Analysis of free vibration of elastically supported functionally graded annular plates


Functionally graded materials have a wide range of applications in the fields of civil, aeronautical, mechanical engineering among others. They are mixtures of ceramics and metals which ensure possibility continuous change of selected material parameters in appropriate directions of structural members.  For the structural elements to function correctly without failures, the variation direction of the properties of the materials is very crucial and hence needs to be taken into consideration.

Functionally graded plates in most cases work together with additional elements, such as the elastic supports, which may influence their mechanical behavior. Therefore, to manage such influence, there is a need for a comprehensive understanding of the distribution of material properties.  The knowledge of effect of material properties distribution on dynamics and stability of plates is very important because it allows us to predict and shape the frequency of plates and to find their optimal parameters.

The research work was conducted at the Bialystok University of Technology, Faculty of Engineering Management in Poland by Dr. Krzysztof Kamil Żur who proposed the use of quasi-Green’s function method for the analysis of free vibrations of functionally graded annular plates in various configurations. His work is published in the journal, Composites Part B: Engineering.

The types of annular plates used by Dr. Żur had clamped, simply supported and free inner edges and a free outer edge, and rested on elastic ring supports. The classical plate theory was used by the author to present the free axisymmetric and non-axisymmetric vibrations of plates. Unlike in the previous studies where the boundary value problem of the free vibrations was solved using complicated techniques, here a simple quasi-Green’s function method is employed.

The obtained solution is very useful for finding multiparametric non-linear characteristics equations of the functionally graded annular plates. This can be used for both classical and non-classical boundary conditions because any number of additional discrete elements does not affect the size of the characteristic matrices of plates in any way.

The author observed that the non-linear characteristic equations obtained by the proposed method depended functionally on a variety of plate and material parameters. They included the volume fraction index, the core radius, the total number of the nodal lines as well as the position and properties of diverse kinds of additional elements.

The technique is much simpler as compared to previous methods. For instance, in computing the above characteristics, neither scaling factors nor the formulation of new boundary value problems is required. Furthermore, well understanding of the effects induced by the discrete elements and distribution material parameters can be applied to predict the dynamic behaviors of such materials thus preventing the occurrence of the resonance phenomenon. Additionally, the presented method and obtained results are the first important step in the design of safe and rational active vibration control system of structural elements.

Free vibration analysis of elastically supported functionally graded annular plates via quasi-Green's function method-Advances in Engineering

About the author

Krzysztof Kamil Żur, PhD, received the MSc degree in mechatronics, robotics, and automation engineering (2010) and the PhD degree in mechanical engineering (2017) from the Faculty of Mechanical Engineering, Bialystok University of Technology, Poland. Currently, he works as an assistant professor at the Faculty of Engineering Management, Bialystok University of Technology.

His research interests include:

  • Vibration and Buckling of Macro, Micro and Nano Composite Beams & Plates
  • Structural Dynamics
  • Structural Mechanics
  • Nonlocal Mechanics
  • Functionally Graded Materials
  • Porous Materials
  • Mathematical Modelling of Discrete-Continuous Dynamical Systems
  • Boundary Value Problem
  • Analytical and Numerical Methods
  • Finite Element Method
  • Partial and Ordinary Differential Equations
  • Integral Equations


Żur, K. (2018). Free vibration analysis of elastically supported functionally graded annular plates via quasi-Green’s function method. Composites Part B: Engineering, 144, 37-55.

Go To Composites Part B, Engineering

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