1(13) 39 57 Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks Abstract The work is focused on the probem of vibrating beams with a variabe cross-section fixed on a rotationa rigid disk. The beam is oaded by a transversa time varying force orthogona to an axis of the beam and simutaneousy parae to the disk s pane. There are many ways of usage of the technica moveabe systems composed of eements with the variabe cross-sections. The main appications are used in numerous types of turbines and pumps. The paper is a kind of introduction to the dynamic anaysis of above mentioned beam systems. The equations of motion of rotationa beams fixed on the rigid disks were derived. After introducing the Coriois forces and the centrifuga forces, the transportation effect in the mathematica mode was considered. This particuar project is the first stage research, where there were proposed certain soutions of probems connected with the inear variabe cross-sections systems. The further investigation considering the noninear systems has been proceeding. The resuts, anaysis and comparison wi be presented in the future works. Keywords variabe cross-section, beam, rotary motion, dynamic fexibiity Sawomir Zokiewski * Siesian University of Technoogy, Institute of Engineering Processes Automation and Integrated Manufacturing Systems, Division of Mechatronics and Designing of Technica Systems, Facuty of Mechanica Engineering, 18a Konarskiego Street, 44-1 Giwice, Poand * Author emai: sawomir.zokiewski@pos.p 1 INTRODUCTION The fundamenta eements of the numerous mechanisms or machines, namey the beams and rods, are most commony considered as the vibrating stationary systems. In these cases they are treated as the immovabe ones without the couping between the oca osciations and the main operationa motion. The mechanica systems are predominanty considered both in the kinematica and dynamica aspects. The main probems resuting from the anaysis are connected with the issues of controing and stabiizing mechanica systems as such. Numerous research can be found in the iterature [1, 3, 4, 7, 9, 14-16, 18-] and so far the soutions have been found by considering the transportation (considered as the main working motion) and the oca osciations separatey [, 3, 5, 6, 8, 1-1, 17]. Mesut Simsek in his paper [14] presents the noninear dynamic anaysis of a functionay graded beam with pinned-pinned supports. Simsek uses the Timoshenko beam theory with the von Karman's noninear strain-dispacement reationships. The beam was anayzed in reation to its Latin American Journa of Soids and Structures 1(13) 39 57
4 Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks variabe thickness of a cross-section according to the power-aw formua. The system of equations of motion is derived by using the Lagrange s equations. Transverse and axia defections and rotation of the cross-sections of the beam were expressed in poynomia forms. The boundary conditions are taken into account by using the Lagrange mutipiers. The Newmark-b method in conjunction with the direct iteration method was used to sove noninear equations of motion. In the work, arge defection, veocity of the moving oad and excitation frequency on the beam dispacements were anayzed. Some detaied cases were compared with each other. Another method of anaysis of highspeed rotating beams is presented in [9] by Gunda. The author introduces a new finite eement for free vibration anaysis of rotating beams. The basis shape functions which appy a inear combination of the soution of the governing static differentia equation of a stiff-string and a cubic poynomia are used. The introduced functions depend on the anguar veocity and position of the anayzed section, what accounts for the centrifuga stiffening effect. In paper [1] the effects of rotary inertia were presented. The author shows the rotation impact on the extensiona tensie force and the eigenvaues of beams. The anayzed beam was rotated uniformy about a transverse axis taking into account ongitudina easticity. The perturbation technique and Gaerkin's method were used. In the exampe of a typica heicopter rotor bade it was indicated that the extensiona tensie force increases up to ten percent, when the rotary inertia contribution is retained in the modeing. Szefer in [15, 16] caimed that the vibrations from fexibiity of eements of the mechanica composition are much smaer than a main disocation of this composition. There is an increased range of veocities and acceerations of such a type systems. A those are caused by using more efficient drives or/and ess and ess weighting materias. In order to constraint power output of drives for motion of mechanica systems, materias with the ower mass density and stiffness are used. A those things are reasons for creating the new modes of designing technica beamike systems. The variabe crosssection beams are appied in many practica appications nowadays. Both the ineary and nonineary variabe cross-sections are used in technica systems. These systems shoud be taken under consideration with respect to a reaistic distribution of mass and fexibiity. Any variations in crosssections cannot be negected and shoud be considered in the mathematica mode assumptions. In spite of many works [e.g. 1, 9-1, 17, ], where their authors anayzes the beams in rotations and beams with variabe cross-sections, this work is devoted to genera anayses both the inear and noninear ones of vibrating transversay beams with the variabe cross-section. It is beieved that new equations and resuts are presented in this paper. MODEL OF THE ANALYZED BEAMS FIXED ON THE ROTATIONAL RIGID DISK In this section the way of modeing of beams in rotationa transportation is presented. The beam was assumed as the uniform one with the variabe cross-section. The beam was fixed on the rotationa disk that was assumed as a rigid eement (Fig. 1). The disk rotates round its axis and the motion was described in two reference frames, the goba stationary reference frame and the oca one. The vibrations occurring in the oca reference frame are transferred into the goba reference frame with taking into consideration so caed the transportation effect [3, 16, 19, ].
Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks 41 Fig. 1 Mode of the anayzed beam on the rotationa disk [3] Fig. Mode of the anayzed beam on the rotationa disk the XZ projection [3] where: ρ mass-density, A(x) cross-section, ength of the beam, x ocation of the anayzed cross-section, ω anguar veocity, Ω frequency, Q rotation matrix, S position vector, F harmonic force, E Young moduus, w vector of dispacement. There is defined a set of generaized coordinates aowing to make an assumption that generaized coordinates are the orthogona projections on the coordinate axes of the goba reference frame and can be written thus: q = r, 1 X (1) q = r, Y () and after differentiating (Eq. 1) and (Eq. ) can be obtained the generaized veocities in the form: q 1 = dq 1 dt = r X = v X, (3) q = dq dt = r Y = v Y, (4) Latin American Journa of Soids and Structures 1(13) 39 57
4 Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks or in the equivaent vector form expressing the dispacements: r = [r X,r Y ] T (5) and the equivaent vector expressing the veocities: r = [ r X, r Y ] T. (6) The cross-section is dependent on the x coordinate and for instance making an assumption that the anayzed beam is a cone (a base with radius equa x/) it can be written as foows: = A 1 x A x = π 1 x, (7) The anguar veocity of the rotationa disk treated as the transportation veocity is defined as: ω = ω T, (8) The ocation of the anayzed cross-section is described by the position vector defined as foows: S = s T. (9) The xyz reference frame is the rotationa one and the matrix aowing expressing vectors is the rotation matrix. The xyz oca reference frame is fixed to the rotating beam and its z-axis coincides with the rotation axis of the rigid disk, thus the rotation matrix is defined as such: cosϕ sinϕ Q = sinϕ cosϕ. (1) 1 Different appications of such a type beam systems may have different boundary conditions and the cross-section can be considered as the ineary or nonineary variabe one. For exampe the beam can be considered as the fixed at one end, the camped-camped one or with other boundary conditions etc. For exampe beams with one end camped and the second free-free are considered (Fig. 1). The beam is fixed on the rotationa rigid disk. An anayzed section of the beam is oaded by a harmonic time varying distributed force with unitary ampitude on a direction perpendicuar to the centre ine of the beam. The forces on origins of the beams are assumed as equa zero and aso the dispacements are equa zero, because the system is camped in one end. Because of a these the specific boundary conditions for this beam shoud be written as foows:
Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks 43 =, w,t w (,t ) =, w (,t ) =, EI x ( w,t ) = F δ( x )cos( Ωt )dx, (11) in every time moment t. For the beams with the ineary variabe cross-section the geometric moment of inertia can be assumed as the function proportiona to the constant initia moment I for exampe as: 4 I ( x ) = I 1 x. (1) After soving the boundary vaue probem, the eigenfunction for the dispacement can be derived in the form: = x n C 1 J n k x X x C Y n ( k x ) C 3 I n ( k x ) C 4 K n ( k x ), (13) where [13]: J n the Besse function of the first kind, Y n the Besse function of the second kind, I n the hyperboic Besse function of the first kind, K n the hyperboic Besse function of the second kind, C 1,,3,4 the integrations constants, where: n is a mode of vibrations of the beam with a variabe cross-section. The k coefficient is a inear approximation of the reaistic eigenvaues of the anayzed systems. 3 EQUATIONS OF MOTION OF THE ANALYZED SYSTEMS The equations of motion of the anayzed beamike systems are obtained with the hep of the cassica method. The derived equations of motion can be presented in the matrix form as foows: Latin American Journa of Soids and Structures 1(13) 39 57
44 Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks ρa x ρa x cosϕ sin ϕ EI x sin ϕ cosϕ 1 cosϕ sin ϕ sin ϕ cosϕ 1 cosϕ sin ϕ sin ϕ cosϕ 1 w ωs ρa x = EI x 3 w 3 cosϕ sin ϕ sin ϕ cosϕ EI x 1 cosϕ sin ϕ sin ϕ cosϕ 1 ω s cosϕ sin ϕ sin ϕ cosϕ 1 4 w 4 w (14) in each point of the range D = {( x,t ),x (, ),t > } the Eq. (3.1) coincides with boundary conditions and initias conditions. The projections of the equations of motion in the goba reference frame are obtained from the Eq. (14) and in the XYZ goba reference frame equations for individua axes can be expressed as: ω s cosϕ ωs sinϕ =, (15) ρa( x ) w EI ( x ) 4 w ω s sinϕ ωs cosϕ =, (16) EI x 4 3 w EI x 3 w =. (17) Third governing equation (17) is a we-known one and represents the equation of motion of the immovabe beam with the variabe cross-section. As we can see the boundary conditions and the oading direction does not provide any new aspects to the dynamica anaysis and the beam (Fig. ) can be treated as equivaent to the stationary one. Much more interesting case of the oading is the beam oaded by a transversa time varying force orthogona to an axis of the beam and simutaneousy parae to the disk s pane. This case (Fig. 3) where the beam with a variabe cross-section A(x) and a variabe moment of inertia I(x) is considered.
Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks 45 Fig. 3 Mode of the anayzed beam with the ineary variabe cross-section fixed on the rotationa disk and oaded by a transversa force orthogona to an axis of the beam and parae to the disk s pane [3] Anaogicay to the previous equations of motion it can be derived by means of the cassica method. Interesting case of the oading is the beam oaded by a transversa time varying force orthogona to an axis of the beam and simutaneousy parae to the disk s pane. The obtained equations of motion are couped ones with the Coriois eements. After oading the beam in the direction of the y axis of the oca reference frame it can be derived the equations of motion of the beam in the matrix form. This system of equations of motion is the fourth order partia differentia equations (PDE) and can be expressed as foows: ρa x ρa x cosϕ sinϕ sinϕ cosϕ 1 cosϕ sinϕ sinϕ cosϕ 1 cosϕ sinϕ = sinϕ cosϕ 1 w ρa x ω w EI x w ρa x in each point of the range D {( x, t), x (, ), t } cosϕ sinϕ sinϕ cosϕ 1 cosϕ sinϕ sinϕ cosϕ 1 ω s ω w ωw ωs = (18) = Œ > the Eq. (18) coincides with boundary conditions and initias conditions. The projection onto the X and Y axes, can be obtained, where the projection onto the X axis of the goba reference frame is as foows: EI x ρa( x ) w X ρa( x ) ωw Y ωs sinϕ w X = ρa x ρa ( x )ω w Y ω ( w X s cosϕ ), (19) Latin American Journa of Soids and Structures 1(13) 39 57
46 Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks and for the Y axis of the goba reference frame, the projection can be written: ρa( x ) w Y EI x ρa( x ) ωw X ωs cosϕ w Y = ρa x ρa ( x )ω w X ω ( w Y s sinϕ ), () where using the rea coordinates the individua dispacements onto the axes of the goba reference frame are: w X = w sinϕ, w Y = w cosϕ. (1) It is obvious that the vibrating beams can be described by different boundary conditions for different oading and fixations types and this case can be treated as the genera one. 4 MATHEMATICAL MODEL OF THE SYSTEM The mathematica mode of the anayzed beam is provided by the orthogonaization of the equations of motion (19) and (). The orthogonaization is provided by the Eqs. (19) and () which are mutipied using the eigenfunction for the dispacements and computing integras from the equations of motion in the beams imits of integration from the origin of the beam (zero) to the end (a ength of the beam). After assuming a constant anguar veocity, an anguar acceeration equas zero and after introducing Eqs. (1), then the projecting equations of motion Eq. (18) onto the axes of the goba reference frame can be written: ρa( x ) w X X ( x )dx EI ( x ) 4 w X X ( x )dx 4 EI x 3 w X 3 X ( x )dx EI x w X X ( x )dx ρ A( x )ω w Y X ( x )dx ρa( x )ω w X X ( x )dx =, () and for the second equation, aso mutipied by the eigenfunction for dispacement and integrated aong the beam: ρa( x ) w Y X ( x )dx EI ( x ) 4 w Y X ( x )dx 4 EI x 3 w Y 3 X ( x )dx EI x w Y X ( x )dx ρ A( x )ω w X X ( x )dx ρa( x )ω w Y X ( x )dx =. (3)
Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks 47 The presented mathematica mode is universa for different functions describing the variabiity of the beam s cross-section. The cross-section of the anayzed beam can be now treated as both ineary and nonineary variabe. If the beam has the nonineary variabe cross-section (Fig. 4-5) the appropriate function describing this variation ought to be adopted. The different oading terms can be considered as we. For exampe the beam is oaded by a transversa force orthogona to an axis of the beam and orthogona to the disk s pane (Fig. 4) or the beam is oaded by a transversa force orthogona to an axis of the beam and in parae to the disk s pane (Fig. 5). Fig. 4 Mode of the anayzed beam with the nonineary variabe cross-section fixed on the rotationa disk and oaded by a transversa force orthogona to an axis of the beam and orthogona to the disk s pane These cases can be considered separatey or combined. According to the scientific iterature the case presented in figure 4 is we-known in the mathematica sense, ikewise the case in figure 1. Fig. 5 Mode of the anayzed beam with the nonineary variabe cross-section fixed on the rotationa disk and oaded by a transversa force orthogona to an axis of the beam and in parae to the pane of the disk It is assumed that eements of the Eqs. () and (3) are the continuousy differentiabe functions. Using the Eq. () and mutipe integrating both sides of this equation by parts, the foowing formua can be obtained: Latin American Journa of Soids and Structures 1(13) 39 57
48 Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks E X ( x )I x 3 w X 3 X ( x )I x w X X ( x )I x E 3 X ( x )I ( x ) w 3 X E I ( x )X IV ( x )w X dx 4E I ( x ) X ( x )w X dx 6E I ( x ) X ( x )w X dx 4E I ( x ) X ( x )w X dx E I IV ( x )X ( x )w X dx ρa( x ) w X X ( x )dx E X x I ( x ) w X X x I ( x ) E I ( x ) X ( x )w X dx 6E I x w X X X x w X I ( x ) ( x )w X dx 6E I ( x ) X ( x )w X dx E I IV ( x )X ( x )w X dx E X x E I x I ( x ) w X X X x I ( x ) w X E I x ( x )w X dx E I IV ( x )X ( x )w X dx X ρ A( x )ω w Y X ( x )dx ρa( x )ω w X X ( x )dx =, w X ( x )w X dx (4) where the variabe cross-section A(x) and the moment of inertia I(x) depend on the geometry of the anayzed beam. There is a beam with the ineary variabe cross-section Eqs. (7) and (1) assumed for further cacuations. The eigenfunction associated with transversa vibrations X(x) of the beam is obtained by appying undamped conditions. Simiary to the first equation, the integration resuts for the second equation (3) can be appied as foows:
Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks 49 E X ( x )I x 3 w Y 3 X ( x )I x w Y X ( x )I x E 3 X ( x )I ( x ) w 3 Y E I ( x )X IV ( x )w Y dx 4E I ( x ) X ( x )w Y dx 6E I ( x ) X ( x )w Y dx 4E I ( x ) X ( x )w Y dx E I IV ( x )X ( x )w Y dx ρa( x ) w Y X ( x )dx E X x I ( x ) w Y X x I ( x ) w Y X x w Y I ( x ) E I ( x ) X ( x )w Y dx 6E I ( x ) X ( x )w Y dx 6E I ( x ) X ( x )w Y dx E I IV ( x )X ( x )w Y dx E X x E I x I ( x ) w Y X X x I ( x ) w Y E I x ( x )w Y dx E I IV ( x )X ( x )w Y dx X ρ A( x )ω w X X ( x )dx ρa( x )ω w Y X ( x )dx =. w Y ( x )w Y dx (5) It is possibe to simpify the obtained equations as shown in the Eq. (6). Additionay, after organizing the eements in order, the Eqs. (4) and (5) can be expressed as: Latin American Journa of Soids and Structures 1(13) 39 57
5 Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks E X ( x )I x 3 w X 3 X ( x )I x w X X ( x )I x w X E 3 X ( x )I ( x ) w 3 X E I ( x )X IV ( x )w X dx E I ( x ) X ( x )w X dx ρa( x ) w X X ( x )dx E I x E X x E X x I ( x ) w X I ( x ) w X X x X x X I ( x ) I ( x ) ( x )w X dx w X w X X x I ( x ) ρ A( x )ω w Y X ( x )dx ρa( x )ω w X X ( x )dx =, w X (6) and the couped second equation: E X ( x )I x 3 w Y 3 X ( x )I x w Y X ( x )I x w Y E 3 X ( x )I ( x ) w 3 Y E I ( x )X IV ( x )w Y dx E I ( x ) X ( x )w Y dx E I ( x ) X ( x )w Y dx ρa( x ) w Y X ( x )dx E X x E X x I ( x ) w Y I ( x ) w Y X x I ( x ) I X x ρa( x )ω w Y X ( x )dx =. ( x ) w Y w Y X x I ( x ) ρ A( x )ω w X w Y X ( x )dx (7) There are introduced the equivaent notation for the eigenfunctions, cross-sections and the moments of inertia as: X ( x ) = X, A( x ) = A, I ( x ) = I. (8)
Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks 51 The Eqs. (6) and (7) after substituting (8) are simpified to: E XI 3 w X X I X I 3 w X w X X I X I X I E ( X I 3 X I 3 X I I X )w X E IX IV w Y dx E I X w dx E I X w dx ρa w X X X X dx E X I w X X I X I w X E X I w X ( X I X I )w X ( X I X I I X )w X ρ Aω w Y X dx ρaω w X X dx = (9) and anaogicay to the second equation (7) for the Y axis of the goba reference frame can be obtained as: E XI 3 w Y X I X I 3 w Y w Y X I X I X I E ( X I 3 X I 3 X I I X )w Y E IX IV w Y dx E I X w dx E I X w dx ρa w Y Y Y X dx E X I w Y X I X I w Y E X I w Y ( X I X I )w Y ( X I X I I X )w Y ρ Aω w X X dx ρaω w Y X dx =. (3) The Eqs. (9) and (3) can be simpified as we and the foowing two equations yied: Latin American Journa of Soids and Structures 1(13) 39 57
5 Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks E XI 3 w X X I X I 3 w X X I w X E ( X I X I )w X E IX IV w X dx E I X w dx E I X w dx ρa w X X X X dx ρ Aω w Y X dx ρaω w X X dx =, (31) the second equation (3) additionay after simpifying the integras: E XI 3 w Y X I X I 3 w Y X I w Y E ( X I X I )w Y E I X w Y dx ρa w Y X dx ρ Aω w X X dx ρaω w Y X dx =. (3) By introducing the acting externa force, it can be noticed: ( ) ( ) EI 3 w X E I w X =, 3 EI 3 w Y E I w Y = F 3 δ( x )cos( Ωt )dx. (33) Based on the boundary conditions (11) it can be written: ( ) w X = w Y =, XI 3 w X 3 XI 3 w Y 3 XI XI w X =, w Y = (34)
Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks 53 and E X I w X X I E X I w Y X I w X w Y =, =. (35) By introducing (34-35) the first equation yieds: E IX IV w X dx E I X w dx E I X w dx ρa w X X X X dx ρ Aω w Y X dx ρaω w X X dx =. (36) Where the searched soutions are defined as dispacements in the goba reference frame and the ones are separabe in time and space domains. The system is symmetric and the motion of the anayzed cross-section is given by the mutipication of the projected harmonic time dependent eigenfunctions with the frequency Ω and the eigenfunctions for dispacements with ampitudes XY. Both the variabes coincide with the responses of two decouped harmonic osciators. In the present case the searched soutions are: w X = n=1 a X X ( x )sin( Ωt ), w Y = a Y X ( x )cos( Ωt ). n=1 (37) The norm for Eqs. (36) is assumed as: γ n = AX dx, (38) for the beam with the ineary variabe cross-section (Fig. 3) the norm is cacuated as integra. Both I 1 and I coefficients are connected with the moments of inertia and the eigenfunctions for dispacements. The I and c (veocity) coefficients are as foows: Latin American Journa of Soids and Structures 1(13) 39 57
54 Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks I 1 = 1 IX dx, γ n I = 1 γ I X dx, n (39) c = E ρ. It can be aso noticed that: X I dx =. (4) The a coefficients (ampitudes) can be obtained as: a X = ργ n c 4 k 4 I I 1 k ωωf X ( ) ( ω Ω ) c k ( I I 1 k )( ω Ω ), (41) a Y = F X ργ n 1 I k c I 1 k 4 ω Ω I k c I 1 k 4 ω Ω 1. (4) Knowing these ampitudes the tota dispacements can be cacuated and the soution in the goba reference frame can be obtained. 5 DYNAMICAL FLEXIBILITIES According to the author the dynamica fexibiity is understood as the ampitude of generaized dispacement in the direction of i generaized coordinate changed by generaized harmonic force with ampitude equas 1 in the direction of k generaized coordinate. This definition can be written in the mathematica form as: 1 s i = Y ik s k. (43) Finay the dynamica fexibiity of the rotating beam with the variabe cross-section is presented as: 1 X ( x )X ( ) Y = ργ n I k c I 1 k 4 ( ω Ω ) X ( x )X ( ). I k c I 1 k 4 ( ω Ω ) (44) n=1
Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks 55 For comparison, the dynamica fexibiity of the rotating beam with a constant cross-section is as foows: X ( x )X ( ) c g k 4 ω Ω Y g =. ρa const γ gn c 4 g k 8 ( ω Ω ) c g k 4 ( ω Ω n=1 ) On the other side the dynamica fexibiity of the immovabe beam is presented as: (45) X ( x )X ( ) Y g =. n=1 ρa const γ gn ( c g k 4 Ω (46) ) where for the Eqs. (44) and (45): and the norm is: c g = γ gn EI ρa const (47) = X dx. (48) Both the derived dynamica fexibiities, the Eq. (44) and Eq. (46), are presented in figures (Fig. 6-7). For numerica exampes the beam of 1m ength is assumed. The beam is the camped-free one and fixed on the rotationa disk. Distinctivey from the beam with a constant cross-section, the beam with the variabe cross-section has different inconstant bifurcations for the individua modes of vibrations (Fig. 7). Fig. 6 Sampe dynamica fexibiity of the anayzed immovabe beam with the variabe cross-section Latin American Journa of Soids and Structures 1(13) 39 57
56 Sawomir Zokiewski / Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks Fig. 7 Dynamica fexibiity of the rotating beam with the variabe cross-section. The disk is being rotated with the anguar veocity equas 1 radians per second Derivations of suitabe dynamica fexibiities are appied to the systems in the transportabe motion both vibrating ongitudinay and transversay. The soutions can be presented on charts of ampitude (dynamica fexibiity) in function of frequency. There are many numerica appications that can be used for this purpose, e.g. GENTA S DynRot or author s program the Modyfit [18]. Concusions The paper concerns a very crucia probem of beams vibrating in transportation with the variabe geometric parameters such as: tapered or coned beams, bades with various shape functions, and others. The anayzed systems coud be appied in many technica impementations, obviousy after proper adjustment of mathematica modes and considering the necessary parameters and conditions. The most popuar appications coud be connected with pumps and rotors; especiay the high speed ones, bades of heicopters and arms of robots. The obtained soutions were aso presented in the form of equations of motion. The dynamica anaysis was provided using the dynamica fexibiity method and the Gaerkin's method. The dynamica fexibiity method appies soutions presentation as the dynamica characteristics of ampitude in the function of frequency. The soutions were determined as the compact mathematica formuae. The dynamica characteristics were presented on charts. The transportation effect and shape functions describing the cross-section variation were taken into consideration. The centrifuga forces and the Coriois forces were taken into consideration in the obtained mathematica mode. The natura frequencies and the minima ampitudes can be easiy read from the dynamica fexibiity characteristics. In these characteristics the beam with the constant cross-section and the beam with the variabe cross-section are put together. The juxtaposition of the rotating beam and the immovabe one is presented. The resuts presented are very significant and enabe controing of such a type rotating systems. In future works more numerica exam-
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