Vibration of statically deformed beams and plates

Vibration of statically deformed beams and plates Huw G. Davies and Saud al Sowayel Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02159 (Received 18 February 1972) The response of beams and plates to fluctuating loads with high mean values is considered. The static deforma- tion is obtained separately; the vibration of the statically deformed structure is then discussed in terms of the induced static in-plane membrane stresses. It is shown that the response of a deformed plate can be predicted by defining an effective uniform tension. Simple expressions are obtained for the loss factor in terms of the impedance of the supports of the beam or plate. Expressions are given for the changes of modal resonance frequencies and loss factors due to the tension, with the assumption of constant boundary impedance. The predicted changes in loss factors agree quite well with some previously published experimental data for very thin plates. The changes for most cases of practical interest, however, are small. Subject Classification: 12.7.1, 12.7.2. A =bh A•, b d D E G C• k LIST OF SYMBOLS perturbation constant (Eqs. 18 and 19) N cross-sectional area of beam p area of plate P,•n (w) width of beam group velocity P phase velocity R (--) modal density t bending stiffness of plate u, v modulus of elasticity U, Y expected value w shear modulus x average value of conductance per unit length y of boundary Z point input admittance of infinite plate a,/• thickness of beam or plate -• moment of inertia (=--I,,) of cross section of e,• beam •' intensity r• imaginary part of (-) 0 wavenumber K m•- /z = modal wavenumber • L II •r p =--(m•.-f-n•.)• L length of beam (side of rectangular plate) w/2•r in-plane force transverse pressure modal component of Fourier transform of pressure perimeter of plate real part of (--) time in-plane displacements static displacement amplitudes Fourier transform of transverse displacement space variable transverse displacement impedance of boundary constants defined by Eqs. 1! (a), 1! (b) absorption coefficient perturbation constant, Eqs. 18 and 19 dashpot constant of end support loss factor angle Poisson's ratio mass associated with end support damping constant of plate power mass per unit length of beam or mass per unit area of plate spring constant of end support frequency INTRODUCTION The vibratory response of structures to fluctuating loadings is usually discussed only for cases when the loading has zero mean value. However, situations exist when the applied loading may have a very large mean value, that is, the structure may experience a large static load as well as a fluctuating load. Large static loadings occur, for example, in heat exchangers in which the fluctuating loadings are generated by high fluid flow rates and by boiling, in aircraft where cabin pressure must be maintained at a level considerably higher than the pressure outside the aircraft, and in ships. This paper discusses the effect of large static transverse loads on the vibratory response of simple structures such as beams and plates. A schematic of a beam both with and without a large static load is shown in Fig. 1. , The Journal of the Acoustical Society of America 1035 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Wed, 26 Nov 2014 08:33:17 DAVIES AND AL SOWAYEL p (x,t) cr cr (a) Po + p(x,t) (b) Fro. 1. Schematic of beam with "almost" simple supports and acted on by (a) a small fluctuating load and (b) by a small fluctuat- ing load together with a large static load. The case where the magnitude of the fluctuating load is much smaller than that of the static load is considered. Thus, if the total loading is treated as a stationary random process, it is required that the vari- ance be much less than the mean square value. It is also assumed that the static component of the load is uniformly distributed over the beam or plate. These requirements correspond to typical cases of interest. The requirement of large static load means that first one can solve separately for the static response of the structure. Large static deflections of the beam or plate are allowed; that is, the static response may be non- linear. The feature of interest here is the stress distribu- tion or, more particularly, the in-plane force distribu- tion caused by the bending together with the stretching of the middle surface of the beam or plate. The vibra- tion problem then can be treated as a linear problem involving a beam or plate with a known in-plane force distribution. The effects of curvature on the vibratory response are neglected except, of course, in so far as the curvature affects the in-plane forces. Although the deflection of the beam or plate may be large at its midpoint, it is assumed that the radius of curvature of the midplarie is always sufficiently large that the cir- cumference of the equivalent cylinder is greater than longitudinal wavelengths in the shell material at the frequencies of interest; that is, attention is restricted to frequencies above the ring frequency. This does not ap- pear to be an overly restrictive requirement. Heckl • has shown that at frequencies above the ring frequency, a cylindrical shell tends to act as a flat plate. A review of some detailed analyses of the effects of curvature mainly involving computations based on finite element methods has been presented recently by Petyt. 2 The vibratory response of the beam or plate is affected both by the bending stiffness and by the in- duced membrane stresses. The structure is effectively stiffer, and there is a corresponding increase in the resonance frequencies of the normal modes of the structure. This aspect of the problem has been dis- cussed recently by Smith, Laura, and Matis? They con- sidered, however, only uniform in-plane loading of a clamped plate. A similar problem involving a stiff string has been discussed by Wolf and Muller. 4 In both the above references, mode shapes and natural fre- quencies of the lower order modes of the system are computed. For the present problem involving a uni- form static load the in-plane forces for the beam are indeed constant along the length of the beam (which follows immediately from the equilibrium condition). However, the in-plane forces for a plate are functions of position. In addition, particularly when dealing with problems of noise, one is interested in the resonant response of the higher order as much as the lower order modes. A second effect caused by the stresses induced by the large static component of the pressure is a change in the effective loss factor of the beam or plate. Energy dissipation in thin vibrating plates is due mainly to losses at the boundaries, either because of transfer of energy to other parts of the system or because of, say, viscous dissipation in the support mechanism. In order to discuss the effect of the induced stresses the same impedance boundary condition for the unstressed and stressed system is assumed. The effective loss factor for a beam is estimated both by perturbing the mode shape to account for the nonideal boundary condition, and also by an energy argument. It is shown that the loss factor is, as might be expected, proportional to the group velocity for propagating waves in the system. This is an extension of a result due to Morse. 5 The stressed system being effectively stiffer exhibits a higher value of the group velocity at any frequency than the unstressed system. The effective loss factor is thus increased by the static stresses. The nonrigid end sup- ports also affect the modal resonance frequencies, al- though in practice it is expected that this effect will be much less than the effect of the static membrane stresses. In what follows the case of a beam is first discussed, and expressions for the modal resonance frequencies and loss factors found in terms of the static load. The response of a plate is then discussed. The induced in- plane forces of the deformed plate are functions of posi- tion. Thus the eigenfunctions of the deformed plate are not the same as those of the undeformed plate. However, an expansion in terms of the undeformed plate 1036 Volume 54 Number 4 1973 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Wed, 26 Nov 2014 08:33:17 VIBRATION OF DEFORMED BEAMS AND PLATES eigenfunctions is-still valid. When there is no modal overlap, it is shown that an effective constant tension can be defined for the deformed plate, and shown furthermore that this effective tension is just the aver- age value over the area of the plate of the in-plane forces. Simple expressions for the modal resonance fre- quencies and loss factors may thus be obtained. Finally, experimental measurements of the loss factors of very thin plates are presented. The predicted changes agree quite well with these data. I. RESPONSE OF A BEAM A schematic of a beam with nonrigid boundary sup- ports and acted on by a transverse force per unit length po+p(x,t) is shown in Fig. l(b). The boundary condi- tions are treated as being almost simply supported, the mass, spring, and dashpot providing a small perturba- tion from the ideal case. It is assumed that as the beam is thin the beam undergoes pure bending together with stretching of the midplane of the beam. The equation of motion is a•y a•.y a•.y EI•--N•-+-p•=poq-p(x,t). (1) Ox 4 Ox 9. Ot 9. The mathematical representation of the boundary con- ditions is discussed later. No damping term is included in Eq. 1; it is assumed that the damping is due only to viscous losses in the supports. The in-plane force distribution N can be expressed in terms of the trans- verse displacement y, and of the in-plane displacement u of the midplane of the beam in the form = øu ,. (2) As the loads are applied transversely the equilibrium requirement ON =0 (3) Ox holds, implying that N is constant along the length of 'the beam. The force is treated as a random process with and $Ep0+ p (x,t) 3= po, (4) poX>>Sg.p:(x,t)-]. (5) The force is thus composed of a large static component P0, together with a small random fluctuating component p(x,t). Equations 4 and 5 suggest that the response may be written as y= yo(x)+.•(x,t), (6) where the mean deflection y0 can be determined solely from the static load P0, and the fluctuating response .•(x,t) can be determined subsequently for the statically deformed beam. The static deflection satisfies the equation a•y0 a•.y0 EI•--N•=po, (7) Ox 4 Ox 9. where N is now given by Eq. 2 with y replaced by y0. It seems reasonable to suppose that y0 satisfies simply supported boundary conditions. This amounts to saying that a static deflection at the boundaries does not alter the impedance of the mass-spring-dashpot supports since it is assumed that the supports behave linearly and elastically. Certainly, plastic deformation may be dis- counted for the types of problem being discussed. As N is constant, Eq. 7 can be solved easily for y0 in terms of the unknown N. Integration of Eq. 2 then leads to an expression for the displacement u in terms of N, and u satisfies the boundary condition u-0 at x-0 and x-L. Finally, putting the symmetry requirement u-0 at x= L/2 in the expression for u gives an equation to be solved for N. The equation so obtained, however, is a fairly involved transcendental equation, and approxi- mate solutions in simple form can not be obtained easily. Instead a simple approximate approach to find N is used that is analogous to that used by Timoshenko 6 for plates. It is assumed that the total load P0 is the sum of a load pl balanced by the bending stresses, and a load p•. balanced by the membrane stresses. Equation 7 can thus be separated into the two equations and ay0 EI• =Pl 02yo N Ox 9. (8) It is further assumed that the shape of the deformed beam is sufficiently well approximated by the first term of the eigenfunction expansion. Thus the static deflec- tions are taken to be and similarly, •'x yo(x) = Y sin--, L 2•-x u(x) = -- U sin. . L (9) A straightforward modal approach, for the first mode only, then yields Pl = Y•EI, 4L 4 p•.= y3 EA, (10) 16L 4 The Journal of the Acoustical Society of America 1037 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Wed, 26 Nov 2014 08:33:17 DAVIES AND AL SOWAYEL Fro. 2. Graphical representation of Eq. 11. and and 1 NL U- ß 2•- EA The main concern is with finding the force N in terms of the applied force per unit length P0. The solution can be expressed in the parametric form 1 2 y 2 (11a) 48 L 4h p0a- p0[•(•-) •-•] = 3(•) aq- (•). (lib) This corresponds to a cubic equation for N. Figure 2 shows a plot of this relation, the numbered points along the curve give the values of ¾/h. Also shown are the asymptotic limits Y N/• = (p0a) 2, for p0a> 1 and hence -->>1. \3/' ' h The fluctuating response y(x,t) satisfies the equation EI•--N•+p•=p(x,t), (13) Ox 4 Ox • Ot • where now the value of N is assumed to be known. At the supports, the fluctuating shear stress and transverse component of the tension together balance the forces due to the mass, spring, and dashpot. The boundary condi- tions can thus be written as Ox a Ox \ Ot • Ot / (14) EI•=O, (15) Ox • the -- sign and the q- sign in Eq. 14 applying at the ends, x=0, and x-L, respectively. Equation 15 arises because it is still assumed that the bending moment is zero at the supports. However, the approach used can be extended to include the effect of a rotational spring and viscous losses due to rotation at the supports. The frequency Fourier transform of the displacement is defined by the relation w(x ,w) = .•(x,t)e-i•tdt. (•6) The boundary conditions for w corresponding to Eq. 14 can be written in terms of an impedance Z (w) which is defined by the expression iwZ= --w•u+iw'•+a. (17) From the form of the problem it is noted that the value of the impedance is large. The form of the boundary conditions suggests that the displacement w be ex- panded in a set of eigenfunctions formed by small perturbations from the eigenfunctions of the ideal case. The form suggested by Morse 5 is used, namely sin•m(X--am), (18) where •m• km(l+•m). (19) The perturbed eigenfunction 18 allows the displacement at the boundaries to be nonzero, and the perturbed eigenvalue Eq. 19 leads to a modal dispersion relation for waves on the beam that takes into account the nonideal supports. am and •m are to be solved for in terms of the impedance Z under the assumptions •mam• VIBRATION OF DEFORMED BEAMS AND PLATES and •m¾¾1. (20) By substituting the function 18 into the Fourier transform of Eq. 14 the expressions iooZ sinkmare= -- km coskmam(Elkm2+ N), at x=O, and iooZ sinkin (L-- am) = -- km coskin (L-- am) (Elkm2+ N), at x=L, are obtained. If second order and higher products of am and em are neglected these equations are satisfied by the relations L•m __(Elkm•+N• am- •- \ i-• /' (21) Equation 21 indicates the order of magnitude of Z required for the perturbation scheme to be valid. It is noted also, a posterJori, that the bending moment at the supports although not zero is small. The modal dispersion relation is defined, from Eq. 13, by the relation tX3m •= Elkm4+ NkmL (22) An expression for &m, the perturbed resonance frequency, is obtained easily by noting that of = Okm =•+k•(c•)•, where corn satisfies the relation tXOm•= Elkm4+ Nkm •, (23) and where (Ca)m represents the group velocity associated with the mth mode. Thus (24) The perturbation includes the factor 1 ---- o ioomZ(oom) The effects of the mass, spring, and dashpot can not be separated, although the mass and spring tend mainly to be associated with a change of modal resonance fre- quency, and the dashpot tends mainly to be associated with an imaginary component of the resonance fre- quency, thus representing a damping term. For pure spring supports the magnitude of the change in reso- nance frequency increases with decreasing spring stiff- ness. This trend is in the proper direction. The pertur- bation term am is negative for pure spring supports and corresponds to a shift in the node from the support to a point just outside the support. The less stiff the spring the more the node is shifted, that is, the modal wave- length is slightly increased, the modal resonance fre- quency correspondingly decreased. The imaginary component of the resonance frequency can be related to the modal loss factor of the system by the expression O•m•m-- 2g(&m). (25) From Eqs. 24 and 25 an expression for the loss factor is obtained in a form similar to that obtained by Heckl. 7 (• 40(Cg)mR • (26) r/m = • •-/. kmL Before discussing the results it is shown that Eq. 26 can be obtained in another way by using the relations between power flow and energy in the system. This latter approach is needed to find the loss factor for a plate. It is supposed that within a narrow band of frequencies many modes contribute to the vibrational energy of the beam. The energy density, that is, energy per unit length, ER is thus fairly uniform. The average power incident on one of the supports is thus one half the product of energy density and group velocity since half the power propagates from right to left and half from left to right. If •, represents the absorption coeffi- cient of a support, then the power absorbed by both supports is ERcg3•. But this is just the dissipated power. The definition of the loss factor for the system thus gives oo•E •L= E•cg3•. (27) Finally, by considering a wave incident on the bound- ary, the absorption coefficient can be related to the boundary impedance by the expression 1-•,= •- 1- 4ocvR , (28) pcv+Z where c• is a phase velocity, and where it is assumed that ,ocv DAVIES AND AL SOWAYEL ..< ..< ..--: I • I I I I 0 I0 20 FIG. 3. Graphical representation of Eq. 33. Po (21.9 E (h/L) 4 ) I absorption at the resonance frequency of the support. The main concern here, however, is with the effect of the in-plane force. Clearly, for any given mode, the modal resonance frequency is increased by the presence of a large static component of the load on the beam, although the effect becomes less marked for high modes. If the dashpot component is supposed large, the modal loss factor is proportional to (co)m, that is, the modal loss factor also increases with increasing values of N. A more interesting question concerns how the loss factor at any given frequency varies with N. The loss factor at any frequency is proportional to (C•)m/km, wtiere the sub- script now refers to a typical mode whose resonance peak includes the frequency of interest. Equation 23 gives the appropriate value of kin. One finds from Eq. 26 that Thus, under the assumption that the boundary imped- ance does not change due to the in-plane force N, the effective loss factor is increased by a factor 1+ N•' •. (30) 4w •.oEI/ Again, the increase is less marked at high frequencies. II. RESPONSE OF A PLATE The response of a plate excited by a random fluctuating pressure field with nonzero mean value can be estimated in a way similar to that in the preceding section. The form of the boundary conditions is somewhat more complicated than that for the beam. For simplicity a square plate with ideal simply supported boundary conditions is considered, and a loss factor included in the equation of motion. The loss factor is subsequently ex- pressed in terms of a given boundary impedance by using again the relations between power flow and energy in the system. Corresponding to Eq. 1 the equation of motion is now of the form O•.y O•.y O•.y DV4y--Nr•--N•.•--2N• ß Oxx •' Ox2 •' Ox•Ox2 Oy O•y +•-+p---=po+p(x,t), (31) Ot Ot •' where N•(x), N•(x), and Nl•(x) are the in-plane forces per unit length. It is assumed that the load satisfies relations similar to Eqs. 4 and 5. The response is thus of the form of Eq. 6, and the static displacement can be solved separately. The approximate analysis given in Timoshenko 6 is followed. The bending and membrane stresses are considered separately and the first term of the eigenfunction expansion used to describe the form of the displacements. Thus, 7I'Xl 7I'X2 yo = Y sin-• sin•, L L and 2•rxl •rx2 u= -- U sin• sin. , L L •rx• 2•rx2 v - -- U sin• sin. L L (32) The amplitudes Y and U are found by the principle of virtual work. For a Poisson's ratio of 0.25 and U =0.294Y•'/L p0 = 21.9E • -•- q-1.41 (33) This last expression together with the asymptotic values is plotted in Fig. 3. The in-plane forces can then be obtained in the usual way from the expressions hE [ Ou Ov 1/Oyo\ •' K(Oyo• N• - •+•+- • •'1' hE [Ov+xOu+l(Oyo• • x(Oyo•], 1--g•LOx• Ox, 2kOx•/ +•k•/ (34) and Ou Ov Ow Ow N•, N•., and N• are all proportional to (Y/h)L But it can be seen that the in-plane force distribution is, in 1040 Volume 54 Number 4 1973 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Wed, 26 Nov 2014 08:33:17 VIBRATION OF DEFORMED BEAMS AND PLATES contrast to that for a beam, a function of position on the plate. The fluctuating response • of the statically deformed plate can now be obtained from Eq. 31 in terms of the presumed known forces N•, N2, and N•2. • satisfies an equation similar to Eq. 13. Now, however, a loss factor is included in the equation of motion and • satisfies simply supported boundary conditions. The Fourier transform of transverse displacement is expanded in terms of the eigenfunctions of the undeformed plate. Thus, oo oo m •r x l 7t •r x 2 w(x,o•) = • • Wr•,•(O•) sin sin---. (35) The eigenfunctions in Eq. 35 do not satisfy the differ- ential equation for the response of the deformed plate, which is in contrast to the case of a simply supported beam, where, as the in-plane force is not a function of position the eigenfunctions of both the deformed and undeformed beam are the same. The use of the ex- pansion 35 thus leads to a set of coupled modal equa- tions for the modal amplitudes Wren. These equations are of the form ( Dkmn4nt-iwT -- pco2)w•,• (co) (36) where the coupling coefficients R• are, of course, functions of N•, N2, and N•.. The complete equations are given in Ref. 8. Coupling occurs only between modes of like parity, for example (odd, even) only with (odd, even). The coefficients in the coupled modal equations describe the wavenumber coupling strength of the modes, that is, the coefficients depend only on the relative shapes of the modes. The degree of coupling between modes is also dependent on the proximity of the modal resonance frequencies. For cases of light damping and low modal density the modal resonance peaks are separate from each other and little modal coupling can occur. The required condition for this to be the case is cortd DAVIES AND AL SOWAYEL PERIMETER Fro. 4. Diagram showing field in one given direction incident on an element of the edge of a plate (from Ref. 9). dissipated power can now be obtained by integrating Expression 39 over all angles of incidence, using Eq. 40 to define •. Thus, Ildiss = 4pCp• R dl cosOdO 2•r .• 2v where P denotes the perimeter of the panel and G represents the average value of the conductance of the boundary. The energy density of a wave on the plate is related to the intensity by the expression dE=d•/co. The total energy of the panel is thus A ,J/co, where A p is the area of the panel. A second expression for the dissipated power is thus Ildiss = 6077. (42) 60 Equations 41 and 42 imply the result 4cpco/ P \_ (43) The term rA pip represents a mean free path between collisions with the boundaries of rays drawn on the plate. This expression is thus analogous to Eq. 26 for the loss factor for a beam where the mean free path is just L. Equation 43 describes the loss factor of any two dimensional system with arbitrarily shaped boundaries in cases where the energy density of the system is fairly uniform. For an arbitrarily shaped plate supporting pure bending waves, Eq. 43 reduces to the simple form r•(w) --- --, (44) ,rAp G• where G• is the point input admittance of an infinite plate. Again, it is noted that • is related to an impedance per unit length, so that • is ind•ed dimensionless. If the boundary impedance is assumed unchanged by a static deformation, then Eq. 43 together with Expres- sion 3 7 can be used to investigate the effect of the static load on the loss factor. For a given mode the loss factor increases as co increases with increased tension. On the other hand, at a given frequency the tension increases the loss factor by a factor which can be written in the form N• ' )• 1+ . (45) 4Dw•.p Here N represents the effective tension, Expression 37. The overall effect of the tension on the vibration level of the plate can be seen best through the use of an example. It is supposed that the fluctuating load p(x,t) represents a homogeneous rain-on-the-roof type of force field with excitation cross-spectral density of the form SP (x ø),x(•'),w) = Sp (w)•/• (x ø)-- x (•')), (46) where xO) and x (" represent two points on the plate and • represents a correlation area. If the spectrum of displacement at a point Sr(x,x,o•) is averaged over narrow bands of frequency and points on the plate, and if the contributions of resonant modes only are included, then 1 fo+/'Sr(x,x,w)dxd w (Area) Aw read w0--Aw/2 7r 5d(wo)SP(wo)A p 2 M•n(w0)w0 • (47) where M represents the total mass of the plate. The effect of the tension appears in the factor where k is the wavenumber of a typical mode resonant in the A• band. The average spectrum is thus decreased by the factor 4D•p/ Expression 48 suggests that it is only the lowest modes that are affected by the tension, and then only when the tension is very high. For high tension the asymptotic value of the static displacement may be used to rewrite Expression 48 in the approximate form X Y 1042 Volume 54 Number 4 1973 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Wed, 26 Nov 2014 08:33:17 VIBRATION OF DEFORMED BEAMS AND PLATES where X represents a typical wavelength on the unde- formed plate at the frequency of interest. III. EXPERIMENTAL RESULTS A number of authors •ø-•3 have reported measured values of the loss factors of plates excited by turbulent flow that are dependent on the flow velocity. The analysis given above appears to explain, at least in part, the data reported by Moore •' and by Leehey. •3 The static pressure acting in this case is the difference be- tween the hydrodynamic pressure inside a wind tunnel duct and the external ambient pressure. The pressure difference was measured for a variety of flow speeds. At each flow speed, two different pressure differences were obtained. These correspond to (i) atmospheric pressure in the reverberant chamber surrounding the wind tunnel test section, and (ii) carefully sealing the door into the reverberant chamber, and partially opening the flow duct to partially reduce the pressure in the chamber. -IO: -2O -3O IO log , I • , I , , I I00 1000 i0 000 -2O 0.006" , I • , [ , , I I oo I ooo io ooo FREOUFNC¾ IN IIERTZ Fro. 5. Experimental values of the loss factor of a steel plate measured in «-oct bands both with and without turbulent bound- ary layer excitation' top, for a 11"X 13"X0.0015" plate, bottom, for a 11"X 13"X0.006"plate (from Ref. 12 and 13). ß Fluid flow speed Pressure difference Key Symbol (m/sec) in inches of water x 0 0 Top 0 30 0.3 O 30 2.3 Bottom x 0 0 [-] 45 2.48 The curves marked A, B, and C are discussed in Sec. IV. Measurements of the loss factor in «-oct bands were made by exciting the plate with an electromagnet, and, when the excitation was switched off observing the decay by using a fotonic sensor displacement gauge connected to a graphic level recorder. Typical data are presented in Fig. 5 for two plates. For the thin plate, the line A is a smooth-curve fit to the data for zero flow velocity. Lines B and C are determined from Eq. 45 for the pressure differences given and for a 12-in. square plate. The agreement is reasonable. The predicted static deflection at a pressure of 2.3 in. of water is Y/h= 72 or Y•0.1 in. This again agrees with the experimental observation, although the static deflection was not measured accurately. Data are also shown for a thicker plate. The maximum predicted static deflection is Y/h= 11 or Y•0.07 inch, and the maximum predicted change in loss factor (at 100 Hz) corresponds to only 0.4 dB. Fitted curves for this data are not shown; however, the magnitude of the predicted change clearly seems to be correct. In the experiment reported by Lyon n there is no static pressure component acting on the ribbon since the ribbon has flow on both sides. The dependence of the loss factor on flow velocity must thus be due to another mechanism, presumably, as has been suggested, in- volving an interaction between the structural vibration and the turbulent flow. This mechanism has not yet been fully explained. Thus, it is not possible to state for which range of parameters a structure-flow interaction is important, and for which range the much more simple explanation involving the effects of membrane stresses is important. IV. CONCLUSION The response of beams and plates excited by fluctu- ating loadings with nonzero mean values has been estimated. For a plate, it has been shown that the average value of the in-plane force distribution acts as an effective tension as far as the response of individual modes is concerned. The results concerning the effects of the tension on the fluctuating response are thus appli- cable either to the uniform static loading case discussed here or to the case of a uniformly applied in-plane loading. A simple expression for the loss factor of a plate has been obtained involving the admittances of the boundaries and of an infinite plate. The loss factor is proportional to the group velocity associated with the system. When the static deflection at the midpoint is less than the thickness of the plate (the case of typical practical interest) the effect on the vibration of the induced membrane stresses is negligible. However, as seen in Fig. 5, their effect on the vibrations of very thin plates with very large deflections can be quite large. In practice it is found usually that the loss factor of a system is very susceptible to slight modifications of the boundary conditions; measurements of loss factors tend The Journal of the Acoustical Society of America 1043 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Wed, 26 Nov 2014 08:33:17 DAVIES AND AL SOWAYEI_, to differ from one set of measurements to the next. Thus the assumption of constant boundary impedance perhaps is not realistic. However, with the assumption of constant boundary impedance the simple analysis given here seems to explain some of the reported dependence of loss factor on flow velocity for systems that are excited by turbulent flow. ACKNOWLEDGMENTS We are pleased to express thanks to Professors Patrick Leehey and William J. Shack for helpful suggestions and discussions. The work was supported in part by the Acoustics Program Branch of the Office of Naval Research, Washington, D. C. •M. Heckl, "Vibrations of Point-Driven Cylindrical Shells," J. Acoust. Soc. Am. 34, 1553 (1962). 2j. Petyt, "Vibration of Curved Plates," J. Sound Vib. 15, 391 (1971). 3G. A. Smith, P. A. Laura, and M. Matis, "Experimental and Analytical Study of Vibrating Stiffened Rectangular Plates," J. Acoust. Soc. Am. 48, 707 (1970). 4D. Wolf and H. Miiller, "Normal Vibration Modes of Stiff Strings," J. Acoust. Soc. Am. 44, 1093 (1968). Sp. M. Morse, Vibration and Sound (McGraw-Hill, New York, 1948), 2nd ed. 6S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), 2nd ed. ?M. Heckl, "Measurements of Absorption Coefficients on Plates," J. Acoust. Soc. Am. 34, 809 (1962). 8S. al Sowayel, "Effects of Distributed Loads on the Vibrations of Thin Plates," M. S. thesis, MIT (1971). 9p. W. Smith and R. H. Lyon, "Sound and Structural Vibration," NASA CR- 160 (1965). •øL. Maestrello, "Use of Turbulent Model to Calculate Vibration Response of a Panel," J. Sound Vib. 5 (1967). •R. H. Lyon, "Response of Strings to Random Noise Fields," J. Acoust. Soc. Am. 28, 391 (1956). •2j. A. Moore, "Response of Flexible Panels to Turbulent Boundary Layer Excitation," MIT A & V Lab. Rep. No. 70208-3 (1969). •3p. Leehey, "Boundary Layer Noise Research," in Aerodynamic Noise (AFOSR-UTIAS Symposium, 1968). 1044 Volume 54 Number 4 1973 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. 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