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Parameters identification of fractional models of viscoelastic dampers and fluids

  • Received : 2016.07.21
  • Accepted : 2017.02.24
  • Published : 2017.07.25

Abstract

An identification method for determination of the parameters of the rheological models of dampers made of viscoelastic material is presented. The models have two, three or four parameters and the model equations of motion contain derivatives of the fractional order. The results of dynamical experiments are approximated using the trigonometric function in the first part of the procedure while the model parameters are determined as the solution to an appropriately defined optimization problem. The particle swarm optimization method is used to solve the optimization problem. The validity and effectiveness of the suggested identification method have been tested using artificial data and a set of real experimental data describing the dynamic behavior of damper and a fluid frequently used in dampers. The influence of a range of excitation frequencies used in experiments on results of identification is also discussed.

Keywords

Acknowledgement

Supported by : National Science Centre, Poland

References

  1. Aprile, A, Inaudi, J.A. and Kelly, J.M. (1997), "Evolutionary model of viscoelastic dampers for structural applications", J. Eng. Mech., 123(6), 551-560. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:6(551)
  2. Bagley, R.L. and Torvik, P.J. (1989), "Fractional calculus-a different approach to the analysis of viscoelastically damped structures", AIAA J., 27, 1412-1417. https://doi.org/10.2514/3.10279
  3. Bucher, C. and Pirrotta, A. (2015), "Dynamic finite element analysis of fractionally damped structural systems in the time domain", Acta Mech., 226(12), 3977-3990. https://doi.org/10.1007/s00707-015-1454-8
  4. Chang, T. and Singh, M.P. (2002), "Seismic analysis of structures with a fractional derivative model of viscoelastic dampers", Earthq. Eng. Eng. Vib., 1(2), 251-260. https://doi.org/10.1007/s11803-002-0070-5
  5. Chang, T.S. and Singh, M.P. (2009), "Mechanical model parameters for viscoelastic dampers", J. Eng. Mech., 135(6), 581-584. https://doi.org/10.1061/(ASCE)0733-9399(2009)135:6(581)
  6. Christopoulos, C and Filiatrault, A. (2006), Principles of passive supplemental damping and seismic isolation, IUSS Press, Pavia, Italy.
  7. Cortes, F. and Elejabarrieta, M.J. (2007), "Finite element formulations for transient dynamic analysis in structural systems with viscoelastic treatments containing fractional derivative models", Int. J. Num. Meth. Eng., 69(10), 2173-219. https://doi.org/10.1002/nme.1840
  8. Couceiro, M. and Ghamisi, P. (2016), Fractional order darwinian particle swarm optimization Applications and evaluation of an evolutionary algorithm, Springer, Heidelberg New York, Dordrecht, London
  9. Di Paola, M., Pirrotta, A. and Valenza, A. (2011), "Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results", Mech. Mater., 43(12), 799-808. https://doi.org/10.1016/j.mechmat.2011.08.016
  10. Enelund, M. and Olsson, P. (1999), "Damping described by fading memory analysis and application to fractional derivative models", Int. J. Solids Struct., 36(7), 939-970. https://doi.org/10.1016/S0020-7683(97)00339-9
  11. Fan, W., Jiang, X. and Qi, H. (2015), "Parameter estimation for the generalized fractional element network Zener model based on the Bayesian method", Physica A, 427, 40-49. https://doi.org/10.1016/j.physa.2015.02.037
  12. Fenander, A. (1996), "Modal synthesis when modeling damping by use of fractional derivatives", AIAA J., 34(5), 1051-1058. https://doi.org/10.2514/3.13186
  13. Frings, C. and De La Llera, J.C. (2011), "Multiphysics modeling and experimental behavior of viscous damper", eds., G. De Roeck, G. Degrande, G. Lambert, G. Muller, The 8th International Conference on Structural Dynamics, (EURODYN 2011), Leuven, Belgium, July 4-6.
  14. Galucio, A.C., Deu, J.F. and Ohayon, R. (2004), "Finite element formulation of viscoelastic sandwich beams using fractional derivative operators", Comp. Mech., 33(4), 282-291. https://doi.org/10.1007/s00466-003-0529-x
  15. Gerlach, S. and Matzenmiller, A. (2005), "Comparison of numerical methods for identification of viscoelastic line spectra from static test data", Int. J. Num. Meth. Eng., 63(3), 428-454. https://doi.org/10.1002/nme.1161
  16. Greco, R., Avakian, J. and Marano, G.C. (2014), "A comparative study on parametr identification of fluid viscous pampers with different models", Arch. Appl. Mech., 84, 117-1134.
  17. Gupta, N., Machida, A., Mutsyoshi, H. and Tanzo, W. (1996), "A fractional derivative hysteretic model for viscoelastic damper", J. Struct. Eng., 42, 697-706.
  18. Gusella, V. and Terenzi, G. (1997), "Fluid viscous device modeling by fractional derivatives", Struct. Eng. Mech., 5(2), 177-191. https://doi.org/10.12989/sem.1997.5.2.177
  19. Hansen, S. (2007), "Estimation of the relaxation spectrum from dynamic experiments using Bayesian analysis and a new regularization constraint", Rheol. Acta, 47(2), 169-178. https://doi.org/10.1007/s00397-007-0225-4
  20. Hatada, T., Kobori, T., Ishida, M. and Niwa, N. (2000), "Dynamic analysis of structures with Maxwell model", Earthq. Eng. Struct. D., 29(2), 159-176. https://doi.org/10.1002/(SICI)1096-9845(200002)29:2<159::AID-EQE895>3.0.CO;2-1
  21. Hayat, T., Nadeem, S. and Asghar, S. (2004), "Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model", Appl.Math. Comp., 151(1), 153-161. https://doi.org/10.1016/S0096-3003(03)00329-1
  22. Hou, C.Y. (2008), "Fluids dynamics and behavior of nonlinear fluid dampers", J. Struct. Eng., 134, 56-63. https://doi.org/10.1061/(ASCE)0733-9445(2008)134:1(56)
  23. Idesman, A., Niekamp, R. and Stein, E. (2001), "Finite elements in space and time for generalized viscoelastic model", Comput. Mech., 27(1), 49-60. https://doi.org/10.1007/s004660000213
  24. Jones, D.I.G. (2001), Handbook of viscoelastic vibration damping. Wiley, New York, USA.
  25. Kennedy, J. and Eberhart, R.C. (1995), "Particle swarm optimization", IEEE International Conference on Neural Networks, 4, 1942-1948.
  26. Khan, M., Anjum, A., Qi, H. and Fetecau, C. (2010), "On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid", Zeitschrift fur Angewandte Mathematik und Physik, 61(1), 133-145. https://doi.org/10.1007/s00033-009-0004-4
  27. Lee, S.H., Son, D.I., Kim, J. and Min, K.W. (2004), "Optimal design of viscoelastic dampers using eigenvalue assignment", Earthq. Eng. Struct. D., 33(4), 521-542. https://doi.org/10.1002/eqe.364
  28. Lewandowski, R. and Chorazyczewski, B. (2010), "Identification of the parameters of the Kelvin-Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers", Comp. Struct., 88(1), 1-17. https://doi.org/10.1016/j.compstruc.2009.09.001
  29. Lewandowski, R., Bartkowiak, A. and Maciejewski H. (2012), "Dynamic analysis of frames with viscoelastic dampers: a comparison of damper models", Struct. Eng. Mech., 41(1), 113-137. https://doi.org/10.12989/sem.2012.41.1.113
  30. Lewandowski, R. and Pawlak, Z. (2011), "Dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractional derivatives", J. Sound Vib., 330(5), 923-936. https://doi.org/10.1016/j.jsv.2010.09.017
  31. Lion, A. (2001), "Thermomechanically consistent formulations of the standard linear solid using fractional derivatives", Arch. Mech., 53(3), 253-273.
  32. Makris, N. (1992), Theoretical and experimental investigation of viscous dampers in applications of seismic and vibration isolation, PhD dissertation, State University of New York at Buffalo.
  33. Makris, N. and Constantinou, M.C. (1991), "Fractional-derivative Maxwell model for viscous dampers", J. Struct. Eng., ASCE, 117(9), 2708-2724. https://doi.org/10.1061/(ASCE)0733-9445(1991)117:9(2708)
  34. Matsagar, V.A. and Jangid, R.S. (2005), "Viscoelastic damper connected to adjacent structures involving seismic isolation", J. Civ. Eng. Manage., 11(4), 309-322. https://doi.org/10.1080/13923730.2005.9636362
  35. Okada, R., Nakata, N., Spencer, B.F., Kasai, K. and Kim, B.S. (2006), "Rational polynomial approximation modeling for analysis of structures with VE dampers", J. Earthq. Eng., 10(1), 97-125. https://doi.org/10.1080/13632460609350589
  36. Palmeri, A., Ricciardelli, F., De Luca, A. and Muscolino, G. (2003), "State space formulation for linear viscoelastic dynamic systems with memory", J. Eng. Mech., 129(7), 715-724. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:7(715)
  37. Park, S.W. (2001), "Analytical modeling of viscoelastic dampers for structural and vibration control", Int. J. Solids Struct., 38(44), 8065-8092. https://doi.org/10.1016/S0020-7683(01)00026-9
  38. Park, J.H., Kim, J. and Min, K.W. (2004), "Optimal design of added viscoelastic dampers and supporting braces", Earthq. Eng. Struct. D., 33(4), 465-484. https://doi.org/10.1002/eqe.359
  39. Perez, R.E. and Behdinan, K. (2007), "Particle swarm approach for structural design optimization", Comp. Struct., 85(19), 1579-1588. https://doi.org/10.1016/j.compstruc.2006.10.013
  40. Pirrotta, A., Cutrona, S., Di Lorenzo, S. and Di Mateo, A. (2015), "Fractional visco-elastic Timoshenko beam deflection via single equation", Int. J. Numer. Meth. Eng., 104(9), 869-886. https://doi.org/10.1002/nme.4956
  41. Podlubny, I. (1999), Fractional differential equations, Academic Press, New York, USA.
  42. Pritz, T. (1996), "Analysis of four-parameter fractional derivative model of real solid materials", J. Sound Vib., 195(1), 103-115. https://doi.org/10.1006/jsvi.1996.0406
  43. Pritz, T. (2003), "Five-parameter fractional derivative model for polymeric damping materials", J. Sound Vib., 265(5), 935-952. https://doi.org/10.1016/S0022-460X(02)01530-4
  44. Schmidt, A. and Gaul, L. (2002), "Finite element formulation of viscoelastic constitutive equations using fractional time derivatives", J. Nonlin. Dyn., 29(1), 37-55. https://doi.org/10.1023/A:1016552503411
  45. Shukla, A.K. and Datta, T.K. (1999), "Optimal use of viscoelastic dampers in building frames for seismic force", J. Struct. Eng., 125(4), 401-409. https://doi.org/10.1061/(ASCE)0733-9445(1999)125:4(401)
  46. Singh, M.P. and Chang, T.S. (2009), "Seismic analysis of structures with viscoelastic dampers", J. Eng. Mech., 135(6), 571-580. https://doi.org/10.1061/(ASCE)0733-9399(2009)135:6(571)
  47. Singh, M.P. and Moreschi, L.M. (2002), "Optimal placement of dampers for passive response control", Earthq. Eng. Struct. D., 31(4), 955-976. https://doi.org/10.1002/eqe.132
  48. Singh, M.P., Verma, N.P. and Moreschi, L.M. (2003), "Seismic analysis and design with Maxwell dampers", J. Eng. Mech., 129(3), 273-282. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:3(273)
  49. Tong, D. and Liu, Y. (2005), "Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe", Int. J. Eng. Sci., 43(3), 281-289. https://doi.org/10.1016/j.ijengsci.2004.09.007
  50. Tsai, M.H. and Chang, K.C. (2002), "Higher-mode effect on the seismic responses of buildings with viscoelastic dampers", Earthq. Eng. Eng. Vib., 1(1), 119-129. https://doi.org/10.1007/s11803-002-0015-z
  51. Yang, P., Lam, Y. and Zhu, K. (2010), "Constitutive equation with fractional derivatives for the generalized UCM model", J. Non-Newtonian Fluid Mech., 165(3), 88-97. https://doi.org/10.1016/j.jnnfm.2009.10.002
  52. Yin, Y. and Zhu, K.Q. (2006), "Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell mode", Appl. Math. Comp., 173(1), 231-242. https://doi.org/10.1016/j.amc.2005.04.001
  53. Welch, S.W.J., Rorrer, R.A.L. and Duren, R.G. (1999), "Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials", Mech. Time-Dependent Mater., 3(3), 279-303. https://doi.org/10.1023/A:1009834317545
  54. Wilke, D.N., Kok, S. and Groenwold, A.A. (2007), "Comparison of linear and classical velocity update rules in particle swarm optimization: Notes on diversity", Int. J. Num. Meth. Eng., 70(8), 962-984. https://doi.org/10.1002/nme.1867
  55. Xu, H. and Jiang, X. (2016), "Creep constitutive models for viscoelastic materials based on fractional derivatives", Comp. Math. Appl., 73(6), 1377-1384.