SOME ASPECTS OF THE EXISTENCE OF COULOMB VIBRATIONS IN A COMPOSITE BAR

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1 SISOM 006, Bucharest 7-9 May SOME ASPECTS OF THE EXISTECE OF COULOMB VIBRATIOS I A COMPOSITE BAR Ştefana DOESCU Techncal Unversty of Cvl Engneerng, Dept. of Mathematcs, emal: stefa05@rdsln.ro. In ths paper, the exstence of the Coulomb vbratons n the moton of a composte bar, modeled as a Daves and Moon lattce, s dscussed. So, the solutons of nonlnear equatons that govern the moton of the composte bar are smply wrtten as a lnear superposton of Coulomb vbratons. Key words. Composte bar, Coulomb vbratons, Daves and Moon lattce.. ITRODUCTIO The Coulomb vbratons are descrbed wth the elementary Coulomb functons F ( μz, η ) wth F 0 ( μz,0) = sn μz, F 0 ( μz,0) = cos μ z (Abramowtz and Stegun []). These functons represent the elementary unts of vbratons used n ths paper to develop the general solutons of a composte bar. The composte bar means a bar wth a double helx structure (MacKay and Aubry []), such as DA (fg..). The varable radus of the composte bar verfes the perodcal relaton r = asn x= asn( π + x), 8 0 x L, a = 0Ǻ, and the length L = 5 0 Ǻ = 5 cm. Fg... The complex bar wth a double helx structure (after MacKay and Aubry []) By extendng the lnear equvalence method (LEM) formulated by Toma [3], t has shown that the Coulomb vbraton s a partcular soluton of the nonlnear system of equatons that descrbes the system moton, for a nonlnear pendulum (Donescu et al. [4], Donescu [5], Munteanu and Donescu [6]). The general solutons are wrtten as a lnear superposton of Coulomb vbratons. In ths paper, we nvestgate the exstence of the Coulomb vbratons for a composte bar modeled as a Daves and Moon lattce for =0, llustrated n fg.. (Munteanu and Donescu [6], Daves and Moon [7]). In the fgure the masses are m, all the sprng constants are, and all the dampng coeffcents are c. The equatons of moton of the lattce model shown n fg.. are c x = (3 x x + x ) x + [ F( x+ x) F( x x )], (.) m m m

2 05 Some aspects of the exstence of Coulomb vbratons n a composte bar wth =,,...,, and boundary condtons of zero dsplacement of the fxed end and zero force on the free end x 0 = 0, x = + x, (.) where m s the bloc mass, s the cantlevered beam stffness, c the dampng coeffcent, F represents the nonlnear force-dsplacement relatonshp for the buclng elements, and the number of masses. The dampng term x n the moton equaton comes about because the dampers between masses and the ones 3 attached between the masses and ground are taen to have dentcal coeffcents. The expresson for the force-dsplacement behavor of the buclng elements s the same as that used by Toda to model the nteracton forces n an atomc lattce γ F( x+ x) = [exp( b( x+ x)) ], (.3) b where γ s the element s lnearzed stffness, and b determnes the strength of the nonlnearty. For the atoms nteracton n a lattce of a sold the Morse nteracton force s vald n quantum mechancs of an electron moton, α beng a constant Fx ( + x) = F0[exp( α( x+ x)) exp( α ( x+ x))]. (.4) Fg... onlnear lattce model. The Morse force of nteracton governs precsely the nearest-neghbor nteracton n an anharmonc lattce of the atoms n a datomc molecule of solds n a contnuum lmt. But, no exact solutons are found for elastcty problems governed by Morse force. For Toda nteracton force there are some exact explct solutons for the model. Wth and c set to zero, equatons (.) and (.3) reduce to a Toda lattce, and the functonal form of the nonlnear solton soluton s gven by mβ + = ln[ sech ( β ) + ]. (.5) x x t b γ Substtuton of (.5) nto (.) and (.3) wth and c set to zero, shows that t solves these equatons f we have λ β = snh. (.6) m If γ > 0 and b < 0, x+ () t x() t < 0 for all values of and t. Ths corresponds to the compressve atomc lattce waves studed by Toda. If γ > 0 and b > 0, then x+ () t x() t > 0, we have the tensle βa waves, whch propagate to the speed c =, where a s the dstance between masses. The speed of ths solton s dependent upon the ampltude. For the case of small dsplacement and zero dampng, (.) and (.3) reduce to the lnear conservatve system defned by γ x + x + ( x+ x + x ) = 0, =,,...,. (.7) m m

3 Şt. DOESCU 06 Substtutng a perodc travelng wave soluton of the form x () t = Aexp[( a ωt)], nto (.7) we obtan the dsperson relaton 4γ sn a ω = ± ( ) +, (.8) m m where ω s the wave frequency, s the wave number and a s the spacng between adacent masses. From (.8) we see that the system allows only n the Brlloun zone of frequences to propagate wthout attenuaton. The equaton (.7) has the soluton wth A are tme-dependent coeffcents. The energy of the Moon [7]) ( ) π x() t = A()sn t, =,,...,, (.9) + = - mode of vbraton s (Daves and + m ( ) ( sn π E = A + A + γ ). (.0) 4 ( + ) If ntal condtons are small the nonlnear model (.) wll have approxmately lnear behavor, and the modal energes gven by (.0) wll be nearly constant n tme. For large ntal condtons, the nonlneartes may lead to complex modal nteractons. Let us analyze the modal nteracton by usng the Toda nteractng equatons. We now that the - solton soluton of a nonlnear equaton can be regarded as a system of - nteractng soltons, each of whch becomes a sngle solton as t ±. When we have sngle soltons ntally apart n space they nteract wth each other and become apart wthout exchangng ther denttes (Donescu and Munteanu [8]). To study the nteracton modal soltons, we consder (.) and (.3) under the form of the orgnal Toda equaton or d V( n) d V( n ) V( n ) V n +V( n ) = + + ( ), (.) d ln[ V( n)] V 0 + Δ =, (.) Here, the operator d s d= / t and n =,...,,,0,,,...,. Δ V( n) = V( n+ ) + V( n ) V( n). (.3). SOLUTIOS Suppose that the soluton Vn ( ) of (.) and (.3) has the form V( n) = d ln f, (.) where the functon f s gven by where I and B are matrces I l = δ, l f( γ, γ,..., γ ) = det[ I + B( n+ )], (.) Bl ( n) = ( n) l ( n) zz φ φ, φ( n) = Cexp γ ( n), γ ( n) = β t α n, (.3) l

4 07 Some aspects of the exstence of Coulomb vbratons n a composte bar z z wth arbtrary real constants α and C,, l, =,,...,, and z = ± exp( α), β =. Let us ntroduce -ndependent tme varables t, =,,...,, and defne as Φ ( n) = C exp Γ ( n), Γ ( n) = β t α n, (.4) where B s a matrx wth elements F( Γ, Γ,..., Γ ) = A( Γ )det[ I + B( n+ )], (.5) Bl ( n) = Φ ( n) Φl ( n),, l, (.6) zz l wth arbtrary functons α ( n) and β ( n). Introduce the operators The soluton of of (.) and (.3) s gven by (.) A( Γ ) = exp[ α ( n) Γ + β( n )], (.7) = n ˆd (,,..., ) ( / ) (,,..., ) t t... F Γ Γ Γ = t F Γ Γ Γ = = = t = t. (.8) ˆ ˆ. (.9) V( n) = d d lnf = d d lnf Here df = dˆ F and ddˆ = dˆ d. We have = = ˆd 0 G =,. (.0) G = d ln[ + ( d ˆ ) F] Δ d lnf l m l m or Ths equaton s satsfed f G s a constant and f Therefore, the soluton becomes The explct form of V ( n) s obtaned as ˆdG = 0 for each. So, we must have, (.) dˆ d ln[ + ( d ˆ ) F] = dˆ Δ d ln F l m l m d(ddˆ ln F) d =Δ (ddˆ ln F). (.) + V( n). (.3) = = V( n) = d dˆ ln F = V ( n) n n n V n A A A A A A ` ` `l 3 r m( ) = + αm + αm l + αm l r = =!, l=, =!!, l, m=,, r= l l m + 4 lmn,,, =, rs,, = l m n α n n n n!! r! s! r s 4 r s ` `l `m `n m AAAA l r n, n n n!! r! r ` `l `m (.4) where α, s =,...,4 are determned from (.). By usng the seres expansons for crcular functons, the s m unnown functons A n, n =,,3,...,6 are determned as

5 Şt. DOESCU 08 A z z A z z B z, z, (.5) + n( ) = {( μ ) n( η) Φ ( μ, η) + ( μ ) n( η) ψ( μ, η)}, η = 0 and = 0,,,3,..., η = 0,,,3,...,6. The functons Φ ( μz, η) and ψ ( μz, η ) are gven by and μ = μη ( ), and max Φ m ( ) ( μz, η) ( μz) Am ( η) m= + 6 = = 6 μ = αλ, α = η +, α 0, =,,3,...,6. l=, ψ μ η = μ η, (.6) m ( ) ( z, ) ( z) Bm ( ) m= + The functon ψ ( μz, η ) s lned to the Φ( μz, η) by d Ψ ( μz, η) = μ Φ ( μz, η) = μφ ( μz, η). (.7) dt By notng F z C z z + ( μ, η) = ( η) ( μ ) Φ( μ, η) (.8) the representaton (.5) becomes A ( z) = { B ( η) F ( μz, η) + C ( η) F ( μz, η)}]. (.9) n n n η, = 0 We have F ( μ z,0) = sn μz, F ( μz,0) = cosμz (.0) 0 0 We see that the functons F ( μz, η have the form of Coulomb wave functons (Abramowtz and Stegun []). These functons F ( μz, η) are called the Coulomb vbratons. So, the solutons of nonlnear equatons that govern the moton of a composte bar are wrtten as a lnear superposton of Coulomb vbratons. Fg.. dsplays the soluton for the composte bar. Fg.. The soluton for the composte bar.

6 09 Some aspects of the exstence of Coulomb vbratons n a composte bar 4. COCLUSIOS In ths paper, a composte bar wth a double helx structure such as DA s consdered. The varable radus of the composte bar verfes the perodcal relaton. We demonstrate the exstence n the bar moton of the Coulomb vbratons, whch are descrbed wth the elementary Coulomb functons F ( μz, η) wth F 0 ( μz,0) = sn μz, F 0 ( μz,0) = cosμ z. These functons represent the elementary unts of vbratons, whch allow the constructon of the solutons for the composte bar. ACKOWLEDGEMET. The authors acnowledge the fnancal support of the atonal Unversty Research Councl (URC-CCSIS) Romana, Grant nr /005, code 5/003, and Grant nr. 7664/005, code60/005. REFERECES. ABRAMOWITZ, M., STEGU, I.A., (eds.), Handboo of mathematcal functons, U. S. Dept. of Commerce, MACKAY, R.S., AUBRY, S., Proof of exstence of breathers for tme-reversble or Hamltonan networs of wealy couple oscllators, onlnearty, 7, pp , TOMA, I., Lnear equvalence method and applcatons, Ed. Flores, Bucharest, DOESCU, St., CHIROIU, V., MUTEAU, L., Applcaton of LISA to the dynamcs of perodcally layered elastc meda, AMSE: Modellng, Measurement and Control, Seres B: Mechancs and Thermcs, 74, 6, pp.37-48, DOESCU, Şt., On the analytcal solutons for a nonlnear coupled pendulum, AMSE: Modellng, Measurement and Control, Seres B: Mechancs and Thermcs, 74, 3, pp. -3, MUTEAU, L., DOESCU, St., Introducton to Solton Theory: Applcatons to Mechancs, Boo Seres Fundamental Theores of Physcs, 43, Kluwer Academc Publshers, DAVIES, M. A., MOO, F. C., Soltons, chaos and modal nteractons n perodc structures, In Dynamcs wth Frcton: Modelng, Analyss and Experment, part II, Seres of Stablty, Vbraton and Control of Systems, World Scentfc Publshng Company, 7, pp.99 3, DOESCU, St., MUTEAU, L., ch.4, The effect of dampng on the stablty of dynamcal systems, n Topcs n Appled Mechancs, vol., Ed. Academe, pp.85-7, 004 (eds. V. Chrou, T. Sreteanu).