Deterministic and Random Response Evaluation of a Straight Beam with Nonlinear Boundary Conditions

  • Zhanchao Huang
  • Yong Wang
  • Weidong Zhu
  • Zhilong HuangEmail author
Original Paper



Deterministic and random responses of one-dimensional continuous structures with nonlinear boundary conditions are evaluated in this article. Dynamic behaviors (i.e., the time-domain response for the deterministic case and mean-square response for the stochastic case) of a straight beam with a cubic nonlinear elastic boundary is investigated numerically.


The nonlinear vibration problem is discretized by a new global spatial discretization method, and its responses are compared with those from the finite element method and assumed mode method.


The effects of selected various types of linear homogeneous boundary conditions and the number of expansion modes on the accuracy of the responses under harmonic excitation are discussed. Detailed discussions on the influence of nonlinear boundary conditions on dynamic behaviors uncover a counter-intuitive phenomenon: with the increase of the nonlinear spring stiffness, the stationary mean-square displacement of the mid-point descends first and then ascends slightly.


Numerical results show that the present procedure possesses high precision for evaluating harmonic responses for both low-frequency and high-frequency excitations. Furthermore, it can be directly generalized to two-dimensional continuous systems and various types of nonlinear boundary conditions.


Straight beam Nonlinear boundary condition Deterministic and random responses New global spatial discretization method Assumed mode method 



This study was supported by the National Natural Science Foundation of China under Grant Nos. 11532011, 11872328, 11621062, and 11772100, and the Fundamental Research Funds for the Central Universities under Grant No. 2018FZA4025.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Amann H (1976) Nonlinear elliptic equations with nonlinear boundary conditions. In: Eckhaus W (ed) North-Holland Mathematics Studies, vol 21. North-Holland, Amsterdam, pp 43–63. CrossRefGoogle Scholar
  2. 2.
    Arrieta JM, Rodríguez-Bernal A (2004) Localization on the boundary of blow-up for reaction–diffusion equations with nonlinear boundary conditions. Commun Partial Differ Equ 29:1127–1148. MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Banks HT, Inman DJ (1991) On damping mechanisms in beams. J Appl Mech 58:716–723. CrossRefzbMATHGoogle Scholar
  4. 4.
    Cai GQ, Zhu WQ (2016) Elements of stochastic dynamics. World Scientific, SingaporeCrossRefGoogle Scholar
  5. 5.
    Carrera E, Giunta G, Petrolo M (2011) Beam structures: classical and advanced theories. Wiley, New YorkCrossRefGoogle Scholar
  6. 6.
    Carrera E, Zozulya V (2019) Carrera unified formulation (CUF) for the micropolar beams: analytical solutions. Mech Adv Mater Struct. CrossRefGoogle Scholar
  7. 7.
    Courant R, Hilbert D (1953) Methods of mathematical physics, vol 1. Wiley, New YorkzbMATHGoogle Scholar
  8. 8.
    Craig RR, Kurdila AJ (2011) Fundamentals of structural dynamics. Wiley, New YorkzbMATHGoogle Scholar
  9. 9.
    Evensen DA (1968) Nonlinear vibrations of beams with various boundary conditions. AIAA J 6:370–372. CrossRefzbMATHGoogle Scholar
  10. 10.
    Fang J, Elishakoff I, Caimi R (1995) Nonlinear response of a beam under stationary random excitation by improved stochastic linearization method. Appl Math Model 19:106–111. CrossRefzbMATHGoogle Scholar
  11. 11.
    Feireisl E (1993) Nonzero time periodic solutions to an equation of Petrovsky type with nonlinear boundary conditions : slow oscillations of beams on elastic bearings. Annali della Scuola Normale Superiore di Pisa Classe di Scienze 20:133–146MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kamali Eigoli A, Ahmadian MT (2011) Nonlinear vibration of beams under nonideal boundary conditions. Acta Mech 218:259–267. CrossRefzbMATHGoogle Scholar
  13. 13.
    Kim C-G, Liang Z-P, Shi J-P (2015) Existence of positive solutions to a Laplace equation with nonlinear boundary condition. Z Angew Math Phys 66:3061–3083. MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Knowles JK (1968) On the dynamic response of a beam to a randomly moving load. J Appl Mech 35:1–6. CrossRefzbMATHGoogle Scholar
  15. 15.
    Ma TF, da Silva J (2004) Iterative solutions for a beam equation with nonlinear boundary conditions of third order. Appl Math Comput 159:11–18. MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mathews JH, Fink KD (1998) Numerical methods using MATLAB. Simon and Schuster Inc, New YorkGoogle Scholar
  17. 17.
    McEwan MI, Wright JR, Cooper JE, Leung AYT (2001) A combined modal/finite element analysis technique for the dynamic response of a non-linear beam to harmonic excitation. J Sound Vib 243:601–624. CrossRefGoogle Scholar
  18. 18.
    Meirovitch L (1975) Elements of vibration analysis. McGraw-Hill, New YorkzbMATHGoogle Scholar
  19. 19.
    Meirovitch L (1997) Principles and techniques of vibrations. Prentice Hall, Englewood CliffsGoogle Scholar
  20. 20.
    Pathak M, Joshi P (2019) High-order compact finite difference scheme for euler–bernoulli beam equation: theory and applications. ICHSA 2018:357–370. CrossRefGoogle Scholar
  21. 21.
    Rao SS (2005) The finite element method in engineering. Elsevier Science, AmsterdamzbMATHGoogle Scholar
  22. 22.
    Roncen T, Lambelin JP, Sinou JJ (2019) Nonlinear vibrations of a beam with non-ideal boundary conditions and stochastic excitations—experiments, modeling and simulations. Commun Nonlinear Sci Numer Simul 74:14–29. MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ryu BJ, Kim HJ, Kim YS (2013) Dynamic response and vibration of a cantilevered beam under an accelerated moving mass. Adv Mater Res 711:305–311. CrossRefGoogle Scholar
  24. 24.
    Spanos PD, Malara G (2017) Random vibrations of nonlinear continua endowed with fractional derivative elements. Procedia Eng 199:18–27. CrossRefGoogle Scholar
  25. 25.
    Tadmor E (2012) A review of numerical methods for nonlinear partial differential equations. Bull New Ser Am Math Soc. CrossRefzbMATHGoogle Scholar
  26. 26.
    Tao LN (1981) Heat conduction with nonlinear boundary condition. Zeitschrift für angewandte Mathematik und Physik ZAMP 32:144–155. MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wang YQ, Huang XB, Li J (2016) Hydroelastic dynamic analysis of axially moving plates in continuous hot-dip galvanizing process. Int J Mech Sci 110:201–216. CrossRefGoogle Scholar
  28. 28.
    Watanabe T (1978) Forced vibration of continuous system with nonlinear boundary conditions. J Mech Design 100:487–491. CrossRefGoogle Scholar
  29. 29.
    Wu K, Zhu WD (2017) A new global spatial discretization method for two-dimensional continuous systems. In: ASME 2017 international design engineering technical conferences and computers and information in engineering conference.
  30. 30.
    Wu K, Zhu WD, Fan W (2017) On a comparative study of an accurate spatial discretization method for one-dimensional continuous systems. J Sound Vib 399:257–284. CrossRefGoogle Scholar
  31. 31.
    Zhu WD, Ren H (2013) An accurate spatial discretization and substructure method with application to moving elevator cable-car systems—part I: methodology. J Vib Acoust. CrossRefGoogle Scholar

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Authors and Affiliations

  • Zhanchao Huang
    • 1
  • Yong Wang
    • 1
  • Weidong Zhu
    • 2
  • Zhilong Huang
    • 1
    Email author
  1. 1.State Key Laboratory of Fluid Power and Mechatronic System, Department of Engineering MechanicsZhejiang UniversityHangzhouPeople’s Republic of China
  2. 2.Department of Mechanical EngineeringUniversity of Maryland, Baltimore CountyBaltimoreUSA

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