Minimizing Machinery Vibration Transmission in a Lightweight Building using Topology Optimization

Transcription

1 1 th World Congress on Structural and Multidiscilinary Otiization May 19-4, 13, Orlando, Florida, USA Miniizing Machinery Vibration ransission in a Lightweight Building using oology Otiization Niels Olhoff, Bin Niu Deartent of Mechanical and Manufacturing Engineering, Aalborg University, DK-9 Aalborg, Denar 1. Abstract his aer deals with the roble of iniizing the vibration transission in a lightweight building fro rotating achinery, by toological design of a flexible base late for the achinery. A odular lightweight building is odeled and used for the analysis of the vibration transission. he design obective is chosen as iniization of the roduct of the frequency resonse in the building transitted fro the achinery via walls and floors, and the volue of aterial used in the design of the base late. Different excitation frequencies are considered. he design and erforance of the otiized achinery base lates are illustrated and discussed via nuerical exales.. Keywords: Machinery, oology otiization, Lightweight building, Vibration, Base late. 3. Introduction Due to the existence of dynaic forces of e.g. echanical, fluid dynaic or electro-agnetic origin, rotating achinery ay be considered a source that excites forced vibration of the foundation, and thereby the floors and walls, etc., of the building. he transission of such vibrations through the building ay result in undesirable sound eission and unsatisfactory cofort conditions for the eole in dwellings and offices of the building. o reedy this, the obective of this wor is to develo and ileent a ethod of toology otiization for the design of a achinery foundation. Hence, this aer studies the couled syste of the achinery installation and the building. We roose that the foundation of the achinery should be a flexible, single-aterial base late that is toology otiized within a quadratic, adissible design doain which is suorted by the floor of a storey or the roof of the building, and iniizes the vibration transission in the building. he reader is referred to the achinery installation indicated on to of Roo -3 in Figure 1 with a close-u in the right-hand side showing the achine and the quadratic, adissible design doain for its base late. he achine is considered to be relatively stiff (aroxiately considered as a rigid body), which is rigidly connected to the base late via four bolts, and the base late is assued to be connected to the elastic roof (or floor) of the building by oent-free hinges at the four corners of the adissible design doain. For the convenience and consistence of ileentation with the other arts, the achine is odeled as a three-diensional elastic body with a higher Young s odulus than that of the base late. he design obective is chosen as iniization of the roduct of the frequency resonse in the building transitted fro the achinery via walls and floors, and the volue of aterial used in the design of the base late. he roduct obective in the toology otiization of vibrating structures is studied in Ref. [1], which is insired and extended fro the wor on extended otiality in Refs. [-4] where Rozvany et al. studied the iniization of the roduct of the static coliance and the volue of aterial used. Figure 1 Scheatic of building and the achinery with the quadratic adissible design doain for the base late 1

2 In the current roduct forulation of siultaneously iniizing the vibration transission in the building and the aount of aterial for the base late, we found it to be exedient to secify a axiu constraint for the static dislaceent of the achine, since the otiization of the base late otherwise attets to achieve iniu frequency resonse (zero) of the building by searating the building fro the vibration source. his trivial solution is, of course, not useful. In this aer, the vibration transission in the building is calculated for a odular lightweight building which is entirely constructed by translation and cobination of three basic odular eleents, i.e., one floor anel and two different wall anels. hese anels are all considered as bea-stiffened double-leaf anels. Furtherore, an artificial seleton is introduced to facilitate the odelling of different connections between odules in the building. More details on the ileentation of the araetric odelling of a odular building are available in our coanion aers [5, 6]. he organization of the rest of the aer is as follows. he otiization forulation is described in Section 4, followed by aterial interolation in Section 5. Section 6 resents the sensitivity analysis of the obective and constraint functions. Nuerical exales are outlined in Section 7 for different excitation frequencies and values of the axiu static dislaceent constraint for the achine. Finally, a section with discussions and conclusions closes the aer. 4. Otiization forulation Since the vibration of the building is alost exclusively deterined by the building alone, the toological design of the base late has little influence on the vibration odes of the building. his eans that the vibration ode of a articular receiving roo cannot be influenced significantly by toological design of the base late. However, different designs of the base late will result in different transitted forces on the installation floor. herefore, the roble of iniizing the vibration transission in the building can be considered as iniizing the agnitudes of the transitted forces on the installation floor while rescribing different frequency-deendent boundary conditions at the four corner suorts at the quadratic design doain of the base late. he frequency-deendent boundary conditions can be deterined fro the forced vibration analysis of the building by alying searately a unit excitation force of the given excitation frequency at each rescribed installation oint. Since the roerties of the building are given, the agnitude of the transitted forces on the installation floor can be equivalently considered by the agnitudes of the dislaceents of the installation oints on the floor. he otiization roble for iniizing the roduct of the vibration transission and the volue of aterial used can be reforulated as iniize U V V subect to Ku Cu Mu f- f, (b) g t K u eq c b t u f, (c) g, (d) 1, 1,, N. (e) d (a) (1) he obective function in Eq. (1a) is defined as the roduct of the squared dislaceent resonse U at the N V installation oints and the volue ratio V between the volue of aterial V v of the base late, where d1 v is the volue of the -th eleent of the base late design, and the volue of aterial V of the adissible design doain. he design variables, 1,, Nd, reresent the voluetric aterial densities of the finite eleents, where N d is the nuber of finite eleents of the adissible design doain. he squared dislaceent resonse U in the obective is evaluated as U 1 1 u u u u, () c c c c Nc Nc c where the vector u denotes the translational dislaceents of the four installation oints, cf. Figure, and N c is the nuber of the translational drees of freedo at the installation oints. he overbar denotes colex conugate of a vector or atrix, and the suerscrit reresents the transose of a atrix or vector. u is extracted fro the dislaceent vector u by introducing a location atrix L, which is a zero atrix with unit values at c

3 the diagonal eleents corresonding to the translational drees of freedo of the nodes of the installation oints. Figure Scheatic of achinery installation in a lightweight building he sybols K, M, and C are the stiffness, the ass, and the daing atrices of the base late with the eq achine, u is the dislaceent vector of the base late with the achine, and K b is the frequency-deendent colex equivalent dynaic stiffness atrix of the building at the installation oints. For different excitation frequencies and different installation ositions in the building, different vibration odes of the building ay be eq excited so that the resonse at the installation oints on the building floor will be different. herefore, K b should be couted via the forced vibration analysis of the building at each given frequency. he sybol f is the excitation force alied at the centre of gravity of the achine, and f t denotes the transitted force at the installation oints on the floor where the achine is installed via the base late. For the four installation oints, the c transitted force f t on the floor and the corresonding dislaceent vector u b are written as f t ft1 ft ft3 f t4, c c c c c b b1 b b3 b4 u u u u u. (3) he forces and dislaceents at the interface of the contiguous base late and the floor are shown in Figure, where the arrows define the ositive directions of the forces and the dislaceents. he couling between the base c c late with the achinery and the building is realized by the coatibility of the dislaceents, u u b. If a fixed oent-free hinge connection is assued at the ounting unction on the floor, only force transission taes lace, and the force vector f t1, see Figure, only has translational coonents in the x-, y-, and z-directions, t1 f x t1, f y t1, f z t1 f. (4) Furtherore, if the in-lane otion of the base late is ignored, only the vertical force in the z-direction should be eq considered. For this case, K b is reduced to a 4 4 colex atrix. Eq. (1d) secifies a constraint on the static ( =) dislaceent of the achine which revents a trivial solution with an unreasonably large value of this dislaceent. he function g u is couted by Eq.(5) as the root-ean-square (RMS) value of the dislaceents of the achine, and g is the uer allowable liit for this RMS value. We have 1 g u u u, (5) N where u denotes the dislaceent vector of all drees of freedo of the achine, and N is the nuber of the drees of freedo of the dislaceents of the achine. he location atrix R is defined for extracting the dislaceent vector u of the achine fro the dislaceent vector u, i.e. u Ru, cf. Sub-section 6.. Eq. (1e) secifies the lower and uer liits and 1. for the design variable. For this toology otiization roble with a single aterial, a sall ositive value lie.1 is rescribed for in order to avoid nuerical singularity. ex i t in the reainder. he sybol We assue tie haronic otion, and oit the tie deendent ter 3

4 denotes the excitation frequency, and i reresents the iaginary unit, i 1. Daing is iortant for the vibration analysis and design of lightweight buildings, but we disrard daing in this aer and assue C =. he otiization forulation in Eq. (1) can be rewritten as iniize U g V V subect to K i C M +K u f, (b) eq b u d g, (c) 1, 1,, N. (d) (a) (6) 5. SIMP odel for toology otiization of late structures A single solid aterial is eloyed in the toology otiization of the base late. As is well nown, a transfored and relaxed continuous roble instead of the discrete /1 design roble is adoted by introducing soe for of continuation and enalization. he classical SIMP (Solid Isotroic Material with Penalization) interolation schee [7, 8] is used in this study. he finite eleent stiffness and ass atrices of the e-th eleent in the design doain can be exressed as resectively, where e and q, (7) e e e e, (8) e e e e e denote the eleent stiffness and ass atrices corresonding to the given solid elastic aterial. Here, the sybols and q are enalty araeters, and continuation of the enalization is adoted in the otiization iterations by gradually increasing and q fro 1. to 3.. he sae factors =3 and q =3 for stiffness and ass atrices are finally used until convergence. In order to suress the nuerical instability and alying a iniu length constraint, the basic density filtering [9] is used in this aer. 6. Design sensitivity analysis he sensitivities of the obective and constraint functions will be derived analytically in Sub-sections 6.1 and 6., resectively Sensitivity of obective function Fro Eq.(1), the sensitivity of the obective function is couted as V U V V V U V U V where the sensitivity of the used aterial volue is couted as V, (9) v, (1) and the sensitivity of the squared dislaceent resonse U is couted via the adoint ethod. he following function is constructed after introducing Lagrangean ultiliers λ and λ where the dynaic stiffness atrix S is written as, cf. Eq. (6b), L U λ Su fλ Su f, (11) S K ic M +K. (1) eq b Furtherore, we can obtain fro the governing equation in Eq. (6b), Su f, (13) 4

5 Su f. (14) he design sensitivity can be analyzed as below if the loading f is assued to be design indeendent L 1 ulu λ λ S u S u Nc Su f Su f λ u S λ u S. (15) After rearrangeent and by setting the coefficient atrix of the ters with the derivative of dislaceent equal to zero, we obtain the sensitivity of the Lagrangean function which is equivalent to Eqs. (11) and (17) give L S S λ uλ u, (16) L λ S u. (17) Re U λ S u. (18) Re he sensitivity of the obective function is easily derived using the result in Eqs. (18) and (9) and the haronic otion assution. he aterial interolation forulas in Eqs. (7) and (8) are used for evaluating the derivatives of the stiffness and ass atrices. 6.. Sensitivity of constraint functions For the constraint on the static dislaceent of the achine, the constraint function g u is defined in Eq. (5), and exressed in ters of u, N and the location atrix R, the sensitivity of the constraint function is couted as 1 uru g 1 1 N 1 g 1 g uru N (19) he adoint ethod is also used for deriving the design sensitivity of the function g, and the following function is constructed after introducing the Lagrangean ultilier for the excitation frequency, eq where b Lg g K K +K, and the design sensitivity can be exressed as α Ku f. () L 1 uru g α K u N Ku f α u K. (1) By setting the coefficient atrix of the ters with the derivative of dislaceent equal to zero, and aing use of of the syetry of the stiffness atrix K and the location atrix R, we obtain Lg K α u. () he sensitivity of the constraint function g is now easily derived fro Eqs. () and (19) g 1 g K α u. (3) 5

6 With these sensitivity results, the otiization roble in Eq. (6) ay be solved by a atheatical rograing ethod, e.g., the Method of Moving Asytotes (MMA) by Svanberg[1]. he convergence criterion is secified as follows. When the axiu relative change of the design variables in two consecutive iterations is saller than a secified value, e.g..1, or the relative change of the values of the obective function in two consecutive iterations is saller than a secified value, e.g. 1e-5, the otiization iterations are stoed. 7. Nuerical exales: iniization of the roduct of the building vibration transission and the volue of base late aterial Building 3-3 as shown in Figure 1 is considered with the diensions of a roo chosen as: length L = 6, height H = 3.6, and width W = 4.8. he base late with the achine is installed on to of Roo -3 through the hinges in the corners of the quadratic design doain for the base late with continuity of only the translational drees of freedo, and the equivalent colex stiffness atrix of the building is couted corresondingly at the four installation oints for each given excitation frequency. he achine is assued to be installed at the centre of the base late by four rigid ounts, where the translational and rotational drees of freedo are fully couled. In the following exales, an excitation loading is alied at the centre of gravity of the achine with the alitude vector f,, 1, i.e., 1 N in the vertical z-direction. he geoetric and aterial araeters of the base late and the achine are listed in able 1 and able, resectively. he base late is eshed into nine-node Mindlin late eleents with three drees of freedo er node, where the in-lane otion is ignored. able 1 Geoetric and aterial roerties of the base late Base late (Aluinu) Diension of the late: Young s odulus Pa Mass density 7 g/ 3 hicness. Meshing nine-node eleents able Material and geoetric araeters of the achine Machine (3D elastic body, ade by steel) Length. Width. Height.1 Young s odulus Pa Mass density 78 g/ 3 Mass = 31. g 7.1. Nuerical exales In this subsection, the toology otiization of the base late is studied by assuing no daing for the base late and the building structures. Free vibration analysis is first erfored for the syste including the building, the achine, and the initial base late. Here, the initial design of the base late is defined by all the (density) design variables of the adissible design doain being equal to.5, which is rarded as a reference design for coarison. he eigenfrequencies and eigenodes of the building syste with the initial base late are resented in Figure 3, where the first eigenfrequency is 9.31 Hz. 6

7 (a) Mode 1, f 1 = 9.31 Hz (b) Mode, f =1.469 Hz (c) Mode 3, f 3 = 13. Hz (d) Mode 4, f 4 =19.89 Hz (e) Mode 5, f 5 =.153 Hz (f) Mode 6, f 6 =1.4 Hz (g) Mode 7, f 7 =1.969 Hz (h) Mode 8, f 8 =.885 Hz Figure 3 Eigenfrequencies and eigenodes of the building syste with the initial base late (a-h). Red colour reresents larger (nuerical) dislaceent agnitude, and blue reresents zero dislaceent. 7

8 hen, design otiization for iniizing the roduct of the vibration transission and the used volue of aterial forulated in Eq. (6) is carried out for different single excitation frequencies, f = 5,, 4 Hz. Different uer liits g of the RMS value of the axiu allowable static dislaceents is studied, e.g., g =3, and g =5. he density filter radius is two ties the length of the late eleent. (a) Excitation frequency f = 5 Hz (b) Excitation frequency f = Hz (c) Excitation frequency f = 4 Hz Figure 4 Otiized toologies of the base late with excitation at different frequencies for iniizing the roduct of the vibration transission in Building 3-3 and the volue of aterial used for the base late. Given uer liit on the static dislaceent of the achine: g =3 : (a) excitation frequency f = 5 Hz, (b) f = Hz, (c) f = 4 Hz able 3 Coarison of the obective, the squared dislaceent resonse at the installation ositions, and the aount of used aterial for the reference and the otiized designs in Figure 4(a-c) U V Relative V decrease of U V obective V Reference Otiized Reference Otiized Reference Otiized Figure 4(a) at f = 1 Hz e e % e-8 1.7e Figure 4(b) at f = Hz e e %.38e e Figure 4(c) at f = 4 Hz e e % e e (a) Excitation frequency f = 5 Hz (b) Excitation frequency f = Hz (c) Excitation frequency f = 4 Hz Figure 5 Otiized toologies of the base late with excitation at different frequencies for iniizing the roduct of the vibration transission in Building 3-3 and the volue of aterial used for the base late. Given uer liit on the static dislaceent of the achine: g =5 : (a) excitation frequency f = 5 Hz, (b) f = Hz, (c) f = 4 Hz 8

9 able 4 Coarison of the obective, the squared dislaceent resonse at the installation ositions, and the aount of used aterial for the reference design and the otiized designs in Figure 5(a-c) U V Relative V decrease of U V V obective Reference Otiized Reference Otiized Reference Otiized Figure 5(a) at f = 5 Hz e e % e e-1.39 Figure 5(b) at f = Hz e e %.38e e Figure 5(c) at f = 4 Hz e e % e-1.317e-1.65 he otiized toologies of the base late at the excitation frequencies f = 5,, and 4 Hz are shown in Figure 4(a-c) for g =3 and Figure 5(a-c) for g =5. For the otiized designs, the constraint on the static dislaceent of the achine is always active. As a coarison, the squared dislaceent resonse U for the reference base late is couted for different cases. he values of the obective function, the volue ratio and the squared dislaceent resonse at different excitation frequencies for the reference and the otiized base lates in Figure 4(a-c) and Figure 5(a-c) are listed in able 3 and able 4, resectively, where it is seen that the obective has generally been decreased quite substantially for the otiized designs relative to the reference design in all the exales. he otiization has decreased the vibration transission into buildings, and siultaneously reduced the aount of used aterial for the excitation frequencies f = 5,, and 4 Hz. 8. Conclusions Vibration transission in lightweight buildings is iniized by design otiization of the base late of achinery at different frequencies. Miniization of the roduct of the vibration transission into the building and the volue of aterial used in the base late is chosen as the design obective. Since the iniization of the vibration transission would lie to searate the building fro the achinery, which results into a trivial solution, a constraint on the static dislaceent of the achinery is introduced in order to rovide a ractically eaningful design. Otiized toologies of the base late are obtained for different excitation frequencies and different static dislaceent constraints. For low frequencies, the designs are stiffness-driven. Since the building floors can be considered relatively stiff in the low frequency range, the otiu toologies are very siilar for different installation ositions in the building. For larger excitation frequencies, the vibration of the building floor has ore influence on the vibration ode of the base late and the vibration transission into the building. For excitation frequencies larger than the first eigenfrequency of the base late, the toologies becoe ore ass-driven and generally exhibit soe concentration of ass in the central area. he influence on the otiu base late toology of aterial daing and of different base late installation ositions in the building, is subect to further study. 9. Acnowledgeents his wor has been funded by the EU InterR Proect Silent Saces and Aalborg University. 1. References [1] B. Niu, N. Olhoff, G. Cheng, oological design for iniization of the roduct of dynaic coliance and aterial cost for structures subected to forced vibration in a range of excitation frequencies, 13, in rearation. [] G.I.N. Rozvany, O.M. Querin, Z. Gasar, V. Poezansi, Extended otiality in toology design, Structural and Multidiscilinary Otiization. 4, 57-61,. [3] G.I.N. Rozvany, A critical review of established ethods of structural toology otiization, Structural and Multidiscilinary Otiization. 37, 17-37, 9. [4] G.I.N. Rozvany, raditional vs. Extended otiality in toology otiization, Structural and Multidiscilinary Otiization. 37, , 9. [5] L.V. Andersen, P.H. Kiraard, K. Persson, N. Kiel, B. Niu, A odular finite-eleent odel for analysis of vibration transission in ulti-storey lightweight buildings, Proceedings of the Eleventh International Conference on Coutational Structures echnology (CS1), Civil-Co Press, Dubrovni, Croatia, 1. [6] B. Niu, L.V. Andersen, N. Kiel, O. Flodén, G. Sandberg, Vibration transission in a ulti-storey lightweight 9

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