Abstract
This paper employs a novel approach to investigating the dynamic behavior of a pipe conveying fluid and the relationship between the variables that influence it, based on causal inference. The pipe is modeled as a beam with Rayleigh beam theory and Hamilton’s variation principle is demonstrated to obtain the equation of motion. Concentrated mass at various locations is introduced using the Dirac delta function. The fluid in the pipe has no compression properties and no viscosity. The non-dimensional equations of motion of the pipe–fluid–mass system are achieved by using the approach of the fluid–structure interaction problem. The non-dimensional partial differential equations of motion are converted into matrix equations and the values of natural frequencies are obtained by using the Finite Differences Method. The relationship between the variables is investigated by causal discovery using the produced natural vibration frequencies dataset. Moreover, the Bayesian Network's probability distribution is fitted to the discretized data using the structural model created through causal discovery, resulting in trustworthy predictions without the need for sophisticated analysis. The findings highlighted that the proposed causal discovery can be an alternative practical way for real-time applications of pipe conveying fluid systems.
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References
Ibrahim, R.A.: Overview of mechanics of pipes conveying fluids—Part I: fundamental studies. J. Press. Vessel Technol. (2010). https://doi.org/10.1115/1.4001271
Hoppmann, W.H.: Forced lateral vibration of beam carrying a concentrated mass. J. Appl. Mech. Trans. ASME 19(3), 301–307 (1952)
Maltbaek, J.C.: The influence of a concentrated mass on the free vibrations of a uniform beam. Int. J. Mech. Sci. 3, 197–218 (1961). https://doi.org/10.1016/0020-7403(61)90004-2
Chen, Y.: On the vibration of a beam or rods carrying a concentrated mass. J. Appl. Mech. 30(2), 310–311 (1963). https://doi.org/10.1115/1.3636537
Pan, H.H.: Transverse vibration of an Euler beam carrying a system of heavy bodies. ASME J. Appl. Mech. 32(2), 434–437 (1965). https://doi.org/10.1115/1.3625821
Sato, K.; Saito, H.K.; Otomi, K.: The parametric response of a horizontal beam carrying a concentrated mass under gravity. ASME J. Appl. Mech. 45(3), 634–648 (1978). https://doi.org/10.1115/1.3424375
Kang, M.G.: Effect of rotary inertia of concentrated masses on the natural vibration of fluid conveying pipes. J. Korean Nuclear Soc. 31(2), 202–213 (1999)
Kang, M.G.: The influence of rotary inertia of concentrated masses on the natural vibrations of a clamped–supported pipe conveying fluid. Nucl. Eng. Des. 196(3), 281–292 (2000). https://doi.org/10.1016/S0029-5493(99)00307-6
Ghayesh, M.H.; Amabili, M.; Païdoussis, M.P.: Thermo-mechanical phase-shift determination in Coriolis mass-flowmeters with added masses. J. Fluids Struct. 34, 1–13 (2012). https://doi.org/10.1016/j.jfluidstructs.2012.05.003
Varol, B.Y.; Sinir, G.B.: The dynamic analysis of a pipe with concentrated masses. In: International Symposium on Computing in Science and Engineering Proceedings 235, (2013)
Zhang, T.; Ouyang, H.; Zhao, C.; Ding, Y.J.: Vibration analysis of a complex fluid-conveying piping system with general boundary conditions using the receptance method. Int. J. Press. Vessels Pip. 166, 84–93 (2018)
ElNajjar, J.; Daneshmand, F.: Stability of horizontal and vertical pipes conveying fluid under the effects of additional point masses and springs. Ocean Eng. 206, 106943 (2020)
Khudayarov, B.A.; Komilova, K.M.; Turaev, F.Z.; Aliyarov, J.A.: Numerical simulation of vibration of composite pipelines conveying fluids with account for lumped masses. Int. J. Press. Vessels Pip. 179, 104034 (2020)
Sunil Kumar, H.S.; Anand, R.B.; Prabhakara, D.L.: Numerical investigation on vibration and stability of cutting fluid delivery viscoelastic conduits. Arab. J. Sci. Eng. 44(6), 5765–5778 (2019)
Goyder H (2015) An experimental ınvestigation of added mass and damping in submerged pipework. ASME 2015 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers. V004T04A031
Bhattacharya, B.; Solomatine, D.P.: Neural networks and M5 model trees in modelling water level—discharge relationship. Neurocomputing 63, 381–396 (2005)
Najafzadeh, M.; Laucelli, D.B.; Zahiri, A.: Application of model tree and evolutionary polynomial regression for evaluation of sediment transport in pipes. KSCE J. Civ. Eng. 21(5), 1956–1963 (2017)
Solomatine, D.P.; Xue, Y.: M5 model trees and neural networks: application to flood forecasting in the upper reach of the Huai River in China. J. Hydrol. Eng. 9(6), 491–501 (2004)
Singh, K.K.; Pal, M.; Singh, V.P.: Estimation of mean annual flood in Indian catchments using backpropagation neural network and M5 model tree. Water Resour. Manag. 24(10), 2007–2019 (2010)
Etemad-Shahidi, A.; Ghaemi, N.: Model tree approach for prediction of pile groups scour due to waves. Ocean Eng. 38(13), 1522–1527 (2011)
Brunton, S.L.; Noack, B.R.; Koumoutsakos, P.: Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508 (2020)
Kochkov, D.; Smith, J.A.; Alieva, A.; Wang, Q.; Brenner, M.P.; Hoyer, S.: Machine learning–accelerated computational fluid dynamics. Proc. Natl. Acad. Sci. (2021). https://doi.org/10.1073/pnas.2101784118
Gul, S.: Machine learning applications in drilling fluid engineering: a review. In: International Conference on Offshore Mechanics and Arctic Engineering, vol. 85208, p. V010T11A007. American Society of Mechanical Engineers (2021, June)
Cheng, L.; Guo, R.; Moraffah, R.; Sheth, P.; Candan, K. S.; Liu, H.: Evaluation methods and measures for causal learning algorithms. In: IEEE Transactions on Artificial Intelligence, https://doi.org/10.1109/TAI.2022.3150264.
Naser, M. Z.: Causality, causal discovery, and causal inference in structural engineering. arXiv preprint arXiv:2204.01543 (2022)
Ombadi, M.; Nguyen, P.; Sorooshian, S.; Hsu, K.L.: Evaluation of methods for causal discovery in hydrometeorological systems. Water Resour. Res. 56(7), 34 (2020). https://doi.org/10.1029/2020WR027251
Naser, M.Z.; Ciftcioglu, A.O.: Causal discovery and causal learning for fire resistance evaluation: incorporating domain knowledge. arXiv preprint arXiv:2204.05311 (2022)
Sharma, A.; Mehrotra, R.: An information theoretic alternative to model a natural system using observational information alone. Water Resour. Res. 50(1), 650–660 (2014)
Wang, Y.; Yang, J.; Chen, Y.; De Maeyer, P.; Li, Z.; Duan, W.: Detecting the causal effect of soil moisture on precipitation using convergent cross mapping. Sci. Rep. 8(1), 1–8 (2018)
Liu, R.; Misra, S.A.: Generalized machine learning workflow to visualize mechanical discontinuity. J. Pet. Sci. Eng. 210, 109963 (2022). https://doi.org/10.1016/j.petrol.2021.109963
Jočković, M.; Radenković, G.; Nefovska-Danilović, M.; Baitsch, M.: Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. Appl. Math. Model. 71, 152–172 (2019)
Han, S.M.; Benaroya, H.; Wei, T.: Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib. 225(5), 935–988 (1999). https://doi.org/10.1006/jsvi.1999.2257
Gouin, H.: Mathematical Methods of Analytical Mechanics. Elsevier, Amsterdam (2020)
Chang, J.R.; Lin, W.J.; Huang, C.J.; Choi, S.T.: Vibration and stability of an axially moving Rayleigh beam. Appl. Math. Model. 34(6), 1482–1497 (2010)
Sınır, B.G.: The mathematical modeling of vibrations in marine pipelines. Doctoral dissertation. DEÜ Institute of Science (2004)
Sınır, B.G.; Demi̇r DD,: The analysis of nonlinear vibrations of a pipe conveying an ideal fluid. Eur. J/. f Mech. B/Fluids 52, 38–44 (2015). https://doi.org/10.1016/j.euromechflu.2015.01.005
Li, B.; Fang, H.; Yang, K.; He, H.; Tan, P.; Wang, F.: Mechanical response and parametric sensitivity analyses of a drainage pipe under multiphysical coupling conditions. Complexity (2019). https://doi.org/10.1155/2019/3635621
Zhang, T.; Ouyang, H.; Zhang YOand Lv BL,: Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses. Appl. Math. Model. 40(17), 7880–7900 (2016). https://doi.org/10.1016/j.apm.2016.03.050
Haberman, R.: Mathematical Models: Mechanical Vibrations. Population Dynamics. and Traffic Flow (Classics in Applied Mathematics), 1st edn. Prentice-Hall Inc., Hoboken (1998)
Jweeg, M.J.; Ntayeesh, T.J.: Dynamic analysis of pipes conveying fluid using analytical. numerical and experimental verification with the aid of smart materials. Int. J. Sci. Res. (2015). https://doi.org/10.13140/RG.2.1.5060.3922
Meirovitch, L.: Analytical Methods in Vibration, 1st edn., p. 275–278. Pearson, London (1967)
Liu, R.; Misra, S.: Monitoring the propagation of mechanical discontinuity using data-driven causal discovery and supervised learning. Mech. Syst. Signal Process. 170, 108791 (2022)
Koller, D.; Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. MIT Press, Cambridge (2009)
Tang, K.; Parsons, D.J.; Jude, S.: Comparison of automatic and guided learning for Bayesian networks to analyse pipe failures in the water distribution system. Reliab. Eng. Syst. Saf. 186, 24–36 (2019)
Liu, M.; Wang, Z.; Zhou, Z.; Qu, Y.; Yu, Z.; Wei, Q.; Lu, L.: Vibration response of multi-span fluid-conveying pipe with multiple accessories under complex boundary conditions. Eur. J. Mech. A/Solids 72, 41–56 (2018)
Dagli, B.Y.; Ergut, A.: Dynamics of fluid conveying pipes using Rayleigh theory under non-classical boundary conditions. Eur. J. Mech. B/Fluids 77, 125–134 (2019)
Hu, G.; Mohammadiun, S.; Gharahbagh, A.A.; Li, J.; Hewage, K.; Sadiq, R.: Selection of oil spill response method in Arctic offshore waters: a fuzzy decision tree based framework. Mar. Pollut. Bull. 161, 111705 (2020)
Atoui, M.A.; Cohen, A.; Verron, S.; Kobi, A.: A single Bayesian network classifier for monitoring with unknown classes. Eng. Appl. Artif. Intell. 85, 681–690 (2019)
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Dagli, B.Y., Ergut, A. & Özyüksel Çiftçioğlu, A. Estimation of Natural Frequencies of Pipe–Fluid–Mass System by Using Causal Discovery Algorithm. Arab J Sci Eng 48, 11713–11726 (2023). https://doi.org/10.1007/s13369-022-07549-z
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DOI: https://doi.org/10.1007/s13369-022-07549-z