A SIMPLE HYBRID APPROACH ON PREDICTING THE VIBRATION TRANSMISSION OF RIB-STIFFENED PLATES

Transcription

1 Lin Ji Marko Jokić Zhenyu Huang Neven Alujević SSN A SMPLE HYBRD APPROACH ON PREDCNG HE VBRAON RANSMSSON OF RB-SFFENED PLAES Summary UDC 534: Accurate redictions of viration transmission of ri-stiffened lates can e quite rolematic in that oth long- and short-wavelength deformations are commonly generated within the same system model. n this case, it is widely acknowledged that hyrid methods which can comine oth the deterministic and statistical modelling techniques are more aroriate than single deterministic or statistical methods. n this aer, a simle hyrid technique, i.e., the so-called mode-ased aroach, is alied to redict the viration transmission of a ri-stiffened late model. he results are then comared with those of the well-estalished hyrid FE/SEA method and the exact modal analysis. t is found that the mode-ased aroach has a etter erformance than the hyrid FE/SEA method regarding oth accuracy and efficiency, although the former is far less generic than the latter at the current stage. Key words: mid-frequency, mode-ased aroach, hyrid FE/SEA method, ri-stiffened late, viration transmission. 1. ntroduction Accurate redictions of viration transmission of ri-stiffened lates can e quite rolematic in that long- and short-wavelength deformations are often generated simultaneously within the same system model. hese rolems are generally defined as the mid-frequency issue [1-2]. t is widely acknowledged [3] that, for the mid-frequency region, instead of a single deterministic method (e.g. the finite element method [4]) or statistical methods (e.g. the statistical energy analysis [5]), hyrid methods are more aroriate, e.g. as those descried in [6-9]. Among the currently existing mid-frequency hyrid methods, one of the mostly develoed and widely alied is the so-called hyrid FE/SEA method [10-12]. he main theoretical ody of the method is so much well develoed that it has een used to generate the mid-frequency module of the commercial software VA-One [13] for viro-acoustic redictions of comlex engineering systems. However, ecause of the way of descriing the susystem interface adoted in the hyrid modelling and the key theoretical assumtion which underins the theoretical develoment, the hyrid FE/SEA method is found to e time- RANSACONS OF FAMENA XXXV-2 (2013) 29

2 L. Ji, M. Jokić, Z. Huang, N. Alujević A Simle Hyrid Aroach on Predicting the Viration ransmission of Ri-Stiffened Plates consuming in the case of a large numer of interface degrees of freedom (DOFs) etween neighouring susystems [9], and also, significant discreancy may e generated etween the hyrid redictions and the measured data [14] if the SEA susystems are of insufficient randomness. n these cases, more efficient and accurate hyrid techniques may e more desirale. With this in mind, in the resent aer another hyrid modelling technique, namely, the mode-ased aroach, which was develoed in [15], is used to redict the viration transmission of a simle ri-stiffened late, as shown in Fig. 1, of which, the eam is assumed to e much stiffer than the late. f an external harmonic force is alied at a articular oint of the eam, the system will exhiit significant mid-frequency viration ehaviour over a large frequency range. he erformance of the mode-ased aroach can then e valued y comaring the results with those from the hyrid FE/SEA method and the exact modal analysis [6, 15]. he aer is organized as follows. First, the main rinciles of the two hyrid modelling techniques are riefly exlained in Section 2, with their major theoretical differences eing discussed. hen in Section 3, a simle ri-stiffened late model with a lineconnection is set u, and the ower transmission of the system model is calculated y oth hyrid techniques and comared to the results from the exact modal analysis. Finally, conclusions are deduced in Section Main theoretical rinciles of the two hyrid modelling techniques n this section, only the main theoretical rinciles of and the major differences etween the two hyrid models are riefly recaitulated. Detailed theoretical derivations and the main results are not given for the revity of the aer, as they can e found very conveniently in Refs. [10] and [15]. Nevertheless, the main theoretical results of the two methods corresonding to a simle eam-late couling model are given so as to enale reading without difficulty. 2.1 Hyrid FE/SEA method n the hyrid FE/SEA method [10], long-wavelength susystems are descried y deterministic FEA models, while the short-wavelength ones are descried y the statistical SEA models. Each individual SEA susystem is further divided into a direct field and a revererant field in that the former accounts for the in-going waves without reflections while the latter for the out-going waves due to the reflections from its oundaries. A diffuse field recirocity relation is then estalished etween the revererant force loading on the deterministic oundaries of the SEA susystem and the energy resonse of the susystem under the assumtion of the non-deterministic oundaries of the susystem eing sufficiently random [11]. his thus allows a statistical relation to e formed etween the dynamic resonse of the FE susystems and the energy resonse of the SEA susystems, uon which the whole set of hyrid FE/SEA equations are uilt-u [12]. t is seen that the key assumtion which underins the hyrid FE/SEA theory is that each SEA susystem needs to e sufficiently random so that the diffuse field recirocity relation can e satisfied. However, there are no exlicit measures on determining the randomness level of a articular susystem. his difficulty can sometimes make the validity range of the hyrid FE/SEA redictions, to a certain degree, amiguous. t is also worth noting that all the interfaces etween susystems (including those etween SEA susystems) are defined as arts of the deterministic models in [10]. When the susystem interfaces contain a large numer of degrees of freedoms (DOFs), e.g. for continuous connections etween the FE and SEA susystems, the matrix size of the 30 RANSACONS OF FAMENA XXXV-2 (2013)

3 A Simle Hyrid Aroach on Predicting the Viration ransmission of Ri-Stiffened Plates L. Ji, M. Jokić, Z. Huang, N. Alujević deterministic model can still e extremely high. n this case, high comutational cost also seems unavoidale [9, 16]. 2.2 Mode-ased method he mode-ased aroach [15] descries susystem comonents in a very similar way to the hyrid FE/SEA theory in that the long-wavelength comonents are descried deterministically while the short-wavelength comonents are aroximated statistically. But in the mode-ased aroach, instead of artitioning each SEA susystem into a direct field and a revererant field, each SEA susystem is simly aroximated as a set of sinusoidal waves with randomly varied hases. Because revious research has shown that the higher ordered mode shaes functions tend to ehave as simle standing waves regardless of the exact oundary conditions and shaes of the structure [17-19], it rovides a solid theoretical foundation to simulate the mode shaes of the short-wavelength receiver as sinusoidal waves. Another significant difference etween the two hyrid techniques is that in the modeased aroach the interfaces etween the long- and short-wavelength susystems are treated aroximately y sets of generalized interface modes [15, 20] rather than as a art of the deterministic model. Previous research reveals that, y choosing the interface mode set roerly, the numer of the interface modes required for a continuous couling interface can e much less than the numer of interface DOFs [20]. herefore, y rojecting the hysical interface DOFs into the sace of generalized interface modes, the size of the generated hyrid model can e dramatically reduced [15, 21]. Desite of the two ovious advantages of the mode-ased aroach over the hyrid FE/SEA method, the former aroach was relatively much less generalized than the latter. 2.3 heoretical alication to a simle ri-stiffened late model n order to demonstrate the alication of the aove resented two hyrid modelling techniques, a simle ri-stiffened late structure, as shown in Fig. 1, is set u in the first instance. A stiff eam (ri) is wholly attached to a flexile late. When external excitations are alied directly onto the eam, the eam-late structure will ossess tyical midfrequency viration ehaviour for a quite large frequency range. n this case, either the hyrid FE/SEA theory or the mode-ased aroach is aroriate to e emloyed to redict the ower transmitted within the system. he relevant equations are given elow Power transmission from the eam to the late ased on the hyrid FE/SEA theory Under the condition of the late susystem eing sufficiently random, a statistical ower alance equation can e set u for the late in the sense of ensemle- and timeaverages [10], as E P. (1) ext, in, n the aove equation, E is the statistical energy resonse of the late, and and, are the daming loss factor of the late and the ower transfer coefficient etween the late ext and the eam, resectively. On the right side of the equation, P in, reresents the ower injected into the late y the external force acting on the eam. hese arameters can e calculated as 2 m 1 ( ) 1*, m, rs tot dir tot n rs D D D D, (2) rs RANSACONS OF FAMENA XXXV-2 (2013) 31

4 L. Ji, M. Jokić, Z. Huang, N. Alujević A Simle Hyrid Aroach on Predicting the Viration ransmission of Ri-Stiffened Plates P 1 1* m D D S D, (3) ext in, dir, rs tot ff tot 2 rs tot dir rs D D D, (4) where, oth r and s are integers, reresenting the degrees of freedom (DOFs) of the eam, ( ) and n and D dir are the modal density and direct field dynamic stiffness matrix of the late, resectively, while D d is the dynamic stiffness matrix of the eam when it is uncouled from the late. he suerscrits and reresent comlex conjugate and matrix transformation, resectively. S in Eq. (3) is the ensemle-averaged cross-sectral matrix of the external excitation force vector f, i.e., where, ff S ff E ff, (5) E reresents ensemle average. Sustituting Eqs. (4)-(5) into Eqs. (2)-(3), then into Eq. (1), the statistical energy resonse of the late E can e determined. hen, y the statistical energy theory descried in [5], the transmitted ower from the eam to the late can e calculated as P E tr, P 1 ext in,, t can e seen clearly from Eq. (4) that the hyrid FE/SEA redicting rocedure can still e quite time-consuming if the eam-late interface contains a large numer of DOFs due to the corresonding large matrix size of D. tot Power transmission from the eam to the late ased on the mode-ased aroach n the mode-ased aroach [15], if the dynamics of the interface etween the source (eam) and the receiver (late) are descried y a set of generalized interface modes k, set of interface modes along the interface x, the corresonding generalized interface forcecoordinates and dislacement coordinates can then oth e written in the matrix forms as 1,k,n e,n f Α Α α Y f, (7) 1,k,n e,n w Α Α Α α Y f. (8) n the aove equations, α and α are oth matrices, whose elements can e determined as x x dx, (9) nk, n, k, V x x dx, (10) mk, m, k, V (6) 32 RANSACONS OF FAMENA XXXV-2 (2013)

5 A Simle Hyrid Aroach on Predicting the Viration ransmission of Ri-Stiffened Plates L. Ji, M. Jokić, Z. Huang, N. Alujević where n, and m, are the mode shae functions of the eam and the late at the local interface locations x and x, resectively, and and reresent transformation matrices which relate the local coordinates of x, x and x with Furthermore, matrices Α and Α are given y,n x x and x x. Α α Y α, (11) Α α Y α. (12),m n the aove two equations, Y,n and Y,m are oth diagonal matrices whose diagonal elements are determined y the corresonding modal recetances of the uncouled susystems Y n, and Y m,, resectively, i.e., Y Y n, m 2 2 n, n, 1 jn, m, m 2 2 m, m, 1 jm, 1 1, (13), (14) where, n, and m, are the n th and the mth natural frequencies of the uncouled source and receiver susystems, resectively, while m n,, n, and m m,, m, are the corresonding modal mass and modal daming loss factors, resectively. Finally, f e,n in Eqs. (7)-(8) is a column vector comosed of the modal forces of the eam. he nth modal force of the eam can e determined as f F x x dx. (15) e e e en, e n, e V For instance, if a unit harmonic force acting at oint of the eam, the corresonding modal force of the eam, y Eq. (15), can e determined as fen, n,, where n, is the nth mode shae function of the eam. he time-averaged ower transmitted from the eam (source) to the late (receiver) can thus e exressed as 1 * Ptr, Re j w, k f, k. (16) 2 k t can e seen clearly from Eqs. (7)-(8) that, y introducing a set of generalized interface modes, the mode-ased aroach can reduce the large interface DOFs into generalized interface coordinates. Meanwhile, y statistically aroximating the receiver as a simle standing wave system, the generated modes are to e easily estimated ased on the wavelength within a susystem, while the natural frequencies are to e estimated from the free wavenumer within the susystem. RANSACONS OF FAMENA XXXV-2 (2013) 33

6 L. Ji, M. Jokić, Z. Huang, N. Alujević A Simle Hyrid Aroach on Predicting the Viration ransmission of Ri-Stiffened Plates 3. Numerical investigations n Fig. 1, a stiff eam is wholly attached to a flexile late with a unit harmonic force acting at the oint ξ of the eam. For simlicity, the eam is assumed to have oth ends free, while the late is assumed to have four edges simly suorted. Moreover, the neutral axis of the eam is assumed to e lying in the mid-lane of the late so that the dynamic effects arising from the longitudinal couling of the system can e taken as relatively very small comared to those of the flexural couling. Consequently, the system viration can e simly treated as involving only flexural wave motions. Firstly, the time-averaged ower transmission from the eam to the late is redicted y the following methods: (1) the exact modal analysis, (2) the mode-ased aroach, and (3) the hyrid FE/SEA method. Comarisons are then made for the three sets of results in terms of oth the accuracy and the comutational cost. j t Fe e w w y y 1 x x 1 Fig. 1 A eam-late model with line-connections For method (1), the modal roerties of oth the (free-free) eam and the (simly suorted thin rectangular) late can e determined y their analytical exressions [18]. his then allows the exact exressions for the frequency resonse functions (FRFs) of the eam and the late to e formed [19]. By means of the dislacement continuity and force equilirium conditions along the interface, the whole resonse of the system model can then e redicted in an exact manner. For methods (2) and (3), the relevant equations used in the calculations have een rovided in Susections and 2.3.2, resectively. Moreover, during the calculations of Methods (1) and (3), the line interface etween the eam and the late is simulated y uniformly distriuted oints which are uniformly saced at no more than a quarter wavelength (of the late) aart along the couling line. However, in Method (2), the line interface is reresented y using the first 20 ending modes of the free-free eam to form the sace of the generalized interface modes, while the mode shae functions of the late are simulated as doule sinusoidal functions ut with aritrary random hases. Here it should e noted that, ecause the hyrid FE/SEA equations were derived in the sense of ensemle averages [10], frequency averages need to e taken from the results of the mode-ased aroach and those of the exact method in order to comare the three sets of results roerly. he andwidth emloyed here for the frequency average is 20Hz. 3.1 Model descrition n Fig. 1, for simlicity, the eam is assumed to have oth ends free, while the late is assumed to have four edges simly suorted. Also, the calculation rocedure only involves the ending motions of the system. he system material roerties are given in ale 1 and the dimensional sizes of the eam and the late as well as their interface locations are listed in ale RANSACONS OF FAMENA XXXV-2 (2013)

7 A Simle Hyrid Aroach on Predicting the Viration ransmission of Ri-Stiffened Plates L. Ji, M. Jokić, Z. Huang, N. Alujević he frequency range of interest is u to 1000 Hz, which comoses aout 6 ending modes of the eam, and 285 and 115 ending modes for the lates with the late thickness of 2 mm and 5 mm, resectively. Oviously, all the three couling cases involved ossess tyical mid-frequency viration ehaviour for a large art of the frequency range of interest. 3.2 Numerical results he statistical ower transmission from the eam to the late calculated y the two hyrid techniques and the exact analytical theory are comared in Figs. 2-3, corresonding to the two couling cases of / 4, 37 and / 2, 76, resectively. n the figures, the real lines are given y the exact modal analysis, and the dotted lines corresond to the results of the mode-ased aroach, while the dashed lines are calculated y the hyrid FE/SEA method. t is seen that most of the significant eaks and troughs of the accurate resonse (real line) of the system have een well-catured y the hyrid solution (the dashed line) in Fig. 2, ut in Fig. 3, the agreement etween the real and the dash lines can e oserved as generally oor. t is thus shown clearly that the redicting accuracy of the hyrid FE/SEA method is much etter in Fig. 2 than in Fig. 3. his is ecause the late in Fig. 2 is much more flexile than that used in Fig. 3, and consequently, the late in Fig. 2 is more ale to meet the sufficient randomness condition than the one in Fig. 3. herefore, one can deduce that, to aly the hyrid FE/SEA theory successfully to ri-stiffened late systems, a late comonent with sufficient randomness is essential. However, comaring the redictions of the mode-ased aroach (dotted line) and the exact solutions (real lines) in oth Fig. 2 and Fig. 3, it can e seen that the mode-ased aroach can make a fairly good rediction of the transmitted ower even for the relatively lower frequency range. For instance, a good agreement etween the dotted and the real lines can e oserved in Fig. 3 from a frequency region lower than 200Hz, where the hyrid FE/SEA solution nearly loses its validity comletely. t is thus clearly revealed that the modeased aroach has a much etter accuracy than the hyrid FE/SEA method, esecially for the frequency region where the late randomness level is generally low. his can e reasonaly exlained in the way that the simle standing-wave aroximation used in the mode-ased aroach for the flexile late susystem is relatively much easier to e sustained than the sufficient randomness assumtion used in the hyrid FE/SEA theory. As a result, the alication condition of the mode-ased aroach can e relatively much more relaxed than that of the hyrid FE/SEA method. Moreover, when calculating the data in Fig. 2, the comutational cost of the modeased aroach (0,4721 min) was found only aout 50% of that of the hyrid FE/SEA method (0,9534 min), and 8% of the exact method (5,6167 mins). But this time-saving advantage of the mode-ased aroach ecomes weaker when the late gets stiffer. When calculating the data in Fig. 3, for instance, the time taken y the mode-ased aroach, the hyrid FE/SEA method and the exact solution are 0,1529 min, 0,2813 min and 1,2068 mins, resectively. n this case, the comutational cost of the mode-ased aroach ecomes 55% and 13% of those of the hyrid FE/SEA method and the exact technique, resectively. One can thus deduce that the mode-ased aroach ecomes more accurate and more efficient as the late wavelengths ecome shorter. ale 1 Material roerties of the eam/late model in Fig. 1 Young s modulus, Poisson s ratio Loss factor Density, kg m -3 GPa 210 0,3 0, RANSACONS OF FAMENA XXXV-2 (2013) 35

8 L. Ji, M. Jokić, Z. Huang, N. Alujević A Simle Hyrid Aroach on Predicting the Viration ransmission of Ri-Stiffened Plates ale 2 Dimensions of the eam-late model and the excitation location on the eam Structure Beam Plates Dimensions, m Length L 2, 0, Length: L x 2, 0, Width: L y 0, 9, Width t 0, 030, hickness: h 0, 002, h 0, 005 Height h 0, 040. x 0, Excitation, m 0, 73 Wavenumer ratio / 4, 37, / 2, Discussion of the results he good agreements in Figs. 2-3 etween the mode-ased aroach and the exact modal analysis illustrate clearly that the mode-ased aroach can rovide very efficient redictions of the ower transmission of ri-stiffened late systems, esecially when the modal density of the late is not high enough so that a sufficient randomness condition cannot e well-satisfied. Comaring with the hyrid FE/SEA solutions in Figs. 2-3, the mode-ased aroach also shows a much etter erformance not only regarding accuracy ut also efficiency. Although the comarisons are only made for the simlest eam-late models, the results show that the mode-ased aroach has a great otential to e a good sulement to the existing hyrid modelling techniques. Fig. 2 ransmitted ower from the eam to the late when the late thickness is 2 mm 36 RANSACONS OF FAMENA XXXV-2 (2013)

9 A Simle Hyrid Aroach on Predicting the Viration ransmission of Ri-Stiffened Plates L. Ji, M. Jokić, Z. Huang, N. Alujević Fig. 3 ransmitted ower from the eam to the late when the late thickness is 5 mm 4. Concluding remarks n this aer, a simle hyrid technique, the so-called mode-ased aroach, is alied to redict the viration transmission of a ri-stiffened late model. he results are then comared with those of the well-estalished hyrid FE/SEA method and the exact modal analysis. t is found that the mode-ased aroach has a much etter erformance than the hyrid FE/SEA method regarding oth accuracy and efficiency. Having said this, however, the mode-ased aroach is far less generic than the hyrid FE/SEA method as the latter has een widely alied to many comlex uilt-u systems in ractical engineering [13]. Desite this fact, the derivation rocedure in [15] has shown that the mode-ased aroach can e extended to generic comlex uilt-u systems with significant local dynamic mismatches in a quite straightforward way. Further research on generalizing the mode-ased aroach is underway. Acknowledgments he authors gratefully acknowledge the financial suort of the Ministry of Science and echnology, China, and the Ministry of Science and echnology, Croatia. REFERENCES [1] C. Soize. A model and numerical method in the medium frequency range for viroacoustic redictions using the theory of structural fuzzy. Journal of the Acoustical Society of America, 94 (1993), [2] W. Desmet. Mid-frequency viro-acoustic modeling: challenges and otential solutions, Proceedings of SMA, 2002, 2: [3] R. Langley. Recent Advances and Remaining Challenges in the Statistical Energy Analysis of Dynamic Systems. 7th Euroean Conference on Structural Dynamics, Southamton, UK, [4] M. Petyt. ntroduction to Finite Element Viration Analysis. Camridge: Camridge University Press, [5] R. H. Lyon, R. G. DeJong. heory and alication of statistical energy analysis. Boston: Butterworth- Heinemann, RANSACONS OF FAMENA XXXV-2 (2013) 37

10 L. Ji, M. Jokić, Z. Huang, N. Alujević A Simle Hyrid Aroach on Predicting the Viration ransmission of Ri-Stiffened Plates [6] R. M. Grice, R. J. Pinnington. A method for the viration analysis of uilt-u structures, art : analysis of the late-stiffened eam using a comination of finite element analysis and analytical imedance. Journal of Sound and Viration, 230 (2000), [7] S. B. Hong, A. Wang, N. Vlahooulos. A hyrid finite element formulation for a eam-late system. Journal of Sound and Viration, 298 (2006), [8] M. Viktorovitch and A. Pratellesi. A hyrid mid-frequency formulation for viro-acoustic redictions. Noise Control Engineering Journal, 56 (2008), [9] K. Vergote, B. V. Genechten, D. Vandeitte, W. Desmet. On the analysis of viro-acoustic systems in the mid-frequency range using a hyrid deterministic-statistical aroach. Comuters and Structures, 89 (2011), [10] P. J. Shorter, R. S. Langley. Viro-acoustic analysis of comlex systems. Journal of Sound and Viration, 288 (2005), [11] P. J. Shorter, R. S. Langley. On the recirocity relationshi etween direct field radiation and diffuse revererant loading. Journal of the Acoustical Society of America, 117 (2005), [12] V. Cotoni, P. Shorter, R. S. Langley. Numerical and exerimental validation of a hyrid finite elementstatistical energy analysis method. Journal of the Acoustical Society of America, 122 (2007), [13] VA-One User s Guide. ES Grou, the Virtual ry-out Sace Comany, [14] R. S. Langley, V. Cotoni. Prediction of the ensemle mean and variance of the resonse of uncertain structures using the hyrid FE/SEA method. Proceedings of SMA 2006, , Belgium. [15] L. Ji, B. R. Mace, R. J. Pinnington. A mode-ased aroach for the mid-frequency viration analysis of couled long- and short-wavelength structures. Journal of Sound and Viration, 289 (2006), [16] A. Peiffer. Comarison of the comutational exense of hyrid FE/SEA and FEM calculation. Proceedings of Noise and Viration: Emerging Methods (NOVEM), Nales: University of Nales, [17] E. Skudrzyk. Virations of a system with a finite or an infinite numer of resonances. Journal of Acoustical Society of America, 30 (1958) [18] A. W. Leissa. he free viration of rectangular lates. Journal of Sound and Viration, 31 (1973), [19] L. Cremer, M. Heckl, E. E. Ungar. Structural-orne Sound. Berlin: Sringer-Verlag, [20] E. Balmes. Use of generalized interface degrees of freedom in comonent mode synthesis. Proceedings of the nternational Modal Analysis Conference, 1996, 1: [21] L. Ji, B. R. Mace. An interface decomosition technique on reducing the size of conventional Comonent Mode Synthesis (CMS) models, Advances in Viration Engineering, 11 (2012), Sumitted: Acceted: Lin Ji School of Mechanical Engineering, Shandong University, Jinan, , China Marko Jokić Faculty of Mechanical Engineering and Naval Architecture, University of Zagre, vana Lučića 5, Zagre, Croatia Zhenyu Huang School of Electronic, nformation and Electrical Engineering, Shanghai Jiao ong University, Shanghai, , China Neven Alujević Production Engineering, Machine Design and Automation (PMA) Section, KU Leuven, Celestijnenlaan ox 2420, 3001 Heverlee, Belgium 38 RANSACONS OF FAMENA XXXV-2 (2013)