Natural Frequency Splitting Research Articles

Experimental and analytical studies on eliminating natural frequency splitting of an axisymmetric structure with cyclically symmetric feature groups

This work analytically and experimentally investigates natural frequency splitting behaviors for an axisymmetric stepped plate with circumferentially periodic groups of mass blocks. The simplified mathematical model of a stepped plate is derived using Hamilton's principle. Solving for the natural frequencies with a perturbation approach and Galerkin discretization yields the same closed-form conditions for natural frequency splitting or non-splitting. These closed-form conditions lead to analytical rules that govern splitting or non-splitting of natural frequencies for two different configurations of identical mass blocks in a group. These rules readily extend to other cyclically symmetric periodic structures of similar configurations. Simulations and experiments with a test rig show splitting or non-splitting of natural frequencies for the two configurations agrees with the analytical rules. Both the analytical and experimental results show the weight of mass blocks has no influence on the splitting or non-splitting of natural frequencies, but larger weight leads to larger differences between split natural frequencies. These tests also demonstrate the validity of these natural frequency splitting elimination rules for significant mode contamination. An experimental evaluation of the natural frequency splitting behaviors with respect to mass block weight and angular position deviations gives that the splitting of natural frequencies is insensitive to these deviations, but the non-splitting can be sensitive (i.e., natural frequencies that do not split without these deviations can split). This study provides guidelines for eliminating or reducing natural frequency splitting for cyclically symmetric systems like gyroscopes.

Frequency splitting approach using wavelet for energy management strategies in fuel cell ultra-capacitor hybrid system

Using frequency splitting, two energy management strategies (EMS) based on Haar wavelet decomposition and Fourier analysis for fuel cell hybrid vehicle (FCHV) are proposed to manage efficiently the power flow between components. The paper aims to discuss the performances of the proposed EMS in terms of dynamic behavior, robustness operation, real time application and fuel economy. For apply this methodology, two EMS approaches are elaborated and successfully tested for parallel Fuel Cell/UC: conventional approach using Fourier Transform analysis (FT) and Wavelet analysis approach allowing natural frequency splitting. Finally, and to evaluate the performance and relevance of the developed approach, a comparison analysis were conducted. The simulation results exhibit the effectiveness of both strategies. Indeed, Wavelet analysis leads to better results in terms of energy flow and dynamic behavior, excellent robustness and stability of system, as well as energy economy improvement. A very relevant strategy is proposed based on Wavelet analysis using digital filtering techniques, which enables a natural frequency splitting to ensure the best global performances. In addition, the approach remains simple and suitable for real time operation.

Crack identification in cyclic symmetric structures based on relative indicators of frequency separation

Abstract. Cyclic symmetric structures are an important class of structures in the fields of civil and mechanical engineering. In order to avoid accidents due to cracks in such structures, an effective method for crack identification is presented in this paper. First, the dynamic model of cyclic symmetric structures with gapless cracks is developed using a structure's sector model and rotation transformation. Then, the effects of cracks on the free vibration characteristics of a cracked cyclic symmetric structure are addressed, with particular interests in the distortion of mode shapes and the shift and split of natural frequencies. On the basis of crack-induced phenomena, an effective method based on relative indicators of frequency separation is developed for quantitative crack identification. Numerical results illustrate that the relative indicators are sensitive to small cracks and insensitive to the predicting model used during analysis. Finally, the method is validated by experiments conducted on an impeller-shaft assembly. The results show the effectiveness of the frequency separation indicators in crack identification in cyclically symmetric structures.

Characterizing Stalagmites’ Eigenfrequencies by Combining In Situ Vibration Measurements and Finite Element Modeling Based on 3D Scans

Broken or deformed speleothems have been used as indicators of paleo-earthquakes since the 1990s; however, a causal link is difficult to prove except for some thin speleothems. In contrast, the presence of intact speleothems permits estimating an upper limit of the level of horizontal ground motions of past seismicity in the area. The natural frequencies of speleothems are fundamental parameters for their response to earthquakes. This study proposes a new method of in situ characterization of these natural frequencies. Tested in the Han-sur-Lesse cave (Belgian Ardennes), the method is based on recording the ambient seismic noise using three-component sensors on a stalagmite and a 3D laser scan of its shape. The ambient seismic noise records allow a precise determination of the eigenfrequencies of the stalagmite. In addition, numerical models based on the 3D scan show good consistency between measured and modeled data. The joint analysis of these two techniques concludes that the shape of the stalagmite (elliptical cross-section and shape irregularities) influence the eigenfrequencies and polarization of the modes while also causing a near-orthogonal split of natural frequencies. The motions recorded on the stalagmite show significant amplification compared to those recorded at the free surface outside the cave, which has a strong impact on seismic hazard assessment based on speleothems.

Расчет расщепления собственной частоты цилиндрического резонатора твердотельного волнового гироскопа на основе численного интегрирования высокой точности

Solid-state wave gyroscopes are widely used in various fields of technology. The principle of their operation is based on the phenomenon of precession of the elastic wave, which rotates at an angular velocity proportional to the angular velocity of device body rotation. The accuracy of the device is significantly affected by splitting the gyroscope resonator frequency. The article considers splitting caused by the distortion of the cylindrical resonator cross-section shape. Since the splitting is very small, the perturbation method is usually used to determine it. The article proposes a new method for calculating the splitting based on the numerical high accuracy integration of equations of dynamics of a cylindrical shell with a non-circular cross section. To search for two very close frequencies, through the difference of which the splitting was found, a linear boundary eigenvalue problem was solved with retention of a large number of decimal places. Examples of determining the natural frequency splitting are presented for various laws of the distribution of the radial deviation of the cavity cross-section shape along the circumferential coordinate. Verification of the results was performed by the finite element method. When using the method each of the two close frequencies was calculated with high accuracy

Analysis of spatial vibrations of coaxial cylindrical shells partially filled with a fluid

This work is devoted to a numerical study of natural vibrations of horizontally oriented elastic coaxial shells with the annular gap completely or partially filled with a compressible viscous fluid. The solution is searched for a three-dimensional formulation of the examined dynamic problem using the finite element method. The fluid motion is described in the acoustic approximation in terms of the velocity potential. The corresponding equations together with the boundary conditions, related to a full fluid-shell contact, are transformed using the Bubnov-Galerkin method. The hydrodynamic forces are determined from the viscous stress tensor. The mathematical formulation of the problem of thin-walled structure dynamics is based on the variational principle of virtual displacements, which includes the normal and tangential components of forces exerted by the fluid on the wetted parts of elastic bodies. Modeling of the shells is based on the assumption that their curvilinear surfaces are simulated quite accurately by a set of flat segments, the strains of which are determined by the relations of the classical thin plate theory. The reliability of the results obtained is substantiated by comparing them with the known data for the case when the annular gap is completely filled with an ideal fluid. The influence of the fluid level and the size of the annular gap on the eigenfrequencies and the corresponding vibration modes of coaxial shells under a variety of boundary conditions is analyzed. It has been demonstrated that partial filling leads to splitting of natural vibration frequencies, while a decrease in the filling level facilitates the growth of their minimum values. It is shown that the appearance of mixed vibrational modes, both in the meridional and circumferential directions depends on the size of the annular gap.

Natural frequency splitting and principal instability of rotating cyclic ring structures

This work develops an analytical method to estimate the natural frequency splitting and principal instability of the rotating cyclic ring structures. An elastic model is built up under the ring-fixed frame by using the energy method. The modeling leads to a partial differential equation with time-variant coefficient. The eigenvalue is formulated to estimate the natural frequency splitting, principal instability and their relationships. The dependence of the basic parameters on the natural frequency splitting and principal instability is demonstrated. The principal instability can occur at the splitting natural frequencies but cannot at the repeating ones. A classical problem regarding the parametric instability of the rotating ring with stationary supports and the inverse problem are examined. The results verify that the natural frequency splitting does not mean unstable for the former problem, but for the latter the splitting implies unstable. Besides, the model is transformed into the support-fixed frame and thus an equivalent time-invariant model is obtained, which is solved by using the general vibration theory. The analytical method is validated through the comparisons with the results in the open literature and especially the comparisons between the results from the two types of frames.

The role of fluid-structure interaction for safety and life time prediction in hydraulic machinery

Reliable assessments of dynamic phenomena in hydraulic machinery require the consideration of fluid-structure interaction (FSI) if natural frequencies or damping properties of submerged structural parts clearly change due to the surrounding water or if the interaction may cause instabilities. Exemplarily, three different applications of strongly coupled FSI are presented which all have a high impact on operational safety or dynamic stresses and fatigue life. At first, flow induced damping effects at a Francis runner exhibit a strong dependency on operating condition and mode shape. Secondly, if a rotating disc is submerged, a splitting of natural frequencies is observed for mode shapes with different spinning direction. Finally, a hydroelastic instabilitiy is identified in a bypass valve at small opening.

Distorted wave response of ultrasonic annular stator incorporating non-uniform geometry

The traveling wave ultrasonic stator is normally fabricated with teeth. The tooth geometry improves the driving speed, but it creates natural frequency splitting and mode contamination, especially a distorted traveling wave. A dynamic model of a stepped-plate periodic stator is developed to examine the distortion. The stator is treated as an annular supported by a thin mid plate, and the support stiffness is formulated by using equivalent energy principle. The effects of the tooth and mid plate on the natural frequency and vibration mode are examined by using the perturbation method. The rules governing the frequency splitting, frequency perturbation as well as mode contamination are also identified. The traveling wave response and elliptical trace on stator surface are obtained by using the mode superposition method and they are proved to be distorted due to the tooth geometry. The response at the repeated doublets becomes coupled forward and backward traveling waves, but that at the split doublets becomes coupled forward traveling, standing and backward traveling waves. The results indicate that the tooth mass instead of the stiffness decreases the vibration amplitude and driving speed of the dominant wave, but their effects are different at the repeated and split doublets. Inspection of the model implies that the distortion can be suppressed by using a suitable combination of the wavenumber, tooth count, tooth height and occupying fraction. Numerical calculations are carried out to demonstrate the tooth geometry effect on the transient waveform, driving speed and elliptical trace. The optimization of the tooth geometry that can help achieve a purer traveling wave is discussed.

Dual-mode frequency splitting elimination of ring periodic structures via feature shifting

Ring periodic structures are widely spread in engineering applications, where natural frequencies can split due to the deviation from axisymmetry. This work aims at the elimination of the dual-mode natural frequency splitting by using feature shifting. A concept of equivalent feature is introduced based on the grouping idea and an analytical model with discrete features is established. The relationships between the group number, feature number, shifting angle and excited circumferential wavenumber are identified as closed-form expressions. The splitting for structures with unshifted standard features depends on the feature number and wavenumber regardless of the grouping patterns. The equivalence between various groupings is verified. Simple rules governing the dual-mode splitting are elaborated, where one splitting is removed by a combination of equivalent feature number and wavenumber, and the other is eliminated by feature shifting. The rules allow immediate estimation and elimination of the dual-mode splitting where the modified structures still hold symmetry. The results can find application in vibratory structures where the frequency splitting is the key concern. The main results are verified by the Finite Element method.

Vibration Characteristics of a Mistuned Bladed Disk considering the Effect of Coriolis Forces

To investigate the influence of Coriolis force on vibration characteristics of mistuned bladed disk, a bladed disk with 22 blades is employed and the effects of different rotational speeds and excitation engine orders on the maximum forced response are discussed considering the effects of Coriolis forces. The results show that if there are frequency veering regions, the largest split of double natural frequencies of each modal family considering the effects of Coriolis forces appears at frequency veering region. In addition, the amplitude magnification factor considering the Coriolis effects is increased by 1.02% compared to the system without considering the Coriolis effects as the rotating speed is 3000 rpm, while the amplitude magnification factor is increased by 2.76% as the rotating speed is 10000 rpm. The results indicate that the amplitude magnification factor may be moderately enhanced with the increasing of rotating speed. Moreover, the position of the maximum forced response of bladed disk may shift from one blade to another with the increasing of the rotational speed, when the effects of Coriolis forces are considered.

Analytical Prediction for Free Response of Rotationally Ring-Shaped Periodic Structures

The in-plane wave motion is analytically examined to address the stationary deflection, natural frequency splitting, and mode contamination of the rotationally ring-shaped periodic structures (RRPS). The governing equation is developed by the Hamilton's principle where the structure is modeled as a thin ring with equally-spaced particles, and the centrifugal effect is included. The free responses are captured by the perturbation method and determined as closed-form expressions. The results imply that the response of stationary RRPS is characterized as standing wave, and the natural frequencies can split when the wave number n and particle number N satisfying 2n/N = int. Also the splitting behavior is determined by the relative angle between the particle and wave antinode. The coefficients of the mode contamination are also obtained. For rotating RRPS, the invariant deflections due to the centrifugal force are estimated at different rotating speeds. It is found that, for certain waves satisfying 2n/N = int, the natural frequency exceeds that of the corresponding smooth ring at the critical speed, and furthermore, the critical speed of the backward traveling wave is lower than that of the forward one. The contamination coefficients of the two kinds of waves are also obtained and they have different magnitudes. All results verify that the splitting and contamination can be determined by the relationship among the mode order, wave number, particle number, and relative position between the particle and antinode. Numerical examples and comparisons with the existing results in the literature are presented.

Vibration Characteristics of Mistuned Bladed Disk by Sensitivity Analysis

It is known that, under appropriate condition, amplitude of vibration of some blades in mistuned bladed disk system become large compared to what would be predicted by an analysis of the perfectly tuned system. In this paper, the effects of the mistuning pattern on the natural frequencies and the frequency response of mistuned bladed disk system are investigated. It is shown that, in the tuned system, the excitation with the wavenumber k excites only the vibration modes with the same wavenumber and that, in the mistuned system, the excitation with the wavenumber k excites all vibration modes. The variations of the natural frequencies due to mistuning pattern are obtained by using sensitivity analysis. The relation between the mistuning patterns and the split of natural frequencies are discussed. It is noted that the mistuning pattern with the circumferential wavenumber 2k effectively splits the natural frequency of the vibration mode with the circumferential wavenumber k of the tuned system and that the amplitude of the responses of the system with the mistuning pattern with the circumferential wavenumber 2k subject to the excitation with the wavenumber k hardly become large.

Vibration of rings on a general elastic foundation

The free vibration eigensolutions of a thin ring on a general elastic foundation are obtained by perturbation and Galerkin analyses. Natural frequencies and vibration modes are determined as closed-form expressions for a ring having a circumferentially varying foundation of very general description. The elastic foundation consists of two orthogonal distributed springs oriented at an arbitrary inclination angle. The foundation stiffnesses vary circumferentially. The simple eigensolution expressions explicitly show the parameter dependencies, lead to natural frequency splitting rules for degenerate unperturbed eigenvalues at both first and second orders of perturbation, and identify which nodal diameter Fourier components contaminate a given n nodal diameter base mode of the free ring. Discrete spring supports are treated as a special case where the natural frequencies are determined by five parameters: nondimensional spring stiffness, stiffness angle, support angle, number of springs, and location of the springs. The predicted effects of these parameters on the natural frequencies are verified numerically. As an application and as the motivating problem for the study, the natural frequencies and vibration modes of a ring gear used in helicopter planetary gears with unequally spaced planets are investigated.

On the statistics of natural frequency splitting for rings with random mass imperfections

This paper considers the statistical distribution of natural frequency splits for an initially perfect ring with different types of random mass imperfection. The analysis used to derive analytical expressions for the natural frequency splits is based on a Rayleigh–Ritz approach, in which it is assumed that the mode shapes of the imperfect rings are identical to those of a perfect ring. The types of random mass imperfection investigated are: (i) random harmonic variations in the mass per unit length around the circumference of the ring; (ii) the attachment of random point masses at random locations on the ring; and (iii) the attachment of random point masses at uniformly spaced positions on the ring. For case (i) it is found that the frequency splits always have a Rayleigh distribution. For case (ii) an expression for the statistical distribution is deduced which tends to a Rayleigh distribution as the number of attached masses increases. For case (iii) it is found that the frequency split distribution is dependent upon the mode considered and the number of attached masses, and that in some situations the frequency splits have a “half Gaussian” (i.e., non-Rayleigh) distribution.

A bidirectional ring fiber laser with a 90° Faraday rotator as the nonreciprocal phase element. I. Theory

The scheme of a bidirectional ring fiber laser with a 90° Faraday rotator in the cavity has been theoretically studied. The Faraday rotator is employed as a nonreciprocal phase element producing splitting of natural frequencies of the laser cavity modes propagating in opposite directions. The theory predicts two possible regimes of bidirectional operation. According to this, the laser generates a pair of modes propagating in opposite directions with either complex-conjugate or orthoconjugate polarization states. In the former case, the frequency shift between the modes propagating in opposite directions is independent of the reciprocal birefringence of the laser cavity and amounts to half of the free spectral band of this cavity.

Detection of Defects in Circular Plates using a Scanning Laser Vibrometer

This paper investigates dynamic characteristics (mode shapes and natural frequencies) and defect detection of circular plates using a scanning laser vibrometer. Analytical dynamic characteristics of a circular aluminum plate having a clamped inner rim and a free outer rim are obtained using two methods; one uses Bessel functions and the other uses a multiple shooting method. An in-house finite element code GESA is also used to analyze the circular plate using the DKT plate element. Numerical results show that high-frequency Operational Deflection Shapes (ODSs) are needed in order to locate small defects. Detection of two defects in the circular aluminum plate is experimentally studied using the distributions of RMS velocities under broadband periodic chirp excitations, RMS velocities of ODSs, symmetry breaking of ODSs, splitting of natural frequencies and ODSs, and a Boundary Effect Detection (BED) method. The BED method is a dynamics-based non-destructive method; it processes experimental ODSs to reveal extra local boundary effects caused by defects to reveal locations of defects. Experimental results show that small defects in circular plates can be pinpointed by these approaches. Moreover, a new concept of using the balance of elastic and kinetic energies within a mode cell for detecting defects in two-dimensional structures of irregular shapes is proposed.

Analytical treatment of uniform multiconductor transmission lines

The authors study a certain class of solutions of the multiconductor-transmission-line (MTL) equations. This class is basically defined by two reasonable but otherwise arbitrary demands: the commutativity between the propagation matrix and the characteristic impedance matrix, and the assumption that the matrices of interest are real matrices times complex-valued functions (e.g., functions of the constitutive parameters of the medium). On the basis of the above requirements, it can be shown that all matrices that are relevant for the MTL equations and for their solutions can be simultaneously diagonalized with only one set of eigenvectors (e.g., the eigenvectors of the per-unit-length inductance matrix). Of course, the sets of eigenvalues corresponding to different matrices are different. The authors investigate the MTL equations and their solutions for different environments and different properties, including lossy lines and lossy media. Mechanisms that can cause splitting of natural frequencies are considered. >