Method For Free Vibration Analysis Research Articles

A meshfree local weak-form method for free vibration analysis of an open laminated cylindrical shell with elliptical section

In this paper, for the free vibration analysis of an open laminated cylindrical shell with elliptical section (OLCSES), under different elastic boundary conditions (BCs), a new meshfree Tchebychev-radial point interpolation method (TRPIM) shape function using the basis function consisted of radial basis and Tchebychev polynomial basis, is proposed. With the help of the first-order shear deformation theory, the local weak-forms of the governing equation and BC of OLCSES are derived. Its BC is modelled as a general BC via an artificial elastic spring technique, and the displacement functions are all approximated by TRPIM shape function in two directions (x, θ) regardless of the BC. The natural frequency of OLCSES is obtained by solving its governing equation and BC equation discretized by TRPIM shape function. The numerical results are in good agreement with those of preceding literatures and finite element method (FEM). The natural frequencies of OLCSES under different BCs including classical and elastic BCs are newly obtained and the effect of some parameters such as its material property and geometry on them is studied. The results can be used as reference data for researchers studying dynamic characteristics of shell structures with elliptical section.

A solution method for free vibration analysis of coupled laminated composite elliptical-cylindrical-elliptical shell with elastic boundary conditions

In this paper, an efficient, convenient and explicit method based on the Haar wavelet discretization approach for analyzing the free vibration of the coupled laminated composite elliptical-cylindrical-elliptical shells (ECESs) with elastic boundary conditions is presented. Two elliptical double curved shells are coupled on both end of cylindrical shell. Based on the first-order shear deformation theory the equations of motion for ECES are derived by means of Hamilton's principle. The separation of variables is first performed; i.e. displacement components and rotations of any point of the ECES are expanded to the Haar wavelet series in the meridian direction and Fourier series in circumferential direction. The constants appearing from the integrating process are determined by boundary conditions, and thus the partial differential equations are transformed into algebraic equations. By solving the characteristic equation, the natural frequencies and mode shapes of coupled laminated composite ECES are obtained. The present results have been compared with those of the published literature. The comparison results show that this method has high accuracy, high reliability and also a higher convergence rate in attaining the frequencies of the coupled laminated composite ECESs. Then, the effects of the main parameters such as material properties, geometrical parameters, and various boundary conditions, on the vibrational behavior of the coupled ECESs, are investigated. Finally, new free vibration analysis results of the coupled laminated composited ECES, which can be used as benchmark data for researchers in this field, are reported through the parameter study.

A meshfree method for free vibration analysis of ply drop-off laminated rectangular plates

The purpose of this paper is to develop a more effective meshfree shape function for free vibration analysis of ply drop-off laminated rectangular plates. The proposed meshfree Tchebychev-radial point interpolation method (TRIPM) shape function uses the Tchebychev polynomials and multi-quadrics (MQ) radial functions as the basis. The first order shear deformation theory (FSDT) is applied to the formulation for free vibration analysis of ply drop-off laminated rectangular plate, and the field functions are approximated by TRIPM shape function. The governing equations and boundary conditions for the substructures of ply drop-off laminated plate are derived, and the equations of the whole system are obtained by combining them using a continuous condition. The boundary and continuous conditions are generalized by the introduction of an artificial spring technique, and the type of boundary conditions is selected according to the spring stiffness. The accuracy and reliability of the proposed method are verified by comparing the results of the literature and the finite element program ABAQUS. The free vibration characteristics including natural frequencies and mode shapes of ply drop-off laminated rectangular plates with various geometrical dimensions and boundary conditions are presented through numerical examples.

Optimal analysis for optimal design of cyclic symmetric structures subject to frequency constraints

In general, the design optimization of large-scale structural systems is an extremely computationally expensive process. However, optimal structural analysis provides efficient and practical methods of structural analysis which can be employed as powerful tools to reduce significantly the computational time of structural analyses performed in the structural optimization process. In this paper, an efficient eigensolution method for free vibration analysis of rotationally repetitive structures is developed for the optimal design of cyclic symmetric structures subject to frequency constraints. Through the present eigensolution method, the initial free-vibration eigenproblem is decomposed into some sub-eigenproblems with much smaller dimensions using an efficient block-diagonalization technique. The main advantage of the present method is that it requires significantly less computational time and memory than the existing classical eigensolution method. Finally, two large-scale numerical examples are given to demonstrate the efficiency and accuracy of the present eigensolution method and compare it with the existing classical method in both terms of computational time and memory requirements. Numerical results indicate that the present eigensolution method not only guarantees the accuracy of the analysis results but also significantly reduces the computational time and memory required compared to the existing classical method. More importantly, the computational time of the optimization process is also significantly reduced. Optimization is carried out by a set-theoretical-based Jaya algorithm, named ST-JA. The proposed ST-JA aims to enhance the diversification and intensification capabilities of the original JA and strike a fine balance between them. Optimization results confirm that the proposed ST-JA outperforms the original JA and has superior or comparable performance to other state-of-the-art optimization algorithms.

Correction to: A Novel Solution Method for Free Vibration Analysis of Functionally Graded Arbitrary Quadrilateral Plates with Hole

Correction to: A Novel Solution Method for Free Vibration Analysis of Functionally Graded Arbitrary Quadrilateral Plates with Hole

A Novel Solution Method for Free Vibration Analysis of Functionally Graded Arbitrary Quadrilateral Plates with Hole

A Novel Solution Method for Free Vibration Analysis of Functionally Graded Arbitrary Quadrilateral Plates with Hole

Application of Haar wavelet discretization method for free vibration analysis of inversely coupled composite laminated shells

In this paper, the free vibration behavior of inversely coupled composite laminated shells (ICCLSs) are studied using Haar wavelet discretization method (HWDM) for the first time. Two doubly-curved shells are inversely coupled on both end of cylindrical shell, in which the doubly-curved shell contains the elliptical, parabolic and hyperbolic shells. The individual shells are coupled by the continuous condition. The boundary condition is modeled by using the artificial spring technique. Based on the first order shear deformation theory (FSDT), the displacement field of coupled shell are established. All displacement functions containing the boundary and continuous conditions are expanded by Haar wavelet series in the meridional direction, and by trigonometric series in the circumferential direction. The boundary and continuous condition functions are contained to satisfy the constants expressed by Haar wavelet series integral form in the function of main system. The convergence and accuracy of current method for the proposed structure are verified by comparison with the finite element method (FEM). The results show that this method is very reasonable for analyzing the free vibration of ICCLSs. Then, the effects of geometric dimensions, fiber orientation angle and lamination type on the natural frequency of ICCLSs are investigated. In end, the new results on natural frequencies and mode shapes of ICCLS are presented, and they will be benchmark data for further research.

Discrete singular convolution–polynomial chaos expansion method for free vibration analysis of non-uniform uncertain beams

This article enhances the discrete singular convolution method for free vibration analysis of non-uniform thin beams with variability in their geometrical and material properties such as thickness, specific volume (inverse of density) and Young’s modulus. The discrete singular convolution method solves the differential equation of motion of a structure with a high accuracy using a small number of discretisation points. The method uses polynomial chaos expansion to express these variabilities simulating uncertainty in a closed form. Non-uniformity is locally provided by changing the cross section and Young’s modulus of the beam along its length. In this context, firstly natural frequencies of deterministic uniform and non-uniform beams are predicted via the discrete singular convolution. These results are compared with finite element calculations and analytical solutions (if available) for the purpose of verification. Next, the uncertainty of the beam because of geometrical and material variabilities is modelled in a global manner by polynomial chaos expansion to predict probability distribution functions of the natural frequencies. Monte Carlo simulations are then performed for validation purpose. Results show that the proposed algorithm of the discrete singular convolution with polynomial chaos expansion is very accurate and also efficient, regarding computation cost, in handling non-uniform beams having material and geometrical variabilities. Therefore, it promises that it can be reliably applied to more complex structures having uncertain parameters.

A simplified method for free vibration analysis of wall-frames considering soil structure interaction

In this study, a method for free vibration analysis of wall-frame systems built on weak soil is proposed. In the development of the method, the wall-frame system that constitutes the superstructure was modeled as flexural-shear beam. In the study, it is accepted that the soil layers are isotropic, homogeneous and elastic, and the waves are only vertical propagating shear waves. Based on this assumption, the soil layer below is modeled as an equivalent shear beam. Then the differential equation system that represented the behavior of the whole system was written for both regions in a separate way. Natural periods were obtained by solving the differential equations by employing boundary conditions. At the end of the study, two examples were solved and the suitability of the proposed method to the Finite Element Method was evaluated.

Application of differential transformation method for free vibration analysis of wind turbine

Application of differential transformation method for free vibration analysis of wind turbine

Application of differential transformation method forfree vibration analysis of wind turbine

In recent years, there has been a tendency towards renewable energy sources considering the damages caused by non-renewable energy resources to nature and humans. One of the renewable energy sources is wind and energy is obtained with the help of wind turbines. To determine the behavior of wind turbines under earthquake loads, dynamic characteristics are required. In this study, the differential transformation method is proposed to determine the free vibration analysis of wind turbines with a variable cross-section. The wind turbine is modeled as an equivalent variable continuous flexural beam and blade weight is considered as a point mass at the top of the structures. The differential equation representing the free vibration of the wind turbine is transformed into an algebraic equation with the help of differential transformation method and the angular frequencies and the mode shapes of the wind turbine are obtained by the help of the differential transformation method. In the study, a sample taken from the literature was solved with the presented method and the suitability of the method was investigated. The same wind turbine example also modeled by finite element modelling software, ABAQUS. Results of the finite element model and differential transformation method are compared with each other and the results are in good agreement.

A unified solution method for free vibration analysis of functionally graded rotating type plates with elastic boundary condition

In this paper, a unified solution method for analyzing the free vibration characteristics of functionally graded rotating type plate (FGRTP) of which the distribution of material is defined by four parameters is presented. It is assumed that the considered functionally graded materials (FGM) are distributed in the thickness direction according to four parameters. The displacement fields of any point on the FGRTP are determined by the first order shear deformation theory (FSDT), and all displacement functions are extended by ultraspherical polynomial. By applying the Ritz method to the energy function of the whole system, the constitutive equation of FGRTP is obtained and the natural frequencies are obtained by solving the eigenvalue problem. The boundary conditions are generalized to arbitrary boundary conditions by artificial elastic technique. The accuracy of the proposed method is verified by comparing with the previous literatures. The effects of different parameters on the free vibration characteristics of FGRTP are studied through some numerical examples.

Dynamic Substructuring by the Craig–Bampton Method Applied to Frames

The Craig–Bampton method is a substructuring technique that reduces the number of internal degrees of freedom of substructures by approximations, using a set of truncated vibration modes. It is useful for structural designs with many degrees of freedom. This work aims to develop substructuring through the Craig–Bampton method for free vibration analysis on planar frames according to the frequency range of interest. The finite-element models of different substructures were assembled. Thus, the degrees of physical freedom were reduced according to the frequency range of interest. The reduced-order models were coupled. Thus, the physical response was obtained. Detailed calculation and the algorithm flowchart are provided. The study of frame dynamics using the present method in different frequency ranges of interest was presented and validated with the results of a commercial software. It was possible to clarify the coupling formulations of the component mode synthesis, as well as the amount of reduction of the number of equations provided by the method, reaching less than 4% of the complete model, for the considered cases.

Matched Interface and Boundary Method for Free Vibration Analysis of Irregular Membranes

Matched interface and boundary (MIB) method is introduced for free vibration analysis of irregular membranes. Two distinct schemes-on-interface and off-interface schemes are used to deal with the topological relations between edges of irregular domains and the Cartesian mesh lines. Different geometric shapes such as triangle and quadrilateral are dealt with by using MIB procedures. A number of examples are chosen to demonstrate the accuracy and convergence of MIB method. Numerical results show that MIB method is an efficient and highly accurate approach to solve free vibration problems of irregular membranes. This study further extends the application of MIB.

A Coupled Improved Element Free Galerkin-Finite Strip (IEFG-FS) Method for Free Vibration Analysis of Plate

In this paper, a coupling of improved element-free Galerkin (IEFG) method and semi-analytical finite strip (FS) method is presented for free vibration analysis of thin plates. This method is very easy to implement and has advantages of both IEFG and FS methods, so that IEFG method is used in sub-domain with complex geometry, and FS method is used for the remaining domains. The use of the FS method considerably reduces the analysis time, and the essential boundary conditions are easily enforced in FS sub-domain. In the IEFG method, the shape function does not have the Kronecker delta function property. Therefore, Lagrange multipliers method is used to satisfy the boundary conditions. Finally, five examples are presented to show the effectiveness of this work.

Dynamic stiffness method for free vibration analysis of thin functionally graded rectangular plates

In this present work, the dynamic stiffness method (DSM) is used to analyze the free vibration of a thin functionally graded rectangular plate. Classical plate theory (CPT) is used to develop the dynamic stiffness matrix of a functionally graded material (FGM) plate. For free vibration analysis, the natural frequencies of the functionally graded material plate are estimated by using DSM with Wittrick-Williams algorithm for different aspect ratios and different boundary conditions. The present research compared the DSM natural frequencies results with those available in the published literature.

Wave Based Method for Free Vibration Analysis of Cross-Ply Composite Laminated Shallow Shells with General Boundary Conditions.

In this paper, a semi-analytical method is adopted to analyze the free vibration characteristics of composite laminated shallow shells under general boundary conditions. Combining two kinds of shell theory, that is, first-order shear deformation shell theory (FSDT) and classical shell theory (CST), to describe the dynamic relationship between the displacement resultants and force vectors, the theoretical formulations are established. According to the presented work, the displacement and transverse rotational variables are transformed into wave function forms to satisfy the theoretical formulation. Related to diverse boundary conditions, the total matrix of the composite shallow shell can be established. Searching the determinant of the total matrix using the dichotomy method, the natural frequency of composite laminated shallow shells is obtained. Through several classical numerical examples, it is proven that the results calculated by the presented method are more accurate and reliable. Furthermore, to discuss the effect of geometric parameters and material constants on the natural frequencies of composite laminated shallow shells, some numerical examples are calculated to analyze. Also, the influence of boundary elastic restrained stiffness is discussed.

Accurate and fast three-dimensional free vibration analysis of large complex structures using the finite element method

Free vibration analysis of large complex structures with improved Lanczos iteration method is investigated through the finite element method. An important step in the traditional Lanczos iteration method for large generalized eigenvalue problems arising from free vibration analysis is that large-scale sparse linear system must be repeatedly solved within each step of the Lanczos iteration, which generally accounts for 70% to 90% of the calculation time of the eigenvalue problems. In the improved Lanczos iteration method given here, three-level preconditioner is proposed and utilized to improve the computational performance of Lanczos iteration. In this way, the multifrontal block incomplete Cholesky (MFBIC) preconditioner and p-type multigrid preconditioner based on higher-order hierarchical basis functions are proposed and utilized to improve the computational performance and decrease the memory requirements. To further reduce the memory consumption, the block Jacobi preconditioner is introduced in the Lanczos iteration, and a new three-level preconditioner that combines MFBIC, p-type multigrid and block Jacobi is proposed. The accuracy and performance of the improved method has been compared with exact solution, basic method and ANSYS. It is shown that this free vibration analysis method can be extremely useful to design structure dynamic characteristics of large complex structures.

A Jacobi polynomial based approximation for free vibration analysis of axially functionally graded material beams

The paper presents an effective approximation for free vibration analysis of axially functionally graded material (AFGM) beams based on the Jacobi polynomial theory. An arbitrary-order derivative of the Jacobi polynomial is expressed as the expression of low-order components, whereas its boundary values are fully defined by the polynomial parameters. The particular feature is used to derive the generalized eigenvalue equation for free vibration analysis of AFGM beams in conjunction with the Euler-Bernoulli, the Timoshenko and the nonlocal strain gradient beam theories. Several numerical examples in the literature are presented to demonstrate potential applications of the Jacobi polynomial approach. A fast convergence of the approximation error for natural frequency results has confirmed high accuracy of the proposed approach. The Legendre and the Chebyshev polynomials are special cases of the Jacobi basis function. This guarantees the flexibility of the presented method for free vibration analysis of AFGM beams with nonuniform geometries and axially varying material properties.

A static and free vibration analysis method for non-prismatic composite beams with a non-uniform flexible shear connection

Steel-concrete composite beams are widely used in practice because of their economic cross-section design. As sustainability becomes more and more important in the construction industry, the design of composite beams must be adapted to meet the requirements of the circular economy. This calls for demountability and reusability of the structural components, as well as optimized use of materials, for example by using non-prismatic beams. Linear-elastic design and the (optimized) use of demountable shear connectors are key in the design of reusable composite structures. In this paper, analytical prediction models for the elastic behaviour and the first eigenfrequency of non-prismatic composite beams with non-uniform shear connector arrangements are derived. The approach is based on 6th and 2nd order differential equations used to define matrix equations for a finite number of linearized composite beam segments. The analytical models are validated using experimental and numerical results obtained with a simply supported tapered composite beam. The analytical models are suitable for comprehensive structural analysis of non-prismatic composite beams with non-uniform shear connection.