Finite Element Vibration Analysis


 Samson Walsh
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1 Finit Elmnt Vibration Analysis Introduction In prvious topics w larnd how to modl th dynamic bhavior of multidof systms, as wll as systms possssing infinit numbrs of DOF. As th radr may raliz, our discussion was limitd to rathr simpl gomtris and boundary conditions, mainly bcaus mor complicatd gomtris tnd to hav progrssivly mor complicatd xact solutions, or vn no xact solutions at all. h vast majority of gomtris which occur in practical nginring applications ar complicatd thrdimnsional continua which cannot b rprsntd adquatly by th simpl closdform mathmatical modls. hs thortical modls also cannot portray th apprciabl nonlinar or anisotropic matrial charactristics which ar oftn mt with in practic. In such situations thrfor, w must rsort to numrical mthods in which th continuum of infinitsimal matrial particls is rprsntd by an approximatly quivalnt assmbly of intrconnctd discrt lmnts which ar ach so simpl that thy can b tratd individually as mathmatical continua. hr ar a numbr of mthods whrby such modls can b analyzd such as numrical solution of diffrntial quations, finit diffrncs, finit lmnts, boundary lmnts, rlaxation tchniqus, and so on. In this topic, w will dmonstrat th Finit Elmnt Mthod (FEM) as a typical powrful approach which can handl vibration analysis. In ssnc, th FE tchniqu is a numrical tchniqu in which a continuous lastic structur, or continuum, is dividd (discrtizd) into small but finit substructurs, known as lmnts. Elmnts ar intrconnctd at nods. In this way, a continuum with infinit numbr of dgrs of frdom can b modld with a st of lmnts having a finit numbr of dgrs of frdom. It is notd that whil ach finit lmnt rprsnts a continuous systm by itslf possssing infinit numbr of DOF, w can choos th siz of th lmnt to b small nough, so that th dformation within th finit lmnt can b approximatd (intrpolatd) by rlativly lowordr polynomials. 1
2 In this topic, w will discuss th application of FE procdurs in vibration analysis, with mphasis on dvloping quations of motion for lmntary gomtris, dvloping FE cods on MAAB and applying commrcial FE softwar to handl mor advancd gomtris. Finit Elmnt Analysis of Rods Considr an lastic uniform rod of total lngth R as shown in Fig.1. Upon applying th FE tchniqu, th rod is discrtzd into a finit numbr of lmnts. As th rod undr invstigation is uniform, it is assumd that all lmnts usd to msh th complt rod ar idntical. In mor advancd problms, of cours, this tchniqu may not b possibl, as th ntity to b mshd oftn has a mor complicatd contour or gomtry, which maks it difficult to mploy idntical lmnts. h tchniqu dscribd hrin, howvr, can b applid to nonidntical lmnts. R i j x Figur 1. Uniform rod. t u(x) dnot th axial displacmnt within an lmnt. It is rcognizd that displacmnts ar also functions of tim, but for th sak of simplicity in writing th quations, th tim t is droppd, but th radr must rmmbr that tim dpndnc is indd rtaind in th upcoming analysis. Not also that th rod lmnt by itslf is tratd as a continuous structural mmbr, maning that axial displacmnt within th lmnt is a continuous function. An ssnc of th finit lmnt tchniqu is that fild variabls, in this cas axial displacmnts, within ach lmnt is intrpolatd (or approximatd) in trms of th nodal quantitis, that is th displacmnts masurd at th two nods bounding th lmnt. In this way, a whol structur is rgardd as a multi
3 DOF systm, with fild variabls valuatd only th nods. Onc ths quantitis ar known, th displacmnt fild within th ntir structur can b valuatd using th intrpolation functions initially assumd for th structur. Svral intrpolation functions can b chosn to proprly satisfy th physics of th problm. For th problm at hand, th displacmnt fild within th rod lmnt is approximatd by a linar function: his can b writtn as: u( x) x, x (1) u( x) x In this way, th vctor of nodal dgrs of frdom is xprssd as: () But w hav: hrfor: from which: u u i j ui u j 1 1 u 1 u j 1 i 1 1 A (3) (4) (5) 1 A Prforming th matrix invrs yilds: Substituting Eq. (7) into () givs: ui 1 1 u j (6) (7) 3
4 or: 1 ui u( x) 1 x 1 1 u j x x ui ux ( ) 1 u j (8) (9) which is writtn as: u( x) N( x) (1) h vctor N( x ) is calld th vctor of intrpolation functions or shap functions, and is typical in all finit lmnt procdurs. Onc again, using this vctor, w can xprss th displacmnt at any point within th rod lmnt in trms of th nodal dgrs of frdom. Svral mthods can b mployd to obtain th quation of motion of rod lmnts. In this sction w will us agrang s quation. In this rgard, w nd to xprss th strain nrgy associatd with th rod lmnt. h radr is rfrrd to any lmntary txtbook in strngth of matrials to xprss th strain nrgy as: 1 u U EA dx x (11) Insrting th shap functions, as dtrmind in (1) and noting that: w obtain: u x Nx Nx 1 U EA Nx Nxdx (1) whr N x Nx ( ) x h strain nrgy can thn b writtn as: 1 U K (13) 4
5 whr K is th lmnt stiffnss matrix, givn by: Prforming th abov calculation yilds: K EA N x N xdx (14) EA 1 1 K 1 1 (15) which is th lmnt stiffnss matrix for a uniform rod lmnt. h lmnt kintic nrgy is thn valuatd for th rod lmnt and can b xprssd as: A u dx 1 (16) Insrting th shap functions, as dtrmind in (1) yilds: 1 A N Ndx (17) which can also b writtn as: 1 M (18) whr M is th lmnt mass matrix, givn by: Prforming th abov calculation yilds: M A N N dx (19) M A (15) which is th lmnt mass matrix for a uniform rod lmnt. Now w can apply agrang s quation, which is in th form: or mor prcisly: d V dt q q q (16) 5
6 d V (17) dt h rsulting quations of motion ar thrfor: K M which is th lmnt quation of motion for fr vibration. Upon assmbly of th lmnt quations of motion (s nxt sction), w can dtrmin th quations of motion for th ntir structur in th form: M K (18) (19) W can asily xtnd th drivation to includ a damping trm and a forcing trm in th quation of motion: M C K f () t () Assmbly of Equations of Motion Assmbly of th lmnt quations of motion can b accomplishd by th routin assmbly procss usd in finit lmnt procdurs. As an xampl, considr a rod consisting of lmnts as shown in Fig Figur. Rod with lmnts. Undr static conditions, w hav for lmnt 1: whras for lmnt : For th whol rod, w hav: F1 EA 1 1u1 F 1 1 u F EA 1 1u F 1 1 u 3 3 (1) () 6
7 F1 1 1 u1 EA F u F 1 1 u 3 3 whr th principl of suprposition has bn applid to obtain th stiffnss matrix for th ntir structur. h sam procdur is usd to assmbl th mass matrix, and for any numbr of lmnts. In this way, th stiffnss and mass matrics ar ssntially bandd matrics, with th proprtis of ach lmnt insrtd as w procd into th assmbly procss. Elmnts having diffrnt matrial proprtis or gomtrical paramtrs (cross sctional aras) can asily b assmbld. (3) Boundary Conditions Inspction of th ovrall stiffnss matrix indicatd in th prvious sction rvals that it is ssntially singular, maning that upon application of an xtrnal forc on cannot solv for th unknown displacmnts (in static analysis, for instanc) using K F by multiplying by th invrs of th stiffnss matrix, sinc it is nonxistnt. hat is so bcaus th structur in this cas is not rstraind, and hnc application of an xtrnal forc causs instability in th analysis. Such an rror is oftn ncountrd in commrcial FE softwar, whr th usr is alrtd of a singular stiffnss matrix rror, which is an indication that th structur has insufficint constraints. Application of boundary conditions can b accomplishd using th sam procdurs typically adoptd in static FE analysis. As an illustrativ xampl, considr th rod shown in Fig.3, whr on nd is rigidly fixd to a wall, whil th othr is subjctd to a forc. 1 F 1 3 Figur 3. Rod with on nd fixd, othr nd loadd. In this cas, w hav: 7
8 F1 1 1 u1 EA F 1 1 u F 1 1 u 3 3 (4) Howvr, F 1 is an unknown raction forc that is yt to b dtrmind. Although w can dirctly obtain th valu of this raction from physical intuition (qual and opposit to th applid tip load), w cannot mak this intrvntion bfor using th FE analysis to solv th problm! Morovr, in mor complicatd cass, such as statically indtrminat problms, w cannot obtain th raction forcs using only th quilibrium quations. As far as displacmnts ar concrnd, w will also rgard u and u 3 as unknown displacmnts. Hnc quation (4) rprsnts thr quations in thr unknowns. In ordr to procd with th solution, w will first liminat th quations prtaining to unknown raction forcs, in this cas th first quation. his is accomplishd by striking th first lmnt in both th load vctor and displacmnt vctor, as wll as striking th first row and column of th stiffnss matrix. o this nd, w will gt : F EA 1u F 1 1 u 3 3 which can b solvd for th unknown displacmnts u u3 (5) as th vctor of xtrnally F applid loads is known. Onc th displacmnts ar known, w can thn substitut F3 into quation (4) with: F1 1 1 EA F 1 1 u F 1 1 u 3 3 in ordr to obtain th unknown raction forc. h procdur outlind in this lmntary xampl can b gnralizd for all problms discussd in this topic. It is worthy to mntion that th sam is applid for th mass matrix, and all calculations ar don on th rducd matrics. (6) 8
9 Vibration Analysis of Rods t us now study th fr and forcd vibration analysis of rods using th finit lmnt procdur outlin in th prvious sctions. For fr vibration, w hav: Imposing harmonic motion in th form: w gt: hnc: M K sint (7) (8) K M (9) 1 M K (3) which is clarly an ignvalu problm that can b solvd to obtain th natural frquncis and mod shaps of th rod undr invstigation. It is prhaps worthy to mntion at this point that such an analysis can b radily mad for any boundary conditions imposd on th structur, which is in clar contrast with xact analytical solutions that largly dpnd on th boundary conditions. For stadystat harmonic analysis, on th othr hand, w hav: with: M K f () t (31) f ( t) F sint (3) h rsulting displacmnt, dscribd by quation (8) is insrtd into (3) yilding: K M F (33) which can b solvd for th stadystat displacmnt amplitud at ach xcitation frquncy: 1 K M F (34) whr th matrix on th right hand sid is commonly known as th dynamic stiffnss matrix. 9
10 As an illustrativ xampl, considr a fixdfr rod with th following proprtis: Modulus of lasticity 8 GPa Crosssctional ara.1 m ngth 8 m Dnsity 78 kg/m 3 abl 1 lists th first thr natural frquncis as w incras th numbr of lmnts usd, togthr with th analytical solution. Convrgnc is sn to tak plac as w incras th numbr of lmnts usd for mshing th rod. abl 1. Natural frquncis and comparison with analytical solution. Numbr of lmnts [Hz] [Hz] [Hz] Analytical Solution Figur 4 shows th first thr mod shaps, as prdictd by th prsnt FE analysis. 1
11 Figur 4. First thr mod shaps of a fixdfr rod. Considr now th cas whn th rod is subjctd to a harmonic forc acting at its tip and having a magnitud of 1N. W can dtrmin th stadystat displacmnt amplitud at any point on th rod in accordanc with th tchniqu outlind prviously. Figur 5 shows such frquncy rspons, whr th rsonanc is shown to occur at th natural frquncis indicatd. Figur 5. Frquncy rspons of harmonically xcitd rod. 11
12 Damping can asily b implmntd in th prsnt modl. Figur 6 shows th frquncy rspons with damping includd in th form of Rayligh (proportional) damping. Figur 6. Frquncy rspons of harmonically xcitd rod with damping. 5.6 Finit Elmnt Analysis of Bams h finit lmnt analysis of bams follows th sam procdur as that dscribd for rods, xcpt for th nrgy xprssions, shap functions and lmnt dgrs of frdom. Figur 7 shows a bam in flxural vibrations, whr th transvrs displacmnt is xprssd as wx. ( ) Onc again, displacmnts ar in fact functions of tim, but th tim t is droppd for brvity in th following analysis. w w(x) x Figur 7. Bam in flxural vibrations. 1
13 For bams, th displacmnt fild within th rod lmnt is approximatd by a cubic polynomial: w( x) x x x, x (35) h choic of this function is basd upon rlations from strngth of matrials govrning th EulrBrnoulli bam thory, in which an lmnt loadd by concntratd forcs at its nods faturs a linar variation in bnding momnt. Sinc bnding momnt is a function of th scond drivativ of transvrs displacmnt, th abov polynomial is justifid. Equation (35) can b writtn as: w( x) 1 x x x h vctor of nodal dgrs of frdom is thn xprssd as: (36) wi i w j j whr w (38) x is th slop. Substituting to gt th valus of th lmnt dgrs of frdom at ach nod, w hav: wi 1 i A w (39) j 3 j 4 from which: (37) 13
14 Substituting (4) into (36) givs: which is writtn as: 1 wi 1 i A w 3 j 4 j wi 1 i wj j 3 w( x) 1 x x x A (4) (41) w( x) N( x) (4) Onc again, th vctor N( x ) is th vctor of intrpolation functions or shap functions for bam lmnts, which is usd to xprss th displacmnt at any point within th rod lmnt in trms of th nodal dgrs of frdom. h strain nrgy of th bam lmnt is xprssd as: w x (43) 1 U EI dx Insrting th shap functions, as dtrmind in (4) yilds: whr N xx N( x) x 1 U EI Nxx Nxxdx (44) h strain nrgy of th bam lmnt can b writtn as: 1 U K (45) whr K is th lmnt stiffnss matrix, givn by: 4 4 K EI Nxx Nxxdx (46) 14
15 Prforming th abov calculation yilds: EI 4 6 K which is th lmnt stiffnss matrix for a uniform bam lmnt. h lmnt kintic nrgy of a bam lmnt is xprssd as: (47) A w dx 1 (48) Insrting th shap functions, as dtrmind in (4) yilds: 1 A N Ndx (49) which can also b writtn as: 1 M (5) whr M is th lmnt mass matrix, givn by: 4 4 Prforming th abov calculation yilds: M A N N dx (51) M A (5) which is th lmnt mass matrix for a uniform bam lmnt. Applying agrang s quation yilds th lmnt quation of motion fro fr vibration as: K M (53) 15
16 Upon assmbly of th lmnt quations of, w can dtrmin th quations of motion for th ntir structur in th form: Or mor gnrally: M K (54) M C K f () t (55) 5.7 Examlpl Considr a cantilvr bam with th following proprtis: Modulus of lasticity 8 GPa Crosssction (width x hight) mm.4 mm ngth. m Dnsity 7 kg/m 3 abl lists th first thr natural frquncis as w incras th numbr of lmnts usd, togthr with th analytical solution. Onc again, convrgnc is sn to tak plac as w incras th numbr of lmnts usd for mshing th rod. 16
17 abl. Natural frquncis and comparison with analytical solution. Numbr of lmnts [Hz] [Hz] [Hz] Analytical Solution Figur 8 shows th first thr mod shaps, as prdictd by th prsnt FE analysis. Figur 8. Cantilvr bam mod shaps. 17
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