Tuned vibration absorbers for control of noise radiated by a panel

Transcription

1 Proceeding of ACOUSICS November 9, Adelaide, Autralia uned vibration aborber for control of noie radiated by a panel Swee S. Kuik, Carl Q. Howard, Colin H. Hanen & Anthony C. Zander School of echanical Engineering, he Univerity of Adelaide, Adelaide, S.A. 55, Autralia ABSRAC A ingle paive tuned vibration aborber (VA) that i pecifically deigned to achieve optimal performance at a particular frequency may not be effective in handling minor change in excitation frequency. One way of taking thi into account i to ue multiple VA tuned to lightly different frequencie. hi paper i concerned with the low- to mid-frequency vibrational behaviour and radiated noie of a panel excited by a point force and controlled uing attached multiple vibration aborber tuned to different frequencie. Finite element analyi i not commonly ued to predict the effect of VA on the ound radiation by the panel. In thi paper, a theoretical model and finite element model (including FEA with and without fluid-tructure interaction) are preented for the calculation of the radiated ound power of a panel with multiple VA attached. A imply upported panel with two VA attached wa ued a an example to compute the theoretical prediction with the finite element analyi reult. 1. INRODUCION uned vibration aborber (VA) have been ued extenively for the control of tonal ound radiation from vibrating tructure (Jolly & Sun 1996, Fuller et al 1997, Fuller & Cambou 1998, arcotte et al 1999, Brennan & Dayou, Wright 3, Griom 3, Eteve and Howard 8). he main reaon for uing VA rather than traditional damping treatment (ead 199, Gentry et al 1997 and Fuller et al ) i that they are light-weight and can be intalled eaily in engineering application uch a tranportation and indutrial equipment. In thi paper, an impedance approach i employed to decribe the effect of VA attached to a imply upported panel and the reult are compared with thoe obtained from finite element analyi. he firt part of thi paper decribe theoretical model to predict the dynamic repone of and ound radiation from a imply upported panel with multiple attached tuned vibration aborber. In the econd part of the paper, reult obtained uing finite element model are compared with thoe obtained from theoretical model.. HEOREICAL BACKGROUND he theoretical analyi preented in thi paper i baed on the work by Wright (3), which i extended here to include the effect of panel damping. An impedance approach i ued to decribe the effect of VA attached to a imply upported panel a illutrated in Figure 1, where, for convenience only a repreentative ingle aborber i hown. In the following ection, analytical expreion for the dynamic repone of a panel with attached VA will be introduced..1 ultiple tuned vibration aborber attached For a thin homogenou panel excited by external force, f ( x, y,t ) and lying in the x-y plane, the two-dimenional bending wave equation in rectangular Carteian coordinate can be written a (Soedel 1993) w w w w EI ρ h = f ( x, y, t) (1) x x y y t where E i Young modulu, I i the econd moment of area of the panel per unit width, 3 I = h /1 ( 1 υ ), υ i Poion ratio, ρ i the denity of the panel, and h i the thickne of the panel. w i the tranvere diplacement of the panel urface at location x, y and time t and i given in modal term by (Soedel 1993) he quantity ( x,y,t) w, () m = 1n= 1 iω t ( x y, t ) = W ( x, y ) e where W i the modal amplitude, i the mode hape at i t location (x, y), e ω repreent the time dependence, and m and n are modal integer. he mode hape of a imply upported panel i given by (Soedel 1993) mπ x nπ y in (3) a b ( x,y ) = in where a and b are the length and width of the panel. he modal amplitude for a panel excited at frequency,ω can be expreed a (Wright 3) Figure 1. A chematic model of a imply upported panel with an attached VA. Acoutic 9 1

2 3-5 November 9, Adelaide, Autralia Proceeding of ACOUSICS 9 W = F ρ h a b ( ω (1+ iη) ω ) where η i the panel lo factor. Eq. () can be written in matrix form a w ( x, y,t)= W ( x,y ) ( x, y ) () i t e ω (5) where the upercript denote the matrix tranpoe, and the mode hape, φ can be expreed a a colu vector and are arranged in order of increaing reonance frequencie, ω, a 11 = 1 Finally, an expreion for the modal amplitude of a panel excited by point force can be written in matrix form a (Wright 3) W, (6) = H F (7) where F i a vector of complex force amplitude and H i a (m x n) diagonal matrix of containing the modal mobilitie, which can be written a (Wright 3) H H = 11 H 1 O H Each element of H with included modal lo factor can be defined a H = (9) ρ h ab( ω (1+ iη ) ω ) which i an extenion of the expreion given by Wright (3) with the lo factor excluded. he reonance frequencie of the panel can be expreed a (Soedel 1993): (8) E I mπ nπ ω = + (1) ρ h a b he mode hape function i evaluated for each mode at the location correponding to the th point force (i.e. =1,, 3 n). Hence, it can be expreed a (Wright 3), = 11,1 1,1,1 11, 1,, 11, 1,, (11) A hown by Wright (3), the external force, F comprie the um of the VA reaction force and diturbance force. Hence, Eq. (7) can be implified to give W = H, j FVAj + H, k k (1) where the diturbance force can be written a a colu vector a F dk 1 = k (13) and, the VA reaction force can be written a a colu vector a F VAj F = F F VA1 VA VA j (1) Here, the complex amplitude of the j th VA reaction force can be expreed a (Wright 3) FVA j = i ω j, j, j (15) Z VA W where the upercript denote the matrix tranpoe. he VA input impedance matrix can be written a ZVA Z = VA j,j 1,1 Z VA, O ZVAj, j (16) where each element in the matrix can be expreed a (Wright 3) iωω + ξva j ω ω VA VA j j Z VA j, j m VA j ( ωva j ω ) iξva j ωωva j = (17) where m VA, ξ VA and ω VA are the ma, critical damping ratio and reonance frequency of the tuned vibration aborber repectively. Alo,, j can be expreed in matrix form a 11,1 =,j 1,1,1 11, 1,, 11, j 1, j, j (18) he matrix φ,k i defined in a imilar way to Eq. (18) in which j i replaced by k. he mode hape function i evaluated for each mode at the location correponding to the th external force (which include force generated by the VA). An expreion for the modal amplitude of the panel with VA attached can be derived from Eq. (1) and i given by Wright (3) a 1 [ I iω H,j ZVA j,j ] H,j,k k W= (19) Finally, Eq. (19) can be ubtituted into Eq. (5) to etimate the dynamic repone w ( x,y,t) of the panel. he mean quare velocity of a panel can be expreed a follow (Wallace 197): Acoutic 9

3 Proceeding of ACOUSICS 9 v 1 H = ω [ w] [ w] () 8 where the upercript H denote the Hermitian tranpoe and w i defined in term of element defined in Eq. (5) a ( ) w x1, y1 (, ) w = w x y i e w( xg, yg ) ωt (1) where the quare of the urface velocity can be obtained by patially averaging the quared velocity a number of g point paced equally over the panel and uing Eq. () to etimate mean quare velocity of a panel.. Sound radiation from a vibrating panel A chematic model of ound radiation from a imply upported panel mounted in an infinite baffle and ubject to a point force excitation i hown in figure. he panel ha a uniform thickne, h, width, a and length, b. he ound preure field radiated by a vibrating urface urrounded by an infinite baffle can be calculated a (Wallace 197) p( r, θ, φ ) = where ( x y) S. i ω ρ a w ( x, y ) e π R i ω t e i k R ds () w, i the component of the complex velocity normal to the urface, ρ a i the denity of the acoutic medium, ω i frequency in rad/, r i the ditance from the obervation point to the coordinate origin. 3-5 November 9, Adelaide, Autralia hu, the total radiated ound power can be written a (Wright 3) π π W = I r in θ d θ d φ () 3. FINIE ELEEN ANALYSIS odelling a complete tructure with attached tuned vibration aborber can be done uing the ANSYS finite element analyi (FEA) oftware package. A finite element model without and with fluid-tructure interaction i a depicted in Figure 3 and Figure repectively. A ignificant iue with conducting finite element analye with fluid-tructure interaction i the long computation time ariing from nonymmetric matrix equation. hi ection preent a comparion between finite element analyi with and without fluidtructure interaction. he reult from the analye compare reaonably well with the theoretical prediction from Section. A finite element analyi without fluid-tructure interaction mean that the finite element model comprie only tructural element. Finite element model a hown in Figure 3 only ha hell (SHELL63), vico-elatic pring-damper (COBIN1), and lumped ma (ASS1) element. A harmonic repone analyi wa conducted uing ANSYS to calculate the normal tructural velocity ditribution due to a point excitation force. he normal fluid velocitie at the tructure urface are aumed to be equal to the normal tructural velocitie. he ANSYS reult of the tructural velocity of the panel (without fluid-tructure interaction) were exported and a ALAB cript wa ued to calculate the total radiated ound power. Figure. Schematic model of ound radiation of a imply upported panel with a point force excitation. Figure 3. An FE model of the plate-mounted VA ytem without fluid-tructure interaction. Panel vibration mode are orthogonal in term of their vibration repone. However, they are not orthogonal when decribing their contribution to the radiated ound field. hi imply mean that the total radiated ound power cannot be evaluated by adding together all of the contribution from each mode. Baically, the integral can be etimated a the um of the field of a ditribution of elemental ource on the radiating urface (each having a complex volume velocity) (Hanen & Snyder 1997). he radiated ound intenity in the far field can be written a (Wright 3) I = p( r, θ, φ ) / (ρ c) (3) o Figure. An FE model of the plate-va mounted ytem with fluid-tructure interaction. Acoutic 9 3

4 3-5 November 9, Adelaide, Autralia Proceeding of ACOUSICS 9 A finite element analyi with fluid-tructure interaction ha both acoutic (FLUID3 and FLUID13) and tructure element. Figure how the finite element model under conideration here that ha a imply upported panel with VA attached, and i urrounded in a emi-infinite acoutic element hemiphere. he ANSYS oftware i ued to calculate the radiated acoutic preure from the panel. hee preure reult were then exported and a ALAB cript wa ued to proce the reult to calculate the total radiated ound power.. DISCUSSION OF RESULS For the analye conidered here, a panel with dimenion (x,y) = (1 mm, 15 mm), thickne of 1 mm and lo factor of.1 wa ued for example purpoe. wo location were elected for the attachment of VA: (x,y) = ( mm, 65 mm) and (5 mm, 6mm). Each VA i intended to target reonance frequencie at 6.36 Hz and repectively. In thi cae, the choen ma ratio between the total added ma of the VA and the panel i 15% (approximately 911. gram in total). hi ma wa equally ditributed amongt the VA. A point force of 1 N wa applied at the location of (x,y) = (3 mm, mm). able 1 how the tuned vibration aborber parameter ued in the analye. Figure 5a how the mean quare velocity of the panel with two VA attached, which i calculated uing the analytical formulation (olid line) and FEA (tar) with tructural repone only. It i oberved that both the theoretical prediction and the FEA reult are almot identical for the frequency range of 1 Hz to Hz. Figure 5b how the ame reult a Figure 5a, only over the frequency range between 5-8Hz. hi figure indicate that there i no mode plitting behaviour at tuning frequencie of the VA. However, the VA alter the repone of the panel by effectively retraining it motion at the connection point. At thee frequencie, the reduction of mean quare velocity i approximately 5.8 db and 5.6 db repectively. Figure 6a how that the correponding radiated ound power for the panel with VA attached. he reult obtained from the theoretical analyi compare reaonably well with the reult from the FEA without fluid-tructure interaction. Figure 6b how that the radiated ound power i reduced near the tuning frequencie of the VA (6.37 Hz ) by approximately 7.5 db and 1.1 db repectively. 6.36Hz able 1. uned vibration aborber parameter a (gram) Frequency (Hz) (x,y) Location (m) (.,.65) (.5,.6) 6.36Hz (a) 6.36Hz (a) 6.36Hz (b) Figure 5 ean quare velocity for a panel with VA attached, excited by a point force with an amplitude of one newton: (a). Comparion between theory and FEA and (b). Zoom - the frequency range of 5 Hz and 8 Hz of part (a). Figure 6 Radiated ound power for a panel with VA attached, excited by a point force with an amplitude of one newton: (a). Comparion between theory and FEA w/o FSI (without fluid-tructure interaction) and (b). Zoom - the frequency range of 5 Hz and 8 Hz of part (a). Figure 7 how a comparion of the reult for calculating the total radiated ound power for three cae: (1) no VA (heory); () with VA (FEA without fluid-tructure interaction); and (3) with VA (FEA with fluid-tructure interaction). he motivation for conducting thi analyi i to confirm that the calculated reult for the finite element analyi without fluid-tructure interaction i imilar to the reult for the prediction where fluid-tructure interaction (b) Acoutic 9

5 Proceeding of ACOUSICS 9 wa included. he reult compare reaonably well over the frequency range of interet. A light deviation can be oberved above 18 Hz, which i mainly due to the ize of the acoutic element. It i recommended that at leat 6 element per wavelength hould be ued for acoutic analye. 6.36Hz Figure 7. Radiated ound power for a panel with VA attached, excited by a point force with an amplitude of one newton - Comparion between FEA with and without FSI. 5. SUARY & CONCLUSION In thi paper, a low- to mid- frequency (1Hz-Hz) analytical model of a panel with multiple VA and finite element model were preented to enable the calculation of the mean quare velocity and radiated ound power. he radiated ound power wa calculated by two method. For the firt method, an FEA wa conducted where the model incorporated fluid-tructure interaction and wa ued to calculate the radiated preure. he radiated ound power wa thu calculated by uing a ALAB cript to calculate radiated intenity, which wa then integrated intenity over a hemiphere to calculate the total radiated ound power. he econd method involved the ue of a finite element model without fluid-tructure interaction to calculate the normal tructural velocity ditribution of the panel. he radiated ound power wa calculated by uing a ALAB cript to determine the far field radiated preure, which wa then ued to calculate the intenity, and then integrated over a hemiphere. 3-5 November 9, Adelaide, Autralia Eteve, S. J., Control of ound tranmiion into payload fairing uing ditributed vibration aborber and Helmholtz reonator, PhD hei Virginia Polytechnic Intitute and State Univerity Fuller, C.R., aillard, J. P., ercadal,. and Flotow, A. H. 1997, Control of aircraft interior noie uing globally detuned vibration aborber, Journal of Sound and Vibration, 3, p Fuller, C. R. and Cambou, P. 1998, An active-paive ditributed vibration aborber for vibration and ound radiation control, he Journal Acout. Soc. Am. 1 (3), p Fuller, C. R., Kidner,. R. F., Li X. And Hanen, C. H., Active-paive heterogeneou blanket for control of vibration and ound radiation, Proceeding of Active99, Williamburg Gentry, C. A., Guigou, C. and Fuller, C. R. 1997, Smart foam for application in paive-active noie radiation control, Journal of Acout. Sco. Am., 11 (), p Griom,. D. 3, Quiet product deign uing optimized broadband vibration aborber, PhD hei, he Pennylvania State Univerity Hanen, C. H. and Snyder, S. D. 1997, Active control of noie and vibration, E & FN Spoon, London Howard, C. Q. 8, ranmiion lo of a panel with an array of tuned vibration aborber, Acoutic Autralia, 36, p Jolly,. R. and Sun J. Q. 1996, Paive tuned vibration aborber for ound radiation reduction from vibrating panel, Journal of Sound and Vibration, 191 (), p arcotte, P., Fuller, C. R. and Cambou, P. 1999, Control of noie radiated from a plate uing a ditributed active vibration aborber, Proceeding of Active 99, p ead, D. J. 199, Plate with regular tiffening in acoutic media: vibration and radiation, he Journal of Acout. Soc. Am. 88, p Soedel, W. 1993, Vibration of hell and plate, arcel Dekker, Inc, New York Wallace, C. E. 197, Radiation reitance of a rectangular panel, he Journal of the Acout. Soc. of Am., 51, p Wright, R.I. 3, A hierarchical noie control ytem uing adaptable tuned vibration aborber, PhD hei, Virginia Polytechnic Intitute and State Univerity he theoretical model preented in thi paper wa verified by comparing theoretical reult with the reult from two finite element model. A imply upported panel with two VA attached wa ued a the tet cae. It wa howed that the reult obtained from FEA compared favourably with the reult from theoretical prediction. Future work that will be invetigated i to conider multipledegree-of-freedom VA attached to the panel for reducing ound radiation. ACKNOWLEDGEEN he firt author would like to thank Karen Adam for her uggetion in preparation of the manucript a well a Izzuddin Zaman for hi uggetion on finite element modelling in thi paper. REFERENCES Brennan,. J. and Dayou, J., Global control of vibration uing a tunable vibration neutralizer, Journal of Sound and Vibration, 3 (3), p Acoutic 9 5