Trigonometric collocation for computation of steady state response of cracked structures

  • Bakeer K. Bakeer

Student thesis: Master's Thesis


Development of vibration-based structural health monitoring techniques requires the use of various computational methods to predict dynamic responses of damaged structures. The method described in this thesis can be used for computation of steady state harmonic responses for structures with fatigue cracks and may have several advantages over alternative techniques. Such advantages include ease of implementation of the method and time saving in computing the steady state response. The steady state response of the system at a given number of time points distributed over one vibration period is represented in terms of Fourier series containing higher frequency harmonics. Equations of motion are formulated in the form that allows for computation of Fourier coefficients for all terms in the series. Iterative procedure is used for determining the time of stiffness change in order to capture bilinear dynamic behavior. The objectives of the study are to implement and validate the method of trigonometric collocation to compute the steady state response of cracked structures with large number of degrees of freedom. Also, the accuracy, efficiency and time savings of this method over direct integration will be assessed. To achieve these objectives, the study included applying the method to a small beam model with a breathing crack represented by 3 nodes. The purpose was to test the developed algorithm of the method by verifying the response of all DOF at all the nodes of the model. Then, the algorithm was applied to large models with crack represented by 3 nodes and 5 nodes. In addition to that, an experiment was conducted with a rectangular cross-section cantilever beam containing a fatigue crack to check the response of the beam experimentally. Also, the response obtained for the models from finite element software Abaqus was compared to the response obtained by the method. The results showed that the method is able to compute the steady state response of a cracked beam, and it yields reasonably accurate approximations for relatively small, localized nonlinearities It takes less time to compute the steady state response compared to the direct integration in Abaqus. Also, the results showed amplification of even harmonics observed when excitation frequency is in vicinity of 1=2 of the natural frequency. This behavior is typical for systems with bilinear stiffness, and the method was able to capture it. On the other hand, several issues were encountered that will need to be addressed in order to provide additional computational advantages. It is concluded that trigonometric collocation technique has a potential to become a valuable computational tool for calculating approximate responses of structures with bilinear stiffness characteristics.
Date of Award2015
Original languageAmerican English
SupervisorOleg Shiryayev (Supervisor)


  • Applied sciences
  • Breathing crack
  • Computational method
  • Finite element analysis
  • Trigonometric collocation
  • Mechanical engineering
  • 0548:Mechanical engineering

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