scholarly journals Development of a Quasi-Exact Dynamic Finite Element (QDFE) Method For the Free Vibration Analysis of Thin Rectangular Multilayered Plates

2021 ◽  
Author(s):  
Heenkenda Jayasinghe
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Free Vibration ◽  
Coefficient Matrix ◽  
Dynamic Coefficient ◽  
Shape Functions ◽  
The Past ◽  
Dynamic Finite Element ◽  
Multilayered Plates ◽  
Governing Differential Equation

The Dynamic Finite Element (DFE) method is a well-established superconvergent semianalytical method that has been used in the past to investigate the vibration behaviour of various beam-structures. Considered as a viable alternative to conventional FEM for preliminary stage modal analysis, the DFE method has consistently proven that it is capable of producing highly accurate results with a very coarse mesh; a feature that is attributed to the fact that the DFE method uses trigonometric, frequency-dependant shape functions that are based on the exact solution to the governing differential equation as opposed to the polynomial shape functions used in conventional FEM. In the past many researchers have contributed towards building a comprehensive library of DFE models for various line structural elements and configurations, which would serve as the building blocks that would help the DFE method evolve into a fullfledged, versatile tool like conventional FEM in the future. However, thus far a DFE formulation has not been developed for plate problems. Therefore, in this thesis an effort has been made for the first time to develop a DFE formulation for the realm of two-dimensional structural problems by formulating a Quasi-Exact Dynamic Finite Element (QDFE) solution to investigate the free vibration behaviour of thin single- and multi-layered, rectangular plates. As a starting point for this work, Hamiltonian mechanics and the Classical Plate Theory (CPT) are used to develop the governing differential equation for thin plates. Subsequently, a unique quasiexact solution to the governing equation is sought by following a distinct procedure that, to the best of the author‘s knowledge, has never been presented before. Through this procedure, the characteristic equation is re-arranged as the sum of two beam-like expressions and then solved for by applying the quadratic formula. The resulting quasi-exact roots are then exploited to form the trigonometric basis functions, which in turn are used to derive the frequency-dependant shape functions; the characteristic feature of the QDFE method. Once developed, the new QDFE technique is applied to determine the vibration behaviour of thin, isotropic, linearly elastic, rectangular, homogenous plates. Subsequently, it is also employed to formulate a Simplified Layerwise Quasi-Exact Dynamic Finite Element solution for the free vibration of thin, rectangular multilayered plates. In addition, the quasi-exact solution to the plate equation is also utilised to develop a Dynamic Coefficient Matrix (DCM) method to investigate the vibrational characteristics of thin, rectangular, homogeneous plates and thin, rectangular, multilayered plates. The Method of Homogenization is used as an alternative procedure to validate the results from the Simplified Layerwise Quasi-Exact Dynamic Finite Element method and the Simplified Layerwise Dynamic Coefficient Matrix method. The results from both the QDFE and DCM methods are, in general, verified for accuracy against the exact results existing in the open literature and those produced by two in-house developed conventional FEM codes and/or ANSYS® software.

scholarly journals Development of a Quasi-Exact Dynamic Finite Element (QDFE) Method For the Free Vibration Analysis of Thin Rectangular Multilayered Plates

2021 ◽  
Author(s):  
Heenkenda Jayasinghe
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Free Vibration ◽  
Coefficient Matrix ◽  
Dynamic Coefficient ◽  
Shape Functions ◽  
The Past ◽  
Dynamic Finite Element ◽  
Multilayered Plates ◽  
Governing Differential Equation

The Dynamic Finite Element (DFE) method is a well-established superconvergent semianalytical method that has been used in the past to investigate the vibration behaviour of various beam-structures. Considered as a viable alternative to conventional FEM for preliminary stage modal analysis, the DFE method has consistently proven that it is capable of producing highly accurate results with a very coarse mesh; a feature that is attributed to the fact that the DFE method uses trigonometric, frequency-dependant shape functions that are based on the exact solution to the governing differential equation as opposed to the polynomial shape functions used in conventional FEM. In the past many researchers have contributed towards building a comprehensive library of DFE models for various line structural elements and configurations, which would serve as the building blocks that would help the DFE method evolve into a fullfledged, versatile tool like conventional FEM in the future. However, thus far a DFE formulation has not been developed for plate problems. Therefore, in this thesis an effort has been made for the first time to develop a DFE formulation for the realm of two-dimensional structural problems by formulating a Quasi-Exact Dynamic Finite Element (QDFE) solution to investigate the free vibration behaviour of thin single- and multi-layered, rectangular plates. As a starting point for this work, Hamiltonian mechanics and the Classical Plate Theory (CPT) are used to develop the governing differential equation for thin plates. Subsequently, a unique quasiexact solution to the governing equation is sought by following a distinct procedure that, to the best of the author‘s knowledge, has never been presented before. Through this procedure, the characteristic equation is re-arranged as the sum of two beam-like expressions and then solved for by applying the quadratic formula. The resulting quasi-exact roots are then exploited to form the trigonometric basis functions, which in turn are used to derive the frequency-dependant shape functions; the characteristic feature of the QDFE method. Once developed, the new QDFE technique is applied to determine the vibration behaviour of thin, isotropic, linearly elastic, rectangular, homogenous plates. Subsequently, it is also employed to formulate a Simplified Layerwise Quasi-Exact Dynamic Finite Element solution for the free vibration of thin, rectangular multilayered plates. In addition, the quasi-exact solution to the plate equation is also utilised to develop a Dynamic Coefficient Matrix (DCM) method to investigate the vibrational characteristics of thin, rectangular, homogeneous plates and thin, rectangular, multilayered plates. The Method of Homogenization is used as an alternative procedure to validate the results from the Simplified Layerwise Quasi-Exact Dynamic Finite Element method and the Simplified Layerwise Dynamic Coefficient Matrix method. The results from both the QDFE and DCM methods are, in general, verified for accuracy against the exact results existing in the open literature and those produced by two in-house developed conventional FEM codes and/or ANSYS® software.

scholarly journals A dynamic coefficient matrix method for the free vibration of thin rectangular isotropic plates

2021 ◽  
Author(s):  
Supun Jayasinghe ◽  
Seyed M. Hashemi
Keyword(s):  
Exact Solution ◽  
Free Vibration ◽  
Free Vibration Analysis ◽  
Flexural Vibration ◽  
Coefficient Matrix ◽  
Dynamic Coefficient ◽  
Rectangular Plates ◽  
Various Boundary Conditions ◽  
Unique Method ◽  
Benchmark Solutions

The free flexural vibration of thin rectangular plates is revisited. A new, quasi-exact solution to the governing differential equation is formed by following a unique method of decomposing the governing equation into two beam-like expressions. Using the proposed quasi-exact solution, a Dynamic Coefficient Matrix (DCM) method is formed and used to investigate the free lateral vibration of a rectangular thin plate, subjected to various boundary conditions. Exploiting a special code written on MATLAB, the flexural natural frequencies of the plate are found by sweeping the frequency domain in search of specific frequencies that yield a zero determinant. Results are validated extensively both by the limited exact results available in the open literature and by numerical studies using ANSYS and in-house conventional FEM programs using both 12- and 16-DOF plate elements. The accuracy of all methods for lateral free vibration analysis is assessed and critically examined through benchmark solutions. It is envisioned that the proposed quasi-exact solution and the DCM method will allow engineers to more conveniently investigate the vibration behaviour of two-dimensional structural components during the preliminary design stages, before a detailed design begins.

scholarly journals New frequency-dependent trigonometric interpolation functions for the dynamic finite element analysis of thin rectangular plates

2021 ◽  
Author(s):  
Supun Jayasinghe ◽  
Seyed M. Hashemi
Keyword(s):  
Finite Element ◽  
Trigonometric Interpolation ◽  
Rectangular Plates ◽  
Structural Components ◽  
Shape Functions ◽  
Element Analysis ◽  
Frequency Dependent ◽  
Analysis And Design ◽  
Dynamic Finite Element ◽  
Semianalytical Method

The Dynamic Finite Element (DFE) formulation is a superconvergent, semianalytical method used to perform vibration analysis of structural components during the early stages of design. It was presented as an alternative to analytical and numerical methods that exhibit various drawbacks, which limit their applicability during the preliminary design stages. The DFE method, originally developed by the second author, has been exploited heavily to study the modal behaviour of beams in the past. Results from these studies have shown that the DFE method is capable of arriving at highly accurate results with a coarse mesh, thus, making it an ideal choice for preliminary stage modal analysis and design of structural components. However, the DFE method has not yet been extended to study the vibration behaviour of plates. Thus, the aim of this study is to develop a set of frequency-dependent, trigonometric shape functions for a 4-noded, 4-DOF per node element as a basis for developing a DFE method for thin rectangular plates. To this end, the authors exploit a distinct quasi-exact solution to the plate governing equation and this solution is then used to derive the new, trigonometric basis and shape functions, based on which the DFE method would be developed.

Vibration Analysis of Axially Loaded Beams Including Rotary Inertia: An Exact Dynamic Finite Element

2004 ◽  
Author(s):  
Seyed M. Hashemi
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Dynamic Stiffness ◽  
Vibrational Analysis ◽  
Single Frequency ◽  
Rotary Inertia ◽  
Shape Functions ◽  
Bending Vibrations ◽  
Dynamic Finite Element ◽  
Axially Loaded

An ‘exact’ basis function Dynamic Finite Element (DFE) for the free vibrational analysis of axially loaded beams and assemblages composed of beams is presented. The shear deformation is neglected but the Rotary Inertia (RI) effects are taken into consideration. The dynamic trigonometric shape functions for bending vibrations of an axially loaded uniform beam element are first derived in an exact sense. Then, exploiting the Principle of Virtual Work together with the nodal approximations of variables based on these dynamic shape functions, leads to a single frequency dependent Dynamic Stiffness Matrix (DSM) that represents both mass and stiffness properties. A Wittrick-Williams algorithm, based on a Sturm sequence root counting technique, is then used as the solution method. The application of the theory is demonstrated by an illustrative example of cantilever beam where the influence of Rotary Inertia (RI) effect and different axial loads on the natural frequencies of the system is demonstrated by numerical results.

Free vibration analysis of a rotating double tapered beam with flexible root via differential quadrature method

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mustafa Tolga Tolga Yavuz ◽  
İbrahim Özkol
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Operating Conditions ◽  
Design Parameters ◽  
Quadrature Method ◽  
Content Type ◽  
Uniform Beam ◽  
Governing Differential Equation

Purpose This study aims to develop the governing differential equation and to analyze the free vibration of a rotating non-uniform beam having a flexible root and setting angle for variations in operating conditions and structural design parameters. Design/methodology/approach Hamiltonian principle is used to derive the flapwise bending motion of the structure, and the governing differential equations are solved numerically by using differential quadrature with satisfactory accuracy and computation time. Findings The results obtained by using the differential quadrature method (DQM) are compared to results of previous studies in the open literature to show the power of the used method. Important results affecting the dynamics characteristics of a rotating beam are tabulated and illustrated in concerned figures to show the effect of investigated design parameters and operating conditions. Originality/value The principal novelty of this paper arises from the application of the DQM to a rotating non-uniform beam with flexible root and deriving new governing differential equation including various parameters such as rotary inertia, setting angle, taper ratios, root flexibility, hub radius and rotational speed. Also, the application of the used numerical method is expressed clearly step by step with the algorithm scheme.

FREE VIBRATION ANALYSIS OF ROTATING FLEXIBLE BEAMS BY USING THE FOURIER p-VERSION OF THE FINITE ELEMENT METHOD

2005 ◽  
Vol 02 (02) ◽  
pp. 255-269 ◽  
Author(s):  
S. M. HAMZA-CHERIF
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
P Element ◽  
The Finite Element Method ◽  
Shape Functions ◽  
Flexible Beams ◽  
Element Method

A p-version of the finite element method is applied to free vibration analysis of rotating beams in conjunction with the modeling dynamic method using the arc-length stretch deformation. In this study the flexible and the rigid body degrees of freedom (d.o.f.) are supposedly uncoupled, the linear equations of motion are derived for flapwise and chordwise bending with the integration of the gyroscopic effect. The hybrid displacements are expressed as the combination of the in-plane and out-of-plane shape functions. These are formulated in terms of linear and cubic polynomial functions used generally in FEM in addition to a variable number of trigonometric shape functions which represent the internal d.o.f. for the rotating flexible beams. The convergence properties of the rotating beam Fourier p-element and the influence of angular speed, boundary conditions and slenderness ratio on the dynamic response are studied. It is shown that using this element the order of the resulting matrices in the FEM is considerably reduced leading to a significant decrease in computational effort.

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