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In this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.

An adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.

The results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.

The proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

This study aimed to solve the engineering problem of free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams. The accurate identification of the crack damage depth, number and location depends on high-precision frequency and vibration mode solutions; therefore, it is critical to obtain these reliable solutions. The high-precision finite element method for the free vibration of cracked beams needs to be developed to grasp and control error information in the conventional solutions and the non-uniform mesh generation near the cracks. Moreover, the influence of multi-crack damage on the natural frequency and vibration mode of a circularly curved beam needs to be detected.

A scheme for cross-sectional damage defects in a circularly curved beam was established to simulate the depth, location and the number of multiple cracks by implementing cross-section reduction induced by microcrack damage. In addition, the h-version finite element mesh adaptive analysis method of the Timoshenko beam was developed. The superconvergent solution of the vibration mode of the cracked curved beam was obtained using the superconvergent patch recovery displacement method to determine the finite element solution. The superconvergent solution of the frequency was obtained by computing the Rayleigh quotient. The superconvergent solution of the eigenfunction was used to estimate the error of the finite element solution in the energy norm. The mesh was then subdivided to generate an improved mesh based on the error. Accordingly, the final optimised meshes and high-precision solution of natural frequency and mode shape satisfying the preset error tolerance can be obtained. Lastly, the disturbance behaviour of multi-crack damage on the vibration mode of a circularly curved beam was also studied.

Numerical results of the free vibration and damage disturbance of cracked curved beams with cracks were obtained. The influences of crack damage depth, crack damage number and crack damage distribution on the natural frequency and mode of vibration of a circularly curved beam were quantitatively analysed. Numerical examples indicate that the vibration mode and frequency of the beam would be disturbed in the region close to the crack damage, and a greater crack depth translates to a larger frequency change. For multi-crack beams, the number and distribution of cracks also affect the vibration mode and natural frequency. The adaptive method can use a relatively dense mesh near the crack to adapt to the change in the vibration mode near the crack, thus verifying the efficacy, accuracy and reliability of the method.

The proposed combination of methodologies provides an extremely robust approach for free vibration of beams with cracks. The non-uniform mesh refinement in the adaptive method can adapt to changes in the vibration mode caused by crack damage. Moreover, the proposed method can adaptively divide a relatively fine mesh at the crack, which is applied to investigating free vibration under various curved beam angles and crack damage distribution conditions. The proposed method can be extended to crack damage detection of 2D plate and shell structures and three-dimensional structures with cracks.

This study aims to provide a reliable and effective algorithm that is suitable for addressing the problems of continuous orders of frequencies and modes under different boundary conditions, circumferential wave numbers and thickness-to-length ratios of moderately thick circular cylindrical shells. The theory of free vibration of rotating cylindrical shells is of utmost importance in fields such as structural engineering, rock engineering and aerospace engineering. The finite element method is commonly used to study the theory of free vibration of rotating cylindrical shells. The proposed adaptive finite element method can achieve a considerably more reliable high-precision solution than the conventional finite element method.

On a given finite element mesh, the solutions of the frequency mode of the moderately thick circular cylindrical shell were obtained using the conventional finite element method. Subsequently, the superconvergent patch recovery displacement method and high-order shape function interpolation techniques were introduced to obtain the superconvergent solution of the mode (displacement), while the superconvergent solution of the frequency was obtained using the Rayleigh quotient computation. Finally, the superconvergent solution of the mode was used to estimate the errors of the finite element solutions in the energy norm, and the mesh was subdivided to generate a new mesh in accordance with the errors.

In this study, a high-precision and reliable superconvergent patch recovery solution for the vibration modes of variable geometrical rotating cylindrical shells was developed. Compared with conventional finite element method, under the challenging varying geometrical circumferential wave numbers, and thickness–length ratios, the optimised finite element meshes and high-precision solutions satisfying the preset error limits were obtained successfully to solve the frequency and mode of continuous orders of rotating cylindrical shells with multiple boundary conditions such as simple and fixed supports, demonstrating good solution efficiency. The existing problem on the difficulty of adapting a set of meshes to the changes in vibration modes of different orders is finally overcome by applying the adaptive optimisation.

The approach developed in this study can accurately obtain the superconvergent patch recovery solution of the vibration mode of rotating cylindrical shells. It can potentially be extended to fine numerical models and high-precision computations of vibration modes (displacement field) and solid stress (displacement derivative field) for general structural special value problems, which can be extensively applied in the field of engineering computations in the future. Furthermore, the proposed method has the potential for adaptive analyses of shell structures and three-dimensional structures with crack damage. Compared with conventional finite element methods, significant advantages can be achieved by solving the eigenvalues of structures with high precision and stability.

The purpose of this paper is to provide a systematic method to perform analysis and test for vibration‐thermal strain behavior of plastic ball grid array (PBGA) assembly by considering thermal and vibration loading mode. Also to investigate the dynamic behavior of PBGA assembly by considering loading modes for design and reliability evaluation of PBGA packaging.

A PBGA assembly prototype with different structure and material parameters is designed and manufactured. Based on investigation of the structural and physical parameters of PBGA sample, the vibration‐thermal strain test is developed to measure the strain distribution at the surface of the BT (bismaleimide triazine) substrates and PCB (printed circuit board) surface under vibration‐thermal cycling loading such as random vibration and the temperature is changed from 0°C to 100°C.

The test results show that the loading modes have different impact on PCB, EMC and substrate, respectively. In the meantime, it is shown that the characteristics of the compound mode is not the linear accumulative result by single vibration mode and single thermal loading mode as forecasted. The nonlinear mechanism for these modes application is the future work for progress.

It is very difficult to set up a numerical approach to illustrate the validity of the testing approach because the complex loading modes and the complex structure of PBGA assembly. The research on an accurate mathematical model of the PBGA assembly prototype is a future work.

It implies a potential design characteristic for future application of PBGA assembly. It also builds a basis for future work for design and reliability evaluation of BGA package.

This paper fulfils useful information about the thermal‐vibration coupling dynamic behavior of PBGA assembly with different structure characteristics, materials parameters.

The purpose of this paper is to propose an effective and novel methodology to determine optimal location of piezoelectric transducers for passive vibration control of geometrically complicated structures and shells with various curvatures. An industry‐standard aircraft leading‐edge structure is considered for the actuator placement analysis and experimental verification.

The proposed method is based on finite element analysis of the underlying structure having a thin layer of piezoelectric elements covering the entire inner surface with pertinent boundary conditions. All the piezoelectric properties are incorporated into the elements. Specifically, modal piezoelectric analysis is performed to provide computed tomography for the evaluations of the electric potential distributions on these piezoelectric elements attributed by the first bending and torsional modes of structural vibration. Then, the outstanding zone(s) yielding highest amount of electric potentials can be identified as the target location for the best actuator placement.

Six piezoelectric vibration absorbers are determined to be placed alongside both of the fixed edges. An experimental verification of the aluminum leading edge's vibration suppression using the proposed method is conducted exploiting two resistive shunt circuits for the passive damping. A good agreement is obtained between the analytical and experimental results. In particular, vibration suppression around 30 and 25 per cent and Q‐factor reduction up to 15 and 10 per cent are obtained in the designated bending and torsional modes, respectively. In addition, some amount of damping improvement is observed at higher modes of vibration as well.

The frequency in the proposed approach will be increased slowly and gradually from 0 to 500 Hz. When the frequency matches the natural frequency of the structure, owing to the resonant condition the plate will vibrate heavily. The vibrations of the plate can be observed by connecting a sensor to an oscilloscope. Owing to the use of only one sensor, not all the modes can be detected. Only the first few modes can be picked up by the sensor, because of its location.

This method can also be used in optimizing not only the location but also the size and shape of the passive vibration absorber to attain maximum amount of damping. This can be achieved by simply changing the dimensions and shape of the piezoelectric vibration absorber in the finite element model on an iterative basis to find the configuration that gives maximum electric potential.

The determination of optimal location(s) for piezoelectric transducers is very complicated and difficult if the geometry of structures is curved or irregular. Therefore, it has never been reported in the literature. Here an efficient FEA‐based electric potential tomography method is proposed to identify the optimized locations for the PZT transducers for passive vibration control of geometrically complicated structures, with minimal efforts. In addition, this method will facilitate the determination of electric potentials that would be obtained at all the possible locations for piezoelectric transducers and hence makes it possible to optimize the placement and configurations of the candidate transducers on complex shape structures.

Moderately thick circular cylindrical shells are widely used as supporting structures or storage cavities in structural engineering, rock engineering, and aerospace engineering. In practical engineering, shells often work with micro-cracks or defects. The existence of micro-crack damage may result in the disturbance of dynamic behaviours and even induce accidental dynamic disasters. The free vibration frequency and mode are important parameters for the dynamic performance and damage identification analysis. In particular, stiffness weakening of the local damage region leads to significant changes in the vibration mode, which makes it difficult for the mesh generated in the conventional finite element method to capture a high-precision solution of the local oscillation.

In response to the above problems, this study developed an adaptive finite element method and a crack damage characterisation method for moderately thick circular cylindrical shells. By introducing the inverse power iteration method, error estimation, and mesh subdivision refinement technique for the analysis of finite element eigenvalue problems, an adaptive computation scheme was constructed for the free vibration problem of moderately thick circular cylindrical shells with circumferential crack damage.

Based on typical numerical examples, the established adaptive finite element solution for the free vibration of moderately thick circular cylindrical shells demonstrated its suitability for solving the high-precision free vibration frequency and mode of cylindrical shell structures. The any order frequency and mode shape of cracked cylindrical shells under the conditions of different ring wave numbers, crack locations, crack depths, and multiple cracks were successfully solved. The influences of the location, depth, and number of cracks on the disturbance of dynamic behaviours were analysed.

This study can be used as a reference for the adaptive finite element solution of free vibration of moderately thick circular cylindrical shells with cracks and lays the foundation for further development of a high-performance computation method suitable for the dynamic disturbance and damage identification analysis of general cracked structures.

It is very important to create a useful cyclic symmetric model for the investigation of the vibration reduction effect of hard-coating blisk. This study aims to develop a cyclic symmetry algorithm which can determine the mode of blisk in the sector coordinate system directly.

Using the exponential and real quasi-equivalent Fourier matrices, the formulas for solving the sector mode were derived, and the relationship between the two kinds of sector modes was also discussed. Based on the proposed cyclic symmetry algorithm, the vibration characteristics of an academic blisk were solved, and the formulas for solving the natural characteristics and vibration responses of the coated blisk were given.

A blisk with NiCrAlCoY+YSZ hard coating on both sides of each blade was chosen as a case to demonstrate the presented method. Based on the verification analysis model, the influences of coating thickness on the vibration reduction effect of the blisk were discussed. The results show that the hard coating has good vibration reduction effect on the blisk even the coating thickness is very thin and the vibration reduction effect of hard coating in the high frequency range is obviously better than that in the low frequency range.

As a large number of reduced order modeling methods of blisk are implemented based on the sector modes, the proposed method which can obtain the sector modes directly will significantly improve the efficiency of dynamic modeling and analysis of the coated blisk structure.

The modes of fuselage vibration that could be excited by jet‐efflux pressure fields are first discussed, and consideration is given to (he initial acoustic and structural damping of the modes. A simplified theory is presented for the acoustic damping of flat (or nearly flat) panels set in a much larger body, such as a fuselage. Using the results of Part I, an estimate is then made of the effect of Aquaplas damping compound on the vibration stresses, amplitudes and rivet loads of a structure subjected to random jet‐efflux excitation. It is assumed that the structure and the damping compound together constitute a linear system. In the two particular cases considered, the maximum possible reduction of rivet load is found to be about 40 per cent and 70 per cent respectively, and it is concluded that this is insufficient to outweigh the possible adverse effects of certain factors which cannot be introduced into a simplified investigation.

This paper aims to present an adaptive approach of the generalized finite element method (GFEM) for transient dynamic analysis of bars and trusses. The adaptive GFEM, previously proposed for free vibration analysis, is used with the modal superposition method to obtain precise time-history responses.

The adaptive GFEM is applied to the transient analysis of bars and trusses. To increase the precision of the results and computational efficiency, the modal matrix is responsible for the decoupling of the dynamic equilibrium equations in the modal superposition method, which is used with only the presence of the problem’s most preponderant modes of vibration. These modes of vibration are identified by a proposed coefficient capable of indicating the influence of each mode on the transient response.

The proposed approach leads to more accurate results of displacement, velocity and acceleration when compared to the traditional finite element method.

In this paper, the application of the adaptive GFEM to the transient analysis of bars and trusses is presented for the first time. A methodology of identification of the preponderant modes to be retained in the modal matrix is proposed to improve the quality of the solution. The examples showed that the method has a strong potential to solve dynamic analysis problems, as the approach had already proved to be efficient in the modal analysis of different framed structures. A simple way to perform h-refinement of truss elements to obtain reference solutions for dynamic problems is also proposed.

To provide an approach to active vibration reduction of flexible spacecraft actuated by on‐off thrusters during attitude control for spacecraft designers, which can help them analysis and design the attitude control system.

The new approach includes attitude controller acting on the rigid hub, designed by using pulse‐width pulse‐frequency modulation integrated with component command technique, and the piezoelectric material elements as sensors/actuators bonded on the surface of the beam appendages for active vibration suppression of flexible appendages, designed by optimal positive position feedback (OPPF) control technique. The OPPF compensator is devised by setting up a cost function to be minimized by feedback gains, which are subject to the stability criterion at the same time, and an extension to the conventional positive position feedback control design approach is investigated.

Numerical simulations for the flexible spacecraft show that the precise attitude control and vibration suppression can be accomplished using the derived vibration attenuator and attitude control controller.

Studies on how to control the on‐off actuated system under impulse disturbances are left for future work.

An effective method is proposed for the spacecraft engineers planning to design attitude control system for actively suppressing the vibration and at the same time quickly and precisely responding to the attitude control command.

The advantage in this scheme is that the controllers are designed separately, allowing the two objectives to be satisfied independently of one another. It fulfils a useful source of theoretical analysis for the attitude control system design and offers practical help for the spacecraft designers.