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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 478389, 18 pages http://dx.doi.org/10.1155/2013/478389 Research Article The Effect of Uncertainty in the Excitation on the Vibration Input Power to a Structure 1 2 A. Putra and B. R. Mace Faculty of Mechanical Engineering, Universiti Teknikal Malaysia Melaka, Hang Tuah Jaya, 76100 Durian Tunggal, Melaka, Malaysia Department of Mechanical Engineering, University of Auckland, Auckland 1142, New Zealand Correspondence should be addressed to A. Putra; azma.putra@utem.edu.my Received 30 April 2013; Accepted 18 July 2013 Academic Editor: Marc om Th as Copyright © 2013 A. Putra and B. R. Mace. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In structural dynamic systems, there is inevitable uncertainty in the input power from a source to a receiver. Apart from the nondeterministic properties of the source and receiver, there is also uncertainty in the excitation. This comes from the uncertainty of the forcing location on the receiver and, for multiple contact points, the relative phases, the force amplitude distribution at those points, and also their spatial separation. This paper investigates qua ntification of the uncertainty using possibilistic or probabilistic approaches. es Th e provide the maximum and minimum bounds and the statistics of the input power, respectively. Expressions for the bounds, mean, and variance are presented. First the input power from multiple point forces acting on an infinite plate is examined. eTh problem is then extended to the input power to a finite plate described in terms of its modes. The uncertainty due to the force amplitude is also discussed. Finally, the contribution of moment excitation to the input power, which is oeft n ignored in the calculation, is investigated. For all cases, frequency band-averaged results are presented. 1. Introduction conditions, and so on in a simple receiver structure, such as a plate, has been described by Langley and Brown [1, 2], where eTh treatment of structure-borne sound sources remains a expressions for the mean vibrational energy and its variance challenging problem. Structural excitation to a building floor, were developed. A closed form solution was presented for for example, by active components like pumps, compressors, the relative variance as a function of modal overlap factor fans, and motors, is an important mechanism of sound andthe nature of theexcitation[1]. In [2], the analysis was generation.Toobtainanaccuratepredictionofthe injected extended to the ensemble average of the frequency band- input power from such sources, both the source and the averaged energy as a function of the frequency bandwidth. receiver must firstly be characterised. However in practical eTh analysis proceeded on the assumption that the natural application, the variability of source and receiver properties frequencies form a random point process with statistics including the lack of knowledge in the excitation force creates governed by, for example, the Gaussian orthogonal ensemble uncertainty in the input power. eTh problem is exacerbated (GOE). The same type of analysis for a more complex system because in practice there will usually be multiple contact has also been proposed [3]. points (typically four) and 6 degrees of freedom (3 for Regardless of the receiving structure, the concept of translation and 3 for rotation) at each, and that force and source descriptor has been proposed to characterise a source moment components at each contact point will contribute based on its ability to deliver power without necessarily to the total input power. er Th efore to assess the uncertainty, knowing any information about the receiver [4, 5]. Here, the some quanticfi ation of the bounds, mean, and variance of the conceptofeeff ctivemobility[ 6, 7], that is, the ratio of the input power is of interest. actual velocity at a point and in one direction, to the con- The uncertainty in vibrational energy due to random tributions of the excitations from all components and points, properties, for example, dimensions, shapes, boundary was used. 2 Advances in Acoustics and Vibration The effective mobility concept was also employed in [ 8]to source mobility is assumed to be much smaller than that of estimatethetotalpowertoaninnfi itebeamthroughfourcon- thereceiver, as is usuallythe case in practice.Theinput power tact points. eTh importance of having knowledge of the force is therefore given by distribution at the contact points was acknowledged. For this purpose, three simple force ratio assumptions were intro- ̃ ̃̃ 𝑃 = Re{F YF}, (1) in duced, and the eeff cts of force position, type of excitation, the 2 structural loss factor, the receiver, and the number of contact 1 2 𝑁 where F =[𝐹 𝑒 𝐹 𝑒 ⋅⋅⋅𝐹 𝑒 ] is the vector of the points were investigated. It was found that the estimation is 1 2 𝑁 complex amplitudes of the time-harmonic forces and where accurate only in the mass-controlled region and when the ∗ denotes the conjugate transpose. The 𝑖 th force has a real system approximated a symmetrical response. In a later paper magnitude 𝐹 and phase 𝜙 .Themobilitiesofthe receiver are [9], the force ratio was defined in terms of its statistical dis- 𝑖 𝑖 represented by an𝑁×𝑁 matrix Y. In this section, only forces tribution. Simple expressions of the distribution at the mass-, are considered. Moment excitation is discussed in Section 6. stiffness-, and resonance-controlled regions were derived. Characterisation of a source in practice can be ap- 2.2. Broadband Excitation. For a broadband excitation over proached by using a reception plate method [10, 11]which afrequency band 𝐵 , the input power should be defined in is based on laboratory measurements, where from measured terms of power spectral density and can be written as mobilities and surface mean-square velocities of the receiver, thefreevelocitiesand mobilities of thesourcecan be 𝑁 𝑁 extracted. ̃ ̃ ̃ 𝑆 = ∑𝑆 Re {𝑌 }+ ∑ Re{𝑆 }Re{𝑌 }, 𝑃 𝑖𝑖 𝑖𝑖 𝑖𝑘 𝑖𝑘 Thispaper focusesonlyonthe uncertaintyinthe excita- in (2) 𝑖 𝑖,𝑘 tionwiththesourceandreceiverassumedtobedeterministic. 𝑖 =𝑘̸ The source may have multiple contact points. eTh uncertainty in input power due to the excitation phase, its location, and where 𝑆 and 𝑆 are the autospectral density and cross- 𝑖𝑖 𝑖𝑘 separation of the contact points is investigated. First some spectral density of the forces, respectively, and𝑌 is the trans- 𝑖𝑘 general comments are made. Broadband excitation is des- fer mobility between points 𝑖 and 𝑘 .Thecross-spectraldensi- cribed, although only time-harmonic excitation is considered ties would oeft n in practice be difficult to measure. However, here with frequency averages subsequently being taken. eTh the coherence relates the auto- and cross-spectral densities of input power from multiple point forces to an infinite plate the excitation. u Th s by assuming that only the autospectra of is examinedtogiveaninsight into thephysicalmechanisms the forces are known, the cross-spectra are such that involved. In practice, the receiver will have modes, although themodal overlapmight be high.Theinput powertoafinite √ (3) 𝑆 =± 𝛾 𝑆 𝑆 , 𝑖𝑘 𝑖𝑖 𝑘𝑘 plate is then analysed, where now the forcing location on the receiver becomes important. eTh mean and the variance of where0≤𝛾 ≤1 is the coherence. This gives maximum and the input power averaged over force positions are investi- minimumboundstothe magnitudeofthe cross-spectral gated. eTh results are also presented in frequency-band aver- density. A rather similar approach is proposed by Evans and ages. Moorhouse [13]for thecaseofarigidbodysource, where eTh uncertaintyinthe inputpower duetouncertainty the real part of the cross-spectral density is predicted using in the force amplitude for multiple contact points is also the available data of the autospectra and the calculated free discussedfor asimplecaseoftwo contactpoints. Rather than velocities at the contact points from the source rigid body dealing with force ratios between contact points, the sum of modes. Hence, this is limited only to the mass-controlled the squared magnitudes of the forces is assumed known. region of the source at very low frequencies. eTh comparison Finally, the inclusion of moments in the excitation is with measured data shows a good agreement. However, investigated and predictions of its contribution to the input neitherofthese approaches areimplemented in this paper. power are made. Allforcesare assumedtobetimeharmonic. 2. Input Power and Uncertainty Assessment 2.3. Uncertainty Quantification. Two approaches are em- ployed to describe the uncertainty in the input power, 2.1. Time-Harmonic Excitation. Consider a vibrating source namely, possibilistic and probabilistic approaches [14]. The connected through a single or 𝑁 contact points to a receiver. possibilistic approach gives an interval description of the For a time-harmonic excitation at frequency 𝜔 ,the input input power, which lies between lower and upper bounds; power is expressed as a function of mobility (or impedance) that is, of thesourceand receiver [10, 12]. Thisrequiresknowledge of both source and receiver mobilities and the so-called 𝑃 ∈[𝑃 𝑃 ], (4) in in in blocked force or free velocity of the source. In general, the mobilities are matrices and the blocked forces or the free where 𝑃 and 𝑃 are the minimum and maximum bounds velocities are vectors, with the elements relating to the various in in translational and rotational degrees of freedoms (DOFs) at and𝑃 is the interval variable. One example has been given in in the contact points. In this paper, however, the analysis is Section 2.2, where the input power can be bounded by using made by assuming that the force excitation is known, and the the spectral coherence data. 𝑗𝜙 𝑗𝜙 𝑗𝜙 Advances in Acoustics and Vibration 3 The probabilistic approach gives information about the likelihood and probability of the input power. The variation is 1.8 specified by a probability density function Π.If Π(𝑧) is a con- 1.6 tinuous function of some variable 𝑧 , the mean or the expected 1.4 value of the input power and its variance are den fi ed by 1.2 𝜇 =𝐸[𝑃 ]=∫ 𝑃 (𝑧 )Π(𝑧 )d𝑧, 𝑃 in in in (5) 0.8 2 2 𝜎 = ∫ 𝑃 (𝑧 )Π(𝑧 )d𝑧−(𝜇 ) . 𝑃 in 𝑃 in in 𝑧 0.6 0.4 3. An Infinite Plate Receiver 0.2 The input power to an infinite plate, as an ideal simple struc- −2 −1 0 1 2 ture, is rst fi investigated. The point and transfer mobilities for 10 10 10 10 10 an inn fi ite plate are given by [ 15] kL Figure 1: The normalised input power to an infinite plate subjected 𝑌 = , to two in-phase (—) and out-of-phase (⋅⋅⋅ ) harmonic unit point 8√𝐵 𝑚 forces (— grey line: max/min bounds at higher frequencies). (6) 2𝑗 (2) ̃ ̃ 𝑌 = 𝑌 [𝐻 (𝑘𝐿 )− 𝐾 (𝑘𝐿 )], 𝑡 𝑝 0 When the structural wavelength is much smaller than 𝐿 (2) (𝜆≪𝐿 ), the input power is where 𝐻 is the zeroth-order Hankel function of the second kind, 𝐾 is the zeroth-order modified Bessel function of the 2 ̃ ̃ 𝑃 =(Re{𝑌 }+Re{𝑌 }cos𝜑)𝐹 . (10) 3 in 𝑝 𝑡 second kind, 𝑚 is the mass per unit area, and 𝐵 =𝐸ℎ /12(1− ] ) is the bending stiffness of the plate having Young’s For small structural wavelength, the maximum and mini- modulus 𝐸 ,thickness ℎ,and Poisson’sratio ].Theinput mum input powers depend on Re{𝑌 }cos𝜑 which indicates point mobility is purely real and independent of frequency, the dependency of the power on the phase and the distance behaving as a damper. In the transfer mobility, the rst fi and between the excitation forces. Now since 𝑘𝐿 ≫ 1 ,the real second terms represent propagating and near-field outgoing part of the transfer mobility in (6) can be expressed in its waves, respectively. asymptotic form [16]as(see Appendix A) Assume an infinite plate is excited by two point forces 2 𝜋 separated by a distance 𝐿 .From(1), the force vector is F = ̃ ̃ (11) Re{𝑌 }= Re{𝑌 } cos(𝑘𝐿 − ). 𝑡 𝑝 1 2 𝜋𝑘𝐿 4 {𝐹 𝑒 𝐹 𝑒 } and the mobility is a2×2 matrix. eTh total 1 2 power is, thus, the sum of the input power at each location The input power can thus be written as which yields 2 𝜋 𝑃 = Re{𝑌 }(1 + cos(𝑘𝐿 − ) cos𝜑) 𝐹 . (12) 2 2 in 𝑝 ̃ ̃ (7) 𝜋𝑘𝐿 4 𝑃 = Re{𝑌 }(𝐹 +𝐹 )+ Re {𝑌 }𝐹 𝐹 cos𝜑, in 𝑝 𝑡 1 2 1 2 The input power at higher frequencies is thus bounded by where 𝜑=𝜙 −𝜙 is the phase difference between the two 1 2 ̃ ̃ ̃ forces.Notethat 𝑌 = 𝑌 = 𝑌 .For thecaseofaninnfi ite 12 21 𝑡 2 𝑃 =(1± ) Re{𝑌 }𝐹 . (13) in 𝑝 structure, the input mobility is the same at any position which 𝜋𝑘𝐿 ̃ ̃ ̃ implies that 𝑌 = 𝑌 = 𝑌 . 11 22 𝑝 Figure 1 shows the total input power when the two forces are in-phase (𝜑=0 )and out-of-phase (𝜑=𝜋 )asafunction 3.1. Dependence on the Contact Points Separation. When the of 𝑘𝐿 . Note that the input power has been normalised with structural wavelength 𝜆 is much larger than the separation respecttothe inputpower from twopoint forces acting of the excitation points (𝜆≫𝐿 ), the transfer mobility is incoherently (equal to two times the power from a single ̃ ̃ approximately equal to the point mobility. eTh refore, 𝑌 ≈ 𝑌 . 𝑝 𝑡 pointforce). It canbeseenthatthe powerfluctuates around For simplicity, if 𝐹 =𝐹 =𝐹 ,the inputpower becomes 1 2 the value it would have if the two forces were applied inde- 𝑃 = Re{𝑌 }𝐹 (1 +cos𝜑). (8) pendently. From (12), the power is minimum and maximum in 𝑝 when 𝑘𝐿 = (2𝑛 + 1)𝜋 for 𝑛 = 0,1, ... for in-phase and out- From (8), the maximum and minimum input power are of-phase forces, respectively. These are when, with respect to the wavelength, the two in-phase forces become out of phase 𝑃 =2𝐹 Re {𝑌 }, for 𝜑=2𝑛𝜋 in 𝑝 and the out of phase forces become in phase. The intersection (9) between the two curves at high 𝑘𝐿 is when𝜑=𝜋/2 .For 𝑘𝐿 < 𝑃 =0, for 𝜑= (2𝑛 + 1 )𝜋, in 1, that is, when the forcing distance is less than half a struc- where 𝑛 is any integer. tural wavelength, the total power is constant with frequency. Normalised power 𝑗𝜙 𝑗𝜙 4 Advances in Acoustics and Vibration 3.2. Random Phase. In practice, accurate information regard- ing the relative phase between the forces is generally not 1.8 known. It might be assumed that all possible relative phases 1.6 have equal probability. eTh probability density function Π of 1.4 the phase is then constant and given by 1.2 Π(𝜑) = ,𝜑= 0,2𝜋 . (14) [ ] 2𝜋 0.8 From (10), themeanpower andits variance using(5)are 0.6 𝜇 = Re{𝑌 }𝐹 , 0.4 𝑃 𝑝 in 0.2 (15) 2 2 𝜎 = (Re{𝑌 }𝐹 ) . 𝑃 𝑡 −2 −1 0 1 2 in 2 10 10 10 10 10 kL It canbeseenthatthe mean powerequalsthatwhich would be given by two forces acting independently. The variance, Figure 2: The normalised input power to an infinite plate subjected hence, arises from interference between the two forces. As to two harmonic unit point forces with various relative phases (grey ̃ ̃ lines): −− mean, −⋅− mean ± standard deviation, — (thick lines) 𝑌 = 𝑌 for𝑘𝐿 ≪ 1 ,from(15),thenormalisedstandarddevia- 𝑡 𝑝 max/min bounds, — mean ± bounds of standard deviation, and ⋅⋅⋅ tion is 𝜎/𝜇 = 1/ 2.For 𝑘𝐿 ≫ 1 , 𝜎 decreases as 𝑘𝐿 increases mean ± bounds of standard deviation due to uncertainty in . (see (11)). Substituting (11)into(15)gives 𝜎 1 𝜋 = ( ) cos (𝑘𝐿 − ) . (16) j𝜙 2 F e 𝜇 𝜋𝑘𝐿 4 j𝜙 F e The bounds of the variance occur when cos (𝑘𝐿 − 𝜋/4) = 1 . The maximum normalised standard deviation is therefore 𝜎/𝜇 = 1/ 𝜋𝑘𝐿 . j𝜙 2 2 F e When the separation distance of the contact points is uncertain, consequently 𝑘𝐿 mod 𝜋 becomes unknown, while j𝜙 1/(𝑘𝐿) is more or less constant. eTh refore by averaging ( 16) F e over all possible 𝑘𝐿 mod 𝜋 ,the variance canbeexpressed as h 2 𝜋/2 𝜎 1 1 𝜋 ⟨ ⟩ =( )[ ∫ cos (𝑘𝐿 − ) d(𝑘𝐿 )] Figure 3: An infinite plate excited by four harmonic point forces 𝜇 𝜋𝑘𝐿 𝜋 4 −𝜋/2 applied at the vertices of a rectangle. (17) = . 2𝜋𝑘𝐿 ̃ ̃ ̃ ̃ ̃ ̃ ̃ shape, therefore, 𝑌 = 𝑌 = 𝑌 = 𝑌 ; 𝑌 = 𝑌 = 𝑌 = 12 21 34 43 13 31 24 ̃ ̃ ̃ ̃ ̃ eTh refore, the normalised standard deviation due to uncer- 𝑌 and 𝑌 = 𝑌 = 𝑌 = 𝑌 .Theproblem canbesimpli- 42 14 41 23 32 tainty in 𝑘𝐿 is 𝜎/𝜇 = 1/ 2𝜋𝑘𝐿 . fied numerically by choosing a force vector whose phases are Figure 2 shows the input power for a number of possible relative to the phase of one reference force. The force vector relative phases of excitation. eTh mean power is a function can be rewritten as of the point mobility which then gives a constant value with ̃ 2 3 4 (18) F = [𝐹 𝐹 𝑒 𝐹 𝑒 𝐹 𝑒 ] , frequency. The normalised deviation of the power ( 1±𝜎/𝜇 ) 1 2 3 4 lies between the maximum and minimum input power and where 𝜓 is the relative phase of 𝐹 with respect to 𝐹 .Thus, 𝑖 𝑖 1 bounded by1±1/ 𝜋𝑘𝐿 . It can be seen that uncertainty in 𝑘𝐿 the total input power is given by increases the deviation of the input power. 2 2 2 2 𝑃 = Re{𝑌 }(𝐹 +𝐹 +𝐹 +𝐹 ) in 𝑝 1 2 3 4 3.3. Four Contact Points. The case of four input excitations is 2 more realistic in practice, for example, a vibrating machine + Re{𝑌 }(𝐹 𝐹 cos𝜓 +𝐹 𝐹 cos(𝜓 −𝜓 )) 12 1 2 2 3 4 3 4 with four feet. Figure 3 shows an infinite plate excited by four harmonic point forces. In this case, from (1), the mobility is a + Re{𝑌 }(𝐹 𝐹 cos𝜓 +𝐹 𝐹 cos(𝜓 −𝜓 )) 13 1 3 3 2 4 2 4 4×4 matrix. eTh diagonal elements are the point mobilities at each location, which for an infinite structure are equal + Re{𝑌 }(𝐹 𝐹 cos𝜓 +𝐹 𝐹 cos(𝜓 −𝜓 )). ̃ 14 1 4 4 2 3 2 3 to 𝑌 . eTh analysis is generally complicated. For simplicity, (19) by assuming that the excitation positions form a rectangular Normalised power 𝑗𝜓 𝑗𝜓 𝑗𝜓 𝑘𝐿 𝑘𝐿 Advances in Acoustics and Vibration 5 Note againin(19), that the input power is the sum of the 4 powers that would be injected by the forces acting individu- 3.5 ally (the first term) and terms that depend on the relative phases of the forces (the remaining terms). u Th s, if equal probability of relative phase is assumed and the phases are uncorrelated, the probability density function 2.5 Π can be expressed as Π(𝜓 ,𝜓 ,𝜓 )=Π(𝜓 )Π(𝜓 )Π(𝜓 ), 2 3 4 2 3 4 1.5 (20) Π(𝜓 )= . 𝑖 1 2𝜋 Using (5) and assuming all the forces have equal amplitudes, 0.5 themeanpower andthe variance aregiven by −1 0 1 ̃ 10 10 10 𝜇 =2 Re{𝑌 }𝐹 , (21) 𝑃 𝑝 in kL 1 2 1 2 2 2 2 ̃ ̃ 𝜎 = (Re{𝑌 }𝐹 ) + (Re {𝑌 }𝐹 ) Figure 4: The normalised input power to an infinite plate subjected 12 13 in 2 2 to four harmonic unit point forces with various phases (grey lines): 1 −− mean,−⋅− mean ± standard deviation, — (thick lines) max/min + (Re{𝑌 }𝐹 ) bounds, — mean ± bounds of standard deviation, and ⋅⋅⋅ mean ± (22) bounds of standard deviation due to uncertainty in . 2 2 1 1 2 2 ̃ ̃ + (Re{𝑌 }𝐹 ) + (Re{𝑌 }𝐹 ) 23 24 2 2 Figure 4 shows the input power with various relative + (Re{𝑌 }𝐹 ) . 2 phases of excitation. As in Figure 2,the powerfluctuates at high 𝑘𝐿 around themeanvalue.Thestandarddeviation As in the case of two point forces in Section 3.2,the variance shown is from (22). The rfi st dip in the standard deviation of the input power depends only on the transfer mobilities corresponds to the frequency where the diagonal 𝐿 equals while the mean power is only a function of the point mobility. half the structural wavelength, and the subsequent dips are In general, the mean and the variance of input power averaged related to 𝐿 and 𝐿 .Equations (25)and (26)again show 1 2 over all possible phases of excitation for 𝑁 contact points for that the standard deviation reduces as the separation of the arectangular separation canbewritten as contact points increases. 𝜇 = ∑ Re{𝑌 }𝐹 , 𝑃 𝑖𝑖 𝑖 in 4. Input Power to a Finite Plate (23) 𝑁−1 𝑁 Forafinitestructure,the mobility and, hence, theinput 1 2 𝜎 = ∑ ∑(Re{𝑌 }𝐹 𝐹 ) . 𝑖𝑘 𝑖 𝑘 𝑃 power can be expressed in terms of a summation over modes in 𝑖=1 𝑘=2 of vibration. The mobility of a ni fi te structure at an arbi- 𝑖<𝑘 trary point (𝑥,𝑦 )subjected to apoint force 𝐹 at (𝑥 ,𝑦 )at 0 0 The bounds for the normalised input power are obtained frequency 𝜔 is given by [10] by inserting the maximum transfer mobility from (11)into (19), which for equal force amplitudes yields ∞ Φ (𝑥 ,𝑦 )Φ (𝑥,𝑦) 𝑛 0 0 𝑛 𝑌=𝑗𝜔 ∑ , (27) 2 2 𝑃 𝜔 (1 + 𝜂𝑗 )−𝜔 in 2 𝑛=1 𝑛 −1/2 −1/2 −1/2 √ (24) =1± (𝐿 +𝐿 +𝐿 ). 1 2 3 where Φ is the 𝑛 th mass-normalised mode shape of the In the same way, substituting (11)into(22), the bounds for the structure, 𝜔 is the 𝑛 th natural frequency, and 𝜂 is damping 𝑛 𝑛 normalised standard deviation are loss factor of the𝑛 th mode. A case oen ft considered is that of a rectangular plate with simply supported boundary conditions 𝜎 1 1/2 −1 −1 −1 (25) =± (𝐿 +𝐿 +𝐿 ) , as this system provides a simple analytical solution. For a 1 2 3 𝜇 2𝜋𝑘 simply supported rectangular plate with dimensions𝑎×𝑏 ,the mode shape and the natural frequency for mode (𝑝, 𝑞) are wherethe positive andnegativesigns arefor themaximum and minimum bounds, respectively. 𝑥 𝑥𝑞𝜋 Following the same method for the case of two point Φ (𝑥,𝑦) = sin( ) sin( ), 𝑝𝑞 forces, the bounds due to uncertainty in 𝑘𝐿 are given by √ 𝑎 𝑏 (28) 2 2 𝜎 1/2 1 𝑞𝜋 −1 −1 −1 = ±0.5 (𝐿 +𝐿 +𝐿 ) . (26) √ 𝜔 = [( ) +( ) ], 1 2 3 𝑝𝑞 𝑚 𝑎 𝑏 Normalised power 𝜋𝑘 𝑝𝜋 𝑝𝜋 𝜋𝑘 𝑘𝐿 6 Advances in Acoustics and Vibration where 𝑀 is the total mass of the plate and 𝑝, 𝑞 = 1,2, 3, ... . The rigid body motion existing in the reception plate method [11] can be approximated by assuming the edges are guided or sliding. The mode shape functions involve cos terms instead of sin terms in (28). Using the corresponding real part of the point mobility in (27), the input power for point excitation is given by 2 2 2 ∞ ∞ −1 𝜂 Φ (𝑥 ,𝑦 ) 𝐹 | | 1 𝑝𝑞 𝑛 𝑝𝑞 0 0 𝑃 = ∑ ∑ . (29) in 2 2 2 2 2 𝑝=1 𝑞=1 (𝜔 −𝜔 ) +(𝜔 𝜂 ) 𝑝𝑞 𝑝𝑞 −2 4.1. Averaging over Force Positions. Equation (29)implies that the input power depends on the forcing location which −3 1 2 3 4 might be uncertain. Figure 5 shows the input power from a 10 10 10 10 single point force for various forcing locations. eTh structure Frequency (Hz) considered is an aluminium plate having dimensions 0.6 × 10 2 Figure 5: The normalised input power to a finite plate subjected to a 0.5 × 0.0015 m, Young’s modulus 7.1 × 10 N/m , 𝜌= single harmonic point force for various possible force positions (grey 2700 kg/m , and damping loss factor 𝜂 = 0.05 , assumed con- lines): — mean and −⋅− mean ± standard deviation. stant for every mode. The input power is normalised with respect to the input power to an infinite plate. The peaks occur at the resonances of the plate. They 2 are distinct at low frequencies, but the modal overlap increases as the frequency increases. Analytically, since 2 1 1/(𝑎𝑏) ∫ ∫ Φ d𝑥 d𝑦 =1/𝑀 , the mean input power aver- 𝑝𝑞 0 0 𝑎 𝑏 aged over all possible force positions is [10] as follows: 2 ∞ ∞ | | 𝑝𝑞 ⟨𝑃 ⟩ = ∑ ∑ . (30) in 𝑥 ,𝑦 2 2 0 0 2𝑀 2 2 2 𝑝=1 𝑞=1 (𝜔 −𝜔 ) +(𝜔 𝜂) −1 𝑝𝑞 𝑝𝑞 It canbeseenthatathighfrequencies,the mean power −2 converges to the same level as the input power to an infinite plate. eTh variance of the input power also decreases at high frequenciesaswithrespect to thespatial variationofthe −3 vibration modes, the forcing location becomes less important 1 2 3 4 10 10 10 10 as the frequency increases. Frequency (Hz) 4.2. Averaging over Frequency Bands. Suppose that the exci- Figure 6: The normalised input power to a finite plate subjected to tation frequency lies between two frequencies 𝜔 and 𝜔 .The 1 2 a single harmonic point force for various possible force positions inputpower canthenbeaveragedoverthisfrequency band averaged over 100 Hz frequency band (grey lines): — mean and−⋅− and can be expressed as mean ± standard deviation. ⟨𝑃 ⟩ = ∫ 𝑃 (𝜔 )d𝜔. (31) in in 𝜔 −𝜔 2 1 2 spacings between successive natural frequencies are statisti- cally independent and have an exponential distribution [17]. Figure 6 shows the average input power for a 100 Hz band- From [1, 2, 10, 17], the mean and variance of the input power width and at 50 Hz centre frequency spacing. For the plate averaged over all possible forcing locations yield example considered, the modal density is 0.065 modes/Hz. eTh refore with this bandwidth, there are on average just over |𝐹 | 𝑛 𝜋 6modes in theband. It canbeseenthatthe mean valueisnow 𝑑 ⟨𝑃 ⟩ = , in (𝜔,𝑥 ,𝑦 ) 0 0 closetotheinnfi iteplatevalueexceptbelow100 Hz,wherethe 4𝑀 (32) modaloverlap is lowand theresponseisstiffnessdominated. |𝐹 | 𝑛 𝜋 ⟨𝜎 ⟩ = , (𝜔,𝑥 ,𝑦 ) 0 0 16𝜂𝑀𝜔 4.3. Prediction of Mean and Variance. The frequency average of the input power in (29) strongly depends on the statistical 2 2 2 2 2 where 𝑛 (𝜔) = 0.276𝑎𝑏/ℎ𝑐 is the modal density of plate and distribution of 𝜔/((𝜔 −𝜔 ) +(𝜔 𝜂) ).For asymmetric 𝑑 𝐿 𝑚𝑛 𝑚𝑛 2 1/2 structure like a rectangular plate, asymptotically the natural 𝑐 =(𝐸/𝜌(1− ] )) is the longitudinal plate wave speed. eTh frequency spacings have a Poisson distribution; that is, the mean power equals the input power to an inn fi ite plate. Normalised power Normalised power 𝜔𝜔 𝜔𝜔 Advances in Acoustics and Vibration 7 2 2 1.5 1.5 1 1 0.5 0.5 0 0 10 10 5 5 0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Frequency (Hz) (a) (b) Figure 7: eTh mean and standard deviation of the input power of a finite plate due to variation in forcing locations (numerical: — mean, −⋅− mean ± standard deviation, and (32): −− mean and ⋅⋅⋅ mean ± standard deviation; (a) 𝜂 = 0.05 and (b) 𝜂 = 0.1 ). Figure 7 shows the normalised standard deviation of the input power averaged over force positions for different damping loss factors together with the modal overlap factor (MOF), where MOF =𝑛 𝜂𝜔 . It can be seen that there is a good agreement between the numerical calculations and the analytical predictions if MOF >1. From (32), therelativestandarddeviation canbedenfi ed as a function of modal overlap factor; that is, −1 ⟨𝜎 ⟩ 1 1 𝑟 = = = . (33) 𝜂𝜔 √ ⟨𝑃 ⟩ √ 𝜋 MOF in 𝑑 −2 For plates with other shapes, the natural frequency spac- ing statistics are not Poisson (e.g., under many circumstances −3 they asymptotetoGaussianorthogonal ensemblestatistics). 0.2 1 10 30 An alternative expression for the variance can then be found kL [1, 2]. Figure 8: The normalised input power to a finite plate subjected 4.4. Four Point Excitations. In this section, results are pre- to four harmonic point forces averaged over all possible excitation sented for the case where there are four rectangularly dis- phases (— mean and −⋅− mean ± standard deviation). tributed point excitations. eTh diagram of the force positions is thesameasin Figure 3 for an infinite plate. The total inputpower is also similarto(19)exceptthatfor afinite structure the leading term is now expressed in terms of 1 × 0.8 × 0.0015 m. eTh simulation is made for forces with the input mobility for each force location. eTh refore by also asquaredistribution, where 𝐿 =𝐿 = 0.1m(seeFigure 1 2 3), and is located around the middle of the plate. eTh result assuming equal force amplitudes and equal probability of all the excitation phases, the mean power for a ni fi te plate shows typical behaviour where the variation is larger at low subjectedtofourpoint forces accordingto(23)isgiven by frequencies and decreases as 𝑘𝐿 increases. For the same force separation, Figure 9 shows the relative ̃ ̃ ̃ 𝜇 = (Re{𝑌 }+Re{𝑌 }+Re{𝑌 } standard deviation 𝑟 for the average input power over all 𝑃 11 22 33 in possible force positions and frequency bands. It can be seen (34) that the standard deviation can be estimated reasonably and + Re{𝑌 }) 𝐹 , accurately by using results for the infinite plate, except at whilst thevarianceisthe same as in (22). very low 𝑘𝐿 .Theresultfrom( 33) is close to the averages Figure 8 shows the normalised mean and the standard of the numerical results. The figure also shows that the dips deviation of the input power to a plate having dimensions clearly correspond to frequencies where the distance between Normalised power MOF Normalised power MOF Normalised power 𝜋𝑛 8 Advances in Acoustics and Vibration 5 5 0 0 −5 −5 −10 −10 𝜎 𝜎 −15 −15 −20 −20 −25 −25 0 5 10152025 0 5 10152025 Frequency (Hz) Frequency (Hz) (a) (b) Figure 9: The relative standard deviation of input power to a finite plate subjected to four harmonic point forces averaged over various force positions and frequency bands: — numerical, −⋅− infinite plate and −− (33) ((a) 𝜂 = 0.1 and (b) 𝜂 = 0.05 ). the excitation points is a multiple of half the structural forces, the maximum variance is obtained when both contact wavelength, 𝑘𝐿 = 𝑛𝜋 .However, Figure 9(b) shows that for points have equal force distribution; that is, 𝐹 =𝐹 ,and 1 2 a smaller damping, the infinite plate result is now an under- the minimum variance is when one of the forces equals the estimate of the numerical values. In this case, the prediction eeff ctive forcewhile theother is zero;thatis, 𝐹 =0; 𝐹 =𝐹 1 2 eff using (33) gives a good agreement, particularly at large 𝑘𝐿 . (see (23)). For a finite plate, the location of the excitation determines the mobility of the receiver. However, if the spatial variation is small (𝑘𝐿 < 𝜋 ), the mean input power also 5. Random Force Amplitude depends only on the receiver mobility as in (36), where 𝑌 = Besides the phases and locations of excitation, the force ̃ ̃ 𝑌 ≈ 𝑌 . 11 22 distribution at the contact points also generates variability in Figure 10(a) shows the distribution of mean and the theinput power. In practice,thiscould be duetothe condi- standard deviation of the input power due to random relative tion of the installation or the nature of the source itself. eTh phases for various force amplitudes in frequency averages, problem is that it is very dicffi ult in practice to have know- 2 2 where 𝐹 =1 N . eTh excitation points are located at (0.25𝑎, eff ledge of its distribution in sufficient detail. The situation 0.6𝑏) and (0.58𝑎, 0.6𝑏) for the same plate dimensions as in becomes more complicated still if moment excitation is also Section 4.1 with damping loss factor 𝜂 = 0.01 .Itcan be seen taken into consideration. u Th s, in theoretical studies oen, ft that at low frequency, the variation is up to 4 dB which is quite for convenience, only translational forces of equal amplitude significant. However, this reduces as the frequency increases. are assumed [8, 11]. The variation can also be reduced by increasing the damping Here, the approach is used to constrain the total square loss factorsasshown in Figure 10(c). magnitude of the forces, that is, to assume The average mean and variance from this results can be 𝑁 obtained by assuming𝐹 is uniformly probable between 0 and 2 2 𝐹 = ∑𝐹 (35) 𝐹 . Within this range, the probability density function Π can eff 𝑖 eff 𝑖=1 be assumed constant; that is, is constant.Thegoalistoobtainthe maximumand minimum Π(𝐹 )= , bounds of themeanand variance of theinput powerprovid- eff (37) ing that only the so-called eeff ctive force 𝐹 is known. From eff 2 2 𝐹 =[0,𝐹 ]. (23)for themeaninput powertoaninnfi ite plateexcited by 1 eff two point forces, Using (5), the mean and the variance of the input power duetorandomphasesaveragedoverrandompoint force 1 1 2 2 2 ̃ ̃ 𝜇 = 𝑌 (𝐹 +𝐹 )= 𝑌 𝐹 . (36) loading for the case of two contact points are given by 𝑃 𝑝 1 2 𝑝 eff in 2 2 ̃ ̃ ⟨𝜇 ⟩ = (Re{𝑌 }+Re{𝑌 })𝐹 , It canbeseenthatfor an innfi iteplate,the mean powerdoes 2 𝑃 11 22 eff in 1 4 not depend on the force distribution but rather on the charac- (38) 1 2 2 2 teristics of the receiver only. However, the variance depends ⟨𝜎 ⟩ = (Re {𝑌 }𝐹 ) . 2 12 𝑃 eff in on the product of the forces at the contact points. For two 1 12 r (dB re 1) r (dB re 1) Advances in Acoustics and Vibration 9 −15 −15 −20 −20 −25 −25 −30 −30 0 5 101520253035 0 5 10 15 20 25 30 35 kL kL (a) (b) −15 −15 −20 −20 −25 −25 −30 −30 0 5 10 15 20 25 30 35 0 5 101520253035 kL kL (c) (d) Figure 10: (a, c) eTh distribution of the mean (dark grey) and mean ± standard deviation (light grey) of the input power to a finite plate with two contact points due to random relative phases for various force amplitudes; (b, d) eTh average mean (—) and average mean ± standard deviation (−⋅−) over random force amplitudes. (a)-(b) 𝜂 = 0.01 ; (c)-(d) 𝜂 = 0.1 . Figures 10(b) and 10(d) show the average mean and 6. The Contribution of Moment Excitation standard deviation of the input power from the results in 6.1.eTh Eeff ctofMomentExcitationontheInputPower. In the Figures 10(a) and 10(c). Here, again it can be clearly seen previous sections, only the translational force is considered as that the deviation decreases with frequency and also as the the driving excitation. However, in principle the motion at a damping loss factor increases. In this example, for𝜂 = 0.1 ,the contact point of a structure-borne source would involve up to deviation of the power from the mean is insignificant above six components where not only forces, but also moments will 𝑘𝐿 = 5 (<1dB). contribute to the total input power. eTh moment excitation Figure 11 shows the mean and variance due to random is oen ft neglected partly because of measurement difficulties phases and force amplitudes for the inn fi ite plate, also with rather than the fact that, in most cases, it gives a small ̃ ̃ ̃ two contact points, where in (38),𝑌 = 𝑌 = 𝑌 .Asymptotic 11 22 𝑝 contribution to the input power [18]. eTh contribution of form of 𝑌 in (11) is used to calculate the maximum and mini- moments is most important at higher frequencies and is less 2 2 2 mum bounds (the case where 𝐹 =𝐹 =𝐹 /2). The range important than that of forces if the source is far away from 1 2 eff betweenthe bounds canbeseentoberoughly 2dBbetween discontinuities or boundaries [19]. 𝑘𝐿 = 4 and16. However, this decreasesasthe frequency Figure 12 illustrates the components of excitation increases. assumedtoact on astructure.Theresponseatthe contact Power (dB re 1 watt) Power (dB re 1 watt) Power (dB re 1 watt) Power (dB re 1 watt) 10 Advances in Acoustics and Vibration −19 −20 −21 −22 M −23 Figure 12: Six components of point excitations. −24 −25 With inclusion of the moment excitation, the mobility matrix for a single contact point is given by −26 0 2 4 6 8 10 12 14 16 18 V𝐹 V𝑀 V𝑀 𝑥 𝑦 ̃ ̃ ̃ 𝑌 𝑌 𝑌 kL ̇ ̇ ̇ [ ] 𝜃 𝐹 𝜃 𝑀 𝜃 𝑀 ̃ ̃𝑥 ̃𝑥 𝑥 ̃𝑥 𝑦 Y = 𝑌 𝑌 𝑌 , (41) [ ] Figure 11: The mean ( −−) and mean ± standard deviation (−⋅−)of ̇ ̇ ̇ 𝜃 𝐹 𝜃 𝑀 𝜃 𝑀 ̃𝑦 ̃𝑦 𝑥 ̃𝑦 𝑦 the input power to an infinite plate with two contact points due to 𝑌 𝑌 𝑌 [ ] random relative phases and random force amplitudes (— max-min mean ± standard deviation, (thick line) max-min bounds). where Y is symmetric. 6.2. Magnitude of Moment. eTh relative contribution to the input power depends of course on the magnitude of the point is a function of point mobilities, transfer mobilities excitation. Force and moment cannot be compared directly as for different axes, and also the cross-mobilities for different they have different units. In a practical situation, they would components. er Th efore, there will be a 6×6 mobility matrix also depend on the nature of the force generation mechanism for each excitation point. eTh problem becomes more com- in the source. The installation condition also has to be plicated for multiple contact points. For 𝑁 contact points, considered. eTh eeff cts of moment excitation for a vibrating the interaction between components will increase the size of machine installed on soft support at the contact points would the system matrix to 6𝑁 × 6𝑁 . be dieff rent to thoseifthe machinewas bolted tightlytothe In this paper, however, the problem is simpliefi d by receiver structure. u Th s, the problem remains of qualifying neglecting thein-planeexcitations,thatis, 𝐹 , 𝐹 ,and 𝑀 . 𝑥 𝑦 𝑧 the relative effects of force and moment. Moorhouse [ 18]pro- eTh refore, the mobility matrix is reduced to a 3×3 matrix posed a dimensionless mobility where, for example, the real for a single contact point. In general, the input power due to a combined point force and moment excitation can be part of a cross-mobility, Re{𝑌 },isnormalisedbythe real rewritten as part of the corresponding point mobilities for both the force V𝐹 √ ̃ ̃ and the moment, Re{𝑌 }Re{𝑌 }. This gives insight into 1 V𝐹 V𝑀 ∗ ̃ ̃ ̃ ̃̃ 𝑃 = (Re{𝑌 } 𝐹 + Re{𝑌 𝑀 𝐹 } therelativecontributiondue to thedieff rentexcitationcom- in ponents. (39) ̇ ̇ 2 ̃ ̃̃ ̃ ̃ + Re{𝑌 𝐹 𝑀 }+ Re{𝑌 } 𝑀 ), 6.2.1. Single Point Excitation. eTh relative importance of force and moment in exciting a structure can be compared only V𝐹 ̃ ̃ where 𝑌 and 𝑌 are the point force and point moment in terms of their input power. However, to calculate the V𝑀 ̃ ̃ power not only the mobilities should be known but also mobilities and 𝑌 and 𝑌 are the cross-mobilities from the magnitudes and the phases of the excitation components force to rotation and from moment to translation, respect- ̇ ̇ V𝑀 V𝑀 ∗ ∗ (see (39)). Petersson [20] introduced the nondimensional ̃ ̃ ̃ ̃̃ ̃ ̃̃ ively. Since𝑌 = 𝑌 ,hence Re{𝑌 𝑀 𝐹 }+Re{𝑌 𝐹 𝑀 }= eccentricity which relates the ratio of magnitude of moment V𝑀 ∗ ∗ V𝑀 ∗ ̃ ̃̃ ̃̃ ̃ ̃̃ Re{𝑌 (𝑀 𝐹 + 𝐹 𝑀 )} = 2Re{𝑌 }Re{𝑀 𝐹 },and (39)can andforce to thestructuralwavenumber. be rewritten as Here, another approach is introduced where the magni- ̃ 1 ̃ 2 tudes of the moments,𝑀=𝑀𝑒 ,and theforce,𝐹=𝐹𝑒 , 1 V𝐹 V𝑀 ∗ ̃ ̃ ̃ ̃̃ 𝑃 = (Re{𝑌 } 𝐹 +2 Re{𝑌 }Re{𝑀 𝐹 } at thecontact pointare relatedbyaneeff ctivelever arm 𝛼 by in (40) ̇ 2 𝑀=𝛼,𝐹 (42) ̃ ̃ + Re{𝑌 } 𝑀 ), where 0<𝛼<∞ .This indicatesthatif 𝛼 is very small, the V𝑀 where, for an infinite plate, Re {𝑌 }=0. structure is excited mainly by force, while if 𝛼 is very large In matrix form, the power can be expressed as in (1)where the structure is driven mainly by a moment. However, for ̃ ̃ ̃ ̃ F =[𝐹 𝑀 𝑀 ] is the vector of the force and moments. convenience, a nondimensional unit is preferred to scale the 𝑥 𝑦 Power (dB re 1 watt) 𝜃𝑀 𝑗𝜙 𝑗𝜙 𝜃𝐹 𝜃𝐹 𝜃𝐹 𝜃𝑀 𝜃𝑀 𝜃𝐹 𝜃𝑀 𝜃𝐹 Advances in Acoustics and Vibration 11 relative input power. eTh total input powers, 𝑃 and 𝑃 ,due 𝐹 𝑀 to a force and a moment on an inn fi ite plate are 1 1 ̇ V𝐹 2 2 1 ̃ ̃ 𝑃 =𝑃 +𝑃 = Re {𝑌 }𝐹 + Re {𝑌 }𝑀 , (43) 10 in 𝐹 𝑀 2 2 and the real parts of the point mobilities are given by V𝐹 Re{𝑌 }= , 8𝐵𝑘 (44) −1 ̇ 𝜔 Re{𝑌 }= , 8𝐵 V𝑀 ̃ ̃ −2 where all the cross-mobilities Re{𝑌 }and Re{𝑌 }are zero. Consequently, the relative phase between the force and the 10 moment is irrelevant. k𝛼 From (42), (43), and (44), the input power from moment Figure 13: eTh normalised input power from force ( −⋅−)and excitation can be scaled in terms of the input power from the moment (−−) excitations at a single contact point and the total force by a nondimensional unit 𝑘𝛼 and is expressed as power (—). 𝑃 = (𝑘𝛼 ) 𝑃 , (45) 𝑀 𝐹 where 𝑘 is the structural wavenumber. Equation (43)can be rewritten as M 𝑃 =((𝑘𝛼 ) +1)𝑃 . (46) in 𝐹 Figure 13 shows the normalised total input power to an infinite plate for a single contact point. It can be seen that M =Msin 𝛿 the power from force excitation is constant with frequency x M =−Mcos 𝛿 whilethe powerfrommomentexcitationisincreasingwith y frequency. Both powers intersect at 𝑘𝛼=1 .For 𝑘𝛼<1 ,the power is dominated by force excitation and for 𝑘𝛼 > 1 ,the power is dominated by moment excitation. Figure 14: eTh lever arm of the moment and force. For a finite plate receiver, the total input power is given as in (40). While the situation is now numerically complicated, (46) can again be used to scale the individual contribution to the input power. situation, the moment about the 𝑥 -axis can thus be expressed in the form 6.2.2. Multiple Point Excitation. Figure 14 shows a diagram of a translational force 𝐹 which generates moment 𝑀 that can be resolved into moments 𝑀 and 𝑀 components. eTh 𝑥 𝑦 𝑥,1 { } moments can be expressed as { } { } { } 𝑥,2 { } 𝑀 =𝐿𝐹𝛽 sin 𝛿 , { } ( ) 𝑀 𝑥 { 𝑥,3 } { } (47) { } 𝑥,4 𝑀 =−𝐹𝛽𝐿 cos(𝛿 ), 𝛼 0−𝐿 𝛽 −𝐿 𝛽 sin𝜃 1 2 3 3 4 where 𝐿 is the lever arm, or the distance from the line of [ ] 0𝛼 −𝐿 𝛽 sin𝜃−𝐿 𝛽 [ ] 2 3 3 2 4 action of 𝐹 to the point attached to the structure, 𝛿 is the = [ ] [ 𝐿 𝛽 𝐿 𝛽 sin𝜃𝛼 0 ] angle between the lever arm and the positive 𝑥 -axis, and 𝛽 2 1 3 2 3 is a dimensionless scaling factor. 𝐿 𝛽 sin𝜃𝐿 𝛽 0𝛼 [ 3 1 2 2 4 ] Equations (42)and (47) can be used to define the relation between force and moment for multiple contact points. { } { } { } Figure 15 shows the forces and moments for a typical four { } × , point contact source, with the points having a rectangular { } { } { 3} 2 2 2 { } distribution, where 𝐿 =𝐿 +𝐿 . eTh reference moment 3 1 2 { } at any contact point might then be considered as a sum of (48) contributions from forces at all the contact points. In this P /P in F 𝜃𝐹 𝜃𝑀 𝜃𝑀 12 Advances in Acoustics and Vibration V𝐹 𝜃 𝑀 z 𝑥 𝑥 ̃ ̃ y + Re{𝑌 }𝐹 𝐹 cos𝜓 + Re{𝑌 }𝑀 𝑀 cos𝜓 1 2 1 𝑥,1 𝑥,2 2 𝑡 𝑡 y,3 M y,4 x 𝜃 𝑀 3 F4 𝑦 𝑦 + Re{𝑌 }𝑀 𝑀 cos𝜓 𝑡 𝑦,1 𝑦,2 3 x,3 M x,4 V𝑀 + Re{𝑌 }(𝐹 𝑀 cos𝜓 +𝐹 𝑀 cos𝜓 ), 𝑡 2 𝑦,1 4 1 𝑦,2 5 F L 2 2 (50) x,1 where 𝑌 denotes the point mobility (the same contact point) M 𝑝 x,2 and 𝑌 denotes the transfer mobility (different contact point). The phase 𝜓 denotes the relative phase between the two y,2 y,1 components at the same or different contact points; for example, 𝜓 is the relative phase between the moment about Figure 15: eTh moment and force directions at source-receiver the 𝑦 -axis and the force at different points. In ( 50), it has interface with four contact points. V𝑀 𝜃 𝐹 𝑦 𝑦 ̃ ̃ been noted that 𝑌 = 𝑌 . Due to the complexity of this 𝑡 𝑡 expression, it is difficult to determine the bounds of input and the moments about the 𝑦 -axis are power analytically. However for simplicity, it is assumed that all the components are in-phase, so that 𝜓 =0 for 𝑖= 𝑦,1 { } 1, 2,3, 4,and 5. By also assuming 𝐹 =𝐹 =𝐹 , 𝛼 =𝛼 and { } 1 2 1 2 { } 𝑦,2 following the same method as in (48)and (49)for a 2×2 { } { } 𝑦,3 { } matrix, thus, 𝑀 =𝑀 =𝑀 =𝛼𝐹 , 𝑀 = (𝛼+𝛽)𝐿 𝐹 , 𝑥,1 𝑥,2 𝑥 𝑦,1 { 𝑦,4 } and 𝑀 =(𝛼−)𝐿𝛽 𝐹 . eTh asymptotic forms of the transfer 𝑦,2 mobility in (50) for this case can be expressed as (see also 𝛼 𝐿 𝛽 0𝐿 𝛽 cos𝜃 1 1 2 3 4 Appendix A) [ ] −𝐿 𝛽 𝛼 −𝐿 𝛽 cos𝜃0 1 1 2 3 3 [ ] [ ] 0𝐿 𝛽 cos𝜃𝛼 𝐿 𝛽 3 2 3 1 4 ̇ V𝑀 2 𝜃 𝑀 𝑦 𝑦 𝑦 ̃ ̃ 𝑌 = Re{𝑌 } −𝐿 𝛽 cos𝜃0 −𝐿 𝛽 𝛼 𝑡 𝑝 3 1 1 3 4 [ ] { } 2 𝜋 3𝜋 { } { } × (sin(𝑘𝐿 − )−𝑗 sin(𝑘𝐿 − )), × . 𝜋𝑘𝐿 4 4 { } { } { } (51a) { } (49) ̇ ̇ 2 2 𝜋 𝜃 𝑀 𝜃 𝑀 𝑥 𝑥 𝑥 𝑥 ̃ ̃ √ (51b) Re{𝑌 }= Re{𝑌 } sin(𝑘𝐿 − ), 𝑡 𝑝 𝑘𝐿 𝜋𝑘𝐿 4 The subsequent sections discuss the effect of moment excita- tion on the input power to infinite and ni fi te plates particu- ̇ ̇ 𝜃 𝑀 𝜃 𝑀 2 𝑦 𝑦 𝑦 𝑦 ̃ ̃ Re {𝑌 }= 2 Re{𝑌 } larly for the multiple point excitation. 𝜋𝑘𝐿 𝜋 1 𝜋 6.3. Inn fi ite Plate Receiver ×(cos(𝑘𝐿 − )− sin(𝑘𝐿 − )). 4 𝑘𝐿 4 6.3.1. Single Contact Point. For a single contact point, (51c) Figure 13 shows the input power as a function of 𝑘𝛼 and the relative phase between the force and moment is not By substituting (51a), (51b), and (51c)into(50)and setting important. However, for multiple contact points, the relative thecos andsin termsequal to unity, themaximum and phases are required as the result of the coupling between minimumboundsoftheinputpowernormalisedwithrespect forces and moments to the response at another contact point. to the input power from translational force (𝑃 )for in-phase excitation are found to be 6.3.2. Multiple Contact Points. As an example, for two contact points there are ee ftfi n relative phases. Assume the distance 𝐿 in (𝑘)𝐿𝛽 =1+ (𝑘𝛼 ) + between the two points is parallel with 𝑥 -axis (𝛿=0 )sothat 2𝑃 2 some transfer moment mobilities about 𝑥 -axis become zero. (52) The transfer moment mobilities are given in Appendix A.In (𝑘)𝐿𝛽 this case, the total input power for the case of two contact ± [1 + (𝑘𝛼 ) −()𝐿𝛽𝑘 + ]. 𝜋𝑘𝐿 𝑘𝐿 points is given by 1 1 ̇ V𝐹 2 2 𝜃 𝑀 2 2 𝑥 𝑥 ̃ ̃ This reduces to ( 13), the case where there is only translational 𝑃 = Re{𝑌 }(𝐹 +𝐹 )+ Re{𝑌 }(𝑀 +𝑀 ) in 𝑝 1 2 𝑝 𝑥,1 𝑥,2 2 2 force excitation, when 𝑘𝛼 ≪ 1 and ≪ 1 . In Section 3.2, assuming random phases with equal prob- 1 𝜃 𝑀 𝑦 𝑦 2 2 + Re{𝑌 }(𝑀 +𝑀 ) 𝑦,1 𝑦,2 ability in (50), the mean and the variance of the input power 𝑘𝛽𝐿 Advances in Acoustics and Vibration 13 2 Again, for 𝑘𝛼 ≪ 1 and ≪ 1 , this yields the standard deviation for force excitation (see (16)). Following the same 1.8 method in(17), the standard deviation due to uncertainty in 1.6 the dimensionless spacing 𝑘𝐿 is given by 1.4 1.2 𝜎 1 𝑘𝛼 ( ) 2 ≈ [1 + +((𝑘𝛼 ) −()𝐿𝛽𝑘 ) 2𝑃 √ 2𝜋𝑘𝐿 (𝑘𝐿 ) 0.8 (56) 1/2 2 2 0.6 +2((𝑘𝛼) +()𝐿𝛽𝑘 )] . 0.4 0.2 Figure 16 shows the mean input power and its standard 0 1 10 10 deviation. The trend is the same as in Figure 2 for transla- tional force excitation, except that the input power tends to kL increase at high frequencies due to moment excitations. Figure 16: eTh normalised input power to an infinite plate subjected to two harmonic unit point forces and two harmonic moments: −− mean, — (thick line) max/min bounds; (52), −⋅− mean ± standard 6.4. Finite Plate Receiver deviation and — mean ± bounds of standard deviation; (55), ⋅⋅⋅ mean ± bounds of standard deviation due to uncertainty in and 6.4.1. Single Contact Point. For a n fi ite plate, the phase dieff r- (56)(𝛼 = 0.003 m, 𝛽 = 0.003 ). ence between the force and moment becomes important as the cross-mobility is not zero. For the same plate dimensions as in Section 4.1, Figure 17 shows the normalised input power against 𝑘𝛼 for a single contact point assuming in-phase force to an inn fi ite plate receiver through 𝑁 contact points can, in and moment. The result in Figure 17(a) shows the increase general, be expressed as of the input power due to moment contribution at 𝑘𝛼 > 0.25 (see also Figure 13). For the case where the excitation 𝜃 𝑀 V𝐹 2 𝑥,𝑦 𝑥,𝑦 2 ̃ ̃ is near to the plate edge in Figure 17(b),the totalpower is 𝜇 = ∑(Re{𝑌 }𝐹 + Re{𝑌 }𝑀 ), (53) 𝑖𝑖 𝑖 𝑖𝑖 𝑥,𝑦 (𝑖 ) in significantly less at low 𝑘𝛼 , because the point mobility for force excitation (which dominates at low frequencies, 𝑘𝛼 ≪ 𝑁−1 𝑁 1) is smaller near the edge. However, when > 0.2 ,theinput 2 V𝐹 𝜎 = ∑ ∑(Re{𝑌 }𝐹 𝐹 ) 𝑃 𝑖𝑘 𝑖 𝑘 in power is the same as that when the excitation position is near 𝑖=1 𝑘=2 𝑘>𝑖 to thecentreofthe platedue to theincreasingpower from the moment, so that it compensates partly for the reducing power 𝑁−1 𝑁 1 𝜃 𝑀 𝑥,𝑦 𝑥,𝑦 from the force. + ∑ ∑(Re{𝑌 }𝑀 𝑀 ) 𝑥,𝑦(𝑖) 𝑥,𝑦(𝑘) 𝑖𝑘 2 Figure 18 shows the normalised input power for various 𝑖=1 𝑘=2 𝑘>𝑖 forcinglocationsontheplate.Theincreaseinthemeanpower (54) due to the contribution of moment excitation can be seen 𝑁−1 𝑁 𝜃 𝐹 𝑥,𝑦 roughly above > 0.35 . + ∑ ∑(Re{𝑌 }𝐹 𝑀 ) 𝑖 𝑥,𝑦(𝑘) 𝑖𝑘 Figure 19 shows the relative standard deviation 𝑟 of the 𝑖=1 𝑘=2 𝜎 𝑘>𝑖 averaged input power for different damping loss factors and magnitudes of moment excitation. For all cases, it can be 𝑁−1 𝑁 1 V𝑀 𝑥,𝑦 ̃ seen that the relative standard deviation, in an average sense, + ∑ ∑(Re{𝑌 }𝑀 𝐹 ) , 𝑥,𝑦(𝑖) 𝑘 𝑖𝑘 agrees reasonably well with that from the translational force 𝑖=1 𝑘=2 𝑘>𝑖 from (33). This indicates that the ratio between the mean and standard deviation is approximately the same even if the where 𝑖 and 𝑘 indicate the 𝑖 th and 𝑘 th contact points, moment excitation is neglected in the calculation of the input respectively. power. Particular attention is focused on the results at large eTh bounds of the normalised standard deviation can be 𝑘𝛼 when the moment starts to contribute substantially to the obtained by substituting (51a), (51b), and (51c)into(54). After total input power. algebraic manipulation, it can be approximated by 6.4.2. Multiple Contact Points. Again, therelativephases 𝜎 1 (𝑘𝛼 ) ≈ [1 + +((𝑘𝛼 ) −()𝐿𝛽𝑘 ) due to coupling between forces and moments are of interest 2𝑃 √ 𝜋𝑘𝐿 (𝑘𝐿 ) for multiple contact points. eTh mean power, assuming the (55) 1/2 relative phases between the excitations are equally probable, is thesameasthatin(53) for an inn fi ite plate. However, for +2((𝑘𝛼 ) +()𝐿𝛽𝑘 )] . afinite plateaforcewillproduce arotationand amoment Normalised power 𝑘𝛼 𝑘𝛼 𝑘𝐿 𝑘𝛽𝐿 14 Advances in Acoustics and Vibration 2 2 10 10 1 1 10 10 0 0 10 10 −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −1 0 −1 0 10 10 10 10 k𝛼 k𝛼 (a) (b) Figure 17: eTh normalised input power of a finite plate subjected to force and moment excitations at a single contact point ((a) the power with (−−) and without (—) moment and (b) the total power for the contact point around the edge (−−) and middle (—) of the plate: 𝛼 = 0.005 m, 𝜂 = 0.1 ). 2 2 10 10 1 1 10 10 0 0 10 10 −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 −1 0 −1 0 10 10 10 10 k𝛼 k𝛼 (a) (b) Figure 18: eTh normalised input power to a finite plate subjected to force and moment excitations at single contact point for various possible forcing locations (grey lines): — mean and −⋅− mean ± standard deviation. (a) discrete frequency and (b) frequency-band average; 𝛼= 0.005 m, 𝜂 = 0.1 . 𝑁 𝑁 will produce a displacement at the same point. er Th efore, the 1 (𝑖𝑘) + ∑ ∑(Re{𝑌 }𝐹 𝑀 ) 𝑖 𝑥,𝑦(𝑘) variance is given by 𝜃 𝐹 𝑥,𝑦 𝑖=1 𝑘=1 𝑘≥𝑖 𝑁 𝑁 (𝑖𝑘) + ∑ ∑(Re{𝑌 }𝑀 𝐹 ) . 𝑁−1 𝑁 V𝑀 𝑥,𝑦(𝑖) 𝑘 2 𝑥,𝑦 2 (𝑖𝑘) 𝑖=1 𝑘=1 𝜎 = ∑ ∑(Re{𝑌 }𝐹 𝐹 ) 𝑃 V𝐹 𝑖 𝑘 in 𝑘≥𝑖 𝑖=1 𝑘=2 𝑘>𝑖 (57) 𝑁−1 𝑁 For four contact points, the mobility matrices are 12 × (𝑖𝑘) + ∑ ∑(Re{𝑌 }𝑀 𝑀 ) 𝑥,𝑦(𝑖) 𝑥,𝑦(𝑘) ̇ 12.Using (53)and (57), Figure 20(a) shows the mean and 𝜃 𝑀 𝑥,𝑦 𝑥,𝑦 𝑖=1 𝑘=2 standard deviation of the input power for damping loss factor 𝑘>𝑖 Normalised power Normalised power Normalised power Normalised power Advances in Acoustics and Vibration 15 5 5 0 0 −5 −5 𝜎 𝜎 −10 −10 −15 −15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k𝛼 k𝛼 (a) (b) −5 −10 −15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k𝛼 (c) Figure 19: eTh relative standard deviation of the input power to a finite plate subjected to force and moment excitations at single contact point averaged over various possible forcing locations and frequency bands; — numerical and −− (33) ((a) 𝛼 = 0.005 m, 𝜂 = 0.1 ;(b) 𝛼 = 0.005 m, 𝜂 = 0.01 ;and (c) 𝛼 = 0.01 m, 𝜂 = 0.01 ). 3 10 2.5 1.5 −5 −10 0.5 0 −15 0 5 10 15 20 25 0 5 10 15 20 25 kL kL (a) (b) Figure 20: (a) The normalised input power to a finite plate subjected to force and moment excitations at four contact points averaged over various possible forcing locations and frequency bands: mean (— numerical calculation, −− infinite plate) and mean ± standard deviation (⋅⋅⋅ numerical calculation, −⋅− infinite plate). (b) The relative standard deviation of the input power; — numerical calculation, −⋅− infinite plate, and −− (33)(𝐿 = 0.14 m, 𝜂 = 0.05 , 𝛼 = 0.005 m, and 𝛽 = 0.005 ). Normalised power r (dB re 1) r (dB re 1) r (dB re 1) r (dB re 1) 16 Advances in Acoustics and Vibration 10 10 8 8 6 6 0 0 𝜎 𝜎 −2 −2 −4 −4 −6 −6 −8 −8 −10 −10 0 2 4 6 8 10 12 0 10 20 30 40 50 60 kL kL (a) (b) Figure 21: eTh relative standard deviation of the input power to a finite plate subjected to force and moment excitations at four contact points averaged over various possible forcing locations and frequency bands; — numerical calculation and −− (33)(𝜂 = 0.01 , 𝛼 = 0.005 m, and 𝛽 = 0.005 ;(a) 𝐿 = 0.07 mand (b) 𝐿 = 0.35 m). 𝜂 = 0.05 . eTh spatial separation of the contact points is again agreement deteriorates and the simple prediction of the mean assumedtoformarectangularshape and 𝐿 is the length and variance can be used [1, 17]. of the diagonal. eTh results agree well with those from the eTh uncertaintyinthe forceamplitude at thecontact infinite plate above 𝑘𝐿 = 10 .Below this,asin Section 4.4,the points has also been discussed. Unless the spatial separation agreement deteriorates due to small damping. This is clearly of the excitation locations is small, the distribution of the shown in the relative standard deviation plotted in Figure force amplitude through the contact points is important to 20(b). However, it can be seen that the numerical result has a obtain accurate estimates of the variation of the mean and good agreement with that from the prediction using (33). standard deviation of the input power, particularly at low Figure 21 shows the relative standard deviation for 𝜂= frequencies. This variation reduces as the damping loss factor 0.01 for different distances between the contact points. Again, is increased. good agreement can be seen using (33), particularly for high The relative effect of moment excitation can be expressed frequencies. In Figure 21(b) differences are seen for 𝑘𝐿 > 25 , in terms of a force and a distance corresponding to a char- but they are less than 1 dB. At low 𝑘𝐿 ,the prediction dieff rs acteristic of the source. It can also be scaled as a function of by 2 dB on average due to the very low-modal overlap. From the input power of the force and the structural wavenumber. the results presented, it shows that (33), which is applicable Thiseeff ct tendstoincreaseasfrequency increases. eTh only for the translational force, can also be used to predict contribution to the total input power can be predicted using the contribution of moment at high frequencies. the simple expression of the relative standard deviation for the force. However in any event, the effects of moment excitation are typically small at low frequency and in any 7. Conclusions event are generally less than the effects of force excitation. The uncertainty in input power to a structure due to uncer- They are typically, thus, of secondary importance. tainty in the excitation has been investigated. For an infinite Finally, there remains the moot point of what uncertainty plate, the distance between the location where multiple is, in practice, acceptable. This is to a large extent dependent forces are applied is not important if it is less than half a on the typical uncertainty of machinery characterisation structural wavelength. eTh variance of the input power due to methods, such as the reception plate method. eTh attempt uncertainty in excitation phase and location tends to decrease here is to quantify to some extent the uncertainty introduced as the nondimensional frequency 𝑘𝐿 increases. For multiple by some details of the excitation, details that would typically point excitation where the relative phases are random, the not be measured. mean power and the variance depend only on the input and transfer mobility, respectively. Appendices As for the infinite plate, the variance of the input power to a n fi ite plate also typically decreases as the frequency A. Force and Moment Transfer Mobilities for increases. The frequency average of the input power over all an Infinite plate possible forcing locations from multiple contact points can be estimated reasonably and accurately by using the infinite Figure 14 defines the force-moment excitation directions. eTh plate result. However, for a very low damping (<5%) the same directions arealsoapplied to theresponseatanother r (dB re 1) r (dB re 1) Advances in Acoustics and Vibration 17 point at distance 𝐿 away from the excitation point. eTh 𝑧 -axis B. Force and Moment Mobilities for is perpendicular to the surface of the plate. The mobility terms a Finite Plate of an inn fi ite plate structure subjected to a harmonic force or For a n fi ite rectangular plate, the mobilities can be written moment point loading are given by [15] in terms of a modal summation. eTh point reference (0,0) is located at the corner of the plate. eTh moment-rotational 𝜔 2𝑗 V𝐹 (2) ̃ velocity transfer mobilities at frequency 𝜔 for a plate with (A.1a) 𝑌 = [𝐻 (𝑘𝐿 )− 𝐾 (𝑘𝐿 )], 8𝐵𝑘 𝜋 damping loss factor 𝜂 are given by [15] ̇ 𝜔 sin𝛿 2𝑗 V𝑀 𝜃 𝐹 (2) 𝑥 𝑥 ̃ ̃ (A.1b) 𝑌 = 𝑌 = [𝐻 (𝑘𝐿 )− 𝐾 (𝑘𝐿 )], 1 1 ∞ ∞ (𝜕/𝜕𝑦)[Φ (𝑥 ,𝑦 )](𝜕/𝜕𝑦)[Φ (𝑥,𝑦)] 8𝐵𝑘 𝜋 𝑝𝑞 0 0 𝑝𝑞 𝜃 𝑀 𝑥 𝑥 𝑌 =𝑗𝜔 ∑ ∑ , 2 2 𝜔 (1+𝑗𝜂)−𝜔 2𝑗 ̇ 𝜔 cos𝛿 𝑝=1 𝑞=1 𝑝𝑞 V𝑀 𝜃 𝐹 (2) 𝑦 𝑦 ̃ ̃ 𝑌 = 𝑌 = [𝐻 (𝑘𝐿 )− 𝐾 (𝑘𝐿 )], (A.1c) 1 1 (B.1a) 8𝐵𝑘 𝜋 ̇ 2𝑗 𝜃 𝑀 2 (2) ∞ ∞ 𝑥 𝑥 ̃ (𝜕/𝜕𝑥 )[Φ (𝑥 ,𝑦 )](𝜕/𝜕𝑥 )[Φ (𝑥,𝑦)] 𝑌 = [sin 𝛿(𝐻 (𝑘𝐿 )+ 𝐾 (𝑘𝐿 )) ̇ 𝑝𝑞 0 0 𝑝𝑞 0 𝜃 𝑀 𝑦 𝑦 8𝐵 𝜋 𝑌 =𝑗𝜔 ∑ ∑ , 2 2 𝜔 (1+𝑗𝜂)−𝜔 𝑝=1 𝑞=1 𝑝𝑞 2 2 + (cos 𝛿− sin 𝛿) (A.1d) (B.1b) 𝑘𝐿 ∞ ∞ (𝜕/𝜕𝑦)[Φ (𝑥 ,𝑦 )](𝜕/𝜕𝑥 )[Φ (𝑥,𝑦)] 2𝑗 ̇ 𝑝𝑞 0 0 𝑝𝑞 (2) 𝜃 𝑀 𝑦 𝑥 ×(𝐻 (𝑘𝐿 )− 𝐾 (𝑘𝐿 ))] , 𝑌 =−𝜔𝑗 ∑∑ , 2 2 𝜔 (1+𝑗𝜂)−𝜔 𝑝=1 𝑞=1 𝑝𝑞 2𝑗 (B.1c) ̇ 𝜔 𝜃 𝑀 2 (2) 𝑦 𝑦 𝑌 = [cos 𝛿(𝐻 (𝑘𝐿 )+ 𝐾 (𝑘𝐿 )) 0 0 8𝐵 𝜋 ∞ ∞ 𝜕/𝜕𝑥 [Φ (𝑥 ,𝑦 )](𝜕/𝜕𝑦)[Φ (𝑥,𝑦)] ( ) 𝑝𝑞 0 0 𝑝𝑞 𝜃 𝑀 𝑥 𝑦 𝑌 =−𝜔𝑗 ∑∑ , 2 2 2 2 + (sin 𝛿− cos 𝛿) (A.1e) 𝜔 (1+𝑗𝜂)−𝜔 𝑝=1 𝑞=1 𝑝𝑞 𝑘𝐿 (B.1d) 2𝑗 (2) ×(𝐻 (𝑘𝐿 )− 𝐾 (𝑘𝐿 ))] , and the force-rotational and the moment-translational veloc- 𝜃 𝑀 ity transfer mobilities are 𝑦 𝑥 ̃ ̃ 𝑌 = 𝑌 𝑀 𝜃 𝑦 𝑥 ∞ ∞ 𝜔 2𝑗 Φ (𝑥 ,𝑦 )(𝜕/𝑦)[ 𝜕 Φ (𝑥,𝑦)] (2) ̇ 𝑝𝑞 0 0 𝑝𝑞 𝜃 𝐹 = [−sin𝛿 cos𝛿(𝐻 (𝑘𝐿 )+ 𝐾 (𝑘𝐿 )) ̃ 0 𝑌 =𝑗𝜔 ∑ ∑ , (B.2a) 2 2 8𝐵 𝜋 𝜔 (1+𝑗𝜂)−𝜔 𝑝=1 𝑞=1 𝑝𝑞 2𝑗 (2) + cos𝛿 sin𝛿(𝐻 (𝑘𝐿 )− 𝐾 (𝑘𝐿 ))], ∞ ∞ 1 1 Φ (𝑥 ,𝑦 ) 𝜕/𝜕𝑥 [Φ (𝑥,𝑦)] ( ) 𝑘𝐿 𝜋 𝑝𝑞 0 0 𝑝𝑞 𝜃 𝐹 𝑌 =−𝜔𝑗 ∑ ∑ , (B.2b) (A.1f) 2 2 𝜔 (1+𝑗𝜂)−𝜔 𝑝=1 𝑞=1 𝑝𝑞 ∞ ∞ (𝜕/𝜕𝑦)[Φ (𝑥 ,𝑦 )]Φ (𝑥,𝑦) where 𝐵 is the bending stiffness, 𝑘 is the structural wavenum- 𝑝𝑞 0 0 𝑝𝑞 V𝑀 𝑌 =𝑗𝜔 ∑ ∑ , (B.2c) (2) 2 2 ber, 𝐻 is the 𝑛 th-order Hankel function of the second 𝜔 (1+𝑗𝜂)−𝜔 𝑝=1 𝑞=1 𝑝𝑞 kind, and 𝐾 is the 𝑛 th-order modified Bessel function of ∞ ∞ the second kind. eTh asymptotic forms of the functions for (𝜕/𝜕𝑥 )[Φ (𝑥 ,𝑦 )]Φ (𝑥,𝑦) 𝑝𝑞 0 0 𝑝𝑞 V𝑀 𝑘𝐿 ≫ 1 are given by [16] 𝑌 =−𝜔𝑗 ∑∑ , (B.2d) 2 2 𝜔 (1+𝑗𝜂)−𝜔 𝑝=1 𝑞=1 𝑝𝑞 (2) −𝑗(𝑘𝐿−(1/2)𝑛𝜋−(1/4)𝜋) where Φ is themassnormalisedmodeshape and 𝜔 is the 𝑝𝑞 𝑝𝑞 √ (A.2) 𝐻 ≈ 𝑒 , natural frequency of the (𝑝, 𝑞) mode as defined in ( 28)for 𝜋𝑘𝐿 a simply supported boundary condition. eTh point mobility −𝑘𝐿 can be obtained by setting𝑥=𝑥 and𝑦=𝑦 . 𝐾 (𝑘𝐿 )≈ 𝑒 0 0 2𝑘𝐿 𝜖−1 (𝜖−1 )(𝜖−9 ) Acknowledgments ×[1+ + (A.3) 8𝑘𝐿 2! 8𝑘𝐿 ( ) The authors gratefully acknowledge the na fi ncial support provided by the Engineering and Physical Sciences Research (𝜖−1 )(𝜖−9 )(𝜖−25 ) + + ⋅⋅⋅], 3 Council (EPSRC) under Grant EP/D002 15X/1. The authors 3!(8𝑘𝐿 ) also acknowledge the valuable discussions with Professor B. M. Gibbsfromthe University of Liverpooland Dr.A.T. where𝜖=4𝑛 .For very large 𝑘𝐿 , 𝐾𝑛(𝑘𝐿) → 0 . Moorhouse from the University of Salford. 18 Advances in Acoustics and Vibration References [18] A. T. Moorhouse, “A dimensionless mobility formulation for evaluation of force and moment excitation of structures,” Jour- [1] R. S. Langley and A. W. M. Brown, “eTh ensemble statistics of nalofthe Acoustical SocietyofAmerica,vol.112,no. 3, pp.972– the energy of a random system subjected to harmonic excita- 980, 2002. tion,” Journal of Sound and Vibration,vol.275,no. 3–5, pp.823– [19] S. H. Yap and B. M. 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