Rayleigh-Ritz Research Articles

Free vibration of functionally graded carbon nanotube‐reinforced composite damping structure based on the higher‐order shear deformation theory

AbstractMulti‐phase functionally graded carbon nanotube‐reinforced composite damping structure (FG‐CNTRCDS) excels other composite structures in that it inherits the excellent performance of the functionally graded carbon nanotube‐reinforced composite, and exhibits an improved damping performance due to the added viscoelastic damping films. Despite its superior properties, investigations on dynamic properties of the FG‐CNTRCDS under clamped boundary conditions are still lacking. Aiming at filling this gap and paving the way to accurately design and analyze the FG‐CNTRCDS, a dynamic analytical model, which is based on the higher‐order shear deformation theory to formulate governing equations for predicting free vibration of FG‐CNTRCDS, has been established. The analytical solution that satisfies clamped boundary conditions is obtained using the Rayleigh–Ritz method. The existing analytical results from the literature have been used to verify the correctness of the theoretical model. In addition, effects of various structural parameters (such as volume fraction and core‐to‐skin thickness ratio) on dynamic performance have been investigated. Here, the dynamic performance refers to the first‐order natural frequency and the first‐order loss factor.

Applicability and Limitations of Ru’s Formulation for Vibration Modelling of Double-Walled Carbon Nanotubes

In this paper, a comparison is conducted between two different formulations of the van der Waals interaction coefficient between layers, as applied to the vibrations of double-walled carbon nanotubes (DWCNTs); specifically, the evaluation of the natural frequencies is achieved through Ru’s and He’s formulations. The actual discrete DWCNT is modelled by means of a couple of concentric equivalent continuous thin cylindrical shells, where Donnell shell theory is adopted to obtain strain-displacement relationships. In order to take into account the chirality effect of DWCNT, an anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and the Rayleigh–Ritz method is used to obtain approximate natural frequencies and mode shapes. A parametric analysis considering different values of diameters and numbers of waves along longitudinal and circumferential directions is performed by adopting Ru’s and He’s formulations. From the comparisons, it is evident that Ru’s formulation provides unsatisfactory results for relatively low values of diameters and relatively high numbers of circumferential waves with respect to the more accurate He’s formulation. This behaviour is observed for every number of longitudinal half-waves. Therefore, Ru’s formulation cannot be used for the vibration modelling of DWCNTs in a large range of diameters and wavenumbers.

Isogeometric analysis-based buckling optimization framework for grid-stiffened shells using asymptotic homogenization method and Rayleigh–Ritz method

Isogeometric analysis-based buckling optimization framework for grid-stiffened shells using asymptotic homogenization method and Rayleigh–Ritz method

Nonlinear free vibration of rotating functionally graded graphene platelets reinforced blades with variable cross-sections

Titanium alloys are widely used for blades due to their favorable combination of high strength, low density and fatigue resistance. Dispersing graphene platelets (GPLs) into titanium alloy in a layer-wise manner to develop novel functionally graded (FG) GPL/titanium alloy blades can achieve further enhanced mechanical performance. This paper first proposes a functionally graded graphene/titanium alloy composite trapezoid plate to investigate the nonlinear free vibration of rotating blades with varying cross-sections. It is assumed that graphene platelets are uniformly dispersed in each layer of titanium alloy, and the weight fraction of GPLs increases from the middle layer to the top layers. The effective Young's modulus, effective Poisson's ratio and mass density are calculated by the modified Halpin-Tsai model and the rule of mixture, respectively. The discretized motion equations of the functionally graded graphene platelets reinforced composite (FG-GPLRC) trapezoid plate are derived by the energy method and Lagrange equations. The Rayleigh-Ritz method and direct iterative process are applied to obtain the linear frequencies and nonlinear frequencies, respectively. A detailed numerical study is carried out to investigate the effects of the material parameters and dimensional parameters on the linear and nonlinear free vibrations of the rotating FG-GPLRC trapezoid plate.

Theoretical and experimental investigations of vibration and damping behaviors of carbon fiber-reinforced composite thin shells with partial bolt looseness constraints

In this paper, the vibration and damping behaviors of fiber-reinforced composite thin shells under partial bolt looseness boundary conditions are investigated both theoretically and experimentally. Based on Love's shell theory, the complex modulus approach, and Hooke's law, a theoretical model of fiber-reinforced composite thin shells is constructed. A non-uniform artificial spring technique is proposed to accurately describe the non-uniform looseness in the bolt constraint positions and their interval locations, in which the two kinds of artificial springs, namely the “main spring” and “secondary spring”, are taken into account separately. After the natural characteristics, damping properties, and vibration responses are solved by the Rayleigh-Ritz method, strain energy approach, and Duhamel integral method, detailed measurements are undertaken on a CA 500 carbon/epoxy composite shell specimen to examine vibration and damping characteristics as well as to validate the proposed model. By comparing the theoretical results with the experimental ones, the maximum calculation errors of natural frequencies, resonant responses, and damping ratios are within an acceptable level. In addition, the influences of partial bolt looseness constraints on the vibration characteristics of such shell structures are also investigated here.

Formation Mechanism and Modeling Method of Wrinkling Defects in Variable Angle Tow Steering Fiber Placement

Variable angle tow steering technology is capable of manufacturing complex aviation parts with a trajectory of intricate curvature planned based on stress or profile characteristics, which greatly improves the forming efficiency, design flexibility and mechanical properties of composite structures. In view of the forming defects such as buckling and wrinkles caused by the lateral bending of fiber prepreg tow, a theoretical buckling model based on the Rayleigh Ritz method, the principle of minimum potential energy and the viscoelastic foundation is established, in which the adhesion coefficient is characterized by the degree of intimate contact to introduce process parameters. On the basis of the contact mechanics analysis, the distribution of the compaction pressure and bending stress is studied to improve the theoretical model, and the critical buckling load and the minimum radius of the tow under the normal and tangential contact conditions are determined precisely. Finally, the finite element models of compaction and variable angle steering placement are proposed, and the theoretical model and simulation model are verified by corresponding trials. It is demonstrated that defects can be effectively suppressed through optimizing process parameters.

Study of the Matrix Shrinkage on a Polymer Matrix Composite under a Drop of Temperature

A polymer composite material consists of two different phases with very different mechanical properties. Thus, there is a shrinkage when a decrease in temperature appears. This paper focuses on the matrix shrinkage of a unidirectional polymer matrix composite under a temperature drop. A Rayleigh-Ritz method is used to rapidly determine the matrix displacement (matrix shrinkage) field of virgin samples (initial state, without thermo oxidation). Additionally, numerical simulations are also carried out. A comparison of maximum matrix shrinkages is carried out among the experiment measurement, the Rayleigh-Ritz method, and the numerical simulation method. The numerical results of the matrix displacement are compared to the experiment and the Rayleigh-Ritz method. There is a good correlation between the results obtained by the two methods. Then, an assessment of the reliability of numerical simulations is given. The numerical simulations are then used to analyze the evolution of stress along the different paths on the sample to predict the damage behavior.

Thickness Configuration Effect on Frequency and Critical Speed of Spinning Variable-Thickness Cylindrical Shells

The thickness configuration effect on natural frequency and critical speed of spinning variable-thickness cylindrical shells is investigated in this paper. Based on Love’s thin shell theory, the eigenfrequency equation is derived by employing the Rayleigh–Ritz method in conjunction with Chebyshev polynomials, considering the effects of Coriolis and centrifugal forces due to rotation. Seven types of thickness configurations and various classical boundary conditions are taken into account. Backward and forward wave frequencies, the critical speed, and the mode shapes of the spinning variable-thickness cylindrical shells are solved. Results show that the thickness configuration not only changes circumferential wave number corresponding to the lowest natural frequency, but also affects the critical speed. However, the influence of thickness configuration on mode shapes of the shell is not obvious. The stronger the boundary constraints, the more obvious effect of slope of thickness variation on the critical speed of spinning cylindrical shells.

Free vibrations of circular and sectorial plates by using the shape functions with trigonometric and hyperbolic terms

In this paper, free vibration of different plates in the polar coordinate is studied by using the Rayleigh–Ritz method and calculating strain and kinetic energies. The investigated plates include annular sectorial and circular, sectorial and circular with fixed center, circular sectorial, and solid circular. In this work, trigonometric and hyperbolic terms are used to express the shape function of each plate so that the shape function approximately satisfies boundary conditions and also regularity conditions for the solid plates. The mentioned plates can have any different types of the boundary conditions and the presented method is versatile and has no limitations. For the solid plates, the shape function is calculated in such a way that it can be used to calculate strain energy of the plate. Some types of the plates are examined as the case studies and the results are also calculated by the ABAQUS CAE for validation of the proposed method.

Multi-objective optimization of frequency and damping of vertical stabilizer skin structure placed with variable-angle tows

The vibration characteristics of composite vertical stabilizer skin structures play a critical role in damping effects designed for overcoming the air disturbances experienced by aircraft structural components during flight. The first-order fundamental frequencies and their corresponding damping characteristics of the vertical stabilizer skin structure tow-steered by automatic fiber placement technique were optimized with the parameterized trajectories and plies as design variables. Firstly, the vibration and damping numerical models were derived based on Kirchhoff laminate theory, the Rayleigh-Ritz method, and the Strain Energy Method. Then the optimization model was developed by adopting the self-adaptive Differential Evolution Multi-objective optimization algorithm and incorporating the solution method of Pareto Front. The constraints of this optimization model considered the experimentally obtained minimum turning radius of prepregs tow-steered in automatic fiber placement process obtained from experimental tests. Finally, the comparison of numerical simulation results was conducted for the optimized trajectories and the conventional straight trajectories under various boundary conditions, and the numerical results were partially validated through damping and frequency tests. The results indicate the vibration characteristics of the composite vertical stabilizer skin structure can be enhanced to a large extent by optimizing fiber trajectories, and the enhancement percentage is affected by the boundary conditions of the actual structure.

Dynamic characteristics of composite damping sandwich open conical shell

In this work, the dynamic characteristics of composite damping sandwich open conical shells are investigated. Considering the superposition effect of rotational inertia and shear deformation during structural vibration, the expressions of kinetic energy and strain energy of composite damping sandwich open conical shell (CDSOCS) are derived based on the first-order shear deformation theory. The Rayleigh-Ritz method is used to establish the free vibration equations for thin and medium-thickness shells. According to the simply supported boundary conditions, the Navier double series method is used to solve the vibration equation. The theoretical solution is compared with the previous results and ANSYS finite element simulation results, and the rationality and effectiveness of the theoretical calculation model are verified. The influence of the thickness change of the sandwich layer under different laying angles and the position change of the sandwich layer under different thickness-diameter ratios on the dynamic characteristics of the CDSOCS is discussed, which provides a reference for the design of orthotropic composite damping sandwich complex structures.

Transference of SH waves in a piezoelectric fiber-reinforced composite layered structure employing perfectly matched layer and infinite element techniques coupled with finite elements

Most commercially used piezoelectric materials suffer from certain drawbacks like low piezoelectric output, brittle nature, etc. Thus, they cannot be utilized to their fullest extent in dynamic applications involving actuators, transducers, Surface Acoustic Wave (SAW) devices, and Love wave sensors, among others. The present work focuses on the transference attributes of anti-plane SH wave in a layered structure composed of a piezoelectric fiber-reinforced composite (PFRC) layer lying over a piezoelectric half-space. The PFRC composed of PZT-5A-epoxy is micro-mechanically modeled using Strength of Materials and Rule of Mixtures, and some of its electro-mechanical advantages over monolithic PZT-5A are discussed. The dispersion relation is obtained by analytical means, as well as two numerical techniques, viz., Semi-Analytical Finite Element - Perfectly Matched Layer (SAFE-PML) technique and Semi Analytical Infinite Element (SAIFE) technique with (1/r) type decay. The Rayleigh–Ritz method and Gauss 3-point quadrature formula are employed to derive the dispersion relation numerically. The effects of the existing parameters, viz. fiber volume fraction, PML length, PML function, etc., on the dispersion curves are graphically demonstrated and discussed. The present work is validated by matching the obtained results with the ones found in the extant literature.

Resonant ultrasound spectroscopy and some extensions

Resonant ultrasound spectroscopy (RUS) is a method in which knowledge of the shape, mass, and some natural frequencies of a sample of solid material may be used to determine the elastic properties of the material. Customarily, the natural frequencies are measured by driving the sample with a sine wave in one transducer and monitoring the sample's response with a second transducer; when the frequency of the drive is swept through a natural frequency, a tuning curve is measured, from which the natural frequency and a quality factor may be determined. Usually, a Rayleigh-Ritz method involving rectangular parallelepiped or cylindrical sample shapes and polynomial basis functions is used to analyze the data to determine the elastic properties. This talk will discuss the customary method and some extensions, including: 1) overcoming a limitation that for some problematic materials (e.g., elastomers), the method can measure only one elastic constant accurately; 2) modifying the method to include piezoelectric materials; 3) overcoming the inaccuracy of the method for samples having sharp interfaces between different materials.

Bending and Free Vibration Analysis of Functionally Graded Sandwich Pile on Elastic Foundation Using Rayleigh-Ritz Method

The present study aims to carry out a parametric investigation on a pile made of different materials, and subjected to static and dynamic loading resulting from the weight of the structure and the shear force acting on the head of the pile. It is worth specifying that the pile is anchored in the ground on a Winkler-Pasternak elastic foundation that is made up of five different layers. In this study, the equilibrium equations were treated using the high-order theory with a new shear deformation shape function of a cylindrical pile made of an isotropic material and a sandwich functionally graded material.The results obtained with a numerical study based on the minimization of energies by the Rayleigh-Ritz method was carried out in order to highlight the influence of the geometric ratio, the volume fraction of the functionally graded material, and the type of loading on the vibration frequencies and the admissible stresses in order to determine the most appropriate material for the pile so that extreme stresses can be absorbed. The precision and the applicability of the method with new solution of the transverse shear deformation are demonstrated through this study with modeling of the layered property of the soil, were then compared with those reported in previous work available in the literature. It turned out that these results are in good agreement with each other, for the different materials of pile and for all boundary conditions considered. The results of pile with sandwich functionally graded material, it is revealed that the role of this material has an influence on the type of loading, especially, in the case statique of a pile subjected to a compound bending weight and the shear effect, in case dynamique the pile with vibration flexible or rigid .

Meter Wave Polarization-Sensitive Array Radar for Height Measurement Based on MUSIC Algorithm.

Obtaining good measurement performance with meter wave radar has always been a difficult problem. Especially in low-elevation areas, the multipath effect seriously affects the measurement accuracy of meter wave radar. The generalized multiple signal classification (MUSIC) algorithm is a well-known measurement method that dose not require decorrelation processing. The polarization-sensitive array (PSA) has the advantage of polarization diversity, and the polarization smoothing generalized MUSIC algorithm demonstrates good angle estimation performance in low-elevation areas when based on a PSA. Nevertheless, its computational complexity is still high, and the estimation accuracy and discrimination success probability need to be further improved. In addition, it cannot estimate the polarization parameters. To solve these problems, a polarization synthesis steering vector MUSIC algorithm is proposed in this paper. First, the MUSIC algorithm is used to obtain the spatial spectrum of the meter wave PSA. Second, the received data are properly deformed and classified. The Rayleigh–Ritz method is used to decompose the angle to realize the decoupling of polarization and the direction of the arrival angle. Third, the geometric relationship and prior information of the direct wave and the reflected wave are used to continue dimension reduction processing to reduce the computational complexity of the algorithm. Finally, the geometric relationship is used to obtain the target height measurement results. Extensive simulation results illustrate the accuracy and superiority of the proposed algorithm.

Analysis of vibration characteristics of stators of electrical machines under actual boundary

This paper presents a general method for the studies of the vibration characteristics of an electrical machine stator with encased construction under practical boundary conditions. In this paper, artificial springs are introduced at both ends of the stator, and arbitrary boundary conditions can be simulated by adjusting the spring stiffness, including the support conditions in practical applications of the stator. In order to cooperate with the solution of the vibration characteristics of the stator with arbitrary edges, the Gram-Schmidt method is employed to construct characteristic orthogonal polynomial series to simulate the mode shape functions under arbitrary boundary conditions. The analysis is based on the Novozhilov shell theory, which is more applicable to the construction and actual operating environment of stators of electric vehicle drive motors, and takes into account moment of inertia neglected in the past. Further, the effects of teeth, windings, frame and cooling ribs are incorporated in the analysis, and the teeth with complicated shapes are considered as longitudinal ribs of unlimited cross-section, with bending and torsional coupling motion and warping deformation considered. The Rayleigh-Ritz method is used to build a unified dynamic analysis model of the stator of an electric machine, which can provide information on the radial, circumferential and axial vibrations of the stator with higher time efficiency, and a series of comparison studies and experiments are performed to demonstrate the general validity and accuracy of the model. On this basis, the possibility of construction optimization of the stator is discussed to improve the vibration and noise performance while reducing the cost.

Vibration Analysis of Rotating Combined Thin-Walled Shells With Multiple Conical Segments

Abstract In this paper, vibration analysis of rotating combined thin-walled shells with multiple conical segments has been carried out. Considering the centrifugal force, Coriolis force, and initial hoop tension due to rotation, the elastic strain energy and kinetic energy of a single rotating conical shell are expressed based on Love’s first approximation theory. The artificial springs are introduced to simulate the connections of adjacent conical shells and the boundaries of the rotating combined thin-walled shells. Taking characteristic orthogonal polynomial series as the admissible functions, the Rayleigh–Ritz method is employed to derive the frequency equations of the combined shell and corresponding vibration characteristics are then obtained. Given that the cylindrical shell and annular plates can be regarded as conical shells with semi-vertex angles of 0 deg and 90 deg, respectively, the solution given is also available for the vibration analysis of rotating combined thin-walled shells comprised of any segments of cylindrical, conical shell, and annular plate. As examples of rotating combined thin-walled shells with two and five segments, vibrations of rotating conical–conical joined shell and cylindrical–conical–cylindrical–conical–cylindrical joined shell are investigated in the paper. Traveling wave frequencies and corresponding mode shapes are shown, and the effects of rotating speed, circumferential wave number, spring stiffness, and semi-vertex angles on the vibration behavior are given in detail.

Development of an analytical model for the flexural vibration of fish bone active camber structures with truncated, variable thickness partitions

The need for smooth, gapless mechanisms that offer superior aerodynamic performance has been the prime mover for a wide range of studies on the morphing concepts from the early years of the twenty-first century. In light of the importance of novel analytic means for inspecting such structures, this article describes a comprehensive dynamic model for the flexural vibration of a special type of morphing structure known as fish bone active camber (FishBAC). In view of the shortcomings of previous models, the developed procedure considers the variable thickness of the central spine of FishBAC as well as the possibility of arbitrary truncations in its numerous partitions to offer more accurate results. A general idea is first proposed based on the assumptions of classical plate theory for such partitioned structures. The equations are derived using the energy method and the assumption of artificial springs to satisfy the continuity of shear force and bending moment along the connecting lines between partitions. A robust Rayleigh-Ritz method along with a set of fast-converging admissible functions is presented to solve the governing equations of motion. Combined with the power of Mathematica and validated by other analytic models and finite element codes, this technique can be used to deal with a plethora of similar systems while paving the path for the dynamic analysis of FishBAC built around various airfoils.

Vibration transmission control of a flexible shaft-bearing system using active/passive support

An active/passive support is proposed to suppress the transmission of vibration from a flexible shaft-bearing system to its foundation. The active/passive support consists of passive isolators and electromagnetic suspension, in which the former bears the static load while the latter provides suspension force for displacement adjustment and vibration suppression. The dynamic model of the shaft-bearing-foundation system is built on the Rayleigh-Ritz method. Based on this model, the adjustment of stiffness, the transmissibility of vibration, the vibration characteristics of the coupled system and the effectiveness of active control are analyzed. An adaptive control method with subband filtering and disturbance reconstruction is adopted and the performance is evaluated. The effectiveness of the active/passive support is also verified by experiment. Numerical and experimental results demonstrate that the transmission of vibration via the active/passive support can be greatly suppressed and the vibration of the foundation can be reduced almost to the same level.

Free vibrational analysis of variable thickness plate made of functionally graded porous materials using internal supports in contact with bounded fluid

This paper investigates the free vibrations of functionally graded porous (FGP) plates with variable thickness and internal supports in contact with a fluid. The FGP plate is the vertical wall of a rigid tank containing fluid. Four different types of thickness changes are considered for the plate, and the modeling of the interaction between the fluid and considered plate is presented for the first time. In addition to having different boundary conditions, the plate with variable thickness has different internal supports, which is less considered in the literature. Three different porosity distributions and four types of variable thickness are considered. The displacement field is estimated using the first-order shear deformation theory (FSDT), and kinetic and strain energies are obtained according to that. The fluid is assumed to be incompressible, inviscid, and irrotational. Natural frequencies of the system are obtained by employing the Rayleigh-Ritz method. Additionally, the natural frequency results of the plate in contact with fluid, the FGP plate, the plate with variable thickness, and the plate with internal supports are separately compared with those reported in the literature. Ultimately, the impact of fluid's depth, porosity coefficient, taper ratio, and the location of internal supports on natural frequencies are studied.