A Meshless Dynamic Finite Element for Beam Vibrations including Rotary Inertia

Hashemi, Soheil; Pereira, Daniel A.

doi:10.4203/ccp.79.127

“…Since its inception, the DFE method has been used in beam, beam-like, and blade dynamic, stability and vibration modelling and analysis. Hashemi and his co-workers have extensively studied the free vibration of isotropic, sandwich, composite and thin-walled beams subjected to various loading configurations, using the DFE method (see, for example, [107][108][109][110][111][112][113].…”

Section: Methods Of Vibration Analysismentioning

confidence: 99%

“…Hashemi and Pereira [112]. Furthermore, Hashemi and Richard [113] also conducted a vibration analysis on an axially loaded bending-torsion geometrically coupled beam using the DFEmethod.…”

Section: Methods Of Vibration Analysismentioning

confidence: 99%

“…From the existing beam vibration results [107][108][109][110][111][112][113], it is evident that the Dynamic Finite Element (DFE) formulation is a highly-convergent, hybrid, frequency-dependent, semi-analytical solution method which is more accurate than the conventional Finite Element Method and, unlike the analytical methods discussed above, is adaptable to many complex configurations. Some advantages of the DFE method, as evident from its previous applications, are highlighted below.…”

Section: Methods Of Vibration Analysismentioning

confidence: 99%

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

Jayasinghe¹

2021

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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.

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“…Since its inception, the DFE method has been used in beam, beam-like, and blade dynamic, stability and vibration modelling and analysis. Hashemi and his co-workers have extensively studied the free vibration of isotropic, sandwich, composite and thin-walled beams subjected to various loading configurations, using the DFE method (see, for example, [107][108][109][110][111][112][113].…”

Section: Methods Of Vibration Analysismentioning

confidence: 99%

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

Jayasinghe¹

2021

Preprint

0

View full text Add to dashboard Cite

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.

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“…Borneman and Hashemi [4] developed a DFE for the free vibration analysis of a bendingtorsion coupled laminated composite wing beam. The effects of shear deformation and rotary inertia were neglected in this study, but were accounted for in another study by Hashemi and Pereira [41].…”

Section: Introductionmentioning

confidence: 99%

Coupled flexural - torsional vibration and stability analysis of pre-loaded beams using conventional and dynamic finite element methods

Jayasinghe¹

2021

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Dynamic Finite Element (DFE) and conventional finite element formulations are developed to study the flexural - torsional vibration and stability of an isotropic, homogeneous and linearly elastic pre-loaded beam subjected to an axial load and end-moment. Various classical boundary conditions are considered. Elementary Euler - Bernoulli bending and St. Venant torsion beam theories were used as a starting point to develop the governing equations and the finite element solutions. The nonlinear Eigenvalue problem resulted from the DFE method was solved using a program code written in MATLAB and the natural frequencies and mode shapes of the system were determined form the Eigenvalues and Eigenvectors, respectively. Similarly, a linear Eigenvalue problem was formulated and solved using a MATLAB code for the conventional FEM method. The conventional FEM results were validated against those available in the literature and ANSYS simulations and the DFE results were compared with the FEM results. The results confirmed that tensile forces increased the natural frequencies, which indicates beam stiffening. On the contrary, compressive forces reduced the natural frequencies, suggesting a reduction in beam stiffness. Similarly, when an end-moment was applied the stiffness of the beam and the natural frequencies diminished. More importantly, when a force and end-moment were acting in combination, the results depended on the direction and magnitude of the axial force. Nevertheless, the stiffness of the beam is more sensitive to the changes in the magnitude and direction of the axial force compared to the moment. A buckling analysis of the beam was also carried out to determine the critical buckling end-moment and axial compressive force.

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A Meshless Dynamic Finite Element for Beam Vibrations including Rotary Inertia

Cited by 3 publications

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

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

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

Coupled flexural - torsional vibration and stability analysis of pre-loaded beams using conventional and dynamic finite element methods

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