Exact Free Vibration of Webs Moving Axially at High Speed


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1 Eact Free Viration of Wes Moving Aially at High Speed S. HATAMI *, M. AZHARI, MM. SAADATPOUR, P. MEMARZADEH *Department of Engineering, Yasouj University, Yasouj Department of Civil Engineering, Isfahan University of Technology, Isfahan IRAN Astract:  Moving wes can e found in a wide range of industrial applications such as paper handling, tetile manufacturing, and magnetic tape recording. The moving we in these applications, which is generally orthotropic, may eperience speeds more than critical speed. Critical speed is defined as that aial speed the system viration has a vanishing eigenvalue and is suject to a uckling instaility. At a supercritical speed, the we may eperience types of instailities and susequently sever out of plane virations. In this paper, ased on thin plate theory, the equation of outofplane motion is derived for an orthotropic we. Then an eact method is employed to evaluate free viration of the we in su and supercritical speeds. This method is in fact the Levytype solution of the equation of motion in a stiffness matri form. The eact viration eigenvalues which are generally comple values, are the roots of stiffness matri determinant. Since the terms of the determinant are comple transcendental functions of eigenvalues, classical eigenvalue solver can not e used. So a suitale algorithm is used here to etract eigenvalues in the twodimensional plane of comple numers. Using a numerical eample, the reliaility of the formulation and the solution procedure is shown. The free viration eigenvalues is etracted for a range of aial speeds. Based on the results, flutter and divergence instailities of the moving we are studied at supercritical speeds. Keywords:  Free viration, Aially moving, Orthotropic we, Supercritical speed, Levytype solution, Eact method, Flutter Instaility, Divergence Instaility. Introduction Mechanical systems comprising moving elastic continua can e found in numerous fields of engineering application e.g. and saw lades, power transmission elts, steel plates in galvanizing line, paper handling and tetile manufacturing. Due to this prevalence, the eamination of moving continua has a long tradition and produced a large amount of pulications, mostly on aially moving slender structures like strings or eams. Comprehensive literature reviews can e found in Marynowski s ook recently pulished y Springer []. Generally, aially moving continua in the form of thin, flat rectangular shape material with small fleural stiffness is called a we. Wes are moving at high speed, for eample, in paper production the paper wes are transported with longitudinal speeds of up to m/min []. Because of their aially speed, moving materials eperience a Coriolis acceleration component which renders such systems gyroscopic []. In a certain critical speed, first natural frequency of the gyroscopic system vanishes and the structure eperience severe virations and ifurcation instaility. In the speeds aove the critical speed, termed supercritical speeds, the structure may eperience divergence or flutter instaility or ecome stale again. In this paper using a two dimensional model, the free viration and staility of wes are eactly solved at su and supercritical aial speeds. The we which generally has orthotropic property is modeled y thin plate theory. The analytical results provide a enchmark for evaluating approimate methods applied to moving continua. Theory The differential equation of motion for a moving orthotropic we is: w w w D + D + D w w w y N y w w w + ρ h( + v + v ) = () t t D = D + D66 The we is assumed to e sujected to a asic state of plane stress which is invariant in direction of we width and simply supported along transport ISSN: ISBN:
2 direction (aial direction). Then in free viration analysis, it can e postulated that, whatever type of viration mode occurs, it is sinusoidal in the y direction. This means that along any line in the we structure parallel to yais, transverse displacement w vary sinusoidally, so w can e written in the form λt w(, y, t) = Wn ( )[ep( ep( ] e () k n =nπ/ is the wave numer of the nth mode in ydirection (n =,, ), is the we width, W n () is shape function along aial direction otaining y solution the equation of motion for nth mode and values of λ etracting from an eigenvalue prolem, are in general comple numers as: λ =σ + iω σ and ω are the real and imaginary parts of eigenvalues λ, respectively. While all eigenvalues λ are pure imaginary (λ =iω), the traveling we is stale and the values of ω are the natural frequencies of the we. By increasing transport speed, the natural frequencies of the we decrease and at a certain speed, the first natural frequency of viration vanishes (ω =) and the we ecomes unstale. This certain speed may e mentioned as critical speed v cr. For aial speeds higher than the critical speed, the we can eperience divergence or flutter instaility that in the unstale conditions the real part of at least one of the eigenvalues λ is nonzero (σ ). The critical speed can also e otained y a static staility analysis ased on the fact that the aially moving we ecomes unstale if multiple equilirium positions eist at any prolem specification. The critical speed is the lowest speed at which multiple equilirium positions eist []. From equation (), the equilirium position in static staility analysis satisfies following equation: w w w D + D + D w w w w + = y N h y ρ v () Also, the displacement function for uckling modes in the eact method can e written as: w(, = Wn ( )[ep( ep( ] () In the present method, the we is divided to a small numer of components that each component fills the domain etween two rollers or a roller and an end oundary. Fig. shows one of these components. The assumption of sinusoidally modes implies that the displacements (w, θ, w, and θ ) and the additional forces (Q, M, Q, and M ), that appear on the edge of every we component due to viration or uckling, vary sinusoidally in the y direction (Fig. ). l z,w y () w,q θ,m θ,m w,q v A N v L N (a) Fig. : (a) A we moving etween some rollers () Nodal displacements and forces for one of the we components. The eact stiffness matri is etracted for each component and y assemling stiffness matrices of all components, overall stiffness matri of the structure will e otained. In the free viration study, the determination of the eact stiffness matri for the component shown in Fig. requires solution of differential equation of motion presented in equation () as satisfy the oundary conditions. Sustituting w from equation () into equation of motion () in the asence of N y and N y, gives an ordinary differential equation with comple terms for nth mode as ISSN: ISBN:
3 Wn Wn D + [ ρ hv D kn ] Wn + ρhλv + ( ρhλ + Dkn ) Wn = (5) in which W ( ) = A ep( r ) (6) n m= A are coefficients deriving from oundary conditions at = and =l and r, the wave numers in direction, can e otained y sustituting equation (6) into equation (5), that yields an polynomial equation for nth mode as D r + ( ρ hv Dkn ) r + ρ hλ r + ( ρhλ + Dkn ) = v (7) Equation (7) has four roots (m = to ) corresponding to each mode (n =,, ), which are comple in general. In the staility study, y using equations (), () and (6), equation (7) is eliminated as the following form r + αr + α = (8) α = ( h N D k ) D, n / = Dk n / D α, and four series of roots are r = ± α ± α α (9) For a component of the moving we shown in Fig., displacement function of equations (6) or (8) satisfies the oundary conditions at y = and y = as at these two sides the transverse displacement w and resultant ending moment per unit length of  direction M y vanish. By satisfying the oundary conditions on two other sides, stiffness matri of the component can e otained. These edges conditions, as shown in Fig., are defined y at =: Q = Q, w = w w M = M, = θ () at =l: Q = Q, w = w w M = M, = θ (), the resultant shear force and ending moment per unit length of ydirection (Q and M respectivel are related to w y the following equations. w w Q = D ( D + D66) w w ( N ρ hv ) + ρhv () t w w M = [ D + D ]. () In the equation of shearing force Q, allowance has een made for the Kirchhoff edge effect and for the components of the inplane loads arising from distortion of the we. The effect of the aial velocity on the edge shear has also een considered. Edge displacements and edge forces vectors can e defined y T { d } = { θ, w, θ, w} () T { p } = { M, Q, M, Q} (5) Using epressions (), (6) and () through (5) edge displacements and edge forces vectors in free viration may e written as: λt { d} = { d}[ep( ep( ] e (6) λt { p} = { p}[ep( ep( ] e (7) r { d n} = A m= r ep( rl) ep( r ) l d } [ X ] { A (8) { n = n n} D r D k n Dr + ( Dkn + D66kn + N ρhv ) r λρhv { pn} = A m= ( Dr Dkn )ep( rl) D r D kn D kn N h r h rl [ ( ρ v ) + λρ v]ep( ) p } [ Y ] { A (9) { n = n n} ISSN: ISBN:
4 Comining equations (8) and (9) y elimination {A n } yields { p n} = [ sn ]{ d n} () [ s ] [ Y ][ X ] () n = n n Here [s n ] is the eact stiffness matri of a component of the we that in general contains comple elements associated with the transport speed v; however, it is Hermitian in form. In the free viration analysis, the elements of the stiffness matri are transcendental functions of the natural frequencies, aial velocity, inplane force, and of the wave numers in and y directions. By assemling stiffness matri of all elements of the moving we and eliminating constrained degrees of freedom, the overall eact stiffness matri of the multispan orthotropic we moving over an elastic foundation [S n ] is otained. By the present method, precise results can e otained y only a few components, leading to small order of overall stiffness matri. The natural frequencies can e determined y vanishing the determinant of [S n ], i.e. det[s n (λ)] = The terms in the determinant are transcendental functions of ω and therefore the conventional techniques of eigensolving cannot e used. At supercritical speeds, the eigenvalues are comple numers and therefore an algorithm is required to search for the eigenvalues in the twodimensional plane of comple numers. In the current study the use is made of such an algorithm as is presented later. As mentioned efore, with a change of aial speed value, staility of the moving we can change to instaility. This happens when one or several eigenvalues λ cross the imaginary ais [5]. The case when a pair of comple conjugate eigenvalues crosses the imaginary ais with a frequency ω =Imλ is known in technical literature as flutter instaility, and the case when a real negative eigenvalue λ crosses the zero and ecomes positive is called divergence instaility. Flutter and divergence are dynamic and static forms of instaility, respectively. Equations (8) through () can e used for static staility analysis too, only y replacement λ with zero. Here, an eigenvalue prolem is also produced, ut its eigenvalue is the aial speed. Thus the critical speed v cr is otained y the following epression. det[s n (v cr )] = () Numerical Results Nondimensional variales used in the results are introduced as ρh π D Ω = ω, ρh π D Γ = σ, L r =, ρh ( c, ccr ) = ( v, vcr ), k = N () π D π D Ω and Γ are dimensionless imaginary and real parts of viration eigenvalues. r is aspect ratio of the we. c and c cr denote nondimensional aial speed and critical aial speed of the we, respectively. k is inwe load parameter along (positive when tensile). D in the relations () is defined as: E D = ( ν yν y ) () In this section, the free viration and staility of an orthotropic we is studied at low and high speeds. It is assumed that the we deflection and rotation at =, L are zero (Fig. a). The we has aspect ratio of r= and sujected to a uniform aial load with k =.. Dimensionless material properties are as follows: E y /E =.5, G y /E =.5, ν y =., ν y /ν y =E /E y E and E y are major and minor elastic modulus of the we, respectively and G y is shear modulus. It is adequate to divide the we into two components, derive the eact stiffness matri of each component and assemle them. Finally we have to equalize the determinant of the overall stiffness matri to zero and compute its roots. The determinant has so a complicated form forces us to utilize a numerical code to determine the roots. Also, since these roots may generally have oth real and imaginary parts, the iterations must e done over the comple plane rather than an ais in simpler ones. Eact values of dimensionless real and imaginary parts of eigenvalues at different dimensionless speed are presented in Tale. This values are relevant the first mode of free viration derived with n=. The normalized critical speed of the we derived y Equation () is c cr=.68. To show how the numerical algorithm is employed to etract the eigenvalues, the variation of the asolute value of the det[s n (λ)] in logarithmic scale over the comple plane for the cases in which the dimensionless speed are equal to and 5 is drawn in Fig.. Also to have a prospect aout the we viration, the qualitative ehaviors (ut not the eact response) of the we at dimensionless speeds are drawn in Fig.. According to this figure, ISSN: ISBN:
5 etween c=.6 which is completely oscillatory and c=.8 which has divergence instaility, we must have a critical speed, in which; the system egins to show instaility. Fig. 5 shows also that etween c= and c=5 we have another instaility since the ehavior of from pure oscillation to a flutter system switches instaility one. To show how the numerical algorithm is employed to etract the eigenvalues, the variation of the asolute value of the det[s n (λ)] in logarithmic scale over the comple plane for the cases in which the dimensionless speed are equal to and 5 is drawn in Fig.. Also to have a prospect aout the we viration, the qualitative ehaviors (ut not the eact response) of the we at dimensionless speeds are drawn in Fig.. According to this figure, etween c=.6 which is completely oscillatory and c=.8 which has divergence instaility, we must have a critical speed, in which; the system egins to show instaility. Fig. 5 shows also that etween c= and c=5 we have another instaility since the ehavior of system switches from pure oscillation to a flutter instaility one. Conclusions The Levytype solusion has een employed to analyze free viration of aially moving orthotropic wes in su and supercritical speeds. The eact stiffness matri of each component of the we has een otained, so precise results can e derived for a multispan moving we, y only a few numers of elements. Terms of this matri are implicit functions of the eigenvalues of free viration, inplane forces act on the we, aial speed and the geometry of the we. The viration eigenvalues are etracted within the domain of comple numers using a suitale algorithm. The free viration eigenvalues of such wes otained y the method can e served as a enchmark for checking the accuracy of other numerical methods. Nevertheless the method has some restrictions aout oundaries and inplane loads. An eample has een presented to eamine the aility of the method for modeling of prolems. In the eample, different types of we instaility (divergence or flutter) were shown in different aial speeds. References: [] K. Marynowski, Dynamics of the Aially Moving Orthotropic We, Springer, 9 [] K. Marynowski, Nonlinear virations of an aially moving viscoelastic we with timedependent tension, Chaos, Solitons & Fractals, Vol., Issue,, pp [] N. Perkins and S.J. Hwang, Dynamics of High Speed Aially Moving Material Systems In Staility of Gyroscopic Systems, edited y A. Guran et al., World Scientific, 999 [] C.C. Lin, Staility and viration characteristics of aially moving plates, Int J Solids Struct, Vol., No, 997, pp [5] A.P. Seyranian and A.A. Mailyaev, Multiparameter Staility Theory with Mechanical Applications In Staility, Viration and Control of Systems, Series A, edited y A. Guran, World Scientific,. Tale : Eact free viration eigenvalues for the first viration mode of the moving orthotropic we Normalized Aial Speed Normalized Eigenvalue c Real Part Imagi nary Part (Γ ) (Ω ) ISSN: ISBN:
6 Γ (a) Ω =.9 Γ=. Ω Γ () Ω Ω =.775 Γ=.868 Fig.. Overall stiffness matri determinant respect to the real and imaginary parts of eigenvalues. (a) c=, () c=5  c =   c =.   c =.   c = c = c =   c = c = 6  Fig.. Lateral deformation of the we center respect to time ISSN: ISBN:
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