In-plane Vibration Characteristics Research Articles

In-plane vibration characteristics of metamaterial plates with locally resonant microstructures

In-plane vibration characteristics of metamaterial plates with locally resonant microstructures

Free in-plane vibration of irregular laminated plate with curved edges based on boundary-type Chebyshev–Ritz method

The free in-plane vibration characteristics of laminated plate with arbitrarily geometrical shape and generally restrained edges, including curved and inclined edges, are investigated based on the boundary-type Chebyshev–Ritz method (BCRM). By adopting the divergence theorem, the double integral encountered in solving the energy equation of irregular laminates is converted to the integral on the boundary edge, which effectively reduces the computational complexity of the involved domain. The Chebyshev polynomials are utilized as test functions and the primitives of domain integrand functions are deduced according to the properties of Chebyshev polynomials. General boundary conditions on straight or even curved edges are achieved by introducing normal and tangential springs and assigning reasonable stiffness values. Natural frequencies and modes of laminated composite plates with straight edges (triangle, square, trapezoid, and hexagon), laminated composite plates with curved edge (ellipse), and irregular laminated composite plates are obtained. The method validation is carried out by comparison of the obtained solutions with literature and finite element results. The effects of variations in layering mode, number of layers, boundary conditions, and geometry on the eigenpairs of composite plate are extensively discussed.

Chebyshev Formulation for In-Plane Vibration Analysis of Arbitrary Laminated Polygonal Plates

This paper proposes a unified solution for in-plane vibration analysis of laminated arbitrary polygonal plates based on the Chebyshev polynomial method. As an innovative point of present work, the arbitrarily shaped polygonal plate is composed of two arbitrarily shaped quadrilateral plates rigidly coupled for simulation. The quadrilateral plate is uniformly mapped into a unit square plate through the four-node coordinate transformation. In the present approach, whatever the shape of the plate and the type of boundary conditions, the in-plane displacement components of the plate after transformation are expanded as Chebyshev polynomial of the first kind. The artificial spring technique is adopted here to simulate the general boundary restraints and the rigid coupling of coupled plates. Based on the Rayleigh–Ritz method, the accurate solution can be obtained through the energy function of the laminated polygonal plate. The in-plane vibration characteristics including the natural frequency and the corresponding mode shapes of the polygonal plate can be easily obtained by solving the eigenvalue problem. A large number of numerical results of the laminated polygonal plate with different shapes and boundary conditions are presented, it is found the present method shows rapid convergence, high computing efficiency, and high accuracy by comparing with those solutions calculated from the finite element method and the results in the experimentation. Besides, comprehensive studies on the effects of geometrical properties and material parameters of rhombic plates, pentagonal plates, and hexagonal plates on their in-plane vibration characteristics are also presented.

In-plane modal studies of arbitrary laminated triangular plates with elastic boundary constraints by the Chebyshev-Ritz approach

Firstly, the Chebyshev-Ritz method is applied to the free in-plane vibration of arbitrary shaped laminated triangular plates with elastic boundary conditions. In order to facilitate the calculation of energy, the arbitrary shaped triangular laminated plate is mapped into a square plate by the coordinate transformation. The constitutive equation of laminated materials is constructed by the equivalent monolayer theory. The displacement functions of the square plate after transformation are generally expressed as two-dimensional Chebyshev polynomials multiplied by coefficients. By means of artificial virtual spring technology, the arbitrary elastic boundary conditions of the plate can be obtained by changing the stiffness values of each spring. The in-plane free vibration characteristics of the triangular laminated plate under different boundary conditions are calculated and the accuracy of this method is verified by comparing with finite element results and experimental results. In this paper, modal experiments of three different triangular plates with free boundary and cantilever support boundary are carried out. The comparison results show that the present method has good convergence, high efficiency and satisfactory actuarial accuracy. Finally, novelty numerical analysis is carried out, the influence of geometrical properties, material parameters and boundary conditions on in-plane modal characteristics of laminated triangular plates are studied in detail.

In-Plane Vibration Analysis of Square Plate with Multiple Cutouts

A theoretical solution method for the in-plane vibration characteristics of the plate with cutouts is proposed in this paper. The energy principle in conjunction with the Rayleigh–Ritz solution technique is adopted for the theoretical modeling of the in-plane vibration of the structure. The energy functions for the plate with cutouts are established by subtracting the energies of the cutout domains from the total energies of the whole plate. To ensure continuity over the entire solution domain, the in-plane displacements are composed of two-dimensional standard Fourier series and supplementary functions. The in-plane eigenmodes of the plates with different square cutouts are compared with the results obtained from the finite element method (FEM), with good agreements. The influences of the cutouts on the in-plane vibration characteristics of the plates with cutouts are investigated by varying the number, size, and position of the cutouts.

Prediction of the in-plane vibration behavior of porous annular plate with porosity distributions in the thickness and radial directions

A semi-analytical model is proposed in this study to investigate the in-plane vibration behavior of porous annular plate with porosity distributions in the thickness and radial directions. A variational method combined with a multi-segment technique is employed to establish the theoretical model of elastic restrained porous annular plate. Admissible functions of each plate segment are expanded in the mixed series form, that is, the Jacobian orthogonal polynomial for radial variables and Fourier series for circumferential variables. The boundary penalty parameters are introduced to implement different boundary restrictions and the coupling penalty parameters are employed for ensuring the continuity conditions between segments. A series of numerical examples are given to discuss the convergence characteristics and numerical stability of the current model. The comparison between modal tests and numerical predictions shows that good prediction capabilities with acceptable errors are achieved by the proposed method. The influences of geometrical and material parameters on the in-plane vibration characteristics of porous annular plate are also discussed in detail.

Effect of bending stiffness on the in-plane free vibration characteristics of a cable network

Stay cables of cable-stayed bridges are connected by cross-ties to form a cable network, or the slings of the same suspension points on a suspension bridge are connected by vibration absorbers to form a sling network. As a result, the cable network’s in-plane free vibration characteristics are different from independent, single cables. However, stay cables or slings have certain bending stiffnesses, which directly affect the cable network’s free vibration characteristics. Therefore, to clarify the real dynamic behavior of a cable network and to improve the accuracy of cable network measurements, a multi-node, cross-tied, multilayer beam analytical model that considers the bending stiffness is proposed. The system equilibrium equation is established and the free vibration frequencies and modes of the system are obtained by analytical algorithm. Then, numerical simulations and field experiments, are carried out to verify the analytical results. In addition, parameter analyses with different configurations are performed on a multilayer stay-cable network and on a double-layer sling network to study the influence of bending stiffness on the cable network’s free vibration characteristics with different parameters.

In-plane vibration characteristics of isotropic plate with elastic edge restraints

It has been well-established that in-plane vibration modes in structures occur at high frequencies. Hence, it is important from the perspective of design consideration to analyse the dynamic behaviour of built up structures subjected to high frequency excitation. In light of the above, the present analysis has been undertaken for analysis of in-plane free vibration characteristics of isotropic rectangular plates using single Fourier series solution for the in-plane plate displacements. The Fourier series representation has been supplemented with auxiliary functions, so chosen, as to remove any potential discontinuities associated with Fourier series function defined over the entire x-y plane. Rayleigh-Ritz procedure has been applied to determine the in-plane natural frequencies of the plate and generation of natural modes. The demonstrated numerical examples provide an excellent accuracy and convergence of resulting solutions and bring out the effects of variation of boundary conditions on the in-plane vibration characteristics.

In-plane vibration characteristics of isotropic plate with elastic edge restraints

In-plane vibration characteristics of isotropic plate with elastic edge restraints

An exact analytical solution for free in-plane vibration of sector plates with simply supported radial edges

In this paper, an exact analytical solution is proposed to investigate the free in-plane vibration of sector plates. The highly complex and coupled differential equations of in-plane vibration motion are decoupled by invoking Helmholtz decomposition. The method of separation of variables is used to solve the resulting uncoupled Helmholtz equations. Applying the boundary conditions at both radial edges, the compatible eigenvalue problems for sector plates are obtained. Then, applying the boundary conditions at inner and outer circumferential edges, the corresponding characteristic equations can be solved to obtain the natural frequencies and the associated mode shapes. The results from the proposed method agree very well with the results from the available literature and the FEM. Significant influence of the boundary conditions and the geometric parameters on the free in-plane vibration characteristics of sector plates is investigated and addressed. The superficial frequency veering “aberration” between neighboring modes is discussed and found in fact to be an eigenvalue crossing phenomenon. It is proved that for a sector plate, only with simply supported radial edges, there exists an exact analytical solution. The proposed exact analytical solution can provide not only the detailed characteristics of the free in-plane vibration of sector plates but also the benchmark for further study.

A new Fourier series method and displacement function for in-plane vibration and power flow characteristics analysis of isotropic rectangular plates

The generation of in-plane vibration in plates is an important issue and frequently occurs due to the presence of excitations in the ship’s hull due to turbulent fluid flows, turbulent airflow excitation on aerospace structures, gear system subjected to axial excitation, assemblies housing piezoelectric crystals and sandwiched plates, and so on. The present analysis aims to establish a universal and numerically efficient method for determination of in-plane vibration characteristics of isotropic rectangular plates both for conventional and general boundary conditions. The new in-plane Fourier series and displacement function of the plate have been developed using beam displacement functions in x and y directions, respectively, under in-plane condition. A modified Fourier series assumption for the in-plane beam displacement has been utilised and further developed as plate displacement function. The computational efficiency of the present method is compared in terms of convergence of natural frequency parameter, speed of execution and manual convenience to reduce human errors with the frequently used Fourier series method by various researchers. Rayleigh–Ritz procedure has been applied to determine the in-plane natural frequencies. The mode shapes for few conventional and generally varying boundary conditions have been presented and analysed. The dynamic response has been obtained and analysed in terms of the in-plane mobility and power flow characteristics of the plate under varying boundary conditions. The validity of results obtained by the current method has shown excellent accuracy and faster convergence with the existing results. The present results can provide a benchmark to analyse the dynamic in-plane response of plate systems being used for built-up structures in real engineering applications.

Free Vibration Analysis of Fiber Metal Laminated Straight Beam

Abstract In recent years, studies on the development of new and advanced composite materials have been increasing. Among these new technological products, Fiber Metal Laminates (FML), and hybrid structures made of aluminium, carbon, glass or aramid fiber, are preferred especially in the aircraft industry due to their high performance. Therefore, free vibration analysis is necessary for the design process of such structures. In this study, the vibration characteristics of FML for clamped-free boundary conditions were investigated experimentally and numerically. Firstly, numerical results were obtained using Finite Element Method (FEM) and then these results were compared with the experimental results. It was seen that the numerical results were in good agreement with the experimental results. As the theoretical model was justified, the effects of various parameters such as number of layers, fiber orientations, and aluminium layer thickness on the in-plane vibration characteristics of the FML straight beam were analysed using FEM. Thus, most important parameters affecting the vibration characteristics of the hybrid structures were determined.

A Modified Fourier–Ritz Formulation for Vibration Analysis of Arbitrarily Restrained Rectangular Plate with Cutouts

A modified Fourier–Ritz method is developed for the flexural and in-plane vibration analysis of plates with two rectangular cutouts with arbitrary boundary conditions, aiming to provide a unified solving process for cases that the plate has various locations or sizes of cutout, and different kinds of boundary conditions. Under the current framework, modifying the position of the cutout or the boundary conditions of the plate is just as changing the geometric parameters of the plate, and there is no need to change the solution procedures. The arbitrary boundary conditions can be obtained by setting the stiffness constant of the boundary springs which are fixed uniformly along the edges of the plate at proper values. The strain and kinetic energy functions of a plate with rectangular cutout are derived in detail. The convergence and accuracy of the present method are demonstrated by comparing the present results with those obtained from the FEM software. In this paper, free in-plane and flexural vibration characteristics of the plate with rectangular cutout under general boundary conditions are studied. From the results, it can be found that the geometric parameters and positions of the cutout and the boundary conditions of the plate will obviously influence the natural vibration characteristics of the structures.

Free in-plane vibration analysis of elastically restrained annular panels made of functionally graded material

In this paper, free in-plane vibration of elastically restrained annular plate made of functionally gradient material (FGM) is investigated by an improved Fourier series method. The material property is assumed to be gradually changed in power law along the radial direction. The energy equation is formulated for the in-plane free vibration description of FGM annular plate, with the admissible functions constructed as the superposition of standard Fourier series and boundary smoothed polynomials to make the spatial differential continuous enough in the entire solving domain. In conjunction with the Rayleigh-Ritz procedure, system characteristic equation in matrix form is straightforwardly derived. Several numerical examples are then presented to validate the proposed model, and study the in-plane vibration characteristics of FGM annular panel with various boundary conditions. Based on the model established, the influence of important parameters, such as FGM power-law exponent and boundary restraints, on the in-plane vibration characteristics of FGM annular panel is addressed and investigated in detail. This work can shed some light for a better understanding on the in-plane dynamic characteristics of such complex structure.

Exact solutions for the free in-plane vibrations of rectangular plates with arbitrary boundary conditions

Though the free in-plane vibration problem of rectangular plates with classical boundary conditions have been studied using various numerical methods like direct separation of variables, the elastic boundary restraint is more general in the practical structural applications compared with the perfect classical boundary condition. Thus, in this paper, an innovative exact solution on the basis of the method of reverberation ray matrix (MRRM) and golden section search (GSS) algorithm is presented to study the exact free in-plane vibration characteristics of rectangular plates with arbitrary boundary conditions. In the present method, the exact natural frequencies and mode shapes can be very easily obtained when two opposite plate edges have either type of the simply supported boundary conditions and the resting plate edges have the arbitrary boundary conditions including the classical case and general elastic restraints. By the comparison with other published exact solutions, it is shown that the present exact solutions have some obvious advantages which are performed in the excellent accuracy, reliability and efficiency. Performing the study, some of the first-known exact results for the free in-plane vibrations of rectangular plates with elastic boundary restraints are also obtained, which can be worked as the benchmark data for the future approximate methods.

원판형 압전 변환기의 면내 방사 진동 특성 측정

이 논문은 원판형 압전 변환기의 면내 방사 진동 특성을 실험적으로 다룬다. 면내 방사 진동은 두께 방향으로 분극이 된 압전 원판에서 푸아송비로 인해 야기되는 것으로서, 레이저 면내 진동 측정기로써 측정되었고, 고유진동수는 임피던스 분석기로써 측정되었다. 실험 결과는 단순화된 이론적 해석과 유한요소 해석으로부터 얻어진 이론적 예측과 비교되었다. 원판형 압전 변환기의 기본 모드는 방사 모드이고, 중심으로부터 가장자리까지의 반경 방향 변위 분포는 단조 증가가 아니라 원주면에서 약간 떨어진 위치에서 최대 값을 나타낸다. 지름 대비 두께 비율이 작은 원판에서 이론적으로 계산된 기본 진동수는 유한요소 결과와 잘 일치하고, 두께/지름 비율 0.4까지 오차 7 % 이내로 정확하다. The paper experimentally deals with the radial in-plane vibration characteristics of disk-shaped piezoelectric transducers. The radial in-plane motion, which is induced due to Poisson's ratio in the piezoelectric disk polarized in the thickness direction, was measured by using an in-plane laser vibrometer, and the natural frequencies were measured by using an impedance analyzer. The experimental results have been compared with theoretical predictions obtained by simplified theoretical and finite-element analyses. It appears that the fundamental mode of a piezoelectric disk transducer is a radial mode and its radial displacement distribution from the center to the perimeter is not monotonic but shows maximum slightly apart from the perimeter. The theoretically-calculated fundamental frequencies agree well with the finite-element results for small thickness-to-diameter ratio, and they are accurate within 7 % error for the ratio up to 0.4.

In-Plane Vibration Analysis of Annular Plates with Arbitrary Boundary Conditions

In comparison with the out-of-plane vibrations of annular plates, far less attention has been paid to the in-plane vibrations which may also play a vital important role in affecting the sound radiation from and power flows in a built-up structure. In this investigation, a generalized Fourier series method is proposed for the in-plane vibration analysis of annular plates with arbitrary boundary conditions along each of its edges. Regardless of the boundary conditions, the in-plane displacement fields are invariantly expressed as a new form of trigonometric series expansions with a drastically improved convergence as compared with the conventional Fourier series. All the unknown expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz technique. Unlike most of the existing studies, the presented method can be readily and universally applied to a wide spectrum of in-plane vibration problems involving different boundary conditions, varying material, and geometric properties with no need of modifying the basic functions or adapting solution procedures. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current solution for predicting the in-plane vibration characteristics of annular plates subjected to different boundary conditions.

A sonic crystal angular rod with ability of multi-dimensional and broadband vibration reduction

Sonic crystal (SC) rod has apparent band gaps in which longitudinal or bending wave can be suppressed. However, the longitudinal wave band gap (LBG) and bending wave band gap (BBG) are located in different frequency ranges generally. Therefore, multi-dimensional vibration in the same frequency band can not be attenuated using single SC rod. For this reason, an SC angular rod is proposed consisting of two single SC rods. The in-plane and out-plane vibration are analyzed using transfer matrix method and finite element simulations are carried out. Our results show that: firstly, the attenuation of LBG and BBG can be fully utilized simultaneously in in-plane vibration owing to the transformation between longitudinal and bending waves and the SC angular rod has the ability of vibration reduction of 3-dimensions and 2-dimensions in BBG and LBG, respectively. Additionally, vibration of low frequency also can be isolated especially when loading angle is close to 90 degrees. Secondly, the performance of 90-degrees SC angular rod is superior to that of other angles such as 45 degrees and 135 degrees. Thirdly, loading angle has certain influence on the characteristics of in-plane vibration. When loading angle closes to 90 degrees the bending vibration becomes dominant and the attenuation in BBG gets stronger. At the same time the transfer peaks of bending wave lying in LBG also get strengthened impairing the attenuation in nearby frequency range. The contrary case is found when the loading angle closes to 0 degree.

Sound radiation from in-plane vibration of thick annular discs with a narrow radial slot

The vibro-acoustic characteristics of in-plane vibration in a thick annular disc containing a narrow radial slot are investigated with a hybrid approach combining numerical or experimental results with pre-developed analytical calculations. Structural eigenfunctions for in-plane vibrations of the disc are approximated as linear combinations of eigenfunctions for a uniform disc that has same material properties and geometric dimensions except for the slot. Acoustic radiations from the normal modes of the disc are calculated using a theoretical solution introduced in previous work for the uniform disc along with the results from the approximations of structural eigenfunctions. Vibration and sound radiation from the disc due to a harmonic force fixed to the disc outer surface are investigated using the mode expansion method. The effects of the circumferential distance between the radial excitation and slot on the vibro-acoustic responses are investigated using the same method. Finally, contributions of sound radiations from two split vibration modes to the overall sound radiation from the sample disc are also investigated. Based on these results, it can be concluded that the procedure suggested in this study has good accuracy in calculating the in-plane vibro-acoustic properties of a thick annular disc with a radial slot.

충만 디스크의 면내 진동 해석을 위한 1차원 환상 평판 요소

We present a one-dimensional annular plate element with which the in-plane vibration of full disks can be analyzed efficiently and accurately by using the FEM. Its elementary mass matrix and stiffness matrix are derived, respectively, from the virtual work by effective forces and the virtual strain energy. The static deformation modes obtained from an integration of the differential equilibrium equations of the annular plate are used as interpolation functions of the one-dimensional annular plate element. The in-plane natural vibration characteristics of a 2-step full disk and a uniform full disk are analysed. Its results are compared with the results obtained by utilizing two-dimensional 8-node quadrilateral plane elements and cyclic symmetry of the disk. And also, by comparing with the theoretical results of previous researchers, the efficiency and accuracy of the presented element are verified.