DESIGN OPTIMIZATION OF FOUNDATION FOR ROTATING MACHINERY AGAINST STANDING-WAVE VIBRATION IN A BUILDING

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1 COMPDYN III ECCOMAS hematic Conference on Computational Metho in Structural Dynamic an Earthquake Engineering M. Paparakaki, M. Fragiaaki, V. Plevri (e.) Corfu, Greece, 5 8 May DESIGN OPIMIZAION OF FOUNDAION FOR ROAING MACHINERY AGAINS SANDING-WAVE VIBRAION IN A BUILDING Bin Niu, Niel Olhoff Department of Mechanical an Manufacturing Engineering, Aalborg Univerity Fibigertraee 6, DK-9 Aalborg Eat, Denmark {bni, no}@m-tech.aau.k Keywor: Machinery Founation, Deign Optimization, Force Vibration, Vibration Iolation. Abtract. hi paper eal with the problem of optimum eign of a founation for rotating machinery on a torey of a builing with a view to minimize the level of taning-wave vibration in the builing. he founation i uually eigne a a bae plate for the machinery, with ome reilient mount fixe to the bottom of the bae plate an upporte by the floor of the torey in orer to provie a uitable level of vibration iolation of the builing. Due to variable ervice pee an the exitence of non-balance mae, the rotating machinery may be coniere a ource that within a given range of excitation frequencie excite force vibration of the founation, an thereby the floor an wall, etc., of the builing. he tranmiion of uch vibration through the builing may reult in uneirable oun emiion an unatifactory comfort conition for the people in welling an office of the builing. o remey thi, the objective of thi work i to evelop an implement a metho of eign optimization to etermine optimum tiffne value of reilient mount ubject to contraint on availability of phyical propertie of material to be ue. he eign objective i choen a minimization of the power tranmitte from the machine to the floor of the builing where the founation for the rotating machinery i mounte. At the current tage of our project, thi problem i only carrie out for a given, quite implifie moel of a builing. However, for thi builing moel, the eign an performance of the optimize machinery founation will be illutrate an icue uing everal numerical example. In the next tage of our work, a multi-material, parameterize builing moel will be evelope with etaile imenion an connection of component, an the current problem will be extene to encompa imultaneou eign optimization of both the builing an the founation for the rotating machinery in orer to minimize the level of taning-wave vibration in the builing.

2 INRODUCION Rotating machinery in builing i uually applie in central heating an ventilation ytem, an larger machinery of thi type incluing a pump i normally mounte on a founation, which i uually eigne a a bae plate for the machinery with ome reilient mount fixe to the bottom of the bae plate an upporte by the floor of the torey. Due to variable ervice pee an the exitence of non-balance mae, the rotating machinery may be coniere a ource that within a given range of excitation frequencie excite force vibration of the founation, an thereby the floor an wall, etc., of the builing. he tranmiion of uch vibration through the builing may reult in uneirable oun emiion an unatifactory comfort conition for the people in welling an office of the builing. Aie from that, vibration increae afety hazar in machinery, builing an intallation. he primary goal of vibration inulation are to retrict the etrimental effect of vibration on people to within reaonable limit, an to protect enitive apparatu an afety ytem from exceive tree from vibration. Problem of eign optimization of machinery founation againt vibration have been mainly tuie from two apect:. Free vibration eign, alo terme a frequency eign. hi aim at keeping the operating frequency a far away a poible from the eigenfrequencie of the machinery mounting ytem by ajuting the mounting ytem in orer to avoi reonance. It i uually realize by maximization of the funamental eigenfrequency or frequency gap between two conecutive eigenfrequencie of the machinery mounting ytem.. Force vibration eign. he machinery mounting ytem i aume to be ubjecte to a time-varying unbalance mechanical loaing, an thi ytem will be eigne by minimizing a choen cot function ecribing the level of vibration repone or tranmiion. he problem of force vibration eign optimization of the intallation ytem of machinery in builing ha been extenively reearche uner the aumption of a rigi upporting tructure [-3]. he eign bae on a rigi upporting tructure moel i reaonable for the intallation of machinery in many real engineering ituation. However, thi rigi upport bae moel may not be appropriate for the problem tuie in the preent paper where the machinery i to be intalle on a relatively flexible floor of the torey in a lightweight builing. Bae on a flexible upport moel, Ahrafiuon [4] tuie eign optimization of aircraft engine-mount ytem for vibration iolation, an Xie et al [5] coniere optimization of the mounting ytem for microelectronic manufacturing equipment uing the receptance matrix metho. In thee work, minimization of the tranmitte force from the vibrating machine to the receiver i choen a the eign objective. In the work [6] it wa uggete to chooe the power flow a the cot function becaue it combine both force an velocitie in a ingle concept. Furthermore, the tranmitte power from the unbalance machine to the floor i cloely relate to the tructural noie emiion from the floor. Power flow i coniere to be a more reaonable meaure of the vibratory tate in vibro-acoutic moeling. he power flow from a vibrating machine to ifferent flexible receiver through reilient mount i preente in the work [6-9]. When rotating machinery i to be mounte on builing floor rather than irectly on a oil founation, uitable reilient mount houl be provie a vibration iolation element uner the machine with a view to reuce the tranmiion of vibration. he eign optimization of machinery mounting ytem i tuie in thi paper. he ytem conit of a rotating, unbalance machine a vibration ource, a mounting ytem a iolator, an a flexible floor a receiver. By auming a imple time harmonic excitation

3 generate within the machine by it operating mechanim, it i convenient to characterize the iniviual ub-tructure by their complex mobilitie evaluate at the interface of contiguou ub-tructure. he mobility i efine a the ratio of the complex amplitue of the velocitie an force at any interface for a given frequency []. he phyical quantitie, uch a velocitie, force an moment at the interface, are olve in term of the mobility matrice of ubtructure. A generalize mathematical moel of mobility power flow i evelope in thi paper for evaluation of the repone of the total ytem ubjecte to a given external excitation. In thi moel, the machine i moele a a rigi ma ubjecte to a harmonically time-varying force. he bae plate i aume to be rigily connecte with the machine, an inclue in the moeling of the machine. he mounting ytem of the machine i eigne a ome reilient upport fixe to the bottom of the machine an upporte by the floor of the torey in orer to provie a uitable level of vibration iolation of the floor. he flexible floor i moele a an elatic uniform plate, an the riving point an tranfer mobilitie of the plate are aopte in the moel. he egree of freeom aociate with flexural vibration of the upporting tructure are of main interet becaue the flexural wave motion i uually ominating the oun raiation compare with the in-plane wave motion. he objective of minimizing vibration tranmiion i realize by izing optimization of the tiffne coefficient of the reilient mount. he eign objective i choen a minimization of the power flow tranmitte to the builing floor through the reilient mount at the excitation frequency of the machinery. he eign an performance of the optimize machinery founation will be illutrate an icue uing everal numerical example. he ret of thi paper i organize a follow. Firt, a generalize mathematical moel of mobility power flow i evelope in thi paper. he formulation of minimization of tranmitte power flow i preente in Section 3. Section 4 preent a implifie parametric example only coniering the vertical flexural motion, an a generalize optimization example with more egree of freeom. In Section 5, a hape optimization problem of a pa that i place on a flexible floor to upport a reilient mount, i preente. Finally, obervation an concluion are rawn bae on the optimization reult. A GENERAL MOBILIY FORMULAION OF HE MACHINERY MOUNING SYSEM A general moel with a rotating, unbalance machine a vibration ource, a mounting ytem a iolator, an a flexible floor a receiver, i evelope for analyi an optimization of vibration tranmiion, ee the moel in Figure. he machine i moele a a rigi boy ubjecte to a harmonically time-varying force. he mounting ytem of the machine i eigne a ome reilient upport fixe to the bottom of the machine an upporte by the floor of the torey in orer to provie a uitable level of vibration iolation of the floor. he flexible floor i moele a an elatic uniform plate, an the riving point an tranfer mobilitie [] of the plate are aopte in the moel. Figure give a repreentation of a vibratory rigi boy reiliently mounte on a four-ege imply upporte plate via multiple reilient mount. he force an velocitie at the interface of the contiguou ub-ytem are hown in Figure, where the arrow efine the poitive irection of the force an velocitie. 3

4 Figure : A machine mounte to a flexible floor via multiple reilient mount Figure : Decription of force an velocitie between three ub-ytem: ource, reilient mount an receiver. Rigi boy A local coorinate ytem xoyz, o o cf. Figure, i aopte with origin in the center of gravity of the machine. It i aume that the machine i excite by a generalize concentrate force vector F with three force an three moment component acting at the center of gravity. he correponing generalize velocity vector at the center of gravity i enote a V. F F, F, F, F, F, F () V V, V, V, V, V, V () where the upercript repreent the tranpoe of a matrix or vector. A number of reilient mount are aume to be attache to the bottom of the machine, an the generalize output force an velocity vector from the rigi boy to the reilient mount at the n junction are aemble in the vector 4

5 F F F F a a a an a a a an, V V V V (3) Correponingly, the generalize force an velocity vector acting at the j-th reilient mount by the machine are F bj bj an V, an each of them inclue ix component, repectively. he ynamic governing equation of the rigi boy [] can be written in term of the mobility matrice, V M M F a= a (4) V M M F where M J, M J R, M R J, M R J R. he ymbol i i i i i the excitation frequency, i repreent the imaginary unit, i, R repreent the location matrix of the reilient mounting junction with repect to the center of gravity of the machine, an J i the general ma matrix. he time epenent term expi t i omitte in the remainer.. Reilient mount he reilient mount are ue a vibration iolator for minimizing vibration tranmiion from the machine to the builing floor. At the mounting junction on the plate, the generalize force an velocity vector at the bottom en of the reilient mount, ee Figure, are aemble in the vector F F F F c c c cn c c c cn, V V V V (5) he relation between the velocity vector an force vector at the two en of the j-th reilient mount i given by uing the four-pole equation [3, 4], bj j j cj F iag iag F bj j j cj (6) V iag iag V j where pq iag, pq,,, mean the iagonal tranmiion ub-matrix of the j-th reilient mount. For generality, in thi tuy, each reilient mount i moelle a ix lumpe tiffne component with negligible ma, where three component with tranlational tiffne an three with rotational tiffne are aume without coupling between the ifferent tiffne component. For example, the p-th component of the j-th reilient mount i ecribe by the following four-pole equation bj cj F p Fp i bj cj V p j Vp p (7) 5

6 where bj F p enote the p-th (p =,, 6) component of the generalize force at the j-th reil- j ient mount, an p enote the tiffne coefficient of the p-th tiffne component in the j-th reilient mount. For all reilient mount, the tranmiion matrix equation i obtaine a b c F F b c V V he conition for force equilibrium an motion compatibility at the junction can be written a a b F F, a b V V, c F F, (8) c V V (9).3 he upporting floor he upporting floor i moele a a thin, elatic uniform plate. When there are no ignificant reflection from the bounarie or from icontinuitie within the receiver, an infinite, uniform plate moel may be aume. Otherwie, a finite plate with uitable bounary conition houl be coniere. he upporting plate floor i excite by the force an moment component at the bottom of each reilient mount. he force an velocity vector at the plate mounting point, ee Figure, are aemble in the vector F F F F n n, V V V V () he force vector at each reilient mount junction j =,, n inclue ix component, j j j j j j j F F, F, F, F, F, F () Accoringly, the velocity vector at each reilient mount junction j =,, n inclue three tranlation component an three rotational component, i.e., j j j j j j j V V, V, V, V, V, V () he mobility equation of the plate can be now written a follow [, 5-8], where Y i the mobility matrix, n V y y y F F n V y y y F F Y n n n nn n n V y y y F F Since the flexural vibration ominate the noie emiion of the builing floor, only the out-of-plane flexural wave will be taken into account. hu, the in-plane hear an longituinal motion inuce by in-plane force, an rilling motion of the plate inuce by twiting moment are neglecte. he out-of-plane flexural wave are inuce by the out-of-plane force an the in-plane moment component. For example, the mobility ub-matrix y can be written a (3) 6

7 ywf y z wm y x wm y y (4) y xf y z xm y x xmy y yf y z ym y x ymy where the mobility term y wf z relate an out-of-plane tranlational velocity V z at the junction no. to an out-of-plane force excitation F z at junction no., where the latter junction i the excitation point, an the former junction i the repone point. he mobility repreentation i a ueful tool to ecribe the ynamic propertie of the upporting floor. For a realitic complicate floor, force an moment mobilitie at the mounting junction can be meaure by experiment. For ome imple infinite an finite tructure, the mobility matrice can be erive analytically. he mobility formulation for an elatic uniform plate with imply upporte ege an an infinite plate can be foun in, e.g., Ref. [,, 5]..4 Power flow he time average power flow tranmitte from the machine to the builing floor can be expree a * * Re Re F V F YF (5) where the ymbol ( ) * repreent conjugate tranpoe of a matrix or vector. he tranmitte force F can be olve from Eq. (4), (8) an (3) by applying the conition of force equilibrium an motion compatibility in Eq. (9). For brevity, the omewhat lengthy erivation i omitte here, a the ame reult can be foun in Ref. [4]. hu one get the following expreion for the tranmitte force F, F M + Y+M Y M F (6) In orer to avoi exceive vibration of the machine, a uitable contraint on the velocity V of the rigi boy may be applie. hu, the velocity V i ometime of interet in eign optimization of machinery mounting ytem. he expreion for the velocity V given in Eq. (7) i obtaine by olving Eq. (4), (8) an (3) with the conition of force equilibrium an motion compatibility in Eq. (9), V =M F M - Y F (7) 3 OPIMIZAION FORMULAION FOR MINIMIZAION OF POWER RANSMISSION 3. Optimization formulation 7

8 he problem of eign optimization of the rotating machinery mounting ytem with the objective of minimizing the total power flow tranmitte from the machine to the builing floor via everal reilient mount, can be formulate a * min F ReYF.t. Given contraint i, i,, n min he preent work aim to realize thi objective by optimizing tiffne coefficient i of reilient mount in a given range between min an max. he lower an upper limit of the tiffne coefficient are uually etermine by phyical propertie of the reilient mount. A reaonable lower limit i important for atifying requirement on the tatic iplacement of the machine, or the motion uring tarting an topping tage. It i well-known that the tiffne of reilient mount i normally frequency epenent. However, for implicity the tiffne will be aume to be inepenent of frequency in the following. In the expreion for the total tranmitte power in (8), F enote the vector of amplitue of the loaing vector acting on the plate with the excitation frequency. he given contraint are pecifie by phyical an geometrical requirement on mounting ytem. he v ymbol n enote the number of eign variable. 3. Deign enitivity analyi he enitivity of the objective function in Eq. (8) with repect to the eign variable can be erive a k * Re F Y F * F Re v k F Y, k =,, n (9) k k where the ymmetric characteritic of the mobility matrix Y have been ue. F In orer to erive the enitivity of the tranmitte force with repect to the eign k variable, Eq. (6) i rewritten a max v (8) M + Y+M Y F M F () By ifferentiating both ie of Eq. () with repect to the eign variable k, coniering the conition that only i epenent on eign variable among the mobility matrice of the machine, reilient mount an the floor plate, an auming that the generalize excitation force be obtaine a F F generate by the rotating unbalance machine i eign inepenent, k M + Y+M Y F, k =,, k F k can v n () 8

9 he accuracy of analytical enitivitie ha been valiate by overall finite ifference enitivity calculation. With thee enitivity reult, the eign problem (8) may be olve by a mathematical programming metho, e.g., MMA by Svanberg [9]. 4 NUMERICAL EXAMPLES 4. A implifie parametric example Firt, a imple pecial cae of the vibratory ytem hown in Figure i coniere. Here, four ientical reilient mount are place ymmetrically with repect to the machine, an mounte on a flexible floor, which i aume to be an infinite elatic plate of contant thickne. hi implification lea to an equivalent ytem of the reilient mount a hown in Figure 3, which conit of four eparate et of a pring of tiffne an a ma m equal to m 4, where the ymbol m repreent the total ma of the machine. he ynamic governing equation i implifie correponingly, i.e., the force acting on all four mounting point are aume to be the ame, that i, F 4, where F enote the vertical excitation force acting on the machine. A imilar implifie moel uing a finite four-ege imply upporte plate a the receiver i tuie in []. Figure 3: A implifie moeling for the machine connecte to a flexible floor via four ymmetrically place mount From the vibration equation of a ytem with a ingle egree of freeom, ee, e.g. [], the force F tranmitte from the ma m to the floor via the j-th pring i obtaine from [, ] j F j F 4 m m i 4 4 where i the tiffne of the pring. Since only the out-of-plane force excitation on the plate i coniere, the mobility matrix of the plate in Eq. (3) can be implifie. By the concept of effective point mobility Y j a a pace average effective mobility over all excitation point [], Y j can be expree a Yj YjYj Yj3 Yj4 for j=,, 3, an 4. he ymbol Y jk i the general mobility element from the excitation point k to the repone point j, where Y jk with k = j i the riving point mobility, while Y jk with k j enote the tranfer mobility. he effective mobility repreent the ratio of the total velocity ue to all applie force to the force acting at the point j. It can be calculate that the effective point mobility of an infinite plate Y j (j=,, 3, 4) are ame, Y j () 9

10 i.e., Y Y Y3 Y4. he riving point an tranfer mobilitie of an infinite plate can be foun in, e.g., [, 5]. he velocity V of the machine i calculate bae on the ame aumption V F i Y 4 m i i Y i 4 i hu, from Eq. (5) the time average power P tranmitte from the machine to the floor can be rewritten a (3) P Re Y F 4 m m i Y 4 4 (4) w A a reference to evaluate the effect of the pring iolator, the power flow P from the machine to the builing floor without pring iolator i given in Eq. (5) below. he moeling without iolation i obtaine by removing the pring iolator from Figure 3, an we fin w that the power P tranmitte from the machine to the floor without pring iolator i given by P w Re Y F 4 m i Y 4 w he velocity V of the machine without pring iolator i calculate a (5) w F Y V 4 m i Y 4 Auming four pring with the ame tiffne coefficient, the epenence of tranmitte power P in Eq. (4) on the tiffne coefficient will be tuie for ifferent excitation frequency an ma value of the machine. In thi ection, no amping i aume for the pring an the plate. For a rigi builing floor, the velocity at the mounting junction on the plate i zero, V =, thu the tranmitte power P =. If the tiffne of the pring i reaonably electe to make the reonance frequency of the ma-pring ytem to be locate far from the ex- m 4 citation frequency, the tranmitte force F i on the floor i relatively mall. When a flexible floor i coniere, the mobility of the plate mut be taken in account, an the velocity of the mounting point on the plate V. hu we houl pay attention to the tranmitte power P, the velocity V an force F. (6)

11 When the bening tiffne of the plate i quite large, the correponing mobility Y i very mall. he concluion i imilar to the cae of a rigi floor above. he tranmitte power P, hown in Figure 4 (a), an force F are very large only in the vicinity of reonance, an in other cae both of them are generally mall. In thi example, the given circular excitation frequency i 5 ra/. 9 A uitably flexible plate of thickne. mae of a material with Young moulu, Poion ratio.3 an ma enity 84, i tuie here. SI unit are aume. ranmitte power for ifferent ma value of machine an excitation frequencie are preente in Figure 4 (b). For a pecifie excitation frequency an a given tiffne coefficient, it i een that when the ma of the machine increae, then the tranmitte power ecreae. Actually, an inertia concrete block i ometime place at the bottom of machine to increae the ma of the vibrating rigi boy. he extra inertia block can be eeme beneficial for reuction of tranmitte power when an infinite plate i the receiver, in which cae no reonant behavior can occur. hough infinite tructure o not exit in reality, the aumption of infinite tructure i applicable in many circumtance where there are no ignificant reflection from icontinuitie or bounarie within the receiver [6]. he only peak of each curve in Figure 4 (b) come from the reonance of the pring-rigi boy ytem. Obviouly, the peak will move to the right when the ma of the machine i increae, an the maximum value at the peak alo ecreae ignificantly. If the ma of the machine i the ame, then the value of the tranmitte power at the peak point i reuce for a higher value of the excitation frequency. hi can be explaine from Eq. (4) where the term my in the enominator increae with increaing excitation frequency, which can be approximately coniere a increaing the effect of amping. w Compare with the power flow P from the machine to the builing floor without pring iolator, it i foun that if a very oft pring i provie uner the machine, the tranmitte power to the floor will be conierably reuce, but it may caue ignificant vibration velocity of the machine, ee Figure 5. If a machine i rigily bolte to the floor in the infinite plate cae, then the vibratory movement of the machine may be reuce, but the tranmitte power to the floor will be relatively large. hu, ome compromie mut be mae between the two requirement, an motivate optimization. ranmitte power m=, =5 without pring m=, =5 ranmitte power 3 x m=, =5 without pring m=, =5 m=5, =5 m=, = m=5, = Stiffne coefficent x 6 (a) Stiffne coefficent x 6 Figure 4: ranmitte power for ifferent mae of machine an excitation frequencie: (a) very tiff floor plate, (b) relatively flexible floor plate (b)

12 7 x m=, =5 without pring m=, =5 m=5, =5 m=, = m=5, = 4 V Stiffne coefficent x 6 Figure 5: Velocity of machine for ifferent mae of machine an excitation frequencie 4. Optimization of the mounting ytem of a machine with ix egree of freeom ubjecte to generalize excitation force he moel hown in Figure i coniere for minimizing the power flow. It i now aume that a machine with ix egree of freeom i excite by a concentrate force vector F acting at it center of gravity. he receiver i moele a a four-ege imply upporte finite plate of uniform thickne., which i mae of the ame material a in the previou example. A lo factor.5 i introuce for the material. he tiffne coefficient of four reilient mount are choen a eign variable, i.e. we v have n 4 when coniering the tiffne component in every irection to be eign variable. he lower an upper limit min an max of the tiffne coefficient are given a an 5, repectively. he unit of tiffne coefficient of tranlational an rotational tiffne F,,,,, i component are N/m an Nm/ra. Firt, a vertical excitation force coniere. he initial value of all eign variable are taken to be 5 4, which provie a convenient reference for evaluation an icuion of the vibration reuction by optimization. At the ame time, the power flow without reilient mount i calculate a a reference to evaluate the effect of the iolator. When coniering vertical force excitation, the eign objective i inepenent on the tiffne component in the irection of the in-plane an twiting motion. However, the rotational tiffne with repect to the x an y irection an the vertical tranlational tiffne will influence the tranmitte power. Excitation frequency Power flow Initial eign Optimize eign Deign without reilient mount e e-5.67 able : Optimize reult for the cae with an excitation force in the z irection, only

13 A tate in able, it i foun that, in comparion with the initial eign an eign without reilient mount, the optimization ha reuce the power flow ignificantly for four ifferent excitation frequencie, =,, 5,. Next, an excitation force F,,,,, with a ifferent irection but the ame magnitue a above, i coniere. When imultaneouly coniering the vertical an horizontal (in the x irection) force excitation, the eign objective epen on the vertical tranlational tiffne, the rotational tiffnee with repect to the x an y irection, an the horizontal tiffne with repect to the x irection. he ame initial eign i ue. Excitation frequency Power flow Initial eign Optimize eign Deign without reilient mount e e able : Optimize reult for cae with excitation force in both the x an z irection he reult hown in able for thi cae alo illutrate that ubtantial reuction of the power flow can be achieve for the excitation frequencie coniere. When the excitation frequency i taken to be or, the eign without reilient mount give relatively lower value of the power flow compare with thoe of the initial eign. hi implie that an inappropriately choen iolator may increae the vibration tranmiion. he important concluion that can be rawn from the reult reporte in able an i that eign optimization i extremely ueful for the bet poible election of an iolator. 5 SHAPE OPIMIZAION OF A SUPPORING PAD PLACED ON A FLEXIBLE FLOOR FOR A RESILIEN MOUN hi ection eal with the hape optimization of a rotationally ymmetric pa (ee Figure 6) which, a illutrate in Figure 7, tranfer a vertical force F from a reilient mount to a itribute preure loaing f x on a flexible wooen floor. One- an two-parameter analytical expreion for the itribute preure loaing are erive in Brunkog an Hammer [3] for the problem of preing a rigi, plane inenter a mall uniform itance into the plane urface of an elatic boy. For implicity an in orer to apply the reult from [3], we hall moel the wooen floor a an infinite, iotropic, elatic plate a coniere in Section 4.. Moreover, we aume that the Young moulu of the pa i large relative to that of the plate, an that the raiu r b of the pa i ufficiently mall uch that the introuction of the pa oe not change the force equilibrium conition F c F an the motion compatibility c conition V V in Eq. (9). Figure 6: Schematic figure of a rotationally ymmetric upporting pa for a reilient mount 3

14 he pa i aume to be a circular truncate cone with a lant bounary ubject to eign. Due to rotational ymmetry of the pa, the eign problem can be formulate a a hape optimization problem of the curve part of the plane raial ection of the pa hown in Figure 7, where the y axi i oriente along the center line, an x inicate the raial irection. he eign bounary i efine by Spline curve evaluate in term of the poition of the given point a an b an three mater noe, i.e., noe, an 3 hown in Figure 7. he coorinate of the mater noe are choen a eign variable. he eign omain i mehe by a number of 4-noe axiymmetric finite element. F c r t b h l 3 y x a e h r f(x) r b Figure 7: he axiymmetric planar eign omain with three mater noe on the eign bounary All imenion of the eign omain are normalize by the raiu r b of the plane bottom urface, i.e., r t t r b, hl lrb, hr rrb. he hape optimization problem i formulate a min C PU x.t. V V x x x L where C i the tatic compliance of the tructure. Auming the point c of action of the force F to be fixe, the compliance C i efine a the calar prouct of the noal force vector P an the vector U of noal iplacement in the y irection at the bottom urface of the pa. Moreover, V enote the volume of the axiymmetric tructure, V i the given upper limit on the volume, an x L an x U enote allowable lower an upper limit of the eign variable which are the coorinate of three mater noe on the eign bounary. he upper limit of the volume i precribe a V =.5 which correpon to 75% of the volume of the circular truncate cone forme by 36 o rotation of the area a-b-c--e along the y axi when rb. he eign variable vector x i expree a 3 3 U (7) x= x x x y y y (8) where ubcript, an 3 repreent noe number of mater noe on the eign bounary. 4

15 A numerical example i preente here with the imenion of eign omain taken to be rb, rt.rb, hl.5rb, hr.5rb. he preure function i aopte from the paper [3] in the form c f x x, 4 c.59 (9) 6 where a mall value i introuce for avoiing ingularity when x equal r b at the outer ege, an the integral of preure over the circular bottom urface i obtaine a r b 5 (3) F f x xx he lower an upper limit of eign variable are choen a.7 x.,.4 x.7, x3.4, hb.5 y.,. y.35, an.35 y3.5 hl. he initial an final eign are compare in Figure 8. he value of the eign variable, volume an compliance of initial an optimize eign are given in able 3. A a reult of the hape optimization, the compliance i reuce from 5.73 to Initial eign Optimize eign x.8.8 x.5.5 x 3..7 y.. y.. y C V.5.5 able 3: Comparion of initial an final eign (V =.5) 6 CONCLUSIONS (a) Initial eign (b) Optimize eign Figure 8: Raial ection of initial an optimize pa eign he problem of optimization of the machinery mounting ytem in a lightweight builing i tuie in thi paper. A general mobility equation i evelope for the ytem coniting of ource, iolator an receiver of vibration by aembling the mobility matrice of each ubytem. he objective of minimizing vibration tranmiion i realize by izing optimization of tiffne coefficient of reilient mount. In comparion with eign without reilient mount, eign with optimize iolator can provie ignificant reuction of the power flow. 5

16 Shape optimization of a upporting pa for a reilient mount i alo tuie. Such a pa may be ue for tranferring a vertical force from a reilient mount to a itribute preure loaing on a flexible floor. Artificial moeling of the reilient mount an the floor of the builing i aopte in thi tuy. Work on more practical moeling of reilient mount an builing tructure i in progre. REFERENCES [] P. Srinivaulu, C.V. Vaiyanathan, Hanbook of machine founation. ata Mcgraw Hill, 976. [] J.A. Snyman, P.S. Heyn, P.J. Vermeulen, Vibration iolation of a mounte engine through optimization. Mechanim an Machine heory, 3, 9-8, 995. [3] R. Alkhatib, G.N. Jazar, M.F. Golnaraghi, Optimal eign of paive linear upenion uing genetic algorithm. Journal of Soun an Vibration, 75, , 4. [4] H. Ahrafiuon, Deign optimization of aircraft engine-mount ytem. Journal of Vibration an Acoutic-ranaction of the Ame, 5, , 993. [5] S. Xie, S.W. Or, H.L.W. Chan, P.K. Choy, P.C.K. Liu, Deign optimization of machinery mounting ytem with an elatic upport tructure. Engineering Optimization, 39, 9-44, 7. [6] H.G.D. Goyer, R.G. White, Vibrational power flow from machine into built-up tructure Part : Introuction an approximate analye of beam an plate-like founation. Journal of Soun an Vibration, 68, 59-75, 98. [7] H.G.D. Goyer, R.G. White, Vibrational power flow from machine into built-up tructure Part 3: Power flow through iolation ytem. Journal of Soun an Vibration, 68, 97-7, 98. [8] J. Pan, J.Q. Pan, C.H. Hanen, otal power flow from a vibrating rigi boy to a thin panel through multiple elatic mount. Journal of the Acoutical Society of America, 9, , 99. [9] L. Sun, A.Y.. Leung, Y.Y. Lee, K. Song, Vibrational power-flow analyi of a mimo ytem uing the tranmiion matrix approach. Mechanical Sytem an Signal Proceing,, , 7. [] F. Fahy, P. Garonio, Soun an tructural vibration, n Eition. Elevier, 7. [] L. Cremer, M. Heckl, B.A.. Peteron, Structure-borne oun: Structural vibration an oun raiation at auio frequencie. Springer, 5. [] C.M. Harri, A.G. Pierol (e.), Shock an vibration hanbook. McGraw-Hill, Lonon,. [3] C.. Molloy, Ue of four-pole parameter in vibration calculation. he Journal of the Acoutical Society of America, 9, , 957. [4] J.I. Soliman, M.G. Hallam, Vibration iolation between non-rigi machine an nonrigi founation. Journal of Soun an Vibration, 8, 39-35, 968. [5] P. Garonio, S.J. Elliott, Driving point an tranfer mobility matrice for thin plate excite in flexure. echnical report no 77, ISVR, Univerity of Southampton, 998. [6] S. Ljunggren, Generation of wave in an elatic plate by a vertical force an by a moment in the vertical plane. Journal of Soun an Vibration, 9, , 983. [7] S. Ljunggren, Generation of wave in an elatic plate by a torional moment an a horizontal force. Journal of Soun an Vibration, 93, 6-87, 984. [8] F. Fahy, J. Walker (e.), Avance application in acoutic, noie an vibration. Spon Pre, New York, 4. 6

17 [9] K. Svanberg, he metho of moving aymptote - a new metho for tructural optimization. International Journal for Numerical Metho in Engineering, 4, , 987. [] C.M. Mak, J.X. Su, A power tranmiibility metho for aeing the performance of vibration iolation of builing ervice equipment. Applie Acoutic, 63, 8-99,. [] S.S. Rao, Mechanical vibration, 4th eition. Prentice Hall, 3. [] B. Peteron, J. Plunt, On effective mobilitie in the preiction of tructure-borne oun-tranmiion between a ource tructure an a receiving tructure, Part : heoretical backgroun an baic experimental tuie. Journal of Soun an Vibration, 8, 57-59, 98. [3] J. Brunkog, P. Hammer, Rigi inenter excitation of plate. Acta Acutica Unite with Acutica, 89, 46-47, 3. 7