The paper discusses an extension to the linear theory of piezoelectricity of the Rayleigh quotient used in the analysis of the properties and the approximate calculation of the natural frequencies of elastic continua. It is shown that for a piezoelectric continuum an infinite number of equivalent expressions can be obtained which generalize the classical Rayleigh quotient. The stationarity conditions of any of these quotients under additional constraints imposed by the Gauss equation of electrostatics and the prescribed natural electrical boundary condition are shown to result in the complete set of the governing equations of the free vibration problem of a piezoelectric continuum. The general results discussed in the paper are illustrated by the approximate calculation of the natural frequencies of a piezoelectric rod by the Rayleigh–Ritz method. Unlike in the case of elastic structures, no monotonic convergence of the approximate frequencies is guaranteed for a piezoelectric continuum, the property which can be explained using the introduced Rayleigh quotients.

1.
Lord
,
Rayleigh
, 1945 (
1st ed.
1877),
Theory of Sound
,
Dover
, New York.
2.
Courant
,
R.
, and
Hilbert
,
D.
, 1937,
Methoden der Mathematischen Physik
,
Springer
, Berlin. (English translation,
Methods of Mathematical Physics
, 1952 (v.1), 1962 (v.2),
Wiley
, New York.)
3.
Mikhlin
,
S. G.
, 1970,
Variational Methods in Mathematical Physics (in Russian)
,
2nd ed.
,
Nauka
, Moscow.
4.
Strang
,
G.
, and
Fix
,
G. J.
, 1973,
An Analysis of the Finite Element Method
,
Prentice–Hall
, New York.
5.
Washizu
,
K.
, 1982,
Variational Methods in Elasticity and Plasticity
,
Pergamon
, Oxford New York.
6.
Cady
,
W. G.
, 1946,
Piezoelectricity
,
McGraw–Hill
, New York.
7.
Mason
,
W. P.
, 1950,
Piezoelectric Crystals and their Application to Ultrasonics
, Van Nostrand, Toronto.
8.
Lawson
,
A. W.
, 1942, “
The Vibration of Piezoelectric Plates
,”
Phys. Rev.
0031-899X,
62
, pp.
71
76
.
9.
Tiersten
,
H. F.
, 1969,
Linear Piezoelectric Plate Vibrations
,
Plenum
, New York.
10.
Allik
,
H.
, and
Hughes
,
T.
, 1970, “
Finite Element Method for Piezoelectric Vibration
,”
Int. J. Numer. Methods Eng.
0029-5981,
2
, pp.
151
157
.
11.
He
,
J.-H.
, 2001, “
Coupled Variational Principles of Piezoelectricity
,”
Int. J. Eng. Sci.
0020-7225,
39
, pp.
323
341
.
12.
Nowacki
,
W.
, 1983,
Electromagnetic Effects in Deformable Solids (in Polish)
,
PWN
, Warsaw.
13.
He
,
J.-H.
, 2001, “
Hamilton Principle and Generalized Variational Principle of Linear Thermopiezoelectricity
,”
ASME J. Appl. Mech.
0021-8936,
68
, pp.
666
667
.
14.
Yang
,
J. S.
, and
Batra
,
R. C.
, 1995, “
Conservation Laws in Linear Piezoelectricity
,”
Eng. Fract. Mech.
0013-7944,
51
, pp.
1041
1047
.
15.
Liu
,
G. R.
, and
Xi
,
Z. C.
, 2002,
Elastic Waves in Anisotropic Laminates
,
CRC
, Boca Raton, FL.
16.
Han
,
X.
, and
Liu
,
G. R.
, 2003, “
Elastic Waves in a Functionally Graded Piezoelectric Cylinder
,”
Smart Mater. Struct.
0964-1726,
12
, pp.
962
971
.
17.
Cupiał
,
P.
, 2005, “
Three Dimensional Natural Vibration Analysis and Energy Considerations for a Piezoelectric Rectangular Plate
,”
J. Sound Vib.
0022-460X,
283
, pp.
1093
1113
.
18.
Reddy
,
J. N.
, 1986,
Applied Functional Analysis and Variational Methods in Engineering
,
McGraw-Hill
, New York.
19.
Landau
,
L. D.
, and
Lifshitz
,
E. M.
, 1993,
Theory of Elasticity (in Polish)
,
PWN
, Warsaw.
20.
Le
,
K. C.
, 1999,
Vibrations of Shells and Rods
,
Springer
, Berlin.
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