ICSV4 Cairs Australia 9- July, 007 THEORETICAL RESEARCH REGARDING ANY STABILITY THEOREMS WITH APPLICATIONS Marcel Migdalovici ad Daiela Bara Istitute of Solid Mechaics, INCAS Elie Carafoli, 5 C-ti Mille street, 0 Iuliu Maiu street, Bucharest, Romaia marcel_migdalovici@yahoo.com Abstract The research is focuses o the theoretical study of the stability i sese of Lyapuov for evolutio of the dyamical systems that deped of parameters. Is prove a origial theorem of separatio betwee the stable ad ustable zoes, i the plae of chose pricipal parameters. I the paper is related, usig this results, a origial method for idetificatio, i the plae of pricipal parameters of the mathematical model of the dyamical system, the stabilities ad istabilities regios of the dyamical system motio. We aalyze also a lot of theorems, as the Floquet stability theorem, about the motio stability for the dyamical systems described by differetial equatio systems with periodical coefficiets. The results are applied to study the motio stability of the couple patograph cotact wire of the electrical locomotive. The parameters of the system cosist of two cocetrated masses, the bedig stiffess, the horizotal tesio, the viscous dampig ad the mass per uit legth of the wire, the other dampig coefficiets ad stiffess elemets of the system ad ay costat speed specified i the model. We study the stabilities ad istabilities regios of the dyamical system motio usig these parameters ad our origial method of idetificatio of stability zoes, i the plae of pricipal parameters of the mathematical model of the dyamical system patograph-cotact wire.. INTRODUCTION Firstly we describe some results about the differetial liear equatios ad systems. Cosider a liear differetial equatio of order for the ukow fuctio y : ( ) ( -)... ' 0 y + a y + + a y + a y = f () where a,..., a, a0, f are fuctios defied o a iterval J R with complex values, ad iitial coditios ( ) 0 0 0 0 y( x ) = y, y'( x ) = y,..., y ( x ) = y.
ICSV4 9- July 007 Cairs Australia ( ) ( ) Usig the otatio w+ = y, w = y,..., w = y ', w = y, the equatio () ca be writte i a matrix form as where: W' = AW + g () 0 0... 0 0 w...... A=, W = 0 0 0... 0 w a a a... a a w 0 ad the iitial coditios are expressed as W( x0) = W0. We preset without proof [4], the followig theorem: 0 y0......, g =, W0 =, 0 y f y Theorem If i matrix equatio () the fuctios f, a0,..., a are cotiuous o the defiitio iterval J R, the equatio () has a uique solutio W( x ), a colum vector, so that W( x0) = W0. Are defied ay liear idepedet solutios of the homogeous system W ' = A W as the fudametal system of solutios. With the liear idepedet vectors, placed oe after the other, oe forms a square matrix W called the fudametal matrix of homogeous system which verifies the matrix equatio W' = AW. Theorem If W is ay fudametal matrix of system W ' = A W, the ay solutio of this system ca be writte as w= Wc where c is a costat vector; if the iitial coditio is wx ( 0) = w0 the the solutio is wx ( ) = W( x)w ( x0) w0. Ay fudametal matrix of system ca be deduced from aother multiplyig at right with a costat matrix. Theorem 3 If W is ay fudametal matrix of the system w' = A w, ad wx ( 0) = w0, 0 the ay solutio of the ihomogeeous system w' = A w+ g, A, g C ( J), is x 0 0 x0 wx ( ) = W( x)w ( x) w + W( x) (W g)( tdt ).. MATRIX FUNCTIONS We preset i the followig paragraphs some details about the matrix fuctios. For the begiig we cosider the polyomial fuctio. If A M with proper values λ, λ,..., λ, k the A M, k N ad p( A ) is: m m m m j pa ( ) = A + ba +... + b A+ b, b C, j=,..., (3)
ICSV4 9- July 007 Cairs Australia For a matrix which admits a diagoal form, that meas that there is a matrix D with o zero values oly o its diagoal, ad a ivertible matrix S with k k A = SD S ; pa ( ) = SpDS ( ), where: - A = SDS the p( λ ) 0... 0 pd ( ) =... 0 0... p( λ ) (4) We exted the matrix fuctio defiitio for differetiable fuctios i a domai i C which cotais the proper values of A. For a closed rectifiable curve γ which icludes iside a poit ς, where g is differetiable, but which does ot iclude a sigularity of g, is kow that g( ς) = ( ς z) g( z) dz. We defie g( A ) as g( A) = ( A zi) g( z) dz, where π i γ πi γ γ is a closed rectifiable curve which icludes the spectrum of A, but does ot iclude ay xz xa sigularity of g. For the expoetial fuctio g( z) = e, x, z C, oe defies g( A) = e xz as g( A) = ( A zi) e dz, where γ is a closed rectifiable curve which icludes the πi γ proper values of A. We differetiate: d d ( e ) = ( ( A zi) e dz) = ( A zi) ze dz = Ae (5) dx dx xa xz xz xa πi γ πi γ xz xa because the last itegral is the matrix fuctio for g( z) = ze. We obtai that e verifies the matrix equatio W' = AW ad for x = 0 we have 0 A e = ( A zi) dz = I, where I is πi γ the uit matrix. xa The matrix e is a fudametal matrix for the differetial system: W ' = A W + g, 0 A, g C ( J) ad the geeral solutio, with the iitial coditios wx ( 0) = w0, is: x ( x x0 ) A xa ta 0 x0 wx ( ) = e w + e e gdt. 3. DIFFERENTIAL EQUATIONS WITH PERIODICAL COEFFICIENTS Cosider the liear homogeous differetial system W ' = A W, A M, A C ( J), J R. We suppose that there is p R + so that A( x+ p) = A( x) for ay x J. The system is periodic with the period p. We metio the followig theorem (Floquet): Theorem 4 If the system W ' = A W is periodic, with the period p > 0, the ay R fudametal matrix W of the system, ca be expressed as W( x) = W ( xe ) x, where 0 3
ICSV4 9- July 007 Cairs Australia W( x) M is a periodical matrix with the period p, ad R M is a costat matrix R = l(c), with costat matrix C defied by W( x + p) = W( x) C, C M. p 4. STABILITY THEORY ASPECTS Cosider the differetial system y' = A y, A M with compoets defied ad cotiuous o I R. Cosider also t 0 I ad y 0 R. From theorem, the solutio y : I R, exists, it is uique, so that yt ( 0) = y 0. Aother solutio y : I R of the system with the iitial coditio yt ( 0) = y0, ad y0 y 0, is called a perturbed solutio of system, reported to y. The solutio y : I R is called Lyapuov stable if for ay ε > 0 exists δ so that, for y0 y 0 < δ the y y < ε for ay t > t0, where y = max{ y( t), y ( t ),..., y ( t ) ; ; t t0 }. If, supplemetary, yj() t y j() t 0, for ay j =,,...,, ad t, the the solutio is called asymptotic stable. Theorem 5(Floquet) If the system W ' = A W is periodic, with the period p > 0, ad W, ay fudametal matrix of the system, expressed as: W( x) = W ( xe ) x, where W( x) M is a periodical matrix with the period p, ad R M is a costat matrix, the, if the proper values of R have egative real part, the solutio of the periodical system is asymptotically stable, ad if at least a proper value of the matrix R is strictly positive, the solutio of the periodical system is ustable. If the proper values of the matrix R have zero real part, the the solutio of the periodical system is udecided ( stable, ustable or periodical). Theorem 6 If y : I R is a stable solutio of the system y' = A y, with matrix of cotiuous compoets, defied by parameters, for fixed parameters, there is a eighbourhood of fixed parameters where the solutio y is also stable. For a ustable solutio of the system we ca formulate aalogue property. Proof: We deote the set of parameters of the system by P ad the solutio of the system y' = A y, for the kow iitial coditios, by ytp (, ). We suppose that the solutio ytp (, ) is stable but there is ot a eighbourhood of fixed parameters where the solutio y of the system is also stable. There is a sequece of parameters P P for which the solutio ytp (, ) is ustable ad for which yt (, P) yt (, P) > ε, where t ad ε are specified values. Because yt (, P) yt (, P) < ε for ay t > t0 ad y0 y 0 < δ, from cotiuity of solutio y with P P, is developed a cotradictio that verify this theorem This theorem is used for separatio of the stable ad ustable zoes i the plae of pricipal parameters by curves of periodical solutios of the system. Determiig the domai of periodic solutios i the two chose parameters plae, oe determies the image of stability zoe i this plae. We use the followig procedure to idetify the boudary of the poits with periodic solutio from the two chose parameters plae. The fixed domai for aalysis, of the R 4
ICSV4 9- July 007 Cairs Australia parameters plae, is covered with a sufficiet fie mesh ad we study the evolutio of the specified displacemet solutio i the mesh poits. I the eighborhood of the periodic poits of the parameters plae oe ca use a refied mesh. 5. APPLICATION The dimesioless system of equatios [6] that specifies the state form of the dyamical system described by patograph ad cotact wire, is: ( µ) y + ω ( y - y ) + ω ( y - y ) + ω y + ω y = 0 3 ςs s 3 s 3 L 3 ς L L 3 si j + 3 + + + j = j µ y (-µ) y Ω ( y - [ Tj ( τ ) wj ] si j τ ) + ω y + ω y = 0 L 3 ς L L 3 4 j. dtj j j + + ( + ) T j = v β dτ EI T - M ( µ y (-µ) y 3 ω L y 3 ς ω L L dt dτ v v = + + + y 3) si j τ ; j=,...,5 (6) m L m m π with v EI = v, v T = v, v β = v, where T is the tesio i the wire ad β is the EI π T β L viscous dampig of the wire ad where we cosider the iitial coditios for the problem: y 3 (0) = y o3, y 3 (0) = y 03, y (0) = y o, y (0) = y 0,T i (0) = T oi, T i (0) = T oi Now we cosider the participatio of the exteral forces by additioal values i the coefficiets of the series developmet of the cotact force betwee patograph ad cotact wire, i the right had of the third equatio from the system. Are deoted by A j, j N the additioal term of the coefficiet for si jτ that itervee i the third equatio of the system (6). I the case of aalysis we cosider the followig fixed values of parameters: Ω = 4.77, ς s = 0.3, M = 0.58, v β = 6.4, µ = 0., ωl = 0.7, ςl = 0.45, v EI = 85.6 The free dimesioless parameters i the plae of parameters are chose, i this case, λ ad v T, where λ = ωl ωs. We aalyse the stability of motio for mass M u with the displacemet y. I fig. is plotted with cotiuous lie the domai of periodic solutios of y i the two chose parameters plae i the case A j = 0 for j N ad with discotiuous lie the domai of periodic solutios of y i the case A = 0.03 ad A j = 0, j. 5
ICSV4 9- July 007 Cairs Australia Figure. Stable ad ustable zoes i the plae of parameters 6. CONCLUSIONS The method of stability aalysis, described by umerical method specified i this paper, has permitted to aalyze the ifluece of exteral forces o the motio of the patograph cotact wire dyamical system, modeled as two sprug superposed masses i cotact with a wire. ACKNOWLEDGEMENTS Thaks to the CNCSIS Bucharest for its fiacial support through the Grat r.33344/04, theme A3/006. REFERENCES [] FLOQUET G., Sur les equatios differetialles lieaires a coefficiets periodiques, A. Ecole Norm. Super., -, 47-88, 883. [] MINORSKY N., Noliear oscillatios, Priceto, New Jersey, New York, Toroto, Lodo, D. Va Nostrad Compay Ic., 96. [3] C. HAYASHI, Noliear Oscillatios i Physical Systems, Ed. McGraw - Hill Book Compay, New York, Sa Fracisco, Toroto, Lodo, 964 [4] N. TEODORESCU, V. OLARIU, Ecuatii diferetiale si cu derivate partiale, vol.i, Editura Tehica, 978. [5] R. VOINEA, I.V. STROE, Itroducere i teoria sistemelor diamice, Ed. Academiei, Bucuresti, 000. [6] M. Migdalovici, D. Bara, About the stability of motio for two sprug superposed masses i cotact with a wire, Proceedigs ICSV0, 47-54, 003 6