Free Vibration Method Research Articles

Vortex-induced vibration characteristics of twin-box girder section in long-span bridge: A numerical evaluation method

The vortex-induced vibration (VIV) characteristics of twin-box girders are critical in the design of long-span bridges. This study aims to investigate the VIV behavior of a twin-box girder section and to deepen the understanding of the mechanisms underlying VIV. A numerical approach that integrates both free and forced vibration analyses is developed to predict VIV performance. The VIV amplitude is evaluated using the free vibration method within the wind speed range identified by the forced vibration method. Comparisons with wind tunnel test results demonstrate the feasibility of this approach. Meanwhile, flow structure and aerodynamic characteristics of the twin-box girder were analyzed in relation to the VIV mechanism. The results show that flow restrictor plate could effectively suppress vertical VIV at a wind attack angle 0° and reduce the VIV amplitude by 50% at +3°wind attack angle. The central opening slot is identified as a significant factor in inducing VIV, as the upper and lower vortices interact with the airflow passing through the respective sections of the slot. The flow restrictor plates positioned at the upper section of the opening slot cause most upper vortices to shift downstream at wind attack angles of 0° and +3°. This shift results in minimal airflow reflux at the slot, which is crucial for optimizing VIV performance. At a wind attack angle of +3°, the vortices at the lower part of the slot persist, albeit with reduced size. These changes in vortex behavior underscore the sensitivity of vortex formation to variations in wind attack angles.

A novel hybrid function-based Ritz method for static and free vibration of axially loaded bi-directional functionally graded porous beams

This paper introduces a novel hybrid function, synthesizing trigonometric and Hermitian polynomials, to analyze the structural responses of bi-directional functionally graded porous (2D-FGP) beams. The displacement field of the beam is based on the higher-order shear deformation theory. A power-law relation governs the material variation along the x- and z-directions, with the porosity manifesting in three distinct distributions. The governing equations are deduced through the application of Lagrange’s equation. The proposed hybrid function is developed to delineate the displacement field. The deflections, stresses, buckling loads, and natural frequencies of 2D-FGP beams are computed for diverse gradation exponents in the x- and z-directions, boundary conditions, porosity type, porosity coefficient, and slenderness ratio. A comparative assessment with finite element method results indicates that the findings agree with available data. It substantiates the accuracy and efficiency of the proposed methodology.

Analytical method for free vibration of steel pipe piles partially submerged in water and penetrated in layered soil

This paper presents a novel analytical method for dynamic characteristics of steel pipe piles submerged in water based on the thin shell theory by using the state space method. The state equation with respect to the length direction is first derived by idealizing the steel pipe pile in water as a thin cylindrical shell, where the soil around the pile is modelled as an elastic foundation with stiffnesses in three directions and the surrounding seawater is considered through added equivalent mass. The governing equations for the free vibration of steel pipe piles are derived after the expansion of Fourier series for the state vectors along the circumferential direction. The frequency equation for free vibration and its corresponding mode shapes are then obtained, which is used to analyze the dynamic characteristics of steel pipe piles in water. Due to the advantage of the state space method, arbitrary boundary conditions of the steel pipe pile are conveniently achieved as well as the effects of seawater, soil, the piling hammer and guiding frame. Finally, the present analytical method is verified and demonstrated in detail by several numerical examples, and results show its simplicity and efficiency.

Damping properties of some tropical wood species determined by free transverse vibration method

ABSTRACT This article describes a non-destructive method for determining the damping properties of wood. It was based on the analysis of the attenuation of the free transverse vibrations of a fixed support-free specimen following excitation. Cameroonian forests abound important species that were much in demand for structural applications and were often subjected to dynamic loads. However, in the literature and in the available databases, very little information was available on the vibration properties of these wood species. To solve this problem, three specimens of twenty-four wood species with a cross-section of 10 × 20 mm, with a length of 1600 mm were prepared and subjected to a free vibration test in the fixed support-free configuration. The accelerations as a function of time were obtained. By transforming the recorded accelerations into displacements, it was allowed to determine the logarithmic decrement by mean the first twenty-one amplitudinal peaks. Thanks to the interrelationships, the viscous damping ratio, loss factor, quality factor and specific damping ratio were deduced. The results obtained were relatively variable between species. Compared with the data available in the literature for the same species, there was a considerable difference.

Testing of wood shear modulus based on cantilevered square plate torsional vibration method

ABSTRACT The application of the energy method has proposed a dynamic testing method for the shear modulus in two mutually perpendicular directions on the principal surface of wood using a cantilevered square plate torsional vibration method. In the study, this method was applied to measure the shear modulus of larch wood in the LT and TL directions, as well as the longitudinal and transverse shear modulus of Laminated Veneer Lumber (LVL) sheets. The validity of the results was confirmed through experimental verification using the asymmetric four-point bending beam method, the free plate torsional vibration method, and the free square plate torsional mode method. Simulation calculations of the principal shear modulus for six species of trees with width-to-thickness ratios ranging from 10 to 25 were also conducted using this method. The conclusions drawn suggest that a width-to-thickness ratio of 15 for the cantilevered square plate provides sufficient accuracy for testing the principal shear modulus of wood. Furthermore, the difference in shear modulus between two mutually perpendicular directions on the principal surface of wood is dependent on the width-to-thickness ratio of the cantilevered square plate, indicating that this method outperforms other dynamic methods for testing wood shear modulus.

Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Метод чисельного визначення динамічних характеристик лопаті повітряного гвинта з композиційного матеріалу

The subject matter of this article is the dynamic characteristics of a composite propeller blade. Determination of the modes and frequencies of natural vibrations is necessary to predict the dangerous resonance modes of aircraft engines and to identify the most stressed local zones of the blade surface. The goal of this study is to develop a verified method for determining the dynamic characteristics of composite rotor parts of aircraft engines based on the known properties of the structural components of the composite material. A general mathematical statement of the problem of elasticity theory for the analysis of the dynamic characteristics of composite structures is described. A complete system of equations that describes the mechanical state of the body within the framework of the continuum mechanics approach was developed. Geometric modeling of the propeller blades was performed. Modeling and numerical investigation of the natural frequencies and mode shapes of the propeller blade were performed using the finite element method using the ANSYS software package. Based on the results of numerical research of the stressed and deformed state of the propeller blade, the first five natural vibration frequencies and the distribution of the local stress field were determined. The most stressed local zones on the blade surface were determined for each form of natural vibration. The propeller blade model was verified using an experimental study of the first five forms and frequencies of natural blade vibrations. The eigenfrequencies of the blade vibration were experimentally determined using the method of free (natural) vibrations. The resonance method was used to experimentally determine the resonant frequencies and vibration modes of the blade. The distribution of the blade deformation field was investigated using the strain gauge method. The highest error in the verification of numerical and experimental research is 4.11% for the fourth vibration frequency. Conclusions. The scientific novelty of the results obtained is that the effective elastic properties of the composite material for calculations should be determined using the procedure of numerical homogenisation of composite materials of different reinforcement structures by the properties of the matrix and fibers. The method does not require experimental determination of the effective elastic constants for the layers of blade components of different weave architecture patterns.

Spectral collocation method for free vibration of sandwich plates containing a viscoelastic core

Viscoelastic constrained layer damping is a simple and efficient method for reducing noise, vibration, and fatigue in metal structures. In this work, a spectral collocation method utilizing a layer-wise plate theory was established to inspect the vibration characteristics of a sandwich plate with a viscoelastic core. The displacements of each layer satisfied the Mindlin plate theory. The core's transverse shear stress remained constant. The three layers’ transverse displacements were assumed to be identical. The displacement fields were reduced to nine variables by utilizing the interfaces’ continuity of the displacements. The equations of motion were obtained by utilizing Hamilton’s principle. The viscoelastic core’s material properties were considered as frequency-dependent. To address the complex eigenvalue problem, an iterative algorithm was used. The theoretical results were compared with published theoretical and experimental data for fully clamped rectangular sandwich plates to validate the proposed method. The modified Oberst beam method was applied by fitting the impulse response function to obtain the adhesive’s frequency-dependent storage modulus and loss factor. The natural frequencies and corresponding loss factors of three sandwich plates with different adhesives were measured by conducting impact tests. The numerical results from the present theoretical method agreed well with the measured results.

An analytical method for free vibrations of the fuel rod with non-uniform mass of small modular reactor

The intricate internal structure of fuel rods results in a non-uniform mass distribution, making it imperative to employ analytical methods for accurate assessment. The study utilizes Euler beam theory to derive the transverse vibration equation for beams with varying mass distribution. The approach involves transforming the non-uniform mass beam into a multi-segment beam with concentrated mass points. Modal function relationships between adjacent uniform segments are established based on continuous conditions at connection points. This transformation leads to the conversion of the variable coefficient differential equation into a nonlinear matrix equation. The Newton-Raphson method is then applied to calculate the characteristic equation and mode shapes, essential for determining natural frequencies. To validate precision, the results obtained are compared with those derived from the finite element method. Furthermore, the developed method is employed to assess the impact of gas plenum location and length on the natural frequency of fuel rods. The proposed methodology serves as a rapid design tool, particularly beneficial during the design phase of fuel rods with non-uniform mass distribution, aiding in configuring structural aspects effectively.

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.

An exact method for free vibration of beams and frameworks using frequency-dependent mass, elastic and geometric stiffness matrices

The frequency-dependent mass, elastic and geometric stiffness matrices of an axially loaded Bernoulli-Euler beam are developed through rigorous application of symbolic computation. These three matrices are related to the corresponding dynamic stiffness matrix so that free vibration analysis of axially loaded beams and frameworks can be carried out in an exact manner by applying the Wittrick-Williams algorithm as solution technique. Representative results from the proposed theory are presented for different boundary conditions of beams and frameworks, carrying tensile and compressive loads. Comparative results from finite element method are also presented. The duality between the free vibration and buckling problems is captured in that when the compressive load in a beam or frame approaches its critical buckling load, the fundamental natural frequency tends to zero and thus, buckling can be thoughtfully interpreted as free vibration at zero frequency. The investigation has opened the possibility of including damping in free vibration analysis of beams and frameworks when applying the dynamic stiffness method.

A novel Trefftz-based meshfree method for free vibration and buckling analysis of thin arbitrarily shaped laminated composite and isotropic plates

A novel Trefftz-based meshfree method for free vibration and buckling analysis of thin arbitrarily shaped laminated composite and isotropic plates

Research progress on dynamic testing methods of wood shear modulus: A review

Wood is a non-homogeneous and orthotropic natural polymer material. It is important to test the wood shear modulus and elastic constants accurately and reliably using dynamic methods. Based on the introduction of the advantages of six common methods for dynamic testing of wood shear modulus, such as free plate torsional mode method, free bar torsional vibration method, and Timoshenko beam iterative method, issues associated with the applicability and accuracy of these methods are also pointed out. Recent methods, such as the free square plate torsional mode method and the square plate static torsional strain method, that were developed to dynamically test the shear modulus of wood and wood composite materials, are presented as effective ways to tackle these issues. These new approaches are expected to provide beneficial technical support for using small specimens, overcoming the size effect of specimens, simplifying the testing procedures, improving the test accuracy, and expanding the application range in the dynamic testing of wood shear modulus. These approaches have practical significance in promoting the industrialization and development of structural engineering, furniture and interior decoration, transportation, military, and musical instrument industries.

Development of spectral element method for free vibration of axially-loaded functionally-graded beams using the first-order shear deformation theory

In this study, the spectral element method to assess free vibration of axially-loaded functionally-graded beams according to the first-order shear deformation theory is developed. In this approach, the underlying equations of motion and related boundary conditions were determined by using Hamilton's principle. Analytical solutions are presented for simply-simply, clamped-clamped, clamped-free, and clamped-simply supported FG beams. The general solutions corresponding to governing equations determined for axially loaded FG beams are presented in the spectral-element matrix. A comparison is performed to validate the proposed formulation and solution between the obtained results with the existing solutions provided in previous studies. Finally, the parametric study is developed to investigate the impacts of slenderness ratio, end supports, gradient parameter, and axial load value on the free vibration characteristics in the axially-loaded FG beams. A comparison of the results of the developed theory and the results of the existing solutions shows that the proposed solution developed based on the spectral element method is the appropriate accuracy and efficiency in determining the dimensionless natural frequencies parameter. Based on the findings, all four parameters are effective in the free vibration of axially-loaded FG beams.

A semi-analytical method for free vibration of semi-closed shells of revolution with variable curvature

This study is concerned with the free vibration analysis of semi-closed shells of revolution with variable curvature, with a focus on the analytical solution to the natural frequency of such shells. A semi-closed shell of revolution with variable curvature is firstly divided into some narrow small shell segments, then the first small shell segment with a closed shell apex is approximated with a semi-closed conical shell, while the remaining small shell segments are approximated with a series of progressively larger truncated conical shells. Since each small conical shell is part of the large semi-closed shell of revolution with variable curvature and has to vibrate in sync with the large shell, the analytical solution to the axisymmetric free vibration of the large shell can be transformed into the analytical solutions to the synchronous free vibration of these small conical shells. The general solutions for the natural frequencies of all small conical shells are analytically derived under the condition that conical shells have zero meridional curvature. The undetermined constants in the general solutions are determined by letting all natural frequencies of these small conical shells be equal. Finally, the analytical solution for the axisymmetric free vibration of semi-closed shells of revolution with variable curvature is presented and is proved to be basically reliable by conducting a confirmatory experiment based on polymer 3D-printing technique.

A novel eddy current damper system for multi-mode high-order vibration control of ultra-long stay cables

Recently, the multi-mode high-order wind-induced vibration of the ultra-long stay cables has been observed on several cable-stayed bridges, which may result in a visual panic of the users and the fatigue of the stay cables at anchorage ends. In order to investigate the multi-mode high-order wind-induced vibration characteristics of ultra-long stay cables, a vibration monitoring system (VMS) of the stay cables is established on the Sutong Bridge (STB). Furthermore, to suppress the in-plane and out-of-plane vibration responses of the stay cables simultaneously, a novel eddy current damper system (ECDS) is proposed in this study. The damping coefficients of the ECDS for multi-mode high-order vibration of the stay cables are determined via a modified method. Moreover, laboratory tests are conducted to evaluate the performance of the developed dampers. Furthermore, the field tests of seven stay cables of the STB are carried out to identify the modal damping ratios of the cable-damper system via the free vibration method. The results show that it is feasible to provide sufficient additional modal damping ratios for multi-mode high-order vibration of ultra-long stay cables applying the ECDS proposed in this study. Finally, the one-year field measurements of the wind-induced vibration responses of the seven stay cables of the STB are carried out to validate the ECDS suppress effects on multi-mode high-order vibration control of ultra-long stay cables. The measurement results show that the multi-mode high-order vibration responses of the monitored stay cables of the STB are effectively suppressed.

An extended separation-of-variable method for free vibration of rectangular Reddy plates

The Reddy plate theory is widely used in the vibration analysis of thick plates, but there are few closed-form analytical solutions for free vibration of rectangular Reddy plates with arbitrary homogenous boundary conditions. This work is to develop the extended separation-of-variable (SOV) method for the free vibration analysis of rectangular Reddy plates. In the present method, the mode functions are in an explicit and closed form, and the natural frequencies corresponding to two-direction eigenfunctions are mathematically assumed to be independent of each other. According to two pairs of boundary conditions, the closed-form mode functions and frequency equations are achieved. The present method is numerically compared with other approximate and analytical methods, and some valuable observations are presented. The present solutions are exact for the plates with at least two opposite sides simply supported, and agree well with the results in literatures for other boundary conditions.

Measuring the natural frequencies of knitted materials for protection against vibration

In this study, the use of knitted structures as potential vibration isolators or parts of protective equipment is proposed. Four types of knitted structures were made from different yarns by varying the stitch depth. By combining these variables, a series of 48 patterns were tested for vibration isolation using the free vibration method to measure their natural frequencies. Knowledge of the natural frequencies of textiles is necessary when designing a material to protect it from vibration, in order to avoid the phenomenon of resonance that occurs when the excitation frequency of an external force overlaps with the natural frequency of the system. From the recorded values of the frequencies, it is possible to infer some properties of the knitted materials. The value of the natural frequency of any material gives information about the stiffness of the material. The higher the natural frequency, the more stiffness can be expected. The shape of the natural frequency curve is relevant to the material’s ability to dampen vibrations. The smoother the shape, the higher the damping capacity of the material. The study shows that the yarn type and the structural parameters of the fabric affect the vibration behaviour. The tightness of the fabric is one of the most important variables affecting the level of natural frequencies of the material. This research is expected to support the development of knitted fabrics as good vibration isolators for humans by also showing their comfort level compared to alternative materials used so far.

A unified Jacobi–Ritz approach for the FGP annular plate with arbitrary boundary conditions based on a higher-order shear deformation theory

In this paper, a unified method for free and forced vibration of functionally graded porous annular and circular plate is proposed, incorporating the Jacobi–Ritz method and the higher-order shear deformation theory. In order to improve the accuracy of calculation, the segment strategy is adopted. Meanwhile, the artificial spring is used to express the connection between each truncated plates and arbitrary boundary conditions. The convergency study and validation are implemented, and parametric study of both free and transient forced vibration are conducted. The present method shows a fast convergence property and high accuracy for thick plate. It has been found that boundary conditions are the dominant influencing factor of the central deflection of functionally graded porous annular plate, and the porous material with vertical gradient change in the plate leads to a better impedance effect than the uniformly distributed porous material.

A Unified Solution Method for Free Vibration of Arbitrarily Shaped Plates without or with Cracks

This paper presents a new approach for analyzing the free vibration of thin plates with arbitrary piecewise smooth curvilinear contour under various boundary conditions. It can also be applied to plates with cracks. This approach is based on the transformation of energy integral expressions and the domain decomposition technique. Furthermore, boundary conditions are modeled by using linear springs to restrain the plate edges. For obtaining the expressions of energy integral, the arbitrarily shaped domain of integration is divided into several trapezoid domains with curved sides by a set of parallel lines passing through the intersection points of contour curve segments. Then, Jacobi orthogonal polynomials are introduced as the admissible functions, so that the repeated integrals in the energy expressions are reduced to definite integrals analytically. At this point, the calculation method of the energy functional is determined by the equation forms of the curve segments of the plate contour. When the equations of the curve segments allow the integrands to have analytic primitive functions, the energy functional has an analytical solution. Otherwise, the Gauss–Legendre method is used to obtain the numerical solution. Accordingly, the arbitrarily shaped plate with cracks is decomposed into several arbitrarily shaped subdomains based on the cracks. Each subdomain is handled according to the above procedure. The continuity conditions at the interconnecting interfaces of the subdomains are realized by linear springs. The plate without or with crack is modeled by setting the spring stiffness to infinity or 0. The accuracy of the proposed method is verified by comparing the obtained results with the published results. Furthermore, the vibration characteristics of plates with various shapes, such as astroid-shaped plates and cracked elliptical plates, are investigated. These new results can serve as benchmarks for further studies on the vibration of plates.