UDC 539.4 Vibrations of a Complex Systems with Damping under Dynamic Loading K. Cabanska-Placzkiewicz Bydgoszcz University, Institute o f Technology, Bydgoszcz, Poland УДК 539.4 Колебания сложных систем с затуханием при динамическом нагружении К. Ц абанска-П лаш кевич Университет, Институт технологии, Быдгощ, Польш а На основе разработанного аналитико-численного метода решения задачи о свободных и вынужденных затухающих колебаниях на примере сложной системы, которая состоит из двух пластин, соединенных вязкоупругим слоем, выполнен численный анализ и изучены новые механические эффекты, обусловленные действием на рассматриваемую систему различного типа динамического нагружения. Расчеты проводились на основе моделей Тимошенко и Кирхгофа-Лява. Introduction. Compound systems coupled together by viscoelastic constraints play an important role in various engineering and building structures. Since 1923, Timoshenko’s model [1] for various compound constructions has been applied. Vibration analysis of laminated plates was presented in [2, 3] and in many other works. The problem of nonaxisymmetric deformation of flexible rotational shells was solved in [4] with the use of the classical Kirchhoff-Love model and improved Timoshenko’s model. The dynamic problem of elastic homogeneous bodies was presented in [5]. Vibration analysis of systems of solid and deformable bodies for complex motion was considered in [6]. Vibrations of elastically connected rectangular double-plate compound systems under moving loading are presented in [7]. Vibration analysis of compound systems with vibration damping is a difficult problem. In the above com plex cases, especially where viscosity and discrete elements occur, it is recomm ended to adopt the m ethod o f solving a dynam ic problem for a system in the domain changing of a complex function [8, 9]. The property of orthogonality of free vibrations of complex types was first described in [8] for discrete systems w ith damping, for discrete-continuous system s w ith damping in [9] and for continuous systems with damping in [10]. The goal o f this paper is to present a method for solving the problem and dynamic analysis o f free and forced vibrations for a com plex system w ith damping, w hich consists o f tw o elastic plates coupled by a viscoelastic interlayer, for various types of dynamic loading. © K. CABANSKA-PLACZKIEWICZ, 2002 82 ISSN 0556-171X. Проблемы прочности, 2002, N 2 Vibrations o f a Complex System Statem ent of the Problem . Let us consider a problem of free and forced vibrations for a complex system with, damping. External layers of the complex system are made from elastic materials as plates coupled by a viscoelastic interlayer (Fig. 1). The elastic plates are simply supported at their ends and described by Timoshenko’s model [1]. The viscoelastic interlayer has the characteristics of a homogenous continuous unidirectional Winkler’s foundation and is described by the Voigt-Kelvin model [11—13]. Fig. 1. Dynamic model of a complex system with damping. In this paper, w e consider two cases. In the first case, small-frequency transverse vibrations of a complex system w ith damping are excited by a stationary dynamic load f j ( x , y , t) at the point x 0 , y 0 w ith the load varying in time t. In the second case, small-frequency transverse vibrations of a complex system w ith damping are exited b y a nonstationary dynam ic load f ̂ x , y , t) varying in tim e t. The phenomenon of small-frequency transverse vibrations for a complex system w ith damping is described by Timoshenko’s model w ith the following non-homogeneous system of conjugate partial differential equations: D, D, S2W 11 + SV u Sx 2 Sy 2 S 2 W12 + S 2 W 12 S x 2 + Sy2 Sx Sx St' 2 , „ t Sw1 , , ) „ S ^ 1 2 „ + H A ^ T + W 12 | — s 1 „ 2 “ 0St S W1 ( o SW 11 SW 12 1 ( S V 1 ^~ 2 T — H 1I V w 1 + ^ T — + l + (w1 — w 2 A k + c — 1 = f 1( x ,y ,0 , D D St S 2 W 21 + S 2 W 21 Sx 2 Sy 2 S 2 W 22 + S 2 W 22 Sx 2 Sx Sy St Sx I TT I Sw2 , , I — S W21 n + H 2^ + W21 l — s 2 ~ - 0 ( 1) Sy Sx St 2Sw 2 1 S y 22 + H 2I ^ + y 22 I - S 2 — ^ - 0, St V 2 2 S w 2 St 2 , SW 21 , SW 22 1 ( - S V w 2 +---------- + 2 Sx Sy — ( w1 — w 2 )\ k + c — I- 0 ISSN 0556-171X. ÏpoôneMbi npounocmu, 2002, № 2 83 K. Cabanska-Placzkiewicz where , H 1 — K [G ih i, H 2 — K 2G 2 h2 ’ dx 2 dy 2 ’ V li —Y li + $ l i , V 2i —Y 2i + $ 2i , i — 1, 2 Here w l — w ^ x ,y , t) and w 2 — w 2 ( x ,y , t) are the transverse deflections of p la tes I and II, respec tive ly ; V 11 —V i i ( x ,y , t ), V l2 —V l2 ( x ,y , t), V 21 — — V 2l( x , y , t), and V 22 — V 22( x , y , t) are the full angles of rectilinear elements of plates I and II turn; E i and E 2 are Young’s moduli of the material of plates I and II, respectively; E is Young’s modulus of the material of the interlayer; Gi and G 2 are the shear moduli of the material of plates I and II, respectively; p i and p 2 are the mass densities of the material of plates I and II; K [ and K 2 are the shear coefficients; k is the coefficient of elasticity of the interlayer; c is the coefficient of viscosity of the interlayer; hi and h 2 are the thickness of plates I and II; h is the thickness of the interlayer; a and b are the dimensions of the plates; y n and y 2i are the angles of the rectilinear element turn due to a cross shear; $ u and $ 2i are the angles of the middle surface turn in plane for a —const and fl — const; v ip and v 2p are Poisson’s ratios; t is time; x , y ,and z are the coordinate axes; f i ( x ,y , t) is the dynamic load acting on the complex system . Separation of Variables. The analytical numerical method is based on the separation of variables. Presenting the solution of the problem considered in the form and substituting (3) in the system of differential equations (l), by the assumption that f i( x ,y , t ) — 0, w e obtain a homogenous system of ordinary differential equations describing complex modes o f free vibrations of a com plex system w ith damping: W i(x , y , t) W i(x , y ) V i i (x ,y , t) ^ n (x ,y ) V n ( x , y , t) W „ ( x , y ) (3) V 21(x ,y , t) ^ 2l (x ,y ) V 22(x ,y , t) ^ 22(x ,y ) (4a) 84 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N2 2 Vibrations o f a Complex System h 11 v 2W + dW ii dW D- D- dx + 12 dy - (Wi - W2 )(k + icv ) + n iWiv 2 = 0, d 2 W21 dx 2 d 2 W 22 + 21 dy 2 dx 2 + 2 W22 + H + W dy 2\ dx + H , I— 2 + W 21 ( dW 2 !2i dx 22 + S 2 W 2iv = ° (4b) dW 2l dW dx + 22 dy + (W 1 - W2 )(k + icv ) + n 2W2v 2 = 0, 2 where i = - 1 . Here W1 = W1( x ,y ) and W2 = W2 ( x ,y ) are the complex transverse vibration modes of plates I and II, v is the complex eigenfrequency of free vibrations of the complex system with damping. Solution of the B oundary Value Problem . The solution of the differential equation (4) is sought in the form [1] Wi ( x , y ) = X i ( x )Yi ( y X Wi i ( x , y ) = @ i i ( x W ii ( y ), _ W12 (x ,y ) = © i2 ( x ) r i2(y X W2 ( x , y ) = X 2 ( x )Y 2 ( y X W 2i (x>y ) = © 2i ( x ) r 2i (y X W 22(x >y ) = © 22(x )T 22(y )■ Searching for a general solution of the system of differential equations (4) in the form Wi ( x , y ) 'A W ii( x , y ) C ii Wi2( x , y ) C i2 W2 ( x , y ) B W2i( x , y ) D 2 i W22( x , y )_ D 22 _ exp( rix )exp( T2 y ), (6) w e obtain a homogeneous system of algebraic equations: H i 2 A T T ri + C i i si = ° D i H 1 o ** A D r2 + C i2s i = 0 (7a) ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 2 85 K. Cabanska-Placzkiewicz where A (r? + r% + p * * ) + Bp* - C n rx - C n r2 = 0, B B H D : H D ' 2 2 r\ + D 2 1s 2 = 0 2 2 r2 + D 22 s 2 = 0, A p 2 + B ( r\ + r2 + p 2 ) D 21 r1 D 22 r2 = 0 * 1 ** 1 p 1 =■— { k + icv ), H 1 p 1 = * 1 ** 1 p 2 = — {k + icv ), H 2 p 2 = * 2 2 H 1 —1 ** s 1 = r1 + r2 D 1 r + ----v , 1 D 1 ’ s1 * 2 2 H 2 —2 ** s 2 = r1 + r2 D 2 r + ---- v , 1 D 2 ’ s 2 2 . 2 H 1 1 + r2 - D 2 2 H 2 —L v 2 , D i (7b) -----r2 + —— V 2.D 2 r2 D 2 (8) Constructing the determinant of the characteristic matrix of the system of equations (7) and equating it to zero H D H D 1 2 ~ ri 1 2 - r2 19 9 *: r1 + r2 + p i 0 0 * p 2 * s 1 0 0 0 0 0 ** s 1 0 0 0 - r 1 - r 2 * p 1 0 0 0 0 H 2 2-----r, D 2 1 * s 2 0 0 0 2 2r 2 2 0 ** s 2 0 0 9 9 ** r1 + r2 + p 2 - r 1 - r2 = 0, (9) w e obtain the characteristic equation in the form of the following algebraic equation: r1 + a 17 r1 + a 16 r1 + a 15 r15 + a 14 r1 + a 13 r1 + a 12 r1 + a 11r1 + r2 + a 27 r2 + --0 (10) with the following roots: r1 j = ( - 1) J ~1 iX1v, r2 j = ( - 1) J ~1 iX2v , j = (2v - 1), 2v , and v = 1, 2, 3, 4. Here a 1 7 , a 16, a 1 5 , a 14 , a 13 , a 1 2 , a n , a 2 7 , a 26, a 2 5 , a 24 , a 23 , a 2 2 , a 2 1 , and a 0 are constant coefficients. 86 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, № 2 Vibrations o f a Complex System Applying Euler’s formulas, w e construct the solution of the system of differential equations (4) in the form of the following system of solutions: Wl (x, y) = 1 (A* sin X lvx + A** cos X lvx )(A*** sinX 2vy + A**** cosX 2v y), V=l 4 V u ( x , y) = 1 (Cu* cosX lvx + Cu* sinXlvx)(C iiV cos X 2v y + C n V \ i n X 2* y), V=l 4 ^ 12 (x y ) = 1 ( C 12 v cos X lvx + Cl2*v sin X lv x)(C l2*v* cos X 2v y + C 12v* sin X 2v y l 4=l (n ) W2 (x, y) = 1 (B v* sinX lvx + B** cosXlvx)(B *** sinX 2vy + B**** cosX 2v y), V=l 4 ^ 2 1 (x> y) = 1 (Dllv cos X lvx + D 21v sin X lvx ')(D21v cos X 2v y + D 2lV* sin X 2v y ), V=l 4 ^ 22 (x y ) = 1 (D22v cos X lvx + D 22v sin X lvx W l * cos X 2v y + D 22* * sin X 2v y )■ V=1 Here Av , Av , Av , Av , B v , B v , B v , B v , C n v , C n v , C n v , C n , C 12v , C !2v , C !2v , C 12v , D 2\v , D 2\v , D 2\v , D 2\v , D 22v , D 22v , D 22v , and D v are constants and X1v = a 1v + ip 1v and X 2v = a 2v + ip 2v are the parameters describing the roots of the characteristic equation (10). In accordance with (7), the following relations exist between the constants of (9) : * « . . . . . . . B v B v* B V ** D B Va v = A* ’ av ** , A v** * C . v ** ** C llv c llv = A * ’A v c llv ** , A v** * C llv ** C llv c l v = A * ’Av* c llv ** , A v** d*lv D llv ' B * ’ d llv D llv = B ** B v d*iv D *iv * , B v* ** d 22v D llv ** B V *** , Av*** *** a v C llv *** , Av*** _*** c llv C llv *** , Av*** *** c llv D llv *** d llv*** , B v D llv d llv*** , B V Av C llv c llv = ' „■ . . . . Av = ^ = C llv c \2v * * * , c \2v * * * * , ( l l ) Av D 21v , d 2lv „ .. . ß . B v D 22v where , d 22v B . B v Xlv X 2v + Pl + i(c llv Xlv + c l2v X 2v ) Pl ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 2 87 K. Cabanska-Placzkiewicz ~ ^ 2iv - i d 1 X 1v + D v 2* *** D \ D 1 ** **** c 1 1v — c 1 1v — c llv — _ ’ c llv — - c 1 1 v - - c 1 1v , _ 1 i2 D 1 K - X 2v - X 2v + D V 2* *** D i D i ** **** c 12v — c 12v — c 12v — _ , c 12v — - c 12v — - c 12v , —1 X \vD 1 2v i2 _ 2 * 2 2 1v i D 1v ^ D ^* *** D 2 D 2 j** j **** d 21v — d 21v — d 21v — _ , d 21v — - d 21v — - d 21v , __2 o2 D 2 * 1v - X \v - * _ ^ X 2v + D 2 v 2 * *** D 2 D 2 i** 7**** d 22v — d 22v — d 22v — _ , d 22v — - d 22v — - d 22v ■ _ 2 ,2 D 2 1 2v (13) On substitution of (12) in (11), the general solution of the system of differential equations (4) takes the following form: 4 2 * ** *** **** (A v sin X 1vx + A v cos X 1vx )(A v sin X 2v y + A v cos X 2v y), V—1 4 ^ 1 1 (x, y ) cUv (A v* cos X 1v x - Av** sin X 1v x ) (A *** cos X 2 v У - Av**** sin X 2 v У1 V—1 4 4 12 (x, У) c12v (A v* cos X 1v x - 4 * sin X 1v x ) (A *** cos X 2v У - 4 * * * sin X 2v У), v —1 4 2 * ** *** **** a v (A v sin X 1vx + A v cos X 1vx ) ( A v sin X 2v у + A v cos X 2v у), v —1 4 4 21(x , У) a v d 21v (4 cos X 1v x - K * sin X 1v x ) (A v*** cos X 2 v У - 4 * * * sin X 2 v У1 v —1 4 4 22(x , У) av d 22v ( A cos X 1v x - A * sin X 1v x )(4 * * cos X 2 v У + 4 * * * sin X 2v У). v —1 (14) In order to solve the boundary value problem , the following boundary conditions are used: 88 ISSN 0556-171X. Проблемы прочности, 2002, N 2 Vibrations o f a Complex System W1 x—0 W1 y—0 dWn — o, = o, dx dWn x—0 dy y—0 W — 0, Wi b — 0,1 y—b — 0, dWii — 0, dx dW i2 dy y—b W21 x—0 = °> W>\ — = 0 — 0, dW 21 dx dW 22 — 0, x—0 dy y—0 W — 0, W2 b — 0,2 y—b dW 2i — 0, dx dW 22 — 0, (15) dy — 0. y—b Substituting (14) in (15), w e obtain a homogenous system of algebraic equations, which in the matrix notation has the following form: Here or YX — 0. X — [Ai , A 2 , A 3 , A 4 , Ai , A 2 , A 3 , A 4 ] X = [Ai , A 2 , A 3 , A 4 , A i , A 2 , A 3 , A 4 ] are the vectors of the unknowns in the system of equations and Y = \Yi* j ]8*g is the characteristic matrix of the system of equations (16). The first four equations (16) are presented in the form 1 1 1 1 **- A 1 - X n c U 1 - X 12 c 112 - X 13 c 113 - X 14 c 114 ** A 2 Ü1 a 2 a 3 a 4 A 3 _- X 11a 1d 211 - X 12 a 2 d 212 - X 13 a 3 d 213 X 14 a 4 d 214 .A 4 (16) (17) — 0 (18) where A 1 = A 2 = A 3 = A 4 = 0. The remaining four equations (16) give the following system of equations: SS11 ss12 ss13 ss14 A* "A{ - X 11c111SS11 - X 12 c112 ss12 - X 13 c113 ss13 - X 14 c114 ss14 * 2 'K a^ssn a 2 ss12 a 3 ss13 a 4 ss14 A * A 3 _- X 11a 1d211ss11 - X 12 a 2 d212 ss12 - X 13 a 3 d213 ss13 X 14 a 4 d214 ss14 A * _ A4 _ — 0, (19) where ssn = sin X 11a , ss12 = s in X 12a , ss13 = sinX 13 a, and ss14 = sinX 14a. The condition of solving the system of equations (19) is vanishing of the characteristic determinant, i.e., ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 2 89 K. Cabanska-Placzkiewicz SSu ssi2 ssi^ ss 4̂ - X i i ci n ssn - X 12cn 2ss12 - X 13 c113ss13 - X 14c114ss14 a 1ssn a 2 ssi2 a 3 ssi3 a 4 ssi4 ~X11a 1d2i i ssii - X 12a2d212ss12 - X 13 a 3d213ss13 - X 14a 4d214ss14 = 0. (20) Expanding determinant (10), w e obtained the following characteristic equation: sinXl l a s inX l2a sin X l3 a sin Xl4a — 0, (2 1 ) where X ll — X l2 — X l3 — X l4 — X]_. The characteristic equation (21) m ay be rewritten in the form sin Xla — 0, (2 2 ) where, in the general case, X l — a l + ifll (23) are complex numbers. Substituting (23) in (22), w e obtain the following equation: sin a l a cosh f l l a + icos a l a sinh f l l a — 0, (24) w hich has the following roots: n ft a n Ji = - 1- , P - = 0, ni = 1 ,2 ,3 , . . . . (25) Taking into account (25) in (23), w e obtain the following identity: X n = a = ~ . (26)a By analogy with equations (16)-(24), w e obtain n 2 J a —2 b , —2 0, П2 1, 2, 3, """, = = n 2 J X n, = a n, = L " (2 1 ) Substituting r1 = iX and r2 = iX n in equation (10) and carrying out all transformations, w e obtain the following equation of frequency: 8 . # 1 . # 6 . # 5 . # 4 . # 3 . # 2 . # . # nv + а ц v + a i6 v + a i5 v + а ы v + a ^ v + a u v + a n v + a 0 = 0 (28) 90 ISSN 0556-171X. Проблемы прочности, 2002, N 2 Vibrations o f a Complex System from w hich a sequence o f com plex eigenfrequencies is determined: v HiH2 = iV HiH2 ~ ® HiH2 , (29) # # # # # # # 1 # •where a 11, a 16, a 15, a 14, a 13, a 12, a 11,a n d a 0 are constant coefficients. By substituting (29) into (13) and (14), the following formulas for the coefficients o f amplitudes are obtained: - ^ 1ri1 H2 — ^ 2»1»2 1 P 1 1 i(c 11»1 n2 ̂ 1U1H2 1 C12H1H2 ̂ 2»1»2 ) a fan, = * 5 1 2 P 1 — ;2 — ; i 2 1n1 n2 1 D 1 1n2 + D 1 V n1n2 where c 11П1П2 H , -__L ;2 D Л\пхпг I 2 — H L 2 + —L 2 2п1 n2 1 d l 2nin2 + D l VnLn2 Cl2nl n2 H — }2D л 2nln2 — }2 — H 2 2 2 Lnl n2 l D nLn2 + D V nLn2 d 2 lnLn2 H 2 2 D ̂ nL n2 H— — ~~~ 2 2 + —2 2 2 nl n2 l D 2nl n2 + d V nLn2 d 22n n = ------------------- --------- ' (30)__2 у 2 D 2 2 nLn2 p* = H1 ( k + iCV nL n2 \ P \* = H f ^ * LV — k — icV nxn1 ), P 2 = H f ( k + iCV nL n2 \ P 2* = H j~ ( * 2V \ * 2 — k — iCV nln2 )- (31) Substituting the sequences к щ , X^ and а ЩПг, С ц ^ , c UniП2, d 2^ , d 22nitli in (14), w e obtain the following six sequences of complex modes of free vibrations for a complex system with damping: ISSN 0556-171X. Проблемы прочности, 2002, N2 2 91 K. Cabanska-Placzkiewicz Wjnjn2 (x ,y ) = sin X ni X sin X ni y , ^ j j n j n2(x , y ) = c jjnjn2 cos X nj x C0s X n2 y > ^12nJn2(x >y ) = c J2nJn2 cos XnJ x cos Xn2 y , W2nJ n2(x , y ) = a nJn2 sin X nJ x sin X n2 y , W 2JnJn2( x ,y ) = a nJn2 d 2JnJn2 cos Xnj x cos Xn2 y , ^ 2 2 nJn2( x , y ) = a nJn2 d 22nxn2 cos X nJ x cos X n2 y- (32) Solution of the Initial Value Problem . In the case of v = v . complex equation of motion T = 0 exp( iv t) can be written in the following form: Tnj n2 = 0 Hj n2 exp ( iv njn2 t); the (33) (34) where O is the Fourier coefficient. Free vibration of a complex system with damping is presented in the form of a Fourier series based on the complex eigenfunctions, i.e.: 2 2 w jnJn2( x ,y ) nj =J n2 =J 00 0C ^ JJnjn2 ( x ,y ) nj =1 n2 =J 0 0 2 2 ^ J2njn2 ( x ,y ) nj =J n2 =J 0 0 2 2 W2nJn2 (x ,y ) nj =J n2 j 0 0 2 2 ^ 2jnjn2 (x ,y ) Wj( x , y , t) V jj( x , y , t) V j 2 ( x , y , t) w 2 ( x ,y , t) V 2j ( x ,y , t) V 22( x ,y , t) nj =j n =j 0 0 2 2 ^ 22nj n2 ( x , y ) 0 njn2 exp (iv njn2 t ) - (35) From the system of equations (4), performing some algebraic transformations, adding the equations together and then integrating them on both sides within the limits from 0 to a, and from 0 to b, w e obtain the property of orthogonality o f eigenfunctions for a com plex system w ith damping using Timoshenko’s model: Hj =j n2 =j 92 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, № 2 Vibrations o f a Complex System a b î î [и ! {Wlm Vln + WlnVlm ) + /Л 2 (W 2m V2n + W2nV2m ) + 0 0 + ^ l (^U m Q n n + ^ l l n Q llm + ^ 12mQ\2n + ^ 12nQ l2m ) + + S 2 (^ 2lmQ 2ln + ^ 2lnQ 2lm + ^ 22mQ 22n + ^ 22nQ 22m ) + + c(Wln - W2n )(Wlm - W2m )]dxdy = NnÔ nm , (36) where a b N n = î î ^ Л lWlnVln + Л 2W2nV2n + “ l (^ llnQ lln + 4*l2nQ l2n ) + 0 0 + “ 2 (^ 2lnQ 2ln + ^ 22nQ 22n ) + c(Wln - W2n Ÿ \dxd y , (37) Vln = iv nWln (x >УX V2n = iv nW2n (x >УX Vlm = iv m Wlm ( x , У X V2m = iv mW2m ( x , У X Q lln = iv n ^ l ln ( x ,У X Q 2ln = iv n ^ 2ln( x ,УX Q llm = iv m ̂ l lm ( x , У ), Q 2lm = iv m ̂ 2lm ( x , УX Q l2n = iv n^ l2n ( x ,У X Q 22n = iv n^ 22n ( x ,У X Q l2m = iv m ̂ l2m (x ’У X Q 22m = iv m ̂ 22m (x ’У X (38) Here d nm is Kronecker’s delta, n = (n 1, n 2 ),a n d m = (m1, m2 ). The following initial conditions are the basis for solving the problem of free vibrations: w l ( x , y ,0) = w ou W2 ( x , y ,0) = w 02 , V 11(x , y ,0) = V 011, V 2l (x , y ,0) = V 02U (39) V 12( x ,y ,0) = XP 012 , V 22( x ,y ,0) = XP 022- By applying conditions (39) in series (35) and taking into account the property of orthogonality (36), the formula for a complex Fourier coefficient is obtained: 1 a b o ^ ni n2 = N S S ( V l (V1ni n2 W01 W1nin2 w 01) N n1n2 0 0 + Л 2 (V 2nln2 w 02 + W2nln2 w 02 ) + О О + “ l (Q llnln2У 0ll + '^ lln ln2 У 0l l + Q l2nln2У 0l2 + ''^l2nln2 У 0l2) + + “ 2 (Q 21n1n2V 021 + ^ 21n1n2 V 021 + Q 22n1n2V 022 + ^ 22пхп2 У 022) + ISSN 0556-171X. Проблемы прочности, 2002, N2 2 93 K. Cabanska-Placzkiewicz + c[(W inin2 - W2n1n2)(Woi - w 02 )]}dxdy. (40) By substituting (32), (34), and (40) into (35) and performing the trigonometric and algebraic transformations, the final form of free vibration of a complex system with damping is obtained: 00 X Wi = 2 2 e n =1 n2 =1 0 0 V ii = 2 2 ' V 1 2 = 2 2 e n =i n =i o , ini n2l o ,ni n2 w iinin2 o ,ni n2 w i2n n cos( (° n̂ 2 t + $ n̂ 2 + 0 iinin2 )’ cos(O ni n2 t + $ ni n2 + 0 i2ni 2̂ ni =i n2 = 0 0 (4 i) w2 = 2 2 e n =i n =i Ini n2' o W cos( O nin2 t + $ nin2 + X 2nin2 l ni =i n2 = 0 0 y 2i = 2 2 i ni =i n2 =i 0 0 V 22 = 2 2 ' lni n2 o ni n2 w 2in n o w 22 ni n2 cos(O nin2 t + $ nin2 + 0 2inin2 X cos(O nin2 t + $ nin2 + 0 22 ni n2 X n =i n =i where W, = v W w iinin2 A i in n + ^ i i n ^2 , w 2inin2 A 2inin2 + Q 2in^ 2 , w i2nin2 w 22nin2 22nin2 + Q 22nin2 , X inin2 arg Winin2 , X 2nin2 argW 2n̂ 2 , 0 iinin2 = arg W iinin2 , 0 2in^2 = arg W 2inin2 = arg Wi2nv = arg W 22n„ o ni n2 A/C ni n2 + D n̂ 2 , $ ni n2 arg 0 nin2 , X in^2 R eW inin2 , Yinin2 lm W inin2 , X 2nin2 R eW 2nin2 , Y2n̂ 2 lm W 2n,n.i n2 A iinin2 Re W iinin2 ’ Q iinin2 Im W iinin2 ’ 94 ISSN 0556-171X. npo6neMbi npouHocmu, 2002, № 2 ni =i n2 =i Vibrations o f a Complex System ^ 12n̂ 2 W j2njn2 ’ ^ 12ntn2 W j2njn2 ’ ^ 22njn2 = W 22njn2 ’ ^ 22njn2 = W 22n j^ ’ C njn2 ^ njn2 ’ D njn2 ^ njn2 ' (42) Solution of the Forced V ibration Problem. In order to solve the differential equations (l), the function of loading is expanded from the operator method [15] 00 X f j (x ’ y ’ t) = jWjnjn 2 ^ ß 2W2njn 2 ^ n, =j n 1 =1 + û j (Wjjnjn2 W j2njn2 ) “ 2 (W2 jnjn2 W 22njn2 ) ] fnjn2 ’ (43) w h e r e Wlnln2 , W llnln2 , V 2 lnln2 , W2nln2 , V 12nln2 , an d V 22nxn2 h a v e b e e n described by equations (32). The function o f the displacement o f a com plex system w ith damping is presented in the form of a Fourier series: Wj V jj V j2 Wj V 2j V 22 0 0 = 2 2 n =j n =j W jnj n2 W jjnjn2 W j2njn2 WoW 2njn2 W 2jnjn2 W 22njn2 T (44) Substituting (43) and (44) into the differential equations (l), w e obtain the following equation o f motion: T njn2 ̂V njn2 Tnj n2 f nj n2 ’ (45) where T are the coefficients of the distribution of the loading function in a Fourier series. Applying the property of the eigenfunction orthogonality (36), w e derive the coefficients for the load distribution, namely: f j n jn 2 N NN j N n jn 2 0 0 ijn 2I f Wjn + W-, + W jjn jn 2 + W j2n jn 2 + ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 2 95 K. Cabanska-Placzkiewicz + ^ 21n1n2 + 'V 22n1n2 V 1(x , y , t )dxdy. The solution of the differential equation (46) has the form [15] (46) T nin2 0 i V n 1n 2 )( t - r ) - W n 1n 2( r )dx. (47) On substituting (46) and (47) into (44), equation (44) can be rewritten in the following form: T n1 n2 COS($ 1n̂ 2 n1^2 ^̂ n1 —1 n2 — 00 X V 11 ^ 11n1 H2 Tn H2 sin( ̂ 11 1̂ H2 ^ U1^2 ) n —1 n2 —1 0 0