Loss factor for a clamped edge circular plate subjected to an eccentric loading

Transcription

1 ndian ounal of Engining & Matials Scincs Vol., Apil 4, pp Loss facto fo a clapd dg cicula plat subjctd to an ccntic loading M K Gupta a & S P Niga b a Mchanical Engining Dpatnt, National nstitut of Tchnology, Kuukshta, ndia b Mchanical & ndustial Engining Dpatnt, ndian nstitut of Tchnology, Rook, ndia Rcivd 3 Fbuay 3; accptd 6 Fbuay 4 Psnt aticl dals with th analytical valuation of odal loss facto of a clapd dg thin cicula plat of unifo thicknss subjctd to an ccntic haonic point load. Stating with th fundantal quation of focd tansvs vibation of a thin lastic plat in ts of Gn s function; th xpssion fo th dflctions is dtind. Stsss a calculatd with th hlp of gnal stss-dflction quations and th odal loss facto is valuatd at sonanc condition. Th nodal cicls adii a also dtind analytically fo ach od shap. PC Cod: nt. Cl. 7 G L / Th atial daping, which is also known as hysttic daping o stuctual daping, plays an ipotant ol in th contol of th vibatoy spons of stuctual bs. t is also of ipotanc in th acoustical fild to contol th sound adiation fo bs lik plats. t is asud in ts of odal loss facto. Lazan has discussd th pocdu fo th valuation of th intnal loss facto of vaious bs. Lissa 3 has viwd th vibation spons of plats. Mos and ngad 4 hav also studid th vibation spons of a clapd, thin cicula plat of unifo thicknss with ccntic point load. Niga, t al. 5 hav valuatd th odal loss facto fo a ctangula plat of vaiabl thicknss. Littl wok is availabl in litatu on analytical valuation of atial daping of thin cicula plats of unifo thicknss with an ccntic haonic point load. Th schatic psntation of a clapd dg, thin, lastic, hoognous, and isotopic cicula plat of unifo thicknss subjctd to an ccntic haonic point load is shown in Fig.. A pola coodinat axis syst is takn fo th analysis. Th classical sall dflction thin plat thoy has bn ployd. To account fo ngy dissipation, igidity has bn takn in coplx fo 6. Stating with th fundantal quation of focd tansvs vibation of thin lastic plat in ts of Gn s function 4 ; th xpssion fo th dflctions is dtind. Onc th dflction is known, th stsss a calculatd with th hlp of gnal stss quations and th odal loss facto is valuatd at sonanc condition. Also, th nodal cicls adii a dtind fo ach od shap. Focd Vibation of Plat Consid a thin cicula plat of adius a clapd at dg subjctd to a haonic point load P as shown in Fig.. Th daping is takn into account by considing flxual igidity to b of coplx fo. Fo a thin cicula plat of isotopic and hoognous atial clapd at dg subjctd to a haonic unit point load, th classical quation of otion givn by Mos and ngad 4 in ts of Gn s function is: 4 jω tb B G+ ρ h G.. δ(. δ( φ φ t wh, ( G G, φ φ, [th dflction at (, φ du to 3 Eh unit load at (, φ ]; B (Flxual igidity; E Modulus of lasticity; ρ Mass dnsity of th 3( ν atial of th plat; ν Poisson atio; h Half of th thicknss of th plat; t Ti; and, ω Fquncy of 4 xcitation.; + +. Lt th haonic point load acting on th plat b givn by: jω t P(, φ, t P(, φ (

2 8 NDAN. ENG. MATER. SC., APRL 4 Lt th haonic spons of th plat du to th haonic xcitation b givn by: G jω t, φ, φ, t G, φ, φ (3 Fo a stady stat condition, fo lations (, and (3, on gts: 4 4 G G δ ( δ ( φ φ (4 wh, 4 3ρω ( ν (5 Eh Lt us consid th hoognous quation fo a cicula plat: 4 4 G G (6 Th solution of hoognous Eq. (6 in ts of Bssls functions 4 is givn by: + B G A (7 wh, A and B a constants, is a fquncy paat, suffics and n in fquncy paat stand fo nodal diat and nodal cicl, spctivly, is th th od Bssls function of fist kind and is th th od odifid Bssls function of fist kind. Fo a clapd dg cicula plat of adius a, bounday conditions a givn by: G at a G at a (8 Fo th fist bounday condition, on gts: B a a A (9 and, th scond bounday condition givs: d a { } d a d a d { } a On solving Eq. (, on gts: ( π a β ( wh, β 3 is givn by: β.5, β.7, β 3 3., β.468, β.483, β3 3.49, β.879, β.99, β 4.. Also, fo n β + n Th chaactistic function is givn by: σ cos φ (, φ sin φ 3 a a ( wh, σ suffix stands fo an vn o an odd function. n cas of an vn function, cos φ has to b considd and fo an odd function, sin φ has to b considd. Th Gn s function can b xpssd as a sis of th chaactistic function σ (,φ, i..: G Aσ σ (3 Fo Eqs (4 and (3, on gts: skl A skl 4 4 (, φ δ ( δ ( φ φ kl skl (4

3 GUPTA & NGAM: LOAD FACTOR OF A CLAMPED EDGE CRCULAR PLATE 8 Considing th popty of th Dlta function and ultiplying Eq. (4 by σ (, φ ddφ and intgating it ov th plat aa, on gts 4 : (, (, φ φ G π (5 Λ wh, π and, dφ σ 4 4 a σ a σ ( σ, (6 { (, φ } d π a Λ [{ ( a } { ( a } ] + Λ (7 d du ( u { ( u } whn whn >,, th stss in th plat will b axiu whn th dflction du to th haonic point load is axiu at th sonanc i.. whn. At sonanc, Eq. (5 ducs to: G Fo a paticula od ( n (, φ (, φ σ σ π a jη 4 Λ (8 wh, η Modal loss facto; B B( + jη, Flxual igidity in coplx fo; G G, Fo (, n od; and, j Phas diffnc btwn th xcitation and th spons and it is nglctd in futh analysis. As, it is a cas of non-sytic vibation; both vn and odd functions hav to b considd in th chaactistic function givn by th lation (. Thus, G ( ( ( cos φ φ 4 πa ηλ wh, a a (9 ( ( ( ( at ( Th dflction at any point (,φ in th plat du to ccntic haonic point load P jω t is givn by: P w (, φ G ( B Substituting th valu of (, on gts: G ( ( fo lation (9 in Eq. P w(, φ cos ( φ φ 4 π a Bη Λ (3 Futh siplifying Eq. (3 by substituting th valus fo Eqs ( and (3 lads to: w C (, φ { R } cos ( φ φ wh, C R η P π a B 4 a a ( Λ Also, odal fquncy fo Eqs (5 and ( will b: a 3ρ ν ( (4 π h E υ ( β (5

4 8 NDAN. ENG. MATER. SC., APRL 4 Loss Facto fo a Plat Th plat is concptually thought to consist of a lag nub of sall lnts of aa d dφ. Stsss σ, σ φ, and τ φ at ach lnt a obtaind fo sipl stss displacnt lations. Fo ths stss valus, pincipal stsss σ a and σ a at th lnt a valuatd. Th a two citia fo dtining th loss facto valus. On is basd on dilatational stain ngy and oth is basd on distotional stain ngy. Th fo givs th upp bound valus and th latt givs th low bound valus fo th odal loss facto. An quivalnt stss σ can b obtaind at th cnt of ach lnt fo th two citia. Th loss facto can thn b obtaind fo 4 : N E σ η (6 π σ wh, is caid though lnts and and N bing th daping constants of th atial. Th quivalnt stss σ basd on th dilatational stain ngy cition can b obtaind as 5 : N ( ν ξ + ξ N ( + N σ σ ξ (7 a Basd on distotional stain ngy cition, th quivalnt stss is givn as : N ( N ν ξ + ξ ( ξ + N σ σ ξ (8 a wh, σ a ξ <. σ a Rsults and Discussion As an illustation, a plat of SAE stl having adius, thicknss, ass dnsity, odulus of lasticity, and Poisson atio as.6,.3, kg/ 3,.68 N/, and.3, spctivly, subjctd to a haonic point load of 5 N has bn considd. Th daping poptis and N fo th abov plat atial a and.86, spctivly. Modal loss facto fo diffnt ods hav bn coputd fo th vaious ccnticity, i.. / a and discussd to suppot th psnt thod of analytical valuation of odal loss facto. Copaing th adii of th nodal cicl(s with th valus givn by Lissa also validats th wok psntd in this pap. Th odal loss facto dpnds on th od shap. Fo stss-displacnt lationship, stss at a point is axiu whn th dflction is axiu and vicvsa. Hnc, fo a paticula od, th odal loss facto will b axiu whn th load acts though an antinod i.. th point of axiu dflction and will b iniu whn it acts though a nod. Tabls and show th vaiation of odal loss facto with th ccnticity fo diffnt ods. Tabl is basd on th distotional stain ngy cition, which givs a low liit of th odal loss facto. Tabl is basd on th dilatational stain ngy cition, which givs an upp liit of th odal loss facto. Tabl Vaiation of odal loss facto with ccnticity basd on distotional stain ngy cition S. No. Eccnticity ( /a Fundantal Mod (, Scond Mod (, Thid Mod (, Modal Loss Facto ( 5 Fouth Mod (, Fifth Mod (, Sixth Mod (, Svnth Mod (,

5 GUPTA & NGAM: LOAD FACTOR OF A CLAMPED EDGE CRCULAR PLATE 83 Tabl Vaiation of odal loss facto with ccnticity basd on dilatational stain ngy cition S. No. Eccnticity ( /a Fundantal Mod (, Scond Mod (, Modal Loss Facto ( 5 Thid Mod Fouth Mod (, (, Fifth Mod (, Sixth Mod (, Svnth Mod (, Tabl 3 Copaison of adii of nodal cicl(s fo diffnt od shaps S. No. Fundantal Mod (, Scond Mod (, Thid Mod (, Radii of Nodal Cicl(s Fouth Mod (, Fifth Mod (, Sixth Mod (, Svnth Mod (,3 Fo Analysis (Figu a a a a.38a a.49a a.55a a.58a.6a Givn by Lissa 3 a a a a.379a a.4899a a.559a a.583a.55a Fig. Vaiation of odal loss facto with ccnticity basd on distotional stain ngy cition Fig. Cicula plat with an ccntic haonic point load Fig. shows th ffct of vaiation of ccnticity on th odal loss facto. t is obsvd that th odal loss factos fo high ods a low than th fundantal od. Th iniu valu of th odal loss facto fo a od givs location of th nodal cicl(s fo that paticula od. A copaison of adii of th nodal cicl(s with th valus givn by Lissa 3 is givn in Tabl 3.

6 84 NDAN. ENG. MATER. SC., APRL 4 dg thin cicula plat subjctd to an ccntic haonic point load. Th dilatational stain ngy cition and th distotional stain ngy cition giv th upp and low valus of th odal loss facto, spctivly fo a clapd dg thin cicula plat subjctd to an ccntic haonic point load. Th actual valu of th odal loss facto fo pactical pupos would b in btwn th two liits. Futh, th iniu valu of th odal loss facto fo a od givs location of th nodal cicl(s fo that paticula od. Fig. 3 Vaiation of thicknss of th plat on odal loss facto basd on dilatational stain ngy cition Futh, th odal loss facto also dpnds upon th thicknss of th plat. Th ffct of vaiation of thicknss of th plat on th odal loss facto is shown in Fig. 3. Th odal loss facto dcass hypbolically with incas in th thicknss of th plat. Conclusions Th psnt wok fo analytical valuation can b usd to obtain an stiat of odal loss facto fo a clapd Rfncs Gupta M K, Study of valuation of atial daping of a clapd dg thin cicula plat with an ccntic point load, M.E. Disstation, Univsity of Rook, Rook, ndia, 99. Lazan B, Daping of atials and bs in stuctual chanics (Pgaon, Nw Yok, Lissa A W, Vibation of Plats, NASA, SP-6, Mos P M & ngad K U, Thotical acoustic (McGaw-Hill Book Copany, Nw Yok, Niga S P, Gov G K & Lal S, AAA, 3 (Spt., 975 No Sodal S, Vibations of shlls and plats (Macl Dkk nc., Nw Yok, 98.