Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods

  • Ehsan Dadkhah
  • Babak ShiriEmail author
  • Hosein Ghaffarzadeh
  • Dumitru Baleanu
Original Research


The dynamic system of a structure utilized with visco-elastic dampers can be modeled by fractional differential equations. All the resulted systems of fractional differential equations can be represented in a state space and can be transformed into a system of multi-term fractional differential equations of order 1. Considering the presence of indeterministic exogenous force like earthquake we need powerful, convergent and reliable numerical methods to simulate the response of this dynamical systems. Therefore, spline collocations method has been proposed and studied for solving system of multi-term fractional differential equations of order 1. A rigorous mathematical analysis is provided to show the efficiency and effectiveness of the method. To this end, we apply a functional analysis framework to obtain convergence and superconvergence properties of the proposed methods on the graded mesh. Some numerical experiments are provided to confirm the theoretical results. Finally, this method is used for simulating the response of a 4-story building under El Centro earthquake excitation.


Visco-elastic dampers Kelvin–Voigt model Spline collocation method \(H_\infty \) norm Large scale structures q-term fractional differential system Caputo-fractional derivative Bagley–Torvik equation Collocation methods Convergent order 

Mathematics Subject Classification

26A33 74D99 



  1. 1.
    Losanno, D., Zinno, S., Serino, G., Londono, J.M.: A design procedure in state-space representation for seismic retrofit of existing buildings with viscous dampers. In: J. Kruis, Y., Tsompanakis, B.H.V., Topping, (eds.) Proceedings of the 15th International Conference on Civil, Structural and Environmental Engineering Computing. Civil-Comp Press, Stirlingshire, UK, Paper 117 (2015)Google Scholar
  2. 2.
    Losanno, D., Spizzuoco, M., Serino, G.: Design and retrofit of multi-story frames with elastic-deformable viscous damping braces. J. Earthq. Eng. 23, 1441 (2017)CrossRefGoogle Scholar
  3. 3.
    Losanno, D., Londono, J.M., Zinno, S., Serino, G.: Effective damping and frequencies of viscous damper braced structures considering the supports flexibility. Comput. Struct. 207, 121 (2018)CrossRefGoogle Scholar
  4. 4.
    Losanno, D., Spizzuoco, M., Serino, G.: An optimal design procedure for a simple frame equipped with elastic-deformable dissipative braces. Eng. Struct. 101, 677–697 (2015)CrossRefGoogle Scholar
  5. 5.
    Gupta, N., Mutsuyoshi, H.: Analysis and design of viscoelastic damper for earthquake-resistent structure. In: Eleventh World Conference on Earthquake Engineering, vol. 1536 (1996)Google Scholar
  6. 6.
    Park, S.W.: Analytical modeling of viscoelastic dampers for structural and vibration control. Int. J. Solids Struct. 38, 8065–92 (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    Escobedo, T.J., Ricles, J.M.: The fractional order elastic-viscoelastic equations of motion: formulation and solution methods. J. Intel. Mater. Syst. Struct. 9, 489–502 (1998)CrossRefGoogle Scholar
  8. 8.
    Kumar, A.M.S., Panda, S., Chakraborty, D.: Piezoviscoelastically damped nonlinear frequency response of functionally graded plates with a heated plate-surface. J. Vib. Control 22, 320–343 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Pawlak, Z., Lewandowski, R.: The continuation method for the eigenvalue problem of structures with viscoelastic dampers. Comput. Struct. 125, 53–61 (2013)CrossRefGoogle Scholar
  10. 10.
    Gemant, A.: A method of analyzing experimental results obtained from elasto-viscous bodies. Physics 7, 311–317 (1936)CrossRefGoogle Scholar
  11. 11.
    Bagley, R.L., Torvi, P.J.K.: On the fractional calculus model of visco-elastic behavior. J. Rheol. 30, 133–155 (1986)CrossRefGoogle Scholar
  12. 12.
    Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, New Jersey (2012)zbMATHCrossRefGoogle Scholar
  13. 13.
    Diethelm, K.: The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Springer, Berlin (2010)zbMATHCrossRefGoogle Scholar
  14. 14.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Limited, Berlin (2006)zbMATHGoogle Scholar
  15. 15.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  16. 16.
    Shiri, B., Baleanu, D.: System of fractional differential algebraic equations with applications. Chaos Solitons Fractals 120, 203 (2019)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Baleanu, D., Shiri, B., Srivastava, H.M., Al Qurashi, M.: A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel. Adv. Differ. Equ. 2018, 353 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambrige (2004)zbMATHCrossRefGoogle Scholar
  19. 19.
    Baleanu, D., Shiri, B.: Collocation methods for fractional differential equations involving non-singular kernel. Chaos Solitons Fract. 116, 136–45 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Shiri, B.: Numerical solution of higher index nonlinear integral algebraic equations of Hessenberg type using discontinuous collocation methods. Math. Model. Anal. 19, 99–117 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Karamali, G., Shiri, B.: Numerical solution of higher index DAEs using their IAE’s structure: trajectory-prescribed path control problem and simple pendulum. Casp. J. Math. Sci. 7, 1–15 (2018)Google Scholar
  22. 22.
    Karamali, G., Shiri, B., Kashfi, M.: Convergence analysis of piecewise polynomial collocation methods for system of weakly singular volterra integral equations of the first kind. Appl. Comput. Math. 7, 1–11 (2017)Google Scholar
  23. 23.
    Pedas, A., Tamme, E.: On the convergence of spline collocation methods for solving fractional differential equations. J. Comput. Appl. Math. 235, 3502–14 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Pedas, A., Tamme, E.: Spline collocation methods for linear multi-term fractional differential equations. J. Comput. Appl. Math. 236, 167–76 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Pedas, A., Tamme, E., Vikerpuur, M.: Piecewise Polynomial Collocation for a Class of Fractional Itegro-Differential Equations, in Integral Methods in Science and Engineering, pp. 471–482. Birkhauser, Cham (2015)zbMATHGoogle Scholar
  26. 26.
    Pedas, A., Tamme, E.: Numerical solution of nonlinear fractional differential equations by spline collocation methods. J. Comput. Appl. Math. 255, 216–30 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Chang and Singh: Seismic analysis of structures with a fractional derivative model of visco-elastic dampers. Earthq. Eng. Eng. Vib. 1, 251–60 (2002)CrossRefGoogle Scholar
  28. 28.
    Lewandowski, R., Pawlak, Z.: Dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractionalderivatives. J. Sound Vib. 330, 923–36 (2011)CrossRefGoogle Scholar
  29. 29.
    Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Bai, Z., Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Daftardar-Gejji, V., Babakhani, A.: Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 293, 511–522 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Daftardar-Gejji, V.: Positive solutions of a system of non-autonomous fractional differential equations. J. Math. Anal. Appl. 302, 56–64 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Debbouche, A., Nieto, J.J.: Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls. Appl. Math. Comput. 245, 74–85 (2014)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Zhou, Y.: Existence and uniqueness of solutions for a system of fractional differential equations. Fract. Calc. Appl. Anal. 12, 195–204 (2009)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Daftardar-Gejji, V., Jafari, H.: An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 316, 753–763 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Jafari, H., Daftardar-Gejji, V.: Solving a system of nonlinear fractional differential equations using Adomian decomposition. J. Comput. Appl. Math. 196, 644–51 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Jafari, H., Seifi, S.: Solving a system of nonlinear fractional partial differential equations using homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 1962–1969 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. J. Appl. Mech. 22(1), 64–69 (2009)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Khalil, H., Khan, R.A.: The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations. Int. J. Comput. Math. 92, 1452–1472 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Colinas-Armijo, N., Di Paola, M., Pinnola, F.P.: Fractional characteristic times and dissipated energy in fractional linear viscoelasticity. Commun. Nonlinear Sci. Numer. Simul. 37, 14–30 (2016)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Di Paola, M., Pinnola, F.P., Zingales, M.: A discrete mechanical model of fractional hereditary materials. Meccanica 48, 1573–86 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Dadkhah. E., Gaffarzadeh, H., Shiri, B.: Design of visco-elastic dampers for structures based on fractional differential equations. In: The First International Conference on Boundary Value Problems and Applications (2018)Google Scholar
  43. 43.
    Lewandowski, R., Chorazyczewski, B.: Identification of the parameters of the Kelvin Voigt and the Maxwell fractional models, used to modeling of viscoelastic dampers. Comput. Struct. 88, 1–17 (2010)CrossRefGoogle Scholar
  44. 44.
    Litewka, P., Lewandowski, R.: Steady-state non-linear vibrations of plates using Zener material model with fractional derivative. Comput. Mech. 60, 333–54 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    El-Sayed, A.M.A., El-Mesiry, A.E.M., El-Saka, H.A.A.: Numerical solution for multi-term fractional (arbitrary) orders differential equations. Comput. Appl. Math. 23, 33–54 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Kolk, M., Pedas, A., Tamme, E.: Modified spline collocation for linear fractional differential equations. J. Comput. Appl. Math. 283, 28–40 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Brunner, H., Pedas, A., Vainikko, G.: Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels. SIAM J. Numer. Anal. 39, 957–982 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Atkinson, K.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)zbMATHCrossRefGoogle Scholar
  49. 49.
    Atkinson, K., Han, W.: Theoretical Numerical Analysis. Springer, Berlin (2005)zbMATHCrossRefGoogle Scholar
  50. 50.
    Torvik, P.J., Bagley, L.R.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294–298 (1984)zbMATHCrossRefGoogle Scholar
  51. 51.
    Mdallal Al, Q.M., Syam, M.I., Anwar, M.N.: A collocation-shooting method for solving fractional boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 15, 3814–3822 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Çenesiz, Y., Keskin, Y., Kurnaz, A.: The solution of the Bagley–Torvik equation with the generalized Taylor collocation method. J. Frankl. Inst. 347, 452–466 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Diethelm, K., Ford, J.: Numerical solution of the Bagley–Torvik equation. BIT Numeri. Math. 42, 490–507 (2002)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Ray, S.S., Bera, R.K.: Analytical solution of the Bagley Torvik equation by Adomian decomposition method. Appl. Math. Comput. 168, 398–410 (2005)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Wang, Z.H., Wang, X.: General solution of the Bagley–Torvik equation with fractional-order derivative. Commun. Nonlinear Sci. Numer. Simul. 15, 1279–1285 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Yüzbaşi, Ş.: Numerical solution of the Bagley–Torvik equation by the Bessel collocation method. Math. Methods Appl. Sci. 36, 300–312 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Fritsch, F.N., Carlson, R.E.: Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17, 238–246 (1980)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Korean Society for Informatics and Computational Applied Mathematics 2019

Authors and Affiliations

  1. 1.Faculty of Civil EngineeringUniversity of TabrizTabrizIran
  2. 2.Faculty of Mathematical ScienceUniversity of TabrizTabrizIran
  3. 3.Department of MathematicsÇankaya UniversityBalgatTurkey
  4. 4.Institute of Space SciencesMagurele, BucharestRomania

Personalised recommendations