Chapter 4 FEM in the Analysis of a Four Bar Mechanism

Transcription

1 Chpter FEM in the nlysis of For r Mechnism he finite element method (FEM) is powerfl nmericl techniqe tht ses vritionl nd interpoltion methods for modeling nd solving bondry vle problem. he method is lso tremendosly sefl for comple mechnism nd strctres with nsl geometric shpes. his method is very systemtic nd modlr. he FEM pproimtes strctre in two seprte wys. he first pproimtion mde in finite element modeling is to divide the strctre into nmber of smll simple prts. hese smll prts re clled finite element. Ech element is generlly simple, sch s bem, br, or plte. ll elements hve n eqtion of motion tht cn be esily solved by nlyticl methods or compter progrmming. he end point of element is clled nodes. he element cn be connected by node. he collection of finite elements nd nodes is clled finite element mesh or finite element grid. he eqtion of motion for ech individl finite element is then formed nd solved. his forms the second level of pproimtion in FEM. he soltions of the element eqtions re pproimted by liner combintion of low order polynomils. Ech of these polynomil soltions is mde compnionble with the djoining soltion t nodes common to two elements. hese soltions re broght together in n ssembly procedre, reslting in globl mss nd stiffness mtrices. his represents the nlysis of strctre s whole. he phrse finite element method is often bbrevited FEM. his bbrevition cn lso denote the phrse finite element model. nother commonly sed bbrevition is FE, which mens finite element nlysis. Sometimes the bbrevition FE is sed to bbrevite finite elements. In FEM, to improve the ccrcy of the compted soltion of eigenvle problem (EVP), two pproches re possible: (i) the mesh cold be refined by.

2 redcing the size of the element nd (ii) the degree of the polynomils over the elements cold be incresed. he first procedre is known s the h-version of the FEM nd second s the p-version. he p-version provides higher rte of convergence thn the h-version. dvntges of FEM inclde the bility to del with strctres with rbitrry loding, inclding spport conditions, nd lso the bility to model strctres of rbitrry geometry. frther dvntge of this method is the possibility of modeling composite strctres comprising different strctrl components. However, for the prpose of dynmic nlysis, n lterntive is to se the ect displcement fnctions rising from the soltion of the governing differentil eqtions for bem vibrtion. his pproch offers certin dvntges bt hs the disdvntge of leding to nonliner eigenvle problem when compting the ntrl freqencies. When the FEM is employed, two stges mst be considered. he first reqires stdy of the individl elements into which the system is divided, while the second involves the stdy of the ssemblge of elements which represent the entire system. hs, the otline of the FE process my be smmrized into five essentil steps which re s follows: Definition of the finite element mesh: he first step involves the process of discretising the strctre into pproprite elements. Selection of displcement models (shpe fnction): In this process, sitble displcement fnction mst be selected for typicl element which wold led to finite nmber of DOF nd wold stisfy the bondry conditions of the system. In order to retin the bonding nd convergence properties inherent in the Ritz procedre, it is necessry tht the element interpoltion fnctions shold inclde the rigid body displcements nd niform strin sttes, nd tht they mintin displcement comptibility long the inter element nd eterior bondries. Formltion the eqtion of motion: he strins t ny point within the element my be epressed in terms of the element nodl displcements. he.

3 sttic eqtions of eqilibrim cn be obtined by sing the principle of sttionry totl potentil energy wheres the dynmic eqtions of motion re obtined by sing Hmilton s principle. Soltion of the eqtions of motion: he soltion of stiffness eqtions led to set of simltneos eqtions wheres the eqtions of motion in the free vibrtion cse led to n EVP. Determintion of the desired properties: Once the nodl displcements hve been determined, strin nd stress cn be clclted from the strindisplcement reltionship nd by Hooke's lw, respectively.. Formltion of stiffness nd mss mtrices of element In dynmic considertions the lods re sddenly pplied, or when the lods re of vrible ntre, the mss nd ccelertion effect come into the ccont. When the mechnism vibrtes minly in one plne, two types of vibrtion mst be considered: (i) the flerl vibrtion nd (ii) longitdinl vibrtion. In mechnism the links re fleible in the plne of motion nd reltively stiffer in the plne perpendiclr to the motion. Here the torsion effect is neglected nd hence it is possible to constrct stiffness nd mss for link by combintion of simple br element nd bem element. he grnge s eqtion writes s below: Π (.) where, is the kinetic energy nd is the potentil energy. he potentil energy cn be defined s: Π U W (.) where, U is internl strin energy nd W is work done by eternl forces. Hmilton s principl for n rbitrry time intervl from t to t, the stte of motion of body etremizes the fnctionl.

4 t I dt (.) t he kinetic energy, potentil energy nd the work done by eternl forces for distribted system re fnction of the il coordinte. t ( U W ) dt δ (.) t et q represent the generlized coordinte, then t [ U ( q) ( q,q ) W ] d dt δ (.5) t W F q (.) where, F is the distribted force. king the vritions of different terms in the Eq. (.5), U δu δq (.7) q δ δq δq q q (.8) δ W F δ q (.9) Sbstitting the vle from Eqs. (.7-.9) in Eq. (.5), we get t U δ q - δ q δ q F q q q δ t q d dt (.) Integrting q term by prts with respect to time, s t t δ q dt q q δ q t t t t d dt δ q dt q (.).

5 t t δ q dt q t t d dt δ q dt q (.) Sbstitting the vles of Eq. (.) in Eq. (.), we get t U q - d dt t q q F δ q d dt (.) herefore grngin eqtion for dynmicl system with two independent prmeters, time t nd one generlized coordinte q is U q d dt q q F (.) Eq. (.) then cn be sed for deriving the elementl properties of br nd bem strctre... he br element [7] he longitdinl vibrtion of br demonstrtes how finite element model is constrcted. he simplest one dimensionl element model is considered s shown in Fig... Fig.. Finite element model of br element o constrct FE model, the following steps re to be considered: Step : Choose the br element which hs two nodes. he nodl displcements re nd s shown in Fig... Step : Define shpe fnction mtri or displcement fields, ssming the displcements t the two nodes re to be known. he displcement of ny point within.5

6 the element is obtined by ssming shpe fnction or interpoltion fnction. In generl, shpe fnction need to stisfy the following: First derivtives mst be finite within n element. Displcements mst be continos cross the element bondry. Inclde representtion of constnt vles of significnt stress or strins. For the il (tensile) displcement of the br element sitble choice of shpe fnction is liner polynomil s below: (.5) where, nd constnts to be determine from the bondry conditions of the element. t nd t (.) y ptting the vle of constnts, Eq. (.5) cn be modified s: ( ) (.7) he Eq. (.7) in mtri form cn be written s: [ N N ] (.8) where, N nd N re shpe fnction of the element. Shpe fnction mtri, N [ N N ] (.9).

7 Displcement mtri, d (.) Eq. (.8) cn be modified by sing Eqs. (.9) nd (.) s: [ N ]{ d } (.) Step : Estblish strin displcement reltions s per the elementry theory. he norml component of strin in il direction cn be clclted s: ε d d [ N, N,] [ ]{ d } (.) Mtri [] gives reltion between the il strin nd the nodl degree of freedom. Step : Determine the strin energy in br element which is given s U E (, ) d (.) y sing Eqs. (. nd.), the Eq. (.) cn be written s: U E d {} [ ] [ ]{} d d (.) Step 5: Constrct the elementl stiffness mtri by considering the potentil energy term from grngin eqtion Eq. (.) cn be written s: U {} d [ ] [ ]{} d d { F} E (.5) Eq. (.5) cn be written in mtri form s.7

8 [ ]{ d } { F} K (.) where, [K] is the stiffness mtri nd is given s: [ K] E [ ] [ ] d (.7) With the help of strin eqtion s mention in Eq. (.), cn be written s: (.8) [ K] E [ ] d y integrtion the Eq. (.8), the stiffness mtri for br element is written s: K (.9) E [ ] Step : Constrct the elementl mss mtri s follow: he totl mss of the br element is ρ, where, ρ is the density of the br element, is the cross section of re nd is the length of the br element. he kinetic energy of br element is given s: ρ d (.) {} [ N] [ N]{} d ρ d d (.) From Eq. (.), considering the potentil energy term only, d dt {} { F} (.) {} From Eqs. (.) nd (.), we get ρ [ ] [ ]{} N N d d.8

9 d dt {} [ ] [ ]{} N N d ρ d (.) From Eqs. (.) nd (.), ρ [ N] [ N ]{} d d { F} [ M ]{ d } { F} (.) where, [M] is the mss mtri of br element s: [ M ] [ N] [ N] ρ d (.5) Sbstitte the vle of shpe fnction from Eq. (.) in Eq. (.5), we get, - ρ d (.) [ ] M y integrtion the bove eqtion, we get M ρ (.7) [ ] he bove mss mtri sed for the distribtion mss, is clled consistent mss mtri of the br element... he bem element [7,7] finite element model of the bem element is shown in Fig... he two nodes nd re presented in elementl coordinte system. o constrct FE model, the following steps re to be considered: Step : Choose the element s shown in Fig.. the element hs two nodes nd. he nodl deflections re nd mesred positive in the direction of y is. he nodl slopes re nd positive in nticlockwise direction..9

10 Fig.. Finite element model of bem element Step : Define shpe fnction Mtri, the displcement nd slopes t both nodes nd of the element is known. he shpe fnction for the trnsverse displcement () is ssmed to be cbic polynomil. ( ) (.8) he bove Eq. (.8) stisfies the governing differentil eqtion of bem. d y EI (.9) d he continity condition of both the displcement nd the slope t the nodes is stisfies the cbic polynomil shpe fnction. y sing the bondry condition of the element t the nodes clclte the vle of for constnt. ( ) ( ) d d ( ) ( ) d d (.) y solving the bove Eq. (.) got the vle of constnt nd sbstitting in Eq. (.9),.

11 ( ) ( ) ( ) ( ) ( ) (.) Collecting terms of nodl degree of freedom nd writing in mtri form [ N ]{ d } (.) where, [ ] [ N N N ] N nd N [ N ] ( ) [ N ] ( ) [ N ] ( ) [ N ] ( ) (.) Step : Estblish the reltions for bending moment displcement, the bending moment in the element is given by d M z ( ) EI zz (.) d Using Eq. (.), Eq. (.) cn be modified s: ( ) EI [ ]{ d } M z zz (.5) where, [ ] [ N N N ],,, N, (.) nd[ ] ( ) N, [ N ] ( ),.

12 [ N ] ( ), [ N ] ( ), nd{ d } (.7) Step : Determine the Strin energy in bem element which is given s: U EI ( ), zz d EI zz {} d [ ] [ ]{} d d (.8) Step 5: Formtion of elementl stiffness mtri by considering the potentil energy term only from Eq. (.), rewrite the Eq. (.8) s: U {} d EI zz [ ] [ ]{} d d { F} (.9) With the help of Eqs. (.7) nd (.9) the stiffness cn be written s: zz N N N N,, [ K] EI [ N N N N ] d,,,,,, (.5) Sbstitting for the derivtives of shpe fnction in bove eqtion nd integrtion, the stiffness mtri s:.

13 . [ ] EI K zz (.5) Step : Constrct the elementl mss mtri s follow: he kinetic energy of bem element is d ρ (.5) y sme method sed in section.. for br element, the mss mtri for element sme s Eq. (.5) s below: [ ] [ ] [ ] d N N M ρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ρ y integrtion the bove eqtion, the mss mtri for the bem element is s follow: [ ] M ρ (.5).. he bem element with si DOF he displcement model, the stiffness mtries nd the mss mtries considered with the il nd bending loding of n element, hve been derived seprtely in section.. nd... In plne motion nlysis, the bem element hs

14 5 Fig.. Finite element model of bem element with si DOF been si DOF s shown in Fig... o derive the stiffness mtri nd mss mtri for the bem hving si DOF ssembling the mtries of br nd bem element. he stiffness mtri cn be written s: K K K (.5) K K [ ] E where,[ ] K nd [ K ] EI zz he mss mtri cn be written s: M M M (.55) M M [ ] 5 ρ M ρ nd[ ] M 5 where, [ ] 5 5 em nd br, both elements follow the clssicl bem theory, mening plne section remin plne nd re cpble of inclding sher deflection sing sher re coefficients (more importnt for short stbby bems or brs) he bem element is cpble of doing more compre to br element s listed below:.

15 em elements cn hve tpered sections, mening one end cn be smller/lrger/wider/ nrrower/ thinner/ thicker thn the other, bt the shpe cnnot be totlly different. em elements re cpble of cconting for lrge deflections nd differentil stiffness de to lrge deflections em elements cn hve three different offsets. One for sher center, one for the netrl is nd one for the nonstrctrl mss is. Wheres br elements hve only one is, ll three re the sme netrl is. For br element the grid points re locted t the section centroidl netrl is. For bem elements they re lwys t the sher center is nd the netrl is is offset from the sher center is. r elements re best for dobly symmetricl sections with lod pplied long centroidl plnes, s they re not cpble of cconting for bending or twisting or wrping of the sections de to il or trnsverse lods. his is only possible with bem elements.. FEM nlysis of for br mechnism he for br plnr mechnism with elstic links is modeled by connecting series of bem element s shown in Fig... he rigid body motion of mechnism is mention by solid lines (CD) nd elstic displcement mechnism by dotted line C Copler Q Crnk P Rocker D Fig.. For br mechnism with elstic links displced.5

16 (PQD) s shown in Fig... hey re connected in sch wy tht it llows the model for devition in the mechnism geometry. complicted link is to be considered s bem element with niform cross section throghot its length. Ech link is to be s one bem element for simplicity of in modeling... Elstic bem in plne motion he link of mechnism is represented s bem element in two reference frmes s shown in Fig..5. he frme (OXY) is the fied frme where, the frme (Oy) is the rotted frme with element. Y y 5 O Fig..5 Rigid nd elstic body with coordinte systems X he rigid body position of bem is shown with solid line nd elsticlly deformed position of bem is shown with dotted line s shown Fig..5. he is of the rotted frme is forever prllel to the rigid body position of the bem element dring its motion. he elstic deflection of the bem element my be specified by si generlized nodl displcement coordintes to. Following reltionships cn be developed from Fig..5..

17 X X cos sin Y Y sin cos (.5) o derive the velocity of node in fied frme, set of Eq. (.5) is differentited with respect to time nd is given s: X X cos sin sin cos Y Y sin cos cos sin (.57) fied frme. Derivtives of Eq. (.57) with respect time give the ccelertion of node in X Y X Y cos sin cos sin sin cos sin cos sin cos sin cos cos sin cos sin (.58) bove Eq. (.58) cn be written with respect to rotting frme with help of following trnsformtion: X cos Y sin y X sin Y cos (.59) y combined nd simplified the Eqs. (.58) nd (.59), we get the ccelertion of node in rotted frme:.7

18 .8 y y (.) Similrly, the ccelertion of node in rotted frme cn be written s: y y (.) where,, y,, y, nd re the kinemtic terms of the rigid body motion of the element. Now defining the following colmn vectors s: { } i y y nd{ } i y y, i,,., (.) From Eqs. (.) (.), we get: r r r r r r (.) he Eq. (.) cn be rewritten s: { } { } { } { } { } { } t c n r (.)

19 Where,{ } n, { } nd { } ccelertion respectively. { } r c s the components{ } n re the norml, Coriolis nd tngentil components of t, { } nd { } nd{ }. he Eq. (.) for ccelertion cn be modified s: { } { } { } r c t re very smll s compred to terms (.5) Similrly for velocity: { } { } { } (.) r.. Element mss nd stiffness mtries he mss nd stiffness mtries of the bem element in locl coordinte frme hve been derived in section... he kinetic energy nd strin (potentil) energy of one bem element (one link) in mtri form re s below: { } [ m]{ } nd {}[ m]{} (.7) where, [ m ] is the mss mtri [], cn be given s: m ρ [ ] (.8).9

20 he stiffness mtri,[ k ] is formed s [], [ k ] E E EI EI EI EI EI EI EI E EI EI EI (.9) From the eqtion of motion s given by Eq. (.) nd from Eqs. (.8) nd (.9), for bem element eqtion of motion is given s: [ ] () t { } [ k ]{ () t } { Q} m (.7).. rnsformtion of mtries to globl coordintes he mss nd stiffness mtries developed in the section.. re epressed in locl or element coordinte system. In prctice, the mechnism or mchine re mde p of nmber of elements with different orienttions. herefore, presenting the displcements in coordinte system prticlr to ech element will crete difficlties in mtching the displcement t given node dring the ssembling process. hs, while hving severl locl coordinte systems, it is reqired to se globl coordinte system for given problem [7]. he generl element with two nodes nd nd two coordinte i.e. locl nd the globl coordinte system is shown in Fig... With the reference of Fig.., the set of eqtions for node is given s: U cos U sin U sin U cos (.7) U.

21 Y 5 U 5 (ocl) U, U U y, U U (Globl) X Fig.. rnsformtion of coordinte from locl to globl coordinte system Similrly for node U cos U 5 sin 5 5 U sin U cos (.7) U [ ] Eqs. (.7) nd (.7) my be written in mtri form s: cos sin sin cos R (.7) cos sin sin cos where, [R] is trnsformtion mtri..

22 . Finite element model of for br mechnism he FEM of for br plnr mechnism with three bem elements is shown in Fig..7. In which ech link is modeled by one finite bem element Fig..7 Finite element model of for br mechnism.. ssembly of the system mtries ssembly of stiffness nd mss mtrices nd the generlized forces vector of individl elements to form the system (overll) mtries for the entire mechnism is chieved by ensring tht the geometric comptibility is stisfied t ll nodes. Eqtion of motion given in Eq. (.7) of the bem element is rewritten in the system coordinte s: { } [] k U () t [ ] U () t { } { Q} m (.7) where, [ m] [ R] [ m][ R], k [ R] [ k ][ R] nd Q [ R] [ Q ] he kinetic energy of link (crnk) s bem elements my be written s: { U } [ m]{ U } (.75).

23 Eq. (.75) cn be epressed s: U U U m m m m m m m m m U U U (.7) Hence, totl kinetic energy of mechnism is given s: (.77) he Eq. (.77) is epressed in mtri form s: { U } [ M ]{ U }, i,,..., 9 i i (.78) Hence, mtri [M] is the totl system mss mtri. Similrly, from strin energy considertion, the totl system stiffness mtri [K] my be derived by sperposing the strin energies of the individl elements s: U { U } [ K]{ U }, i,,...,9 i i (.79).. Eqtion of motion he eqtion of motion for mechnism my be written in mtri form s described in Eqs. (.8 nd.8) [-]: [ M ]{ U } [ K ]{ U} { Q} (.8) If the strctrl dmping mtri for the mechnism is denoted by [C], then by inclding the dmping forces, the eqtion of motion becomes: [ M ]{ U } [ C]{ U } [ K ]{ U} [ M ]{ } (.8) U r Here, the coefficient mtries [M], [C] nd [K] re the fnction of the mechnism geometry nd chnge with chnge in crnk ngle nd lso{ U r } is represent the rigid body ccelertion vector..

24 .. Dmping in mechnism In ctl mechnism some energy dissiption is lwys present. Mesrement nd modeling of the mteril dmping of system generlly proves to be difficlt problem tht reqires frther reserch. It is therefore necessry to ssme n pproimte form for the mteril dmping. proportionl viscos dmping form is cstomrily ssmed de to the ese in which it cn be incorported into the eqtion of motion, nd lso to ensre tht the eqtions of motion cn be ncopled... Stress clcltion il forces within link re generted de to its own longitdinl vibrtion, the foreshortening de to its trnsverse vibrtions nd the elstic effect of the other links trnsmitted throgh the pins t its ends dring the internl. Strin nd stress re clclted s bellow: he il strins t the netrl is is ε () t () t (.8) ( ) he il stress is t σ E (.8) () t..5 Method of soltion In nmericl methods, this continos motion is replced by nmber of discretized steps. he concept is nlogos to finite element theory, where the elstic medim itself is discretized [7]. Dring ech time step, the system prmeters (mss, dmping, nd stiffness) re ssmed to remin constnt in solving the eqtion of motion. his prodces is only n pproimte soltion, while the tre soltion is pproched s the step size tends to zero. Mny methods hve been sggested by reserchers to find ot the soltion of eqtion of motion re listed below: Direct integrtion method.

25 Modl nlysis Forier series method Newmrk method Rnge-Ktt method.5