Rayleigh Method Research Articles

Buckling load of heavy columns with variable cross section and flexible end restraints

AbstractEffective and economical design of heavy columns, taking into account their own self weight is a common problem in engineering. The degree of restraint provided at the ends of the column should also be taken into account and it is addressed by adding rotational and/or translational springs to simulate several levels of fixity. In this work results are given for five different combinations of boundary conditions at the two ends, and for several restraint rigidities at the supports. An approximate solution using Rayleigh's method is presented and compared to numerical finite element results and to the exact solution previously derived for general cases of variable cross section columns with variable axial force.

Operator-pencil treatment of multi-interval Sturm-Liouville equation with boundary-transmission conditions

This paper is devoted to a new type of boundary-value problems for Sturm-Liouville equations defined on three disjoint intervals (−π,−π+d),(−π+d,π−d) and (π−d,π) together with eigenparameter dependent boundary conditions and with additional transmission conditions specified at the common end points −π+d and π−d, where 0<d<π. The considered problem cannot be treated by known techniques within the usual framework of classical Sturm-Liouville theory. To establish some important spectral characteristics we introduced the polynomial-operator formulation of the problem. Moreover, we develop a new modification of the Rayleigh method to obtain lower bound of eigenvalues.

Free Vibration Analysis of a Simply Supported Axial Functionally Graded Beam Using the Rayleigh Method

Free Vibration Analysis of a Simply Supported Axial Functionally Graded Beam Using the Rayleigh Method

A Method for Determining the Fundamental Site Period and the Average Shear Wave Velocity

The soil–structure interaction plays a crucial role in determining the displacement and internal forces of multi-story buildings subjected to strong ground motion. One of the critical dynamic characteristics influencing soil–structure interaction is the fundamental site period and the average shear wave velocity associated with it. This study introduces an original equation to determine these parameters. In addition, for the first time in the literature, the version of the Rayleigh method used for finding the fundamental periods of buildings is used to find the fundamental site period. The soil is modeled as an equivalent shear beam to obtain the proposed equation. The peak displacement is obtained by acting the soil mass as an external load on the equivalent shear beam. For single-layer soil, the fundamental site period is proportional to the square root of the peak displacement of the equivalent shear beam. The least squares method generalizes the proposed relation for single-layer soils to multi-layer soil profiles. Modified Finite element Transfer matrix method is used for calibration in the least squares method. The equations used in the literature and earthquake codes for determining the fundamental site period and average shear velocity are tested on various examples, and it is shown that the method proposed in this study, along with the Rayleigh method, gives better results than these equations. The performances of these two methods and the five commonly used equations are tested and compared on different soil profiles. Transfer functions, Finite Element Method (SAP200) and Modified Finite Element Transfer Matrix Method are used for verification. For all soil profiles, the results obtained from the transfer function, Finite Element Method (SAP200) and Modified Finite Element Transfer Matrix Method are found to be in agreement. The true percent relative error found in the results obtained with the proposed method is 4.47%.

Investigating Static Deflection of Axial Functionally Graded Non-Prismatic Beams Using the Rayleigh Method

Investigating Static Deflection of Axial Functionally Graded Non-Prismatic Beams Using the Rayleigh Method

Analytical insight into local defect resonance using a two-step method

Local defect resonance (LDR) has gained increasing attention in the field of non-destructive testing due to its capacity to selectively amplify vibrations within the defect regions and to enable defect-selective and high-resolution imaging. How-ever, the conventional theory based on vibration method adopted for analyzing the fundamental frequencies of LDR deviates from experimental results when dealing with relatively thick defects. To tackle this deficiency, this study presents a two-step analytical methodology designed to precisely evaluate the local reso-nance effects of thick planar defects. In the first step, the normal-mode expansion method is employed to calculate the phase shift of the reflected elastic wave. In the second step, by using the standing wave's wavelength obtained in the first step, the Rayleigh method is taken into account to calculate LDR frequencies for various defect shapes. The refined theory is subsequently validated through finite element analysis and experimental measurement. The results clearly demonstrate that the LDR frequencies calculated using the refined theory exhibit a better agreement with experimental and simulated data.

Inextensional vibrations of thin spherical shells using strain gradient elasticity theory

We study inextensional vibrations of the spherical shell using strain gradient elasticity theory under Kirchhoff-Love hypotheses. For admissibility of the inextensional deformations the shell is punctured by making a tiny hole. The shell is assumed to be made of linearly elastic, homogeneous, and isotropic material. The closed form expression for the frequencies of inextensional modes of vibration of spherical caps of various polar angles and that of nearly complete spherical shell are derived using Rayleigh's method. Towards this, the displacement and velocity fields are obtained by setting the membrane strains to zero. Further, to facilitate the existence of inextensional vibrations, boundaries are assumed to be traction free. The frequencies are computed for the positive and negative signs in the strain gradient term of the constitutive law. The computed frequencies, employing negative sign are found to be higher than those computed using classical elasticity theory. The opposite effect is seen when the frequencies are computed with the positive sign. Further, the frequencies are found to saturate when the positive sign is used in the constitutive law. Parametric studies viz. variation in frequencies with the circumferential wavenumbers for different values of length scale parameter, variation in frequencies with circumferential wavenumbers for different values of polar angles and a given length scale parameter, variation of frequency with increasing polar angle for a given circumferential wavenumber, variation of saturation wavenumber with increasing polar angle, and increasing length scale parameter are carried out. The frequencies of nearly a spherical shell with increasing circumferential wavenumber are also computed.

Feasibility study on quench detection methods of Raman and Rayleigh scattering techniques in optical fiber encapsulated HTS coils

Quench detection is a key issue to be solved for safe operation of high temperature superconducting (HTS) magnets. Recently, distributed optical fiber sensors based on Raman and Rayleigh scattering techniques are proposed for quench detection in HTS applications. However, sensing and demodulating principles of these two techniques are different, and they may show different performances in quench detection. In this paper, a 13 m long optical fiber encapsulated HTS coil (OF-coil) was fabricated. Two sensing devices based on Raman and Rayleigh techniques, which are named Raman-device and Rayleigh-device, were prepared. Overcurrent and heater induced quench detection experiments in the OF-coil were carried out and the above two devices were used for quench detection. The results show that both Raman-device and Rayleigh-device can measure temperature changes of the OF-coil in the overcurrent experiment. And their quench response time is 36 s and 14 s, respectively. In heater induced quench detection experiment, Rayleigh-device can successfully acquire local temperature rise of the OF-coil, and the quench response time is 1.7 s. For Raman-device, its spatial resolution is too large to realize local quench detection.

AXISYMMETRIC DYNAMIC BEHAVIOUR OF THIN OBLATE SHELLS

This paper presents a theoretical investigation of the axisymmetric free vibrations of an isotropic thin oblate spheroid shell. The analysis depends on the Rayliegh – Ritz's method. The non – shallow shell theory is used for the analysis. The analysis based on considering the oblate spheroid as a continuous system constructed from two spherical shell elements matched at the continuous boundaries.Throughout the results, it is shown that when the eccentricity reaches zero, an exact thin sphere solution is emerged and when the eccentricity equals one an exact thin circular plate solution is emerged. Therefore, the eccentricity of an oblate shell at medium value lies between these two values.It was found that the Rayleigh – Ritz's method is suitable for all eccentricities, while the literature showed that the Rayleigh's method is suitable for eccentricities less than 0.6.

Natural frequencies of bars with variable cross-sections

This paper presents the possibility of determining the natural frequency of flexural bars treated as Bernoulli-Euler beams using the Rayleigh method. It is assumed that the shape of the bar axis during vibration is the same as the shape of the deflection line of this bar under continuous load. By comparing the potential energy in the deflected position with the kinetic energy in the undeflected position, it is possible to determine the frequency and period of natural vibration as a function of Young's modulus and material density, as well as the parameters describing the geometric shape of the bar. Examples of solutions for truncated cone and truncated wedge-shaped bars are shown, as well as a solids of revolution with the generatrix described by linear and curvilinear (parabolic, exponential) forms. It applies to both solid and hollow bars, and different types of fixing the ends of the bar. The accuracy of the obtained results (compared with the literature and with the results of FEM calculations) is sufficient for practical applications. The authors also showed the possibility of extending the method to higher frequencies of vibration.

Improved analytic method for outrigger structures considering floor stiffness

SummaryCurrent analytical models for outrigger structures in super high‐rise buildings tend to oversimplify by not considering the stiffness of individual floors. This paper introduces a more refined calculation model, based on the substructure method, which takes floor stiffness into account. To verify our proposed approach, we derived a mathematical algorithm and developed a finite element model using ANSYS. When compared to traditional methods that only account for outrigger stiffness, our model, which incorporates both outrigger and floor stiffness, provides improved accuracy in calculating vertex displacement. It also suggests an upward shift in the optimal position for the outrigger and bolsters the overall building's lateral stiffness. To further our analysis, we introduced an equivalent stiffness calculation formula, using the Bayesian parameter estimation method. When applied to dynamic analysis, this formula aligns closely with the results from the finite element simulations. Furthermore, the suggested algorithm for determining the best position for the outrigger is consistent with theoretical calculations. By considering the contribution of regular floors to the overall structure, we found that the fitted equivalent core tube stiffness offers a reliable reflection of structural stiffness. Lastly, when this equivalent stiffness was applied to a dynamic analysis based on Rayleigh's energy method, there was a noticeable reduction in computational effort. This yields not only more efficient calculations but also precise results, rendering it particularly valuable during the initial design phases of high‐rise buildings.

ЗАВИСИМОСТЬ ОСНОВНОЙ ЧАСТОТЫ СОБСТВЕННЫХ КОЛЕБАНИЙ ПЛОСКОЙ ШПРЕНГЕЛЬНОЙ ФЕРМЫ ОТ ЧИСЛА ПАНЕЛЕЙ

A scheme of a statically determined planar truss with downward braces and truss reinforcement is proposed. The problems of the lower and upper boundaries of the main natural frequency of free oscillations of the structure are solved. A model with masses concentrated at the nodes is adopted. To find the formula for the dependence of the oscillation frequency limits on the number of panels, the Dunkerley and Rayleigh methods are used. The expressions for the forces in the truss rods are found from the solution of the system of linear equilibrium equations for all nodes. To obtain the matrix of the system of equations, Maple computer mathematics operators are used. The rigidity of the structure is calculated using the Maxwell – Mohr formula. The sequence of solutions obtained for trusses with different numbers of panels is generalized by induction to an arbitrary case. For the coefficients of the formula in the desired solution, homogeneous linear recurrent equations are compiled and solved. The obtained solution is compared with the numerical solution of the problem of the oscillation spectrum of a structure with many degrees of freedom. It is shown that the first frequency of the spectrum is close to its analytical estimate. With an increase in the number of panels in the truss, the accuracy of the derived formulas increases.

Dynamic characteristics identification of RC beams subjected to impact damage

Structural dynamic characteristics are essential parameters for detecting structural damage. This study conducted impact and static loading tests on six reinforced concrete (RC) beams, and the plastic energy dissipation of the reinforced concrete beams under both loading conditions is be essentially consistent. The random decrement technique and Hilbert-Huang transform (RDT-HHT) combined approach is used to identify the first three mode dynamic characteristics (frequency and damping ratio) of impact-damaged reinforced concrete beams. An approach based on finite element (FE) plastic damage distribution is proposed to obtain the continuous flexural stiffness of damaged reinforced concrete beams. Both the global stiffness and Rayleigh method are used to predict the natural frequency. The results show that the first three mode frequencies of the damaged reinforced concrete beams decrease, whereas their damping ratios increase. Specifically, the first mode frequency reduction of reinforced concrete beams under impact damage is slightly lower than that under static damage, while the second and third mode frequencies exhibited a greater decrease. The increase in damping of the damaged reinforced concrete beams is mainly related to the damage extent, damage distribution, and damage location. Furthermore, the first and third mode damping ratios of reinforced concrete beams increase more significantly than the second mode damping ratio after damage, primarily because the most severe damage occurred at the location (mid-span) with the maximum curvature of the first and third mode modal shapes. Due to the more concentrated distribution and more severe local damage of impact damaged reinforced concrete beam, the increase in the first and third mode damping ratios after impact damage is more pronounced compared to static damage. The global stiffness method overestimates the frequency of damaged reinforced concrete beams. Whereas the Rayleigh method utilizes a reasonable first mode modal shape function and the continuous flexural stiffness to predict the structural frequency after damage, the first mode frequency predicts after damage with an error of less than 3%. It is recommended to consider changes in both frequency and damping to assess the extent of impact damage in structural components.

Vibration Analysis of Rotating Elements of A Motor

Abstract: With the industrial revolution, the range of applications for rotating machinery in daily life as well as for industrial use in manufacturing and processing, nuclear power plants, automobiles, oil & gas refineries, etc., has greatly expanded. And for all this rotating machinery, the problems brought on by bearing faults significantly contribute to machine failure. Machine Design courses usually teach students how to design a system consisting of a shaft and its bearings under rotating, bending, transverse, axial, and torsional loads. Most textbooks cover Rayleigh's and Holzer's methods, which do not cover other important dynamic effects in shaft and bearing system design. The diameter of the shaft is calculated and then the deflection of the shaft is evaluated by using static deflection formulas. However, this assumption can cause serious errors in shafts' deflection due to eccentric and nonaligned gears mounted on the shaft. The authors addressed this potential manufacturing defect issue in a Machine Design class as a term project, which required students to transfer and apply content knowledge from their dynamics and vibrations’ courses. This paper presents all aspects of this successful experience of implementing ABET strategies in the engineering classroom to maximize its reach and potential impact.

Random vibration fatigue behavior of directionally solidified superalloy: Experiments and evaluation of life prediction methods

Vibration is closely related to structural dynamics, and the study of random vibration fatigue loads is typically conducted in the frequency domain. In this study, various frequency domain methods were employed to predict the random vibration fatigue life of DZ125L directionally solidified superalloy under different vibration signal intensities. Experimental analysis was used to examine the fatigue life and random vibration response of DZ125L alloy specimens under different vibration signal intensities. The S-N curve derived from sinusoidal vibration tests was used to replace the traditional S-N curve obtained from tension-compression tests in random vibration fatigue analysis, leading to improved life prediction accuracy. Additionally, an intensity function fξ was established using the spectral width parameters α1 and α2 of the random process and the vibration signal intensity ξ, which was used to modify the traditional Rayleigh method. A new vibration fatigue life prediction model based on frequency domain method is obtained. The results indicate that the intensity function fξ emphasizes the elastic or plastic stage of the specimen during the random process, leading to highly accurate life prediction values using the Modified Rayleigh method across different vibration signal intensities, with all prediction points are within 1.5 times the scatter band.

Crack identification driven by the fusion of mechanism and data for the Variable-Cross-Section cantilever beam

Crack identification for variable-cross-section cantilever structures is a main means of avoiding sudden failure caused by cracking and realizing the prognostics and health management (PHM) of such structures. This paper proposes an inverse crack identification method driven by mechanism and data for variable-cross-section cantilever beams. First, the Rayleigh method is revised for the fast, accurate calculation of the natural frequencies of a healthy cantilever beam, which is regarded as a basic criterion. Subsequently, for the rapid, accurate expression of the relationship between crack information and natural-frequency variations, a data-driven method based on the radial basis function is proposed to calculate the response variations accurately according to the crack information. Then, the obtained acceleration of the cantilever beam is decomposed to determine its natural frequencies, and the variations in these natural frequencies are obtained and compared with the results of the mechanism model. Finally, crack positions, lengths, and widths are identified based on the forward model and decomposition results. Particle swarm optimization is employed to minimize the discrepancy in the changes between the monitored and calculated natural frequencies. The proposed forward model is compared with other methods, and the superiority of the crack identification method is verified experimentally. Results indicate that the proposed method accurately identifies cracks on variable-cross-section cantilever beams and provides an effective PHM approach for beam-type structures.

An Application of Six Sigma for Optimality of Medium Density Fiberboard Production

During the production process of MDF, there is a high level of internal bond (IB) variation. This results in the waste of out-of-standard IB values that account for 0.38 % with damage value over 1 million baht/year. The company required products with fewer volatile compounds from formaldehyde adhesives, focusing on reducing the amount of adhesive but still being strong according to IB-specification which will reduce the cost of production by about 20 − 30 million baht/year. The results of wood sampling and IB testing were divided into 6 areas, namely IB1-IB6. It was found that most of the data were symmetrical except for the IB5 data as the area where the most variation occurs. The distributions of the IB1 and IB6 data showed relatively low variability compared to data from other areas. IB1 - IB6 values were normal distribution, expect for IB5. Process capacity in IB2 was relatively high compared to IB from other areas. From the Correlation Matrix and Correlation Map, it was found that the variables that influenced the IB were Press Factor, % Dosing Glue, Heat Circuit1, Primary Circuit Intel and % Mc After Gluing. To conduct the experiment and find the best variable conditions by 25-2 - Factorial Design (Resolution: III). It was found that Glue = 7.4, Heat1 = 234.4, and Press = 6.5 would give IB = 0.88 which was closest to target (0.7). Glue = 7.1, Heat1 = 233.2, and Press = 6.48 would give IB = 1.15 which was the highest value. Results of production conditions at optimum or maximum that can be generalized from Rayleigh Method Dimensional Analysis was found that at the levels of 7.85, 254.28 and 257.70 of Glue, Heat1 and PrimCirIn, the target response (IB) was 0.7. and at the levels of 8.07, 233.35 and 281.60 of Glue, Heat1 and PrimCirIn resulted in a response value (IB) of 1.27.

Retraction Note: Application of correction factors in Rayleigh method in calculation of the fundamental frequency of vibration Cylindrical shell with rectangular cross section

This paper has been formally retracted on ethical grounds because the article contains extensive and repeated instances of plagiarism. The authors are responsible for this mistake and apologise for it. Request approved by the proceedings Editor and the Publisher on September 13, 2023.

Retracted Article: Application of correction factors in Rayleigh method in calculation of the fundamental frequency of vibration Cylindrical shell with rectangular cross section

This paper has been formally retracted on ethical grounds because the article contains extensive and repeated instances of plagiarism. The authors are responsible for this mistake and apologise for it. Request approved by the proceedings Editor and the Publisher on September 13, 2023.

Modelling and Parameter Estimation of Cartridge Probes for Atomic Force Microscopy

A critical challenge for industrial Atomic Force Microscopes (AFM) is automatic selection of appropriate probes required for the scanning and measurement tasks (or sequences of tasks). This challenge can be addressed by specially assigned multi-probe cartridge holders with replaceable probes. Cantilevers with different spring constants and resonant frequencies are required to make specific AFM measurements. In this paper the classical PDE of lateral vibration of bars of variable cross-sections to model dynamics of the AFM cantilevers and calculate their spring constants is used. In addition, utilizing Rayleigh method, resonant frequency can be estimated. Using software based on these models, probe designer can interactively select appropriate geometric and material parameters to achieve desired spring constants and resonant frequencies of the cantilevers to be assembled in the multi-probe cartridge holders.