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

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.

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

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.

ACOUSTIC RESONANCE IN AN AXIAL MULTISTAGE COMPRESSOR LEADING TO NON-SYNCHRONOUS BLADE VIBRATION

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 non-coherent 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 as observed in the presented experiments.

Shear horizontal circumferential wave EMAT design for pipeline inspection based on FEM

The electromagnetic acoustic transducer (EMAT) is widely used in In-Line Inspection of gas pipeline. However, the transduction efficiency of the EMAT transmitter is lower compared with the traditional piezoelectric transducer and needs improving. To enhance the amplitude of the received signal, an improved periodic-permanent-magnet EMAT is provided as well as its construction method to generate shear horizontal guided waves propagating in the circumferential direction. By designing a racetrack coil and magnetic array with the same radian of the annular pipeline, the EMAT is more suitable to generate the circumferential wave in the pipeline. The performance of the EMAT transmitter has been assessed with the validated finite element model.

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

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.

Nonlinear Free Vibrations Analysis of Delaminated Composite Conical Shells

This paper examines the nonlinear free vibration of laminated composite conical shells throughout the circumferential delamination. First, based on the energy method, the governing equation of motion for the shell was derived. To simplify the analysis, the nonlinear partial differential equations were reduced into a system of coupled ordinary differential equations using Galerkin’s method. Consequently, the results were obtained by the numerical methods. Finally, the effects of delamination, variations in the delamination length, conical shells characteristics, materials property and circumferential wave number on the nonlinear response of delaminated composite conical shells were examined. The results show that the presence of delamination leads to increase in the amplitude of oscillations for the shells. Besides, the increase in the delamination length and decrease of the circumferential wave number, number of layers, and half vertex angle of the cone and orthotropy bring about a decrease in the nonlinearity of delaminated composite conical shells. However, an increase of the middle surface radius of the shell leads to a reduction of the nonlinearity as well as an increase of the amplitude.

On dispersion relations in smart laminated fiber-reinforced composite membranes considering different piezoelectric coupling effects

Wave propagation characteristics are determined for smart laminated fiber-reinforced composite cylindrical membrane shells with different piezoelectric coupling effects. Wave motion equations are derived using the membrane shell model. By solving an eigenvalue problem, dispersion curves of the wave motion are obtained for different axial and circumferential wave numbers. The effects of piezoelectric coupling, fiber orientation, stacking sequence, and material properties of the host shell on wave behaviors are investigated. The results of this paper can be used for studies on dynamic stability of piezoelectric coupled laminated fiber-reinforced composite shell structures and in design of smart structures with the piezoelectric materials for the applications of damage detection and structural health monitoring.