STABILITY OF THERMOELASTIC LAYERED COMPOSITE IN AXIAL MOVEMENT

6th European Conference on Computationa Mechanics (ECCM 6) 7th European Conference on Computationa Fuid Dynamics (ECFD 7) 11-15 June 18, Gasgow, UK STABILITY OF THERMOELASTIC LAYERED COMPOSITE IN AXIAL MOVEMENT NIKOLAY BANICHUK 1, EVGENY MAKEEV, PEKKA NEITTAANMAKI 3 AND TERO TUOVINEN 4 1 Ishinsky Institute for Probems in Mechanics, Russian Academy of Sciences, Moscow, Russian Federation Prospekt Vernadskogo, 11-1, Moscow 11956 Russia banichuk@ipmnet.ru Ishinsky Institute for Probems in Mechanics, Russian Academy of Sciences, Moscow, Russian Federation Prospekt Vernadskogo, 11-1, Moscow 11956 Russia makeev@ipmnet.ru 3 Facuty of Information Technoogy, University of Jyvaskya, Finand Mattianniemi, 414 Jyvaskya pekka.neittaanmaki@jyu.fi 4 Facuty of Information Technoogy, University of Jyvaskya, Finand Mattianniemi, 414 Jyvaskya tero.tuovinen@jyu.fi Key words: Composite, Layered panes, Stabiity, Thermoeastic materia, Axia movement Abstract. In this paper we study mechanica mode for a ayered thermoeastic composite and its stabiity in high-speed axiay movement. We consider statica forms of the oss of stabiity and appy some approaches, based on averaging of the thermomechanica characteristics of ayered composite materias and find some effective moduus of the considered composite structures. 1 INTRODUCTION In our previous studies (see e.g., Banichuk et a. [1], [], [3], [4], [5], [6]), we have considered many aspects of mathematica modeing of axiay moving materias. Exampes incude the processing of paper or stee, fabric, rubber or some other continous materia, and ooping systems such as band saws and timing bets. In this artice, we have wi extend our studies focusing now on ayered composites and their thermoeastic stabiities in movement. Frequenty used modes for systems of axiay moving materia have been traveing fexibe strings, membranes, beams, and pates. The dynamic and stabiity aspects discussed in this paper were first reviewed in the artice by Mote [7]. Natura frequencies are

commony anayzed together with the stabiity. It was reaized eary on that the vibration probem for an axiay moving continuum is not the a conventiona one. Because of the ongitudina continuity of the materia, the equation of motion for transverse vibration wi contain additiona terms, representing a Coriois force and a centripeta force acting on the materia. As a consequence, the resonant frequencies wi be dependent on the ongitudina veocity of the axiay moving continuum, as was noted by Archibad and Emsie [8], as we as Swope and Ames [9], Simpson [1], and Mujumber and Dougas [11]. Traveing beams have been further anayzed by e.g. Parker [1] in his study on gyroscopic continua, and by Kong and Parker [13], where an approximate anaytica expression was derived for the eigenfrequencies of a moving beam with sma fexura stiffness. Response predictions have been made for particuar cases where the excitation assumes specia forms, such as harmonic support motion, see Miranker [14], or a constant transverse point force as presented by Chonan [15]. Arbitrary excitations and initia conditions were anayzed with the hep of a moda anaysis and the Green s function method in the artice by Wickert and Mote [16]. As a resut, the critica speeds for traveing strings and beams were expicity determined. Traveing strings and beams on an eastic foundation have been investigated by, e.g., Bhat et a. [17], Perkins [18], Wickert [19] and Parker []. The oss of stabiity was studied with an appication of dynamic and static approaches in the artice by Wickert [1]. It was shown by means of numerica anaysis that in a cases instabiity occurs when the frequency is zero and the critica veocity coincides with the corresponding veocity obtained from static anaysis. The dynamica properties of moving pates have been studied by Shen et a. [] and by Shin et a. [3], and the properties of a moving paper web have been studied in the two-part artice by Kuachenko et a. [4, 5]. Critica regimes and other probems of stabiity anaysis have been studied e.g. by Wang [6] and by Syguski [7]. The resuts indicating that axiay moving beams experience divergence instabiity at a sufficienty high beam veocity have been obtained aso for beams interacting with externa media; see, e.g., study by Chang and Moretti [8]. In a study by Banichuk et a. [9], the authors extended these ideas to a two-dimensiona mode of the web, considered as a moving pate under homogeneous tension but without externa media. The most straightforward and efficient way to study stabiity is to use a inear stabiity anaysis. In an artice by Hatami et a. [3], the free vibration of a moving orthotropic rectanguar pate was studied at sub- and supercritica speeds, and its futter and divergence instabiities at supercritica speeds. The study is imited to simpy supported boundary conditions at a edges. For the soution of equations of orthotropic moving materia, many necessary fundamentas can be found in the work by Marynowski et. a (see e.g. [31] and [3]). In the present study, we wi imit our focus to moving ayered composites. We consider the statica forms of the oss of stabiity and appy some approaches, based on averaging of the thermomechanica characteristics of ayered composite materias and find some effective moduus of the considered composite structures. We wi perform the studies mainy using anaytica approaches. The artice is structured in the foowing manner.

First, we wi formuate equations for the instabiity of a homogeneous thermoeastic continuous pane in the Section. In the Section 3, we consider a ayered thermoeastic composites. Finay, in the Section 4 we wi draw concusions. The summarizing resut of this artice is that in the case of the considered mutiayered composite, we have reaized the technique of averaging of thermomechanica properties of different ayers and found the effective composite characteristics. This has been done using basic materia properties and taking into account that there is no siding and discrepancies between ayers. By using the found averaged effective moduus incuding combined effective characteristics, the required vaues were obtained and used in the fina formuas for critica veocities and temperatures of a moving composite. STABILITY OF THE HOMOGENEOUS THERMOELASTIC CONTIN- UOUS PANEL IN AXIAL MOVEMENT The homogeneous panes are mechanicay simpe supported at the infow (x = ) and out-fow (x = ) boundaries of the panes. The panes are traveing at a constant veocity V in the x- direction of the rectanguar goba coordinate system and are oaded by axia tension T and therma oads. The ength and the tota thickness H are supposed to be given, whie < x < and H/ < z < H/. Free transverse vibrations of a homogeneous pane axiay moving with constant veocity and oaded by axia tension and heated by some temperature are described by the foowing equation for transverse dispacement w and simpy supported boundary conditions ( w m t + V w x t + V ( ) w (w) x= =, x x= w x ) = ( T EH 1 ν ε θ =, (w) x= =, ) w x D 4 w (1) x 4 ( ) w = () x where m, E, ν, D are, respectivey, the mass per unit area, Young s moduar, Poisson ratio, bending rigidity (D = EH/1(1 ν )) and the deformation ε θ is defined as x= ε θ = α θ θ, θ = θ a θ. (3) Here α is a inear expansion coefficient, θ is the temperature discrepency, θ is the temperature of zero deformation, θ a is the actua temperature. In a stationary case, when w t = w t = (4) the transverse dispacement w = w(x) satisfies the equation d 4 w dx 4 + λd w dx = (5) where parameter λ (eigenvaue) is given by the expression λ = 1 ( mv + EH ) D (1 ν) α θθ T = f ( V, θ ). (6) 3

If we introduce a new unknown variabe Ψ(x) as we obtain the foowing spectra probem Ψ(x) = d w, x (7) dx d Ψ + λψ = (8) dx Ψ ( ) =, Ψ () =. (9) Here the vaue λ pays the roe of an eigenvaue. A nontrivia soution to the formuated eigenvaue probem can be represented as ( ( )) ( ( )) λ x + λ x + Ψ(x) = C 1 sin + C cos (1) with two arbitrary coefficients C 1 and C and an unknown vaue λ. Taking into account (9) and (1) we wi have C = and ( ) jπ λ =, j = 1,,... (11) ( ) jπ Ψ(x) = C 1 sin (x + ) (1) with arbitrary constant C 1. Thus, for given probem parameters D, E, ν, H,, T, m, V, α θ we obtain the critica temperature θ div of instabiity (divergence of bucking) θ div = (1 ν) EHα θ { ( π D ) + T mv (13) and ( π ) λ min = (14) corresponding the minima j = 1 in the equation (11). Anaogousy we find the critica instabiity veocity (squared) (V { ( ) V div 1 ( π = D m ) + T EHθα θ (1 ν) ) div as: (15) where D, E, ν, H,, θ, α θ, m are considered as a given positive parameters. The safety domain for stabiity in the vaues (θ, V ) is defined by the inequaity f ( V, θ ) ( π ) < λ min = (16) 4

V A C /C v C /C θ B θ Figure 1: Safety domain OAB. that is reduced to the condition F ( V, θ ) 1 D ( ) f ( V, θ ) = C v V + C θ θ C < (17) π where ( ) 1. C θ = C v = m D C = T D π EHα θ D(1 ν) ( ) + 1 π ( ) (18) π The safety domain of the vaues V, θ has a trianguar shape OAB shown in the Figure 3 LAYERED THERMOELASTIC COMPOSITE Consider the ayered pane that is symmetricay composed with respect to a midde pane (see Figure ) and consist of n + 1 (odd number) thermoeastic ayers characterized by mass per unit area m i, Young s moduus E i, Poisson ratio ν i, coefficient (α θ ) i and distances h i from the symmetry of interna pane structure, i.e. E(z) = E( z), ν(z) = ν( z), α θ (z) = α θ ( z) (19) 5

and derive the expressions for effective modui D ef, ν ef, ε ef θ and mef. To this end we appy the formuas for stresses and strains and the expression for bending moment H/ H/ σ x zdz = ( H/ z E(z)dz 1 (ν(z)) ) w x = Def ( w Thus we find the expression for effective bending rigidity in the form x ). () H/ D ef z E(z) = dz. (1) 1 (ν(z)) Using mechanica and geometric characteristics of the pane ayers E i, ν i, h i we evauate the integra in the equation (1). We wi have the foowing formua E n+1 D ef = h 3 3 1 νn+1 n+1 + 3 E i 1 ν i ( ) h 3 i h 3 i+1. () In an anaogous manner, we derive the formuas for an effective Poisson s ratio ν ef and for effective therma deformation ε ef θ of a nonhomogeneous isotropic ayered pane. We have ν ef = D ef H/ [ = ν n+1 E n+1 h 3 n+1 + 3D ef 1 νn+1 z ν(z)e(z) 1 (ν(z)) dz = E i ν i 1 ν i ( ) ] h 3 i h 3 i+1 (3) ε ef θ = H H/ α θ (z)θdz = H { (α θ ) n+1 θ n+1 h n+1 + (α θ ) i θ i (h i h i+1 ) (4) Besides that we have the foowing expression for m ef : m ef = m n+1 + We derive aso the corresponding formua for a joint expression in the foowing form a ef = H H/ a(z)dz = m i. (5) a(z) = HE(z) 1 ν(z) ε θ(z) = HE(z) 1 ν(z) α θ(z)θ(z) (6) { (αθ ) n+1 E n+1 θ n+1 1 ν n+1 h n+1 + 6 (α θ ) i E i θ i 1 ν i (h i h i+1 ). (7)

The ast formua contains many particuar cases. Thus, if the Poisson s ratio and the temperature are the same for a materias, i.e. then we wi have a ef = ν 1 = ν = ν 3 = = ν n+1 = ν, (8) θ 1 = θ = θ 3 = = θ n+1 = θ (9) { θ (α θ ) 1 ν n+1 E n+1 h n+1 + (α θ ) i E i (h i h i+1 ) If besides (8), (9) the Young s moduus are equa for a ayers, i.e. (3) E 1 = E = E 3 = = E n+1 = E (31) then we obtain { a ef = Eθ (α θ ) 1 ν n+1 h n+1 + (α θ ) i (h i h i+1 ). (3) To use the obtained resuts (18)-(3) for a stabiity anaysis we wi take (13), (15) and suppose that D = D ef, ν = ν ef, a = a ef, m = m ef (33) in the equations (13), (15). We wi have where b ef = ( V ) div = 1 m ef θ div = 1 b ef { ( π ) D ef + T a ef { ( π ) D ef + T a ef ( ) { ef EHαθ (αθ ) = n+1 E n+1 + 1 ν 1 ν n+1 4 SOME NOTES AND CONCLUSIONS (α θ ) i E i 1 ν i (h i h i+1 ) In this paper the stabiity probems have been studied for homogeneous thermoeastic panes and for nonhomogeneous ayered composites that perform axia movement. The critica veocities of instabiity and critica temperatures of bucking have been presented in an anaytica form. The safety domains of principa parameters were obtained and presented in the paper. This resut can be used in the engineering practice for tuning the parameters of mechanica systems to optimize the efficiency of production processes. In the case of the considered mutiayered composite we reaized the technique of averaging of thermomechanica properties of different ayers and found the effective composite 7 (34) (35) (36)

n + 1 n + 1 h n+1 h 3 x... h h 1 H/ 3 1 z Figure : Cross-sectiona of the pane modeed as a ayered continuous composite. characteristics. This has been done using basic materia properties and taking into account that there is no siding and discrepancies between ayers. Using the found averaged effective moduus incuding combined effective characteristics the required vaues were obtained and used in the fina formuas for critica veocities and temperatures of a moving composite. Acknowedgements: The work was supported by the Russian Science Foundation (project no 17-19-147). REFERENCES [1] N. Banichuk, M. Kurki, P. Neittaanmäki, T. Saksa, M. Tirronen, and T. Tuovinen. Optimization and anaysis of processes with moving materias subjected to fatigue fracture and instabiity. Mechanics Based Design of Structures and Machines: An Internationa Journa, 41():146 167, 13. [] N. Banichuk, J. Jeronen, P. Neittaanmäki, T. Saksa, and T. Tuovinen. Theoretica study on traveing web dynamics and instabiity under non-homogeneous tension. Internationa Journa of Mechanica Sciences, 66C:13 14, 13. [3] N. Banichuk, J. Jeronen, M. Kurki, P. Neittaanmäki, T. Saksa, and T. Tuovinen. On the imit veocity and bucking phenomena of axiay moving orthotropic membranes and pates. Internationa Journa of Soids and Structures, 48(13):15 5, 11. [4] N. Banichuk, J. Jeronen, P. Neittaanmäki, and T. Tuovinen. Static instabiity anaysis for traveing membranes and pates interacting with axiay moving idea fuid. Journa of Fuids and Structures, 6():74 91, 1. 8

[5] N. Banichuk, J. Jeronen, P. Neittaanmäki, T. Saksa, and T. Tuovinen. Mechanics of moving materias, voume 7 of Soid mechanics and its appications. Springer, 14. ISBN: 978-3-319-1744- (print), 978-3-319-1745-7 (eectronic). [6] N. Banichuk and S. Ivanova. Optima Structura Design: Contact Probems and High-Speed Penetration. Water de Gruyter GmbH & Co KG, 17. [7] C. D. Mote. Dynamic stabiity of axiay moving materias. Shock and Vibration Digest, 4(4): 11, 197. [8] F. R. Archibad and A. G. Emsie. The vibration of a string having a uniform motion aong its ength. ASME Journa of Appied Mechanics, 5:347 348, 1958. [9] R. D. Swope and W. F. Ames. Vibrations of a moving threadine. Journa of the Frankin Institute, 75:36 55, 1963. [1] A. Simpson. Transverse modes and frequencies of beams transating between fixed end supports. Journa of Mechanica Engineering Science, 15:159 164, 1973. [11] A. S. Mujumdar and W. J. M. Dougas. Anaytica modeing of sheet futter. Svensk Papperstidning, 79:187 19, 1976. [1] R. G. Parker. On the eigenvaues and critica speed stabiity of gyroscopic continua. ASME Journa of Appied Mechanics, 65:134 14, 1998. [13] L. Kong and R. G. Parker. Approximate eigensoutions of axiay moving beams with sma fexura stiffness. Journa of Sound and Vibration, 76:459 469, 4. [14] W. L. Miranker. The wave equation in a medium in motion. IBM Journa of Research and Deveopment, 4:36 4, 196. [15] S. Chonan. Steady state response of an axiay moving strip subjected to a stationary atera oad. Journa of Sound and Vibration, 17:155 165, 1986. [16] J. A. Wickert and C. D. Mote. Cassica vibration anaysis of axiay moving continua. ASME Journa of Appied Mechanics, 57:738 744, 199. [17] R. B. Bhat, G. D. Xistris, and T. S. Sankar. Dynamic behavior of a moving bet supported on eastic foundation. Journa of Mechanica Design, 14(1):143 147, 198. [18] N. C. Perkins. Linear dynamics of a transating string on an eastic foundation. Journa of Vibration and Acoustics, 11(1): 7, 199. [19] J. A. Wickert. Response soutions for the vibration of a traveing string on an eastic foundation. Journa of Vibration and Acoustics, 116(1):137 139, 1994. [] R. G. Parker. Supercritica speed stabiity of the trivia equiibrium of an axiaymoving string on an eastic foundation. Journa of Sound and Vibration, 1():5 19, 1999. 9

[1] J. A. Wickert. Non-inear vibration of a traveing tensioned beam. Internationa Journa of Non-Linear Mechanics, 7(3):53 517, 199. [] J. Shen, L. Sharpe, and W. McGiney. Identification of dynamic properties of pateike structures by using a continuum mode. Mechanics Research Communications, (1):67 78, January 1995. [3] C. Shin, J. Chung, and W. Kim. Dynamic characteristics of the out-of-pane vibration for an axiay moving membrane. Journa of Sound and Vibration, 86(4-5):119 131, September 5. [4] A. Kuachenko, P. Gradin, and H. Koivurova. Modeing the dynamica behaviour of a paper web. Part I. Computers & Structures, 85:131 147, 7. [5] A. Kuachenko, P. Gradin, and H. Koivurova. Modeing the dynamica behaviour of a paper web. Part II. Computers & Structures, 85:148 157, 7. [6] X. Wang. Instabiity anaysis of some fuid-structure interaction probems. Computers & Fuids, 3(1):11 138, January 3. [7] R. Syguski. Stabiity of membrane in ow subsonic fow. Internationa Journa of Non-Linear Mechanics, 4(1):196, January 7. [8] Y. B. Chang and P. M. Moretti. Interaction of futtering webs with surrounding air. TAPPI Journa, 74(3):31 36, 1991. [9] N. Banichuk, J. Jeronen, P. Neittaanmäki, and T. Tuovinen. On the instabiity of an axiay moving eastic pate. Internationa Journa of Soids and Structures, 47(1):91 99, 1. [3] S. Hatami, M. Azhari, M. M. Saadatpour, and P. Memarzadeh. Exact free vibration of webs moving axiay at high speed. In AMATH 9: Proceedings of the 15th American Conference on Appied Mathematics, pages 134 139, Stevens Point, Wisconsin, USA, 9. Word Scientific and Engineering Academy and Society (WSEAS). Houston, USA. [31] K. Marynowski and T. Kapitaniak. Dynamics of axiay moving continua. Internationa Journa of Mechanica Sciences, 81:6 41, apr 14. [3] K. Marynowski. Dynamics of the Axiay Moving Orthotropic Web, voume 38 of Lecture Notes in Appied and Computationa Mechanics. Springer Verag, Germany, 8. ISBN 978-3-54-78988-8 (print), 978-3-54-78989-5 (onine). 1