Cyclic Symmetric Structures Research Articles

Reduction and coupling of substructures via Gram–Schmidt Interface modes

The paper presents a novel reduction method in the class of component mode synthesis techniques. The method is here called Gram–Schmidt Interface (GSI), since it is based on the Gram–Schmidt algorithm. The GSI reduction method addresses problems where the Craig–Bampton reduced order model of a substructure has still a large number of interface degrees of freedom that should be further reduced. Starting from the pre-existing reduction method based on the Characteristic Constraint Modes, the GSI technique goes further, being suitable to reduce a Craig–Bampton model that retains as a master a wide variety of boundary DoFs (interface, active, etc.). The GSI method is here proved to be effective when assembling adjacent substructures having non-compatible meshes at the interfaces. This is possible by performing the interfaces coupling in the modal space of the GSI modes instead in the space of physical degrees of freedom. The technique is here applied to a turbomachinery test case consisting of a blade coupled with a disk’s sector treated in cyclic symmetry conditions. It is also demonstrated how the GSI method appears useful in the reduction of cyclic symmetric structures, representing a valid alternative of the method developed by Tran.

Night panoramic image stitching algorithm based on cyclic symmetric structure

Aiming at the problem that the current stitching image has poor visual effect and low stitching efficiency when performing the night panoramic image stitching task, a night panoramic image stitching algorithm based on cyclic symmetric structure is proposed. In this algorithm, appropriate image acquisition method is used and the image obtained by the acquisition is smoothed, edge sharpened, geometrically corrected, etc. The position of the key point of the preprocessed image and the corresponding scale are determined by Gaussian function, and the direction value is assigned to the feature point to generate the feature descriptor, so as to realize the image edge feature extraction; on this basis, the edge feature points are hierarchically segmented according to the topological structure of the edge feature point set of target image, and the edge feature point registration of night panoramic image is completed; the cyclic symmetric structure is used to make the discrete Fourier transform of the target image, in order to reduce the amount of calculation; according to the gradient value and correlation coefficient of each pixel neighborhood in the overlapping region, the stitching line is searched to overcome the ghost image of the image after registration and realize seamless stitching of night panoramic image. The experimental results show that the proposed algorithm can effectively preserve the texture information of the original image while eliminating the smooth transition of the stitching seam, and at the same time, it has better stitching effect and stitching efficiency for the images under different conditions.

A Mode-Accelerated XXr (MAX) method for complex structures with large blends

Various reduced order models have been proposed for characterizing cyclic symmetric structures with complex geometry and varying material properties that are subject to complex boundary or loading conditions. Small variations can be represented as small mass or stiffness mistuning. Techniques developed to handle such variations rely on the fact that the modes of the system can be accurately approximated using a linear combination of modes of the healthy/nominal system. Such approximations are valid in regions of high modal density, but they break down when variations are large or geometric changes are present. To address this challenge, a novel method is presented to predict the vibration response of cyclic symmetric structures with large geometric changes due to damage in the form of missing material (blends). The central idea of the new approach is an extension of the XXr method for cracked structures. The XXr method was developed for modeling small mistuning and large cracks. That method is extended in this work to handle large blends. In addition, a specialized component mode synthesis is combined with the extended XXr method to maintain accuracy. Also, unique to the proposed novel method is a technique to accelerate the convergence of the order reduction, and thus obtain very low order models. These low order models provide excellent computational speed and effectiveness while maintaining accuracy. Therefore, the method can be applied to highly refined, realistic models of industrial size. To demonstrate the proposed mode-accelerated XXr method, the effects of large blends on the response of a bladed disk are investigated.

A piecewise partitioning Scaled Boundary Finite Element algorithm to solve viscoelastic problems with cyclic symmetry

Scaled Boundary Finite Element Method (SBFEM) and a temporally adaptive algorithm are combined to solve viscoelastic problems. By expanding variables at a discretized time interval, a spatially and temporally coupled viscoelastic problem is decoupled into a series of recursive spatial problems, which are solved by SBFEM, the computing accuracy in the time domain is controlled via a self-adaptive process. For the cyclic symmetric structures, the cyclic symmetry is exploited to reduce the computational expense of SBFEM, both the eigenvalue and system equations of SBFEM are partitioned into a number of smaller independent problems, which are solved by a partitioning algorithm. Two numerical examples are given to verify and illustrate the proposed approach.

Finite Elements in Rotordynamics

Equation of motion of rotating structures may be derived referring to co-rotating as well as fixed reference systems. Correspondingly, different additional matrices must be taken into account in finite element idealization. Symmetry is very important since axisymmetric as well as cyclic-symmetric structures after Coleman transformation may be investigated in fixed coordinates whereas unsymmetric rotors with isotropic support are usually studied in co-rotating coordinates. In the most general case one gets a system of equation with periodic coefficients which may be solved using Floquet's theory.

Special Issue of Applied Mechanics Reviews in Collaboration With the Journal of Vibration and Acoustics

Special Issue of Applied Mechanics Reviews in Collaboration With the Journal of Vibration and Acoustics

Correlation Correction for Doublet Mode Shapes of Cyclic Symmetric Structures

Correlation Correction for Doublet Mode Shapes of Cyclic Symmetric Structures

Vibration Analysis of Aeroengine Turbine Blade-Disc Coupling System

This paper introduces the FEA method for cyclic symmetric structures and makes an vibration characteristics analysis to a certain type of aeroengine turbine blade-disc coupling system based on this method. According to the analysis results, the dynamic characteristics of the blade-disc coupling system are discussed. The analysis method and results in this paper can be used for further study on optimal design and vibration safety verification for the turbine blade-disc coupling system.

Vibration Analysis of Aeroengine Compressor Blade-Disc Coupling System

This paper introduces the FEA method for cyclic symmetric structures and makes an vibration characteristics analysis to a certain aeroengine compressor blade-disc coupling system based on this method. According to the anslysis results the dynamic characteristics of the blade-disc coupling system are discussed. The analysis method and results in this paper can be used for further study on optimal design and vibration safety verification for the blade-disc coupling system.

The Gas Phase HO-Initiated Oxidation of Furan: A Theoretical Investigation on the Reaction Mechanism

The reaction mechanism of the gas-phase HO-initiated oxidation of furan has been investigated by means of high level theoretical methods. The reaction is a complex process that begins with the formation of a pre-reactive hydro- gen bonded complex, previous to the addition of the HO radical to furan, forming the 2 and 3-HO-adducts. In the pre- reactive complex, the hydrogen bond is formed by interaction between the hydrogen of the hydroxyl radical and the π sys- tem of furan and its stability is computed to be 1.6 kcal·mol -1 (including the BSSE corrections). The 2 and 3-HO-adducts are computed to be 30.5 and 12.5 kcal·mol -1 respectively, more stable than the reactants. The transition state leading to the formation of the 2-HO-adduct lies below the energy of the separate reactants (0.5 kcal·mol -1 ) and the transition state pro- ducing the 3-HO-adduct that is computed to lie 3.4 kcal·mol -1 above the sum of the energies of furan and hydroxyl radical. There are four reaction paths for the ring-opening of the 2-HO-adduct, leading to the formation of different conformers of 4-hydroxybutenaldehyde radical. The most stable of these conformers presents a cyclic symmetric (C 2V ) structure and can be characterized as a low-barrier hydrogen bond. The geometry optimizations and characterizations done in this work were carried out at MP2/6-311G(d,p), MP2/6-311+G(2df,2p) and QCISD/6-311G(d,p) levels of theory, whereas the rela- tive energies are obtained at CCSD(T)/cc-pVTZ level of theory.

Resonant vibrations in harmonically excited weakly coupled mechanical systems with cyclic symmetry

The resonant vibrations in weakly coupled nonlinear cyclic symmetric structures are studied. These structures consist of weakly coupled identical nonlinear oscillators. A careful bifurcation analysis of the amplitude equations is performed in the fundamental resonance case for an illustrative example consisting of a three particle system. In case of a uniformly distributed excitation, a localized response is identified in which one of the particles exhibits large amplitude motions compared to those of the other particles. In case of single-particle excitation, it is found that for very small coupling strength and large external mistuning, a large stable localized periodic response coexists with an extended small response. With an increase in the coupling strength, multiple extended solutions arise near the exact external resonance via saddle-node bifurcations. Further increase in coupling strength and a decrease in damping results in isolated asymmetric solution branches, which bifurcate from the symmetric solutions via symmetry-breaking bifurcations. The role of coupling strength in creating/destroying localized solutions is discussed.

Aeroelastic Vibrations and Stability in Cyclic Symmetric Domains

This paper deals with rotating cyclic symmetric structures, immersed in light fluid flow. Firstly, general and usual cyclic symmetric properties are recovered from the Floquet theorem for differential equations, having space periodic coefficients in conjunction with a discrete space Fourier series development. The approach for aeroelastic problem having cyclic symmetry is then formulated based on the twin mode approach. In addition to modal one, rotating and stationary wave bases are introduced to derive the equilibrium equations for a non‐dissipative system subjected to aerodynamic loading. Rotating wave basis is the natural one, and it also permits consistently to prescribe the aerodynamic pressure on the boundary between the fluid and the structure. The aerodynamic load is then derived from a harmonic analysis of the fluid flow extending to turbomachinery as in the case of aeroplane wing. In this way, aerodynamic forces may be obtained as general as required, depending on successive time derivatives of degrees of freedom in addition to themselves. Finally, some special cases are given and stability is studied for a cyclic periodic blade assembly, even when mistuning between sectors can occur.

Comparison of methods for free-vibration analysis of large cyclic symmetric problems

The solution techniques to the free-vibration problems associated with cyclic symmetric structures with generalized loading are discussed. Three distinct approaches to solve the problem are explained. The three approaches and their variations are compared with respect to core and time required. Guidelines are given for designers to select suitable methods, keeping in mind industrial problems, the core available in a PC and ease of organization of the program.

SUBHARMONIC OSCILLATIONS IN HARMONICALLY EXCITED MECHANICAL SYSTEMS WITH CYCLIC SYMMETRY

The third-order subharmonic oscillations in weakly non-linear cyclic symmetric structures with multiple degrees of freedom are studied. These strongly coupled cyclic structures, in their linear approximation, are known to possess pairwise double-degenerate natural frequencies with orthogonal normal modes. The asymptotic method of averaging is used to study the nonlinear interactions between the pairs of modes with nearly identical natural frequencies when the external excitation is nearly three times the natural frequency of the modes being excited. A careful local bifurcation analysis of the averaged equations is conducted to study the effects of frequency mistuning and excitation amplitudes, as well as the modal damping in the system. Subharmonic standing and travelling wave type solutions, Hopf bifurcation from travelling wave solutions to quasiperiodic responses, period-doubling bifurcations, and Silnikov type chaos are found to exist in the averaged system.

Pseudo element method (PEM) for solving matrix with elementary operation and its application

A pseudo element method (PEM) is presented in this paper for solving the matrix with elementary operation arising from the static analysis of cyclic symmetric structures. Three kinds of pseudo elements are devised to consider the transformed terms in the stiffness matrix during the elementary operation. Thus, the overall stiffness matrix may not be assembled by using the frontal solution technique. This can need relatively small computer memory and relatively short CPU time. The centrifugal displacements and stresses of a ring structure are calculated by introducing the cyclic symmetric conditions. The calculated results show, in comparison with theoretical ones, that the present method and its program are effective and can be conveniently used on a personal computer. At the end of this paper, the centrifugal fillet stresses of left-handed helical gear structures with different rim and web sizes are evaluated.

Modal methods for the analysis of cyclic symmetric structures

In this paper, the modal displacement and modal acceleration methods have been extended to the cyclic symmetry analysis in the semi-complex domain. The study of the influence of forcing frequency on the relative accuracy of the two modal methods is also made. It is shown that only few modes are sufficient to get accurate stresses in the case of modal acceleration method.

On the Use of the Ritz-Wilson Method for the Dynamic Response Analysis of Cyclic Symmetric Structures

The Ritz-Wilson technique is extended for the dynamic analysis of cyclic symmetric structures in the semi-complex domain. The load participation factor, eigenspectrum and force error norm for the present technique are studied, and the accuracy and efficiency are compared with those for the modal displacement technique.

SOME STUDIES ON FREE VIBRATION ANALYSIS OF CYCLIC SYMMETRIC STRUCTURES USING SUBSPACE METHOD

The free vibration analysis of cyclic symmetric structures is considered as a Hermitian eigenvalue problem in semi‐complex domain using subspace iteration method is presented. The trial vectors are selected using a modified Ritz vector scheme. A modified convergence criterion which gives true error estimates which is suited for clustered eigenvalue problems is presented. Also the effect of purification of trial vectors on convergence is considered.

Application of block Lanczos method for analysis of cyclic symmetric structures

AbstractIn this study the free vibration of circumferentially periodic structures has been analysed using the block Lanczos method in a semi complex domain for extracting real natural frequencies and complex mode shapes. The influence of the number of vectors in the block and also the effect of renumbering of nodes, depending on whether they are internal, adjacent or boundary nodes, on the CPU time are studied. The efficiency of the present method is compared with the subspace iteration schemes.

Free‐vibration analysis of cyclic symmetric structures

AbstractA subspace iteration scheme for the Hermitian eigenvalue problem arising from analysis of cyclic symmetric structures is presented. This operates in the semi‐complex domain and its computational efficiency is demonstrated. A modified convergence checking criterion which gives better error estimates, and a new trial vector selection scheme which accelerates convergence for the Hermitian eigenvalue problem based on the Ritz method, are presented.