Rayleigh-Ritz Approach Research Articles

Structural and aeroelastic analyses of a wing with tip rotor

Based on parametric studies, this paper studies the structural and aerodynamic properties of a typical wing-rotor system by considering additional coupling effects from the tip rotor. The governing equations are established by virtue of Yamanaka’s theory and Hamilton’s principle in an Euler–Bernoulli precision. In the derivation process, complex mode theory and the Rayleigh–Ritz approach are employed to obtain the analytical solution and the numerical solution, respectively, and a set of trial functions are generated for the sake of reliability. Two aeroelastic models are generated via steady and unsteady aerodynamic models to investigate the divergence and flutter performance, respectively, where the computational p-k method is adopted because of the gyros matrix. In case studies, the effectiveness of the numerical solution is validated through comparative analysis; thereafter, the aeroelastic solutions are further verified. The results demonstrate that the gyro effect and the thrust effect have significant influences on the fundamental structural and aeroelastic performances, and parametric designs have great potential to postpone aeroelastic instabilities.

Nonlinear vibrations of thin plates with variable thickness: Application to sound synthesis of cymbals.

Geometrically nonlinear vibrations of thin plates and shells with variable thickness are investigated numerically with the purpose of synthesizing the sound of cymbals. In cymbal making, taper refers to the gradual change in thickness from the centre to the rim and is known to be a key feature that determines the tone of the instrument. It is generally used in conjunction with shape variations in order to enable the cymbal to play a bell-like sound when hit near its centre, or a crash sound when struck close to the edge. The von Kármán equations for thin plates with thickness and shape variations are derived, and a numerical method combining a Rayleigh-Ritz approach together with a Störmer-Verlet scheme for advancing the problem in time is detailed. One main advantage of the method is its ability to implement easily any frequency-dependent loss mechanism which is a key property for sound synthesis. Also, the accuracy of the computation of the nonlinear restoring force is especially preserved. The method is employed to synthesize the sounds of cymbal-like instruments. The impact of taper is addressed and the relative effects of both thickness and shape variations, are contrasted.

Stochastic Response Analysis of a Built-Up Vibro-Acoustic System with Parameter Uncertainties

In this paper, the quantification of uncertainty effects on stochastic responses of interior vibro-acoustic interaction systems with moderate geometry complexities and uncertain design parameters is investigated. A variational-based stochastic model is developed to predict the vibro-acoustic responses submitted to probabilistic parameters, and it is illustrated by application to a built-up system consisting of an irregular acoustic cavity backed up a plate assembly. The model is derived from the combination of the multi-domain Rayleigh–Ritz approach, used to solve the deterministic structural–acoustic equations, together with the generalized polynomial chaos expansion (gPCE) to represent propagation of uncertainty and estimate the statistical characteristics of the responses. Benchmark comparisons are made with the Monte Carlo simulations (MCS) to demonstrate the tremendous computational advantage of the present methodology. Uncertainty analysis is performed to ascertain the influence of random parameters on responses. The results reveal that system uncertainty is significant enough to affect the vibro-acoustic behaviors and hence the consideration of input uncertainties is necessary in analyses and designs to ensure the sustainable system performance.

An innovative determination approach to tooth compliance for spiral bevel and hypoid gears by using double-curved shell model and Rayleigh–Ritz approach

In high-performance sophisticated flank design for the spiral bevel and hypoid gears, tooth compliance is always an important input for identifying the tooth mechanical properties. Distinguished with the conventional modeling, a new accurate finite element structure model is established to determine tooth compliance by using elasticity-based deformation solution. In full consideration of the flank flexural behavior characteristic, a double-curved shell model with varying thickness is established to get a more accurate representation of geometric shape for face-milling or face-hobbing gear than a beam or plate model. In determination of tooth compliance, the high order shear theory of Bhimaraddi shell is used to set the displacement assumption, and the Rayleigh–Ritz approach having algebraic polynomial trail functions is performed to obtain an expression for the transverse deflection and shear rotation by considering the geometric boundary conditions. A given numerical result for the face-milling spiral bevel and hypoid gear is provided to verify the proposed methodology by comparing with the loaded tooth contact analysis (LTCA) using well-known finite element method (FEM).

Vibration-based identification of mechanical properties of orthotropic arbitrarily shaped plates: Numerical and experimental assessment

Abstract An innovative procedure is introduced for the identification of the mechanical parameters of orthotropic plates of arbitrary shape, under various boundary conditions, based on vibration data. The method employs a combination of a convenient Rayleigh-Ritz approach and Particle-Swarm Optimization to estimate elastic constants of the orthotropic material in a straightforward manner, without requiring computationally demanding iterative Finite Element analyses. Specifically, the pb-2 Rayleigh-Ritz procedure is extended and applied to deal with orthotropic plates, simplifying the approach to more easily treat generic plate shapes, taking advantage of the Green's theorem. The method is then appropriately combined with the Particle-Swarm Optimization procedure to expeditiously identify material parameters based on available vibration data. Several numerical applications are presented to show the reliability of the approach, and comparisons with pertinent results available in the literature demonstrate the efficiency and accuracy of the proposed procedure. The study is then supplemented by experimental tests developed in the Laboratory of Experimental Dynamics at the University of Palermo, Italy. In this context, because of the obvious relevance for modern additive manufacturing processes, vibration tests are performed on several 3D printed stiffened plates. Numerical vis-a-vis experimental data are examined, showing that the proposed procedure accurately capture equivalent orthotropic parameters of the stiffened plates.

A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels

A Rayleigh-Ritz solution approach for generally restrained multilayered variable angle tow stiffened plates in postbuckling regime is presented. The plate model is based on the first order shear deformation theory and accounts for geometrical nonlinearity through the von Kármán’s assumptions. Stiffened plates are modelled as assembly of plate-like elements and penalty techniques are used to join the elements in the assembled structure and to apply the kinematical boundary conditions. General symmetric and unsymmetric stacking sequences are considered and Legendre orthogonal polynomials are employed to build the trial functions. A computer code was developed to implement the proposed approach and to establish its applicability and its features for investigating variable angle tow structures. The proposed solution is validated by comparison with literature and finite elements results. Original results are presented for postbuckling of variable angle tow stiffened plates showing the potentialities of the method.

Wavelet Decompositions for High Frequency Vibrational Analyses of Plates

A wavelet-decomposed Rayleigh–Ritz model for 2D plate vibration analyses is proposed in this work. For an elastically-supported rectangular plate under Love–Kirchhoff theory, 2D Daubechies wavelet scale functions are used as the admissible functions for analyzing the flexural displacement in an extremely large frequency range. For constructing the mass and stiffness matrices of the system, the 2D wavelet connection coefficients are deduced. It is shown that by inheriting the versatility of the Rayleigh–Ritz approach and the superior fitting ability of the wavelets, the proposed method allows reaching very high frequencies. Validations are carried out in terms of both eigen-frequencies and the forced vibration responses for cases which allow analytical solutions. Effects of the wavelet parameters on the calculation accuracy and convergence are also studied.

Effects of boundary conditions on bistable behaviour in axisymmetrical shallow shells.

Multistable shells are thin-walled structures that have more than one stable state of self-stress. We consider isotropic axisymmetrical shallow shells of arbitrary polynomial shapes using a Föppl-von Kármán analytical model. By employing a Rayleigh-Ritz approach, we identify stable shapes from local minima in the strain energy formulation, and we formally characterize the level of influence of the boundary conditions on the critical geometry for achieving bistable inversion-an effect not directly answered in the literature. Systematic insight is afforded by connecting the boundary to ground through sets of extensional and rotational linear springs. For typical cap-like shells, it is shown that bistability is generally enhanced when the extensional spring stiffness increases and when the rotational spring stiffness decreases, i.e. when boundary movements in-plane are resisted but when their rotations are not; however, for certain other shapes and large in-plane stiffness values, bistability can be enhanced by resisting but not entirely preventing edge rotations. Our predictions are furnished as detailed regime maps of the critical geometry, which are accurately correlated against finite-element analysis. Furthermore, the suitabilities of single degree-of-freedom models, for which solutions are achieved in closed form, are evaluated and compared to our more accurate predictions.

Sound Transmission and Radiation from a Plate–Cavity System in Supersonic Flow

Noise transmission into and radiation from a rectangular cavity through a flexible structure in a noisy/thermal environment at supersonic flow is studied. Most works in acoustic fluid/solid interaction have focused on linear models in the preflutter domain (stable region). In this paper, the pre-/postflutter regions are considered in a nonlinear study. The governing equations of a flexible panel are constructed using the von Kármán theory, the first-order piston theory, and a modal cavity formulation; also, a stationary-homogenous turbulent boundary layer is modeled based on experimental data through stochastic modeling. The thermal environment and external incident random noise are also considered, and the far-field sound is predicted with a boundary integral method. The Rayleigh–Ritz approach is adopted to discretize the partial differential equations of the acoustic fluid/solid interaction system, and the resulting ordinary differential equations are solved numerically by the fourth-order Runge–Kutta method. In the postflutter domain, the acoustic pressure in the cavity is nearly spatially uniform and has the highest sound pressure levels in the lower-frequency regions. In-plane thermal loading may increase the dominant frequency of the sound field, but it may decrease or increase the sound pressure level. Higher cavity modes are excited by the turbulent boundary layer and the incident random load in the preflutter domain.

Analysis of nonlinear aeroelastic characteristics of a trapezoidal wing in hypersonic flow

Nonlinear aeroelastic behavior of a trapezoidal wing in hypersonic flow is investigated. The aeroelastic governing equations are built by von Karman large deformation theory and the third-order piston theory. The Rayleigh–Ritz approach combined with the affine transformation is formulated and employed to transform the equations of a trapezoidal wing structure, modeled as a cantilevered wing-like plate, into modal coordinates. And then the modal equations are solved by numerical integrations. Several typical cases are studied to validate the capability of the proposed method for linear and nonlinear aeroelastic analysis of trapezoidal cantilever plate in hypersonic flow. The effects of Rayleigh–Ritz mode truncation for various wing-plate geometrical characteristics, i.e., sweep angle of leading edge, taper ratio and span, are examined to determine the appropriate mode number for accurate modeling and fast calculation. Meanwhile, the effects of various geometries of trapezoidal cantilever plates on the flutter stability are investigated. The nonlinear dynamic behaviors of the model with three typical geometries, namely, the rectangular, parallelogram and trapezoidal wing-like plate, are simulated numerically. Furthermore, complex dynamic behaviors are observed and identified via the phase plot, the Poincare map and the largest Lyapunov exponent. The results demonstrate that geometrical parameters of trapezoidal wing have significant effects on the nonlinear aeroelastic behaviors of wing structure. In particular, the evolution processes of chaos exhibit remarkable difference for these three wing configurations.

On the equivalent mass-spring parameters and assumed mode of a cantilevered beam with a tip mass

In the Rayleigh-Ritz approach to the mathematical model of a cantilevered beam with a tip mass, the proper selection of basis functions is critical in representing the original system by an equivalent mass-spring system. Although the fundamental bending mode shape of a beam varies for a different tip mass magnitude, the numerical values of 33/140 and 3 have been conventionally employed as those of the normalized dimensionless equivalent mass and spring constants, respectively, which correspondingly yield errors in its calculated natural frequencies. This work firstly proposes a method to evaluate more accurate values of the equivalent mass and spring for a wide range of the tip mass-to-beam mass ratio by direct use of a fundamental mode, and then proposes a new basis function as a linear combination of two polynomials, which represent static deflection shapes of a beam under a tip force and a uniformly distributed force, respectively, yielding natural frequencies fairly close to those by the continuous beam equation.

Energy Harvesting with a Piezoelectric Thunder

In this work the energy harvesting performance of a piezoelectric curved energy generator (THUNDER) is studied via experimental and analytical methods. The analytical model of the THUNDER is created based on the linear mechanical electrical constitutive law of the piezoelectric material, the linear elastic constitutive law of the substrate, and the Euler-Bernoulli beam theory. With these linear modal functions, the Rayleigh-Ritz approach was used to then obtain the reduced mechanical electrical coupled modulation equations. With above analytical model, two types of energy harvest circuit are proposed and compared: 1) directly charging mode at low level excitation, and 2) memory stored optimal duty cycle step-down converter mode at high level excitation. The value of the optimal duty cycle is determined based on the characteristics of the vibration signals of the ambient vibration source. To reduce the energy consumption of the microcontroller, the optimum duty cycle values are stored in the microprocessor instead of doing onsite computing during energy harvesting process. For the purpose of designing a low power cost mechanical switch to control the operation of both modes, the threshold voltage between these two operation modes is converted into threshold displacement.

Influence of imperfect end boundary condition on the nonlocal dynamics of CNTs

Imperfections that unavoidably occur during the fabrication process of carbon nanotubes (CNTs) have a significant influence on the vibration behavior of CNTs. Among these imperfections, the boundary condition defect is studied in this investigation based on the nonlocal elasticity theory. To this end, a mathematical model of the non-ideal end condition in a cantilever CNT is developed by a strongly non-linear spring to study its effect on the vibration behavior. The weak form equation of motion is derived via Hamilton’s principle and solved based on Rayleigh–Ritz approach. Once the frequency response function (FRF) of the CNT is simulated, it is found that the defect parameter injects noise to the FRF in the range of lower frequencies and as a result the small scale effect on the FRF remains undisturbed in high frequency ranges. Besides, in this work a process is introduced to estimate the nonlocal and defect parameters for establishing the mathematical model of the CNT based on FRF, which can be competitive because of its lower instrumentation and data analysis costs. The estimation process relies on the resonance frequencies and the magnitude of noise in the frequency response function of the CNT. The results show that the constructed dynamic response of the system based on estimated parameters is in good agreement with the original response of the CNT.

A weighted extended B-spline solver for bending and buckling of stiffened plates

The weighted extended B-spline method (Höllig, 2003) is applied to bending and buckling problems of plates with different shapes and stiffener arrangements. The discrete equations are obtained from the energy contributions of the different components constituting the system by means of the Rayleigh-Ritz approach. The pre-buckling or plane stress is computed by means of Airy's stress function. A boundary data extension algorithm for the weighted extended B-spline method is derived in order to solve for inhomogeneous Dirichlet boundary conditions. A series of benchmark tests is performed touching various aspects influencing the accuracy of the method.

Energy harvesting efficiency optimization via varying the radius of curvature of a piezoelectric THUNDER

In this work the energy harvesting performance of a piezoelectric curved energy generator (THin layer UNimorph DrivER (THUNDER)) is studied via experimental and analytical methods. The analytical model of the THUNDER is created based on the linear mechanical electrical constitutive law of the piezoelectric material, the linear elastic constitutive law of the substrate, and the Euler–Bernoulli beam theory. With these linear modal functions, the Rayleigh-Ritz approach was used to obtain the reduced mechanical–electrical coupled modulation equations. The analytical model is verified by the experimental results. Both the experimental and analytical results of the THUNDER’s AC power output, DC power output with Rectifier Bridge and a capacitor, as well as the power output with a microcontroller energy harvesting circuit are reported. Based on the theoretical model, the analytical solution of the DC power is derived in terms of the vibration amplitude, frequency, and the electrical load. To harvest energy from low-frequency vibration source by a piezoelectric generator requires the piezoelectric device possessing low resonance frequency and good flexibility. The THUNDER developed by Langley Research Center exhibits high power when it is used as an energy generator and large displacement when it is used as an actuator. Compared to the less flexible PZT, although THUNDER is more difficult to model, THUNDER has better vibration absorption capacity and higher energy recovery efficiency. The effect of the THUNDER’s radius of curvature on energy harvesting efficiency is mainly investigated. We set the THUNDER’s radius of curvature as a dynamic tuning parameter which can tune the piezoelectric generators’ frequency with the source excitation frequency.

Application of the Craig-Bampton model order reduction method to a composite structure: MACco, COMAC, COMAC-S and eCOMAC

Abstract The Craig-Bampton model order reduction (CBMOR) method based on the Rayleigh-Ritz approach was applied in a previous work to simulate dynamic behavior of a composite structure (CFRP) using the modal assurance criteria (MAC) and cross orthogonality (XOR) to validate the correlation. Different coordinate modal assurance criteria are applied to complement and verify the eigenfrequencies and eigenvectors obtained of the full and reduced models using substructures (super-elements). An improvement is observed per paired mode-sensor with the MAC per coordinates criterion (MACco) in a CFRP once the stiffness parameters are updated in the full model applying a mix-numerical experimental technique (MNET) using a design of experiments (DOE). The coordinate modal assurance criteria (COMAC) and the scaleCOMAC (COMACS) results of the full models display the best results respect to the reduced model. Furthermore, slight improvement of the enhanced COMAC (eCOMAC) results are observed in the reduced model despite having lower MAC performance. This approach complements the results of the previous work using several COMAC techniques, and demostrates the feasibility to achieve low COMACs results in the reduced finite element model once the stiffness parameters of the full element model are updated. The example was prepared and solved with MSC/NASTRAN SOL103 and SDTools-MATLAB for comparative purposes.

On the efficacy of the wavelet decomposition for high frequency vibration analyses

This paper reports the extraordinary ability of the wavelet decomposition for vibration analyses under the framework of Rayleigh–Ritz method. Using a beam as an example, Daubechies wavelet scale functions are used as admissible functions for decomposing the flexural displacement of the structure, along with the artificial springs at the boundary, to predict vibration of an Euler–Bernoulli beam in an extremely large frequency range. It is shown that the use of wavelet basis allows reaching very high frequencies, typically covering more than 1000 modes using conventional computational facility within the available numerical dynamics of the computers with no particular care needed for round-off errors. As a side benefit, the use of spring boundary also allows handling any elastic boundary conditions through a dynamic contribution in the Hamiltonian of the beam. The wavelet decomposed approach combines the flexibility of the global methods and the accuracy of local methods by inheriting the versatility of the Rayleigh–Ritz approach and the superior fitting ability of the wavelets. Numerical results on both free and forced vibrations are given, in excellent agreement with predictions of classical methods.

Influence of ring support on free vibration of sandwich functionally graded cylindrical shells with middle layer of isotropic material

In this vibration analysis, influence of ring supports on functionally gradient threelayeredsandwich cylindrical shells is presented with middle layer fabricated ofisotropic material. The ring support is introduced beside radial direction of the shell.Love’s first order thin shell theory is used for strain- and curvature-displacementsrelationship. Rayleigh–Ritz approach is employed to form the shell frequency equation.Axial modal dependence is approximated by characteristics beam functions. Study iscarried out for placing ring support in different position of the shell, for differentconfiguration of the functionally graded material’s layers to investigate the naturalfrequencies of the cylindrical shells under different boundary conditions for a numberof physical parameters. Results obtained are validated with the previous publishedworks in the open literature.

Theoretical Study on Vibration of Non-homogeneous Tapered Visco-Elastic Rectangular Plate

A temperature-thickness coupling problem of non-homogeneous rectangular plate is analyzed. Here authors considered bi-parabolic variation in temperature i.e. parabolic in x-direction and parabolic in y-direction while variation in plate’s thickness is assumed bi-linear i.e. linear in x-direction and linear in y-direction. To characterize the non-homogeneity present in plate’s material, authors considered that poisson ratio and density of the plate’s material vary exponentially and linearly in one direction respectively. Along with, material of the plate is considered isotropic and visco-elastic. Rayleigh Ritz approach has been applied to find the time period and deflection for first two modes of vibration for diverse values of non-homogeneity constant, taper constant, density parameter, aspect ratio and thermal gradient. Results for both modes of vibration are shown in tabular form.

Post-Buckling Analysis of Damaged Multilayered Composite Stiffened Plates by Rayleigh-Ritz Method

A Rayleigh-Ritz approach for the analysis of buckling and post-buckling behavior of cracked composite stiffened plates is presented. The structure is modeled as the assembly of plate elements modeled by the first order shear deformation theory and taking geometric nonlinearities into account through the von Karman’s theory assumptions. Continuity along the plate elements connected edges and the enforcement of rigid and elastic restraints of the plate boundaries are obtained by using penalty techniques, which also allow to straightforwardly implement efficient crack modeling strategies. General symmetric and unsymmetric stacking sequences are considered and numerical procedures have been developed and used to validate the present solution by comparison with FEA results. Original results are presented for post-buckling solution of multilayered stiffened plates with through-the-thickness cracks, showing the effects of large displacements on the cracked plate post-buckling behavior.