DESIGN ANALYSIS OF A RUBBER MOUNT SYSTEM FOR A PUSH- TYPE CENTRIFUGE Carl Nel

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1 ICSV4 Carns Australa 9-2 Jul, 27 DESIGN ANALYSIS OF A RUBBER MOUNT SYSTEM FOR A PUSH- TYPE CENTRIFUGE Carl Nel School of Mechancal Enneern, North-West Unverst Potchefstroom, South Afrca carlbnel@netactve.co.a Abstract The lack of balance, whch occurs n all rotatn machnes, could cause troublesome vbraton f the attachments to the foundaton of the machne are unsutable. Ths s partcularl true for centrfues, where uneven loadn of the basket, for eample, due to temporar rreulartes n the feed or laers of salt sedment on the basket, ma lead to partcularl bad balancn. The am of ths nvestaton was to evaluate the use of rubber mounts and to predct feasble mount stffness coeffcents to be used. These mounts have to produce ood vbraton solaton to protect the supportn buldn from dnamc forces transmtted, but must also lmt the statc and dnamc mount dsplacements to allow acceptable moton of the centrfue assembl. It was therefore necessar to use a s-deree of freedom mathematcal model, mplemented n computer prorams to compute these statc and dnamc mount dsplacements and mount forces, for 2 possble desns. An objectve functon as a measure of vbraton transmtted, was also used. In addton, the mount sstem natural frequences were also computed, to prevent resonance wth the normal basket anular speed and also the forced frequenc of the aal pusher mechansm.. INTRODUCTION Ths nvestaton reardn vbraton solaton was done for centrfues located at the frst level of supportn renforced concrete buldn. A s-deree of freedom model was used. A drect approach n terms of forces transmtted to the support structure s consdered. 2. MATHEMATICAL MODEL A mount sstem model smlar to that emploed b Nel and Hens s consdered [3], [5], [6]. Hence the centrfue assembl (centrfue and base plate) s dealsed as a rd bod of mass m attached to a rd support structure b means of an arbtrar number n of elastc mounts

2 wth arbtrar postons and orentatons wth respect to the centrfue assembl lobal coordnate sstem. The orn of the fed lobal co-ordnate sstem s located at, the center of ravt (c..) of the centrfue assembl as shown n fure. It s assumed that the stffness coeffcent k, k, k n the three co-ordnate drectons are ndependent of each other f the rotatonal stffness of the mount s nelected. For elastomerc materals, as are usuall used at rubber mounts, a hsteretc dampn model wth a comple stffness matr s assumed here [2], [4], [5], see also fure, so that j k j k j k ) ( ) ( ) ( η η η () where j =, and η, η and η are the mount loss factors n each of the three co-ordnate drectons. A transformaton s requred to relate the translatonal dsplacements of each mount to translatonal and rotatonal dsplacements of the centrfue assembl c.. Assumn small centrfue dsplacements, G s defned b u = = = = G U (2) where u s the translatonal dsplacement matr at mountn pont and U are the centrfue assembl c.. dsplacements.,, are translatonal and,, rotatonal dsplacements. Wth F vector comprsn the three forces and the three moments due to mount actn on the centrfue assembl alon, and and about, and respectvel, t now follows that F = - [ ] T G K G U (3) Assumn aan small dsplacements, the rd bod equatons of moton [] s M Ü = F (4) wth F the matr of resultant total forces and moments on the centrfue assembl. The sstem mass matr s M = I I I m m m (5)

3 where m s the total centrfue assembl mass and I, I and I are the elements of centrfue assembl nerta epressed n terms of the lobal co-ordnate sstem. The centrfue assembl c.. acceleraton matr s Ü = && && && && && && (6) where & &, & &, & & are translatonal acceleratons and &, &, & anular acceleratons. Addn toether the effects of all the mounts of the centrfue assembl, t follows that n F = - n T G K G U (7) = = = - K U where K s s a comple dnamc matr whch ncludes hsteretc loss effects at the centrfue c.. From ths and equaton (4) t now follows that M U & + K U = F e (8) where F e represents all forces and moments on the centrfue assembl other than the mount reacton forces. In order to take advantae of the frequenc doman approach, F e s assumed to comprse of v snusodal forces and moments ncluded b F wth correspondn frequences ω h and phase anles a h h,2,..., v, =. It ma then be shown that dh v U = h= 2 [ K ] ω h M [ d ] h F (9) where U and F dh represent dsplacement and force phasors respectvel [2]. The centrfues assembl shakn forces and moments are F F e = e F F M M M e e F F = M M M e e e e e e e e () The model ma be emploed to determne an objectve functon. The dnamc dsplacements at each mount n the, and drectons are computed n the tme doman. It s thus possble to compute for each force component at each mount the averae value of the

4 sum of the dnamc forces at each tme ncrement dt over the perod T. As crteron of vbraton transmtted, an objectve functon s defned b T n T ϕ = O ( f ( X ) + f ( X ) + f ( X )) dt = () n whch X s a vector of desn varables chosen here to comprse stffness coeffcents. It s now requred to mnme wth respect to varables X, subject to nequalt constrants w = U ma( U ) U (2) s + d c where U c s a vector of specfed mamum acceptable mount translatonal dsplacement ampltudes. The mamum dnamc translatonal dsplacement elements n U d are n the same drecton as the statc dsplacement elements n U s. These constrants also ve an ndcaton of acceptable centrfue assembl moton and acceptable mount dsplacements. 3. CHARACTERISATION OF PARAMETERS The manufacturer suppled the centrfue and base plate mass and centre of mass propertes. These propertes were used n the computaton of the moments of nerta for the total centrfue and base plate assembl for each lobal as. The total prncpal moments of nerta are , and km 2 correspondn to the centrfue lobal, and aes as shown n fure. The total centrfue assembl mass s 98 k. The lobal coordnates for each of the mounts and those for the basket centre were determned. The centrfue assembl s supported b 4 mounts postoned at the follown lobal co-ordnates: Table 4. Mount lobal co-ordnates [m]. Mount The lobal co-ordnates of the basket centre are: Table 5. Basket centre lobal co-ordnates [m]. c c c The basket radus s 4 mm and the reatest possble ecess mass actn on ths radus s 2.3 k (provded b the manufacturer). The normal runnn speed of the basket s 2 rev/mn. The stffness coeffcents of the 2 estn ver soft ppe compensators used were nelected, because these mantudes were rearded as ver small compared to the stffness

5 coeffcents of tpcal rubber mounts. The dnamc mount stffness coeffcents of standard mounts avalable were chosen for 2 dfferent possble desns A and B as shown n Table 6. Table 6. Mount stffness coeffcents [kn/m]. Mount stffness coeffcents [kn/m] Desn A Desn B (Resonance) (Feasble) k, k 2, k 3, k 4 2 k, k 2, k 3, k 4 2 k, k 2, k 3, k Snce mount loss factors n the rane of.5 to.5 do not apprecabl affect the dnamc behavour, and rubber mounts normall have mount loss factors n ths rane, mount loss factor values of. were assumed for all the mounts n all 3 orthoonal drectons. 4. COMPUTER IMPLEMENTATION The mathematcal model was mplemented n computer prorams n a MATLAB envronment. The mantudes of the characterstcs of the parameters descrbed above were used as nput data. The centrfue assembl shakn forces and moments were computed frst, and then used n the computaton of dnamc and statc mount dsplacements, dnamc and statc mount forces transmtted to the support structure throuh the mounts and also objectve functon values. 5. CENTRIFUGE ASSEMBLY SHAKING FORCES AND MOMENTS The mantudes of the parameters requred for computaton of the centrfue assembl shakn forces and moments and mathematcal equatons mplemented n computer prorams were also used, to compute these forces and moments. Fure 2 shows ths force and moment tme hstores for the centrfue anular speed at 2 rev/mn. The frequenc observed from these wave forms, corresponds to the centrfue basket anular speed. These tme doman forces and moments were Fourer analsed b usn the MATLAB fft.m functon, and the Fourer coeffcents at correspondn frequences and phase anles were then used n the computaton of the responses n the, and drectons at each mount b usn the frequenc doman computer proram (see for eample fure 3). Table 7 shows the force and moment ampltudes at phase anles. The force mantude of the aal pusher mechansm was consdered relatvel small compared to centrfual forces, and therefore nelected. The mantude of ths ver low forced frequenc force s mportant, and was taken nto account when the mount sstem natural frequences were evaluated (see Table 9), to avod resonance. Table 7. Shakn force [N] and moment [Nm] ampltudes at phase anles [rad] and anular speed of rad/s. F e F e M e M e M e Ampltudes [N] and [Nm] Phase anle [rad]

6 6. RESULTS AND EVALUATION As crteron or measure of vbraton transmtted to the support structure, the objectve functon (equaton ) was used. Ths snle objectve functon value was then computed for each of the 2 dfferent desns A and B and also a rd mounted sstem desn, as shown n Table 8 for a basket speed at 2 rev/mn. Lare objectve functon values ndcate bad vbraton solaton, and lower objectve functon values mproved vbraton solaton [5], [6]. Table 8. Objectve functon [kn] at 2 rev/mn. Desn A Desn B Rd desn (Resonance) (Feasble) Objectve functon ψ [kn] B usn the MATLAB e.m functon, the eenvalues and eenvectors were also computed for the 2 dfferent desns A and B. Table 9 shows these natural frequences and mode shapes. The stffness coeffcents used for desns A and B are shown n Table 6. These natural frequences are mportant n order to avod resonance wth the basket normal anular speed (2 H), but also wth the aal pusher mechansm force frequenc ( to.25 H). Table 9. Natural frequences [H] and correspondn mode shapes. Natural frequences [H] and mode shapes Desn A Desn B CONCLUSIONS The use of the ver low stffness coeffcents renders the best vbraton solaton, but has the larest statc mount dsplacements. The use of ver low stffness coeffcents also results n low sstem natural frequences, whch could easl correspond to the pusher mechansm force frequenc ( to.25 H), thus resonance. The use of ver low stffness coeffcents also results n unacceptable centrfue assembl moton, thus lare mount dnamc dsplacements. Desn A wth the moderatel hh stffness coeffcents renders the poorest vbraton solaton (see Table 8 for comparson of objectve functon mantudes), but also has smaller statc mount dsplacements compared to desn B, but results n the larest dnamc mount dsplacements. Ths vbraton solaton s unacceptable. The use of hher stffness coeffcents n eneral tends to create natural frequences whch could easl correspond, or be near to the centrfue basket operaton speed, see also Table 9, whch s the case for desn A. When one of the mount sstem natural frequences corresponds to the forced basket frequenc, then resonance takes place, whch s undesrable and danerous and could lead to falures and mount durablt problems. The objectve functon used has the larest values when resonance takes place (see Tables 8 and 9), whch ndcates that ts use s effectve for the measure of vbraton transmtted.

7 Desn B renders the best compromse between vbraton solaton and acceptable mount dsplacements. The natural frequences of desn B are also far awa enouh from the basket s normal operatonal speed, and also the forced frequenc of the pusher mechansm (see Table 9). The vbraton solaton obtaned for desn B s also snfcantl better compared to a desn where mount stffness coeffcents were chosen to represent a centrfue whch s rdl attached (see Table 8 for comparson of objectve functon mantudes). REFERENCES [] A.F. D Soua and V.K. Gar, Advanced dnamcs Modeln and analss, Prentce-Hall, 984. [2] D.J. Ewns, Modal Testn: Theor and practce, Research Studes Press, 984. [3] P.S. Hens, C.B. Nel, and J.A. Snman, Optmsaton of enne mountn confuraton. Proceedns of ISMA 9. Tools for Nose and Vbraton Analss, Volume II. Katholeke Unverstet Leuven, Belum, September 994. [4] C.B. Nel, and P.S. Hens, An optmsaton approach to mountn charactersaton, Proceedns of Nose and Vbraton 95, Unverst of Pretora, November 995. [5] C.B. Nel and P.S. Hens, Epermental verfcaton of an optmsaton proram for a front-wheel-drve enne mount sstem, Proceedns of ISMA2, Nose and Vbraton Enneern, Voume. III, Katholeke Unverstet, Leuven, Belum, September 996. [6] C.B. Nel, Optmsaton of enne mount sstems for multple operatonal condtons at front-wheel-drve vehcles, Proceedns of ISMA25, Internatonal Conference on Nose and Vbraton Enneern, Katholeke Unverstet, Leuven, Belum, September 2.

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