasingly more accurate. For clamped square plates, closed-form expressions are reported in algebraic form for the first time. These expressions are developed by tracing the path of plane waves as they reflect from edges while accounting for phase changes over a total trip. This change includes phase addition/subtraction due to edge reflections. A natural frequency is identified as a frequency in which three phase changes (in the plate's horizontal, vertical, and path directions) each sum to an integer multiple of $2π$, enforcing phase closure along each direction. A solution of the subsequent equations is found in closed form, for multiple boundary conditions, such that highly convenient algebraic expressions result for the plate natural frequencies. The expressions are exact for the case of all sides simply supported, while for other boundary conditions, the expressions are semi-exact. For the practically important and difficult case of a fully clamped plate, the expressions for a square plate yield the first 20 nondimensional natural frequencies to within 0.06% of their exact values.

<>

## References

Love, A. E. H. , 1888, “ The Small Free Vibrations and Deformation of a Thin Elastic Shell,” Philos. Trans. R. Soc., A, 179(0), pp. 491–546.
Kirchhoff, G. R. , 1850, “ Uber Das Gleichgewicht Und Die Bewegung Einer Elastischen Scheibe,” J. Reine Angew. Math., 40, pp. 51–88.
Timoshenko, S. , Woinowsky-Krieger, S. , and Woinowsky-Krieger, S. , 1959, Theory of Plates and Shells, McGraw-Hill, New York.
Szilard, R. , 1974, Theory and Analysis of Plates: Classical and Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ.
Reddy, J. N. , 2006, Theory and Analysis of Elastic Plates and Shells, CRC Press, Boca Raton.
Leissa, A. W. , 1973, “ Free Vibration of Rectangular-Plates,” J. Sound Vib., 31(3), pp. 257–293.
Leissa, A. W. , 1969, “ Vibration of Plates,” National Aeronautics and Space Administration, Washington, DC, Technical Report No. NASA SP-160.
Warburton, G. , 1954, “ The Vibration of Rectangular Plates,” Proc. Inst. Mech. Eng., 168(1), pp. 371–384.
Janich, R. , 1962, “ Die Näherungsweise Berechnung Der Eigenfrequenzen Von Rechteckigen Platten Bei Verschiedenen Randbedingungen,” Die Bautech., 3, pp. 93–99.
Xing, Y. , and Liu, B. , 2009, “ New Exact Solutions for Free Vibrations of Rectangular Thin Plates by Symplectic Dual Method,” Acta Mech. Sin., 25(2), pp. 265–270.
Keller, J. B. , 1958, “ Corrected Bohr–Sommerfeld Quantum Conditions for Nonseparable Systems,” Ann. Phys., 4(2), pp. 180–188.
Keller, J. B. , and Rubinow, S. , 1960, “ Asymptotic Solution of Eigenvalue Problems,” Ann. Phys., 9(1), pp. 24–75.
Chen, G. , Coleman, M. P. , and Zhou, J. X. , 1991, “ Analysis of Vibration Eigenfrequencies of a Thin Plate by the Keller–Rubinow Wave Method. 1. Clamped Boundary-Conditions With Rectangular or Circular Geometry,” SIAM J. Appl. Math., 51(4), pp. 967–983.
Gupta, G. S. , 1970, “ Natural Flexural Waves and Normal Modes of Periodically-Supported Beams and Plates,” J. Sound Vib., 13(1), pp. 89–101.
Gupta, G. S. , 1971, “ Natural Frequencies of Periodic Skin-Stringer Structures Using a Wave Approach,” J. Sound Vib., 16(4), pp. 567–580.
Gupta, G. S. , 1972, “ Propagation of Flexural Waves in Doubly-Periodic Structures,” J. Sound Vib., 20(1), pp. 39–49.
Mead, D. J. , 1975, “ Wave-Propagation and Natural Modes in Periodic Systems. 1. Mono-Coupled Systems,” J. Sound Vib., 40(1), pp. 1–18.
Mead, D. J. , 1975, “ Wave-Propagation and Natural Modes in Periodic Systems. 2. Multi-Coupled Systems, With and Without Damping,” J. Sound Vib., 40(1), pp. 19–39.
Mead, D. J. , and Parthan, S. , 1979, “ Free Wave-Propagation in 2-Dimensional Periodic Plates,” J. Sound Vib., 64(3), pp. 325–348.
Mead, D. J. , Zhu, D. C. , and Bardell, N. S. , 1988, “ Free-Vibration of an Orthogonally Stiffened Flat-Plate,” J. Sound Vib., 127(1), pp. 19–48.
Faulkner, M. G. , and Hong, D. P. , 1985, “ Free-Vibrations of a Mono-Coupled Periodic System,” J. Sound Vib., 99(1), pp. 29–42.
Brillouin, L. , 1946, Wave Propagation in Periodic Structures, McGraw-Hill, New York.
Graff, K. F. , 1975, Wave Motion in Elastic Solids (Oxford Engineering Science Series), Clarendon Press, Oxford, UK.
Bolotin, V. , 1961, “ Dynamic Edge Effect in the Elastic Vibrations of Plates,” Inzh. Sb., 31(1), pp. 3–14 (in Russian).
Mace, B. , 1984, “ Wave Reflection and Transmission in Beams,” J. Sound Vib., 97(2), pp. 237–246.

## Figures

Fig. 2

Wave paths and reflections associated with waves characterized by ζ2=ξ

Fig. 1

Wave reflection and conversion at a supported plate edge (e.g., clamped)

Fig. 4

Wave paths and reflections associated with waves characterized by qζ2=pξ: (a) q=2, p=3, (b) q=3, p=4, and (c) q=4, p=5

Fig. 5

Wave paths and reflections for the case p=4, q=1 in a rectangular plate characterized by b=2a

Fig. 6

Percent error in predicted nondimensional natural frequency as a function of mode number for a square, thin plate

Fig. 3

Wave paths and reflections associated with waves characterized by ζ2=pξ: (a) p=2, (b) p=3, and (c) p=4

Fig. 7

Phase paths for generating the first mode (left) and the resulting real part of the mode shape predicted using the semi-exact approach (right)

Fig. 8

Corrected first mode

Fig. 9

Wave paths and reflections associated with waves characterized by qζ2¯=pξ¯ for an example rectangle with b=2a: (a) q=1, p=1, (b) q=1, p=2, (c) q=2, p=1, and (d) q=3, p=2

Fig. 10

Percent error in predicted natural frequencies as a function of mode number for an example rectangular, thin plate with aspect ratio 2:1

## Discussions

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