# Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity
###### Abstract  
In many practical engineering situations, a source of vibrations may excite a large and flexible structure such as a ship's deck, an aeroplane fuselage, a satellite antenna, a wall panel. To avoid transmission of the vibration and structure-borne sound, radial or polar periodicity may be used. In these cases, numerical approaches to study free and forced wave propagation close to the excitation source in polar coordinates are desirable. This is the paper's aim, where a numerical method based on Floquet-theory and the FE description of a finite slice of the radial periodic structure is presented and verified. Only a small slice of the structure is analysed, which is approximated using piecewise Cartesian segments. Wave characteristics in each segment are obtained by the theory of wave propagation in periodic Cartesian structures and Finite Element analysis, while wave amplitude change due to the changes in the geometry of the slice is accommodated in the model assuming that the energy flow through the segments is the same. Forced response of the structure is then evaluated in the wave domain. Results are verified for an infinite isotropic thin plate excited by a point harmonic force. A plate with a periodic radial change of thickness is then studied. Free waves propagation are shown, and the forced response in the nearfield is evaluated, showing the validity of the method and the computational advantage compared to FE harmonic analysis for infinite structures. periodic structures; polar coordinates; wave propagation; forced response of plates and shells; finite element analysis; unbounded structures +
Footnote †: journal: Applied Sciences  
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Footnote 1: Departmento di Ingegneria e Architettura, Universita di Parma, 43124 Parma, Italy; [EMAIL_ADDRESS]  
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Footnote 1: Departmento di Ingegneria e Architettura, Universita di Parma, 43124 Parma, Italy; [EMAIL_ADDRESS]  
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Footnote 2: Departmento di Ingegneria e Architettura, Universita di Parma, 43124 Parma, Italy; [EMAIL_ADDRESS]  
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Footnote 3: Departmento di Ingegneria e Architettura, Universita di Parma, 43124 Parma, Italy; rinaldo. neering structures has been devoted to developing theoretical and numerical approaches due to the importance of understanding and characterising wave propagation behaviour. [PERSON] [9] and his coworkers at the University of Southampton [10] gave a substantial contribution to methods for predicting and analysing wave motion in periodic engineering structures. Among the numerical approaches proposed, those based on Finite Element theory have shown large versatility and applicability to modelling physical structures. Two different main approaches are typically applied. The firsts implement absorbing boundary conditions/layers that mimick the infinity domains through artificial non-reflective boundaries [11; 12]; the advantage of this approach relies on the capability of standard FEA to model waveguides as a part of more complex structures. However, the computational cost remains a common issue in such an analysis, especially at high frequencies, when the mesh must be refined to reach sufficient accuracy in studies of the performance of a waveguide. It implies a substantial increase in the number of DOFs of the model, which leads to prohibitive computational time at higher frequency. The seconds are numerical methods based on FE analysis of a unit cell of the structures and the theory of wave propagation in periodic structures, e.g., [13; 14; 15]. Compared to FE modelling with absorbing boundary conditions/layers, these numerical methods allow high accuracy up to high frequency at very low computation cost, and they can be the preferable choice when the requirement is the prediction of wave motion in 1-dimensional and 2-dimensional single waveguides, e.g., beam-like structures, panels and shells, cylindrical waveguides, etc. Some FE commercial software have recently implemented new specific modules for studying Cartesian periodic structures exploiting Bloch-Floquet theory.