Comparison of Analytical and Numerical Results in Modal Analysis of Multispan Continuous Beams with LS-DYNA

Transcription

1 th Iteratoal S-N Users oferece Smulato Techology omparso of alytcal ad Numercal Results Modal alyss of Multspa otuous eams wth S-N bht Mahapatra ad vk hatteree etral Mechacal Egeerg Research Isttute, urgapur 79, Ida. bstract Ths paper deals wth the study of atural frequeces of vbrato of cotuous beams supported o hged ed supports wth ad wthout overhag. The paper llustrates the aalytcal formulato of the atural frequeces ad correspodg modes of a -spa cotuous beam by usg a geeral soluto for the Euler-eroull dfferetal equato. Usg two dfferet approaches, amely the aalytcal method ad the umercal method some typcal cotuous beams are aalyzed ad the coformace of the FEM solver S-N s tested. The obectve s to test the correlato betwee appromated aalytcal ad umercal methods adopted for ths partcular study. The computed results are gve tabular form. Itroducto I dyamc aalyss of structures the computato of atural frequeces ad mode shapes s mportat, mostly, the desg of structures subected to vbratory loadg. I desg of multspa cotuous beams, t s essetal that accurate determato of the atural frequeces ad mode shapes are doe sce the structures subected to dyamc loads s depedet o both the lower ad the hgher modes. I ths paper, based o the aalytcal models, the atural frequeces ad assocated mode shapes of the vbratg system are obtaed drectly from the dfferetal equato of moto for the udamped free trasverse vbrato of the cotuous beam wth assumptos that each spa of the cotuous beam s a uform Euler-eroull beam. I S-N each of the cases s solved.e. the cotuous beam havg beam elemet shell elemet ad sold elemets ad frequeces obtaed are compared wth the aalytcal. The agreemet betwee the two approaches.e. aalytcal ad beam S-N usg beam elemet has bee foud to be ecellet. Multspa otuous eam o Smply Supports -spa cotuous beam [] has bee aalytcal solved for the two cases preferably, smply supported eds wthout overhag smply supported eds wth overhag as show the Fg.a & b. The beam s assumed to be a Euler-eroull beam wth costat stffess EI ad uform mass dstrbuto m. Each spa of a cotuous beam s treated as a dvdual. The partal dfferetal equato of moto for the udamped free trasverse vbrato of a uform Euler-eroull beam whe the effects of traverse shear deformato ad rotary erta are eglected s as follows: -

2 Smulato Techology th Iteratoal S-N Users oferece y y EI, t m, t t s the aal posto of a pot o the beam, y,t s the trasverse dsplacemet respose of the beam at posto ad tme t, m s the mass per ut legth of the beam, E s the elastc modulus of the materal, I s the momet of erta of the cross-secto. a y b Fg.. -spa cotuous beam layout a wthout overhag b wth overhag Whe the system performs harmoc free trasverse vbrato, oe has ωt y, t e s the ampltude of y,t, ω s the agular frequecy of the whole system ad. Substtutg Eq. Eq. yelds the well kow fourth order dfferetal equato, '''' m ω EI The geeral soluto of the fourth order dfferetal Eq. for trasverse vbrato of beams may be wrtte the followg form, cosh sh cos s 5 Eq.5 s the dsplacemet fucto for each beam segmet betwee ay two adacet statos of a multspa beam. It s also kow as the correspodg mode shape. The costat,, ad ca be evaluated from the boudary codtos of the spas of the cotuous beam. etermato of atural frequeces ad modes shapes For a termedate spa as show Fg., Eq. 5 ca be rewrtte as cosh sh cos s 6 It should be oted that s are gve by Eq. ad that -

3 th Iteratoal S-N Users oferece Smulato Techology - 7 I a smlar maer, for spa s cos sh cosh 8 The frst ad secod dervatves of Eq. 6 w.r.t are cos s cosh sh 9a s cos sh cosh 9b cos s cosh sh 9c The frst ad secod dervatves of Eq. 8 w.r.t are cos s cosh sh a s cos sh cosh b cos s cosh sh c Wthout Overhag The boudary codtos s at supports are: a,b The cotuty codtos ca be wrtte for ay two cosecutve spas &. These codtos suggest that the rotatos ad momets of the two spas at support should be equal.e. c d pplyg s Eq. to Eqs. 8, 9 ad we get the followg goverg equatos for each par of adacet spas after smplfyg, No Spa Equato & & Fg.. Two cosecutve spas ad Table

4 Smulato Techology th Iteratoal S-N Users oferece coth cot a csch csc b The product,, ca be epressed terms of as per Eq. 7. Wth overhag The boudary codtos at the free eds.e at of spa are zero bedg momet a zero shear force b at of spa are zero bedg momet c zero shear force d The boudary codtos at - supports,,,,..., - a,b,,,,..., - c,,,,..., - d pplyg s Eqs. & to Eqs. 8, 9 ad we get the followg goverg equatos as show Table, for each par of adacet spas after smplfyg, Table No Spa Equato - α & & α α cosh cos 5a sh cos cosh s α cosh cos 5b sh cos cosh s -

5 th Iteratoal S-N Users oferece Smulato Techology -5 No Trval Soluto For a o trval soluto, the determat of the coeffcets of,, the Table must be equal to zero.e., 6 Smlarly, case of Table, α α 7 The determats so obtaed.e. Eq. 6 & Eq. 7 are called the frequecy determats whch o epaso yelds the frequecy equato, F 8 No smple epresso for the roots of Eq. 8 s avalable. smple way to determe the values of that satsfy the frequecy equato s to graphcally plot versus F for varous values of. The roots of the equato are,, p p p such that I p,.the correspodg atural frequeces f ad egevalues e of the beam are respectvely gve by the epressos, π ω f 9 ad e ω m EI ω Illustratve Eamples Two cases of the cotuous beam cofgurato have bee llustrated here. The beam has a uform rectagular cross-sectoal area over a legth of 7.5 meters ad s hged at uequal tervals as show Fg. &.

6 Smulato Techology th Iteratoal S-N Users oferece a 6 uequal spa cotuous beam wthout overhag as show Fg Fg.. 6-spa cotuous sgle beam wthout overhag b 6 uequal spa cotuous beam wth overhag as show Fg Table Parameters 5 Fg.. 6-spa cotuous sgle beam wth overhag Value Elastc modulus of the materal, E GPa esty of the materal, ρ Kg/m 786 rea of cross secto, m.5 wdth.8 depth egth of cotuous beam, m 7.5 Momet of erta of the cross-secto, I m. E-7 Mass per ut legth, m Kg/m. Posso s rato. The above eamples are solved aalytcally as well as usg FEM code S-N. The aalytcal formulato has bee solved the commercal code MT [] to obta a graphcal plot of vs F ad subsequetly the frequeces. I S- N, dfferet types of elemets have bee used to solve the problem.e. the cotuous beam havg beam elemets shell elemets amely quads ad sold brck elemets. The cotuous beam s made of staless steel. Table shows some of the mportat parameters requred to llustrate the eample. The materal property of the beam has bee mplemeted S-N [], usg MT # *MT_ESTI. The odes at the hged pots are costraed.e. traslatoal costrat, y & z drecto ad rotatoal costrat & y drecto. The computer tme s relatvely short for beam ad shell elemet model sce t has fewer degrees of freedom as for sold elemets the computato tme s relatvely more due to more umber of elemets ad odes the model. Each secod smulato requred about 5- secods of the PU o a SGI ONX IRIX Workstato. The values of atural frequeces trasverse drecto obtaed by aalytcal method are compared wth the FE Model aalyzed S-N as show Table. Fg. 5 & 6 shows the mode shapes trasverse drecto of the 6-spa cotuous beam wth ad wthout overhag. -6

7 th Iteratoal S-N Users oferece Smulato Techology a b c Fg. 5. Eghth mode shape trasverse drecto of 6-spa beam wthout overhag a eam elemet b Shell elemet ad c Sold brck elemet Scale Factor for Ege Vector mato s -7

8 Smulato Techology a th Iteratoal S-N Users oferece b c Fg. 6. Sth mode shape trasverse drecto of 6-spa beam wth overhag a eam elemet b Shell elemet ad c Sold brck elemet Scale Factor for Ege Vector mato s -8

9 th Iteratoal S-N Users oferece Smulato Techology Table. omparso of Frequeces trasverse drecto of mult-spa cotuous beam Two cases metoed above Wthout Overhag Wth Overhag alytcal eam Elemet EFORM elystschko -Schwer full cross-secto tegrato Nodes :5 Elemets:75 Nodal mass.9e Kg wthout overhag Nodal mass.e Kg wth overhag Frequeces Hz S-N Shell Elemet EFORM elystschko Tsay Nodes : 56 Elemets: 75 Nodal mass.9e Kg wthout overhag Nodal mass.e Kg wth overhag Sold Elemet EFORM Fully tegrated quadratc 8 ode sold elemet wth odal rotatos. Nodes : 55 Elemets: Nodal mass.9e Kg wthout overhag Nodal mass.e Kg wth overhag ocluso Table shows that the frequeces obtaed by the FEM solver S-N, usg sold brck elemets somewhat devates from that of the aalytcal eact method. lthough quadratc 8 ode sold elemets have o more tegrato pots tha a lear elemet, t yelds better results at less epese tha the lear elemets. ut t gves less accurate results compare to beam ad shell elemets due to varatos stffess whle usg dfferet formulatos frequeces deped o stffess ad mass. The degrees of freedom are dfferet for beam, shell ad sold elemets. The bedg deformato of the model havg quadratc 8 ode sold elemets s stff ad ot more accurate tha shell or beam elemets. So the frequeces of the model wth sold elemets are hgher. O the other had, the agreemet betwee the two approaches.e. aalytcal ad beam S-N usg beam elemet has bee foud to be ecellet. I geeral, the -9

10 Smulato Techology th Iteratoal S-N Users oferece atural frequeces of the cotuous beams ca be calculated more accurately by the use of beam elemets rather tha usg sold brck elemets S-N. ut t s ot always possble to model a comple geometry usg beam elemets. The results also showed the relablty ad capablty of the dyamc code S-N whch ca be further utlzed to solve problems of bgger ad comple ature. Ths study provded a bass for studyg the modal aalyss of -rod type Rado Frequecy Quadruple RFQ structure mplemetg boudary codtos as actual scearo. ut the above topc s beyod the scope of dscusso ths paper. Refereces [] Ferts emeter G., yamcs ad Vbrato of Structures, Joh Wley & Sos, Ic.,97. [] Usg Matlab Verso 7, The Mathwork, Ic., Natck M 76. [] S-N Keyword User s Maual Verso 97, vermore Software Techology orporato, vermore, 955, prl. -