Nonlinear Vibration Analysis of Uniform and Functionally Graded Beams with Spectral Chebyshev Technique and Harmonic Balance Method

Dedekoy D., CİĞEROĞLU E., Bediz B.

40th Conference and Exposition on Structural Dynamics (IMAC), Florida, United States Of America, 7 - 10 February 2022, pp.119-129 identifier identifier

  • Publication Type: Conference Paper / Full Text
  • Doi Number: 10.1007/978-3-031-04086-3_18
  • City: Florida
  • Country: United States Of America
  • Page Numbers: pp.119-129
  • Keywords: Nonlinear vibration, Functionally graded beam, Forced response, Harmonic balance method, AMPLITUDE FREE-VIBRATIONS
  • Middle East Technical University Affiliated: Yes


In this paper, nonlinear forced vibrations of uniform and functionally graded Euler-Bernoulli beams with large deformation are studied. Spectral and temporal boundary value problems of beam vibrations do not always have closed-form analytical solutions. As a result, many approximate methods are used to obtain the solution by discretizing the spatial problem. Spectral Chebyshev technique (SCT) utilizes the Chebyshev polynomials for spatial discretization and applies Galerkin's method to obtain boundary conditions and spatially discretized equations of motions. Boundary conditions are imposed using basis recombination into the problem, and as a result of this, the solution can be obtained to any linear boundary condition without the need for re-derivation. System matrices are generated with the SCT, and natural frequencies and mode shapes are obtained by eigenvalue problem solution. Harmonic balance method (HBM) is used to solve nonlinear equation of motion in frequency domain, with large deformation nonlinearity. As a result, a generic method is constructed to solve nonlinear vibrations of uniform and functionally graded beams in frequency domain, subjected to different boundary conditions.