Free Vibration Modes Research Articles

Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

Preliminary findings on the experimental investigation of the small-strain behaviour of Pizzoli silty sand

The dynamic properties of soils play a crucial role in solving many geotechnical problems with special attention to earthquake engineering. In particular, the small-strain soil behavior should be accurately reproduced in geotechnical modelling to allow quantifying of the earthquake-induced site response. If the determination of the small-strain shear modulus can be easily inferred from in-situ measurements of shear wave velocity, the small-strain damping ratio of soils is rarely obtained from in-situ tests and it is commonly defined through cyclic or dynamic laboratory tests. This paper describes preliminary findings obtained from a laboratory investigation performed to measure the small-strain dynamic properties of the silty sand deposit of the Pizzoli site (L’Aquila, Italy). Due to the remarkable seismic hazard of the considered area, demonstrated by several seismic events, such as recently the 2009 L’Aquila and the 2016-2017 Central Italy earthquakes, and in the past, the 2 February 1703 earthquake, a specific investigation program including boreholes, geophysical and geotechnical in-situ tests was carried out. Resonant column tests have been also performed at the Geotechnical Laboratory of the University of L’Aquila in both forced and free vibration modes. The interpretation of the results has been used to identify the small-strain shear modulus and damping ratio. The shear modulus as obtained from the laboratory has been compared with that obtained via the existing in-situ shear wave velocity measurements. In contrast, the damping ratio has been compared with the value estimated with a literature relationship proposed for soil deposits of Central Italy.

Numerical Study of the Mechanical Behavior of a Composite Material Plate

Composite materials are playing an increasingly significant role in the field of mechanics, such as aeronautics, aerospace, ships, and modern cars. The objective of this work was to investigate the mechanical properties and behavior of a composite plate (carbon/epoxy) through various tests, including tension, compression, and bending, using numerical analysis. Additionally, static analysis of mechanical behavior was performed, as well as the natural vibration of undamped structures. This analysis aimed to determine stresses, displacements, natural frequencies, and modes of free vibration. The obtained results illustrated that the composite material has a significant impact on improving the quality of mechanical properties and increasing the lifetime of materials, making it a valuable option for design and repair techniques. Furthermore, the results obtained in this work align well with previous literature studies. By examining these parameters, this study aimed to enhance our understanding of how composite materials perform under different mechanical loads and dynamic conditions, thus contributing valuable data to advance the field of composite mechanics.

Natural frequencies of prismatic and tapered composite girders with corrugated steel webs

The present study proposed an analytical formula to predict the natural frequencies of prismatic and tapered composite girders with corrugated steel webs (CSWs), and rigorously provides the derivation process. It is experimentally and numerically verified that the proposed formula is more applicable for predicting the natural frequencies of girders with CSWs, in comparison to the traditional formula based on Euler-Bernoulli beam theory. The study demonstrates that the traditional formula used for determining the natural frequencies of prismatic girders with CSWs leads to significant overestimation, mainly because it ignores the effects of shear deformation. The finite element (FE) simulation suggests that the free vibration mode shapes of tapered girders with CSWs exhibit sinusoidal patterns. Consequently, an equivalent stiffness method has been developed for accurately and conveniently estimating the natural frequencies of tapered girders with CSWs. This paper establishes a theoretical foundation for modifying the natural frequency calculation formula of prismatic and tapered girders with CSWs in current specifications, the research findings hold significant implications for the structural dynamic analysis and design of such girders.

Free vibration analysis of thin plates with side cracks by the weak form quadrature element method

The weak form quadrature element method is applied to vibration analysis of thin plates with side cracks. Eigenfunctions are introduced into the inner square domain around the crack tip, while the rest of the problem domain is analyzed by the weak form quadrature element method (QEM). The continuity of displacements and slopes on the interface between the inner and outer domains is enforced. A generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first ten nondimensional frequency parameters for several typical problems are obtained and compared with those available in the literature. The corresponding modes of free vibration are plotted, and the influence of the crack on vibration frequencies and modes is discussed at length.

Parameter optimization analysis of rotary electromagnetic vibration energy harvester for performance enhancement under free vibration

In this paper, three new important aspects of rotary electromagnetic vibration energy harvesting technology (RE-VEH) are concerned and investigated: (i)vibro-electric coupling mechanism of the RE-VEH system is studied through theoretical modeling; (ii) quantitative analysis of system parameters based on numerical simulation method is carried out for the optimal design of RE-VEH; and (iii) dynamic power output performance of the RE-VEH system in free vibration is discussed. The parameter adjusting methods of the RE-VEH system in free vibration mode are obtained through theoretical analysis and numerical simulation. The experimental results show that the power output performance of RE-VEH in free vibration mode matches the numerical simulation results. The simulation and experimental results show that the maximum voltage output and power output of the RE-VEH with different structure parameters under free vibration can be up to the level of 100∼101 V/watt. The above results indicate that RE-VEH in a free vibration environment has significant energy output performance.

Dynamic response of laminated functionally graded carbon nanotube-reinforced composite viscoelastic plates

The dynamic response of viscoelastic plates reinforced with functionally graded carbon nanotube-reinforced composite material (FG-CNTRC) under dynamic loads was examined in this work. This research investigates the dynamic analysis of CNTRC plates by single-walled CNTs (SWCNTs). SWCNTs were considered to be straight and aligned, with a homogeneous pattern. Carbon nanotube (CNT) configurations were investigated, inclusive of uniform, and three varieties of FG distributions of CNTs across the thickness. The equation of motion of composite plates was obtained using Hamilton’s principle. The time-dependent equations were derived by applying the Navier solution method to the equation of motion. These equations were converted into the Laplace domain. The Modified Durbin procedure was used to convert the resultant computations from the Laplace field to the time field. The resulting findings were compared to other methods. It was demonstrated that the findings are consistent with those of other methods. Then, sensitivity analysis revealed that CNT volume fractions, damping ratios, and FG distributions had a substantial influence on the quasi-static and dynamic behavior of viscoelastic FG-CNTRC plates. Results show that with the help of the Laplace technique no need to use free vibration frequencies or modes, it is possible to solve the problem quite precisely, effectively, and easily.

Çelik Bina Modellerinin Kolonlarında Molibden Profil Kullanımının Modal Parametrelere Etkisi

From past to present, building designs and materials used are developing. Especially against the destructive effects of ground movements and free vibrations on structures, many structural system designs and composite structure designs have been developed. The purpose of the composite structure design is to choose different types of materials according to the structural load-bearing system stress, in short, to choose the most advantageous material type according to the cross-sectional stresses or to eliminate the negative aspects of one material with the positive aspects of another material. It is a known fact that the dynamic performance of steel structure carrier systems is high under the influence of ground movements and free vibrations. However, in cases where the section geometry cannot be changed due to architectural concerns due to architectural design difficulties, there are cases where the rigidity of the structure is not sufficient. In such cases, profiles made of different materials other than steel can be used in order to increase the rigidity of the structure, especially in the columns, which are a very important component of the structural load-bearing elements. Therefore, in this study, the effect of using molybdenum profile instead of steel profile on modal parameters in model steel structure columns was investigated. In the light of the information obtained, a decrease of approximately 23.72 percent was observed in the period value in the 1st free vibration mode of the steel-molybdenum structure model. Thus, it is understood that the rigidity of the model steel structure system increases. In cases where it is not possible to change the architectural design in steel structures, it is recommended to use column profiles as molybdenum profiles instead of steel profiles in order to provide the necessary rigidity and increase rigidity.

Terahertz-infrared spectroscopy of Ge2Sb2Te5 films on sapphire: Evolution of broadband electrodynamic response upon phase transitions

Phase-change alloy Ge2Sb2Te5 (GST) forms a favorable material platform for modern optics, photonics, and electronics thanks to a pronounced increase in conductivity with thermally induced phase transitions from amorphous (a-GST) into cubic (c-GST) and then hexagonal (h-GST) crystalline states at the temperatures of ≃150 and ≃300°C, respectively. Nevertheless, the data on broadband electrodynamic response of distinct GST phases are still missing, which hamper the design and implementation of related devices and technologies. In this paper, a-, c-, and h-GST films on a sapphire substrate are studied using broadband dielectric spectroscopy. For all GST phases, complex dielectric permittivity is retrieved using Drude and Lorentz models in the frequency range of 0.06–50 THz or the wavelength range of ≃5000–6 μm. A contribution from the free charge-carriers conductivity and vibrational modes to the broadband response of an analyte is quantified. In this way, the Drude model allows for estimation of the static (direct current—DC) and dynamic (at 1.0 THz) conductivity values, caused by motions of free charges only, which are as high as σDC≃15 and 40 S/cm and σ1.0THz≃8.8 and 28.6 S/cm for the c- and h-GSTs, respectively. This overall agrees with the results of electrical measurements of GST conductivity using the four-point probe technique. The broadband electrodynamic response models obtained for the three GST phases are important for further research and developments of GST-based devices and technologies.

Effect of Graphene Reinforcement on Free Vibration and Material Properties of the FG-GPLRC Porous Cantilever Torsional Plate

This paper researches the free vibrations and corresponding material parameters of a functional gradient graphene platelets-reinforced composite (FG-GPLRC) cantilever torsional plates with both pore and graphene variations in the transverse direction. Utilizing the feature of closed-cell cellular solids, the mixture rule, and the modified Halpin–Tsai model, the material parameters of the composite are determined for different volume fractions of component materials. Then, using the classical plate theory (CLPT), the Rayleigh–Ritz technique and polynomials, the dynamic equation that can be used to obtain the free vibration mode shapes and frequencies of the rotating cantilever torsional plate is given. Comparison studies with the previous calculation results from available literature and the finite element (FE) models of cantilever plates are conducted, and the correctness of the present theoretical formulation and numerical calculation is verified. Finally, the effects of graphene platelet (GPL) distribution, porosity distribution (PD), GPL content, rotational speed, and average geometric size of GPL on free vibrations of the system are studied in depth.

Nonlinear system identification of a UAV model with distributed aerodynamics and flexible structure

This paper presents the nonlinear system identification of a slightly flexible 25 kg fixed-wing UAV in the time-domain using a computationally efficient distributed aerodynamics model and a linear structural dynamics representation. The equations of motion are formed by making use of the free vibration modes of the structure and the mean axes formulation. The structural modes and mode shapes are determined from ground vibration tests. The distributed aerodynamics, accounting for elastic deformations, are modeled using a quasi-steady stability and control derivative approach and by applying strip theory. Initial distributions for the derivatives are obtained from vortex-lattice-method calculations. For matching the model response to the measured response, parameters for scaling the initial derivative distributions are introduced. The flexible model is subsequently identified based on flight test data using the output error method in the time-domain and maximum-likelihood estimation. A good overall identification result is achieved with a close match of the fast aircraft dynamics. Finally, an evaluation is given on the suitability of the identified model for real-time simulation, loads’ estimation, and active load control law design.

Structural Damage Identification Using the First-Order Vibration-Mode-Based Frequency-Shift Flexibility Sensitivity Algorithm

The diagnosis of structural damage usually belongs to a mathematical inverse problem. This work presents a novel frequency-shift flexibility sensitivity algorithm for structural damage assessment using only the first-order vibration mode to achieve the goal of successfully identifying structural damage with fewer modal parameters. The core idea of the proposed method is to make the first-order vibration mode contribute the most to a structural flexibility matrix through the frequency-shift operation. A high-precision flexibility matrix can be obtained after the frequency-shift operation, which only needs the first mode of structural free vibration. Through this special advantage, structural damage coefficients can be accurately calculated by the frequency-shift flexibility sensitivity equation. Thus, a reliable identification result can be obtained according to the values of the calculated damage coefficients. In some engineering applications, another advantage of the proposed method is that it does not require a complete finite element modeling process, as long as a few lower-frequency vibration modes of the intact structure are measured. A truss structure and a beam structure are used as two numerical examples to demonstrate the proposed approach. The results show that the proposed method has higher calculation accuracy than the ordinary flexibility sensitivity method by using only the first-order vibration mode. The proposed method can overcome possible misdiagnosis of the ordinary flexibility sensitivity method. It also has been shown that the proposed method may have the potential to identify minor damage in a structure. Using the experimental data of a steel frame structure, the effectiveness and reliability of the proposed method have been further verified. The proposed method provides a simple way for structural damage identification with only a few vibration modal data.

Techniques for adjusting qualities of aircraft structural models for more effective aeroelastic flutter analyses

To solve numerically the problem of aeroelastic flutter, two virtual components are required: a structural finite element model (FEM) and an aerodynamic model. MSC Software package containing Nastran with embedded flutter solution enables to perform calculations with the use of an integrated aeroelastic model. For data block including the model and the set of entry parameters, flutter equation is being solved. On the basis of postprocessed output data curves of run for resulting parameters can be plotted which gives information of flutter critical point.Numerical results may be subject to some uncertainty. Result deviation could occur as effects of various factors resulting from inaccurate geometry or poor structural qualities of mapped aircraft. Structural defects affecting mass or stiffness matrices of FEM are often hard to identify. Aspects important for effective flutter solutions are strictly connected with the coincidence of structural properties of the real aircraft and its model. In the paper, the method of numerical flutter analysis was presented with the use of techniques available in MSC Nastran. Three assessment methods were proposed to identify structural parameters usable for validating FEMs. The method of mass correction relies on adjustment of model mass values and tuning the location of the center of mass of the model to the coordinates of the location of the center of mass of the real aircraft. For adjusting stiffness of model parts, very useful are results obtained from strength test of the real airframe. The completely reliable but costly method is performing ground vibration tests (GVT) to measure precisely resonant frequencies and natural modes of free vibrations. The results obtained from experiments should be treated as the referenced, but tuning FEM to them requires both adjusting numerical eigenfrequencies and normal modes.

Numerical Evaluation of Frequencies and the Modes of Free Vibrations of Isosceles Triangular Plates with Free Edges

Free vibrations are analyzed for isotropic plates in the form of isosceles triangles. We deduce the formula for the evaluation of the frequencies of free vibrations for plates of regular triangular shape with free edges and compute the coefficients of vibration mode and boundary conditions. The frequencies and modes of free vibrations of the isotropic thin isosceles triangular plates with free edges and different apex angles are computed by the finite-element method.

In-plane and out-of-plane free vibration analysis of thin-walled box beams based on one-dimensional higher-order beam theory

In this paper, the accurate free vibration characteristics of thin-walled box beam are discussed and the kinematic model is established by using fine shear deformation theory. The model adopts the in-plane and out-of-plane displacement fields including extension, torsion, warping and distortion, as well as the transverse shear due to bending and warping due to torsion. One-dimensional high-order beam theory is applied to the dynamic solution of thin-walled box beam, and the analytical solution of high free vibration modes of multi-deformation coupled modes under different boundary conditions is derived. The results show that warping, distortion and shear deformation play an important role in the free vibration characteristics of thin-walled box beam, and the validity of the model is verified. In addition, finite element software (ANSYS) is used for finite element simulation. The application of vibration mode in the structure design of thin-walled box beam is summarized, especially in the case of higher natural frequency. The calculation method is in good agreement with the finite element results.

Assessing the coupling strength between subsystems in (hybrid deterministic-)statistical energy analysis

Statistical energy analysis (SEA) is a standard approach to high-frequency vibro-acoustic analysis that relies on a conceptual division of the system into subsystems that are assumed to carry a diffuse field and to be weakly coupled. The weak coupling assumption means that the exchange of energy between subsystems can be described in terms of their uncoupled free vibration modes. In this work, a criterion is derived for assessing the coupling strength in the general case where the subsystems are rigidly coupled and/or via deterministic linear dynamic components. The criterion is elaborated such that it can be directly evaluated from quantities that appear in the SEA power balance. In this process, the hybrid deterministic-SEA approach is employed such that subsystems and connections of arbitrary complexity can be tackled in a rigorous way. In one of its approximate forms, the proposed general coupling strength criterion reduces to the gamma criterion that has appeared in the literature for assessing some special cases of coupling. The criterion is validated with a numerical example involving two diffuse plate subsystems connected via a deterministic beam, whose dynamic behaviour influences the coupling strength.

EQUILIBRIUM FORMS OF AN INITIALLY CURVED BERNOULLI–EULER BEAM IN ELECTRIC AND THERMAL FIELDS

On the basis of a geometrically nonlinear model of a Bernoulli-Euler beam, the influence of the amplitude and shape of the initial curvature on the nature and stability of the static equilibrium positions of the system under conditions of electrostatic and thermal effects has been studied. With the help of numerical methods of the theory of bifurcations, a complete parametric analysis of the diagrams of evolution of equilibrium positions for beams with initial curvature in terms of symmetric and asymmetric lower modes of free vibrations is performed. The regions in the parameter space of the values of the axial force of a mechanical or temperature nature and the strength of the electrostatic field, in which the elastic system has the property of bistability, are determined. The prospect of using the considered class of structures with projected geometry perturbations as sensitive elements of high-precision microelectromechanical sensors of various physical quantities is noted.

FREE VIBRATION ANALYSIS OF ORTHOTROPIC RECTANGULAR PLATES USING THE RITZ NUMERICAL METHOD

The main objective of this paper is to contribute to a better understanding of the dynamic response of FRP rectangular plates through modal analysis. The Ritz’s approximation method originated from the principle of minimum potential energy and was chosen to formulate the Eigenvalue problem. The natural frequencies and mode shapes are determined for the first 36 free vibration modes by assuming the double Fourier series functions for the transverse displacement. The most common Beam functions are used as a functional basis. Results are shown for the specific plate problem of Angle ply rectangular plate clamped at one edge, supported at two adjacent edges, and accessible at another edge (CSSF). It is observed that maximum natural frequencies are obtained under the clamped boundary of the plate on all edges of (CCCC) for the rectangular laminates.

Effect of Seismic Isolation on the Performance of High-Rise Buildings with Torsional Instability

Seismic bearings have been used to mitigate the harmful effect of the earthquakes. Torsion mode, one of the most important irregularities, generally increases the shear forces to the vertical members such as columns and shear walls in turn this may results in brittle failure of the reinforced concrete (RC) members. Accordingly, it is vital to eliminate the torsion failure mode or switch to the higher modes with lower mass contribution. This study has evaluated the seismic performance of a high-rise building with torsion mode through push-over analysis including nonlinear time history analyses. The damage conditions of RC structural members are defined considering the Eurocode definitions and general performance assessments of the building have been evaluated accordingly. Lead rubber bearings have been used for base isolation system. By using enough number of rubber bearings, the dominant torsion mode (first free vibration mode) has been shifted to higher modes. Various earthquake records have been used in non-linear dynamic analysis to evaluate the positive effects of the bearings. The results revealed that proper arrangement of rubber bearings in structural plan of ground floor can effectively improve dynamic behavior of a high rise building with torsional instability to achieve better seismic performance.

Free Vibration Analysis of Tapered Composite Aircraft Wing via the Finite Element Method

The responses of the structures used in engineering applications under the effects of static and dynamic forces are significant in the design phase. Determination of the response of dynamic forces for a structure is initially performed by the evaluation of free vibration characteristics that are mode shape of the structure and vibration frequencies. This paper presents modal analyses of tapered aircraft wing structures that consist of NACA4415 design and different common materials used in the aviation industry. Furthermore, the effect of winglets on natural frequencies is examined. The main wing structures as ribs and shells are drawn using CATIA and imported to ANSYS Workbench. Analyses have been carried out considering the aircraft wing as a three-dimensional cantilever beam by fixing one end (root chord) of the aircraft wing while the other end (tip chord) is free. The first ten modes of free vibration with their respective natural frequencies and mode shapes of the wing structures of the aircrafts are obtained. The results show that the winglets decrease the natural frequency noticeably and the shell material as Carbon Epoxy UD has been observed to have higher natural frequency compared with Kevlar Epoxy.