Three-Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders.

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Three-Dimensional Investigation of the Propagation of Waves in HollowCircular Cylinders. I. Analytical Foundation

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  • THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 31, NUMBER 5 MAY, 1959

    Three-Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders. I. Analytical Foundation

    DENOS C. GAZI$ Research Laboratories, General Motors Corporation, Detroit 2, Michigan

    (Received November 28, 1958)

    The propagation of free harmonic waves along a hollow circular cylinder of infinite extent is discussed within the framework of the linear theory of elasticity. A characteristic equation appropriate to the circular hollow cylinder is obtained by use of the Helmholtz potentials for arbitrary values of the physical parameters involved. Axially symmetric waves, the limiting modes of infinite wavelength, and a special family of equi- voluminal modes are derived and discussed as degenerate cases of the general equations.

    INTRODUCTION

    HE propagation of free harmonic waves in an infinitely long cylindrical rod has been discussed

    on the basis of the linear theory of elasticity by Poch- hammer and Chree? Similar waves in a hollow circular cylinder have been investigated, under the restriction of axial symmetry of motion, by McFadden, a Ghosh, 4 and Herrmann and Mirsky. 5 A three-dimensional solu- tion of the more general problem of wave propagation in a hollow cylinder without the stipulation of axial symmetry is desirable, not only as a further contribution to the elastic theory, but also as a means of estimating the range of applicability of various shell theories.

    This paper contains the analytical foundation for the investigation of the most general type of harmonic waves in a hollow circular cylinder of infinite extent. A characteristic equation has been obtained, in the framework of the linear theory of elasticity, for the eigenmodes of an isotropic continuum bounded by two concentric cylindrical surfaces. This equation appears, in general, rather intractable, but its evaluation can always be achieved numerically by the use of a modern high-speed electronic computer. In this manner the frequency spectrum has been completely determined for a wide range of the physical parameters that are involved.

    The paper is divided into two parts. In Part I the frequency equation is obtained for an arbitrary number of waves around the circumference n, Poisson's ratio v, ratio of wall thickness to internal radius h/a, and longitudinal wave number (Fig. 1). The Helmholtz displacement potentials are used for the sake of clarity of presentation. An added advantage of this presenta- tion is that it demonstrates the mechanism of coupling of dilatational and equivoluminal motion; it also facilitates the derivation and discussion of some simple degenerate cases, namely the cases of axial symmetry and/or infinite wavelength, and the case of the Lamb-

    type equivoluminal modes. A derivation and discussion of these degenerate cases is included in Part I.

    Part II contains the numerical results obtained by the use of an IBM 704 digital computer, and a com- parison with the corresponding results of a shell theory.

    FREQUENCY EQUATION

    The equations of motion for an isotropic elastic medium are, in invariant form,

    tV2uq-(Xq-t)VV.u=p(Ou/Ot'), (1) where u is the displacement vector, p is the density, X and u are Lam(?s constants, and W' is the three-dimen- sional Laplace operator.

    The vector u is expressed in terms of a dilatational scalar potential and an equivoluminal vector potential H according to

    u= v,/,+vXH (2) with

    V-H=F(r,t). (3) In Eq. (3) F is a function of the coordinate vector r

    and the time, which can be chosen arbitrarily due to the gauge invariance 6 of the field transformation de- scribed by Eq. (2). The displacement equations of motion are satisfied if the potentials and H satisfy the

    L. Pochhammer, J. ftir Math. (Crelle) 81, 324-336 (1876). C. Chree, Quart. J. Math. 21, 287-298 (1886). a j. A. McFadden, J. Acoust. Soc. Am. 26, 714-715 (1954). 4 j. Ghosh, Bull. Calcutta Math. Soc. 14, 1, 31-40 (1923-1924). s G. Herrmann and I. Mirsky, Trans. Am. Soc. Mech. Engrs.

    78, 563-568 (1956).

    FIo. 1. Reference coordinates and dimensions.

    6 See, for example, P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Part 1, p. 297.

    568

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  • PROPAGATION IN CYLINDERS. I 569

    wave equations

    where

    Let

    ,'-V% = 0%/0t" (4) ,-V-H = 0-H/Ot ,

    ..= (X+ 2u)/, (S) 2 2 ---

    =/(r) cosn0 cos(t+z) tit= gr(r) sinn0 sin(cot-+-z) (6) Ho= go(r) cosno sin(cotq-z) tIz= ga(r) sinn0 cos(cot+ z);

    then Eqs. (4) yield

    (7) (V -- 1/r+/v)H, -- (2/r ) (OHo/OO)=0 (w- (odoo)=0.

    Furthermore, using the differential operator notation

    t Ox x Ox

    one obtains, from Eqs. (6) and (7),

    . ,[ga3= 0 (8) .[g-go=O

    where

    (9) The general solution of Eqs. (9) is given in terms of

    the Bessel functions 7 J and Y, or the modified Bessel functions I and K of the arguments ar= l arl and r= !r], depending on whether a and , as determined by Eqs. (9), are real or imaginary. The proper selection of Bessel functions to be used is shown in Table I.

    TABLE I. Bessel functions used at different intervals of the frequency co.

    Interval Functions used

    vd/

  • 570 DENOS C. GAZIS

    derivatives,

    X(a2q-2)f q-2[f"q--nr(ga'- g)q-g']} X cosnO cos (wt+

    aro= -- _ _ (2ga"--152ga)

    /n+l --[----g--g')} sinn0 cos(cot+ /z) (17)

    ={--2f'-nr[g'q-(n71--15=q-)g]--a} X cosno sin (cotq- z).

    Substitution of Eqs. (17) into Eqs. (16) yields the characteristic equation, formed by the determinant of the coefficients of the amplitudes A, B, A, B, As, and Ba, as follows:

    [c,il =0, (i, j=l to 6), (18) where i identifies the row and j the column of the determinant. The first three rows of this determinant are given by

    11-- [2n(n- 1)- (if'- 2)a']Z(ma)q - 2XalaZn+l ca.= 25lSxaZ (15xa)- 2a(nq- 1)Za ($aa) caa= - 2n(n- 1)Z (15aa) q- 2X nlSaaZx (15xa) c14 = [2n(n-- 1)-- (tt -- )a']W(ma)q - 2maW,+a (ala) ca= 2X2/51a'W (Ba)-- 2 (n-t- 1)aW,4 c16= -- 2n(n- 1) W (gaa) q- 2ntSaaW,,+a (gaa) c.= 2n(n- 1)Z (aa) - 2X nmaZ+ (ma) c..= -- lSaaZ (5aa) q- 2a (n q- 1)Z a ($aa) c.a= -- [ 2n (n- 1) -- lSa'JZ (/51a)- 2X gSaaZ, a (19) c.4= 2n(n-- 1) W (aa)-- 2naxaW+ (aa) c= -- XlSa'W (taa) q- 2a (n-i- 1) W ($a) c.6= -- [2n (n- 1)--15a2']W (15a) - 215aW,,+ (15a) a -- 2ncelZn (ota) -- 2Xla2Zn+l

    ntaz, (tt'- caa= -- naZ,, (151a) c,= 2naW (ala) - 2aaW (ma) ca,: MnlSaW,, (15a)-- (15 ca6= -- nliaW,, (15a). The remaining three rows are obtained from the first three by substitution of b for a. In the foregoing Eqs. (19) X and M are parameters which are introduced in order to account for the differences in the recursion and differentiation formulas between the different kinds of

    Bessel functions. The value of these parameters is 1 when J and Y functions are used, and - 1 when I and K functions are used.

    By reference to Table I it is seen that Xi vary as follows:

    v/

  • PROPAGATION IN CYLINDERS. I

    and longitudinal shear modes are the cutoff frequencies of waves with the same number of circumferencial waves n in a diagram of the frequency as a function of the wave number.

    For the longitional shear vibrations, it may be ascer- tained that the displacement field is derived from a potential g alone and is given by

    Ur = 40 -- 0

    I,g,=[AI[Jn([F)'JFBI[IZn([F)3 GOS/'/0 sint. (24) The frequency equation is

    J,'Oa)Y,'(b)-J'(b)Y,'(a)=O (25) and the amplitude ratio

    A 1/B= -- Y' (a)/J' (a). (25a) The lowest mode of this type corresponds essentially

    to a shearing of the cylinder as a whole across its diameter; the displacement u, is zero along 2n radial planes corresponding to the zeros of the function cos. The second and all higher longitudinal shear modes involve, in addition, a number of concentric nodal cylindrical surfaces, and correspond essentially to a shearing of the cylinder across its thickness.

    Equation (25) may be used for the determination of the frequency ratio

    for any arbitrary ratio h/a. A numerical computation of the longitudinal shear cutoff frequencies is included in Part II. A brief investigation of Eq. (25) for limiting values of h/a is given in the following.

    For thin cylindrical shells, that is h/al and b>>l. Accordingly, using the Hael-Kirchhoff asymp- totic approximations 1 for the Bessel functions, one obtains

    sinh-[(4n+3)h/8ab cosh=0 (26) and finally

    v l+8(q) , q=1,2,3.... (27) The preceding approximations are valid for all the longitudinal shear modes except the lowest one for which h0 as h/aO. As is to be expected, for h/aO the frequencies of the second and higher modes tend to the frequencies of the simple thickness-shear modes of a plate of thickness h.

    When a/hO, that is for an almost solid cylinder, Eq. (25) tends asymptotically to the corresponding frequency equation for a solid cylinder of radius h,

    nJ,(h)-hJ (h) = 0. (28) See, for example, reference 7, p. 194.

    MOTION INDEPENDENT OF 0

    For motion independent of the angular coordinate O, (n=0), the determinantal Eq. (18) breaks into the product of subdeterminants

    D)4= 0, (29) where

    11 1 14 15

    Da= Sl S9. 84 G85 and D4= ca c0 (30) co co c4 co

    and the terms co are given by Eqs. (19) with n=0.

    Longitudinal Waves

    The frequency equation

    Da=0 (31) corresponds to longitudinal waves, i.e., waves involving displacements up and u which are independent of 0. A frequency equation equivalent to Eq. (31) has been given by J. Ghosh, 4 who also derived a simplified equation for thin cylindrical shells, and the correspond- ing frequency equation for a cylinder which is rigidly clamped along one of its boundaries and free along the other.

    The displacement field is derived from a dilatational potential f and an equivoluminal potential g. As may be seen from Eqs. (17) the dilatational and equivolu- minal parts of the solution of the wave equations are in general coupled through the boundary conditions. However, some pure equivoluminal modes may exist uncoupled and are discussed in the following.

    Equivoluminal Lam6-Type Modes A particular type of equivoluminal waves, analogous

    to the ones first discussed by Lam( 8 for the infinite isotropic plate, may be obtained in the following manner.

    For n=0, the boundary conditions (17) are satisfied if /= > 0 f=g3=0 (32)

    g' (la)= g' (lb)=O, where

    g (ttr) = A J (lr) + Br Hence, the frequency equation

    and the amplitude ratio

    A /Bi= -- Y' (la)/J' (ta).

    (33)

    (34)

    (35)

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  • 572 DENOS C. GAZIS

    The complete solution is ur=//[-A 1J1 (r)nLBiYl(lgr) cos (wt-q- Uz= -- [A iJo(lr)q-BiYo(lr) sin(wt-l-z) (36) a,= -- 2u[A 121' (Br)q-BiY' (Br) cos(wt+z) gtO -- O'rO : O'rz: O.

    A superposition of two waves of this type, of the same amplitude and traveling in opposite directions yields standing equivoluminal waves with traction-free planes z-constant at intervals 2r/. A number of cylindrical surfaces r=cj may also be traction free, the cj being determined by

    Jl'(i)Yl'(a)--Jl'(ct)I/"l'(j)---O. (37) Equation (34) is of the same form as the frequency

    equation (25) of the longitudinal shear vibrations, for n-1. The results of the investigation of the latter are directly applicable to the case of the Lam(-type equivoluminal modes, insofar as the determination of the frequency ratio/h is concerned. Accordingly, for the limiting values of the ratio h/a one obtains the following:

    For thin cylindrical shells, i.e., sin/gh-- (71h/81g2ab) cos/gh=0 (38)

    and, finally, by virtue of the first of Eqs. (32)

    qr [ 7 (!)21 wV2--v. lq- , q-l, 2, 3.... (39) 8(qr) ' For h/a--O, Eq. (39) yields the frequencies of the

    straight-crested Lam( modes s of a plate of thickness h. When a/h-->O, Eq. (34) tends asymptotically to the

    frequency equation for the Lam(-type modes of a solid cylinder of radius h

    Jl' (lh) = O. (40) One last remark on the frequency equation (34): for

    some specific values of the ratio of the inner to the outer diameter of the hollow cylinder, a/b, it is possible to obtain a/g such as to make Jl' (a) and Jl' (/b) vanish simultaneously. As a consequence the amplitude B1 [Eq. (33)-] vanishes and one obtains the equivoluminal modes obtained by Goodman n for sections of an infinite plate. The appropriate ratios a/b are the ratios of any

    TABLE II.

    dJ (x) /dx -- 0 d Y (x) /dx - 0

    1.841185 3.683025 5.331445 6.941504 8.536320 10.123409

    11. 706009 13.285762 14.863590 16.440059

    n L. E. Goodman, "Circular-crested vibrations of an elastic solid bounded by two parallel planes," Proceedings of the 1st U.S. National Congress of Applied Mechanics, p. 70 (1951).

    two roots of Eq. (40). Similarly, ratios of any two roots of the equation

    dI/rl (X) /dx- O, (41) if set equal to a/b, make it possible to obtain solutions with zero amplitude A1 [Eq. (33)-] and hence another family of free contour surfaces analogous to those obtained by Goodman. It may be seen that the ratios a/b corresponding to the preceding modes are inde- pendent of the elastic constants of the material. The first five roots of Eqs. (40) and (41) are given in Table II.

    Torsional Waves

    For f- gl= 0 and D4-- 0, (42)

    one obtains motion involving displacements u only, i.e., torsional modes. Equation (42) may be reduced to

    J.(a)Y.(lb)--J.(Bb)Y.(lga)=O, (43) where/g is given by (9). It may be ascertained that no roots of Eq. (42) exist for/'

  • PROPAGATION IN CYLINDERS. I 573

    There is no dispersion for waves of this type, both the phase velocity and the group velocity being equal to v2.

    SUMMARY

    The displacement field is derived from a dilatational potential f and two equivoluminal ones g3 and g, all three of them periodic in 0 and sinusoidal in z. When the number of waves around the circumference n and the longitudinal wave number are both zero, i.e., for axially symmetric motion and infinite wavelength, the three potentials generate three uncoupled families of modes identified as plane-strain extentional, plane-strain shear, and longitudinal shear modes, respectively.

    For =0 and na0 the potentials f and g3 are coupled through the boundary conditions and generate the nonaxially symmetric plane-strain modes. An un- coupled family of longitudinal shear modes is again derived from g alone.

    For a0 and n=0 the potentials f and g are similarly

    coupled yielding the axially symmetric longitudinal modes, which are uncoupled from the g3-generated torsional modes. A special family of the longitudinal modes are the Lam(-type equivoluminal modes which are derived from g alone for some discrete values of the frequency and wavelength given by Eqs. (37) and (38).

    Finally, for a0 and na0 all three potentials are coupled through the boundary conditions. Thus non- axially symmetric waves may be considered as the result as coupling of motion in the plane r, 0, analogous to the motion in plane-strain and longitudinal shear motion. Alternatively, they may be considered as the result of coupling of modes analogous to the longitudinal and torsional modes but periodic in 0. This may provide an approach for obtaining the coupled frequency spectrum with the aid of the spectra of two uncoupled families of modes. A more direct approach is a numerical computa- tion of the frequency spectrum such as given in Part II of this paper.

    THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 31, NYMBER 5 MAY, 1959

    Three-Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders. II. Numerical Results

    DENOS C. GAZlS Research Laboratories, General Motors Corporation, Detroit 2, Michigan

    (Received November 28, 1958)

    The results are given of a numerical evaluation of a characteristic equation derived in Part I, appropriate to free harmonic waves propagated along a hollow cylinder of infinite extent. This equation is evaluated for some representative cylinders covering the entire range from thin shells to solid cylinders, and the results are compared with the corresponding results of a shell theory. Observations are made regarding the variation of the frequency spectrum with the physical parameters, as well as the range of applicability of shell theories.

    INTRODUCTION

    HE frequency equation for the propagation of free harmonic waves along a hollow cylinder of infinite extent is given in Part I of this paper, together with a discussion of some degenerate cases of simple motion.

    Part II contains the results of a numerical evaluation of the complete frequency spectrum which was obtained, for various sets of the physical parameters involved, by means of an IBM 704 digital computer. It also includes a comparison with the corresponding results obtained by Mirsky and Herrmann on the basis of their Timo- shenko-type shell theory.

    The reference coordinates and dimensions of the cylinder are shown in Fig. 9, and the covered range of parameters is as follows:

    Poisson's ratio, v=0.30; number of circumferential waves, n= 1 and 2; ratio of thickness to mean radius, re=h/R= 1/30,

    1/4, 1 and 2; ratio of thickness to wavelength, 0