Mindlin R.D., Yang J. An Introduction to the Mathematical Theory of Vibrations of Elastic Plates

Подождите немного. Документ загружается.

Intermediate Approximations

171

we obtain

to =

?b ^—— (6.0420)

P/Ki-v

)

This is the frequency for long wave-lengths of the lowest mode in the

three-dimensional theory. Equation (6.0420) is plotted in Fig. 5.071 as

the curve marked "Classical". It may be seen that the approximation is

very good for

cola>

<0.1.

6.05 Moderately-High-Frequency Vibrations of Thin Plates

In the reduction of the equations of Section 6.02 to those of Section

6.04, the range of the flexural equations is reduced to

cola>

<0.1 while

the extensional equations are usable up to

OJ/CO

=1.

It is worthwhile to

form equations in which the flexural part has a wider range. These have

applications to flexure of isotropic plates, and they are especially useful

in crystal plates where extension and flexure are coupled.

The equations are obtained from those of Section 6.02 simply by

dropping the remaining terms of the first-order kinetic energy. That is,

we drop the rotatory inertia terms but retain the thickness-shear strains.

In so doing, we lose the correction factor K

just as we did in the case of

reduction to the classical theory of thin plates (Section 6.04). In the

present case somewhat better results are obtained, especially for the

flexural part, if the correction factor is transferred to the

strain-energy-density, as described in Section 5.02. Then the

stress-displacement relations (6.025) become, for the triclinic plate,

172

Mathematical Theory of Vibrations of Elastic Plates

T^=2b

(i)

•

+ u

(1)

=2b

8x,

-31

+ C

(0)

dx-,

•

+ KC

•

+ u

(i)

/- (0) ^

•

(1)

_ 0w[

(0)

=2fticc

(0)

•

+ c.

dx,

+ /CC

+ M

(1)

+ C

—1_+—i—

^ a*i

3 J

+ /CC,

- +

(1)

-26

du™

-51

+ /CC

3x, "" dx

3x; dx

'<'

V &i

+M[

(1)

(0)

= 26/c!

du\

_ a«r

'- + C

3— -r-^64

OX,

-61

-+KC.

dx,

(1)

- + u

(1)

Intermediate Approximations

173

(X)

-d)

ftl

a*!" an,

)

—

+ /'l3 - —

dx,

+ ^33

+ /53

dx-,

to®

dx-,

+ ri5

+ r

du?

fix,

'to®

'duf

3",

(1)

——

dx-,

3»,

(1)

——

dx-,

a«,

(1)

+——

3 J

(6.051)

It will be observed that these are the same as (6.025) except that the

constants c are multiplied by K if one subscript is even and by K

if both subscripts are even. Except for this alteration of constants and the

omission of the accelerations ii

and ii^p, the equations of motion

(6.026) remain unchanged.

In the isotropic case, the extensional part reduces to the zero-order

approximation (Chapter 4) and the flexural part becomes (see (5.053) and

(5.065))

(0) _

2bK

T^=2bK

du^

dx,

(i)

+ u

(i)

duP

——

dx-,

3 J

du? du

]

——

v—

dx-, dx

(i)

(l-v)

1 J

duf

du[

3 J

(6.052)

174 Mathematical Theory of Vibrations of Elastic Plates

2bK

fi{V]ufUe^)

2bpuf

(l-v)Vf

+(l

2„0)

(l-v)V,w^+(l

deV

dx-,

-2bK

ij.

-2bK

fdu

f')

+ F!

(1)

=0 (6.053)

««

+ F

>=0

Upon eliminating w,

(1)

and u^ from (6.053) we obtain, for comparison

with (6.0418),

DVyv+lpbu^-^-V

(6.054)

K•

•v F;

To compare the frequency-wave-length relations given by (6.054) and

(6.0418) we set the face-tractions equal to zero and

,(°)

u?>

=Asin£

We obtain, from (6.054),

a> =

3p(l-v

)

2.2

+ -

2£

3/c

(\-v)

-1/2

(6.055)

When this relation is plotted on Fig.

5.071,

the curve is indistinguishable

from the corresponding curve for the exact solution.

References

Benscoter, S. U., Review of "Timoshenko's shear coefficient for flexural vibrations of

beams," by R. D. Mindlin and H. Deresiewicz, Applied Mechanics Review, Vol. 8,

(1995) pp.

510-511.

Bresse, M., "Cours de M^canique Applique^," Mallet-Bachelier, Paris, (1859) p. 126.

Carrier, Q R, "The propagation of waves in orthotropic media," Quarterly of Applied

Mathematics, Vol. 4, (1946) pp. 160-165.

Cauchy, A. L., "Sur I' 6quilibre et le mouvement d'une plaque solide," Exercices de

Mathematique, vol. 3, (1828) pp. 328-355. Also, Todhunter and Pearson, Vol. I, p.

336.

Cauchy, A. L., "Sur I' 6quilibre et le mouvement d'une plaque dlastique dont I' elasticity

n'est pis la meme dans tous les sens," Exercies de Mathematique, Vol. 4, (1829) pp.

1-14.

Ekstein, H., "High frequency vibrations of thin crystal plates," Physical Review, Vol. 68,

(1945) pp.

11-23.

Germain, Sophie, "Recherches sur la Theorie des Surfaces Elastiques," Courcier, Paris,

(1821).

Goodman, L. E., "Circular-crested vibrations of an elastic solid bounded by two parallel

planes," Proceedings of the First U. S. National Congress of Applied Mechanics,

(1951) pp.

65-73.

American Society of Mechanical Engineers, New York, N. Y.

Holden, A. N., "Longitudinal modes of elastic waves in isotropic cylinders and slabs,"

Bell System Technical Jouranl, Vol. 30, No. 4, Part 1, (1951) pp. 956-969.

Institute of Radio Engineers, "Standards on Piezoelectric Crystals," (1949); Proceedings

of the I.R.E., Vol. 37, No. 12, December, (1949).

Kane, T. R., and Mindlin, R. D., "High-frequency extensional vibrations of plates,"

Journal of Applied Mechanics, Paper No. 55-A-50, (1955).

175

176 Mathematical Theory of Vibrations of Elastic Plates

Kellogg, O. D., "Foundations of Potential Theory," Julius Springer, Berlin, (1929).

Kirchhoff,

G, "fiber das Gleichgewicht und die Bewegung einer elastischen Scheibe,"

Crelles Journal, Vol. 40, (1850) pp. 51-88. (Todhunter and Pearson, Vol. II, Part II,

pp.

39-50.)

Knott, C. G, "Reflection and refraction of elastic waves," Philosophical Magazine (5) Vol.

48,

(1899) pp. 64-97.

Koga, Issac, "Thickness vibrations of piezoelectric oscillating crystals," Physics, Vol. 3,

(1932) pp. 70-80.

Lamb.

H., "On waves in an elastic plate," Proceedings of the Royal Society, London,

Series A, Vol. 93, (1917) pp. 114-128.

Lame, M. G, "Lecons sur la Theorie Mathematique de I'Elasticity des Corps Solides,"

Gauthier-Villars, Paris, Second Edition, (1866).

Love, A. E. H., "A Treatise on the Mathematical Theory of Elasticity," Cambridge

University Press, London, (1927).

Mason, W. P., "Quartz Crystal Applications," Charter I, "Quartz Crystals for Electrical

Circuits," edited by Raymond A. Heising, D. Van Nostrand Co., New York, N. Y.,

(1946) pp. 11-56.

Mason, W. P., "Piezoelectric Crystals and Their Application to Ultrasonics," D. Van

Nostrand Co., New York, N. Y, (1950).

Mindlin, R. D., "Influence of rotatory inertia and shear on flexural vibrations of isotropic,

elastic plates," Journal of Applied Mechanics, Vol. 18, (Trans. ASME, Vol. 73)

(1951) pp. 31-38. (A)

Mindlim, R. D., "Thickness-shear and flexural vibrations of crystal plates," Journal of

Applied Physics, Vol. 22, (1951) pp. 316-323. (B)

Mindlin, R. D., "Forced thickness-shear and flexural vibrations of piezoelectric crystal

plates,"

Journal of Applied Physics,

Vol.

23, (1952)

pp.

83-88.

Poisson, S. D., "Memoire sur I' equilibre et le mouvement des corps elastiques," Mem,

Acad. Sci., Ser. 2, Vol. 8, Paris, (1829) pp. 357-570.

Rayleigh, Lord, "On waves propagated along the plane surface of an elastic solid,"

Proceedings of the London Mathematical Society, Vol. 17 (1887) pp. 4-11.

Rayleigh, Lord, "On the free vibrations of an infinite plate of homogeneous isotropic

elastic matter," Proceedings of the London Mathematical Society, Vol. 20, (1889) pp.

References

177

225-234.

Reissner, E., "The effect of transverse shear deformation on the bending of elastic

plates,"

Journal of Applied Mechanics, Vol. 12, (Trans. ASME, Vol. 67) (1945) pp.

69-77.

Reissner, E., "On bending of elastic plates," Quarterly of Applied Mathematics, Vol. 5,

(1947) pp. 55-68.

Sokolnikoff,

I. S., "Mathematical Theory of Elasticity," McGraw Hill Book Co., New

York, N.Y., (1946).

Sommerfeld, Arnold, "Spezielle Losungen des Problems der elastischen

Eigenschwingungen beim Quader und Wiirfel," S.-B. Math. -Nat. Abt. Bayer. Akad.

Wiss.

1945-46, (1947) pp. 51-88.

Sternberg, E., and Eubanks, R. A., "On the concept of concentrated loads and an

extension of the uniqueness theorem in the linear theory of elasticity," Journal of

Rational Mechanics and Analysis, Vol. 4, (1955) pp. 135-168.

Sykes, R. A., "Modes of Motion in Quartz Crystals, the Effects of Coupling and Methods

of Design," Chapter VI, "Quartz Crystals for Electrical Circuits," edited by

Raymond

Heising, D. Van Nostrand Co., New York, N. Y., (1946) pp. 205-248.

Timoshenko, S., "On the correction for shear of the differential equation for transverse

vibrations of prismatic bars," Philosophical Magazine, Series 6, Vol. 41, (1921) pp.

744-746.

Timoshenko, S., "Theory of Plates and Shells," McGraw Hill Book Co., New York, N. Y,

(1940).

Timoshenko, S., "On transverse vibrations of bars of uniform cross-section,"

Philosophical Magazine, Series 6, Vol. 43, (1922) pp.

125-131.

Timoshenko, S., and Goodier, J. N., "Theory of Elasticity," McGraw-Hill Book Co., New

York, N. Y, Second Edition, (1951).

Todhunter, I., and Pearson, K., "A History of the Theory of Elasticity," Cambridge

University Press, London, (1886).

Uflyand, Ya. S., "The propagation of waves in the transverse vibrations of bars and

plates,"

Akad. Nauk, SSSR, Prikl. Mat. Mekh., Vol. 12, (1948) pp. 287-300.

Voigt, W., "Lehrbuch der Kristallphysik," B. G. Teubner, Leipzing, (1928); Edwards

Brothers, Inc., Ann Arbor, Michigan, (1946).

Appendix

Applications of

the

First-Order Approximation

Mindlin, R. D., "Influence of rotatory inertia and sher on flexural vibrations of isotropic,

elastic plates," Journal of Applied Mechanics, Vol. 18, (Trans. ASME, Vol. 73"

(1951) pp. 31-38.

Mindlin, R. D., "Thickness-shear and flexural vibrations of crystal plates," Journal of

Applied Physics, Vol. 22, (1951) pp. 316-323.

Mindlin, R. D., "Forced thickness-shear and flexural vibrations of piezoelectric crystal

plates,"

Jouranl of Applied Physics, Vol. 23, (1952) pp. 83-88.

Mindlin, R. D., and Forray, M., "Thickness-shear and flexural vibrations of contoured

crystal plates," Journal of Applied Physics, Vol. 25, (1954) pp. 12-20.

Mindlin, R. D., and Deresiewicz. H., "Thickness-shear vibrations of piezoelectric crystal

plates with incomplete electrodes," Journal of Applied Physics, Vol. 25, (1954) pp.

21-24.

Mindlin, R. D., and Deresiewicz, H., "Suppression of overtones of thickness-shear and

flexural vibrations of crystal plates," Journal of Applied Physics, Vol. 25, (1954) pp.

25-27.

Mindlin, R. D., and Deresiewicz, H., "Thickness-shear and flexural vibrations of a

circular disk," Journal of Applied Physics, Vol. 25, (1954) pp. 1329-1332.

Mindlin, R. D., and Deresiewicz, H., "Timoshenko's shear coefficient for flexural

vibrations of beams," Proceedings of the Second U. S. National Congress of Applied

Mechanics, (1954) pp. 175-178. American Society of Mechanical Engineers, New

York, N. Y.

Deresiewicz, H., and Mindlin, R. D., "Axially symmetric flexural vibrations of a circular

disk," Journal of Applied Mechanics, Vol. 22, (Trans. ASM, Vol. 77) (1955) pp.

86-88.

179

180 Mathematical Theory of Vibrations of Elastic Plates

Mindlin, R. D., and Deresiewicz, H., "Thickness-shear and flexural vibrations of

rectangular crystal plates," Journal of Applied Physics, Vol. 26, (1955) pp.

1435-1442.

Kane, T. R., and Mindlin, R. D., "High-frequency extensional vibrations of plates,"

Journal of Applied Mechanics, Paper

No.

55-A-50, (1955).

Mindlin, R. D., Schacknow, A., and Deresiewicz, H., "Flexural vibrations of rectangular

plates,"

Journal of Applied Mechanics, Paper

No.

55-A-78, (1955).

Deresiewicz, H., "Symmetric flexural vibrations of a clamped circular disk," scheduled

for publication in the Journal of Applied Mechanics.