Modal Identification of Motorcycle Tires Using Circumferential Wave Number Decomposition

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.

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

Volume 10A: Structures and Dynamics ◽

10.1115/gt2020-16011 ◽

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.

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Two-Dimensional Shell Vibration of Microtubule in Living Cell

ASME 2010 7th International Symposium on Fluid-Structure Interactions, Flow-Sound Interactions, and Flow-Induced Vibration and Noise: Volume 3, Parts A and B ◽

10.1115/fedsm-icnmm2010-30636 ◽

2010 ◽

Author(s):

E. Ghavanoo ◽

F. Daneshmand ◽

M. Amabili

Keyword(s):

Living Cell ◽

Shell Theory ◽

Coupled System ◽

Eukaryotic Cell ◽

Wave Number ◽

Orthotropic Materials ◽

Two Dimensional ◽

Polynomial Eigenvalue Problem ◽

Shell Vibration ◽

Circumferential Wave

The mechanical behavior of a eukaryotic cell is mainly determined by its cytoskeleton. Microtubules immersed in cytosol are a central part of the cytoskeleton. Cytosol is the viscous fluid in living cells. The microtubules permanently oscillate in the cytosol. In this study, two-dimensional vibration of a single microtubule in living cell is investigated. The Donnell’s shell theory equations for orthotropic materials is used to model the microtubule whereas the motion of the cytosol is modeled as Stokes flow characterized by a small Reynolds number with no-slip condition at microtubule-cytosol interface. The stress field in the cytosol induced by vibrating microtubule is determined analytically and the coupled vibrations of the microtubule-cytoplasm system are investigated. A coupled polynomial eigenvalue problem is developed in the present study and the variations of eigenvalues of coupled system with cytosol dynamic viscosity, microtubule circumferential Young’s modulus and circumferential wave number are examined.

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Free Vibrations of Thick Hollow Circular Cylinders From Three-Dimensional Analysis

Journal of Vibration and Acoustics ◽

10.1115/1.2889692 ◽

1997 ◽

Vol 119 (1) ◽

pp. 89-95 ◽

Cited By ~ 58

Author(s):

Jinyoung So ◽

A. W. Leissa

Keyword(s):

Three Dimensional ◽

Free Vibrations ◽

Circular Cylinders ◽

Wave Number ◽

Algebraic Polynomials ◽

Cylindrical Wall ◽

Local Coordinates ◽

Significant Figures ◽

Circumferential Wave ◽

Outside Diameter

A three-dimensional (3-D) method of analysis is developed for the free vibration frequencies of hollow circular cylinders of elastic material. The method is based upon local coordinates whose origin is attached to the center of cylindrical wall. It assumes for the three displacement components a Fourier series in the circumferential (θ) direction and algebraic polynomials in the radial (q) and axial (z) directions. Convergence studies for completely free cylinders show that the analysis can yield frequencies which are exact to five or six significant figures. These accurate frequencies are compared with those from other 3-D analyses available for free hollow circular cylinders having various length-to-outside diameter (L/Do) and inside-to-outside diameter (Di/Do) ratios. Extensive, accurate data are presented for the first 10 frequencies of each circumferential wave number 0 through 5 for hollow circular cylinders having Di/Do of 0.1, 0.5, and 0.9, with L/Do = 0.2, 1 and 5 and a Poisson’s ratio (v) = 0.3.

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The Free Vibration of a Cylindrical Shell on an Elastic Foundation

Journal of Vibration and Acoustics ◽

10.1115/1.2893828 ◽

1998 ◽

Vol 120 (1) ◽

pp. 63-71 ◽

Cited By ~ 25

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.

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Discontinuities in the Sensitivity Curves of Laminated Cylindrical Shells

Journal of Applied Mechanics ◽

10.1115/1.1748341 ◽

2004 ◽

Vol 71 (3) ◽

pp. 418-420 ◽

Cited By ~ 3

Author(s):

Yiska Goldfeld ◽

Izhak Sheinman

Keyword(s):

Parametric Study ◽

General Procedure ◽

Cylindrical Shells ◽

Buckling Analysis ◽

Wave Number ◽

Sensitivity Curves ◽

Post Buckling ◽

Circumferential Wave ◽

Laminated Cylindrical Shells

The discontinuity in the sensitivity of laminated cylindrical shells is investigated via the initial post-buckling analysis. A general procedure for sensitivity, based on Koiter’s parameters and using the Donnell and Sanders shell theories, is developed and used for parametric study of the discontinuity phenomenon. It was found that the discontinuity occurs at points of change of the circumferential wave number.

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

Science Progress ◽

10.1177/00368504211032550 ◽

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.

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Vibration Analysis of a Floating Roof Taking Into Account the Nonlinearity of Sloshing

Journal of Applied Mechanics ◽

10.1115/1.2912739 ◽

2008 ◽

Vol 75 (4) ◽

Cited By ~ 18

Author(s):

M. Utsumi ◽

K. Ishida

Keyword(s):

Linear Approximation ◽

Nonlinear Effect ◽

Vibration Analysis ◽

Numerical Results ◽

Roof Displacement ◽

Wave Number ◽

Nonlinear Sloshing ◽

Modal Component ◽

Liquid Depth ◽

Circumferential Wave

The vibration of a floating roof hydroelastically coupled with nonlinear sloshing is analyzed. Influences of the nonlinearity of sloshing on the magnitude of stresses arising in a floating roof are investigated. Numerical results show that (i) neglecting the nonlinearity of sloshing significantly underestimates the magnitude of the stresses, even when the nonlinear effect is small for the roof displacement; and (ii) the underestimation associated with the use of the linear approximation becomes more marked with the decrease in the liquid depth. The reasons for these results are explained based on the fact that in the nonlinear sloshing, the modal component with circumferential wave number 2 is excited.

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

Journal of Vibration and Acoustics ◽

10.1115/1.2889653 ◽

1996 ◽

Vol 118 (2) ◽

pp. 227-236 ◽

Cited By ~ 9

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.

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Dynamic Stability of Truncated Conical Shells Under Pulsating Torsion

Journal of Applied Mechanics ◽

10.1115/1.3157628 ◽

1981 ◽

Vol 48 (2) ◽

pp. 391-398 ◽

Cited By ~ 7

Author(s):

J. Tani

Keyword(s):

Dynamic Stability ◽

Practical Importance ◽

Low Frequency ◽

Instability Region ◽

Wave Number ◽

Vibration Modes ◽

Conical Shells ◽

Instability Regions ◽

The Galerkin Method ◽

Circumferential Wave

The dynamic stability of clamped, truncated conical shells under periodic torsion is analyzed by the Galerkin method in conjunction with Hsu’s results. The instability regions of practical importance are clarified for relatively low frequency ranges. Numerical results indicate that under the purely periodic torsion only the combination instability region exists but that with an increase in the static torsion the principal instability region becomes most significant. The relative openness of the instability regions is found to depend sensitively on the circumferential phase difference of two vibration modes excited simultaneously at the resonance with the same circumferential wave number.

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