Improved eigenfrequencies and eigenmodes in free vibration analysis

Wiberg, Nils‐Erik; Baušys, Romualdas; Hager, Patrik

doi:10.1016/s0045-7949(98)00265-x

Cited by 19 publications

(9 citation statements)

References 25 publications

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“…In the finite element computation, there are superconvergent points (Zienkiewicz et al, 2015) with a higher convergence order compared with the current mesh solution. The accuracy of the current finite element solution can be improved by using superconvergent points combined with element patching and high-order shape function interpolation technology, and a global superconvergent solution (Wang et al, 2018;Wiberg et al, 1999aWiberg et al, , 1999b) can be obtained. In this study, the finite element solution of the vibration mode (displacement) under the current mesh is obtained for the free vibration problem of a circularly curved beam, and the superconvergent solution of the vibration mode is obtained using the superconvergent patch recovery method.…”

Section: Local Mesh Refinement Techniques and Proceduresmentioning

confidence: 99%

“…The accuracy of the current finite element solution can be improved by using superconvergent points combined with element patching and high-order shape function interpolation technology, and a global superconvergent solution (Wang et al. , 2018; Wiberg et al ., 1999a, 1999b) can be obtained. In this study, the finite element solution of the vibration mode (displacement) under the current mesh is obtained for the free vibration problem of a circularly curved beam, and the superconvergent solution of the vibration mode is obtained using the superconvergent patch recovery method.where, P=false[center1x…xpfalse] represents the given function vector, and a=[a1a2…am]T denotes the undetermined coefficient vector, which will be determined through that the value of Equation (12) at the node is the same as the node displacement.…”

Section: Local Mesh Refinement Techniques and Proceduresmentioning

confidence: 99%

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Free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams

Wang

2023

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PurposeThis study aimed to solve the engineering problem of free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams. The accurate identification of the crack damage depth, number and location depends on high-precision frequency and vibration mode solutions; therefore, it is critical to obtain these reliable solutions. The high-precision finite element method for the free vibration of cracked beams needs to be developed to grasp and control error information in the conventional solutions and the non-uniform mesh generation near the cracks. Moreover, the influence of multi-crack damage on the natural frequency and vibration mode of a circularly curved beam needs to be detected.Design/methodology/approachA scheme for cross-sectional damage defects in a circularly curved beam was established to simulate the depth, location and the number of multiple cracks by implementing cross-section reduction induced by microcrack damage. In addition, the h-version finite element mesh adaptive analysis method of the Timoshenko beam was developed. The superconvergent solution of the vibration mode of the cracked curved beam was obtained using the superconvergent patch recovery displacement method to determine the finite element solution. The superconvergent solution of the frequency was obtained by computing the Rayleigh quotient. The superconvergent solution of the eigenfunction was used to estimate the error of the finite element solution in the energy norm. The mesh was then subdivided to generate an improved mesh based on the error. Accordingly, the final optimised meshes and high-precision solution of natural frequency and mode shape satisfying the preset error tolerance can be obtained. Lastly, the disturbance behaviour of multi-crack damage on the vibration mode of a circularly curved beam was also studied.FindingsNumerical results of the free vibration and damage disturbance of cracked curved beams with cracks were obtained. The influences of crack damage depth, crack damage number and crack damage distribution on the natural frequency and mode of vibration of a circularly curved beam were quantitatively analysed. Numerical examples indicate that the vibration mode and frequency of the beam would be disturbed in the region close to the crack damage, and a greater crack depth translates to a larger frequency change. For multi-crack beams, the number and distribution of cracks also affect the vibration mode and natural frequency. The adaptive method can use a relatively dense mesh near the crack to adapt to the change in the vibration mode near the crack, thus verifying the efficacy, accuracy and reliability of the method.Originality/valueThe proposed combination of methodologies provides an extremely robust approach for free vibration of beams with cracks. The non-uniform mesh refinement in the adaptive method can adapt to changes in the vibration mode caused by crack damage. Moreover, the proposed method can adaptively divide a relatively fine mesh at the crack, which is applied to investigating free vibration under various curved beam angles and crack damage distribution conditions. The proposed method can be extended to crack damage detection of 2D plate and shell structures and three-dimensional structures with cracks.

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Section: Local Mesh Refinement Techniques and Proceduresmentioning

confidence: 99%

Section: Local Mesh Refinement Techniques and Proceduresmentioning

confidence: 99%

Free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams

Wang

2023

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“…It originates [14,17] from the idea of fitting an improved stress distribution field to a set of so-called superconvergent points, when they exist, where stresses are calculated with a higher accuracy. Wiberg et al [18] fitted the polynomial to displacements instead of stresses (SPRD) in order to improve the calculated value of natural frequencies, which depend not only on the displacement derivatives, but also on the displacements themselves.…”

Section: Introductionmentioning

confidence: 99%

“…(R e : element rotation matrix) Calculate ϕ 3 , ϕ r in Equations ( 6) and ( 19) Calculate K p , M p in Equations ( 17) and (18) Obtain ϕ p as the first eigenvector of Equation ( 16) Evaluate V ce , T ce by applying Equations ( 24)- (26) The element-corrected quadratic forms of the whole structure can be collected in Rayleigh's quotient to obtain an improved value of the structure's natural frequency ω c…”

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confidence: 99%

Improved FEM Natural Frequency Calculation for Structural Frames by Local Correction Procedure

Urruzola,

Garmendia

2024

Buildings

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The accurate calculation of natural frequencies is important for vibration and earthquake analyses of structural frames. For this purpose, it is necessary to discretize each beam or column of the frame into one or more smaller elements. The required number of elements per member increases when the frame’s modal shapes have wavelengths similar to the beam lengths. This paper presents a method that reduces the number of elements needed for a precise calculation. This is achieved by implementing a straightforward local correction to the kinetic and elastic energy of certain elements, resulting in a substantial decrease in error. The validity of this method is demonstrated through a range of examples, from simple canonical cases to more realistic ones. Additionally, the paper discusses the unique features of this method and examines its relationship with other approaches.

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“…Nevertheless, a sequence of solutions on progressively more reÿned nested grids starting from a very coarse one can constitute the basis for an ultimate e cient and accurate partial eigensolution to the problem of interest. An alternative way to ameliorate the coarse grid solution is proposed by Wilberg et al [15], who suggest to improve the initial eigenpair estimate by a preconditioned conjugate gradient (PCG) via the super-convergent recovery technique. This approach relies on the existence of certain mesh points, named super-convergent points, where the solution has an accuracy superior to that of the original (relatively coarse) FE mesh.…”

Section: Introductionmentioning

confidence: 99%

3-D nested eigenanalysis on finite element grids

Pini¹,

Gambolati

2006

Commun. Numer. Meth. Engng.

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SUMMARYThe computation of several of the leftmost eigenpairs of generalized symmetric eigenproblems in 3-D ÿnite element (FE) discretizations is addressed using the nested iteration de ation accelerated conjugate gradient (NI-DACG) method previously studied and improved by the authors in a 2-D framework. Numerical results from the ow (di usion type) equation integrated over tetrahedral FE shows speedups relative to DACG ranging between 2 up to more than 10 depending on the aspect ratio of the elements used. NI-DACG also compares favourably with the performance of IRAM, i.e. the implicit restarted Arnoldi method, implemented in the eigenvalue package ARPACK.

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Improved eigenfrequencies and eigenmodes in free vibration analysis

Cited by 19 publications

References 25 publications

Free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams

Free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams

Improved FEM Natural Frequency Calculation for Structural Frames by Local Correction Procedure

3-D nested eigenanalysis on finite element grids

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