Advertisement

Meccanica

, Volume 54, Issue 9, pp 1307–1326 | Cite as

A novel statistical linearization solution for randomly excited coupled bending-torsional beams resting on non-linear supports

  • Andrea BurlonEmail author
  • Giuseppe Failla
  • Felice Arena
Stochastics and Probability in Engineering Mechanics
  • 66 Downloads

Abstract

A novel statistical linearization technique is developed for computing stationary response statistics of randomly excited coupled bending-torsional beams resting on non-linear elastic supports. The key point of the proposed technique consists in representing the non-linear coupled response in terms of constrained linear modes. The resulting set of non-linear equations governing the modal amplitudes is then replaced by an equivalent linear one via a classical statistical error minimization procedure, which provides algebraic non-linear equations for the second-order statistics of the beam response, readily solved by a simple iterative scheme. Data from Monte Carlo simulations, generated by a pertinent boundary integral method in conjunction with a Newmark numerical integration scheme, are used as benchmark solutions to check accuracy and reliability of the proposed statistical linearization technique.

Keywords

Statistical linearization Coupled bending-torsional vibrations Non-linear supports Random loads Monte Carlo simulations 

Notes

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Timoshenko S, Young DH, Weaver WJR (1974) Vibrations problems in engineering. Wiley, New YorkGoogle Scholar
  2. 2.
    Friberg PO (1983) Coupled vibrations of beams-an exact dynamic element stiffness matrix. Int J Num Method Eng 19:479–493CrossRefzbMATHGoogle Scholar
  3. 3.
    Dokumaci E (1987) An exact solution for coupled bending and torsion vibrations of uniform beam having single cross-sectional symmetry. J Sound Vib 119(3):443–449ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Hallauer WL, Liu RYL (1982) Beam bending-torsion dynamic stiffness method for calculation of exact vibrations modes. J Sound Vib 85(1):105–113ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Banerjee JR (1989) Coupled bending-torsional dynamic stiffness matrix for beam elements. Int J Num Methods Eng 28:1283–1298CrossRefzbMATHGoogle Scholar
  6. 6.
    Banerjee JR (1999) Explicit frequency equation and mode shapes of a cantilever beam coupled in bending and torsion. J Sound Vib 224(2):267–281ADSCrossRefGoogle Scholar
  7. 7.
    Hashemi SM, Richard MJ (2000) A Dynamic Finite Element (DFE) method for free vibrations of bending-torsion coupled beams. Aerosp Sci Technol 4:41–55CrossRefzbMATHGoogle Scholar
  8. 8.
    Eslimy-Isfahany SHR, Banerjee JR (2000) Use of generalized mass in the interpretation of dynamic response of bending-torsion coupled beams. J Sound Vib 238(2):295–308ADSCrossRefGoogle Scholar
  9. 9.
    Eslimy-Isfahany SHR, Banerjee JR, Sobey AJ (1996) Response of a bending-torsion coupled beam to deterministic and random loads. J Sound Vib 195(2):267–283ADSCrossRefGoogle Scholar
  10. 10.
    Bishop RED, Cannon SM, Miao S (1989) On coupled bending and torsional vibration of uniform beams. J Sound Vib 131:457–464ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Bercin AN, Tanaka M (1997) Coupled flexural-torsional vibrations of Timoshenko beams. J Sound Vib 207(1):47–59ADSCrossRefGoogle Scholar
  12. 12.
    Tanaka M, Bercin AN (1997) Finite element modeling of the coupled bending and torsional free vibration of uniform beams with an arbitrary cross-section. Appl Math Model 21(6):339–344CrossRefzbMATHGoogle Scholar
  13. 13.
    Banerjee JR, Guo S, Howson WP (1996) Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping. Comput Struct 59:613–621CrossRefzbMATHGoogle Scholar
  14. 14.
    Banerjee JR, Williams FW (1994) Coupled bending torsional dynamic stiffness matrix of an axially loaded Timoshenko beam element. Int J Solids Struct 6:749–762CrossRefzbMATHGoogle Scholar
  15. 15.
    Hashemi MS, Richard MJ (2000) Free vibration analysis of axially loaded bending-torsion coupled beams: a dynamic finite element. Comput Struct 77:711–724CrossRefGoogle Scholar
  16. 16.
    Jun L, Wanyou L, Rongying S, Hongxing H (2004) Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli–Euler beams. Mech Res Commun 31:697–711CrossRefzbMATHGoogle Scholar
  17. 17.
    Jun L, Wanyou L, Rongying S, Hongxing H (2004) Coupled bending and torsional vibration of axially loaded Bernoulli–Euler beams including warping effects. Appl Acoust 65:153–170CrossRefzbMATHGoogle Scholar
  18. 18.
    Jun L, Rongying S, Hongxing H, Xianding J (2004) Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Int J Mech Sci 46:229–320zbMATHGoogle Scholar
  19. 19.
    Adam C (1999) Forced vibrations of elastic bending-torsion coupled beams. J Sound Vib 221(2):273–287ADSCrossRefGoogle Scholar
  20. 20.
    Han H, Cao D, Liu L (2017) Green’s functions for forced vibration analysis of bending-torsion coupled Timoshenko beam. Appl Math Model 45:621–635MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sapountzakis EJ, Tsiatas GC (2007) Flexural-torsional vibrations of beams by BEM. Comput Mech 39:409–417CrossRefzbMATHGoogle Scholar
  22. 22.
    Oguamanam DCD (2003) Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling. Int J Mech Sci 45:963–979CrossRefzbMATHGoogle Scholar
  23. 23.
    Gokdag H, Kopmaz O (2005) Coupled bending and torsional vibration of a beam with in span and tip attachments. J Sound Vib 287:591–610ADSCrossRefGoogle Scholar
  24. 24.
    Burlon A, Failla G, Arena F (2017) Coupled bending and torsional free vibrations of beams with in-span supports and attached masses. Eur J Mech-A/Solids 66:387–411MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Burlon A, Failla G, Arena F (2018) Exact frequency response of two-node coupled bending-torsional beam element with attachments. App Math Model 63:508–537MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Burlon A, Failla G, Arena F (2018) Exact stochastic analysis of coupled bending-torsion beams with in-span supports and masses. Probab Eng Mech 54:53–64CrossRefzbMATHGoogle Scholar
  27. 27.
    Burlon A, Failla G, Arena F (2018) Coupled bending-torsional frequency response of beams with attachments: exact solutions including warping effects. Acta Mechanica 229(6):2445–2475MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gurgoze M (1995) On the eigenfrequencies of a cantilevered beam, with a tip mass and in-span support. Comput Struct 1:85–92ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Wang J, Qiao P (2007) Vibration of beams with arbitrary discontinuities and boundary condition. J Sound Vib 308:12–27ADSCrossRefGoogle Scholar
  30. 30.
    Caddemi S, Caliò I, Cannizzaro F (2015) Influence of an elastic end support on the dynamic stability of Beck’s column with multiple weak sections. Int J Nonlinear Mech 69:14–28CrossRefGoogle Scholar
  31. 31.
    Colajanni P, Falsone G, Recupero A (2009) Simplified formulation of solution for beams on Winkler foundation allowing discontinuities due to loads and constraints. Int J Eng Educ 25(1):75–83Google Scholar
  32. 32.
    Pakdemirli M, Boyaci H (2003) Non-linear vibrations of a simple-simple beam with a non-ideal support in between. J Sound Vib 268:331–341ADSCrossRefzbMATHGoogle Scholar
  33. 33.
    Ghayesh MH, Kazemirad S, Reid T (2012) Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: a general solution procedure. Appl Math Model 36:3299–3311MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Failla G (2016) Stationary response of beams and frames with fractional dampers through exact frequency response functions. J Eng Mech 143:5Google Scholar
  35. 35.
    Papoulis A, Pillai SU (1991) Probability, random variables, and stochastic processes. McGraw Hill, New YorkGoogle Scholar
  36. 36.
    Elishakoff I, Crandall SH (2017) Sixty years of stochastic linearization technique. Meccanica 52(1–2):299–305MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Roberts JB, Spanos PD (1991) Random vibration and statistical linearization. Wiley, New YorkzbMATHGoogle Scholar
  38. 38.
    Socha L (2008) Linearization methods for stochastic dynamic system. Springer, BerlinCrossRefzbMATHGoogle Scholar
  39. 39.
    Spanos PD, Kougioumtzoglou IA (2012) Harmonic wavelets based statistical linearization for response evolutionary power spectrum determination. Prob Eng Mech 27:57–68CrossRefGoogle Scholar
  40. 40.
    Kougioumtzoglou IA (2013) Stochastic joint time-frequency response analysis of nonlinear structural systems. J Sound Vib 332:7153–7173ADSCrossRefGoogle Scholar
  41. 41.
    Kougioumtzoglou IA, Spanos PD (2016) Harmonic wavelets based response evolutionary power spectrum determination of linear and nonlinear oscillators with fractional derivative elements. Int J Non-Linear Mech 80:66–75CrossRefGoogle Scholar
  42. 42.
    Fragkoulis V, Kougioumtzoglou IA, Pantelous A (2016) Statistical linearization of nonlinear structural systems with singular matrices. J Eng Mech 142(9):1–11CrossRefGoogle Scholar
  43. 43.
    Kougioumtzoglou IA, Fragkoulis V, Pantelous A, Pirrotta A (2017) Random vibration of linear and nonlinear structural systems with singular matrices: a frequency domain approach. J Sound Vib 404:84–101ADSCrossRefGoogle Scholar
  44. 44.
    Petromichelakis I, Psaros AF, Kougioumtzoglou IA (2018) Stochastic response determination and optimization of a class of nonlinear electromechanical energy harvesters: a Wiener path integral approach. Prob Eng Mech 53:116–125CrossRefGoogle Scholar
  45. 45.
    Herbert RE (1965) On the stresses in a nonlinear beam subject to random excitation. Int J Solids Struct 1(2):235–242CrossRefGoogle Scholar
  46. 46.
    Seide P (1975) Nonlinear stresses and deflections of beams subjected to random time dependent uniform pressure. J Eng Ind 98:1014–1020CrossRefzbMATHGoogle Scholar
  47. 47.
    Spanos PD, Malara G (2014) Nonlinear random vibrations of beams with fractional derivative elements. J Eng Mech 140:9CrossRefGoogle Scholar
  48. 48.
    Fang J, Elishakoff I (1991) Nonlinear response of a beam under stationary random excitation by improved stochastic linearization method. Appl Math Model 19(2):106–111CrossRefzbMATHGoogle Scholar
  49. 49.
    Honerkamp J (2013) Statistical physics: an advanced approach with applications web-enhanced with problems and solutions. Springer, BerlinzbMATHGoogle Scholar
  50. 50.
    Katsikadelis JT, Tsiatas GC (2003) Large deflection analysis of beams with variable stiffness. Acta Mech 164:1–13CrossRefzbMATHGoogle Scholar
  51. 51.
    Shinozuka M, Deodatis G (1991) Simulation of stochastic processes by spectral representation. Appl Mech 44:191–204MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Civil, Environmental, Energy and Materials Engineering (DICEAM)University “Mediterranea” of Reggio CalabriaReggio CalabriaItaly

Personalised recommendations