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Resonant oscillations of fluid-loaded struts
Journal of Sound and Vibration (1983) 87(3), 429437
RESONANT
OSCILLATIONS
OF FLUID-LOADED
STRUTS
D. G. CRIGHTON Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, England (Received 5 April 1982)
A simplified model is used to obtain a description of the way in which radiation damping limits the resonant oscillations of a cantilever or strut. It is shown that radiation damping is equivalent, in the low frequency limit, to a fictitious internal dissipation, with loss factor nlrad= (a/2)[m’/(m +m’)](k,,a)‘, where 2a is the strut thickness, m its mass per unit length and m’ the virtual fluid mass per unit length. Typical values of n)7,.dappear to be slightly higher than internal loss factors, showing that the resonant amplitude is limited only by radiation loss, even for thin struts for which kOa c 1. When kOa >>1 it is found that grad is much larger than any internal loss factor, and therefore that high frequency resonances are heavily damped, regardless of internal dissipation. The pressure field near the strut is also examined, and in the case k,a CC1 is found at resonance to exceed the driving pressure field by the large factor (kOa)-*. The far field pressure at resonance may, under some conditions, also greatly exceed the off-resonance scattered field, and this great scattering efficiency of a thin strut at resonance is demonstrated by detailed examination of a particular case.
1. INTRODUCTION The purpose of this work is to determine the vibration amplitude of a cantilever or strut system at resonance in a fluid environment under the action of an incident pressure field. The total pressure field may be regarded as composed of an incident wave, a wave scattered from the strut as if at rest, and a wave generated by motion of the strut as if no incident wave were present. Call the sum of the first two of these fields the driving pressure. It is a function of whatever sources are present in the fluid and of the geometry of the strut at rest, to be determined from the usual scattering theories. The third field above represents a back-reaction on the strut due to its motion through the fluid environment, and the purpose of this paper is to find its form in a simple way, so that it may be explicitly included in the left side, as it were, of a modified equation of motion of the strut. The case of principal interest here involves a strut whose thickness is small compared with the acoustic wavelength of the incident field-a prime requirement of a strut often being that its scattering cross-section should be as small as possible. That requirement, however, is the root of the problem which arises at resonance. Because the strut is an inefficient scatterer, it will give rise to a radiation loss which at resonance is insufficient to prevent large response occurring. Moreover, the pressure field near the strut at resonance will also be large, and may greatly exceed the driving levels. Further, under some circumstances this great increase of near field pressure will be propagated away without substantial cancellation effects, resulting in a far field level much in excess of what would be expected from consideration of the static scattering cross-section of the strut. These ideas are quite general, and do not need a very elaborate model to support them-which is fortunate in view of the fact that a satisfactory solution to the problem 429 0022-460X/83/070429
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of scattering by a rigid thin strut of finite length is still not in sight. In the next section a simple situation will be examined in which all the basic features will be exposed. The methods to be used are hardly novel; for example, the equation of motion of an oscillating fluid-loaded cylinder (to be derived in section 2, below) is in essence given on p. 163 of reference [l] in a discussion of the free waves on a fluid-loaded string, while some aspects of the radiation from a finite cylinder oscillating in a normal mode u,(z) = b, sin (n7rz/l) of the string or beam type have been discussed in Chapter 8 of reference [2]. What is believed to be novel is the application of these methods to the specific problem of a finite elastic strut at resonance, and to a comparison of resonant and off-resonant scattered pressure levels, both in the near field and in the very distant field. 2. EQUATIONS OF MOTION UNDER FLUID LOADING The strut or cantilever will be taken as circular in section, with radius a, and immersed in compressible fluid of density p and sound speed co. Where convenient one can also suppose the strut to be infinite in length by the device of continuing the real finite strut into an infinite one with the additional lengths held at rest (cf. [2] page 227). This avoids difficulties with the end faces of the strut-which one can argue are always negligible anyway for a thin strut, that being the case of principal interest. Take cylindrical co-ordinates (r, 8, z), and let the velocity of the strut at any axial station t be u(z) exp (-iwt) in the direction 8 = flo. It is assumed that torsional and compressional motion of the strut does not occur, and the discussion is confined to pure bending motion in the plane 8 = oo. Then if there is no incident field, the velocity potential 4 in the fluid satisfies (V2+k&
=o,
(2.1,2.2)
~(a,8,Z)=U(Z)COS(B-Bo),
with a radiation condition at infinity, k. = w/co being the acoustic wavenumber. A Fourier integral solution is easily found in the form 4 (r, f?,z) = & /;a @(r, 6, a) exp (-icuz) da, 00 @(r, 8,
(Y)=
U(CX)cos (e -
eo) HI(w)
K
u(z)
HI’(KU)’
(2.3) exp (icuz) dz.
Here H1 = HI”’ denotes the Hankel function of the first kind and of order unity, while with a suitable choice of branches. (k&a*)“*, From equation (2.3) one finds that the force on the strut at section z and in the direction 8 = e. is K =
-P2
iwa
U(a)Hl(Ka)
e-i”’
da
=
u(z')G(z
-2’)
dz,
(2.4)
K &‘(~a)
WY)=-p$l__
+co HI(KU) KHicKQJe
_.
‘aYda.
(2.5)
Suppose next that the strut is subjected to a driving force f in the direction 8 = Bo, and that the strut has mass m per unit length and a stiffness coefficient B. These are, by definition, such that the equation of motion of the strut in a vacuum would be B (a4u/az4) - mw = -iof under the same driving force f (2).Then obviously the equation of motion for the strut in the fluid environment is obtained from the vacuum equation
*u
OSCILLATIONS
OF
FLUID-LOADED
431
STRUT’S
simply by writing u(z’)G(z -z’> dz’
fm+j_I in place of f(z), to give +cO
4
Bs-mo’u
+io
I -cc
u(z’)G(t
-z’) dz’= -iwf(z),
(2.6)
If one has a cantilever system in which, say, the end x = 0 is built-in, so that u(O) = u’(0) = 0, while the end x = 1 is free, so that u”(Z) = u”(l) = 0, then equation (2.6) is to be solved for u(z) subject to these four boundary conditions. A way of doing this formally would be to expand u(z) and f(z) as unknown and known series, respectively, of the eigenfunctions of the equation B(a4u/~z4) - rn~‘u = 0, with the same boundary conditions. In general, however, this leads to great algebraic difficulties because of the form of G(y), and it is preferable to make some approximations before making the eigenfunction expansion. Suppose that u(z) varies only slowly with axial distance z. In the cases considered later, for example, u is a function of (nrz/l) for integral values of n. Then one requires that the scale l/nlr on which u changes should be large compared with the scale on which G(y) changes. The maximum scale for changes in G is clearly the larger of k<’ and a, so that in the low frequency limit koa <<1 one must require k,$/nm >>1 in order for u to be slowly varying. In the high frequency case, k,g >>1, the requirement is l/a >>nrr, and for small n this must in any case be satisfied if bending motion is to dominate torsional and compressional motion. It must be stressed here that the terms “low” and “high” frequency are being used to indicate whether the acoustic wavelength is large or small, respectively, compared with the strut radius. In both cases the frequency must be high enough for satisfaction of the appropriate slow variation condition, and it will be seen below in equation (2.7) that this always involves the approximation of K by k. for all wavenumbers (Ysignificantly present in the strut vibration. But this, of course, is equivalent to requiring that the frequency lie above the “coincidence” frequency at which a wave on the strut in a vacuum would have sonic phase speed. The basic approximation of this paper is thus one of high frequency in the sense that the frequency must exceed the coincidence frequency, and only then can one further distinguish between low and high frequencies. When the variation of u(z) is slow in the sense described above, one has the approximation u(z’)G(z -z’) dz’=-u(z)+ou
Hl(koa
1
koH;(koa)’ and the equation of motion is B a4u/dz4-
mw*u +q.m2u[H~(kou)/koH~(kou)]u
= -hf.
(2.7)
Clearly the approximation here is that at any axial station the motion is identical with that produced by an infinite cylinder oscillating everywhere with the amplitude and phase actually existing at that station. Further simplification can be achieved in the low and high frequency limits. Take kou c 1, for example, and expand the Hankel functions for small argument, keeping only the dominant real term and the dominant imaginary term since these have physically distinct interpretations. Thus Hi(x)/H;(x) =x(-l + 7rx2/2i)
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D. G. CRIGHTON
and then equation (2.7) becomes B(a4u/az4) - (m + m’)w*z4 -[im’7r(k0a)*/2]f.0*c4 = -iwf,
(2.8)
in which the virtual fluid mass is m’ = 7ru *p per unit length. One observes that the effect of fluid loading is to increase the effective cylinder mass from m to m +m’, and to introduce a dissipative term. By including further terms in the expansion of Hi(x) one can derive small compressibility corrections to the (incompressible) virtual mass m’, but such small effects are of no interest here-and in any case are not significant in view of the approximations already made. In the high frequency limit, koa >>1, the Hankel functions can be expanded for large argument, with the result that B(t14u/t3z4)-(m +a)u*u -i(pah/2)o*u
= -iwf,
(2.9)
where A = 2?rkO’ is the acoustic wavelength, and the virtual mass is now u = ph */8~. In practical terms, the effect of the virtual mass on resonance frequencies is very substantial when kou <<1, but negligible when k,g >>1. 3. RESONANT OSCILLATIONS As a typical case, consider the built-in cantilever, with u(O) = u’(0) = 0 and u”(l) = u”‘(l) = 0 as conditions at the clamped and free ends, respectively. In the low frequency limit, kou c 1, it can be anticipated that in equation (2.8) the virtual mass term will be large, with the dissipative term generally small, and it is therefore appropriate to solve equation (2.8) by expanding in terms of the eigenfunctions of the equation B(a4u/az4) (m +m’)o*u = 0, with the above boundary conditions. It is easy to show that if q4 = (m + m ‘)o*/B, then the eigenvalues are given by q” = ?lT/l and that the corresponding
(n = 1,2,. . .)
eigenfunctions
u,, = N,{sinh nr[cosh
are
(nrz/l)
-cos (n7z/l)]
- (cash n7~+ cos nr)[sinh
(nr.z/l) -sin (~.z/l)]},
where the N,, are chosen so that the eigenfunctions
I
(3.1)
are orthonormal
(3.2) over (0, I): i.e.,
r
u,(z)u,(z)
dr = S,,.
0
Write u(z)=
-iwf(z)=
5 a”&(r),
g b,u,(z),
(3.3)
II=1
n=l
where the a, are to be found from equation (2.8), while the 6, are, in principle, known in the form b, = -iw
‘f(z)un(z)
I0
dz.
(3.4)
Then one finds at once from equation (2.8) that 00
u(z)=n~l
ban (2) [B(nr/l)4-Bq4-$im’r(kou)2w2]’
(3.5)
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Consider now the resonant case, in which o is such that equation (3.1) is satisfied for some IZ: i.e., the nth mode resonantly excited. Then, provided that with this value of W, 2~b,,~/rn’~(k~a)*~~ is large compared with the values of Jb,,l/[B(n’~/l)4-Bq4] for all values of n’ other than II, the resonant n th mode will be the only mode appreciably excited, and its velocity amplitude will be 2ib,u,(z)/m’~k~a2w2.
(3.6)
Compare this result with the corresponding result obtained by ignoring damping due to fluid loading, but including in its place a structural damping term. Introduction of a conventional mechanical loss factor n in place of radiation damping results in B(l-iq
sgnw)a4u/az4--Bq4u
=-iwf,
(3.7)
and then the amplitude of the n th mode at resonance is ib,u,(z)/Bq
(n~/l)~.
(3.8)
Thus radiation damping is equivalent to a fictitious internal loss factor given by q& = m’~k~a2u2/2B(n~/l)4, where one must remember that w is given by equation (3.1) and that koa is necessarily small, so that qrad
=
(T/2)
[m’/h
+m’)]
(koa)‘.
(3.9)
This shows clearly that radiation damping is very small, even at resonance, under conditions designed to minimize the off-resonance scattering cross-section of the cantilever. However, small though it is, radiation damping appears to be more effective in limiting resonances than the normal internal mechanisms for energy loss. For a steel bar with thickness 5 cm and a resonance frequency around 5 kHz in water one sees, for example, that qrad- 5 x lo-‘, a value rather higher than the mechanical loss factor for steel. The high frequency limit, koa >>1, can be treated in a similar manner. The amplitude of the n th mode at resonance is 2ib,u,(z)/pahw’,
(3.10)
and the effective radiation loss factor is Trad
=
br~*/~)(ko~)-‘.
(3.11)
For the 5 cm thick steel bar in water with a resonance around 20 kHz, the high frequency approximation is probably adequate, and then one has n& - 6 x lo-‘, which is again larger than any mechanical loss factor likely to be obtained in practice. Thus, except at very high frequencies indeed (in water), radiation damping seems in the high frequency limit to be sufficiently strong that large amplitude resonance cannot occur. 4. THE LOCAL PRESSURE FIELD AT RESONANCE In the low frequency limit, the motion of the strut is governed by equation (2.8), in which the driving pressure field is dictated by the incident field and the geometrically scattered field alone. Suppose that the driving pressure has a typical level PO at the strut surface, and that the driving pressure is distributed along the strut in a way which generally resembles the n th resonant mode shape of expression (3.2). Then only the n th mode will be significantly excited, and its velocity amplitude will be given by equation (3.6). That velocity field in turn drives a pressure field which is determined by equations (2.1) and (2.2), with expression (3.6) for u(z). The amplitude of this pressure field at
434
D. G. CRIGHTON
the strut surface r = a is easily seen to exceed the driving amplitude by a factor of order (k&Y -a very large increase under the assumed conditions. This opens up a serious problem-that design of struts to minimize off-resonance scattering necessarily brings large increases of local pressure levels at resonance. It cannot, however, be concluded that the far field pressure at resonance will be increased in the same manner, for it was required in section 2 that when koa <<1, then k,,l/tnr D 1, in order for equation (2.8) to hold. Thus for moderate values of n the acoustic wavelength must nof be large compared with the strut length, and in that case the far field is determined by a delicate balance of cancellation effects along the strut. To see just what sort of balance is achieved, an examination will be given in the next section of a case which exposes the essential results with a minimum of algebra. 5. THE FAR FIELD PRESSURE FROM RESONANT
SCATTERING
The eigenfunctions (3.2) for the cantilever with one free end and one built-in end lead to very heavy algebra, so matters will be simplified a little by considering instead a strut with pinned (simply held) ends at z = OJ. The eigenfunctions then satisfy a4u/& 4 - q4u = 0, with u(0) = u”(0) = u(I) = u”(l) = 0 and with q4 = (m +m’)w’/B. The normalized eigenfunctions for this case are
U, = (2/1)“2 sin (~.z/l),
(5.1)
and the velocity amplitude of the n th mode at resonance is exactly as given in equation (3.6)-since the eigenvalues are still q,, = m/l as for the cantilever, though the eigenfunctions are of course different in the two cases. Whenever necessary the reader must imagine the strut to be continued to infinity in both directions by infinite lengths of strut held at rest. As before, the strut will be taken to be circular in section. The z-axis lies along the strut, (x, y) and (r, 8) are Cartesian and polar co-ordinates in planes perpendicular to the strut, while (R, 8, $) will be taken as spherical polar co-ordinates based on any convenient origin on the strut, with the angle $ measured from the z-axis. Suppose that the strut is excited by a plane incident wave of potential 4i = exp (-ikoz
cos t,bo-
the time factor exp (-iwt) being understood. is the solution of
ikox sin &J,
Then the geometrically
4s = exp (-ikOz cos &)4(x, y), (a2/8x2+a2/ay2+K2)4 a(& + di)/ar = 0
(5.2) scattered field & = 0,
on r = U, all 8, 2,
with a radiation condition at infinity and with K = k. sin IJ~~.This field is easily calculated, and near the surface of a thin strut one finds 4i+4s
-
-iK(r+u2/r)cos8
exp (-ikoz cost,h)
(for kg cc 1, kg<< 1). From this, the force f(z) on unit length of the cylinder in the direction 8 = 0 is found to be f(z) = -27rpKou2 exp (-ikOz cos &,), whose eigenfunction (5.1), are
(5.3)
coefficients b,, with respect to the functions u,(z) of expression
6, =iw(2/1)1’22~pKou2nd[e-ikofCoS”ocos
nr-lJ/(k~12cos2
&-n27r2).
(5.4)
OSCILLATIONS
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STRUTS
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If the frequency is such that the it th mode is resonantly excited, its velocity amplitude at resonance is given by equations (3.6), (5.1) and (5.4) as 8pK nr[e-ik~’m$0cos n7r
u(z)=-,‘k2
0
-
(k3zcos2I&n2?r2)
nllz l] sin _ 1 *
Under the further assumption that other modes are then only negligibly excited, and bearing in mind the assumption that u(z) = 0 outside the segment (0, I), one finds the Fourier transform of the velocity along the z-axis is simply
In particular, one finds U(ff =
8pKn2r21 -ko
cm
$I=
-
m,k2 0
[eeikoL‘08 ‘O cos mr - l] [e-iko’cosllrco,s nrr - l] > (kzo12cos* &l- n27r2) (&$* co? IJ - n27r2) *
(5.5)
Equation (2.3) now determines the potential & of resonance scattering as
where cP(r, 8, C-X)= U(a) cos 13 Hr(Kr)/KHi(Ka), K = (kg -cx~)~‘~ and V(cr) is given in equation (5.5). At distances R much greater than either the wavelength ko’ or the length 1 of the strut, this Fourier integral may be evaluated by expanding Hi(M) for large Kr and then performing a standard steepest descent calculation. The result involves a knowledge of V(a) only for (Y= -k. cos (jl, as given in equation (5.5), and after some reduction one finds, remembering that koa CC1, Ce-W ~0sGocos ,r,r _ I] [,-W ~0s+ cos n,r _ 11 $,-4’(g) (kol)(n~>2 sin * sin CL0cos 8 (ko2,2 cos*&J-n2?T2) (k~12c0s2f+t2**)~
(5.6) If the propagation vector makes a general angle B. with the (y, z) plane, and not the special angle e. = 0 chosen in expression (5.2), then one has to replace 8 in equation (5.6) by (8 -eo). Then it is evident that expression (5.6) is symmetric in the angles (8, eo) and in (I,+,tie). This is a satisfactory state of affairs in that it shows that the large number of approximations made has not violated the fundamental Reciprocity Theorem [3, p, 1951. The 8 dependence of expression (5.6) is simply that of a dipole, reflecting the fact that the strut is compact relative to the wavelength (koa CC1) in its transverse dimension, In the axial direction the strut is not compact (in fact in the derivation of expression (5.6) it has been assumed that kol >>n7~), and consequently the $-dependence is that of a typical long array, with a rapidly varying pattern of lobes arising from the factor exp (-ikol cos 9). The total power scattered at resonance can be expressed in closed form from expression (5.6), but the result is too cumbersome to be useful here. Next, a rough comparison will be attempted between expression (5.6) and the field scattered (geometrically) from a finite rigid strut of large length (kol >>1) and small thickness under the same incident field (5.2). The field scattered from the small circular ends of the strut will be ignored and the assertion will be made that the far field can be expressed in terms of an integral along the strut of the net force per unit length exerted on the fluid by the strut. That force is in the direction 8 = 0, and is equal to -f(z) as
436
D. G. CRIGHTON
given in equation (5.3). The radiated density then follows from a well-known expression [4, p. 621 as 1 a ‘f(z’, t - f’/Co) dz p’(x, t)=-7 r’ 47rc0ax I 0
(5.7)
where f(z, t) = f(z ) exp (-iwt) and r’= {x2 + y2 + (z - z ‘)2}1’2.Carrying out appropriate far field simplifications, one finds P’--
ipwKu 2 exp (-iot) 2c:koR
a
eikoR
-ik,l(cos
llo+cos
a)
_
.
11
ax 1(cos (I,+ cos $0) e
I
Differentiation of exp (ika) is the only process which gives a term of order R-',the derivatives of cos $ contributing only terms at most 0(Re2). Hence one ends up with a far field scattered potential & from the strut at rest given by ik,R
“-?
i e (--> k&
e
(koa)2 sin 4 sin &, cos 8 (
-ik,l(cos
~+cos
a&) _
cos~+cos&J
1 >*
(5.8)
Note again, that if allowance is made for a non-zero angle & for the incident field, the expression (5.8) is then symmetric under interchange of 8 and 6+,or of 4 and $O. Expressions (5.6) and (5.8) contain rather similar rapidly varying directional factors like exp [-ikJ(cos I,++ cos I,@]. In the case of the rigid strut, I plays ho part in determining the field amplitude, whereas in the case of the resonant strut, the thickness a plays no role. Since the field q& decreases like (kou)2 while 4, increases as (kol) (at any rate for angles 4, &, near to 7r/2), it is certain that the resonant field will exceed the geometrically scattered field in some angular ranges at least. For example, if tJ and &, are close to 1r/2 (corresponding to observation of the back-scattered field from a plane wave incident broadside on the strut) one has & = O(kJ) and q& = U(k$z’k,,l), so that &/& = 0((kou)-2). In this case the distant field at resonance exceeds the geometrically scattered field by (kou)-2-exactly as in the near field, discussed in section 4. In other situations the distant field need not be so dramatically increased at resonance. For example, if IJ and 40 are both small, 4, = 0((kJm3) and & = G((kOu)2), and the ratio &/I$~ may be either large or small depending on kou and the value of I/u. But at any rate it is clear that the near field amplification at resonance, discussed in section 4, will be maintained into the far field, at least within certain angular ranges. 6. CONCLUSIONS The analysis of section 5 shows that the field scattered geometrically by a thin long rigid strut decreases as (kou)2, and that the length 1 serves mainly to determine the angular distribution. Thus the scattered field in general is best minimized by reduction of kou. If, however, the strut is not rigid, this reduction of kou reduces the radiation damping which is the only significant dissipative mechanism controlling resonant oscillations. The thin strut therefore executes large oscillations at resonance, and generates a nearby pressure field in excess of the geometrically scattered field. According to section 4, .the surface pressure amplitude at resonance exceeds that on a perfectly rigid strut by the large factor (kou)-2. According to section 5 this increase will generally be transmitted to the distant field also, particularly if the strut is long, since the distant resonant field increases with 1 (at least within some angular ranges) while the geometrically scattered amplitude is largely independent of I. There are two options open for reducing the importance of the resonant increases. One can increase (kou) (not a very appropriate
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step, since it only reduces the resonant field relative to the static scattered field, the latter being increased), or one can introduce internal loss mechanisms in the strut and thereby eliminate radiation loss as the basic controller of resonance. ACKNOWLEDGMENT The author is grateful to a referee for bringing to his attention references [5] and [6]. Of these, reference [5] has much in common with the present work, dealing as it does with the scattering of a plane wave by a thin circular rod, with the rod displacement expanded in terms of natural modes. The aim of reference [5] was, however, to show that strong scattering is to be observed in a direction opposite to that of the incident sound wave, whereas the aim of the present work is to show rather generally the intense scattering from a thin resonant strut. Reference [6] also deals with scattering of an acoustic field by a thin strut, this time with the incident field being due to a point source within a rectangular chamber. The expansion in eigenmodes again has points in common with the present approach, but the specific problem addressed is quite different. The author wishes also to thank Dr M. C. Junger for a helpful discussion.
REFERENCES 1. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill Book Company. 2. M. C. JUNGER and D. FEIT 1972 Sound, Structures and Their Interaction. Cambridge, Massachussetts: M.I.T. Press. 3. A. D. PIERCE 1981 Acoustics: An Introduction to its Physical Principles and Applications. New York: McGraw-Hill Book Company. 4. J. LIGHTHILL 1978 Waves in Fluids. Cambridge, England: Cambridge University Press. 5. L. M. LIAMSHEV 1958 Soviet Physics-Acoustics 4, 50-58. Scattering of sound by a thin bounded rod. 6. S. N. YOUSRI and F. J.FAHY 1976 Journal of Sound and Vibration 45, 583-594. Acoustically induced vibration of, and sound radiation from, beams inside an enclosure.
RESONANT
OSCILLATIONS
OF FLUID-LOADED
STRUTS
D. G. CRIGHTON Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, England (Received 5 April 1982)
A simplified model is used to obtain a description of the way in which radiation damping limits the resonant oscillations of a cantilever or strut. It is shown that radiation damping is equivalent, in the low frequency limit, to a fictitious internal dissipation, with loss factor nlrad= (a/2)[m’/(m +m’)](k,,a)‘, where 2a is the strut thickness, m its mass per unit length and m’ the virtual fluid mass per unit length. Typical values of n)7,.dappear to be slightly higher than internal loss factors, showing that the resonant amplitude is limited only by radiation loss, even for thin struts for which kOa c 1. When kOa >>1 it is found that grad is much larger than any internal loss factor, and therefore that high frequency resonances are heavily damped, regardless of internal dissipation. The pressure field near the strut is also examined, and in the case k,a CC1 is found at resonance to exceed the driving pressure field by the large factor (kOa)-*. The far field pressure at resonance may, under some conditions, also greatly exceed the off-resonance scattered field, and this great scattering efficiency of a thin strut at resonance is demonstrated by detailed examination of a particular case.
1. INTRODUCTION The purpose of this work is to determine the vibration amplitude of a cantilever or strut system at resonance in a fluid environment under the action of an incident pressure field. The total pressure field may be regarded as composed of an incident wave, a wave scattered from the strut as if at rest, and a wave generated by motion of the strut as if no incident wave were present. Call the sum of the first two of these fields the driving pressure. It is a function of whatever sources are present in the fluid and of the geometry of the strut at rest, to be determined from the usual scattering theories. The third field above represents a back-reaction on the strut due to its motion through the fluid environment, and the purpose of this paper is to find its form in a simple way, so that it may be explicitly included in the left side, as it were, of a modified equation of motion of the strut. The case of principal interest here involves a strut whose thickness is small compared with the acoustic wavelength of the incident field-a prime requirement of a strut often being that its scattering cross-section should be as small as possible. That requirement, however, is the root of the problem which arises at resonance. Because the strut is an inefficient scatterer, it will give rise to a radiation loss which at resonance is insufficient to prevent large response occurring. Moreover, the pressure field near the strut at resonance will also be large, and may greatly exceed the driving levels. Further, under some circumstances this great increase of near field pressure will be propagated away without substantial cancellation effects, resulting in a far field level much in excess of what would be expected from consideration of the static scattering cross-section of the strut. These ideas are quite general, and do not need a very elaborate model to support them-which is fortunate in view of the fact that a satisfactory solution to the problem 429 0022-460X/83/070429
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of scattering by a rigid thin strut of finite length is still not in sight. In the next section a simple situation will be examined in which all the basic features will be exposed. The methods to be used are hardly novel; for example, the equation of motion of an oscillating fluid-loaded cylinder (to be derived in section 2, below) is in essence given on p. 163 of reference [l] in a discussion of the free waves on a fluid-loaded string, while some aspects of the radiation from a finite cylinder oscillating in a normal mode u,(z) = b, sin (n7rz/l) of the string or beam type have been discussed in Chapter 8 of reference [2]. What is believed to be novel is the application of these methods to the specific problem of a finite elastic strut at resonance, and to a comparison of resonant and off-resonant scattered pressure levels, both in the near field and in the very distant field. 2. EQUATIONS OF MOTION UNDER FLUID LOADING The strut or cantilever will be taken as circular in section, with radius a, and immersed in compressible fluid of density p and sound speed co. Where convenient one can also suppose the strut to be infinite in length by the device of continuing the real finite strut into an infinite one with the additional lengths held at rest (cf. [2] page 227). This avoids difficulties with the end faces of the strut-which one can argue are always negligible anyway for a thin strut, that being the case of principal interest. Take cylindrical co-ordinates (r, 8, z), and let the velocity of the strut at any axial station t be u(z) exp (-iwt) in the direction 8 = flo. It is assumed that torsional and compressional motion of the strut does not occur, and the discussion is confined to pure bending motion in the plane 8 = oo. Then if there is no incident field, the velocity potential 4 in the fluid satisfies (V2+k&
=o,
(2.1,2.2)
~(a,8,Z)=U(Z)COS(B-Bo),
with a radiation condition at infinity, k. = w/co being the acoustic wavenumber. A Fourier integral solution is easily found in the form 4 (r, f?,z) = & /;a @(r, 6, a) exp (-icuz) da, 00 @(r, 8,
(Y)=
U(CX)cos (e -
eo) HI(w)
K
u(z)
HI’(KU)’
(2.3) exp (icuz) dz.
Here H1 = HI”’ denotes the Hankel function of the first kind and of order unity, while with a suitable choice of branches. (k&a*)“*, From equation (2.3) one finds that the force on the strut at section z and in the direction 8 = e. is K =
-P2
iwa
U(a)Hl(Ka)
e-i”’
da
=
u(z')G(z
-2’)
dz,
(2.4)
K &‘(~a)
WY)=-p$l__
+co HI(KU) KHicKQJe
_.
‘aYda.
(2.5)
Suppose next that the strut is subjected to a driving force f in the direction 8 = Bo, and that the strut has mass m per unit length and a stiffness coefficient B. These are, by definition, such that the equation of motion of the strut in a vacuum would be B (a4u/az4) - mw = -iof under the same driving force f (2).Then obviously the equation of motion for the strut in the fluid environment is obtained from the vacuum equation
*u
OSCILLATIONS
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431
STRUT’S
simply by writing u(z’)G(z -z’> dz’
fm+j_I in place of f(z), to give +cO
4
Bs-mo’u
+io
I -cc
u(z’)G(t
-z’) dz’= -iwf(z),
(2.6)
If one has a cantilever system in which, say, the end x = 0 is built-in, so that u(O) = u’(0) = 0, while the end x = 1 is free, so that u”(Z) = u”(l) = 0, then equation (2.6) is to be solved for u(z) subject to these four boundary conditions. A way of doing this formally would be to expand u(z) and f(z) as unknown and known series, respectively, of the eigenfunctions of the equation B(a4u/~z4) - rn~‘u = 0, with the same boundary conditions. In general, however, this leads to great algebraic difficulties because of the form of G(y), and it is preferable to make some approximations before making the eigenfunction expansion. Suppose that u(z) varies only slowly with axial distance z. In the cases considered later, for example, u is a function of (nrz/l) for integral values of n. Then one requires that the scale l/nlr on which u changes should be large compared with the scale on which G(y) changes. The maximum scale for changes in G is clearly the larger of k<’ and a, so that in the low frequency limit koa <<1 one must require k,$/nm >>1 in order for u to be slowly varying. In the high frequency case, k,g >>1, the requirement is l/a >>nrr, and for small n this must in any case be satisfied if bending motion is to dominate torsional and compressional motion. It must be stressed here that the terms “low” and “high” frequency are being used to indicate whether the acoustic wavelength is large or small, respectively, compared with the strut radius. In both cases the frequency must be high enough for satisfaction of the appropriate slow variation condition, and it will be seen below in equation (2.7) that this always involves the approximation of K by k. for all wavenumbers (Ysignificantly present in the strut vibration. But this, of course, is equivalent to requiring that the frequency lie above the “coincidence” frequency at which a wave on the strut in a vacuum would have sonic phase speed. The basic approximation of this paper is thus one of high frequency in the sense that the frequency must exceed the coincidence frequency, and only then can one further distinguish between low and high frequencies. When the variation of u(z) is slow in the sense described above, one has the approximation u(z’)G(z -z’) dz’=-u(z)+ou
Hl(koa
1
koH;(koa)’ and the equation of motion is B a4u/dz4-
mw*u +q.m2u[H~(kou)/koH~(kou)]u
= -hf.
(2.7)
Clearly the approximation here is that at any axial station the motion is identical with that produced by an infinite cylinder oscillating everywhere with the amplitude and phase actually existing at that station. Further simplification can be achieved in the low and high frequency limits. Take kou c 1, for example, and expand the Hankel functions for small argument, keeping only the dominant real term and the dominant imaginary term since these have physically distinct interpretations. Thus Hi(x)/H;(x) =x(-l + 7rx2/2i)
432
D. G. CRIGHTON
and then equation (2.7) becomes B(a4u/az4) - (m + m’)w*z4 -[im’7r(k0a)*/2]f.0*c4 = -iwf,
(2.8)
in which the virtual fluid mass is m’ = 7ru *p per unit length. One observes that the effect of fluid loading is to increase the effective cylinder mass from m to m +m’, and to introduce a dissipative term. By including further terms in the expansion of Hi(x) one can derive small compressibility corrections to the (incompressible) virtual mass m’, but such small effects are of no interest here-and in any case are not significant in view of the approximations already made. In the high frequency limit, koa >>1, the Hankel functions can be expanded for large argument, with the result that B(t14u/t3z4)-(m +a)u*u -i(pah/2)o*u
= -iwf,
(2.9)
where A = 2?rkO’ is the acoustic wavelength, and the virtual mass is now u = ph */8~. In practical terms, the effect of the virtual mass on resonance frequencies is very substantial when kou <<1, but negligible when k,g >>1. 3. RESONANT OSCILLATIONS As a typical case, consider the built-in cantilever, with u(O) = u’(0) = 0 and u”(l) = u”‘(l) = 0 as conditions at the clamped and free ends, respectively. In the low frequency limit, kou c 1, it can be anticipated that in equation (2.8) the virtual mass term will be large, with the dissipative term generally small, and it is therefore appropriate to solve equation (2.8) by expanding in terms of the eigenfunctions of the equation B(a4u/az4) (m +m’)o*u = 0, with the above boundary conditions. It is easy to show that if q4 = (m + m ‘)o*/B, then the eigenvalues are given by q” = ?lT/l and that the corresponding
(n = 1,2,. . .)
eigenfunctions
u,, = N,{sinh nr[cosh
are
(nrz/l)
-cos (n7z/l)]
- (cash n7~+ cos nr)[sinh
(nr.z/l) -sin (~.z/l)]},
where the N,, are chosen so that the eigenfunctions
I
(3.1)
are orthonormal
(3.2) over (0, I): i.e.,
r
u,(z)u,(z)
dr = S,,.
0
Write u(z)=
-iwf(z)=
5 a”&(r),
g b,u,(z),
(3.3)
II=1
n=l
where the a, are to be found from equation (2.8), while the 6, are, in principle, known in the form b, = -iw
‘f(z)un(z)
I0
dz.
(3.4)
Then one finds at once from equation (2.8) that 00
u(z)=n~l
ban (2) [B(nr/l)4-Bq4-$im’r(kou)2w2]’
(3.5)
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Consider now the resonant case, in which o is such that equation (3.1) is satisfied for some IZ: i.e., the nth mode resonantly excited. Then, provided that with this value of W, 2~b,,~/rn’~(k~a)*~~ is large compared with the values of Jb,,l/[B(n’~/l)4-Bq4] for all values of n’ other than II, the resonant n th mode will be the only mode appreciably excited, and its velocity amplitude will be 2ib,u,(z)/m’~k~a2w2.
(3.6)
Compare this result with the corresponding result obtained by ignoring damping due to fluid loading, but including in its place a structural damping term. Introduction of a conventional mechanical loss factor n in place of radiation damping results in B(l-iq
sgnw)a4u/az4--Bq4u
=-iwf,
(3.7)
and then the amplitude of the n th mode at resonance is ib,u,(z)/Bq
(n~/l)~.
(3.8)
Thus radiation damping is equivalent to a fictitious internal loss factor given by q& = m’~k~a2u2/2B(n~/l)4, where one must remember that w is given by equation (3.1) and that koa is necessarily small, so that qrad
=
(T/2)
[m’/h
+m’)]
(koa)‘.
(3.9)
This shows clearly that radiation damping is very small, even at resonance, under conditions designed to minimize the off-resonance scattering cross-section of the cantilever. However, small though it is, radiation damping appears to be more effective in limiting resonances than the normal internal mechanisms for energy loss. For a steel bar with thickness 5 cm and a resonance frequency around 5 kHz in water one sees, for example, that qrad- 5 x lo-‘, a value rather higher than the mechanical loss factor for steel. The high frequency limit, koa >>1, can be treated in a similar manner. The amplitude of the n th mode at resonance is 2ib,u,(z)/pahw’,
(3.10)
and the effective radiation loss factor is Trad
=
br~*/~)(ko~)-‘.
(3.11)
For the 5 cm thick steel bar in water with a resonance around 20 kHz, the high frequency approximation is probably adequate, and then one has n& - 6 x lo-‘, which is again larger than any mechanical loss factor likely to be obtained in practice. Thus, except at very high frequencies indeed (in water), radiation damping seems in the high frequency limit to be sufficiently strong that large amplitude resonance cannot occur. 4. THE LOCAL PRESSURE FIELD AT RESONANCE In the low frequency limit, the motion of the strut is governed by equation (2.8), in which the driving pressure field is dictated by the incident field and the geometrically scattered field alone. Suppose that the driving pressure has a typical level PO at the strut surface, and that the driving pressure is distributed along the strut in a way which generally resembles the n th resonant mode shape of expression (3.2). Then only the n th mode will be significantly excited, and its velocity amplitude will be given by equation (3.6). That velocity field in turn drives a pressure field which is determined by equations (2.1) and (2.2), with expression (3.6) for u(z). The amplitude of this pressure field at
434
D. G. CRIGHTON
the strut surface r = a is easily seen to exceed the driving amplitude by a factor of order (k&Y -a very large increase under the assumed conditions. This opens up a serious problem-that design of struts to minimize off-resonance scattering necessarily brings large increases of local pressure levels at resonance. It cannot, however, be concluded that the far field pressure at resonance will be increased in the same manner, for it was required in section 2 that when koa <<1, then k,,l/tnr D 1, in order for equation (2.8) to hold. Thus for moderate values of n the acoustic wavelength must nof be large compared with the strut length, and in that case the far field is determined by a delicate balance of cancellation effects along the strut. To see just what sort of balance is achieved, an examination will be given in the next section of a case which exposes the essential results with a minimum of algebra. 5. THE FAR FIELD PRESSURE FROM RESONANT
SCATTERING
The eigenfunctions (3.2) for the cantilever with one free end and one built-in end lead to very heavy algebra, so matters will be simplified a little by considering instead a strut with pinned (simply held) ends at z = OJ. The eigenfunctions then satisfy a4u/& 4 - q4u = 0, with u(0) = u”(0) = u(I) = u”(l) = 0 and with q4 = (m +m’)w’/B. The normalized eigenfunctions for this case are
U, = (2/1)“2 sin (~.z/l),
(5.1)
and the velocity amplitude of the n th mode at resonance is exactly as given in equation (3.6)-since the eigenvalues are still q,, = m/l as for the cantilever, though the eigenfunctions are of course different in the two cases. Whenever necessary the reader must imagine the strut to be continued to infinity in both directions by infinite lengths of strut held at rest. As before, the strut will be taken to be circular in section. The z-axis lies along the strut, (x, y) and (r, 8) are Cartesian and polar co-ordinates in planes perpendicular to the strut, while (R, 8, $) will be taken as spherical polar co-ordinates based on any convenient origin on the strut, with the angle $ measured from the z-axis. Suppose that the strut is excited by a plane incident wave of potential 4i = exp (-ikoz
cos t,bo-
the time factor exp (-iwt) being understood. is the solution of
ikox sin &J,
Then the geometrically
4s = exp (-ikOz cos &)4(x, y), (a2/8x2+a2/ay2+K2)4 a(& + di)/ar = 0
(5.2) scattered field & = 0,
on r = U, all 8, 2,
with a radiation condition at infinity and with K = k. sin IJ~~.This field is easily calculated, and near the surface of a thin strut one finds 4i+4s
-
-iK(r+u2/r)cos8
exp (-ikoz cost,h)
(for kg cc 1, kg<< 1). From this, the force f(z) on unit length of the cylinder in the direction 8 = 0 is found to be f(z) = -27rpKou2 exp (-ikOz cos &,), whose eigenfunction (5.1), are
(5.3)
coefficients b,, with respect to the functions u,(z) of expression
6, =iw(2/1)1’22~pKou2nd[e-ikofCoS”ocos
nr-lJ/(k~12cos2
&-n27r2).
(5.4)
OSCILLATIONS
OF FLUID-LOADED
STRUTS
435
If the frequency is such that the it th mode is resonantly excited, its velocity amplitude at resonance is given by equations (3.6), (5.1) and (5.4) as 8pK nr[e-ik~’m$0cos n7r
u(z)=-,‘k2
0
-
(k3zcos2I&n2?r2)
nllz l] sin _ 1 *
Under the further assumption that other modes are then only negligibly excited, and bearing in mind the assumption that u(z) = 0 outside the segment (0, I), one finds the Fourier transform of the velocity along the z-axis is simply
In particular, one finds U(ff =
8pKn2r21 -ko
cm
$I=
-
m,k2 0
[eeikoL‘08 ‘O cos mr - l] [e-iko’cosllrco,s nrr - l] > (kzo12cos* &l- n27r2) (&$* co? IJ - n27r2) *
(5.5)
Equation (2.3) now determines the potential & of resonance scattering as
where cP(r, 8, C-X)= U(a) cos 13 Hr(Kr)/KHi(Ka), K = (kg -cx~)~‘~ and V(cr) is given in equation (5.5). At distances R much greater than either the wavelength ko’ or the length 1 of the strut, this Fourier integral may be evaluated by expanding Hi(M) for large Kr and then performing a standard steepest descent calculation. The result involves a knowledge of V(a) only for (Y= -k. cos (jl, as given in equation (5.5), and after some reduction one finds, remembering that koa CC1, Ce-W ~0sGocos ,r,r _ I] [,-W ~0s+ cos n,r _ 11 $,-4’(g) (kol)(n~>2 sin * sin CL0cos 8 (ko2,2 cos*&J-n2?T2) (k~12c0s2f+t2**)~
(5.6) If the propagation vector makes a general angle B. with the (y, z) plane, and not the special angle e. = 0 chosen in expression (5.2), then one has to replace 8 in equation (5.6) by (8 -eo). Then it is evident that expression (5.6) is symmetric in the angles (8, eo) and in (I,+,tie). This is a satisfactory state of affairs in that it shows that the large number of approximations made has not violated the fundamental Reciprocity Theorem [3, p, 1951. The 8 dependence of expression (5.6) is simply that of a dipole, reflecting the fact that the strut is compact relative to the wavelength (koa CC1) in its transverse dimension, In the axial direction the strut is not compact (in fact in the derivation of expression (5.6) it has been assumed that kol >>n7~), and consequently the $-dependence is that of a typical long array, with a rapidly varying pattern of lobes arising from the factor exp (-ikol cos 9). The total power scattered at resonance can be expressed in closed form from expression (5.6), but the result is too cumbersome to be useful here. Next, a rough comparison will be attempted between expression (5.6) and the field scattered (geometrically) from a finite rigid strut of large length (kol >>1) and small thickness under the same incident field (5.2). The field scattered from the small circular ends of the strut will be ignored and the assertion will be made that the far field can be expressed in terms of an integral along the strut of the net force per unit length exerted on the fluid by the strut. That force is in the direction 8 = 0, and is equal to -f(z) as
436
D. G. CRIGHTON
given in equation (5.3). The radiated density then follows from a well-known expression [4, p. 621 as 1 a ‘f(z’, t - f’/Co) dz p’(x, t)=-7 r’ 47rc0ax I 0
(5.7)
where f(z, t) = f(z ) exp (-iwt) and r’= {x2 + y2 + (z - z ‘)2}1’2.Carrying out appropriate far field simplifications, one finds P’--
ipwKu 2 exp (-iot) 2c:koR
a
eikoR
-ik,l(cos
llo+cos
a)
_
.
11
ax 1(cos (I,+ cos $0) e
I
Differentiation of exp (ika) is the only process which gives a term of order R-',the derivatives of cos $ contributing only terms at most 0(Re2). Hence one ends up with a far field scattered potential & from the strut at rest given by ik,R
“-?
i e (--> k&
e
(koa)2 sin 4 sin &, cos 8 (
-ik,l(cos
~+cos
a&) _
cos~+cos&J
1 >*
(5.8)
Note again, that if allowance is made for a non-zero angle & for the incident field, the expression (5.8) is then symmetric under interchange of 8 and 6+,or of 4 and $O. Expressions (5.6) and (5.8) contain rather similar rapidly varying directional factors like exp [-ikJ(cos I,++ cos I,@]. In the case of the rigid strut, I plays ho part in determining the field amplitude, whereas in the case of the resonant strut, the thickness a plays no role. Since the field q& decreases like (kou)2 while 4, increases as (kol) (at any rate for angles 4, &, near to 7r/2), it is certain that the resonant field will exceed the geometrically scattered field in some angular ranges at least. For example, if tJ and &, are close to 1r/2 (corresponding to observation of the back-scattered field from a plane wave incident broadside on the strut) one has & = O(kJ) and q& = U(k$z’k,,l), so that &/& = 0((kou)-2). In this case the distant field at resonance exceeds the geometrically scattered field by (kou)-2-exactly as in the near field, discussed in section 4. In other situations the distant field need not be so dramatically increased at resonance. For example, if IJ and 40 are both small, 4, = 0((kJm3) and & = G((kOu)2), and the ratio &/I$~ may be either large or small depending on kou and the value of I/u. But at any rate it is clear that the near field amplification at resonance, discussed in section 4, will be maintained into the far field, at least within certain angular ranges. 6. CONCLUSIONS The analysis of section 5 shows that the field scattered geometrically by a thin long rigid strut decreases as (kou)2, and that the length 1 serves mainly to determine the angular distribution. Thus the scattered field in general is best minimized by reduction of kou. If, however, the strut is not rigid, this reduction of kou reduces the radiation damping which is the only significant dissipative mechanism controlling resonant oscillations. The thin strut therefore executes large oscillations at resonance, and generates a nearby pressure field in excess of the geometrically scattered field. According to section 4, .the surface pressure amplitude at resonance exceeds that on a perfectly rigid strut by the large factor (kou)-2. According to section 5 this increase will generally be transmitted to the distant field also, particularly if the strut is long, since the distant resonant field increases with 1 (at least within some angular ranges) while the geometrically scattered amplitude is largely independent of I. There are two options open for reducing the importance of the resonant increases. One can increase (kou) (not a very appropriate
OSCILLATIONS
OF FLUID-LOADED
STRUTS
437
step, since it only reduces the resonant field relative to the static scattered field, the latter being increased), or one can introduce internal loss mechanisms in the strut and thereby eliminate radiation loss as the basic controller of resonance. ACKNOWLEDGMENT The author is grateful to a referee for bringing to his attention references [5] and [6]. Of these, reference [5] has much in common with the present work, dealing as it does with the scattering of a plane wave by a thin circular rod, with the rod displacement expanded in terms of natural modes. The aim of reference [5] was, however, to show that strong scattering is to be observed in a direction opposite to that of the incident sound wave, whereas the aim of the present work is to show rather generally the intense scattering from a thin resonant strut. Reference [6] also deals with scattering of an acoustic field by a thin strut, this time with the incident field being due to a point source within a rectangular chamber. The expansion in eigenmodes again has points in common with the present approach, but the specific problem addressed is quite different. The author wishes also to thank Dr M. C. Junger for a helpful discussion.
REFERENCES 1. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill Book Company. 2. M. C. JUNGER and D. FEIT 1972 Sound, Structures and Their Interaction. Cambridge, Massachussetts: M.I.T. Press. 3. A. D. PIERCE 1981 Acoustics: An Introduction to its Physical Principles and Applications. New York: McGraw-Hill Book Company. 4. J. LIGHTHILL 1978 Waves in Fluids. Cambridge, England: Cambridge University Press. 5. L. M. LIAMSHEV 1958 Soviet Physics-Acoustics 4, 50-58. Scattering of sound by a thin bounded rod. 6. S. N. YOUSRI and F. J.FAHY 1976 Journal of Sound and Vibration 45, 583-594. Acoustically induced vibration of, and sound radiation from, beams inside an enclosure.