Dynamics of Nonlinear Beam on Elastic Foundation

Transcription

1 Proceedings of he World Congress on Engineering Vol II WCE, June - July,, London, U.K. Dynamics of Nonlinear Beam on Elasic Foundaion Salih N Akour Absrac Simply suppored nonlinear beam resing on linear elasic foundaion and subjeced o harmonic loading is invesigaed. Parameric sudy is carried ou in he view of he linear model of he problem. Hamilon s principle is uilized in deriving he governing equaions. Well known forced duffing oscillaor equaion is obained. The equaion is analyzed numerically using Runk-Kua echnique. Three main parameers are invesigaed: he damping coefficien, he naural frequency, and he coefficien of he nonlinear erm. Sabiliy regions are unveiled. Index Terms Elasic Foundaion, Nonlinear Beam, Parameric Sudy. I. INTRODUCTION There are many applicaions for beam on elasic foundaion mainly in mechanical and civil engineering e.g. disc brake pad, shafs suppored on ball, roller, or journal bearings, vibraing machines on elasic foundaions, nework of beams in he consrucion of floor sysems for ships, buildings, and bridges, submerged floaing unnels, buried pipelines, railroad racks ec. The elasic foundaion for he beam par is supplied by he resilience of he adjoining porions of a coninuous elasic srucure. More deails of he applicaions of his concep are discussed by Heenyi []. Beams on elasic foundaions received grea aenion of researches due o is wide applicaions in engineering. Heenyi [] and Timoshenko [] presened an analyical soluion for beams on elasic suppors using classical differenial equaion approach, and considering several loading and boundary condiions. I is well known in engineering ha a beam suppored by discree elasic suppors spaced a equal inervals acs analogously o a beam on an elasic foundaion and ha he appropriaeness of ha analogy depends on he flexural rigidiy of he beam as well as he siffness and spacing of he suppors. Ellingon invesigaed condiions under which a beam on discree elasic suppors could be reaed as equivalen o a beam on elasic foundaion []. Beams resing on elasic foundaions have been sudied exensively over he years due o he wide applicaion of his sysem in engineering. This sysem according o he lieraure can be divided a leas ino hree caegories. The firs caegory is linear beam on linear elasic foundaion. Example of his ype can be found in references []-[]. The applicaions in his caegory include bu no limied o Euler - Bernoulli beam, Timoshenko beam, Winkler foundaion, Pasernak foundaion, ensionless foundaion, Mechanical and Indusrial Engineering Deparmen, College of Engineering, Sulan Qaboos Universiy, PO Box, Al Khoud, Musca, Oman, Fax: +9, Phone: +9-7, or, +9- -, akour@squ.edu.om. On Leave from Mechanical Engineering Deparmen, Faculy of engineering and Technology, Universiy of Jordan, Amman 9, Jordan. ISBN: ISSN: 7-9 (Prin); ISSN: 7-9 (Online) single parameer or wo parameer foundaion, saic loading, harmonic loading and moving loading. The second caegory is linear beam on nonlinear elasic foundaion []-[]. In his caegory he foundaion is considered o have nonlinear siffness. Also his ype includes differen boundary and loading condiions according o he engineering applicaion. The hird caegory is nonlinear beam on linear elasic foundaion []-[7]. Usually he beam nonlineariy means large deflecions. Mos of he sudies relaed o his caegory have analyzed he sysem eiher using boundary elemen mehod or boundary inegral equaion mehod. Similar o he above wo caegories, here is wide variey of boundary and loading condiions being applied o such sysem according o he applicaion. Nonlinear beam subjeced o harmonic disribued load resing on linear elasic foundaion is invesigaed in his research. The sudy is carried ou in he view of he linearized model of he sysem. Well known duffing equaion is obained using Hamilon s principle. Three main parameers are invesigaed: he damping coefficien, he naural frequency, and he coefficien of he nonlinear erm. The effec of hese parameers on he sysem sabiliy is unveiled. Up o he auhor s knowledge, his work is no published in he lieraure. II. PROBLEM STATEMENT Nonlinear beam resing on elasic foundaion ha is shown in Fig. is subjeced o he following condiions:. The beam maerial properies are linear.. The damping () and siffness (k f ) of he foundaion are linear.. The beam is slender and prismaic.. The beam is simply suppored (pin-pin ends). The load applied is harmonic and disribued over he lengh of he beam. Fig. : schemaic drawing of he beam on elasic foundaion. III. MATHEMATICAL FORMULATION A. Kineic Energy The roary ineria of he beam will be negleced since he beam is slender. WCE

2 Proceedings of he World Congress on Engineering Vol II WCE, June - July,, London, U.K. Where : maerial densiy, A: beam cross secional area, L: beam lengh, w=w(x,): beam ransverse displacemen (in y- direcion). B. Poenial Energy: The poenial energy due o bending can be calculaed as he following: Where The formulaion of he due o sreching poenial energy can be cased as he following []: The elasic foundaion is assumed o have consan linear spring modulus. This resuls in, () () () () The load is uniform along he lengh of he beam and varies harmonically wih respec o ime. Therefore,,.,..., Where P: ampliude of exciaion and e : frequency C. Derivaion of governing equaion The lagrangian is defined as he following: sin By applying Hamilon s principle () exciaion sin Denoe he firs and he second inegral by F and F respecively. This gives Inegraing he firs and he second erm by pars wih respec o x he resul is he following equaion: sin Since is arbirary, he following can be concluded from he above equaion: The governing equaion comes from seing he expression wihin he brackes in () equal o zero. Upon carrying ou he indicaed differeniaions, he governing can be rewrien as, sin (7) where, I is obvious ha (7) is he duffing oscillaor equaion. This equaion is going o be recased ino a more familiar form in he nex secion. The boundary and iniial condiions can be obained from he remaining erms in (). The boundary condiions a x= and x=l are Eiher is zero or w is prescribed (a) Eiher is zero or w is prescribed (b) Eiher is zero or w is prescribed (c) Boundary condiions (a) correspond o end momens and slopes respecively. In (b), w corresponds o end displacemen, and in (c) he firs condiion corresponds o () ISBN: ISSN: 7-9 (Prin); ISSN: 7-9 (Online) WCE

3 Proceedings of he World Congress on Engineering Vol II WCE, June - July,, London, U.K. pre-sreching. For he pinned ends, he boundary condiions are: These boundary condiions mus be saisfied by he mode shapes of he sysem. This fac will be used in he following secions as he crieria for selecing he form of he mode shape equaion. Finally he iniial condiions for are Eiher is zero or is prescribed In his case, i will be assumed ha he sysem sars from res i.e. he iniial displacemen and velociy is zero. D. Discreizaion and linearizaion The following expression is used for w(x, ) in order o discreize he problem, (9) For simpliciy he limis of he above summaion, he subscrip of w, and he ime dependence of w will be implied in he equaions ha follow. I is eviden from (9) ha he pinned ends boundary condiion (a) are saisfied since ransverse displacemens a and L are zero, and he end slopes are free (implying zero bending momens a he ends). Equaion (9) represens series summaion of N modes each has ime dependen ampliude response, wih spaial sine funcion. Subsiuing (9) ino he original inegral expressions for he kineic and poenial energy of () hrough () hen applying he Lagrangian and uilizing he orhogonaliy, he following equaion comes ou: () Lagrangian s equaion for each mode can be wrien as he following:,,, () Subsiuing () in () and carry ou he differeniaion yields, A simplified form of () resuls afer rearranging he coefficiens and defining some new coefficiens. The concise form and he coefficien definiions are w n+ ω n + () Where ω ω f ω ω f () Wriing () for a single mode and insering he linear damping erm gives, w n+ μ ω n + () Where µ is he damping coefficien ISBN: ISSN: 7-9 (Prin); ISSN: 7-9 (Online) This makes i clear ha he above equaion represen unforced damped duffing oscillaor. Recasing () ino he following: + μ () Where ω n + In order o linearize he sysem for he firs mode (n=) he sysem is convered ino firs order ordinary differenial equaions by he following subsiuion Applying his o (), μ, μ From he above equaions i is obvious ha (, ) is he only criical poin for he sysem. So he equivalen linear sysem is obained by expanding he above equaion using Taylor series abou (, ), so he remaining linear erms are, μ The corresponding Jacobi marix is μ So he Eigenvalues of J are,, μ μ The following can be said abou (, ): Sable and aracive as long as. Sable if. Unsable if. A node as long as A spiral poin if A cener if The general soluion of he linearized unforced sysem is Applying he iniial condiions, he consans of inegraion are going o be as he following: E. Simulaion of he nonlinear sysem μ sin () where ω n + ω f ω, ω,, ω f, I is obvious ha he srengh of he nonlineariy is inversely proporional o he square of he radius of gyraion of he beam. This indicaes ha he nonlineariy remains weak as long as he beam is relaively slender as assumed in his sudy. Finally, he frequency equaion can be simplified o ω ω f WCE

4 Proceedings of he World Congress on Engineering Vol II WCE, June - July,, London, U.K. The apparen naural frequency of he sysem is he square roo of he sum of he squares of he naural frequencies of he beam and he elasic foundaion. The nonlinear second order ordinary differenial equaion is convered ino a sysem of firs order ordinary differenial equaions. This is suiable for numerical sudy using Runge- Kua Techniques. sin μ IV. RESULTS AND DISCUSSION The resuls for simply suppored beam on elasic foundaion are presened in Tables I and II. Table I represens sample phase diagrams of he sudied ranges. Table II shows he ime response for oward chaos cases ha are presened in Table I. The phase diagram and he ime response are colleced afer long period of ime o be sure ha he sysem has passed he ransien range. The duffing () is solved using MATLAB package by uilizing he Runga-Kua ODE (Ordinary Differenial Equaion) solver. The equaion which represens he sysem under invesigaion is of cubic nonlineariy wih harmonic exciaion. The sample resuls presen he effec of he damping when he sysem has weak, medium and srong nonlineariy for exciaion frequencies below, a and above resonance. The whole sudy is considering weak nonlineariy ha does no exceed =. and hose levels of weak, medium and srong wihin ha range. Only he firs mode is considered in his sudy. The parameers range covered in his invesigaion are for =. hrough., =. hrough. and naural frequency =.7 hrough.. I can be seen from Table I ha when here is no damping he sysem is ending oward chaos however when lile damping is applied he sysem is ending owards limi cycle. I is obvious ha he damping and he nonlineariy are he mos effecive parameers in conrolling he chaoic behavior of he sysem. As long as he radius of gyraion for he beam under consideraion is large i.e., he beam is more owards slender, he nonlineariy is going o be weak. This means ha he conribuion of he sreching energy o he behavior of he sysem is going o be low. The damping sysem dissipaes he oscillaing energy and provides a conrol over he sysem behavior. For he linear sysem, as long as damping coefficien is posiive, he ransien response is going o decrease exponenially and he homogenous response (due o he forcing exciaion) is bounded even a resonance. For undamped linear sysem homogenous response is no bounded and he response is increasing wih ime. For he nonlinear sysem, he response is ending oward chaos as i can be seen in able for =.. However when he damping increases he sysem is ransferring from chaos o limi cycle. For he resonance case of no damping i.e., he exciaion frequency equals he naural frequency, he linear sysem has increasing ampliude response whereas he nonlinear sysem is ending oward chaos wih bounded ampliude. Also for he nonlinear case (wihin he range of invesigaion) as long as here is no damping he sysem is ending oward chaos. V. CONCLUSION The behavior of nonlinear beam on elasic foundaion is unveiled. I is found ha he sysem is sable and conrollable as long as he damping coefficien is non zero ISBN: ISSN: 7-9 (Prin); ISSN: 7-9 (Online) and posiive. As he nonlineariy increases more damping is required o preven i from moving owards chaos. For firs mode shape he naural frequency could be calculaed as square roo of he sum of squares of boh naural frequency of he beam and he foundaion. The srengh of he nonlineariy is inversely proporional o he square of he radius of gyraion, i.e. as long as he beam more owards slender he nonlineariy is weaker. The sreching poenial energy is responsible for generaing he cubic nonlineariy in he sysem. VI. REFERENCES [] M. Heenyi, Beams on elasic foundaions. Ann Arbor, MI: Universiy of Michigan Press; 9, 9. [] S. Timoshenko Srengh of maerials, Par II, advanced heory and problems. Third ed. Princeon, NJ: Van Nosrand; 9. [] J.P. Ellingon, The beam on discree elasic suppors, Bullein of he Inernaional Railway Congress Associaion, vol., no., 97, pp [] Sao Moohiro, Kanie Shunji and Mikami Takashi, Srucural modeling of beams on elasic foundaions wih elasiciy couplings, Mechanics Research Communicaions, vol., 7, pp. 9. [] C. Miranda and K. Nair, Finie beams on elasic foundaion, ASCE Journal of Srucure Division, vol. 9, 9, pp.. [] T. M. Wang and J. E., Sephens Naural frequencies of Timoshenko beams on Pasernak foundaions, Journal of Sound and Vibraion, vol. 9, 977, pp. 9. [7] T. M. Wang and L. W., Gagnon Vibraions of coninuous Timoshenko beams on Winkler-Pasernak foundaions, Journal of Sound and Vibraion, vol., 977, pp. 9. [] M. Eisenberger and J. Clasornik, Beams on variable woparameer elasic foundaion, Compuers and Srucures, vol., 9, pp.. [9] Wang C M, Lam K Y, He X O. Exac soluion for Timoshenko beams on elasic foundaions using Green s funcions. Mechanics of Srucures & Machines, 99,. [] T. M. Wang and J. E. Sephens, Naural frequencies of Timoshenko beams on Pasernak foundaions, Journal of Sound and Vibraion, vol. 9, 977, pp. 9. [] T. M. Wang and L. W. Gagnon, Vibraions of coninuous Timoshenko beams on Winkler-Pasernak foundaions, Journal of Sound and Vibraion, vol., 977, pp. 9. [] M. Eisenberger and J. Clasornik, Beams on variable woparameer elasic foundaion, Compuers and Srucures, vol., 9, pp.. [] R. H. Guierrez, P. A. Laura and R. E. Rossi, Fundamenal frequency of vibraion of a Timoshenko beam of non-uniform hickness, Journal of Sound and Vibraion, vol., 99, pp.. [] W. L. Cleghorn and B. Tabarrok, Finie elemen formulaion of apered Timoshenko beam for free laeral vibraion analysis, Journal of Sound and Vibraion, vol., 99, pp. 7. [] Faruk Fıra Çalım, Dynamic analysis of beams on viscoelasic foundaion, European Journal of Mechanics A/Solids, vol., 9, pp. 9 7 [] L. SUN, A Closed-Form Soluion Of A Bernoulli-Euler Beam On A Viscoelasic Foundaion Under Harmonic Line Loads, Journal of Sound and vibraion, vol., no.,, pp. 9-7 [7] Seong-Min Kim, Sabiliy and dynamic response of Rayleigh beam columns on an elasic foundaion under moving loads of consan ampliude and harmonic variaion, Engineering Srucures, vol. 7,, pp. 9. [] Garinei, Vibraions of simple beam-like modelled bridge under harmonic moving loads, Inernaional Journal of Engineering Science, vol.,, pp [9] MO Yihua, OU Li and ZHONG Hongzhi, Vibraion Analysis of Timoshenko Beams on a Nonlinear Elasic Foundaion, Tsinghua Science And Technology, vol., No., June 9, pp.. [] F. W. Beaufai and P. W. Hoadley, Analysis of elasic beams on nonlinear foundaions, Compuers and Srucures, vol., 9, pp [] Birman V., On he effecs of nonlinear elasic foundaion on free vibraion of beams, ASME Journal of Applied Mechanics, vol., 9, pp. 77. WCE

5 Proceedings of he World Congress on Engineering Vol II WCE, June - July,, London, U.K. [] N. R. Naidu and G. V. Rao, Free vibraion and sabiliy behavior of uniform beams and columns on non-linear elasic foundaion, Compuers & Srucures, vol., 99, pp. -. [] N. Rajasekhara Naidu and G. Venkaeswara Rao, Free Vibraion And Sabiliy Behaviour Of Uniform Beams And Columns On Nonlinear Elasic Foundaion, Compuers & Srucures, vol., No., 99, pp.. [] Pellicano and F. Masroddi, Nonlinear Dynamics of a Beam on Elasic Foundaion, Nonlinear Dynamics, vol., 997, pp.. [] Ashraf Ayoub, Mixed formulaion of nonlinear beam on foundaion elemens, Compuers and Srucures, vol.,, pp.. [] J. T. Kasikadelis, and A. E. Armenakas, Analysis of clamped plaes on elasic foundaion by boundary inegral equaion mehod, Transacions of he ASME, vol., 9, pp. 7. [7] J. Puonen and P. Varpasuo, Boundary elemen analysis of a plae on elasic foundaions Inernaional Journal of Numerical Mehods in Engineering, vol., 9, pp. 7. [] T. Horibe, An analysis for large deflecion problems of beams on elasic foundaions by boundary inegral equaion mehod, Transacion of Japan Sociey of Mechanical Engineers (JSME), vol., No. 7, A, 97, pp. -9. [9] E. J. Sapounakis and J. T. Kasikadelis, Unilaerally suppored plaes on elasic foundaions by Boundary elemen mehod, Transacions of American Sociey of Mechanical Engineers (ASME), vol. 9, 99, pp.. [] Horibe, T., Boundary Inegral Equaion Mehod Analysis for Beam-Columns on Elasic Foundaion, Transacion of Japan Sociey of Mechanical Engineers (JSME), vol., No., A, 99, pp 7-7. [] Tadashi Horibe and Naoki Asano, Large Deflecion Analysis of Beams on Two-Parameer Elasic Foundaion Using he Boundary Inegral Equaion Mehod, JSME Inernaional Journal, A, vol., No.,, pp. [] N. Kamiya, and Y. Sawaki, An Inegral Equaion Approach o Finie Deflecion of Elasic Plaes, Load-deflecion curves of beam on Pasernak foundaion (C-C, y=) Inernaional Journal Non-Linear Mechanics, vol. 7, No., 9, pp [] S. Miyake, M. Nonaka, and N. Tosaka, Geomerically Nonlinear Bifurcaion Analysis of Elasic Arch by he Boundary-Domain Elemen Mehod, Boundary Elemens XII, vol., 99, pp.. [] Ashraf Ayoub, Mixed formulaion of nonlinear beam on foundaion elemens, Compuers and Srucures, vol.,, pp.. [] S. Lenci and A. M. Taranino, Chaoic Dynamics of an Elasic Beam Resing on a Winkler-ype Soil, Chaos, Soluions & Fracals, vol. 7, No., 99, pp., [] B. Kang and C.A. Tan, Nonlinear response of a beam under disribued moving conac load, Communicaions in Nonlinear Science and Numerical Simulaion, vol.,, pp.. [7] I Coskun and H. Engin, Non-Linear Vibraions of a Beam on an Elasic Foundaion, Journal of Sound and Vibraion, vol., no., 999, pp.. [] I. Shames, and C. Dym, Energy and finie elemen mehods in srucural Mechanics, Hemisphere Publishing, New York, 9. ISBN: ISSN: 7-9 (Prin); ISSN: 7-9 (Online) WCE

6 Proceedings of he World Congress on Engineering Vol II WCE, June - July,, London, U.K. Table I: Presenaion of sample phase rajecories for he parameers under invesigaion. ώ. ώ ώ. ώ. ώ.7 ώ. = =. =. = =. = = =. =.7 µ=. =. P= Toward Chaios Toward Chaios Toward Chaios. =. =. =. =. =. =. =. =. =.7. µ=. =. P= Towards Limi Cycle Towards Limi Cycle Towards Limi Cycle. =. =. =. =. =. = =. =. =.7. µ=. =. P= Limi cycle Limi cycle Limi cycle = =. =. = =. = = =. =.7 µ=. =. P= Toward Chaios Toward Chaios Toward Chaios. =. =. =. =. =. =. =. =. =.7. µ=. =. P= Towards Limi Cycle Towards Limi Cycle Towards Limi Cycle. =. =. =. =. =. = =. =. =.7. µ=. =. P= Limi cycle Limi cycle Limi cycle ISBN: ISSN: 7-9 (Prin); ISSN: 7-9 (Online) WCE

7 Proceedings of he World Congress on Engineering Vol II WCE, June - July,, London, U.K. = =. =. = =. = = =. =.7 µ=. =. P= Toward Chaios Toward Chaios Toward Chaios =. =. =.. =. =. = =. =. =.7... µ=. =. P= Towards Limi Cycle Towards Limi Cycle Towards Limi Cycle =. =. =.. =. =. = =. =. =.7... µ=. =. P= Limi cycle Limi cycle Limi cycle Table II: Illusraion of ime response of he chaoic cases. ώ. ώ ώ.ώ. ώ.7 ώ. = =. =. = =. = = =. =.7 µ=. =. P= = =. =. = =. = = =. =.7 µ=. =. P= = =. =. = =. = = =. =.7 µ=. =. P= ISBN: ISSN: 7-9 (Prin); ISSN: 7-9 (Online) WCE