scholarly journals Free vibration and damping analysis of the cylindrical shell partially covered with equidistant multi-ring hard coating based on a unified Jacobi-Ritz method

2021 ◽  
Vol 104 (3) ◽  
pp. 003685042110325
Author(s):  
Jian Yang ◽  
Hua Song ◽  
Dong Chen ◽  
Yue Zhang
Keyword(s):  
Cylindrical Shell ◽  
Loss Factor ◽  
Ritz Method ◽  
Rectangular Pulse ◽  
Hard Coating ◽  
Wave Number ◽  
Thickness Variation ◽  
Ring Number ◽  
Wave Numbers ◽  
Circumferential Wave

In this study, the aim was to evaluate the vibration suppression performance of the partially covered equidistant multi-ring hard coating damping treatment for the cylindrical shell structure in aviation power equipment. A continuous rectangular pulse function was presented to describe the local thickness variation of arbitrary coating proportion and arbitrary number of coating rings. A semi-analytical unified solution procedure was established by combining the rectangular pulse function, the generalized Jacobi polynomials, and the Rayleigh-Ritz method. The stiffness coefficient k = 1013 N/m2 and the truncation number N = 8 were found to be large enough to achieve an accurate and efficient solution of the vibration analysis of the shell. The modal loss factor generally increased with the increase of the coating proportion ranging from 0.0 to 1.0 for all the circumferential wave numbers. The modal loss factor increased roughly linear with the coating proportion for all the circumferential wave numbers. And the modal loss factor was increased with the circumferential wave number, and the greater the number of circumferential waves, the greater the rate of change. The increase of the ring number was not always beneficial for vibration reduction of the shell, while the modal loss factor increased roughly linear with the coating proportion. The increased ring number and coating proportion tend more to exhibit an obvious incremental damping effect under larger circumferential wave number.

The Free Vibration of a Cylindrical Shell on an Elastic Foundation

1998 ◽  
Vol 120 (1) ◽  
pp. 63-71 ◽  
Author(s):  
D. N. Paliwal ◽  
Rajesh K. Pandey
Keyword(s):  
Cylindrical Shell ◽  
Elastic Foundation ◽  
Circular Cylindrical Shell ◽  
Frequency Equation ◽  
Radial Mode ◽  
Wave Number ◽  
Wave Parameter ◽  
Longitudinal Modes ◽  
Foundation Parameters ◽  
Circumferential Wave

The frequency equation for a thin circular cylindrical shell resting on an elastic foundation is developed by using the first order shell theory of Sanders and eigenfrequencies are calculated. These eigenfrequencies are plotted against the axial wave parameter. Effects of the axial wave parameter, circumferential wave number, non-dimensional thickness and foundation parameters on eigenfrequencies are investigated. It is found that the foundation modulus chiefly affects the radial mode eigenfrequency and has no effect on torsional and longitudinal modes. On the otherhand, shear modulus does have influence on radial as well as tangential modes of vibrations. Though the effect on radial mode frequency is more pronounced.

Vibration of a Spinning Cylindrical Shell With Internal/External Ring Stiffeners

1996 ◽  
Vol 118 (2) ◽  
pp. 227-236 ◽  
Author(s):  
Shyh-Chin Huang ◽  
Lin-Hung Chen
Keyword(s):  
Cylindrical Shell ◽  
Mass Density ◽  
Wave Number ◽  
External Ring ◽  
Spin Speed ◽  
Receptance Method ◽  
Frequency Changes ◽  
Spinning Speed ◽  
Stiffening Effect ◽  
Circumferential Wave

The paper presents an approach to the vibration analysis of a spinning cylindrical shell with internal, symmetric, or external ring stiffeners. A modified receptance method for spinning structures is employed in this analysis. Various numerical examples are demonstrated and the results are compared with the existing data. The effects of types, numbers of stiffeners and of spin speed on the shell frequencies are extensively discussed. The results show that for no spin the ring stiffeners stiffen only then > 1 modes (n–circumferential wave number), and the stiffening effect become more significant with the increasing n number. With spin, the rings stiffen the forward modes in a way similar to the non-spin cases. The backward modes are however all stiffened by the attached rings for all n values. Among the three types of rings, on backward modes, the internal rings always have a better stiffening effect, then the symmetric and the external rings. As to the forward modes, as spinning speed increases, the external rings raise the shell’s frequencies faster than the others due to the largest centrifugal force. At last, the effects of the ring’s location, stiffness, and mass density on the frequency changes are examined. Numerical results show that the sensitivity of the shell’s frequencies to these parameters increases with the spin speed. Among the shell modes, the lower n modes are affected more by these parameters.

Vibration-damping characteristic analysis of constrained stand-off layer damping cylindrical shell using Rayleigh-Ritz method

2019 ◽  
Vol 37 (1) ◽  
pp. 93-119
Author(s):  
Bijuan Yan ◽  
Huijun Liang ◽  
Minjie Jin ◽  
Zhanlong Li ◽  
Yong Song
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Dynamic Model ◽  
Loss Factor ◽  
Ritz Method ◽  
Accurate Solution ◽  
Vibration Damping ◽  
Content Type ◽  
Damping Characteristics ◽  
Classical Boundary

Purpose In the vibration reduction field, constrained stand-off layer damping cylindrical shell plays an important role. However, due to the lack of accurate analysis of its damping characteristics, this hinders its further research and application. Therefore, the purpose of this paper is concerned with an accurate solution for the vibration-damping characteristics of a constrained stand-off-layer damping cylindrical shell (CSDCS) under various classical boundary conditions and conducts a further analysis. Design/methodology/approach Based on the Rayleigh–Ritz method and the Hamilton principle, a dynamic model of CSDCS is established. Then the loss factor and the frequency of CSDCS are obtained. The correctness and convergence behavior of the present model are verified by comparing the calculation results with the literature. By using for various classical boundary conditions without any special modifications in the solution procedure, the characteristics of CSDCS with S-S, C-C, C-S, C-F and S-F boundaries are discussed. Findings The Rayleigh–Ritz method is effective in handling the problem of CSDCS with different boundaries and an accurate solution is obtained. The boundary conditions have an important influence on the vibration and damping behavior of the CSDCS. Originality/value Based on the Rayleigh–Ritz method and Hamilton principle, a dynamic model of CSDCS is established for the first time, and then the loss factor and frequency of CSDCS are obtained. In addition, the effectiveness of adding the stand-off layer between the base shell and the viscoelastic layer is confirmed by discussing the characteristics of CSDCS with S-S, C-C, C-S, C-F and S-F boundaries.

Infrared spectra and substituent effects in 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones

1983 ◽  
Vol 48 (2) ◽  
pp. 586-595 ◽  
Author(s):  
Alexander Perjéssy ◽  
Pavol Hrnčiar ◽  
Ján Šraga
Keyword(s):  
Substituent Effects ◽  
Phenyl Group ◽  
Wave Number ◽  
Vibrational Coupling ◽  
Wave Numbers ◽  
Stretching Vibration ◽  
Stretching Vibrations ◽  
The Relationship ◽  
Derivatives Of ◽  
Effect Of Structure

The wave numbers of the fundamental C=O and C=C stretching vibrations, as well as that of the first overtone of C=O stretching vibration of 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones and 1,3-cycloheptanedione were measured in tetrachloromethane and chloroform. The spectral data were correlated with σ+ constants of substituents attached to phenyl group and with wave number shifts of the C=O stretching vibration of substituted acetophenones. The slope of the linear dependence ν vs ν+ of the C=C stretching vibration of the ethylenic group was found to be more than two times higher than that of the analogous correlation of the C=O stretching vibration. Positive values of anharmonicity for asymmetric C=O stretching vibration can be considered as an evidence of the vibrational coupling in a cyclic 1,3-dicarbonyl system similarly, as with derivatives of 1,3-indanedione. The relationship between the wave numbers of the symmetric and asymmetric C=O stretching vibrations indicates that the effect of structure upon both vibrations is symmetric. The vibrational coupling in 1,3-cycloheptanediones and the application of Seth-Paul-Van-Duyse equation is discussed in relation to analogous results obtained for other cyclic 1,3-dicarbonyl compounds.

Free vibration analysis of three-layer thin cylindrical shell with variable thickness two-dimensional FGM middle layer under arbitrary boundary conditions

2021 ◽  
pp. 109963622110204
Author(s):  
Xue-Yang Miao ◽  
Chao-Feng Li ◽  
Yu-Lin Jiang ◽  
Zi-Xuan Zhang
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Admissible Function ◽  
Free Vibration Analysis ◽  
Middle Layer ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions

In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.

scholarly journals The band spectrum of hydrogen

1924 ◽  
Vol 106 (735) ◽  
pp. 69-82 ◽  
Keyword(s):  
Molecular Hydrogen ◽  
Hydrogen Molecule ◽  
Spectral Lines ◽  
Wave Number ◽  
Band Spectrum ◽  
Wave Numbers ◽  
Experimental Side ◽  
The Subject ◽  
The Moment ◽  
Theoretical Side

Many attempts have been made to detect regularities amongst the numerous lines which constitute the secondary or many-lined spectrum of hydrogen. The extreme complexity of the spectrum may be realised from the fact that in the Bakerian Lecture of 1922 Merton and Barratt record some 750 lines in the interval between Hα (wave-number v = 5233.216) and Hβ ( v = 20564.793). Three methods of investigation may be employed in the search for regularities. (1) The lines may be classified according to their physical characteristics, such as intensity or mode of excitation, as in the tables of Merton and Barrat ( loc. cit .). (2) Lines may be grouped together by the discovery of relations between their wave-lengths or wave-numbers, as in the important groups of lines which have been arranged in bands by Fulcher. (3) Lastly, the question may be attacked from the theoretical side, and a model of the hydrogen molecule may be imagined, which will give rise to the emission of certain characteristic spectral lines. Thus Sutherland, working on the foundation of the classical mechanical laws, more than twenty years ago, came to the conclusion that spectral series must arise from kinematical considera­tions, and explained them by considering the nodal sub-divisions of a circle. At the present time we may expect more successful results to follow from the application of the quantum theory, and in this paper an endeavour will be made to examine the secondary spectrum of hydrogen, and more particularly the Fulcher bands, from this standpoint. I may add that my interest in the subject was aroused when attempting to construct a model of the hydrogen molecule, for it seemed that the most likely method of obtaining reliable information from the experimental side as to the moment of inertia of the molecule would be from a study of the spectrum of molecular hydrogen.

An hp-version adaptive finite element algorithm for eigensolutions of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Yongliang Wang ◽  
Jianhui Wang
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Cylindrical Shells ◽  
Higher Order ◽  
Wave Number ◽  
Adaptive Finite Element ◽  
Content Type ◽  
Circular Cylindrical Shells ◽  
Geometrical Factors ◽  
Circumferential Wave

PurposeThis study presents a novel hp-version adaptive finite element method (FEM) to investigate the high-precision eigensolutions of the free vibration of moderately thick circular cylindrical shells, involving the issues of variable geometrical factors, such as the thickness, circumferential wave number, radius and length.Design/methodology/approachAn hp-version adaptive finite element (FE) algorithm is proposed for determining the eigensolutions of the free vibration of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation. This algorithm first develops the established h-version mesh refinement method for detecting the non-uniform distributed optimised meshes, where the error estimation and element subdivision approaches based on the superconvergent patch recovery displacement method are introduced to obtain high-precision solutions. The errors in the vibration mode solutions in the global space domain are homogenised and approximately the same. Subsequently, on the refined meshes, the algorithm uses higher-order shape functions for the interpolation of trial displacement functions to reduce the errors quickly, until the solution meets a pre-specified error tolerance condition. In this algorithm, the non-uniform mesh generation and higher-order interpolation of shape functions are suitable for addressing the problem of complex frequencies and modes caused by variable structural geometries.FindingsNumerical results are presented for moderately thick circular cylindrical shells with different geometrical factors (circumferential wave number, thickness-to-radius ratio, thickness-to-length ratio) to demonstrate the effectiveness, accuracy and reliability of the proposed method. The hp-version refinement uses fewer optimised meshes than h-version mesh refinement, and only one-step interpolation of the higher-order shape function yields the eigensolutions satisfying the accuracy requirement.Originality/valueThe proposed combination of methodologies provides a complete hp-version adaptive FEM for analysing the free vibration of moderately thick circular cylindrical shells. This algorithm can be extended to general eigenproblems and geometric forms of structures to solve for the frequency and mode quickly and efficiently.

Acoustic Resonance in an Axial Multistage Compressor Leading to Non-Synchronous Blade Vibration

2020 ◽  
Author(s):  
Anne-Lise Fiquet ◽  
Agathe Vercoutter ◽  
Nicolas Buffaz ◽  
Stéphane Aubert ◽  
Christoph Brandstetter
Keyword(s):  
Acoustic Waves ◽  
High Speed ◽  
Numerical Study ◽  
Acoustic Resonance ◽  
Structural Vibration ◽  
Wave Number ◽  
Blade Vibration ◽  
Acoustic Modes ◽  
Blade Vibrations ◽  
Circumferential Wave

Abstract Significant non-synchronous blade vibrations (NSV) have been observed in an experimental three-stage high-speed compressor at part-speed conditions. High amplitude acoustic modes, propagating around the circumference and originating in the highly loaded Stage-3 have been observed in coherence with the structural vibration mode. In order to understand the occurring phenomena, a detailed numerical study has been carried out to reproduce the mechanism. Unsteady full annulus RANS simulations of the whole setup have been performed using the solver elsA. The results revealed the development of propagating acoustic modes which are partially trapped in the annulus and are in resonance with an aerodynamic disturbance in Rotor-3. The aerodynamic disturbance is identified as an unsteady separation of the blade boundary layer in Rotor-3. The results indicate that the frequency and phase of the separation adapt to match those of the acoustic wave, and are therefore governed by acoustic propagation conditions. Furthermore, the simulations clearly show the modulation of the propagating wave with the rotor blades, leading to a change of circumferential wave numbers while passing the blade row. To analyze if the effect is self-induced by the blade vibration, a noncoherent structural mode has been imposed in the simulations. Even at high vibration amplitude the formerly observed acoustic mode did not change its circumferential wave number. This phenomenon is highly relevant to modern compressor designs, since the appearance of the axially propagating acoustic waves can excite blade vibrations if they coincide with a structural eigenmode, as observed in the presented experiments.

scholarly journals Small-Scale Solar Magnetic Fields

1976 ◽  
Vol 71 ◽  
pp. 69-99 ◽  
Author(s):  
J. O. Stenflo
Keyword(s):  
Magnetic Fields ◽  
Magnetic Flux ◽  
Solar Surface ◽  
Life Time ◽  
Wave Number ◽  
Small Scale ◽  
Solar Magnetic Fields ◽  
Field Amplification ◽  
Wave Numbers ◽  
Diffuse Form

The observed properties of small-scale solar magnetic fields are reviewed. Most of the magnetic flux in the photosphere is in the form of strong fields of about 100–200 mT (1–2 kG), which have remarkably similar properties regardless of whether they occur in active or quiet regions. These fields are associated with strong atmospheric heating. Flux concentrations decay at a rate of about 107 Wb s-1, independent of the amount of flux in the decaying structure. The decay occurs by smaller flux fragments breaking loose from the larger ones, i.e. a transfer of magnetic flux from smaller to larger Fourier wave numbers, into the wave-number regime where ohmic diffusion becomes significant. This takes place in a time-scale much shorter than the length of the solar cycle.The field amplification occurs mainly below the solar surface, since very little magnetic flux appears in diffuse form in the photosphere, and the life-time of the smallest flux elements is very short. The observations further suggest that most of the magnetic flux in quiet regions is supplied directly from below the solar surface rather than being the result of turbulent diffusion of active-region magnetic fields.

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