Free Vibration Frequencies Research Articles

Measurement and Analysis of the Vibration Responses of Piano Soundboards with Different Structures.

The effect of structure on the vibration response was explored for four piano soundboards with different but commonly adopted structures. The vibration response was obtained using the free-vibration method, and the values of the dynamic modulus of elasticity and dynamic shear modulus obtained using the free-vibration frequency method (EF and GF) were compared with the dynamic modulus of elasticity obtained using the Euler beam method (EE) and dynamic shear modulus obtained using the free-plate torsional vibration method (GT), respectively. It was found that the soundboards with different structures had different vibration modes and that excitation at different locations highlighted different vibration modes. For all the soundboards analyzed, the EE and GT were higher than EF and GF by 2.2% and 24.3%, respectively. However, the trends of the results of these methods were the same. The four piano soundboards with different structures possessed varying dynamic moduli of elasticity and dynamic shear moduli. These rules are consistent with the grain directions of the soundboards and the anisotropy of the wood (the direction of the units of the soundboards). The results show that the vibration mode of the piano soundboard is complex. The dynamic elastic modulus of the soundboard can be calculated using the Euler beam method. The results provide a reference for studies on the vibration response, material selection, production technology, and testing of piano soundboards.

Частоты собственных колебаний призматических тонких оболочек

Частоты собственных колебаний призматических тонких оболочек

Nonlinear dynamical characteristics of carbon nanotube-reinforced composite beams with piezoelectric actuators and elastically restrained ends under thermo-electro-mechanical loads

Nonlinear free vibration and dynamical responses of carbon nanotube (CNT) reinforced composite beams with surface-bonded piezoelectric layers and tangentially restrained ends under thermo-electro-mechanical loads are investigated in this paper. The properties of constitutive materials are assumed to be temperature-dependent and effective properties of nanocomposite are estimated using an extended rule of mixture. Unlike previous studies, the present work considers the effects of tangentially elastic constraints of two ends on the nonlinear dynamic characteristics of hybrid beams. Motion equation is established within the framework of Euler-Bernoulli beam theory taking into account von Kármán nonlinearity. Analytical solution is assumed to satisfy simply supported boundary conditions and Galerkin procedure is employed to obtain a time ordinary differential equation including both quadratic and cubic nonlinear terms. This differential equation is numerically solved employing fourth-order Runge-Kutta scheme to determine the frequencies of nonlinear free vibration and nonlinear transient response. Parametric studies are executed to examine numerous influences on the nonlinear dynamical characteristics of hybrid nanocomposite beams. The study reveals that tangential constraints of ends substantially effect the frequencies and dynamic response of the beam, especially at elevated temperatures. The results also indicate that nonlinear dynamic responses can be controlled effectively by means of piezoelectric actuators and elasticity of tangential constraints of ends should be considered in design of piezo-CNTRC beams.

Finite element based dynamic analysis of a porous exponentially graded shaft system subjected to thermal gradients

Abstract In modern applications, demand for the rotor shafts made of FGM has increased in aerospace, gas turbine and turbo jet engines due to its improved dynamic characteristics at elevated temperatures. The dynamic vibration response of a functionally graded Jeffcott rotor-bearing system with induced porosities has been studied for non-uniform porosity distributions. The functionally graded (FG) shaft’s radial direction exhibits continuous variation in the material qualities. The FG rotor’s cross-sectional material properties have been graded using the exponential law. The temperature gradients have been considered to be varied across the cross-section based on exponential temperature distribution. For the first time, using the FE method, it has been possible to study the impact of induced porosities on the free vibration frequencies and steady-state vibration responses of the functionally graded rotor-bearing system. This has led to the development of a two-node porous functionally graded rotor element with uneven porosity distributions using the Timoshenko beam theory. It has been observed that the free vibration frequencies and the FG rotor-bearing system critical speeds are affected by increased porosity and thermal gradients. The frequency responses demonstrate that as temperature gradient and volume fraction of porosity rise, the FG rotor-bearing system’s critical speeds shift to the left.

Formulas for Calculating Deformations and Natural Frequency of Free Vibrations of a Hexagonal Tower

Statement of the problem. A scheme of a six-sided prismatic statically determinate spatial girder is proposed. The task is set by induction to derive formulas for the dependence of the deflection of the structure and the lower limit of the main frequency of natural vibrations on the number of panels along the height of the prism. Materials and methods. The forces in the rods along with the reactions of the supports are found in an analytical form by means of the method of cutting nodes in the Maple symbolic mathematics system. One of the nodes at the base of the truss has a spherical support, one has a cylindrical support, the remaining four supports are racks. The top deflection is determined by the Maxwell-Mohr formula. From the analysis of the sequences of coefficients in the formulas for individual structures with a different number of panels, their common members are determined, which are included in the desired calculation formula. The Dunkerley method is used to estimate the first frequency of free oscillations. Results. For various types of loads, formulas for the dependence of girderf deflections on the number of panels are obtained. The coefficients in the solution are polynomial in the number of panels. The derived analytical dependence of the first frequency on the number of panels in comparison with the numerical solution has a small error, which decreases with increasing number of panels. Conclusions. A design of an axisymmetric statically determinate tower-type girder has been developed, which allows analytical solutions to the problem of deflection and the problem of the first natural frequency for an arbitrary number of panels. The resulting formulas can be used to assess the accuracy of numerical solutions and for preliminary calculations of models of structures of this type.

Egg-Clutch Biomechanics Affect Escape-Hatching Behavior and Performance.

Arboreal embryos of phyllomedusine treefrogs hatch prematurely to escape snake predation, cued by vibrations in their egg clutches during attacks. However, escape success varies between species, from ∼77% in Agalychnis callidryas to just ∼9% in A. spurrelli at 1 day premature. Both species begin responding to snake attacks at similar developmental stages, when vestibular mechanosensory function begins, suggesting that sensory ability does not limit the hatching response in A. spurrelli. Agalychnis callidryas clutches are thick and gelatinous, while A. spurrelli clutches are thinner and stiffer. We hypothesized that this structural difference alters the egg motion excited by attacks. Since vibrations excited by snakes must propagate through clutches to reach embryos, we hypothesized that the species difference in attack-induced hatching may reflect effects of clutch biomechanics on the cues available to embryos. Mechanics predicts that thinner, stiffer structures have higher free vibration frequencies, greater spatial attenuation, and faster vibration damping than thicker, more flexible structures. We assessed clutch biomechanics by embedding small accelerometers in clutches of both species and recording vibrations during standardized excitation tests at two distances from the accelerometer. Analyses of recorded vibrations showed that A. spurrelli clutches have higher free vibration frequencies and greater vibration damping than A. callidryas clutches. Higher frequencies elicit less hatching in A. callidryas, and greater damping could reduce the amount of vibration embryos can perceive. To directly test if clutch structure affects escape success in snake attacks, we transplanted A. spurrelli eggs into A. callidryas clutches and compared their escape rates with untransplanted, age-matched conspecific controls. We also performed reciprocal transplantation of eggs between pairs of A. callidryas clutches as a method control. Transplanting A. spurrelli embryos into A. callidryas clutches nearly tripled their escape success (44%) compared to conspecific controls (15%), whereas transplanting A. callidryas embryos into different A. callidryas clutches only increased escape success by 10%. At hatching competence, A. callidryas eggs are no longer jelly-encapsulated, while A. spurrelli eggs retain their jelly coat. Therefore, we compared the hatching response and latency of A. spurrelli in de-jellied eggs and their control, jelly-encapsulated siblings using manual egg-jiggling to simulate predation cues. Embryos in de-jellied eggs were more likely to hatch and hatched faster than control siblings. Together, our results suggest that the properties of parentally produced egg-clutch structures, including their vibration biomechanics, constrain the information available to A. spurrelli embryos and contribute to interspecific differences in hatching responses to predator attacks.

Opto-electro-thermo-mechanical behaviours of perovskite plates

Metal-halide perovskites are rapidly emerging as promising candidates for next-generation solar cells and photoelectric technologies. However, their structural behaviours considering opto-electro-thermo-mechanical properties in multi-physics fields have not yet been fully understood, thus limiting their engineering applications. This paper proposes a new opto-electro-thermo-elastic model taking into account photostriction, photothermal effect, electrostriction, and piezoelectricity for lead halide perovskites and investigates their static and dynamic opto-electro-thermo-mechanical behaviours. The governing equations of perovskite plates are derived within the theoretical framework of first-order shear deformation theory and are numerically solved using the Chebyshev-Ritz method. Comprehensive parametric studies are carried out to examine the influences of multi-physics fields on the critical light intensity, critical electric field, static bending, elastic buckling, free vibration, as well as dynamic responses of perovskite plates. Numerical results show that light illumination, photoinduced temperature rise, and applied electric field play an important role in bending deformation, dynamic deflection, critical buckling load, and free vibration frequency hence need to be taken into consideration for the design of perovskite-based photoelectric devices.

Assessment of machine learning methods predicting the axial vibration frequencies of microbars

AbstractMicrobars are one of the important components of microelectromechanical systems. With the recent increase in their applications, the importance of understanding their mechanical response has become an important topic. In this study, for the first time, the mechanical behavior of microbars based on the strain gradient theory is investigated using a machine learning (ML) approach. Four distinct ML models, namely artificial neural network (ANN), support vector regression (SVR), decision tree regression (DTR), and random forest regression (RFR), are developed for microbars with clamped boundary conditions. The performance of these models is individually assessed using five different metrics: coefficient of determination (R2), Mean Absolute Error (MAE), root mean square error (RMSE), mean absolute percentage error (MAPE) and Nash‐Sutcliffe efficiency coefficient (NSE). The best‐performing model is selected based on these comparisons. Additionally, a semi‐analytical approach is employed to determine the natural frequencies of microbars under general elastic boundary conditions using the Fourier sine series and Stokes' transform. While the R2 value for all four models indicated a good fit of 0.999, the percentage difference in MAE and RMSE values between the training and testing data for DTR and RFR models was relatively higher as compared to ANN and SVR. The results showed that ANN and SVR models exhibit the best performance in predicting the natural frequencies on both training and testing data across all three metrics. Finally, a study on the free axial vibration frequencies of microbar under various effects was conducted.

Investigation of the vibration problem of sandwich heterogeneous orthotropic plates

In this study, the solution of the free vibration problem of sandwich plates consisting of heterogeneous orthotropic layers is presented. First, the Kirchhoff-Love theory (KLT) for homogeneous plates is extended to sandwich plates composed of heterogeneous orthotropic layers. Within the framework of Donnell theory, after establishing the basic relations of multilayer plates, governing equations are derived based on Airy stress and deflection functions. The solution of the governing equations is carried out by the Galerkin method and the analytical expression is found for the linear frequency of sandwich plates consisting of heterogeneous orthotropic layers. Finally, the effects of various factors such as heterogeneity, number of layers and their arrangement on the free vibration frequency of sandwich plates are examined.

Residual stress dynamic detection and vibration characteristics of wire saw busbar

Diamond wire saw cutting is the primary method for processing single crystal silicon carbide (SiC) and silicon (Si) in the photovoltaic and semiconductor industries, the diamond wire saw is formed by busbar electroplate diamond particles, and the diamond wire saw busbar is drawn from steel wires, and the residual stress generated during the drawing process is a critical factor to determine whether the busbar is easy to break. This paper proposes a method to assess residual stress by measuring the free vibration frequency of the busbar, which detects residual stress of busbars with extremely fine and ultra-long (lengths of up to 250 km and diameters as small as 0.04 mm) for real-time, accurate, non-destructive way. First, the residual stress distribution of the drawn busbar is obtained by Abaqus simulation, according to the simulation results, this paper analyzes the mechanical mechanism underlying different types of free bending in the busbar, and develops the relationship between the free bending circle diameter and the residual stresses; Then, the residual stresses are equated with the axial tensile stresses of the busbar by numerical methods, the equivalent mathematical model of the stresses is developed, and the correctness of the model is verified by experiment. Finally, the dynamic model of the moving busbar with residual stresses is established, the effects of different tensions and velocity on the free vibration characteristics of the busbar under residual stresses are analyzed, and the correctness of the detection method is verified by numerical calculations and experimental comparisons. The results show that the residual stress of busbar drawing can be equivalent to tensile force numerically, and the larger the residual stress, the smaller the circle diameter of the busbar, and the larger the first-order natural frequency of free vibration.

Nonlinear vibration analysis of a double curved shallow sandwich shell in which the core made of three-phase nanocomposite and the two-outer layer of electromagnetic materials

This paper investigates the nonlinear vibrations of a double curved shallow sandwich shell with a three-phase nanocomposite core and two outer electromagnetic layers considering the thermal environment's influence. Relying on the Halpin–Tsai model, the core is assumed to be made of some composite plies incorporating three constituents: carbon nanotubes (CNTs) or graphene nanoplatelets (GPLs) as nanocomposites, fibers and polymeric matrix (resin). Governing equations calculating the nonlinear vibration of the double curved shallow shell were obtained by combining Reddy's first-order shear deformation theory (FSDT) and the Von Kármán geometrical nonlinearity. Later, this system of equations is transformed into the nonlinear ordinary differential expressions by utilizing the Galerkin procedure. Thus, the effects of material properties, temperature, geometrical properties, elastic foundations to the frequency of linear and nonlinear free vibration, and amplitude–frequency relations of the double curved shallow sandwich shell were discussed. The results demonstrate that the free vibration frequency of the shell increases if the magnetic field enhances, whereas that decreases when the voltage expands in both models as CNTs and GPLs. Simultaneously, the shallow shell has a better ability to force load compared to a plate with the same dimensions. These outcomes can make a profit in the design and manufacturing of new smart structures applied in energy harvesting, and medicine.

Advanced Frequency Study of Thick FGM Cylindrical Shells by Using TSDT and Nonlinear Shear

For the advanced frequency study of thick functionally graded material (FGM) circular cylindrical shells, it is interesting to consider the extra effects of nonlinear coefficient term in third-order shear deformation theory (TSDT) of displacements on the calculation of varied shear correction coefficient. The formulation for the advanced nonlinear shear correction coefficient are based on the energy equivalence principle. The values of nonlinear shear correction coefficient are usually functions of nonlinear coefficient term of TSDT, power-law exponent parameter and environment temperature. The free vibration frequencies of thick FGM circular cylindrical shells are investigated with the simply homogeneous equation by considering that simultaneous effects of the TSDT, the nonlinear shear correction coefficient of transverse shear force and the two direction of mode shapes. The novelty is more important and reasonable for the thick FGM circular cylindrical shells, especially for the ratio of length to thickness is five by considering the effects of advanced nonlinear shear correction coefficient and nonlinear coefficient term of TSDT on the advanced calculation of fundamental first natural frequencies.

Free Vibration Frequency Prediction to Design Optimum Adhesively Bonded Composite Double Lap Joint

Free Vibration Frequency Prediction to Design Optimum Adhesively Bonded Composite Double Lap Joint

Axisymmetric thermal vibration analysis of graded porous circular plates based on physical neutral surface and Reddy’s third-order shear theory

In this article, the free vibration of functionally graded porous (FGP) circular plate with various boundary conditions are investigated by means of the differential quadrature method (DQM) based on Reddy’s theory. Performance of the materials varies continuously in the whole thickness and two cosine forms of nonuniform porosity distribution along its thickness are considered. Hamilton’s principle is adopted to derive the governing equation of the system, which effectively considers the effects of thermal stress. It is worth noting that due to the introduction of physics neutral surface, reducing stretching-bending coupling effect, so the complex governing equations have been appropriately simplified. Then, using the DQ method, the natural frequencies of free vibration of FGP circular plates subjected to thermal environments with various boundary conditions were obtained. Convergence and comparative research are performed to prove the convergence, reliability, and accuracy of the DQ method. The effects of various factors such as porosity coefficient, porosity distribution pattern, temperature rise, thickness-diameter ratio, and boundary conditions on the natural frequencies are discussed in detail.

Magneto-thermo-elastic coupled free vibration and nonlinear frequency analytical solutions of FGM cylindrical shell

The magneto-thermo-elastic coupled free vibration of functionally graded materials (FGM) cylindrical shell is investigated. Based on the physical neutral surface theory and Kirchhoff-Love theory, the expressions of internal force and internal torque are obtained by considering the constitutive relationship that includes thermal stress. According to the electromagnetic elasticity theory, the model of Lorentz forces of the shell in magnetic field are derived. The expressions of potential energy, kinetic energy and its variational operators are given by introducing geometric nonlinearity, respectively. Based on Hamilton principle, the magneto-thermal-elastic coupled vibration equation of FGM cylindrical shell in multi-physical field is established. The displacement functions under simply supported boundary conditions are solved, which are combined with Galerkin method for derivation of nonlinear ordinary differential equations. The second-order approximate analytical solution of natural frequency is obtained by applying the multi-scale method. The characteristic curves of natural frequency with different parameters are drawn through numerical examples. The results show that reducing the volume fraction index and increasing the initial amplitude can lead to the increase of the natural frequency under transient conditions. With the increase of temperature and magnetic induction intensity, the natural frequency decreases. In addition, the natural frequency is also influenced by the shell size and volume fraction index.

Dynamic response of composite materials subjected to low velocity impacts in water: A numerical study

The damage tolerant design of marine composite structures for impacts is dependent on empirical data to ensure continuing seaworthiness. Impact studies on composite laminates are typically performed according to ASTM D7136 which was originally developed by the aerospace industry. The marine operating environment, however, is vastly different. Impact characterisation and damage prediction of underwater composite structures is hindered by the limited understanding on the effects of fluid-structure interaction of a surrounding water medium with the composite structure and impactor. A coupled Eulerian-Lagrangian (CEL) numerical model is presented in this work with the aim of advancing fundamental understanding of the effects of a fluid on the transient dynamic response of composite plates subjected to foreign body impact. The added mass effect of the water increased the peak impact force while reducing the free vibration frequency and peak displacement. The virtual added mass factor also reduced significantly with increasing laminate thickness. This work will pave the way in improving the damage tolerant design of underwater composite structures as will computationally efficient methods of simulating such events.

Analysis of isotropic stiffened plate structure

The analysis of stiffened plate has been carried out using finite element method. The study is divided into static and dynamic analyses. Primarily, free vibration frequencies of the stiffened plate have been determined followed by determination of deflection and von Mises stress for the stiffened plate subjected to the unit uniformly distributed load (1 kN/m2). Further, deflection and von Mises stress are determined, when the plate structure is subjected to hydrostatic loading. The analyses, static and dynamic, have been done using ANSYS Workbench 15.0. A single flat stiffener is used to stiffen the plate structure and the results are evaluated by varying the stiffener geometry, keeping the volume of material almost same as the unstiffened plate. Stiffened plate structure is very popular in the structural engineering domain and has a wide range of engineering applications from ship to aerospace structures. The material used to model stiffener is extracted from the primary plate for keeping the same volume of material. The plate structure is stiffened to reduce the out-of-plane bending even if the same (approximately) amount of material is utilised. Eight-noded plate bending and two-noded beam elements are used to model the plate and stiffener, respectively. The obtained results are compared with the published results and the effect of varying stiffener geometry is further examined. It may be concluded that the response of the stiffened plates are better compared to unstiffened plate even if the same or even less quantity of material is utilised in stiffening. The present findings are useful for the designers when the plate structure is influenced by the surrounding fluid.

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.

Consideration of supports stiffness at torsional and longitudinal vibrations of the rod

The study of the impact of the stiffness of the supports on the value of the first natural frequency of torsional and longitudinal free vibrations of the spring-hinged rod was carried out. In both cases of vibrations the mathematical formulation of the problem of determining the first natural frequency of vibrations reduces to compiling and solving the corresponding differential equation under given boundary conditions. The solution of this complex mathematical problem is performed by the numerical method in a specially created program for many combinations of supports stiffness. To obtain an analytical solution, a quadratic approximation of the obtained values was performed, which made it possible to obtain simple dependencies convenient for engineering applied calculations. The quadratic approximation also allowed solving the inverse problem of determining the required stiffness of supports for the required value of the first natural frequency of vibrations. The obtained solutions were compared with known solutions, which showed that the approximation error is less than 5% if the stiffness of the rod supports of the same order. The obtained results can be used in engineering applied calculations of natural frequencies of torsional and longitudinal vibrations of rods with elastic supports.

Asymptotic behavior of resonant frequencies in resonant ultrasound spectroscopy.

Resonance ultrasound spectroscopy is a non-destructive technique used to assess materials' elastic and anelastic properties. It involves measuring the frequencies of free vibrations in a carefully prepared sample to extract material properties. In this paper, we investigate the asymptotic behavior of eigenfrequencies. Our primary focus is on analyzing the asymptotic behavior of eigenfrequencies, aiming to understand their rate of growth and convergence. We also make observations regarding the impact of elastic constants on eigenfrequencies.