Number Of Natural Frequencies Research Articles

Suitability of Polymers for 3D-Printing Laboratory Models for Shaking Table Experiments: Discussion and Indications.

This paper presents a comprehensive assessment of the suitability of seven commercially available polymers for crafting laboratory models designed for dynamic shaking-table tests using 3D-printing technology. The objective was to determine whether 3D-printed polymer models are effective for dynamic assessments of structures. The polymers underwent experimental investigations to assess their material properties, i.e., the elastic modulus, the mass density, and the limit of linear-elastic behaviour. The following methodology was applied to obtain the correct values of elasticity moduli and yield points of the polymers: (1) the uniaxial tensile test, (2) the compression test, and (3) the three-point loading test. The filament density was determined as the ratio of sample mass to its volume. The results indicate substantial variations in stiffness, density, and elasticity limits among them. For the similarity analysis, an existing reinforced concrete chimney 120 m high was chosen as a prototype. A geometric similarity scale of 1:120 for a laboratory mock-up was adopted, and a numerical model of the mock-up was created. The similarity scales were calculated for mock-ups made of each filament. Based on these scales, numerical calculations of natural frequencies and dynamic performance under a strong earthquake were carried out for models made of different polymers. Assessment of the polymers' suitability for laboratory models revealed positive outcomes. The agreement between field experiments, shaking-table tests, and numerical predictions in terms of natural frequencies was observed. Maximum stresses resulting from the earthquake indicated the satisfactory performance of the model below the linear-elastic limit. Despite differences in material properties, the selected polymers were deemed suitable for 3D-printing models for shaking-table tests. However, the discussion raised some important considerations. The upper frequency limit of the shaking-table imposes restrictions on the number of natural frequencies that can be determined. Numerical assessments of natural frequencies are recommended to prevent underestimation and to assess the feasibility of their determination. Additionally, resonance during natural frequency determination may lead to exceeding the linear-elastic limit, affecting filament properties, and making the similarity criteria invalid. Practically, this research contributes insights for planning shaking-table tests, aiding in selecting the most suitable filament and highlighting crucial considerations to ensure reliable and accurate dynamic assessments.

Sound insulation performance and modal analysis of asymmetrical insulating laminated glass

Windows are commonly the primary way for noise to enter the building environment. Most windows achieve higher sound insulation targets by simply increasing glass thickness or laminate. However, the glass thickness cannot be increased freely due to the window frame’s limitation. This research selects eleven kinds of asymmetric insulating laminated glass (glass + polyvinyl butyral + glass + air + glass) within 27 mm, which are the typical thicknesses of window frames. The three pieces of glass have equal thicknesses. The half-space acoustics method investigated the acoustic properties and modal behavior of the glass with finite thickness. The results show that when PVB or three glass increases simultaneously, and the sound insulation of each glass can be improved by 0.9∼3.4 dB or 3.7∼21.8 dB at first resonant frequencies, respectively. The glass with the best sound insulation is 7 mm glass + 0.76 mm polyvinyl butyral + 7 mm glass + 5 mm air + 7 mm glass (7 + 0.76 + 7 + 5A + 7 for short). Its sound transmission loss (TL) is 38.5 dB from 125 to 4000 Hz and 16.8 dB in the first resonant frequency (200 Hz). In addition, its first resonant frequency is the highest among eleven kinds of glass, and the number of natural frequencies in the low frequency range is only 9. These findings can provide a basis for designing asymmetric insulating laminated glass to improve the low frequency noise reduction effect effectively.

An optimal form of the finite element mass matrix in the analysis of longitudinal vibrations of rods

In this paper, an attempt is made to find the optimal form of the mass matrix of a rod finite element, which allows one to obtain the smallest errors in the longitudinal frequency determination of natural vibrations of any boundary conditions within the whole range of determined frequencies. It is assumed that the mass matrix can be treated as a linear combination of the consistent and diagonal matrices. Based on analytical considerations, the optimal values of certain weights for creating a linear combination of the mentioned matrices have been determined. As a result, a mass matrix has been obtained which allows the determination of natural frequencies with the smallest mean error within the possible spectrum of frequencies. It is also shown that the value of the weight coefficient changes depending on the number of natural frequencies within the spectrum one wants to determine. • Determination of the optimal form of a rod finite element mass matrix in the context of minimal errors in calculation of natural frequencies for longitudinal vibration. • Calculation of certain values of weights in order to determine the natural frequencies with the smallest mean error in the range of analysed frequencies. • Determination of optimal values of weight factors.

Free Vibrations of Cylindrical Shells by The Nodal Line Finite Difference Method-Part-I Theory.(Dept.C)

In this paper, the nodal line finite difference method (NLFDM) is extended to the analysis of a thin circular cylindrical shell, as shown in Fig. 1, undergoing free vibrations [1]. The material of the circular cylindrical shell is assumed to be linear elastic isotropic. The complexity of a theoretical analysis of a vibration problem depends largely on the number of degrees of freedom of the structural system in question. Since a thin circular cylindrical shell is a continuous system it has an infinite number of degrees of freedom. The NLFDM [10] transforms this continuous system into a system having a finite number of degrees of freedom. While carrying out the dynamic (or static) analysis by the NLFDM, the number of degrees of freedom depends on the number of the used nodal lines. The natural frequencies and modes are treated in more detail because they are basic to understanding the dynamic response under any kind of excitation. It will be shown that the number of natural frequencies and that of normal modes, for axial wave number different from zero, are each equal to three times the number of nodal lines used. The orthogonality relationships of the normal modes obtained by the nodal line finite difference method are also included. It should be remarked that these relationships are slightly different from those obtained by the finite element method; in fact the overall matrix of coefficients replaces the stiffness matrix, and any two different modes are orthogonal with resp ect to the unit matrix instead of the mass matrix.

An improved generalized flexibility matrix approach for structural damage detection

ABSTRACTStructural damage detection is a key part of structural health monitoring. The generalized flexibility matrix approach is an effective method for damage detection. Compared with the flexibility matrix approach, it requires less number of natural frequencies and corresponding mode shapes. In this paper, the approach is improved by considering the nonnegativity of the damage extent. The error bound of two solutions calculated by the improved approach using natural frequencies and corresponding mode shapes without and with noises is derived. Numerical examples with different damage scenarios show that compared with the original generalized flexibility matrix approach, our proposed method works better.

A Vibration-Based Structural Health Monitoring System for Transmission Line Towers

In this paper, we present a vibration-based transmission tower structural health monitoring system consisting of two parts that identifies structural changes in towers. An accelerometer group realizes vibration response acquisition at different positions and reduces the risk of data loss by data compression technology. A solar cell provides the power supply. An analyser receives the data from the acceleration sensor group and calculates the transmission tower natural frequencies, and the change in the structure is determined based on natural frequencies. Then, the data are sent to the monitoring center. Furthermore, analysis of the vibration signal and the calculation method of natural frequencies are proposed. The response and natural frequencies of vibration at different wind speeds are analysed by time-domain signal, power spectral density (PSD), root mean square (RMS) and short-time Fouier transform (STFT). The natural frequency identification of the overall structure by the stochastic subspace identification (SSI) method reveals that the number of natural frequencies that can be calculated at different wind speeds is different, but the 2nd, 3rd and 4th natural frequencies can be excited. Finally, the system was tested on a 110 kV experimental transmission line. After 18 h of experimentation, the natural frequency of the overall structure of the transmission tower was determined before and after the tower leg was lifted. The results show that before and after the tower leg is lifted, the natural frequencies of each order exhibit obvious changes, and the differences in the average values can be used as the basis for judging the structural changes of the tower.

An Exact Dynamic Stiffness Formulation for Predicting Natural Frequencies of Moderately Thick Shells of Revolution

An exact dynamic stiffness formulation is proposed for calculating the natural frequencies of shells of revolution based on the first-order Reissner-Mindlin theory. Equations of motion are reduced to be one-dimensional, should the circumferential wave number is specified, and then are rewritten in Hamilton form. Therefore, a shell of revolution with a moderate thickness is analysed as a special skeletal structure with five degrees of freedom at element ends. Dynamic stiffnesses are constructed directly from the one-dimensional governing vibration equations using classic skeletal theory. Number of natural frequencies below a specific value is counted by applying the Wittrick-Williams (W-W) algorithm, and exact eigenvalues can be simply obtained using bisection method. A solution to the number of clamped-end frequencies J0 in the W-W algorithm is also proposed and proven to be reliable. Numerical examples on a variety of shells with different boundary conditions are investigated, and results are well compared and validated. Influences of a variety of shell parameters on natural frequencies of both cylindrical and spherical shells are discussed in detail, demonstrating that the proposed dynamic stiffness formulation is applicable to analyse natural frequencies of moderately thick shells of revolution with high accuracy.

Structure of the Basic Software and Hardware Complex of the System for Seismometric Monitoring of the Technical Condition of Buildings and Structures

The objective of the present study is to design a basic software and hardware complex (SHC) of a seismometric monitoring system for buildings and structures recommended for serial production. To do this, the authors used modern methods and principles of designing software and hardware systems. The system is methodologically based on the engineering-seismometric method. This method uses recording of spatial vibrations of an object as a result of microseisms of natural and anthropogenic origin. Next, dynamic and elastic characteristics of the structure are determined by vibrations that make it possible to evaluate its technical condition. As a result of conducted studies, the authors defined the main technical and functional requirements for the monitoring system for a wide range of applications. A three-level structure of a modular technical condition monitoring system is proposed for discussion. This structure makes it possible to create an extensible open system in which the number of measuring channels can be easily increased by increasing the number of plugin standard modules. In addition, the system can both record seismic events, earthquakes, and perform seismometric monitoring by microseisms. In order to represent the main processes and structure of the proposed basic complex of the system in the case of its full operation, the authors developed a functional model of the system. The model is based on a data flow diagram that describes the processes of collecting, processing, storing, analyzing, and presenting seismometric monitoring data. The functioning of the proposed complex is briefly described. The input data of the complex are the vibrations of the structure recorded at observation points using three-component geophones. Then, the seismic signal recorder collects, amplifies, digitizes, and transmits data to the server. The server either records seismic events and then evaluates their impact on the structure or carries out a planned recording of microseisms in order to monitor the technical condition of the structure and stores vibration data in the corresponding files. If an earthquake is detected, the system notifies the responsible personnel. Records of microseisms are used by the data processing software to compute the statistical parameters of vibrations and complex transfer functions according to the spectra of which the operator manually selects the values of natural frequencies. Seismic monitoring data processing software evaluates elastic characteristics by a number of natural frequencies using a mathematical design model of the structure vibrations. On the basis of the analysis of the change in the obtained dynamic and elastic characteristics and taking into account the effect of external factors, the software generates information for monitoring the technical condition of the structure. These results, as well as data of the evaluation of the event impact on the structure, are the output data of the system.

Experimental study of extracting artificial boundary condition frequencies for dynamic model updating

In the field of dynamic measurement and structural damage identification, it is generally known that modal frequencies may be measured with higher accuracy than mode shapes. However, the number of natural frequencies within a measurable range is limited. Accessing additional forms of modal frequencies is thus desirable. The present study is concerned about the extraction of artificial boundary condition (ABC) frequencies from modal testing. The ABC frequencies correspond to the natural frequencies of the structure with a perturbed boundary condition, but they can be extracted from processing the frequency response functions (FRF) measured in a specific configuration from the structure in its existing state without the need of actually altering the physical support condition. This paper presents a comprehensive experimental investigation into the measurability of the ABC frequencies from physical experiments. It covers the testing procedure through modal testing, the data processing and data analysis requirements, and the FRF matrix operations leading to the extraction of the ABC frequencies. Specific sources of measurement errors and their effects on the accuracy of the extracted ABC frequencies are scrutinised. The extracted ABC frequencies are subsequently applied in the damage identification in beams by means of finite element model updating. Results demonstrate that it is possible to extract the first few ABC frequencies from the modal testing for a variety of artificial boundary conditions incorporating one or two virtual pin supports, and the inclusion of ABC frequencies enables the identification of structural damages without the need to involve the mode shape information.

Bird mating optimizer for structural damage detection using a hybrid objective function

Abstract A structural damage detection approach based on bird mating optimizer (BMO) in time-frequency domain is proposed in this paper. A hybrid objective function is introduced by minimizing the discrepancies between the measured and calculated natural frequencies and correlation function vector of acceleration of damaged and intact structures. Then the BMO algorithm with a disturbance procedure is developed to solve the objective function. Benefited from the hybrid objective function, only a few number of natural frequencies are needed in the detection process. And the disturbance procedure designed in this paper can enhance the precision of identification. The efficiency and robustness of the proposed method are verified by a planar truss and a frame, a three connected shear buildings and an experimental work. The studies in numerical simulations validate that the proposed objective function and disturbance procedure are helpful to improve the precision of identification. The experimental work shows that the proposed method has the potential of practical application. In addition, comparison among the proposed method and other optimization algorithms, i.e. GA, ABC, L-SHADE and HCLPSO, reveals the superiority of the proposed method in structural damage detection.

Restoration of the linear potential in the Sturm-Liouville problem

The problem of identifying the variable coefficient of elasticity of a medium with respect to natural frequencies of a string oscillating in this medium is considered. A method for solving the problem is found, based on the representation of linearly independent solutions of the differential equation in the form of Taylor series with respect to two variables, substituting them into the frequency equation, and determining the unknown coefficients of the linear function from this frequency equation. An analytical method has also been developed that allows one to prove the uniqueness or nonuniqueness of the restored polynomial coefficient of elasticity of a medium by a finite number of natural frequencies of oscillations of a string, and also to find a class of isospectral problems, that is, boundary value problems for which the eigenvalue spectra coincide. The latter is based on the method of variation of an arbitrary constant. We consider examples of finding isospectral classes, and also unique boundary value problems having a given spectrum.

A lattice spring model for dynamic analysis of damaged beam-type structures under moving loads

Abstract A discrete model is presented to study the dynamic behaviour of beam-like structures under moving loads in the presence of surface defects. We utilise the lattice spring model (LSM) to describe the uniform Timoshenko beam supported by a viscoelastic foundation as a one-dimensional (1D) assembly of particles interacting through shear and rotational springs. The forced vibratory response of the system using both the discrete and continuum models has been studied to establish a link between the connecting springs at the discrete level and the elastic properties of the matter at large scale. The effects of edge cracks are also depicted as weak structural links in the lattice model by expressing them in terms of localised loss of bending stiffness and replacing them with lumped massless rotational springs. The total applied loading path in the LSM is divided into a number of subintervals, and thus the generalised discrete system of differential equations can be solved sequentially where the resulting displacements and velocities computed at each time-step are used as the initial conditions of the following step. To illustrate the efficiency and fidelity of the proposed approach, it is applied for solving three different numerical examples: short beams with a surface defect, 1D structures resting on a foundation with abrupt change in the stiffness, and damaged beams under successive moving loads. The numerical examples demonstrate the efficiency of the present discrete formulation in modelling complex systems where a large number of natural frequencies is required for convergence of the modal expansion technique.

Exact solution based finite element formulation of cracked beams for crack detection

In this study, a new finite element formulation is presented for straight beams with an edge crack, including the effects of shear deformation, and rotatory inertia. The main purpose of the study is to present a more accurate formulation to improve the beam models used in crack detection problems. Stiffness matrix, consistent load vector, and mass matrix of a beam element is obtained using the exact solution of the governing equations. The formulation for frame structures is also presented. Crack is modelled utilizing from the concepts of linear elastic fracture mechanics. Several numerical examples existing in the literature related to the vibrations of such structures are solved to validate the proposed model. Additionally, an experimental modal analysis is performed to see the superiority of the present method for high modes of vibration, which are generally not taken into account in crack detection problems. The inverse problem is also solved using a well–known optimization technique called genetic algorithms. Effects of shear deformation, rotatory inertia, and number of natural frequencies considered, on the accuracy of the estimation of crack parameters are investigated. It is found that considering more number of frequencies yields better estimation of crack parameters, but require a better modelling of the dynamics of the beam. Therefore, the present formulation is found to be an essential tool in crack detection problems.

Reduction of surface waves in finite element analysis of oscillations of elastic liquid-containing bodies

An efficient method of exclusion of the degrees of freedom connected with high-frequency oscillations of the free liquid surface from the equations of the finite element model of a deformed liquid-containing structure is presented. The number of natural frequencies specified by the shortwave oscillations of the free surface which do not significantly affect the dynamics of loading of the bearing structure grows sharply, with a refinement of the finite element partitioning of the hydroelastic system. The higher frequencies of this subspectrum alternating with natural frequencies specified by elasticity of the structure during interaction with its liquid complicate the analysis of its dynamic characteristics and the construction of mathematical models of behavior under the action of external dynamic factors.

종류별 이륜차 프레임에 대한 구조해석

This study aims to analyze durability by comparing displacement on vibration at driving bicycle frame models of 1, 2, 3 and 4. Among maximum equivalent stresses at 4 kinds of models, model 1 has highest value with 410.39 MPa and becomes 30 times than model 4 with lowest value. The natural frequency number at Model 4 increases more than the other models. Among four models, the number of frequency at model 1 becomes lowest at harmonic vibration with real loading condition. In cases of four kinds of models, the maximum stress is shown near the assembly of rear wheel and the maximum displacement is shown near saddle assembly at this harmonic condition. The structural result about this study can be effectively utilized on the design of bicycle frame by investigating durability and prevention against its damage.

A Modified Wave Approach for the Calculation of Natural Frequencies and Mode Shapes of the Rods with Variable Cross Section

Analytical solutions for vibration analysis of the rods with variable cross section are in general complex and in many cases impossible. On the other hand, approximate methods such as the weighted residual, Rayleigh-Ritz and finite difference methods also have their own shortcomings such as a limited number of natural frequencies and accuracy. Using the wave propagation method, the structure is partitioned into several continuous segments with constant cross-section, for which there exists an exact analytical solution. Waves in positive and negative directions at the entrance of each segment are obtained in terms of waves at the initial segment. Then, by satisfying the boundary conditions, the characteristic equation is obtained and all natural frequencies are calculated. By adding waves in positive and negative directions at each point, the shape modes are obtained. To verify this modified method, the frequencies and mode shapes of a rod with polynomial cross section, which has an exact analytical solution, are compared and have proven to be of highly accuracy. Therefore, this method can also be used to calculate the natural frequencies and its mode shapes of the rods with variable cross section for which no analytical solution is available.

A modified wave approach for calculation of natural frequencies and mode shapes in arbitrary non-uniform one-dimensional waveguides

In this article, using the wave propagation method, the natural frequencies and mode shapes of an arbitrary non-uniform one-dimensional waveguide are calculated. The non-uniform rods and beams are partitioned into several continuous segments with constant cross-sections, for which there exists an exact analytical solution. At the end of each segment, waves in positive and negative directions are obtained in terms of waves at initial segment and subsequently, the calculations of the mode shapes become simple. By satisfying the boundary conditions, the characteristic equation is obtained and natural frequencies are calculated for both the arbitrary non-uniform rod and beam. Also, by adding waves in positive and negative directions at the end point of each segment, the mode shapes are obtained. To verify the modified wave method presented here, the frequencies and mode shapes of the rod and the beam with a polynomial cross-section having an exact analytical solution are compared and have been proven to be of high accuracy. Besides, comparisons of finite element method are also included. Therefore, this method can also be used to calculate the natural frequencies and mode shapes of rods and beams with any arbitrary variable cross-section for which no analytical solution is available. For the ‘Modified Wave Approach’ developed here, dimensions of transmission matrix remain constant if the number of segments is increased, while in general wave propagation method, dimensions of transmission matrix increase upon increasing the number of segments. Besides this novelty, this method has the advantage that it gives all the natural frequencies and mode shapes, unlike other approximate methods such as weighted residual, Rayleigh–Ritz, and finite difference methods which have their own shortcomings such as limited number of natural frequencies. Also, since each segment has an exact analytical solution, in contrast to other approximate methods, much higher accuracy is obtained even with only a few number of partitions.

슬롯을 갖는 회전 원판의 주파수 수렴 현상에 대한 연구

A circular plate is the main part of a circular saw blade, which is widely used in industrial cutting tools. In experiments using a circular saw containing slots, we observed that a number of natural frequencies are simultaneously emerged in specific frequency regions. The interesting phenomenon is called the frequency convergence. Effects of design parameters such as the number, the length and the width of slots, on the frequency-convergence phenomenon are analyzed by FEM simulations using ALGOR. The frequency-convergence phenomenon will be intensified when the number, length and width of slots increase.

Assessing how changes to a structure can create gaps in the natural frequency spectrum

Structures vibrate with discrete natural frequencies, which divide the frequency spectrum into frequency-free ranges, or spectral gaps. We may need a structure to have a particular spectral gap, and if this range is found to contain frequencies then the structure must be changed. The extent of the changes depends on the number of natural frequencies that are contained in the required spectral gap – the rank of the bracing must be at least the number of frequencies to be removed. This paper first deals with bracing of this minimal rank, connecting to the least number of freedoms possible, and later generalises this to higher rank bracing, with connections to any number of freedoms. The question is thus If a required spectral gap contains n natural frequencies, can changes to the structure stiffness of rank r ⩾ n, connecting to c ⩾ r freedoms i 1, i 2,… , i c remove them? In all cases, a simple criterion is developed for answering this question, and if the answer is yes, all successful changes are identified as mappings from more fundamental sets. The cases are developed separately even though later cases imply earlier ones, for clarity of the argument.

Dynamic characteristics of a beam and distributed spring-mass system

This paper studies dynamic characteristics of a beam with continuously distributed spring-mass which may represent a structure occupied by a crowd of people. Dividing the coupled system into several segments and considering the distributed spring-mass and the beam in each segment being uniform, the equations of motion of the segment are established. The transfer matrix method is applied to derive the eigenvalue equation of the coupled system. It is interesting to note from the governing equations that the vibration mode shape of the uniformly distributed spring-mass is proportional to that of the beam at the attached regions and can be discontinuous if the natural frequencies of the spring-masses in two adjacent segments are different. Parametric studies demonstrate that the natural frequencies of the coupled system appear in groups. In a group of frequencies, all related modes have similar shapes. The number of natural frequencies in each group depends on the number of segments having different natural frequencies. With the increase of group order, the largest natural frequency in a group monotonically approaches the natural frequency of corresponding order of the bare beam from the upper side, whereas the others monotonically move towards those of the independent spring-mass systems from the lower side. Numerical results show that the frequency coupling between the beam and the distributed spring-mass mainly occurs in the low order of frequency groups, especially in the first group. In addition, vibratory characteristics of the coupled system can be approximately represented by a series of discrete multi-degrees-of-freedom system. It also demonstrates that a beam on Winkler elastic foundation and a beam with distributed solid mass are special cases of the proposed solution.