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Eigenmodes and added-mass matrices of hydroelastic vibrations of complex structures

Published online by Cambridge University Press:  30 August 2023

A.A. Korobkin Open the ORCID record for A.A. Korobkin [Opens in a new window] ,
T.I. Khabakhpasheva Open the ORCID record for T.I. Khabakhpasheva [Opens in a new window]  and
K.A. Shishmarev Open the ORCID record for K.A. Shishmarev [Opens in a new window]
A.A. Korobkin
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
T.I. Khabakhpasheva*
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
K.A. Shishmarev
Affiliation:
Department of Differential Equations, Altai State University, Barnaul 656049, Russia
*
†Email address for correspondence: t.khabakhpasheva@uea.ac.uk
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Abstract

The eigenmodes and eigenfrequencies of two-dimensional elastic structures in contact with a liquid are investigated within the linear theory of hydroelasticity. The shapes of the structural vibrations and the hydrodynamic loads acting on the structure are calculated at the same time. The wet modes are obtained as superpositions of the dry modes of the structure, where the coefficients are solutions of a matrix equation with the hydrodynamic loads being represented by an added-mass matrix. The added-mass matrix of a homogenous elastic plate is calculated analytically through Bessel functions. Added-mass matrices of complex structures are obtained using representations of the dry modes through dry modes of the homogeneous plate of the same length. Relations between wet and dry modes and their frequencies depending on parameters of the problems are studied. The structure could be made of several plates, connected or not, completely or partially wetted, and of any variable thickness and rigidity. The main contribution to a wet mode comes from the corresponding dry mode. The wet frequency is below the corresponding dry frequency with their ratios being weakly dependent on the properties of the structure. This finding is new. The obtained added-mass matrices are suggested to use in problems of hydroelastic slamming for any geometry of impacting elastic body and any distributions of its elastic characteristics. The matrices can be also used to design an interface between hydrodynamic and structural solvers in numerical analysis of hydroelastic slamming.

JFM classification

Mathematical Foundations: General fluid mechanics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 970 , 10 September 2023 , A14
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Abramowitz, M. & Stegun, I.A. (Eds) 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55. US Government Printing Office.Google Scholar
Batyaev, E.A. & Khabakhpasheva, T.I. 2022 Hydroelastic waves in an ice-covered channel with linearly varying ice thickness. Fluid Dynamics 57 (3), 281–294.CrossRefGoogle Scholar
Chladni, E.F.F. 1787 Entdeckungen über die Theorie des Klanges. Zentralantiquariat der DDR.Google Scholar
Faltinsen, O.M. 1997 The effect of hydroelasticity on ship slamming. Phil. Trans. R. Soc. Lond. A: Math. Phys. Engng Sci. 355 (1724), 575–591.CrossRefGoogle Scholar
Faltinsen, O.M., Kvålsvold, J. & Aarsnes, J.V. 1997 Wave impact on a horizontal elastic plate. J. Mar. Sci. Technol. 2 (2), 87–100.CrossRefGoogle Scholar
Feng, S., Zhang, G., Wan, D., Jiang, S., Sun, Z. & Zong, Z. 2021 On the treatment of hydroelastic slamming by coupling boundary element method and modal superposition method. Appl. Ocean Res. 112, 102595.CrossRefGoogle Scholar
Fu, X. & Qin, Z. 2014 Calculation of the added mass matrix of water impact of elastic wedges by the discrete vortex method. J. Fluids Struct. 44, 316–323.CrossRefGoogle Scholar
Gakhov, F.D. 2014 Boundary Value Problems. Elsevier.Google Scholar
Gradstein, I.S. & Ryzhik, I.M. 1965 Tables of Integrals, Sums, Series, and Products. Academic Press.Google Scholar
Ionina, M.F. 1999 Penetration of an elastic circular cylindrical shell into an incompressible liquid. J. Appl. Mech. Tech. Phys. 40 (6), 1163–1172.CrossRefGoogle Scholar
Ionina, M.F. & Korobkin, A.A. 1999 Water impact on cylindrical shell. In Proceedings of the 14th International Workshop on Water Waves and Floating Bodies, Port Huron, MI, USA. (ed. R. Beck & W.W. Schultz). University of Michigan.Google Scholar
Kvålsvold, J. & Faltinsen, O. 1993 Hydroelastic modelling of slamming against wetdecks. In Proceedings of the 8th International Workshop on Water Waves and Floating Bodies, Saint Johns, Canada. (ed. J. Pawlowski). Institute for Marine Dynamics.Google Scholar
Kvålsvold, J. & Faltinsen, O.M. 1995 Hydroelastic modeling of wet deck slamming on multihull vessels. J. Ship Res. 39 (3), 225–239.CrossRefGoogle Scholar
Khabakhpasheva, T.I. & Korobkin, A.A. 1997 Wave impact on elastic plates. In Proceedings of the 12th International Workshop on Water Waves and Floating Bodies, Carryle-Rouet, France. (ed. B. Molin). Ecole Superieure d'Ingenieurs de Marseille.Google Scholar
Khabakhpasheva, T.I. & Korobkin, A.A. 2003 Approximate models of elastic wedge impact. In Proceedings of the 18th International Workshop on Water Waves and Floating Bodies, Le Croisic, France. (ed. A.H. Clement & P. Ferrant). Ecole Centrale de Nantes.Google Scholar
Khabakhpasheva, T.I. & Korobkin, A.A. 2013 Elastic wedge impact onto a liquid surface: Wagner's solution and approximate models. J. Fluids Struct. 36, 32–49.CrossRefGoogle Scholar
Khabakhpasheva, T.I. & Korobkin, A.A. 2021 Blunt body impact onto viscoelastic floating ice plate with a soft layer on its upper surface. Phys. Fluids 33 (6), 062105.CrossRefGoogle Scholar
Korobkin, A. 1995 Wave impact on the bow end of a catamaran wet deck. J. Ship Res. 39 (4), 321–327.CrossRefGoogle Scholar
Korobkin, A. 1996 a Water impact problems in ship hydrodynamics. In Advances in Marine Hydrodynamics (ed. M. Ohkusu), pp. 323–371, chap. 7. Computational Mechanics Publications.Google Scholar
Korobkin, A., Gueret, R. & Malenica, S̆. 2006 Hydroelastic coupling of beam finite element model with Wagner theory of water impact. J. Fluids Struct. 22 (4), 493–504.CrossRefGoogle Scholar
Korobkin, A.A. 1996 b Elastic effects on slamming. NAOE-96-39, Department of NAOE, University of Glasgow, 134 pp.Google Scholar
Korobkin, A.A. 1998 Wave impact on the center of an Euler beam. J. Appl. Mech. Tech. Phys. 39 (5), 770–781.CrossRefGoogle Scholar
Korobkin, A.A. & Khabakhpasheva, T.I. 1998 Plane problem of asymmetrical wave impact on an elastic plate. J. Appl. Mech. Tech. Phys. 39 (5), 782–791.CrossRefGoogle Scholar
Korobkin, A.A. & Khabakhpasheva, T.I. 1999 a Plane linear problem of the immersion of an elastic plate in an ideal incompressible fluid. J. Appl. Mech. Tech. Phys. 40 (3), 491–500.CrossRefGoogle Scholar
Korobkin, A.A. & Khabakhpasheva, T.I. 1999 b Periodic wave impact onto an elastic plate. In Proceedings of the 7th International Conference on Numerical Ship Hydrodynamics Nantes.Google Scholar
Korobkin, A.A. & Khabakhpasheva, T.I. 2006 Regular wave impact onto an elastic plate. J. Engng Maths 55 (1–4), 127–150.CrossRefGoogle Scholar
Li, Q.S. 2000 Vibration analysis of flexural-shear plates with varying cross-section. Intl J. Solids Struct. 37 (9), 1339–1360.CrossRefGoogle Scholar
Meyerhoff, W.K. 1965 Die Berechnung hydroelastischer Stöße, Schiffstechnik, 12, pp. 18–30, 49–64.Google Scholar
Pegg, M., Purvis, R. & Korobkin, A. 2018 Droplet impact onto an elastic plate: a new mechanism for splashing. J. Fluid Mech. 839, 561–593.CrossRefGoogle Scholar
Reinhard, M. 2013 Free elastic plate impact into water. PhD thesis. University of East Anglia, UK.Google Scholar
Reinhard, M., Korobkin, A.A. & Cooker, M.J. 2013 Water entry of a flat elastic plate at high horizontal speed. J. Fluid Mech. 724, 123–153.CrossRefGoogle Scholar
Scolan, Y.M. 2004 Hydroelastic behaviour of a conical shell impacting on a quiescent-free surface of an incompressible liquid. J. Sound Vib. 277 (1–2), 163–203.CrossRefGoogle Scholar
Timoshenko, S. & Young, D.H. 1955 Vibration Problems in Engineering, 3rd edn. D. Van Nostrand Co., Inc.Google Scholar
Vega-Martinez, P., Rodriguez-Rodriguez, J., Khabakhpasheva, T.I. & Korobkin, A.A. 2019 Hydroelastic effects during the fast lifting of a disc from a water surface. J. Fluid Mech. 869, 726–751.CrossRefGoogle Scholar