Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer
An analytical method of solution of the free vibration problem for composite system of two viscoelastic beams coupled by a viscoelastic interlayer has been proposed. The phenomenon of free vibration has been described using a homogenous system of conjugate partial differential equations. After the s...
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Інститут гідромеханіки НАН України
1999
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| Цитувати: | Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer / K. Cabanska-Placzkiewicz // Акустичний вісник — 1999. —Т. 2, № 1. — С. 3-10. — Бібліогр.: 18 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1409672025-02-09T16:58:21Z Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer Вiльнi зв'язанi коливання системи двох в'язко-пружних балок, зв'язаних в'язкопружним шаром Свободные связанные колебания системы двух вязко-упругих балок, связанных при помощи вязко-упругого слоя Cabanska-Placzkiewicz, K. An analytical method of solution of the free vibration problem for composite system of two viscoelastic beams coupled by a viscoelastic interlayer has been proposed. The phenomenon of free vibration has been described using a homogenous system of conjugate partial differential equations. After the separation of variables in the differential equations the boundary problem has been solved and two complex sequences have been obtained: the sequence of frequencies and the sequence of modes of free vibration. The property of orthogonality of complex modes of free vibration has been demonstrated. Polyharmonic free vibration has been expanded into the complex Fourier series with respect to complex eigenfunctions. The coeffcients at the eigenfunctions are determined by the initial conditions. Запропоновано аналітичний метод розв'язку задачі про вільні коливання системи, складеної з двох в'язко-пружних балок, сполучених за допомогою в'язко-пружного проміжного шару. Вільні коливання описувались з використанням однорідної системи спряжених диференційних рівнянь у частинних похідних. Після розділення змінних у диференційних рівняннях було розв'язано граничну задачу і отримано дві комплексні послідовності: послідовність частот та послідовність мод вільних коливань. Продемонстровано властивість ортогональності комплексних мод вільних коливань. Полігармонічні вільні коливання розкладались у комплексні ряди Фур'є відносно компплексних власних функцій, коефіцієнти при яких визначаються початковими умовами. Предложен аналитический метод решения задачи о свободных колебаниях системы, состоящей из двух вязко-упругих балок, связанных при помощи вязко-упругого промежуточного слоя. Свободные колебания были описаны с использованием однородной системы сопряженных дифференциальных уравнений в частных производных. После разделения переменных в дифференциальных уравнениях была решена граничная задача и получены две комплексные последовательности: последовательность частот и последовательность мод свободных колебаний. Продемонстрировано свойство ортогональности комплексных мод свободных колебаний. Полигармонические свободные колебания раскладывались в комплексные ряды Фурье относительно комплексных собственных функций, коэффициенты при которых определяются начальными условиями. 1999 Article Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer / K. Cabanska-Placzkiewicz // Акустичний вісник — 1999. —Т. 2, № 1. — С. 3-10. — Бібліогр.: 18 назв. — англ. 1028-7507 https://nasplib.isofts.kiev.ua/handle/123456789/140967 534.11 en application/pdf Інститут гідромеханіки НАН України |
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An analytical method of solution of the free vibration problem for composite system of two viscoelastic beams coupled by a viscoelastic interlayer has been proposed. The phenomenon of free vibration has been described using a homogenous system of conjugate partial differential equations. After the separation of variables in the differential equations the boundary problem has been solved and two complex sequences have been obtained: the sequence of frequencies and the sequence of modes of free vibration. The property of orthogonality of complex modes of free vibration has been demonstrated. Polyharmonic free vibration has been expanded into the complex Fourier series with respect to complex eigenfunctions. The coeffcients at the eigenfunctions are determined by the initial conditions. |
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Cabanska-Placzkiewicz, K. |
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Cabanska-Placzkiewicz, K. Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer |
| author_facet |
Cabanska-Placzkiewicz, K. |
| author_sort |
Cabanska-Placzkiewicz, K. |
| title |
Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer |
| title_short |
Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer |
| title_full |
Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer |
| title_fullStr |
Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer |
| title_full_unstemmed |
Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer |
| title_sort |
free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer |
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Інститут гідромеханіки НАН України |
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1999 |
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https://nasplib.isofts.kiev.ua/handle/123456789/140967 |
| citation_txt |
Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer / K. Cabanska-Placzkiewicz // Акустичний вісник — 1999. —Т. 2, № 1. — С. 3-10. — Бібліогр.: 18 назв. — англ. |
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2025-11-28T05:18:38Z |
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2025-11-28T05:18:38Z |
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ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 1. �. 3 { 10UDC 534.11 FREE VIBRATION OF THE SYSTEMOF TWO VISCOELASTIC BEAMSCOUPLED BY VISCOELASTIC INTERLAYERK. C A B A�N S K A - P L A C Z K I E W I C ZPedagogical University in Bydgoszcz, PolandReceived 29.09.98 � Revised 28.01.99An analytical method of solution of the free vibration problem for composite system of two viscoelastic beams coupled by aviscoelastic interlayer has been proposed. The phenomenon of free vibration has been described using a homogenous systemof conjugate partial di�erential equations. After the separation of variables in the di�erential equations the boundaryproblem has been solved and two complex sequences have been obtained: the sequence of frequencies and the sequenceof modes of free vibration. The property of orthogonality of complex modes of free vibration has been demonstrated.Polyharmonic free vibration has been expanded into the complex Fourier series with respect to complex eigenfunctions.The coe�cients at the eigenfunctions are determined by the initial conditions.� ¯à®¯®®¢ ® «÷â¨ç¨© ¬¥â®¤ ஧¢'離㠧 ¤ ç÷ ¯à® ¢÷«ì÷ ª®«¨¢ ï á¨á⥬¨, ᪫ ¤¥®ù § ¤¢®å ¢'離®-¯àã¦¨å¡ «®ª, ᯮ«ãç¥¨å § ¤®¯®¬®£®î ¢'離®-¯à㦮£® ¯à®¬÷¦®£® è àã. �÷«ì÷ ª®«¨¢ ï ®¯¨á㢠«¨áì § ¢¨ª®à¨áâ אַ¤®à÷¤®ù á¨á⥬¨ á¯à殮¨å ¤¨ä¥à¥æ÷©¨å à÷¢ïì ã ç áâ¨¨å ¯®å÷¤¨å. �÷á«ï ஧¤÷«¥ï §¬÷¨å ã ¤¨ä¥à¥-æ÷©¨å à÷¢ïïå ¡ã«® ஧¢'ï§ ® £à ¨çã § ¤ çã ÷ ®âਬ ® ¤¢÷ ª®¬¯«¥ªá÷ ¯®á«÷¤®¢®áâ÷: ¯®á«÷¤®¢÷áâì ç áâ®ââ ¯®á«÷¤®¢÷áâì ¬®¤ ¢÷«ì¨å ª®«¨¢ ì. �த¥¬®áâ஢ ® ¢« á⨢÷áâì ®à⮣® «ì®áâ÷ ª®¬¯«¥ªá¨å ¬®¤ ¢÷«ì¨åª®«¨¢ ì. �®«÷£ ମ÷ç÷ ¢÷«ì÷ ª®«¨¢ ï ஧ª« ¤ «¨áì ã ª®¬¯«¥ªá÷ à廊 �ãà'õ ¢÷¤®á® ª®¬¯¯«¥ªá¨å ¢« á¨åäãªæ÷©, ª®¥ä÷æ÷õ⨠¯à¨ ïª¨å ¢¨§ ç îâìáï ¯®ç ⪮¢¨¬¨ 㬮¢ ¬¨.�।«®¦¥ «¨â¨ç¥áª¨© ¬¥â®¤ à¥è¥¨ï § ¤ ç¨ ® ᢮¡®¤ëå ª®«¥¡ ¨ïå á¨á⥬ë, á®áâ®ï饩 ¨§ ¤¢ãå ¢ï§ª®-ã¯àã£¨å ¡ «®ª, á¢ï§ ëå ¯à¨ ¯®¬®é¨ ¢ï§ª®-ã¯à㣮£® ¯à®¬¥¦ãâ®ç®£® á«®ï. �¢®¡®¤ë¥ ª®«¥¡ ¨ï ¡ë«¨ ®¯¨á ëá ¨á¯®«ì§®¢ ¨¥¬ ®¤®à®¤®© á¨á⥬ë ᮯà殮ëå ¤¨ää¥à¥æ¨ «ìëå ãà ¢¥¨© ¢ ç áâëå ¯à®¨§¢®¤ëå. �®á«¥à §¤¥«¥¨ï ¯¥à¥¬¥ëå ¢ ¤¨ää¥à¥æ¨ «ìëå ãà ¢¥¨ïå ¡ë« à¥è¥ £à ¨ç ï § ¤ ç ¨ ¯®«ãç¥ë ¤¢¥ ª®¬¯«¥ªá-ë¥ ¯®á«¥¤®¢ ⥫ì®áâ¨: ¯®á«¥¤®¢ ⥫ì®áâì ç áâ®â ¨ ¯®á«¥¤®¢ ⥫ì®áâì ¬®¤ ᢮¡®¤ëå ª®«¥¡ ¨©. �த¥¬®-áâà¨à®¢ ® ᢮©á⢮ ®à⮣® «ì®á⨠ª®¬¯«¥ªáëå ¬®¤ ᢮¡®¤ëå ª®«¥¡ ¨©. �®«¨£ ମ¨ç¥áª¨¥ ᢮¡®¤ë¥ ª®«¥-¡ ¨ï à ᪫ ¤ë¢ «¨áì ¢ ª®¬¯«¥ªáë¥ àï¤ë �ãàì¥ ®â®á¨â¥«ì® ª®¬¯«¥ªáëå ᮡá⢥ëå äãªæ¨©, ª®íää¨æ¨¥âë¯à¨ ª®â®àëå ®¯à¥¤¥«ïîâáï ç «ì묨 ãá«®¢¨ï¬¨.INTRODUCTIONStrings and systems of beams coupled together byviscoelastic constraints play an important role in var-ious engineering and building structures. They arebeing used in railway and tram tractions with liveload [1, 2]. Such kind of structures can also be foundin some ski lifts and cable car systems. Beams canwork together with strings, slabs and membranes invarious structures. Light roof structure of the sportarena is an example of matching of strings and mem-branes.Analysis of vibration of complex structural systemswith damping posesses a di�cult problem. In theabove mentioned cases, especially where the viscosi-ty and discrete elements occur, it is recommended tosolve the dynamic problem by representing the ampli-tudes as the complex functions of real variable [3, 4].For the �rst time the property of orthogonality ofthe complex modes of free vibration has been demon-strated in paper [3] for discrete systems with damp-ing, and in paper [4] for discrete { continuous systemswith damping. With the use of complex functions thedescription of free vibration of the beam supported
on viscoelastic continuous Winkler's foundation [5, 6]has been developed in papers [7 { 10]. In the pa-per [11] the dynamic problem for complex continuoussystem has been solved by classical method [12] ac-cording to complete theory of non-damped vibration.In paper [10] the uniform method of solving the freevibration problems for complex continuous one- andtwo-dimensional structures with damping for variousboundary conditions and di�erent initial conditionshas been presented. Veri�cation of this method hasbeen carried out for the system of two strings in thecase where no damping occurs [11]. The results fornatural frequencies and coe�cients of amplitude de-rived by method presented in paper [10] agree withthe results obtained with classical method [11].The goal of this paper is to carry out the math-ematical analysis of solution of the free vibrationproblem for composite structure consisting of twoviscoelastic beams coupled by viscoelastic interlay-er. According to proposed method the solution foreigenforms of the system is presented through the setof complex functions. Phase characteristics of theeigenmodes are studies as the fucntions of their or-ders.c
K. Caba�nska-P laczkiewicz, 1999 3
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 1. �. 3 { 101. FORMULATION OF THE PROBLEMThe physical model of the system under consid-eration consists of two parallel homogenous beamsof equal length coupled together by viscoelastic in-terlayer (�g. 1). The Bernoulli { Euler's beams aresimply supported at the ends. We assume that thebeams are made of viscoelastic materials and theirphysical properties are described in scope of Voigt {Kelvin's model [13{ 15]. We assume that the vis-coelastic properties of interlayer also can be describedby the Voigt { Kelvin's model [13{ 15]. As to the elas-tic properties, the interlayer is considered as classicalone-directional Winkler's foundation [6].Stated assumptions allow to express the forces ofinteraction between the beams in terms of the de
ec-tions of beams and physical characteristics of Win-kler's foundation. The small transverse vibrations ofconsidered structure are described with the system ofthe following conjugate partial di�erential equations:E1I1�1 + c1 @@t� @4w1@x4 + �1 @2w1@t2 ++ c @@t (w1 �w2) + k(w1 � w2) = 0;E2I2�1 + c2 @@t� @4w2@x4 + �2 @2w2@t2 ++ c @@t (w1 �w2) + k(w1 � w2) = 0; (1)where w1=w1(x; t) and w2=w2(x; t) are the de
ec-tions of the beams (hereafter the subscripts \1" and\2" mean the beam I and the beam II respectively);E1 and E2 are Young's modules of materials of thebeams; I1 and I2 are the moments of inertia of cross-sections of the beams; �1 and �2 are the masses ofthe beams per unit of length; c1 and c2 are relativecoe�cients of viscosity in the beams; c is the coe�-
Fig. 1. Dynamic model of the system of two viscoelasticbeams coupled by viscoelastic interlayer
cient of viscosity of the interlayer; k is the coe�cientof elasticity of the interlayer; l is the length of thebeams.2. SEPARATION OF VARIABLESThe goal of this paper is to describe the free vibra-tion of complex visco-elastic system. The process ofvibration is not harmonical one because of dampingexisting in the system. However, in scope of the con-sidered model it is possible to express the time evolu-tion of the system introducing the notion of complexfrequency �. This approach allows to assume the de-sired de
ections to be the harmonical functions oftime.Substituting the expressionw1 = W1(x) exp(i�t);w2 = W2(x) exp(i�t); (2)for w1 and w2 in the system of di�erential equa-tions (1) we obtain the homogenous system of con-jugate ordinary di�erential equations describing thecomplex modes of vibration of the beams:d4W1dx4 ��E1I1(1+c1�)��1���(�1�2�k�ic�)W1+(k+ic�)W2�=0;d4W2dx4 ��E2I2(1+c2�)��1���(�2�2�k�ic�)W2+(k+ic�)W1�=0; (3)where W1(x), W2(x) is the complex mode of vibrationof the beams; � is the complex frequency of vibrationof the beams; t is the time.3. SOLUTION OF THE BOUNDARY VALUEPROBLEMSearching for particular solution of system of dif-ferential equations (3) in the following form [16]:W1 = Aerx; W2 = Berx; (4)we obtain the homogenous system of linear algebraicequationsA�R1(1 + ic1�)r4 � �1�2 + k + ic����B�k + ic�� = 0;A�k + ic����B�R2(1 + ic2�)r4 � �2�2 + k + ic�� = 0; (5)4 K. Caba�nska-P laczkiewicz
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 1. �. 3 { 10where R1=E1I1, R2=E2I2. This system has a non-trivial solution only if its determinant is equal to zero.This condition is in essence the dispertion equationlinking the wavenumber r and the complex frequency�. In doing so, we obtain the four-quadratic equationr8 � ��1�2 � k � ic�R1(1 + ic1�) + �2�2 � k � ic�R2(1 + ic2�) � r4++�2�1�2�2 � (�1 + �2)(k + ic�)R1R2(1 + ic1�)(1 + ic2�) = 0; (6)with the following roots:rj = �i�v ; rj = ��v ;j = 1; 2; 3; 4; v = 1; 2; (7)where�v = 4s12��1�2�k�ic�R1(1+ic1�) + �2�2�k�ic�R2(1+ic2�) ��p�; (8)and� = ��1�2 � k � ic�R1(1 + ic1�) � �2�2 � k � ic�R2(1 + ic2�) �2 ++ 4(k + ic�)2R1R2(1 + ic1�)(1 + ic2�) > 0; (9)is the discriminate of the four-quadratic equation (6).After applying the Euler's formulas the general so-lution of the system of di�erential equations (3) canbe represented through the fundamental system ofsolutions:W1(x) = 2Xv=1A�v Sh�vx + A��v Ch�vx++ A���v sin�vx + A����v cos �vx;W2(x) = 2Xv=1B�v Sh�vx + B��v Ch�vx++ B���v sin�vx + B����v cos �vx; (10)where A�v, A��v , A���v , A����v , B�v , B��v , B���v , B����vare constants.In agreement with (5) there exist the following re-lations between the constants from (10):av = B�vA�v = B��vA��v = B���vA���v = B����vA����v ; (11)whereav = R1(1 + ic1�v)�4v � �1�2v + k + ic�vk + ic�n == k + ic�nR2(1 + ic2�v)�4v � �2�2v + k + ic�v : (12)
After incorporating the representation (11) in (10)the general solution of the system of di�erential equa-tions (3) takes the formW1(x) = 2Xv=1A�v Sh�vx + A��v Ch�vx++ A���v sin�vx + A����v cos �vx;W2(x) = 2Xv=1 av�A�v Sh�vx + A��v Ch�vx++ A���v sin�vx + A����v cos �vx�: (13)To determine the amplitudes of the eigenmodes ofthe system one should specify some kind of boundaryconditions. In this scope we assume the ends of thebeams to be simply supported:W1(0) = 0; W1(l) = 0;d2W1dx2 (0) = 0; d2W1dx2 (l) = 0 (14)for beam I andW2(0) = 0; W2(l) = 0;d2W2dx2 (0) = 0; d2W2dx2 (l) = 0 (15)for beam II. Substituting the general solution (13)into boundary conditions (14), (15) we obtain the ho-mogenous system of linear algebraic equations, whichin the matrix notation has the following form:YX = 0; (16)whereX = [A�1; A��1 ; A���1 ; A����1 ; A�2; A��2 ; A���2 ; A����2 ]Tis the vector of the amplitude coe�cients andY = [Yij]8�8 (17)is the characteristic matrix of the system of equa-tions (16). The elements of this matrix are presentedin table 1, whereSS1 = Sh�1l; SS2 = Sh�2l;CC1 = Ch�1l; CC2 = Ch�2l;ss1 = sin�1l; ss2 = sin�2l;cc1 = cos �1l; cc2 = cos�2l;ll1 = �21; ll2 = �22:K. Caba�nska-P laczkiewicz 5
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 1. �. 3 { 10Table 1. Matrix of coe�cients Yiji 1 2 3 4 5 6 7 8j1 0 0 1 1 0 0 1 12 0 0 ll1 ll2 0 0 �ll1 �ll23 0 0 a1 a2 0 0 a1 a24 0 0 a1ll1 a2ll2 0 0 �a1ll1 �a2ll25 SS1 SS2 CC1 CC2 ss1 ss2 cc1 cc26 ll1SS1 ll2SS2 ll1CC1 ll2CC2 �ll1ss1 �ll2ss2 �ll1cc1 �ll2cc27 a1SS1 a2SS2 a1CC1 a2CC2 a1ss1 a2ss2 a1cc1 a2cc28 a1ll1SS1 a2ll2SS2 a1ll1CC1 a2ll2CC2 �a1ll1ss1 �a2ll2ss2 �a1ll1cc1 �a2ll2cc2The condition of solvability for the system of equa-tions (16) is the vanishing of the characteristic deter-minant, i. e. detY = 0: (18)The identity A�1=A�1=A��1 =A��1 =A����1 =A����1 =0,obtained from the system (16) leads to the reductionof the characteristic equations (18) to the followingform: ������ sin�1l sin�2la1 sin�1l a2 sin�2l ������ = 0; (19)where �1 = �1 + i�1;�2 = �2 + i�2 (20)are in the general case the complex numbers.Vanishing of the determinant in (19) is equivalentto the following transcential equation:sin(�1 + i�1)l sin(�2 + i�2)l = 0; (21)having the roots�1n = �2n = �n = s�l ;�1n = �2n = �n = 0;s = 1; 2; 3; : : : ; (22)where n = 2s� �n;(2s�1); (23)and �n;(2s�1) is the Kronecker's number.Substitution of (22) in (20) leads to the equality�1n = �2n = �n = �n = s�l : (24)Substituting r4=�4 in the equation (6) and carry-ing out all the transformations we readily obtain the
following equation with respect to frequency:�4�(�1�2)�1�ic(�1+�2)�3++ ��R1(1+ic1�)�4n+k��2++ �R2(1+ic2�)�4n+k��1��2�� ic�R1(1+ic1�)+R2(1+ic2�)��4n�+� ��R1R2(1+ic1�)(1+ic2�)��4n++ k�R1(1+ic1�)+R2(1+ic2�)���4n�=0; (25)from which the sequence of complex eigenfrequenciescan be determined:�n = i�n � !n: (26)In fact the equation (25) has four roots. In (26) weleft only two of them that correspond to physicallyconsistent vibration of the system decaying with time.Using the representation (26) in the expression forav (12) we obtain the �nal formulas for coe�cientsof amplitudes that are in fact the relative ratios ofamplitudes of the two beams vibrating on the certainmode:an = R1(1 + ic1�n)�4n � �1�2n + k + ic�nk + ic�n == k + ic�nR2(1 + ic2�n)�4n � �2�2n + k + ic�n : (27)Incorporation of the sequences of �n and an into (13)gives two sequences of modes of free vibration forbeams I and II:W1n(x) = sin�nx;W2n(x) = an sin�nx: (28)6 K. Caba�nska-P laczkiewicz
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 1. �. 3 { 104. SOLUTION OF THE INITIAL VALUE PROB-LEMThe complex equation of motionT = � exp(i�t); (29)in the case of �=�n can be written asTn = �n exp(i�nt); (30)where �n is the Fourier coe�cient.It is natural to present the free vibration of beamsin form of the Fourier series based on complex eigen-functions, i. e.w1n(x; t) = 1Xn=1W1n(x)�n exp(i�nt);w2n(x; t) = 1Xn=1W2n(x)�n exp(i�nt): (31)When omitting the damping in the beams, from thesystem (3) after performing the algebraic transfor-mations of its equations, adding them together andintegration of the �nal expression in limits from 0 to lwe obtain the property of orthogonality of eigenfunc-tions [8 { 10].lZ0 ��1(W1mV1n+W1nV1m)+�2(W2mV2n+W2nV2m)++2��(W1n�W2n)(W1n�W2m)��dx=Nn�mn; (32)where �nm is Kronecker's delta,Nn = 2 lZ0 ��1W1nV1n + �2W2nV2n++�(W1n �W2n)2�dx; (33)V1n(x)= i�nW1n(x); V2n(x)= i�nW2n(x);V1m(x)= i�mW1m(x); V2m(x)= i�mW2m(x);�1=�1=�; �2=�2=�; �=�1=(2�): (34)The problem on free vibration of beams is solved byapplying the following conditions:w1(x; 0) = w01; _w1(x; 0) = _w01;w2(x; 0) = w02; _w2(x; 0) = _w02: (35)
Application of conditions (35) in the series (31) andtaking into consideration the property of orthogonal-ity (32) leads to the formula for the Fourier coe�-cient [8 { 10]:�n = 1Nn lZ0 ��1(V1nw01 + W1n _w01)++�2(V2nw02 + W2n _w02)++2��(W1n �W2n)(w01 �w02)��dx: (36)After the substitution of (28), (30) and (36) into (31)and obvious transformations �nally one can obtainthe expressions for free vibration of beams:w1n= 1Xn=1 e��tj�njW1n(x)�cos(!nt+'n)++ i sin(!nt+'n)�;w2n= 1Xn=1 e��tj�njW2n(x)�cos(!nt+'n)++ i sin(!nt+'n)�; (37)where 'n=arg �n.5. NUMERICAL RESULTS AND DISCUSSIONTwo beams coupled by viscoelastic interlayerhave been considered. Calculations have been car-ried out for the following data: E1=1010 N m�2,E2=1010 N m�2, I1=4:5 � 10�4 m4, I2 == 8:9 � 10�4 m4, �1=1:2 � 102 kg m�1, �2 == 1:75 � 102 kg m�1, k=2 � 105 N m�2, l=10 m,c=0:75, c1=0, c2=0. From teh above values it iseasy to see that in calculations both the beams wereregarded as elastic with no internal damping.The Fourier coe�cients �n (36) were derived forthe following initial conditions:w01 = As sin(�x=l); _w01 = As!1 sin(2�x=l);w02 = 0; _w02 = 0; As = 0:01l:For the system of two beams coupled by viscoelas-tic interlayer the calculations were performed in theprogram \MATHEMATICA" [17, 18]. The viscoelas-tic Bernoulli { Euler's beams coupled together by vis-coelastic material can be assumed to be the thinbeams, so that there don't occur the angles of ro-tation of cross-sections of the beams. As to the vis-coelastic one-directional Winkler's interlayer, it wasK. Caba�nska-P laczkiewicz 7
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 1. �. 3 { 10Table 2. Natural frequencies and ratios of amplitudesfor lower modes of vibrations of the systemNumber of mode �n ans=1 n=1 21:0245 + 0:000002:84721i 0:953953 + 0:00000355862in=2 56:8328 + 0:00526501i �1:391128� 0:0000140292is=2 n=3 81:93308 + 0:000635179i 0:479134 + 0:0000975824in=4 99:4811 + 0:00463268i �0:698737� 0:000172791is=3 n=5 176:268 + 0:00242943i 0:110562 + 0:0000740459in=6 203:96 + 0:00283843i �0:161222� 0:000123323i
a bFig. 2. Modes ofvibration of beam II for s=1, (n=1; 2):a { n=1; b { n=2considered as one of smallthic kness, in which the lon-gitudinal interaction between the interlayer and theBernoulli { Euler's beam don't occur. Table 2 con-tains the complex natural frequencies �n and com-plex coe�cients of modes offree vibration an for thesystem with damping. Theresults are giv en for s=1,(n=1; 2), s=2, (n=3; 4), s=3, (n=5; 6).Investigation of complex modesfor beams I and IIhas shown that relative amplitude of vibration ofbeam II (normalized to that of beam I) decreaseswith the increase of s and number of mode n, re-specively. So, if at s=1 (n=1; 2) the amplitudes ofvibrations of the both beams are of the same order,with the increase s to 3 (n=5; 6) their ratio decreasesto 10{ 15%.The another peculiarity is some phase shift be-tween the modes with corresponding numbers for
beams I and II. Obviously, this phenomenon is con-ditioned by presence of viscousity in the elements ofconsidered structure. This relative phase shift canbe expressed by the existence of nonzero imaginaryparts of spatial modes of beam II while for all modesof beam I the imaginary partcan be regarded as iden-tical to zero. Eigenmodes for the both beams withs=1; 2; 3 are represented on �g. 2 {�g. 4.Presence of the above-mentioned phase shift be-tween the de
ections of the beams is natural forvisco-elastic systems, and its the value stronly de-pends on the material parameters of components ofsystem. The analysis shows that for lower modesof vibration this shift of phases is extremely low,but demonstrating the trend to growth: for n=1max jImw2n=Rew2nj is less then 5 � 10�4 % and forn=6 it grows to 0.077%.8 K. Caba�nska-P laczkiewicz
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 1. �. 3 { 10
a bFig. 3. Modes ofvibration of beam II for s=2, (n=3; 4):a { n=3; b { n=4
a bFig. 4. Modes ofvibration of beam II for s=3, (n=5; 6):a { n=5; b { n=6Also, it should be noted that amplitude ratios ofmodes for bothbeamssigni�can tly decrease with theincrease of s. It is interesting that modes for beams Iand II with odd numbers are excited almostin phase, but modes having even numbers are counterphaseones.To substantiate the reliability and marketability ofthe o�ered method of the solution the veri�cationK. Caba�nska-P laczkiewicz 9
ISSN 1028 -7507 �ªãáâ¨ç¨© ¢÷ᨪ. 1999. �®¬ 2, N 1. �. 3 { 10of numerical results for the system of two Bernoulli {Euler's beams without damping has been carried out.The results for natural frequencies and ratios of am-plitudes derived by means of the presented method,have been compared with the results obtained usingthe classical method [11]. The reasonable agreementbetwen both mentioned results has been noted.For the solution of the problem with damping thenew analytical uniform method presented in this pa-per may be used. The basis of this method is theproperty of orthogonality of the complex modes offree vibration of two Bernoulli { Euler's beams cou-pled together by viscoelastic interlayer (32). Thisproperty is identical to that of system of two stringscoupled by viscoelastic interlayer [10].CONCLUSIONS1. The analytical method of the solution of prob-lems on free vibration of continuous system oftwo viscoelastic beams coupled by viscoelasticinterlayer is introduced in this paper. Accordingto this method the set of complex modes of freevibration forms has been found. Mentioned setcan be regarded as functional basis for represen-tation of arbitrary vibration of the system.2. The calculations for presented system have beencarried out for two complex sequences: the se-quence of frequency, and the sequence of modesof free vibration.3. Spatial modes of vibration of beams I and IIare slightly shifted in phase. With the increaseof number of mode n it is oserved the decrease ofrelative amplitude of vibration of beam II withrespect of that of beam I. The absolute valuesof amplitude ratios for both beams are also de-creasing with the increase of n.4. The method presented in this paper can be ap-plied to solutions of free vibration of di�erent en-gineering structures consisting of two viscoelasticbeams coupled by viscoelastic interlayer (girder,road or railway bridges).
1. Szcze�sniak W. The selection of problems of beamsand shells subjecting the inertial moving load //Building Engineering.{ Pub. of the Warsaw Univ.of Tech., 1994.{ 125.2. Szcze�sniak W. The selection of railway problems //Building Engineering.{ Pub. of the Warsaw Univ.of Tech., 1995.{ 129.3. Nizio l J., Snamina J. Free vibration of the discrete-continuous system with damping // J. Theor.and Appl. Mech.{ Warsaw, 1990.{ 28, N 1{2.{�. 149{160.4. Tse F., Morse I., Hinkle R. Mechanical vibrationstheory and applications. Boston: Allyn and Bacon,1978.5. Nowacki W. The building mechanics. Warsaw:Arkady, 1976.6. Winkler E. Die Lehre von der Elasticitat und Fes-tigeit. Prag: Dominicus, 1867.7. Caba�nska-P laczkiewicz K. Description of free vibra-tion of beams by helping complex functions // Sim-ulation in Investigation and Development.{ JeleniaGora, 1997.{ P. 57{58.8. Caba�nska-P laczkiewicz K. Dynamics of the systemof two Bernoulli {Euler's beams with a viscoelas-tic interlayer // XXXVII-th Symposium of Model.in Mech., 7.{ Silesian Univ. of Tech., Gliwice, 1998.{P. 49{54.9. Caba�nska-P laczkiewicz K. Free vibration of the sys-tem of string-beam with a viscoelastic interlayer //Theoretic. Found. in Civil Engineer.{ 6.{ Warsaw,1998.{ �. 59{68.10. Caba�nska-P laczkiewicz K. Free vibration of the sys-tem of two strings coupled by a viscoelastic interlay-er // J. Eng. Transact.{ 46, N 2.{ Warsaw, 1998.{�. 217{228.11. Oniszczuk Z. Vibration analysis of the compoundcontinuous systems with elastic constraints // Pub.of the Rzeszow Univ. of Tech.{ Rzeszow, 1997.12. Osi�nski Z. The theory of vibration. Warsaw: PWN,1978.13. Kasprzyk S. Dynamics of the continuous system //Pub. of AGH.{ Krakow, 1989.14. Nowacki W. The building dynamics. Warsaw:Arkady, 1972.15. Osi�nski Z. Damping of the mechanical vibration.Warsaw: PWN, 1979.16. Gutowski R. The ordinary di�erential equations.Warsaw: WNT, 1971.17. Drwal G., Grzymkowski R., Kapusta A., Slota D.\MATHEMATICA" for everybody. Skalmierskiego,Gliwice: WPKJ, 1996.18. Janiak W. The entrance to \MATHEMATICA".Warsaw: WPLJ, 1994.
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