Abstract
This chapter is devoted to the analysis of the travelling panel, submerged in axially flowing fluid. In order to accurately model the dynamics and stability of a lightweight moving material, the interaction between the material and the surrounding air must be taken into account somehow. The light weight of the material leads to the inertial contribution of the surrounding air to the acceleration of the material becoming significant. In the small displacement regime, the geometry of the vibrating panel is approximately flat, and hence flow separation is unlikely. We will use the model of potential flow for the fluid. The approach described in this chapter allows for an efficient semi-analytical solution, where the fluid flow is solved analytically in terms of the panel displacement function, and then strongly coupled into the partial differential equation describing the panel displacement. The panel displacement, accounting also for the fluid–structure interaction, can then be solved numerically from a single integrodifferential equation. In the first section of this chapter, we will set up and solve the problem of axial potential flow obstructed by the travelling panel. In the second section, we will use the results to solve the fluid–structure interaction problem, and give so me numerical examples.
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Allen MB III, Herrera I, Pinder GF (1988) Numerical modeling in science and engineering. Wiley Interscience, New York
Anderson JD Jr (1985) Fundamentals of aerodynamics. McGraw-Hill, New York
Ashley H, Landahl M (1985) Aerodynamics of wings and bodies. Dover, New York
Bisplinghoff RL, Ashley H (1962) Principles of aeroelasticity. 2nd edn. Dover Publications, New York
Brenner SC, Scott LR (2010) The mathematical theory of finite element methods, texts in applied mathematics. vol 15, 3rd edn. Springer, New York
Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral methods in fluid dynamics. Springer, New York
Chang YB, Moretti PM (1991) Interaction of fluttering webs with surrounding air. TAPPI J 74(3):231–236
Chang YB, Fox SJ, Lilley DG, Moretti PM (1991) Aerodynamics of moving belts, tapes and webs. In: Perkins NC, Wang KW (eds) ASME DE, vol 36, presented in ASME Symposium on Dynamics of Axially Moving Continua, Miami, Florida, September 22–25, pp 33–40
Christou MA, Christov CI (2007) Fourier-Galerkin method for 2D solitons of Boussinesq equation. Math Comput Simul 74:82–92
Ciarlet PG (1978) The finite element method for elliptic problems. Studies in mathematics and its applications, North-Holland, Amsterdam
Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, ISBN: 0-471-49666-9
Eloy C, Souilliez C, Schouveiler L (2007) Flutter of a rectangular plate. J Fluids Struct 23(6):904–919
Evans LC (1998) Partial differential equations. American Mathematical Society, ISBN 0-8218-0772-2
Frondelius T, Koivurova H, Pramila A (2006) Interaction of an axially moving band and surrounding fluid by boundary layer theory. J Fluids Struct 22(8):1047–1056
Garziera R, Amabili M (2000) Damping effect of winding on the lateral vibrations of axially moving tapes. ASME J Vibr Acoust 122:49–53
Golub GH, van Loan CF (1996) Matrix Computations, 3rd edn. Johns Hopkins, ISBN 0-8018-5414-8
Gresho PM, Sani RL (1999) Incompressible flow and the finite element method: advection-diffusion and isothermal laminar flow. Wiley, reprinted with corrections. ISBN 0 471 96789 0
Guo CQ, Païdoussis MP (2000) Stability of rectangular plates with free side-edges in two-dimensional inviscid channel flow. ASME J Appl Mech 67:171–176
Hughes TJR (2000) The finite element method. Linear static and dynamic finite element analysis. Dover Publications, Mineola, USA. ISBN 0-486-41181-8
Jeronen J (2011) On the mechanical stability and out-of-plane dynamics of a travelling panel submerged in axially flowing ideal fluid: a study into paper production in mathematical terms. PhD thesis, Department of Mathematical Information Technology, University of Jyväskylä, http://julkaisut.jyu.fi/?id=978-951-39-4596-1, Jyväskylä studies in computing 148. ISBN 978-951-39-4595-4 (book), ISBN 978-951-39-4596-1 (PDF)
Johnson C (1987) Numerical solution of partial differential equations by the finite element method. Cambridge University Press, reprint by Dover, 2009
Kornecki A, Dowell EH, O’Brien J (1976) On the aeroelastic instability of two-dimensional panels in uniform incompressible flow. J Sound Vibr 47(2):163–178
Kreyszig E (1993) Advanced engineering mathematics, 7th edn. Wiley, ISBN 0-471-59989-1
Krizek M, Neittaanmäki P (1990) Finite element approximation of variational problems and applications. Longman Scientific & Technical, Harlow, copubl. Wiley, New York
Kulachenko A, Gradin P, Koivurova H (2007) Modelling the dynamical behaviour of a paper web. Part II Comput Struct 85:148–157
Lighthill J (1986) An informal introduction to theoretical fluid mechanics. Oxford Science Publications, ISBN 0-19-853630-5
Nehari Z (1952) Conformal mapping. Dover, ISBN 0-486-61137-X
Païdoussis MP (2004) Fluid-structure interactions: slender structures and Axial flow, vol 2. Elsevier Academic Press, ISBN 0-12-544361-7
Païdoussis MP (2005) Some unresolved issues in fluid-structure interactions. J Fluids Struct 20(6):871–890
Païdoussis MP (2008) The canonical problem of the fluid-conveying pipe and radiation of the knowledge gained to other dynamics problems across applied mechanics. J Sound Vibr 310:462–492
Parker RG (1999) Supercritical speed stability of the trivial equilibrium of an axially-moving string on an elastic foundation. J Sound Vibr 221(2):205–219
Pramila A (1986) Sheet flutter and the interaction between sheet and air. TAPPI J 69(7):70–74
Pramila A (1987) Natural frequencies of a submerged axially moving band. J Sound Vibr 113(1):198–203
Qian J, Lin WW (2007) A numerical method for quadratic eigenvalue problems of gyroscopic systems. J Sound Vibr 306(1–2):284–296, http://dx.doi.org/10.1016/j.jsv.2007.05.009
Sedov LI (1972) A course in continuum mechanics, vol 3. English edn. Wolters-Noordhoff Publishing, Groningen, Netherlands, ISBN 90-0179682-6
Sherman DI (1952) On the stress distribution in partitions, an elastic heavy medium which is weakened by elliptic holes. Izvestiya Akademii Nauk SSSR. Otdelenie Tekhnicheskikh Nauk (OTN) 7:992–1010
Silberman I (1954) Planetary waves in the atmosphere. J Meteorol 11:27–34
Strang WG, Fix GJ (1973) An analysis of the finite element method. Wellesley Cambridge Press, ISBN 978-0961408886
Tadmor E (1987) Stability analysis of finite-difference, pseudospectral and Fourier-Galerkin approximations for time-dependent problems. SIAM Rev 29(4):525–555
Wang Y, Huang L, Liu X (2005) Eigenvalue and stability analysis for transverse vibrations of axially moving strings based on Hamiltonian dynamics. Acta Mech Sinica 21:485–494
Wickert JA, Mote CD (1990) Classical vibration analysis of axially moving continua. ASME J Appl Mech 57:738–744
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Banichuk, N., Jeronen, J., Neittaanmäki, P., Saksa, T., Tuovinen, T. (2014). Travelling Panels Interacting with External Flow. In: Mechanics of Moving Materials. Solid Mechanics and Its Applications, vol 207. Springer, Cham. https://doi.org/10.1007/978-3-319-01745-7_6
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DOI: https://doi.org/10.1007/978-3-319-01745-7_6
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