Vibration Eigenvalues Research Articles

Modeling and mathematical approach for mechanical natural frequencies of structural plates with free edges

The current study aims to demonstrate an easy mathematical procedure and simple to be used by engineers to maintain the mechanical natural frequencies (Eigen values of free vibration) of thin rectangular structural plates with free edges included in boundary conditions. The reason for this easiness and simplicity is that new method of analysis depends mainly on using mechanical minimum energy concept represented in the Rayleigh–Ritz method with appropriate real polynomials as admissible functions. The new procedure guides these functions to satisfy the boundary conditions at the middle points on the free edges of the plate beside the nodal points of the vibration modes along each edge direction. Therefore, there are no complicated or imaginary terms in the developed functions while they lead to surprisingly acceptable and even accurate results when compared with other works and Ansys program models.

Output-only modal identification with recursive dynamic mode decomposition for time-varying systems

The time-varying modal identification is a prominent analytical means but challenging task to grasp the dynamic characteristics of time-varying systems, which is important guarantee for real-time control and on-line health monitoring of many engineering structures. In this paper, a data-driven approach for time-varying modal identification without requiring parametric modeling about underlying system is presented. Firstly, the fundamental theory for Dynamic Mode Decomposition (DMD) is reviewed combining the Koopman operator. Subsequently, the DMD incorporated into operational modal identification is examined from the perspective of linear system, and the inner connection between modal parameters of vibration system and eigenvalues and eigenmodes of DMD is established. On the basis of that, a weighted recursive strategy integrated with DMD is employed to investigate the capability of time-varying modal identification, and the main identification procedures are elaborated. Finally, the effectiveness and practicability of the proposed method in extracting modal properties of time-varying systems are investigated by a numerical and experiment example. The results demonstrate that, compared to other methods, the proposed approach has high computational efficiency while ensuring the identification accuracy, exhibiting a good engineering application prospect.

Geometrically Nonlinear Aeroelastic Flutter Characteristic of Laminated Composite Shell Panels Under Supersonic Flow

The nonlinear aeroelastic flutter characteristics of laminated composite curved (flat and cylindrical) panels are reported in this paper by obtaining the finite element (FE) solutions under the supersonic flow. The panel model has been derived from the third-order shear deformation theory framework including full geometrical nonlinearity via Green–Lagrange strain. In contrast, the flutter due to aerodynamic loading is included via the first-order piston theory. The current predicted solution accuracy and their validations have been demonstrated by relating the free vibration frequency, coalescence frequency, and critical aerodynamic pressure with the available numerical data. The initial free vibration eigenvalue responses are compared with in-house experimental values. Finally, a few numerical examples are presented by varying parameters like the effect of fiber orientation, flow angle, end boundary conditions, aspect ratio, modular ratio, thickness ratio, and amplitude ratio on the supersonic flutter boundaries of shell panels. The critical aerodyanamic pressure for simply supported cross-ply flat panels increases by 42.85% when the amplitude ratio increases from 0 to 0.75.

Concurrent optimization method of shape and topology for natural vibration problem of shell structures

In this paper, a concurrent optimization method of shape and topology for natural vibration design of shell structures is presented. The fundamental vibration eigenvalue is maximized under volume constraints for shape and topology optimization. The free-form optimization method for shells and the solid isotropic material with penalization (SIMP) method are respectively employed and combined effectively for shape and topology design, in which shape and fictitious homogenized-density variations are used as design variables and simultaneously determined. In other words, the optimal topology is determined in the variable design surface optimized by shape optimization. The optimal design problem is formulated as a distributed-parameter optimization problem, and the sensitivity functions with respect to shape and density variations are theoretically derived. Both the optimal shape and density variations are determined with the unified H1 gradient method, where the sensitivity functions are respectively applied as the Robin condition to the design surface in order to determine the optimal shape and topology. We also show an improved method for the spurious (or localized) mode problem of shell structures, which often occurs in the lower density region in topology optimization for vibration design. With the proposed method, shell structure for light weight and improved low-order vibration characteristics can be obtained without any design parameterization, free of numerical instabilities such as checkerboard pattern and zigzag-shape problems.

Prediction of Fluid Accumulation in Engine Disk Cavity Based on Encoder-Decoder

A neural network model based on the encoder and decoder is proposed for aero-engine fluid accumulation. The model uses the attention mechanism to adjust the output weight of the encoder and then uses the decoder to predict the fluid volume. The vibration eigenvalues were obtained by fast Fourier transform of the original vibration signals, and the eigenvalues of vibration data were used as the input model to predict the volume of fluid in the disk cavity. To evaluate the effectiveness of the model, an aero-engine disk cavity fluid simulation experiment was built, and the data obtained from the experiment were used to test the method and the model. The prediction accuracy of the proposed model for the volume of engine disk cavity fluid was over 95%, which has an important engineering value for the reliable operation of an aero-engine.

Identification of a symmetric mass density in a rectangular membrane from finite eigenvalue data

In this paper we seek to recover the mass density of a rectangular taut membrane from the first eigenvalues of the small transverse vibration under supported boundary conditions, provided the mass is a small perturbation of the constant density and is symmetric with respect to both midlines. The proposed method allows for the determination of the generalized Fourier coefficients of the mass density change evaluated on a suitable basis of functions, which naturally arises from the linearized Taylor expansion of the eigenvalues. The reconstruction is based on a sequence of successive linearizations of the inverse problem in a neighborhood of the uniform membrane, and numerical results indicate improvements over other techniques available in the literature. The method has been tested also on mass densities that fall outside the range of small perturbations of the constant density, showing some potential for the reconstruction.

Concurrent multiscale optimization method for natural vibration design of porous structures

This paper proposes a concurrent multiscale optimization method of macrostructure topology and microstructure shapes for porous structures, aiming at maximizing the vibration eigenvalue. The multi-material distribution of the macrostructure and the shape of the microstructures are optimized by topology and shape optimization respectively. The homogenized properties of the porous materials are calculated using the homogenization method, and the homogenized elastic tensor and density are applied to the macrostructure. The optimum distribution of the porous material in the macrostructure is determined by a multi-material topology optimization using the GSIMP method. The KS function is introduced to solve the repeated-root problem that lies in the maximization of vibrational eigenvalues. The area-constrained multiscale optimization problem is formulated as a distributed parameter optimization problem, and the sensitivity functions are derived using the Lagrange multiplier method and the adjoint variable method. Based on the obtained sensitivity functions, the design variables are updated using the H1 gradient method. The effectiveness of the proposed method for maximizing vibration eigenvalues is confirmed through numerical examples.

Control of Vibrations of a Beam with Nonlocal Boundary Conditions

The control problem the first eigenvalues of the Euler-Bernoulli beam vibration with hinged and clamped fixings at both ends is considered. The control is based on Kanguzhin’s algorithm through integral perturbations of the boundary conditions of the original problem. Conditions for the boundary parameters for controlling the first eigenvalues are found.

Analysis of vibration in rotating pretwisted functionally graded graphene platelets reinforced nanocomposite laminated blades with an attached point mass

This work presents an investigation on the free vibration behavior of rotating pre-twisted functionally graded graphene platelets reinforced composite (FG-GPLRC) laminated blades/beams with an attached point mass. The considered beams are constituted of [Formula: see text] layers which are bonded perfectly and made of a mixture of isotropic polymer matrix and graphene platelets (GPLs). The weight fraction of GPLs changes in a layer-wise manner. The effective material properties of FG-GPLRC layers are computed by using the modified Halpin-Tsai model together with rule of mixture. The free vibration eigenvalue equations are developed based on the Reddy’s third-order shear deformation theory (TSDT) using the Chebyshev–Ritz method under different boundary conditions. After validating the approach, the influences of the GPLs distribution pattern, GPLs weight fraction, angular velocity, the variation of the angle of twist along the beam axis, the ratio of attached mass to the beam mass, boundary conditions, position of attached mass, and geometry on the vibration behavior are investigated. The findings demonstrate that the natural frequencies of the rotating pre-twisted FG-GPLRC laminated beams significantly increases by adding a very small amount of GPLs into polymer matrix. It is shown that placing more GPLs near the top and bottom surfaces of the pre-twisted beam is an effective way to strengthen the pre-twisted beam stiffness and increase the natural frequencies.

Vibration Properties of a Concrete Structure with Symmetries Used in Civil Engineering

The paper aims to study a concrete structure, currently used in civil engineering, which has certain symmetries. This type of problem is common in engineering practice, especially in civil engineering. There are many reasons why structures with identical elements or certain symmetries are used in industry, related to economic considerations, shortening the design time, for constructive, simplicity, cost or logistical reasons. There are many reasons why the presence of symmetries has benefits for designers, builders, and beneficiaries. In the end, the result of these benefits materializes through short execution times and reduced costs. The paper studies the eigenvalue and eigenmode properties of vibration for components of the constructions’ structure, often encountered in current practice. The identification of such properties allows the simplification and easing of the effort necessary for the dynamic analysis of such a structure.

Reverse Engineering and Topology Optimization for Weight-Reduction of a Bell-Crank

This paper describes a new design method that was developed to achieve an optimal design method for weight reduction of a bell crank, sourced from a Louis Christen Road Racing F1 Sidecar. The method involved reverse engineering to produce a 3D model of the mechanical part. The 3D bell crank model was converted to a finite element (FE) model to characterize the eigenvalues of vibration and responses to excitation using the Lanczos iteration method in Abaqus software. The bell crank part was also tested using a laser vibrometer to capture its natural frequencies and corresponding vibration mode shapes. The test results were used to validate the FE model, which was then analysed through a topology optimization process. The objective function was the weight and the optimization constraints were the stiffness and the strain energy of the structure. The optimized design was converted back to a 3D model and then fabricated to produce a physical prototype for design verification and validation by means of FE analysis and laboratory experiments and then compared with the original part. Results indicated that weight reduction was achieved while also increasing the natural frequency by 2%, reducing the maximum principal strain and maximum von Mises stress by 4% and 16.5%, respectively, for the optimized design when compared with the original design. The results showed that the proposed method is applicable and effective in topology optimization to obtain a lightweight (~3% weight saving) and structurally strong design.

Free vibration characteristics of nonlocal viscoelastic nano‐scaled plates with rectangular cutout and surface effects

AbstractIn the present study, the viscoelastic free vibration behavior of nano‐scaled plates is studied by employing a finite element method based on the two‐phase nonlocal integral theory. Various boundary conditions, surface effects and cutouts have been assumed. The principle of total potential energy is used for developing the nonlocal finite element method, and the classical plate theory is assumed to derive the formulations. By numerically solving the eigenvalue problem, which has been obtained by the variational principle, the complex eigenvalues of free vibration of the viscoelastic nano‐scaled plates are acquired. The current results are compared with those available in the literature and those obtained by commercial finite element software, and the influences of the nonlocal parameter, viscoelastic parameter, geometrical parameters (e.g. cutout and size), surface effects and different boundary conditions on the complex eigenvalues are studied. It is noted that the current method is able to handle quite versatile boundary conditions and geometries like cutouts, which are rather difficult (or impossible) to be tackled by employing other methods available in most researches.

Particle Swarm Optimization-Support Vector Machine Model for Machinery Fault Diagnoses in High-Voltage Circuit Breakers

According to statistic data, machinery faults contribute to largest proportion of High-voltage circuit breaker failures, and traditional maintenance methods exist some disadvantages for that issue. Therefore, based on the wavelet packet decomposition approach and support vector machines, a new diagnosis model is proposed for such fault diagnoses in this study. The vibration eigenvalue extraction is analyzed through wavelet packet decomposition, and a four-layer support vector machine is constituted as a fault classifier. The Gaussian radial basis function is employed as the kernel function for the classifier. The penalty parameter c and kernel parameter δ of the support vector machine are vital for the diagnostic accuracy, and these parameters must be carefully predetermined. Thus, a particle swarm optimization-support vector machine model is developed in which the optimal parameters c and δ for the support vector machine in each layer are determined by the particle swarm algorithm. The validity of this fault diagnosis model is determined with a real dataset from the operation experiment. Moreover, comparative investigations of fault diagnosis experiments with a normal support vector machine and a particle swarm optimization back-propagation neural network are also implemented. The results indicate that the proposed fault diagnosis model yields better accuracy and efficiency than these other models.

A Semianalytical Vibration Analysis of Partially Wet Square Cantilever Plate With Numerical and Experimental Verification: Partially Wet Modeshapes

Abstract A semianalytical study of a uniform homogenous partially submerged square cantilever plate vibration is presented. The structure is assumed to be a Kirchhoff's plate, clamped on one edge and free on the other edges. The lengthwise section of the plate is a cantilever clamped-free (CF) beam, while the widthwise section is a free-free (FF) beam. The plate modeshape is a weighted superposition of the product of the beam modeshapes, with unknown weights. The CF beam has only flexural modes. The FF beam has two rigid-body modes, i.e., translational and rotational modes. Rayleigh–Ritz method (RRM) is used to set up the free vibration eigenvalue problem. The eigenvector gives the unknown weights. The modeshapes generated are further used in the boundary element method (BEM) to calculate the fluid inertia, which participates in the vibration and leads to a consistent drop in frequencies. The dependence of this reduction on the submergence level is studied for the first six frequencies of the plate. The frequencies are also experimentally generated by the impact hammer test, both in the dry state, and under three distinct levels of submergence: 25%, 50%, and 75% from the free edge opposite to the clamped edge. The frequencies and modeshapes are also verified through numerical analysis using the commercial code ansys 16.0. Conclusions are drawn regarding the influence of fluid inertia distribution on the final plate modeshape, leading to insights into sound structural designs.

The effect of uncertain elastic modulus on eigenvalue in the free vibration of a functionally graded plate

Abstract This research deal with investigation of eigenvalue in stochastic free vibration problem of functionally graded (FG) plates involving uncertain elastic modulus. The governing equations of FG plate are derived using classical plate theory. The stochastic isogeometric analysis is established using the standard isogeometric analysis in conjugation with perturbation technique. An uncertain elastic modulus is hypothesized to be Gaussian random field. The result predicted by present method is validated with Monte Carlo simulation show good agreement. The effect of uncertain elastic modulus on variability in the eigenvalue of free vibration of plate is significant.

Dynamic analysis of partial-interaction Kant composite beams by weak-form quadrature element method

In this paper, the Kant higher-order beam theory is applied to model each segment of the partial-interaction composite beams, aiming to capture as possible fidelity as the plane stress model. On this basis, the weak-form equation is obtained through the principle of virtual work. Besides, the weak-form quadrature element (WQE), as a counterpart of the conventional finite element (CFE), is derived and implemented to more efficiently solve problems, including free vibration eigenvalue analysis and dynamic responses prediction to moving loads. After the verification of all the programs developed, a series of numerical examples are given to investigate the WQE’s superiority on convergence rate and numerical smoothness over the CFE. At the end of the paper, the influences of structural damping and loads’ moving speed on impact factor of two-span continuous beams are analyzed. Numerical results show that the proposed WQE, due to the variable-order interpolation of the element, possesses overwhelmingly higher computational efficiency than the CFE, and the numerical smoothness problem in the internal force analysis is significantly alleviated by WQE method.

GBT-Based Vibration Analysis Using the Exact Element Method

Generalized Beam Theory (GBT), intended to analyze the structural behavior of prismatic thin-walled members and structural systems, expresses the member deformed configuration as a combination of cross-section deformation modes multiplied by the corresponding longitudinal amplitude functions. The determination of the latter, often the most computer-intensive step of the analysis, is almost always performed by means of GBT-based “conventional” 1D (beam) finite elements. This paper presents the formulation, implementation and application of the so-called “exact element method” in the framework of GBT-based elastic free vibration analyses. This technique, originally proposed by Eisenberger (1990), uses the power series method to solve the governing differential equations and obtains the vibration eigenvalue problem from the boundary terms. A few illustrative numerical examples are presented, focusing mainly on the comparison between the combined accuracy and computational effort associated with the determination of vibration solutions with the exact and conventional GBT-based (finite) elements. This comparison shows that the GBT-based exact element method may lead to significant computational savings, particularly when the vibration modes exhibit large half-wave numbers.

Form II of Mindlin's second strain gradient theory of elasticity with a simplification: For materials and structures from nano- to macro-scales

Abstract The fundamental equations for Form II of Mindlin's second strain gradient elasticity theory for isotropic materials are first derived. A corresponding simplified formulation is then proposed, with six and two higher-order material parameters for the strain and kinetic energy, respectively. This simplified model is still capable of accounting for free surface effects and surface tension arising in second strain gradient continua. Within the simplified model, at first, surface tension effects appearing in nano-scale solids near free boundaries are analyzed. Next, a thin strip under tension and shear is considered and closed-form solutions are provided for analyzing the free surface effects. Expressions for effective Poisson's ratio and effective shear modulus are proposed and found to be size-dependent. Most importantly, for each model problem a stability analysis is accomplished disallowing non-physical solutions (befallen but not exclusively disputed in a recent Form I article). Finally, a triangular macro-scale lattice structure of trusses is shown, as a mechanical metamaterial, to behave as a second strain gradient continuum. In particular, it is shown that initial stresses prescribed on boundaries can be associated to one of the higher-order material parameters, modulus of cohesion, giving rise to surface tension. For completeness, a numerical free vibration eigenvalue analysis is accomplished for both a fine-scale lattice model and the corresponding second-order continuum via standard and isogeometric finite element simulations , respectively, completing the calibration procedure for the higher-order material parameters. The eigenvalue analysis confirms the necessity of the second velocity gradient terms in the kinetic energy density .

Uncertainty qualification for the free vibration of a functionally graded material plate with uncertain mass density

This work studies the response variability in the eigenvalue of free vibration of functionally graded material (FGM) plates with uncertain mass density. The mass density of plates is assumed to be a Gaussian random field on the plane of the plate. The formulation of free vibration of FGM plates is derived using isogeometric analysis based on higher-order shear deformation theory. The stochastic isogeometric analysis is established based on standard isogeometric analysis and a perturbation approach. Also, the formulation for variation of the eigenvalue of free vibration of the plate is issued. The present approach is validated with Monte Carlo simulation (MCS) based on a spectral representation method. The numerical examples investigated show good agreement between the present approach and Monte Carlo simulation. Based on the numerical results, the effect of a parameter of uncertain mass density on variability in the natural frequencies is drawn with respect to FGM plates.

Natural Period Formula of a Reinforced Concrete Shear Wall Structure Considering Flange Wall Effect

In this study, natural period formular is presented for a RC shear wall structure with H-, T-, and L-shaped wall sections. The natural period formular proposed by Goel and Chopra and adopted in ASCE 7-10 was modified by using the ratio of the flange and web wall area. The natural periods of structures with H-shaped wall were numerically obtained, the results indicated that the ASCE 7-10 could not consider the natural period variation according to the length of the flange wall, but the proposed formula could do. Especially, ASCE 7-10 estimated much longer periods than eigenvalue analysis, and this implies that conservative seismic design is difficult. The periods by eigenvalue analysis exist between the upper and lower bounds given by the proposed formula, and conservative design is possible by using the proposed lower bound value. In order to verity the effectiveness of the proposed method, actual residential buildings with various types of flange walls are considered. Ambient vibration tests, eigenvalue analyses, and nonlinear dynamic analyses were conducted and the periods were compared with the values by ASCE 7-10 and the proposed formula. The results showed that the proposed formula could estimate more accurately the periods than ASCE 7-10.