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The purpose is to explore the free vibration behaviour of elastic foundation-supported porous functionally graded nanoplates using the Rayleigh-Ritz approach. The goal of this study is to gain a better knowledge of the dynamic response of nanoscale structures made of functionally graded materials and porous features. The Rayleigh-Ritz approach is used in this study to generate realistic mathematical models that take elastic foundation support into account. This research can contribute to the design and optimization of advanced nanomaterials with potential applications in engineering and technology by providing insights into the influence of material composition, porosity and foundation support on the vibrational properties of nanoplates.

A systematic methodology is proposed to evaluate the free vibration characteristics of elastic foundation-supported porous functionally graded nanoplates using the Rayleigh-Ritz approach. The study began by developing the mathematical model, adding material properties and establishing governing equations using the Rayleigh-Ritz approach. Numerical approaches to solve the problem are used, using finite element methods. The results are compared to current solutions or experimental data to validate the process. The results are also analysed, keeping the influence of factors on vibration characteristics in mind. The findings are summarized and avenues for future research are suggested, ensuring a robust investigation within the constraints.

The Rayleigh-Ritz technique is used to investigate the free vibration properties of elastic foundation-supported porous functionally graded nanoplates. The findings show that differences in material composition, porosity and foundation support have a significant impact on the vibrational behaviour of nanoplates. The Rayleigh-Ritz approach is good at modelling and predicting these properties. Furthermore, the study emphasizes the possibility of customizing nanoplate qualities to optimize certain vibrational responses, providing useful insights for engineering applications. These findings expand understanding of dynamic behaviours in nanoscale structures, making it easier to build innovative materials with specific features for a wide range of industrial applications.

The novel aspect of this research is the incorporation of elastic foundation support, porous structures and functionally graded materials into the setting of nanoplate free vibrations, utilizing the Rayleigh-Ritz technique. Few research have looked into this complex combo. By tackling complicated interactions, the research pushes boundaries, providing a unique insight into the dynamic behaviour of nanoscale objects. This novel approach allows for a better understanding of the interconnected effects of material composition, porosity and foundation support on free vibrations, paving the way for the development of tailored nanomaterials with specific vibrational properties for advanced engineering and technology applications.

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.

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.

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.

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.

This paper presents the generalized theory of the most important energy principles in structural analysis. All derive from two basic complementary theorems denoted as the principles of virtual displacements and virtual forces. Both exact and approximate methods are discussed and particular attention is paid to the derivation of upper and lower limits. The theory is not restricted to linearly elastic bodies but includes ab initio the effect of non‐linear stress‐strain laws and thermal strains. Finally the basic principles are illustrated on a number of simple examples in preparation for the more complex ones to appear in Parts II and III.

A variational procedure is developed, in the form of an extension of the Rayleigh‐Ritz method, leading to a rapid estimation of the flapwise vibration modes and frequencies of a helicopter rotor blade. The initial data required are the blade mass and stiffness distribution and the angular velocity of the rotor blade. The normal modes and frequencies are subsequently used to determine blade shapes in flight. The aerodynamic forces only enter at a late stage of the analysis, and the effect of differing flight conditions is readily assessed. The method makes extensive use of matrix formulation and particularly lends itself to electronic computation techniques. A numerical example is given for the special case of constant spanwise blade mass distribution, although the method may readily be extended to cover this restriction. The bending moment distribution is also worked out, and flexible and rigid blades are compared.

This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method.

The proposed method uses a GCR solver for the outer iteration and the generalized minimal residual (GMRES) with deflated restarting in the inner iteration. Approximate eigenpairs are evaluated at the end of each inner GMRES restart cycle. The approach determines the number of vectors to be deflated from the spectrum based on the number of negative Ritz values, k∗.

The authors show that the approach restores convergence to cases where GMRES with restart failed and compare the approach against standard GMRES with restarts and deflated restarting. Efficiency is demonstrated for a 2D NACA 0012 airfoil and a 3D common research model wing. In addition, numerical experiments confirm the scalability of the solver.

This paper proposes an extension of dynamic deflated restarting into the traditional GCRO method to improve convergence performance with a significant reduction in the memory usage. The novel deflation strategy involves selecting the number of deflated vectors per restart cycle based on the number of negative harmonic Ritz eigenpairs and defaulting to standard restarted GMRES within the inner loop if none, and restricts the deflated vectors to the smallest eigenvalues present in the modified Hessenberg matrix.

The purpose of the study is to obtain and analyze vibro-acoustic characteristics.

A unified analysis model for the rotary composite laminated plate and conical–cylindrical double cavities coupled system is established. The related parameters of the unified model are determined by isoparametric transformation. The modified Fourier series are applied to construct the admissible displacement function and the sound pressure tolerance function of the coupled systems. The energy functional of the structure domain and acoustic field domain is established, respectively, and the structure–acoustic coupling potential energy is introduced to obtain the energy functional. Rayleigh–Ritz method was used to solve the energy functional.

The displacement and sound pressure response of the coupled systems are acquired by introducing the internal point sound source excitation, and the influence of relevant parameters of the coupled systems is researched. Through research, it is found that the impedance wall can reduce the amplitude of the sound pressure response and suppress the resonance of the coupled systems. Besides, the composite laminated plate has a good noise reduction effect.

This study can provide the theoretical guidance for vibration and noise reduction.

The purpose of this paper is to demonstrate the simplicity and versatility of micro‐cantilever based sensors and to present the influence of added mass and stress on the frequency response of the sensor in order to determine the most suitable sensing domain for a given application.

The frequency response of micro‐cantilevers depends not only on the applied mass and surface stress, but also on the mass position. An interpretation of the theoretical frequency results of the 1st and 2nd natural frequencies, for added mass, identifies a nodal point for the 2nd natural frequency which demonstrates mass invariance. Hence, at this nodal point, the frequency response remains constant regardless of mass and may be used for identifying purely induced surface stress influences on the micro‐cantilever's dynamic response. The Rayleigh‐Ritz energy method is used for the theoretical analysis. Theoretical results are compared with experimental results.

A graph of the 2nd natural frequency of micro‐cantilevers with added mass demonstrates the variability of the frequency with mass position on the micro‐cantilever. Of particular interest is the nodal point at which mass independence is revealed. This nodal point may be exploited to investigate purely stress‐related influences on the dynamic characteristics of micro‐cantilever sensors, thereby eliminating such effects as reactant evaporation from the micro‐cantilever sensor surface. In this regard, the nodal point of the 2nd natural frequency response is used to decouple mass‐stress influences.

Owing to the micro‐scale size of the micro‐cantilevers, it may not be possible to apply mass or stress directly at the nodal point and to concentrate its influence there. Hence, a certain amount of influence due to mass‐stress coupling may remain in the frequency responses observed.

Silicon micro‐cantilevers can be easily shaped and sensitized to a variety of influences. These qualities are highly regarded for sensor applications. The work presented herein, contributes to the optimization of micro‐cantilever sensors' dynamic response as a function of mass and surface stress influences. The main criterion for choosing one or the other is based on the time for the surface reaction to take place between the sensing material and the target material. The results presented contribute to the performance optimization of micro‐cantilever based medical and bio‐sensors.

Surface stress effects are generally of much smaller magnitude than mass influences; hence, through an investigation of the stress effects at the nodal point of the 2nd natural frequency it is possible to eliminate the mass influence completely. At this position mass and stress influences are decoupled and the sensor response can be uniquely quantified as a function of the applied stress. This is important for bio‐medical and health monitoring applications in which changes to the applied mass or surface stress on a micro‐cantilever sensor, may be readily observed through changes to the natural frequency response of the micro‐cantilever.

SOME four or five years ago, the Wright Air Development Centre, Wright‐Patterson Air Force Base, Ohio, began to receive an increasing number of complaints of corrosion of wheel bearings on both operational aircraft and aeroplanes which had been inactive for extended periods. Many instances of wheel bearing corrosion were also reported by the U.S. Navy, mostly on stored aircraft. The greases in question were those meeting Specification AN‐G‐15.

The purpose of this paper is to focus on the changing dynamics of the ultrasonic consolidation (UC) process due to changes in substrate geometry. Past research points to a limiting height to width ranging from 0.7 to 1.2 on build features.

Resonances of a build feature due to a change in geometry are examined and then a simple non‐linear dynamic model of the UC process is constructed that examines how the geometry change may influence the overall dynamics of the process. This simple model is used to provide estimates of how substrate geometry affects the differential motion at the bonding interface and the amount of energy emitted by friction change due to build height. The trends of changes in natural frequency, differential motion, and frictional energy are compared to experimental limits on build height.

The paper shows that, at the nominal build, dimensions of the feature the excitation caused by the UC approach two resonances in the feature. In addition trends in regions of changes of differential motion, force of friction, and frictional energy follow the experimental limit on build height.

This paper explores several aspects of the UC process not currently found in the current literature: examining the modal properties of build features, and a lumped parameter dynamic model to account for the changes in of the substrate geometry.

The potential flow field for the transitional region in a constant radius, two‐dimensional duct was studied. The potential velocity and pressure profiles in various duct geometries with different turning angles and duct heights were investigated. A generalized solution of potential flow in a transitional bend was obtained.