Vibration of a rectangular plate with a central power-law profiled groove by the Rayleigh–Ritz method

The prediction of natural frequencies of rectangular plates where a profiled indentation is present is made using a Rayleigh–Ritz variational energy method. Panels with holes are often found for access cables and access gaps, and it is shown that the application of damping to the profile leads to a more efficient method of reducing vibration than covering a whole rectangular plate, which is advantageous where weight saving is necessary.     

An efficient general approach to modal analysis of frame resonators with applications to support loss in microelectromechanical systems

The design of high-Q resonators such as Xylophone Bar Resonators (XBRs) capable of being fabricated using Micro-Electro-Mechanical Systems (MEMS) processes is of considerable interest in light of the widespread and rapidly growing use of systems dependent on their availability and performance. This paper is concerned with vibration analysis and Q optimisation of an XBR, with the method extending directly to other planar frames and straightforwardly to more complex structures. The Rayleigh–Ritz method is discussed in some detail, first treating the discrete case, followed by developing and applying a kinematical procedure to an L-frame structure. Attention is given to geometric interpretation of the Rayleigh–Ritz procedure and to developing an intuitive understanding the method before turning to the XBR case. Having developed an approximation for system dynamics, the results are used in conjunction with an analytical model of elastic wave propagation in the substrate to obtain an estimate for the support Q factor. Natural frequencies, mode shapes, and support Q values are presented and compared to Finite Element models of the same problem, with excellent agreement observed at substantially lower computational cost. For the first time in the literature, the geometric impedance tuning principle underlying the XBR design is validated and quantified, including sensitivity to manufacturing error.     

Optimal locations and orientations of piezoelectric transducers on cylindrical shell based on gramians of contributed and undesired Rayleigh–Ritz modes using genetic algorithm

In this study, the active vibration control and configurational optimization of a cylindrical shell are analyzed by using piezoelectric transducers. The piezoelectric patches are attached to the surface of the cylindrical shell. The Rayleigh–Ritz method is used for deriving dynamic modeling of cylindrical shell and piezoelectric sensors and actuators based on the Donnel–Mushtari shell theory. The major goal of this study is to find the optimal locations and orientations of piezoelectric sensors and actuators on the cylindrical shell. The optimization procedure is designed based on desired controllability and observability of each contributed and undesired mode. Further, in order to limit spillover effects, the residual modes are taken into consideration. The optimization variables are the positions and orientations of piezoelectric patches. Genetic algorithm is utilized to evaluate the optimal configurations. In this article, for improving the maximum power and capacity of actuators for amplitude depreciation of negative velocity feedback strategy, we have proposed a new control strategy, called “Saturated Negative Velocity Feedback Rule (SNVF)”. The numerical results show that the optimization procedure is effective for vibration reduction, and specifically, by locating actuators and sensors in their optimal locations and orientations, the vibrations of cylindrical shell are suppressed more quickly.     

Structural–acoustic model of a rectangular plate–cavity system with an attached distributed mass and internal sound source: Theory and experiment

In this paper three approaches are combined to develop a structural–acoustic model of a rectangular plate–cavity system with an attached distributed mass and internal sound source. The first approach results from a recently presented analysis based on the Rayleigh–Ritz method and is used to circumvent the difficulties in obtaining the natural frequencies and mode shapes of a plate with an attached, distributed mass. Furthermore, different plate boundary conditions can be accommodated. The resulting mode shapes are defined as continuous functions; this is advantageous as they can be directly used in the second approach, i.e., the classic modal-interaction approach in order to obtain the coupled equations of the system. Finally, in the third approach a group of point sources emitting a pressure pulse in the time domain is used to model an internal sound source. For the validation of the developed model an experiment was conducted in two configurations using a simply supported aluminium plate and a clamped plate coupled with a plexiglas box containing a loudspeaker. Good agreement was found between the analytical and experimental data.     

Extension of Rayleigh–Ritz method for eigenvalue problems with discontinuous boundary conditions applied to vibration of rectangular plates

The Rayleigh–Ritz (R–R) method is extended to eigenvalue problems of rectangular plates with discontinuous boundary conditions (DBC). Coordinate functions are defined as sums of products of orthogonal polynomials and consist of two parts, each satisfying the BC in its respective region. These parts are matched by minimizing the mean square error of functions and their x-derivatives at the interface between regions. Matching defines a positive definite 2N²×2N² matrix Q whose eigenvectors form the orthogonal coordinate functions. The corresponding eigenvalues measure the matching error of the two parts at the interface. When applying the R–R method, the total error is the sum of the matching error and that arising from the finite number of coordinate functions. Although most of the coordinate functions correspond to the zero eigenvalue, these do not suffice and additional functions corresponding to small but finite eigenvalues must be included. In three examples with discontinuous BC of the clamped, simply supported and free kind, the calculated frequencies match closely those from a finely discretized finite element method.     

A variable kinematic Ritz formulation for vibration study of quadrilateral plates with arbitrary thickness

A new variable kinematic Ritz method applied to free vibration analysis of arbitrary quadrilateral thin and thick isotropic plates is presented. Carrera's unified formulation and the versatile pb-2 Ritz method are properly combined to build a powerful yet simple modeling and solution framework. The proposed technique allows to generate arbitrarily accurate Ritz solutions from a large variety of refined two-dimensional plate theories by expanding so-called Ritz fundamental nuclei of the plate mass and stiffness matrices. Theoretical development of the present methodology is described in detail. Convergence and accuracy of the method are examined through several examples on thin, moderately thick, and very thick plates of rectangular, skew, trapezoidal and general quadrilateral shapes, with an arbitrary combination of clamped, free and simply supported edges. Present results are compared with existing three-dimensional solutions from open literature. Maximum and average differences of various higher-order plate theories and three-dimensional results are also presented with the aim of providing useful guidelines on the choice of appropriate plate theory to get a desired accuracy on frequency parameters.     

Low-frequency linear vibrations of single-walled carbon nanotubes: Analytical and numerical models

Low-frequency vibrations of single-walled carbon nanotubes with various boundary conditions are considered in the framework of the Sanders–Koiter thin shell theory. Two methods of analysis are proposed. The first approach is based on the Rayleigh–Ritz method, a double series expansion in terms of Chebyshev polynomials and harmonic functions is considered for the displacement fields; free and clamped edges are analysed. This approach is partially numerical. The second approach is based on the same thin shell theory, but the goal is to obtain an analytical solution useful for future developments in nonlinear fields; the Sanders–Koiter equations are strongly simplified neglecting in-plane circumferential normal strains and tangential shear strains. The model is fully validated by means of comparisons with experiments, molecular dynamics data and finite element analyses obtained from the literature. Several types of nanotubes are considered in detail by varying aspect ratio, chirality and boundary conditions. The analyses are carried out for a wide range of frequency spectrum. The strength and weakness of the proposed approaches are shown; in particular, the model shows great accuracy even though it requires minimal computational effort.     

An investigation into the maximum entropy production principle in chaotic Rayleigh–Bénard convection

The hypothesis is made that the temperature and velocity fields in Rayleigh–Bénard convection can be expressed as a superposition of the active modes with time-dependent amplitudes, even in the chaotic regime. The maximum entropy production principle is interpreted as a variational principle in which the amplitudes of the modes are the variational degrees of freedom. For a given Rayleigh number, the maximum heat flow for any set of amplitudes is sought, subject only to the constraints that the energy equation be obeyed and the fluid be incompressible. The additional hypothesis is made that all temporal correlations between modes are zero, so that only the mean-squared amplitudes are optimising variables. The resulting maximal Nusselt number is close to experimental determinations. The Nusselt number would appear to be simply related to the number of active modes, in particular the number of distinct vertical modes. It is significant that reasonable results are obtained for the optimised Nusselt number in that the dynamics (the Navier–Stokes equation) is not used as a constraint. This suggests grounds for optimism that the maximum entropy production principle, interpreted in this variational manner, can provide a reasonable guide to the dynamic steady states of non-equilibrium systems whose detailed dynamics are unknown.     

Study of a vibrating plate: comparison between experimental (ESPI) and analytical results

Real-time electronic speckle pattern interferometry (ESPI) was used for tuning and visualization of natural frequencies of a trapezoidal plate. The plate was excited to resonant vibration by a sinusoidal acoustical source, which provided a continuous range of audio frequencies. Fringe patterns produced during the time-average recording of the vibrating plate—corresponding to several resonant frequencies—were registered. From these interferograms, calculations of vibrational amplitudes by means of zero-order Bessel functions were performed in some particular cases. The system was also studied analytically. The analytical approach developed is based on the Rayleigh–Ritz method and on the use of non-orthogonal right triangular co-ordinates. The deflection of the plate is approximated by a set of beam characteristic orthogonal polynomials generated by using the Gram–Schmidt procedure. A high degree of correlation between computational analysis and experimental results was observed.     

Group theoretical analysis of a quantum-mechanical three-dimensional quartic anharmonic oscillator

This paper illustrates the application of group theory to a quantum-mechanical three-dimensional quartic anharmonic oscillator with O_h$O_h$ symmetry. It is shown that group theory predicts the degeneracy of the energy levels and facilitates the application of perturbation theory and the Rayleigh–Ritz variational method as well as the interpretation of the results in terms of the symmetry of the solutions. We show how to obtain suitable symmetry-adapted basis sets.     

Vibration analysis of circular cylindrical double-shell structures under general coupling and end boundary conditions

In this paper, the free and forced vibration analysis of circular cylindrical double-shell structures under arbitrary boundary conditions is presented. This is achieved by employing the improved Fourier series method based on Hamilton’s principle. In the formulation, each displacement component of the cylindrical shells and annular plates is invariantly expanded as the superposition of a standard Fourier series with several supplementary functions introduced to remove the potential discontinuities of the original displacement and its derives at the boundaries. With the introduction of four sets of boundary springs at the coupling interfaces and end boundaries of the shell–plate combination, both elastic and rigid coupling and end boundary conditions can be easily obtained by assigning the stiffnesses of the artificial springs to certain values. The natural frequencies and mode shapes of the structures as well as frequency responses under forced vibration are obtained with the Rayleigh–Ritz procedure. The convergence of the method is validated by comparing the present results with those obtained by the finite element method. Several numerical results including natural frequencies and mode shapes are presented to demonstrate the excellent accuracy and reliability of the current method. Finally, a number of parameter studies concerning various end and coupling boundary conditions, different dimensions of shells and annular plates are also performed.     

On the symmetry of three identical interacting particles in a one-dimensional box

We study a quantum-mechanical system of three particles in a one-dimensional box with two-particle harmonic interactions. The symmetry of the system is described by the point group D_3d$D_{3d}$. Group theory greatly facilitates the application of perturbation theory and the Rayleigh–Ritz variational method. A great advantage is that every irreducible representation can be treated separately. Group theory enables us to predict the connection between the states for the small box length and large box length regimes of the system. We discuss the crossings and avoided crossings of the energy levels as well as other interesting features of the spectrum of the system.     

Free vibration of rectangular nanoplates using Rayleigh–Ritz method

Vibration analysis of isotropic rectangular nanoplates based on the classical plate theory in conjunction with Eringen's nonlocal elasticity theory is considered. Nanoplates are one of the structural units that are used in nanoscale applications. In this study, Rayleigh–Ritz method with algebraic polynomial displacement function is used to solve the vibration problem of isotropic rectangular nanoplates subjected to different boundary conditions. The advantage of the method is that one can easily handle the specified boundary conditions at the edges. A comparison of the results with those available in the literature has been made. The proposed method is also validated by convergence studies. Frequency parameters are given for different nonlocality parameters, length of nanoplates and boundary conditions. The study highlights that nonlocality effects increase with the increase in mode number and the influence of nonlocal effects becomes increasingly pronounced for higher order vibration modes. Three-dimensional mode shapes for the specified nanoplates have also been presented.     

Effect of interactive damping on vibration sensitivities of a scanning near-field optical microscope probe

The effect of interactive damping on the sensitivity of flexural and axial vibration modes of scanning near-field optical microscope (SNOM) with a tapered optical fiber probe has been analyzed. The interaction of the SNOM probe with a sample surface is modeled by a combination of a spring and a dashpot in the flexural direction and a similar combination in the axial direction. An approximate form for the sensitivities of both modes was derived by using the Rayleigh–Ritz method. The results show that the interactive damping will decrease the sensitivities of both flexural and axial vibration modes when the contact stiffness is low. The more the damping effect, the lower the sensitivities are. In addition, when the contact stiffness was low, the flexural sensitivity of the tapered probe slightly increased as the tapered angle decreased. However, the axial sensitivity apparently decreased as the tapered angle decreased. When the contact stiffness became higher, the sensitivities of both flexural and axial vibration modes increased as the tapered angle increased. PACS 68.35.Ja; 07.79.Fc; 61.16.Ch     

Vibrational energy flow models for the Rayleigh–Love and Rayleigh–Bishop rods

Energy Flow Analysis (EFA) has been developed to predict the vibrational energy density of the system structures in the medium-to-high frequency range. The elementary longitudinal wave theory is often used to describe the longitudinal vibration of a slender rod. However, for relatively large diameter rods or high frequency ranges, the elementary longitudinal wave theory is inaccurate because the lateral motions are not taken into account. In this paper, vibrational energy flow models are developed to analyze the longitudinally vibrating Rayleigh–Love rod considering the effect of lateral inertia, and the Rayleigh–Bishop rod considering the effect not only of the lateral inertia but also of the shear stiffness. The derived energy governing equations are second-order differential equations which predict the time and space averaged energy density and active intensity distributions in a rod. To verify the accuracy of the developed energy flow models, various numerical analyses are performed for a rod and coupled rods. Also, the EFA results for the Rayleigh–Love and Rayleigh–Bishop rods are compared with the analytical solutions for these models, the traditional energy flow solutions, and the analytical solutions for the classical rod.     

3-D vibration analysis of circular rings with sectorial cross-sections

The free vibration characteristics of circular rings with sectorial cross-section are studied based on the three-dimensional (3-D), small strain, linear elasticity theory. The complete vibration spectrum has been obtained by using the Ritz method. A set of three-dimensional orthogonal coordinates composing of the polar coordinates (r,θ) at the origin of the sectorial cross-section and circumferential coordinate ? around the ring is developed to describe the variables in the analysis. Each of the displacement components is taken as a triplicate series: two Chebyshev polynomial series, respectively, about the r and θ coordinates, and a trigonometric series about the ? coordinate. Frequency parameters and vibration mode shapes are computed by means of the displacement-based extremum energy principle. Upper bound convergence of the first eight frequency parameters accurate to at least five significant figures is presented. The effect of radius ratio, subtended angle, and initial slope angle on frequency parameters is investigated in detail. All major modes such as flexural modes, thickness-shear modes, stretching modes, and torsional modes, etc. are presented in the paper. The present results may serve as a benchmark reference to validate the accuracy of various approximate theories and other computational techniques for the vibration of circular rings.     

ON PROPER ORTHOGONAL CO-ORDINATES AS INDICATORS OF MODAL ACTIVITY

Proper orthogonal decomposition (POD) is applied for examining modal activity. The extraction of proper orthogonal modal co-ordinates (POCs) is outlined. The proper orthogonal values (POVs) are the mean squared values of the POCs. The number of POVs above the noise floor provides a bound on the number of significant modes based on POVs above the noise floor. The ideas are illustrated on a linear cantilevered beam experiment. The displacement ensembles are obtained by processing six strain measurements. Coherent proper orthogonal modes (POMs) and POVs above the noise floor together confirm that six active modes are detected in the system. The distribution of modal components in POCs is discussed. The characteristics of the POMs, POVs and POCs are then examined in the presence of added noise.     

Free vibration of continuous polar orthotropic annular and circular plates

The free vibration of a polar orthotropic annular plate supported on concentric circles is analyzed by the Ritz method with use of Lagrange multipliers. A trial function for the deflection of the plate is expressed in terms of simple power series, and a frequency equation for the plate is derived by the condition for minimizing the total potential energy with the constraint equations included. In the numerical examples it is also shown that the method can directly yield quite accurate frequency values for a solid circular plate. Natural frequencies of annular and circular plates are calculated for wide ranges of the support location and orthotropic parameters.     

Vibration of scanning near-field optical microscope probe with laser-induced thermal effect using Timoshenko beam theory

The Timoshenko beam theory, including the effects of rotary inertia and shear deformation, is used to analyze the resonant frequency of lateral vibration of scanning near-field optical microscope (SNOM) tapered probe with a laser-induced thermal effect. In the analysis, the thermal effect can be considered as an axial force and is dependent of temperature distribution of the probe. The Rayleigh–Ritz method is used to solve the vibration problem of the probe. According to the analysis, the frequencies of the first three vibration modes increase when the thermal effect is taken into account. The effects of shear deformation and rotary inertia on the frequency ratio of a Timoshenko beam to an Euler beam increase when the mode number increases and the contact stiffness decreases. In addition, the frequency of mode 1 increases with increasing taper angle and coating thickness of the probe. Comparison of the frequency of a SNOM probe coated with Al, Ag, or Au, the highest is with Al coating, and the lowest is with Au coating.     

Theoretical study on liquid crystal cyanobiphenyls: Phase stability and phase behavior

This article presents a theoretical study on liquid crystalline materials in homologous series of 4'-n-alkyl-4-cyanobiphenyl (nCB) with propyl (3CB), pentyl (5CB), and heptyl (7CB) groups. The atomic net charge and dipole moment components at each atomic center have been evaluated using the complete neglect differential overlap (CNDO/2) method. The modified Rayleigh–Schrodinger perturbation theory along with the multicentered-multipole expansion method has been employed to evaluate the long-range intermolecular interactions, while a ‘6-exp’ potential function has been assumed for short-range interactions. Further, these interaction energy values have been used as input to calculate the translational entropy, and free energy of nCB (n=3, 5, and 7) molecules during the stacking, and in-plane interactions. The observed results have been correlated with the mesogenic behavior and phase stability based on the thermodynamic parameters introduced in this article. Further, an attempt has been made to elucidate the flexibility of a configuration at a particular temperature, which has a direct relation with phase transition property of the molecules.