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Sound radiation from flexible blades
Journal ofSound
and Vibration (1985) 98(2), 171-182
SOUND
RADIATION
FROM
FLEXIBLE
BLADES
S. A. L. GLEGG Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 5NH, England (Received 19 October 1983)
The sound radiation from fluctuating hydrodynamic loads on flexible cantilever beams is discussed. It is shown that when the beam moves in response to the applied load, the sound radiation can be significantly changed. For a light beam in a heavy fluid, the radiation at the resonant frequencies of the beam is enhanced and limited only by damping. Between resonant frequencies the radiation may be reduced; when each mode of vibration has equal energy, this reduction is a maximum of 4, 7 and 8.5 dB between the first four resonant peaks respectively. Beams of denser materials have lower radiation at resonance than those of lighter materials with the same stiffness, and for a typical propeller blade in a light fluid no change in radiation is expected due to blade flexibility.
1. INTRODUCTION The sound radiated by low Mach number flows over blades with acoustically compact chords may be modelled by a dipole source distribution whose strength is equal to the fluctuating force applied to the fluid surrounding the blade. For rigid blades this can be related to the aerodynamic or hydrodynamic loads caused by unsteady inflows or blade boundary layers. However, if the blade is sufficiently flexible, then it will move in response to these fluctuating loads and this motion will modify the blade/fluid interaction. The objective of this paper is to investigate the implications of this effect. An example of modified radiation due to “recoil” effects of this type has been described by Lighthill [l] when discussing sound radiation by insects and “creatures in the ocean”. He showed that a compact body of volume V moving through a fluid of density p0 produces an acoustic far field equivalent to that of a dipole of strength Oi( t) = -Gi( t) +PoVtii(t), where Gi(t) is the fluctuating force applied to the body by the fluid and ii(t) is the acceleration of the body. However, if the body is of density P,,,, then it will move under the effect of the unsteady load with an acceleration of lii( t) = Gj( t)/p,V. Thus the dipole strength becomes oi(t)=-Gi(t)(l-p,lp,).
(1)
For insects in air P,,, >>pO, so the body motion will have a negligible effect, but in water creatures will be neutrally buoyant (p,,, = pO), so the sound radiation will be very much reduced. In this analysis it is assumed that the fluctuating force Gi(t) is determined only by fluid dynamic loads which are very much greater than the virtual mass loads due to body motion. However, when the virtual mass of the body, M, is of the same magnitude as the displaced mass, this assumption is no longer valid. To account for this, the force Gi( t) is defined as the sum of the fluid dynamic load Ri( t) and the virtual mass reaction -Moti, SO that Oi( t) = -Ri( t) + (M, + p. V)tii. The acceleration of the body is now determined by the applied force Ri( t) and its fluid loaded mass (M, + pm V) such that ti, = 171 0022-460X/85/020171+12 %03.00/O @ 1985Academic Press Inc. (London) Limited
172 R(t)l(Mo+pmV),
s.
A. I_. GLEGG
and
This shows that even for bodies with significantly different densities from that of the surrounding fluid, the sound radiation is reduced if M, is of the same magnitude as p,,,V. For example, a disc of radius Q and thickness h has a virtual mass of 8/3p,a3 and a body Thus if pa/p, is of the same order or greater than h/a, then Di( t) will mass of p,m’h. be reduced. In the discussion so far it has been assumed that the body is limp. If, however, the motion is resisted by a set of springs of stiffness Ki, then for harmonic motion at the frequency w, the acceleration will be given by tii=Rie-i”‘/(Mu+p,V-Ki/~2)
and the dipole source strength becomes Q(t)
= -Ri
(Pm-PCl)V-Kilw2
M”+p,V-Ki/w’
This indicates that at frequencies well below the fluid loaded resonance, where the body is effectively rigid, the dipole source strength equals the applied force Di(t) = -Ri exp (-iwt); at frequencies well above the fluid loaded resonance one obtains the limp body result (2). However, the most important feature of this result is that at the fluid loaded resonance the source strength is determined by the damping of the system, which is not specifically included in expression (3). In the following sections the principles outlined above will be applied to the particular case of sound radiation from cantilever beams which have acoustically compact chords and are excited by fluid dynamic loads along their span. In particular, the change in radiated sound power due to the motion of the beam will be evaluated and the effects of damping on the sound radiation will be described.
2.
THE INFINITE BEAM
Before discussing the radiation from a cantilever beam, the method of analysis will be introduced by considering the radiation from an infinite rectangular beam as illustrated in Figure 1. It will be assumed that the chord of the beam is acoustically compact so that the source distribution may be modelled by a line dipole of strength d,(y,) exp (-iot) per unit length, where the axis of the dipole is normal to the chord of the beam. The sound pressure field is then given by e-iw(r-r/c,)
p(x,) e-‘-l = -$
” I
5
m
4
M
4m
dy,.
In all that follows the time dependence exp (-iwr) will be implicitly assumed, and so, by using the far field approximations given in Figure 1, the complex pressure amplitude is
(4)
SOUND
RADIATION
FROM
FLEXIBLE
BLADES
173
Figure I. Illustration of an infinitely long fluid loaded beam with velocity displacement u,(y,) exp (-iwl) and driving forcef,(y,) exp (-ior) radiating an acoustic field p(q) exp (-ior). Note that I may be approximated by r = r, - x,yJ r,, = r,, - y, cos 0 in the acoustic far field.
where the overbar is used to represent the wavenumber spectrum of a function (see the list of symbols in the Appendix). Therefore, to evaluate the radiated acoustic field only the wavenumber spectrum of the dipole source distribution need be considered. To evaluate this, one may use the force per unit length applied to the beam by the fluid, F,(y,) exp (-iwt), and the displaced mass. If the thickness of the beam is h and the chord 2a, then (5)
4(y3) = -F,(y,)-iop,(2ah)u,(y,).
If the beam is excited by a fluid dynamic driving force f,(y3) then the force applied to the vibrating beam is oil Fl(Y3) =“fl(y3) -
I -cr
Z(Y,, Q4~J
dz,,
(6)
where Z(y,, z3) is a function which describes the reaction force of the fluid at y, due to a velocity displacement at z3. In the case of an infinite beam this can only be a function of the difference between yJ and z3, and so the integral in equation (6) will be of the convolution type. The relationships between F, and f, may be obtained by using the wave equation for flexural motion of the beam: E184u,/8yl- 02mtl, = -iwF,. By taking the wavenumber spectrum of this with respect to the wavenumber k (see the list of symbols in the Appendix) and using expression (6) one finds (EIk4-w2m)fi,(k)
= -i&,(k)
= -iwf,(k)+iwZ(k)ti,(k).
(7)
The solutions for E,(k) and F,(k) are then -iwf,(k)
c,(k) =(EIk4_iwZ_w2m)
and
F,(k) =f,(k)
-
EIk4 - w2m
( EIk4-id-02m
(8)
174 The sound given as
S. A. L. GLEGG
radiation
is then described
d,(k) = -f(k)
by expression
(4) with the dipole
EIk4-w*(2ah)(p,
source
-po)
EIk4-iwz-w’(2ah)p,
strength
(9)
which should be evaluated with k = w cos O/co. This result shows that at frequencies well below the coincidence frequency of the beam, where k4= (w cos f.l/c,)*<< Ii&?+ w’ml/El, the source strength is reduced to
To illustrate the significance of this effect, consider the case when 2 is given by the radiation impedance [2] as
(10) so that when ma/c,,<< 1 one has
This result is similar to that given by equation (2) and demonstrates that when pm = p,, or p,Jp, is of the same order or greater than the thickness to chord ratio of the beam then the radiated sound will be less than predicted simply from the fluctuating loads. However, of more importance is that at frequencies above coincidence there will always be a radiation angle for which k4= (w cos 0/c,)“={~0*m+Re
iwz)}/EI,
which is the wavenumber where the displacement is a maximum. At this angle ZiI >>f and so the force F is dominated by the beam motion, which is resisted by the surrounding fluid. Using equations (9) and (10) gives the maximum dipole strength as
d,(k) = -f(k)
Re (iwz)
+ w2p,(2ah)
i Im (-ioZ)
=
-f(k)
( ;;ro+s~r;;;0;2)-
This shows that the radiation
is increased by a factor of more than (oa/c,)-* at this angle, and this increase does not depend on the relative densities of the beam and the surrounding fluid. 3. THE FINITE
BEAM
To model a beam of finite length realistically with the correct end conditions, it may be assumed to be part of an infinite beam [3], as illustrated in Figure 2. This model does
t-~
L
Figure 2. Illustration of cantilever beam of length L as part of an infinite beam.
SOUND
RADIATION
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BLADES
175
not allow for leakage flow at the ends, but this is likely to have only a small effect on the motion of the beam. The velocity displacement in the region 0 < yS < L may then be described by a set of orthogonal normal modes [4], such that q(y3)
=
n;,
whl(kd3)9
0 <
y3 <
.v3 <
I 0,
0,
L,
y3
>
L.
The wave equation which describes the beam motion may then be written as ;
n=I
(EZk:-
~*~)wA(k,yJ
= -ioFl(y3).
(11)
To solve this equation for the beam modal displacement velocities u,, one makes use of the orthogonality properties of the mode shapes, multiplying both sides by ~j(kjy,) and integrating over y,, so that the left-hand side of the equation is reduced to ( EZkP- w*m) UjL. Before applying the same method to the right-hand side, the loading Fl(y3) must first be expressed in the form given by equation (6) and then as L S ICl,(k,Z3)(C’l(kjY3)Z(Y3, ZX)dz3 dy,. -iw ‘fi(y3Mi(kjy3) dy,+iw ,4, u, Is0 -X’ I0 The first of these integrals is defined in terms of a modal loading, such that it becomes -iwf;L, and the second in terms of a modal impedance, such that it becomes iwnjZjL. At low frequencies, wa/cO<< 1, the resistive part of the impedance is important only close to resonant frequencies, and the reactive part is approximately equivalent to an attached mass, po7ra2, uniformly distributed over the length of the beam. This approximation breaks down when k,a > 0.5 (see references [2] and [6]), and eventually when k,a >>1 the attached mass will be negligible. If the motion is dominated by a single mode, then this impedance may be defined by using the wavenumber spectrum of the terms in the second integral above: i.e.,
(l-2) where Z is the wavenumber spectrum of the impedance for the infinite beam, given for example by equation (lo), and this form of the definition will be of particular use in subsequent sections. Using these results enables the modal displacements to be evaluated as uj = -iwjj/(EZkT - iwZj - o*m), and hence by using equations (11) and (5) the dipole source distribution is obtained as EZk:-w*(p,-pJ2ah dl(Y3)
= -
lf
“=,
fn
EZkt - iwZ, - w2p,2ah
4n(kny,).
(13)
The interpretation of this result is slightly different from that for the infinite beam case given in equation (9). Below the resonant frequency of the first mode the terms in brackets are dominated by EZki and so, due to the orthogonality of the modes, d,(y,) = --x f,& (k,,y,) = -f (y3), which is equivalent to the rigid beam case. However, at a resonance the source strength will be determined by the beam motion, which depends on the resistive part of the impedance of that mode. At frequencies above the resonance of a particular mode the mass terms become important and, as described in earlier sections, the radiation from that mode will be reduced by a factor of (p, - pO)/ (p, + iwZJ2ah). In a heavy fluid this reduction may be quite significant and the radiation from that mode can be effectively
176
S. A. L. GLEGG
“cut off”. The amount of reduction obtained will depend on the distribution of energy between the modes, but if one particular mode is dominant then quite significant effects can be predicted.
4.
THE SOUND POWER RADIATED BY A LOCALLY CORRELATED LOADING DISTRIBUTION
In the early studies on sound radiation by blades, Sharland [5] demonstrated that power spectral density in the acoustic far field could be obtained by summing the contribution of a number of uncorrelated sources over the span of the blade. This involves the assumption that both the blade chord and the spanwise correlation length scale of the source strength are acoustically compact. The sound pressure spectral density is then given by S,(w) = (wx,/47r&J2
IL&,(w),
(14)
where S,,(w) is the mean power spectral density of the dipole source strength d,(y3). In the case of the rigid beam this is the power spectral density of the fluctuating load Sr(w), but for the resonant beam the excitation of a particular mode will result in finite cross correlation of source strength fluctuations over the span. To allow for this, equation (14) is written as
(15) where C,(y3, zj) is the cross-spectral density of the dipole source strength fluctuations at y, and z3. If it is assumed that the loading which excites the beam is only locally correlated (k,l<< 1) and is uniform over the span, then the modal loadings used in equation (13) will be incoherent and each mode will have the same power, with a mean square level of (I/L)S,(o). Then, using equation (13), one obtains
Cd(Y3,z3)=; S,(w) f
EIk4, - w’(p, - po)2ah
II=1 Elk: - iwZ,, - 02p,2ah
*
$n(knd(Cln(knzd.
(16)
Therefore equation (14) may be rewritten, by using equations (15) and (16), in the form S,(w) = (wx,/4*r~c~)‘~~~,(w)H(wc0s
e/c,),
(17)
The function H(e) describes the correction factor which should be applied to allow for blade flexibility when calculating the sound field from rigid blade results. As before, it may be shown that below the first resonant frequency H = 1, and above this frequency a reduction in the value of H may occur in heavy fluids. The sound power spectral density is obtained by integrating S,(o)/p& over a spherical surface of radius r, in the acoustic far field. This gives S,(w) = {w*~LS,(@)ll2P,c;]
N(w),
where the sound power correction factor is given by m EIk:-o*(p, -po)2ah * T m / lo sin301&(~)12d0. -w*p 2ah n I EIk4-i& n n
N(o)=& c
(19)
SOUND
RADIATION
FROM
FLEXIBLE
177
BLADES
The integral in this equation may be rewritten as
where kd = (w/co)‘- k2 and the functions S,(e) are those tabulated in reference [2] for beams with different end conditions, including cantilevers. Thus one obtains
0
N(w)=6 ~
EIk4-w2(p
-po)2ah
‘E,IEIk’_iwZ mwZP n
n
(20)
m
2ah
is the principal result of this section, which will be evaluated in section 5 for different materials in typical use.
This
5. PARAMETRIC STUDIES 5.1. METHOD OF EVALUATION To evaluate equation (20), it is first necessary to find a suitable method of obtaining the functions S,(o/k,co). These are most difficult to compute in the region where w/ k,c, = 1, while at frequencies well above or below this they are well represented by their asymptotic forms. In what follows only cantilever beams in water are considered, for which the region of interest is the lower frequency range where @L/co < 5 < ooL/c,,, w. being the critical (coincidence) frequency. In this case the function S,(o/k,c,) is well approximated by its low frequency asymptote at and below resonance. At frequencies above resonance the contribution from a particular mode is effectively “cut off’ in a heavy fluid, and so the precise evaluation of S, is not important. For this reason, the computation has been simplified by approximating S, by two straight lines corresponding to the low and high frequency asymptotes, as shown in Figure 3.
Approximation to S,(w/k,cc)
1
)
w/k,&,
(~k,LV~~ Figure
3.
The
approximation
used for S,(w/k,c,);
the gradient
of the low frequency
asymptote
as illustrated
is (16/3)&(k,L).
5.2.
PERTINENT
PARAMETERS
The important non-dimensional parameters in equation (20) have been found to be the following: the non-dimensional frequencies k,L and p = wL/co; the blade aspect ratio A = L/2a; the material to medium density ratio pm/pa; the modal impedance which is
178
S.
A.
L. GLEGG
non-dimensionalized as Z,,/wm = -_(n, +iP,,), where n,, is the loss factor and &, the attached mass ratio; and a stiffness to weight parameter sO= (~/LX,,) dEl/m, which is equal to the inverse of the non-dimensional critical frequency s0 = (w,L/ c,)~ ‘. For a beam of rectangular cross section sO= (h/L)c,/Jl:! c0 and for most metallic materials in water (i.e., cP= 5000 m/s and cO= 1500 m/s), the value of s,, is given approximately by the thickness to length ratio (h/L). By using these parameters, equation (20) can be rewritten as
kL)4d-~2U -~ol~m) ’
N(p)=6p-3 “:I(k,L)4s2-p2(1+P 0
n
-iv
n
Sn(p/kL).
)
(21)
For the lower order modes of a high aspect ratio beam it is found that k,a < 0.5 and (n -f)7r/2A, so the attached mass will be equal to pova2. However, since k,a = k,L/2Ait is found that for low aspect ratio blades the condition k,a < 0.5 is not always met by the first and second order modes. The reduction in attached mass for higher order modes has been evaluated by Blake [6], who showed that this may be quite accurately estimated by using P,, = &,B,, where PO= rpa’/ m and k,L < A,
1, Bn = { (0*5+ k,L/2A)-‘,
k,L > A.
An example of N(w) as a function of frequency is shown in Figure 4 for a brass cantilever beam in water with an aspect ratio of 2 and a stiffness to weight parameter
-1oL
Figure 4. Sound power correction factor N(w) as a function of wL/c, density ratio pm/p,,= 8.5, loss factor 1)= 0.1 and s,, = 0.05.
for beam with aspect ratio A = 2,
so = 0.05. It should be noted that at the higher frequencies illustrated here the compactness condition walcocc 1 begins to break down, and the sound power output may be reduced still further. The important features illustrated in this example are the strong radiation at resonances and the reduction in radiated power between resonances. At the resonant frequencies the radiation correction factor is given by N,,, which is the correction factor for a single mode at resonance, N, = {2a,(P,
+P,lPoMknLhn~2.
(22)
(Note that 6pP3S,(p/ k,,L) = 4af,/( k,L)* at frequencies below coincidence.) This is strongly dependent on the loss factor and attached mass ratio, and these will be discussed in more detail in the next section. Above its resonance the contribution from a mode is reduced by a factor of [( 1 - pJp,)/( 1 + &)I and so when the attached mass ratio is large the radiation from this mode is effectively “cut off”. To estimate the maximum reduction which may occur between resonances N(w) can be estimated from the sum of the modes excited below
SOUND
RADIATION
their resonant frequency.Therefore N and N+ 1 is at least
FROM
FLEXIBLE
179
BLADES
the level between the resonant frequencies of modes N= 1,
[ -4.1 dB,
I etc. Consequently, in this situation where all the modes have been excited equally, only a limited reduction is possible; however, if one particular mode is dominant, then greater reductions may occur. 5.3. DAMPING MECHANISMS In the previous example a loss factor of n = 0.1 has been assumed to apply uniformly over the spectrum. However, in practice the beam motion will be resisted by a number of forces which result in different loss mechanisms. The most important of these are radiation, viscous and hydrodynamic loads and the loss factors associated with each of these are as follows: radiation loss factor [2]: viscous loss factor [7]: hydrodynamic
7)R
t7r=
=
Po(~*/2~*MnWkL);
~(Po/P~)
loss factor [8]:
J+h*;
A>> 1,
1, 77!f=P0 Lil. x wa 0.5, ()I
A-
1.
U is the flow velocity. The relative significance of each of these mechanisms is shown in Figure 5 for the beam considered in the previous section. This illustrates that hydrodynamic and radiation damping are the most significant mechanisms at realistic flow speeds, and reducing the chord of the beam will tend to increase the ratio of the hydrodynamic to the radiation damping. The levels at resonance may be assessed from equation (22) by assuming that hydrodynamic damping dominates over the other mechanisms. This is obtained by using
n=l
0.1
t-l=2
n=3
I
1
1.0
10
WL/C,
Figure
5. Loss factors
for beam with A = 2, s,, = 0.05, &, =
I.
180 P,, +PO/P,,,
S. A. = Po(&
P = sO(knL)2/Jl
+47/7r),
L. GLEGG
where T is the thickness to chord ratio of the beam, and
+ P,,, at resonance. Thus
(23) This result shows that the levels at resonance are controlled by the stiffness to weight parameter so and the aspect ratio of the beam. For a given planform and stiffness, EI, changing the material of the beam will alter the levels at resonance, and lighter materials will result in a larger response. However, for very light materials, /I,, >>1, equation (23) reduces to N =I2~,0,+4r/~)W* n r&l
EI PO u2’
(24)
which is controlled only by the flow speed and stiffness of the beam. In the alternative extreme, a very heavy material in a light medium, Pn <<1, equation (23) reduces to N,, ={on(B,+4~/~)k,,L}*(EI/L2mU2A2) Considering a typical example of a propeller blade in air, with an aspect ratio of A = 10, and thickness to chord ratio of r = 0.1 with a flow speed greater than 34 m/s, gives N, < (4k,L/ loo)*. Consequently all the lower order modes will be overdamped, and the correction factor (20) will be determined by the non-resonant modes. Therefore in air one does not expect a large response at resonance because of the “hydrodynamic” damping.
6. CONCLUSIONS
It has been shown that the sound radiation from fluctuating loads on a cantilever beam in a heavy fluid is strongly influenced by the motion of the beam. At resonant frequencies the radiation is increased significantly and limited by some form of damping. In many practical situations this will be hydrodynamic damping and will increase in proportion to the flow speed and the aspect ratio of the blade. It appears that blades of a heavy material will limit the resonant response better than blades of a lighter material with the same stiffness. Between resonances the radiation is reduced, and if all the modes have equal power of excitation, the maximum reductions will be 4, 7 and 8.5 dB between the first and second, second and third, and third and fourth modes respectively. However, for the heavier blades (for instance brass blades in water) these reductions may not be obtained because the mode will not be completely “cut off”. For heavy propeller blades in a light fluid the effects are quite different. First, modes are not “cut off” because the attached mass is insignificant and, second, in a typical application the response at a resonant frequency is sufficiently damped for it to have a negligible effect on the overall radiated sound power. ACKNOWLEDGMENT
This work has been supported by the Ministry of Defence (Procurement Executive). The author would also like to thank Dr A. Robins and Dr F. J. Fahy for their useful comments on the original manuscript.
SOUND
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REFERENCES 1. M. J. LIGHTHILL 1978 Waves in Fluids. Cambridge: Cambridge University Press. 2. S. A. L. GLEGG 1983 Journal of Sound and Vibration 87, 637-642. Sound radiation from beams at low frequencies. Mass.: 3. M. C. JUNGER and D. FEIT 1972 Sound, Structures and their Interaction. Cambridge, M.I.T. Press. 4. R. E. D. BISHOP and D. C. JOHNSTON 1960 The Mechanics of Vibration. Cambridge: Cambridge University Press. 1964 Journal of Sound and Vibration 1,302-333. Sourcesof noise in axial flow 5. I. J. SHARLAND fans. 6. W. K. BLAKE 1974 Journal of Sound and Vibration 33, 427-450. The radiation from free-free beams in air and water. I. W. K. BLAKE 1972 Shock nnd Vibration Bulletin 42,41-55. On the damping of transverse motion of free-free beams in dense stagnant fluids. 8. W. K. BLAKE and L. J. MAGA 1975 Journal of the Acoustical Sodiety of America 57, 610-625. On the flow excited vibrations of cantilever struts in water. 1. Flow-induced damping and vibration.
APPENDIX: A
Cd Q
E F, G! H(k) I K, L MU N R R,, S,,, Sp, S, S,,( )
u V X Z a % ; i i j k 1 m n P r, r. s
t w x : Y,
LIST
OF SYMBOLS
aspect ratio ( = L/2a ) cross-spectral density dipole strength Young’s modulus force applied to beam per unit length force applied to body correction to power spectral density second moment of inertia spring stiffness length of beam virtual mass sound power correction driving force applied to body resistive part of impedance power spectral density radiation function flow velocity volume reactive part of impedance impedance radius or semi-chord speed of sound dipole strength per unit length driving force per unit length applied to beam thickness J-=i; = 1, 2, 3, tensor notation mode order wave number correlation length scale mass per unit length of beam, = p,2ah mode order sound pressure propagation distance coincidence parameter time displacement velocity observer position
182
S. A. L. GLEGG
Yi Yi 2; zi
source position source position
Tensor notation Overbar
3 = (xI, x27x3) f(k) =j?wf(y) emikr dy attached mass ratio loss factor angle from source to observer = wL/ c, kinematic viscosity density parameter used in reference [2] for definition of S,(s) thickness to chord ratio modal displacement radian frequency
and Vibration (1985) 98(2), 171-182
SOUND
RADIATION
FROM
FLEXIBLE
BLADES
S. A. L. GLEGG Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 5NH, England (Received 19 October 1983)
The sound radiation from fluctuating hydrodynamic loads on flexible cantilever beams is discussed. It is shown that when the beam moves in response to the applied load, the sound radiation can be significantly changed. For a light beam in a heavy fluid, the radiation at the resonant frequencies of the beam is enhanced and limited only by damping. Between resonant frequencies the radiation may be reduced; when each mode of vibration has equal energy, this reduction is a maximum of 4, 7 and 8.5 dB between the first four resonant peaks respectively. Beams of denser materials have lower radiation at resonance than those of lighter materials with the same stiffness, and for a typical propeller blade in a light fluid no change in radiation is expected due to blade flexibility.
1. INTRODUCTION The sound radiated by low Mach number flows over blades with acoustically compact chords may be modelled by a dipole source distribution whose strength is equal to the fluctuating force applied to the fluid surrounding the blade. For rigid blades this can be related to the aerodynamic or hydrodynamic loads caused by unsteady inflows or blade boundary layers. However, if the blade is sufficiently flexible, then it will move in response to these fluctuating loads and this motion will modify the blade/fluid interaction. The objective of this paper is to investigate the implications of this effect. An example of modified radiation due to “recoil” effects of this type has been described by Lighthill [l] when discussing sound radiation by insects and “creatures in the ocean”. He showed that a compact body of volume V moving through a fluid of density p0 produces an acoustic far field equivalent to that of a dipole of strength Oi( t) = -Gi( t) +PoVtii(t), where Gi(t) is the fluctuating force applied to the body by the fluid and ii(t) is the acceleration of the body. However, if the body is of density P,,,, then it will move under the effect of the unsteady load with an acceleration of lii( t) = Gj( t)/p,V. Thus the dipole strength becomes oi(t)=-Gi(t)(l-p,lp,).
(1)
For insects in air P,,, >>pO, so the body motion will have a negligible effect, but in water creatures will be neutrally buoyant (p,,, = pO), so the sound radiation will be very much reduced. In this analysis it is assumed that the fluctuating force Gi(t) is determined only by fluid dynamic loads which are very much greater than the virtual mass loads due to body motion. However, when the virtual mass of the body, M, is of the same magnitude as the displaced mass, this assumption is no longer valid. To account for this, the force Gi( t) is defined as the sum of the fluid dynamic load Ri( t) and the virtual mass reaction -Moti, SO that Oi( t) = -Ri( t) + (M, + p. V)tii. The acceleration of the body is now determined by the applied force Ri( t) and its fluid loaded mass (M, + pm V) such that ti, = 171 0022-460X/85/020171+12 %03.00/O @ 1985Academic Press Inc. (London) Limited
172 R(t)l(Mo+pmV),
s.
A. I_. GLEGG
and
This shows that even for bodies with significantly different densities from that of the surrounding fluid, the sound radiation is reduced if M, is of the same magnitude as p,,,V. For example, a disc of radius Q and thickness h has a virtual mass of 8/3p,a3 and a body Thus if pa/p, is of the same order or greater than h/a, then Di( t) will mass of p,m’h. be reduced. In the discussion so far it has been assumed that the body is limp. If, however, the motion is resisted by a set of springs of stiffness Ki, then for harmonic motion at the frequency w, the acceleration will be given by tii=Rie-i”‘/(Mu+p,V-Ki/~2)
and the dipole source strength becomes Q(t)
= -Ri
(Pm-PCl)V-Kilw2
M”+p,V-Ki/w’
This indicates that at frequencies well below the fluid loaded resonance, where the body is effectively rigid, the dipole source strength equals the applied force Di(t) = -Ri exp (-iwt); at frequencies well above the fluid loaded resonance one obtains the limp body result (2). However, the most important feature of this result is that at the fluid loaded resonance the source strength is determined by the damping of the system, which is not specifically included in expression (3). In the following sections the principles outlined above will be applied to the particular case of sound radiation from cantilever beams which have acoustically compact chords and are excited by fluid dynamic loads along their span. In particular, the change in radiated sound power due to the motion of the beam will be evaluated and the effects of damping on the sound radiation will be described.
2.
THE INFINITE BEAM
Before discussing the radiation from a cantilever beam, the method of analysis will be introduced by considering the radiation from an infinite rectangular beam as illustrated in Figure 1. It will be assumed that the chord of the beam is acoustically compact so that the source distribution may be modelled by a line dipole of strength d,(y,) exp (-iot) per unit length, where the axis of the dipole is normal to the chord of the beam. The sound pressure field is then given by e-iw(r-r/c,)
p(x,) e-‘-l = -$
” I
5
m
4
M
4m
dy,.
In all that follows the time dependence exp (-iwr) will be implicitly assumed, and so, by using the far field approximations given in Figure 1, the complex pressure amplitude is
(4)
SOUND
RADIATION
FROM
FLEXIBLE
BLADES
173
Figure I. Illustration of an infinitely long fluid loaded beam with velocity displacement u,(y,) exp (-iwl) and driving forcef,(y,) exp (-ior) radiating an acoustic field p(q) exp (-ior). Note that I may be approximated by r = r, - x,yJ r,, = r,, - y, cos 0 in the acoustic far field.
where the overbar is used to represent the wavenumber spectrum of a function (see the list of symbols in the Appendix). Therefore, to evaluate the radiated acoustic field only the wavenumber spectrum of the dipole source distribution need be considered. To evaluate this, one may use the force per unit length applied to the beam by the fluid, F,(y,) exp (-iwt), and the displaced mass. If the thickness of the beam is h and the chord 2a, then (5)
4(y3) = -F,(y,)-iop,(2ah)u,(y,).
If the beam is excited by a fluid dynamic driving force f,(y3) then the force applied to the vibrating beam is oil Fl(Y3) =“fl(y3) -
I -cr
Z(Y,, Q4~J
dz,,
(6)
where Z(y,, z3) is a function which describes the reaction force of the fluid at y, due to a velocity displacement at z3. In the case of an infinite beam this can only be a function of the difference between yJ and z3, and so the integral in equation (6) will be of the convolution type. The relationships between F, and f, may be obtained by using the wave equation for flexural motion of the beam: E184u,/8yl- 02mtl, = -iwF,. By taking the wavenumber spectrum of this with respect to the wavenumber k (see the list of symbols in the Appendix) and using expression (6) one finds (EIk4-w2m)fi,(k)
= -i&,(k)
= -iwf,(k)+iwZ(k)ti,(k).
(7)
The solutions for E,(k) and F,(k) are then -iwf,(k)
c,(k) =(EIk4_iwZ_w2m)
and
F,(k) =f,(k)
-
EIk4 - w2m
( EIk4-id-02m
(8)
174 The sound given as
S. A. L. GLEGG
radiation
is then described
d,(k) = -f(k)
by expression
(4) with the dipole
EIk4-w*(2ah)(p,
source
-po)
EIk4-iwz-w’(2ah)p,
strength
(9)
which should be evaluated with k = w cos O/co. This result shows that at frequencies well below the coincidence frequency of the beam, where k4= (w cos f.l/c,)*<< Ii&?+ w’ml/El, the source strength is reduced to
To illustrate the significance of this effect, consider the case when 2 is given by the radiation impedance [2] as
(10) so that when ma/c,,<< 1 one has
This result is similar to that given by equation (2) and demonstrates that when pm = p,, or p,Jp, is of the same order or greater than the thickness to chord ratio of the beam then the radiated sound will be less than predicted simply from the fluctuating loads. However, of more importance is that at frequencies above coincidence there will always be a radiation angle for which k4= (w cos 0/c,)“={~0*m+Re
iwz)}/EI,
which is the wavenumber where the displacement is a maximum. At this angle ZiI >>f and so the force F is dominated by the beam motion, which is resisted by the surrounding fluid. Using equations (9) and (10) gives the maximum dipole strength as
d,(k) = -f(k)
Re (iwz)
+ w2p,(2ah)
i Im (-ioZ)
=
-f(k)
( ;;ro+s~r;;;0;2)-
This shows that the radiation
is increased by a factor of more than (oa/c,)-* at this angle, and this increase does not depend on the relative densities of the beam and the surrounding fluid. 3. THE FINITE
BEAM
To model a beam of finite length realistically with the correct end conditions, it may be assumed to be part of an infinite beam [3], as illustrated in Figure 2. This model does
t-~
L
Figure 2. Illustration of cantilever beam of length L as part of an infinite beam.
SOUND
RADIATION
FROM
FLEXIBLE
BLADES
175
not allow for leakage flow at the ends, but this is likely to have only a small effect on the motion of the beam. The velocity displacement in the region 0 < yS < L may then be described by a set of orthogonal normal modes [4], such that q(y3)
=
n;,
whl(kd3)9
0 <
y3 <
.v3 <
I 0,
0,
L,
y3
>
L.
The wave equation which describes the beam motion may then be written as ;
n=I
(EZk:-
~*~)wA(k,yJ
= -ioFl(y3).
(11)
To solve this equation for the beam modal displacement velocities u,, one makes use of the orthogonality properties of the mode shapes, multiplying both sides by ~j(kjy,) and integrating over y,, so that the left-hand side of the equation is reduced to ( EZkP- w*m) UjL. Before applying the same method to the right-hand side, the loading Fl(y3) must first be expressed in the form given by equation (6) and then as L S ICl,(k,Z3)(C’l(kjY3)Z(Y3, ZX)dz3 dy,. -iw ‘fi(y3Mi(kjy3) dy,+iw ,4, u, Is0 -X’ I0 The first of these integrals is defined in terms of a modal loading, such that it becomes -iwf;L, and the second in terms of a modal impedance, such that it becomes iwnjZjL. At low frequencies, wa/cO<< 1, the resistive part of the impedance is important only close to resonant frequencies, and the reactive part is approximately equivalent to an attached mass, po7ra2, uniformly distributed over the length of the beam. This approximation breaks down when k,a > 0.5 (see references [2] and [6]), and eventually when k,a >>1 the attached mass will be negligible. If the motion is dominated by a single mode, then this impedance may be defined by using the wavenumber spectrum of the terms in the second integral above: i.e.,
(l-2) where Z is the wavenumber spectrum of the impedance for the infinite beam, given for example by equation (lo), and this form of the definition will be of particular use in subsequent sections. Using these results enables the modal displacements to be evaluated as uj = -iwjj/(EZkT - iwZj - o*m), and hence by using equations (11) and (5) the dipole source distribution is obtained as EZk:-w*(p,-pJ2ah dl(Y3)
= -
lf
“=,
fn
EZkt - iwZ, - w2p,2ah
4n(kny,).
(13)
The interpretation of this result is slightly different from that for the infinite beam case given in equation (9). Below the resonant frequency of the first mode the terms in brackets are dominated by EZki and so, due to the orthogonality of the modes, d,(y,) = --x f,& (k,,y,) = -f (y3), which is equivalent to the rigid beam case. However, at a resonance the source strength will be determined by the beam motion, which depends on the resistive part of the impedance of that mode. At frequencies above the resonance of a particular mode the mass terms become important and, as described in earlier sections, the radiation from that mode will be reduced by a factor of (p, - pO)/ (p, + iwZJ2ah). In a heavy fluid this reduction may be quite significant and the radiation from that mode can be effectively
176
S. A. L. GLEGG
“cut off”. The amount of reduction obtained will depend on the distribution of energy between the modes, but if one particular mode is dominant then quite significant effects can be predicted.
4.
THE SOUND POWER RADIATED BY A LOCALLY CORRELATED LOADING DISTRIBUTION
In the early studies on sound radiation by blades, Sharland [5] demonstrated that power spectral density in the acoustic far field could be obtained by summing the contribution of a number of uncorrelated sources over the span of the blade. This involves the assumption that both the blade chord and the spanwise correlation length scale of the source strength are acoustically compact. The sound pressure spectral density is then given by S,(w) = (wx,/47r&J2
IL&,(w),
(14)
where S,,(w) is the mean power spectral density of the dipole source strength d,(y3). In the case of the rigid beam this is the power spectral density of the fluctuating load Sr(w), but for the resonant beam the excitation of a particular mode will result in finite cross correlation of source strength fluctuations over the span. To allow for this, equation (14) is written as
(15) where C,(y3, zj) is the cross-spectral density of the dipole source strength fluctuations at y, and z3. If it is assumed that the loading which excites the beam is only locally correlated (k,l<< 1) and is uniform over the span, then the modal loadings used in equation (13) will be incoherent and each mode will have the same power, with a mean square level of (I/L)S,(o). Then, using equation (13), one obtains
Cd(Y3,z3)=; S,(w) f
EIk4, - w’(p, - po)2ah
II=1 Elk: - iwZ,, - 02p,2ah
*
$n(knd(Cln(knzd.
(16)
Therefore equation (14) may be rewritten, by using equations (15) and (16), in the form S,(w) = (wx,/4*r~c~)‘~~~,(w)H(wc0s
e/c,),
(17)
The function H(e) describes the correction factor which should be applied to allow for blade flexibility when calculating the sound field from rigid blade results. As before, it may be shown that below the first resonant frequency H = 1, and above this frequency a reduction in the value of H may occur in heavy fluids. The sound power spectral density is obtained by integrating S,(o)/p& over a spherical surface of radius r, in the acoustic far field. This gives S,(w) = {w*~LS,(@)ll2P,c;]
N(w),
where the sound power correction factor is given by m EIk:-o*(p, -po)2ah * T m / lo sin301&(~)12d0. -w*p 2ah n I EIk4-i& n n
N(o)=& c
(19)
SOUND
RADIATION
FROM
FLEXIBLE
177
BLADES
The integral in this equation may be rewritten as
where kd = (w/co)‘- k2 and the functions S,(e) are those tabulated in reference [2] for beams with different end conditions, including cantilevers. Thus one obtains
0
N(w)=6 ~
EIk4-w2(p
-po)2ah
‘E,IEIk’_iwZ mwZP n
n
(20)
m
2ah
is the principal result of this section, which will be evaluated in section 5 for different materials in typical use.
This
5. PARAMETRIC STUDIES 5.1. METHOD OF EVALUATION To evaluate equation (20), it is first necessary to find a suitable method of obtaining the functions S,(o/k,co). These are most difficult to compute in the region where w/ k,c, = 1, while at frequencies well above or below this they are well represented by their asymptotic forms. In what follows only cantilever beams in water are considered, for which the region of interest is the lower frequency range where @L/co < 5 < ooL/c,,, w. being the critical (coincidence) frequency. In this case the function S,(o/k,c,) is well approximated by its low frequency asymptote at and below resonance. At frequencies above resonance the contribution from a particular mode is effectively “cut off’ in a heavy fluid, and so the precise evaluation of S, is not important. For this reason, the computation has been simplified by approximating S, by two straight lines corresponding to the low and high frequency asymptotes, as shown in Figure 3.
Approximation to S,(w/k,cc)
1
)
w/k,&,
(~k,LV~~ Figure
3.
The
approximation
used for S,(w/k,c,);
the gradient
of the low frequency
asymptote
as illustrated
is (16/3)&(k,L).
5.2.
PERTINENT
PARAMETERS
The important non-dimensional parameters in equation (20) have been found to be the following: the non-dimensional frequencies k,L and p = wL/co; the blade aspect ratio A = L/2a; the material to medium density ratio pm/pa; the modal impedance which is
178
S.
A.
L. GLEGG
non-dimensionalized as Z,,/wm = -_(n, +iP,,), where n,, is the loss factor and &, the attached mass ratio; and a stiffness to weight parameter sO= (~/LX,,) dEl/m, which is equal to the inverse of the non-dimensional critical frequency s0 = (w,L/ c,)~ ‘. For a beam of rectangular cross section sO= (h/L)c,/Jl:! c0 and for most metallic materials in water (i.e., cP= 5000 m/s and cO= 1500 m/s), the value of s,, is given approximately by the thickness to length ratio (h/L). By using these parameters, equation (20) can be rewritten as
kL)4d-~2U -~ol~m) ’
N(p)=6p-3 “:I(k,L)4s2-p2(1+P 0
n
-iv
n
Sn(p/kL).
)
(21)
For the lower order modes of a high aspect ratio beam it is found that k,a < 0.5 and (n -f)7r/2A, so the attached mass will be equal to pova2. However, since k,a = k,L/2Ait is found that for low aspect ratio blades the condition k,a < 0.5 is not always met by the first and second order modes. The reduction in attached mass for higher order modes has been evaluated by Blake [6], who showed that this may be quite accurately estimated by using P,, = &,B,, where PO= rpa’/ m and k,L < A,
1, Bn = { (0*5+ k,L/2A)-‘,
k,L > A.
An example of N(w) as a function of frequency is shown in Figure 4 for a brass cantilever beam in water with an aspect ratio of 2 and a stiffness to weight parameter
-1oL
Figure 4. Sound power correction factor N(w) as a function of wL/c, density ratio pm/p,,= 8.5, loss factor 1)= 0.1 and s,, = 0.05.
for beam with aspect ratio A = 2,
so = 0.05. It should be noted that at the higher frequencies illustrated here the compactness condition walcocc 1 begins to break down, and the sound power output may be reduced still further. The important features illustrated in this example are the strong radiation at resonances and the reduction in radiated power between resonances. At the resonant frequencies the radiation correction factor is given by N,,, which is the correction factor for a single mode at resonance, N, = {2a,(P,
+P,lPoMknLhn~2.
(22)
(Note that 6pP3S,(p/ k,,L) = 4af,/( k,L)* at frequencies below coincidence.) This is strongly dependent on the loss factor and attached mass ratio, and these will be discussed in more detail in the next section. Above its resonance the contribution from a mode is reduced by a factor of [( 1 - pJp,)/( 1 + &)I and so when the attached mass ratio is large the radiation from this mode is effectively “cut off”. To estimate the maximum reduction which may occur between resonances N(w) can be estimated from the sum of the modes excited below
SOUND
RADIATION
their resonant frequency.Therefore N and N+ 1 is at least
FROM
FLEXIBLE
179
BLADES
the level between the resonant frequencies of modes N= 1,
[ -4.1 dB,
I etc. Consequently, in this situation where all the modes have been excited equally, only a limited reduction is possible; however, if one particular mode is dominant, then greater reductions may occur. 5.3. DAMPING MECHANISMS In the previous example a loss factor of n = 0.1 has been assumed to apply uniformly over the spectrum. However, in practice the beam motion will be resisted by a number of forces which result in different loss mechanisms. The most important of these are radiation, viscous and hydrodynamic loads and the loss factors associated with each of these are as follows: radiation loss factor [2]: viscous loss factor [7]: hydrodynamic
7)R
t7r=
=
Po(~*/2~*MnWkL);
~(Po/P~)
loss factor [8]:
J+h*;
A>> 1,
1, 77!f=P0 Lil. x wa 0.5, ()I
A-
1.
U is the flow velocity. The relative significance of each of these mechanisms is shown in Figure 5 for the beam considered in the previous section. This illustrates that hydrodynamic and radiation damping are the most significant mechanisms at realistic flow speeds, and reducing the chord of the beam will tend to increase the ratio of the hydrodynamic to the radiation damping. The levels at resonance may be assessed from equation (22) by assuming that hydrodynamic damping dominates over the other mechanisms. This is obtained by using
n=l
0.1
t-l=2
n=3
I
1
1.0
10
WL/C,
Figure
5. Loss factors
for beam with A = 2, s,, = 0.05, &, =
I.
180 P,, +PO/P,,,
S. A. = Po(&
P = sO(knL)2/Jl
+47/7r),
L. GLEGG
where T is the thickness to chord ratio of the beam, and
+ P,,, at resonance. Thus
(23) This result shows that the levels at resonance are controlled by the stiffness to weight parameter so and the aspect ratio of the beam. For a given planform and stiffness, EI, changing the material of the beam will alter the levels at resonance, and lighter materials will result in a larger response. However, for very light materials, /I,, >>1, equation (23) reduces to N =I2~,0,+4r/~)W* n r&l
EI PO u2’
(24)
which is controlled only by the flow speed and stiffness of the beam. In the alternative extreme, a very heavy material in a light medium, Pn <<1, equation (23) reduces to N,, ={on(B,+4~/~)k,,L}*(EI/L2mU2A2) Considering a typical example of a propeller blade in air, with an aspect ratio of A = 10, and thickness to chord ratio of r = 0.1 with a flow speed greater than 34 m/s, gives N, < (4k,L/ loo)*. Consequently all the lower order modes will be overdamped, and the correction factor (20) will be determined by the non-resonant modes. Therefore in air one does not expect a large response at resonance because of the “hydrodynamic” damping.
6. CONCLUSIONS
It has been shown that the sound radiation from fluctuating loads on a cantilever beam in a heavy fluid is strongly influenced by the motion of the beam. At resonant frequencies the radiation is increased significantly and limited by some form of damping. In many practical situations this will be hydrodynamic damping and will increase in proportion to the flow speed and the aspect ratio of the blade. It appears that blades of a heavy material will limit the resonant response better than blades of a lighter material with the same stiffness. Between resonances the radiation is reduced, and if all the modes have equal power of excitation, the maximum reductions will be 4, 7 and 8.5 dB between the first and second, second and third, and third and fourth modes respectively. However, for the heavier blades (for instance brass blades in water) these reductions may not be obtained because the mode will not be completely “cut off”. For heavy propeller blades in a light fluid the effects are quite different. First, modes are not “cut off” because the attached mass is insignificant and, second, in a typical application the response at a resonant frequency is sufficiently damped for it to have a negligible effect on the overall radiated sound power. ACKNOWLEDGMENT
This work has been supported by the Ministry of Defence (Procurement Executive). The author would also like to thank Dr A. Robins and Dr F. J. Fahy for their useful comments on the original manuscript.
SOUND
RADIATION
FROM
FLEXIBLE BLADES
181
REFERENCES 1. M. J. LIGHTHILL 1978 Waves in Fluids. Cambridge: Cambridge University Press. 2. S. A. L. GLEGG 1983 Journal of Sound and Vibration 87, 637-642. Sound radiation from beams at low frequencies. Mass.: 3. M. C. JUNGER and D. FEIT 1972 Sound, Structures and their Interaction. Cambridge, M.I.T. Press. 4. R. E. D. BISHOP and D. C. JOHNSTON 1960 The Mechanics of Vibration. Cambridge: Cambridge University Press. 1964 Journal of Sound and Vibration 1,302-333. Sourcesof noise in axial flow 5. I. J. SHARLAND fans. 6. W. K. BLAKE 1974 Journal of Sound and Vibration 33, 427-450. The radiation from free-free beams in air and water. I. W. K. BLAKE 1972 Shock nnd Vibration Bulletin 42,41-55. On the damping of transverse motion of free-free beams in dense stagnant fluids. 8. W. K. BLAKE and L. J. MAGA 1975 Journal of the Acoustical Sodiety of America 57, 610-625. On the flow excited vibrations of cantilever struts in water. 1. Flow-induced damping and vibration.
APPENDIX: A
Cd Q
E F, G! H(k) I K, L MU N R R,, S,,, Sp, S, S,,( )
u V X Z a % ; i i j k 1 m n P r, r. s
t w x : Y,
LIST
OF SYMBOLS
aspect ratio ( = L/2a ) cross-spectral density dipole strength Young’s modulus force applied to beam per unit length force applied to body correction to power spectral density second moment of inertia spring stiffness length of beam virtual mass sound power correction driving force applied to body resistive part of impedance power spectral density radiation function flow velocity volume reactive part of impedance impedance radius or semi-chord speed of sound dipole strength per unit length driving force per unit length applied to beam thickness J-=i; = 1, 2, 3, tensor notation mode order wave number correlation length scale mass per unit length of beam, = p,2ah mode order sound pressure propagation distance coincidence parameter time displacement velocity observer position
182
S. A. L. GLEGG
Yi Yi 2; zi
source position source position
Tensor notation Overbar
3 = (xI, x27x3) f(k) =j?wf(y) emikr dy attached mass ratio loss factor angle from source to observer = wL/ c, kinematic viscosity density parameter used in reference [2] for definition of S,(s) thickness to chord ratio modal displacement radian frequency