Discrete Singular Convolution Method Research Articles

Static and dynamic response of sector-shaped graphene sheets

ABSTRACTNonlocal elasticity theory is presented for the free vibration and bending analysis of nano-scaled graphene sheets having a sector shape. An eight-node curvilinear domain is used for transformation of the governing equation of motion of sector graphene from physical region to computational region in conjunction with the Kirchhoff plate theory. The discrete singular convolution method is employed for numerical solutions of resulting nonlocal governing differential equations and related boundary conditions. Then, the effects of nonlocal parameters, mode numbers, sector angles, and radius ratios on the static and vibration results of nano-scaled sector-shaped graphene sheets are discussed.

Dynamic Stability Analysis of Beams with Shear Deformation and Rotary Inertia Subjected to Periodic Axial Forces by Using Discrete Singular Convolution Method

AbstractThis paper presents dynamic stability analysis of beams with shear deformation and rotary inertia subjected to periodic axial forces by employing a discrete singular convolution (DSC) algorithm with regularized Shannon kernel. The shear effect has been taken into account based on Engesser’s and Haringx’s theories, respectively. A modified algorithm is proposed to solve the governing equations of beam motion. The influence of rotary inertia, shear deformation, and damping on dynamic instability regions has been investigated. The obtained numerical results are compared with those of the existing method. Numerical results indicate that the modified algorithm is a reliable approach for the solutions of this kind of problem in this investigation. The differences between dynamic instability regions of beams based on Engesser’s and Haringx’s theories are also presented.

Thermal buckling of temperature dependent FG-CNT reinforced composite conical shells

In this research, linear thermal buckling of a composite conical shell made from a polymeric matrix and reinforced with carbon nanotube fibres is investigated. Distribution of reinforcements across the shell thickness is assumed to be uniform or functionally graded. Thermomechanical properties of the constituents are temperature dependent. Under the assumption of first order shear deformation shell theory, Donnell kinematic assumptions and von Kármán type of geometrical nonlinearity, the complete set of equilibrium equations and boundary conditions of the shell are obtained. A linear membrane analysis is carried out to obtain the pre-buckling thermal stresses of the shell. Adjacent equilibrium criterion is implemented to establish the stability equations associated with the buckling state. The resulting equations are discreted by means of trigonometric expansion through the circumferential direction and discrete singular convolution method through the shell length. The established eigenvalue problem is solved iteratively to obtain the critical buckling temperature and critical mode number. Parametric studies are presented to explore the influences of semi-vertex angle, volume fraction of CNTs, distribution pattern of CNTs and boundary conditions. It is shown that, conical shells with intermediate carbon nanotube volume fraction do not have, necessarily, intermediate critical buckling temperature.

Discrete Singular Convolution Method for Dynamic Stability Analysis of Beams under Periodic Axial Forces

AbstractIn the present study, the discrete singular convolution (DSC) method is used to analyze the dynamic stability of beams subjected to different boundary conditions under periodic axial forces. A unified DSC algorithm is established to solve the governing equations of beam motion under different boundary conditions. The regularized Shannon’s delta kernel is selected as singular convolution to illustrate the present algorithm. The matched interface and boundary method is applied to the treatment of elastic restraint boundary conditions. The effects of a constant term in the periodic axial force and damping on dynamic instability regions are also investigated. The obtained numerical results are compared with those of Bolotin’s method. Numerical results indicate that the DSC method is a reliable method for dynamic stability analysis of beams.

Dynamic Instability Analysis of a Rotating Ship Shaft under a Periodic Axial Force by Discrete Singular Convolution

Dynamic instability of a rotating ship shaft subjected to a periodic axial force is studied by using discrete singular convolution (DSC) with regularized Shannon’s delta kernel. The excitation frequency is related to the spinning speed and the number of blades on the propeller. Effects of number of blades, constant term in the periodic force, and damping on dynamic instability regions are investigated. The results have shown that the increase of number of blades and damping could improve the dynamic stability of rotating shaft with damping. The increase of constant term in the periodic force leads to dynamic instability regions shifting to lower frequencies, making the shaft more sensitive to periodic force. Those dynamic instability regions obtained by DSC method have been compared with those by Floquet’s method to verify the application of DSC method to dynamic instability analysis of rotating ship shaft.

Free vibration analysis of circular thin plates with stepped thickness by the DSC element method

Novel formulation is presented by using the discrete singular convolution (DSC) for free vibration analysis of circular thin plates with uniform and stepped thickness. Different from the commonly used ones in literature, regularity conditions are not needed at the circular plate center point to avoid singularity. DSC circular and annular thin plate elements are established. For the DSC circular plate element with radius of R1, the stiffness equation is first formulated in region [−R1, R1] with even number of nodes and then reduced to region [0, R1] by using either symmetric or anti-symmetric conditions. The proposed DSC circular and annular plate elements are used for obtaining frequencies of uniform/stepped circular thin plates or annular thin plates with different boundary conditions. Comparison of the present DSC results to existing analytic and numerical solutions verifies the proposed formulations. The present research extends the DSC method to free vibration of circular thin plates with stepped thicknesses.

Static analysis of laminated conical shells by Discrete Singular Convolution (DSC) approach

Static analyses of laminated conical and cylindrical shell are presented. The governing equation is derived using Love’s first approximation thin shell theory for conical geometry. Then, the method of Discrete Singular Convolution (DSC) is applied to the solution of the resulting governing equation and boundary conditions for bending. Numerical results for stress and deflections of laminated conical and cylindrical shells are presented for different geometric and material parameters. The numerical results show that the present method is quite easy to implement, accurate and efficient for the problems considered.

Vibration analysis of stepped rectangular plates by the discrete singular convolution algorithm

A novel scheme in using discrete singular convolution (DSC) algorithm is proposed for analyzing the challenging vibration problem of thin plates with geometric discontinuity . An n th-order interpolation polynomial is employed on each side of a step. Then two jump conditions together with two continuous conditions at the step are established to correlate the two polynomials. Detailed formulations and solution procedures are given. Several numerical examples are analyzed to demonstrate the applicability of the proposed method to the free vibration analysis of the stepped rectangular plates . Results obtained by the modified DSC are compared with existing solutions to verify the proposed method. For lower order mode frequencies, DSC results agree well with existing analytical and numerical solutions or data obtained by the finite element method (FEM). Besides, the DSC can also yield relatively accurate frequencies for higher order modes. This research extends the application range of the DSC method. • A modified DSC is presented for analyzing vibration of stepped rectangular plates. • Jumped conditions at the step are enforced using the interpolation method. • The ability of the modified DSC to obtain accurate higher order frequencies is demonstrated.

Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix

Free vibration analysis of micro-scaled annular sector and sector shaped graphene located on an elastic matrix are studied via nonlocal elasticity theory. An eight-node curvilinear element is used for transformation of the governing equation of motion of annular sector graphene from physical region to computational region in conjunctions with the thin plate theory. Elastic matrix is modeled via two-parameters which are Winkler–Pasternak elastic foundations. The discrete singular convolution (DSC) method is employed for numerical solution of resulting nonlocal governing differential equations and related boundary conditions. Then, the effects of nonlocal parameter, mode numbers, sector angles and foundation parameters on the frequency response of micron-scaled annular sector and sector graphene are discussed.

Stability and non-stationary vibration analysis of beams subjected to periodic axial forces using discrete singular convolution

Dynamic instability of beams subjected to periodic axial forces is studied using the discrete singular convolution (DSC) method with the regularized Shannon's delta kernel. The principal regions of dynamic instability under different boundary conditions are examined in detail, and the non-stationary vibrations near the stability-instability critical regions have been investigated. It is found that the results obtained by using the DSC method are consistent with the analytical solutions, which shows that the DSC algorithm is suitable for the problems considered in this study. It was found that there is a narrow region of beat vibration existed in the vicinity of one side (<TEX>${\theta}/{\Omega}$</TEX> > 1) of the boundaries of the instable region for each condition.

Modal and response bound predictions of uncertain rectangular composite plates based on an extreme value model

An extreme value (EV) model is constructed to predict statistical bounds of low, mid and high frequency free and forced vibration responses of symmetrically-laminated rectangular composite plates with random orientation angles, boundary conditions and damping ratios. Sample data fed to the EV model is generated by the discrete singular convolution (DSC) method through performing Monte Carlo simulations. The DSC has already proved itself as an accurate method particularly at high frequency region in which many numerical methods suffer from computational expense and efficiency. In the study, the DSC method is initially verified by comparison with classical laminated plate theory for simple supported, six-layered, specially-orthotropic composite plates for modal and response analysis throughout a broad frequency range. The uncertainties in plate rigidity, internal damping and assembling conditions of the composite plate are modelled using normal distribution of orientation angles, damping loss factors and boundary conditions. Modal and frequency response bounds are then predicted based on an EV based model considering a small number of samples of simulated uncertain data.

Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation

In the present study, free vibration analysis of nano-sized annular sector plate is analyzed using the nonlocal continuum theory. The method of discrete singular convolution (DSC) is used for numerical computations. Firstly, equation of motion of thin plates is formulated via nonlocal elasticity. Then, irregular physical domain is transformed into a rectangular domain by using geometric coordinate transformation. The DSC procedures are then applied to discretization of the transformed set of governing equations and related boundary conditions. The effects of nonlocal parameter, mode numbers, sector angle and radius ratio on the vibration frequencies are investigated in detail. It is seen that the size effects are significant in vibration analysis of nano-scaled annular sector plates and need to be included in the mechanical model.

Numerical determination of statistical energy analysis parameters of directly coupled composite plates using a modal-based approach

Statistical energy analysis (SEA) parameters such as average modal spacing, coupling loss factor and input power are numerically determined for point connected, directly coupled symmetrically laminated composite plates using a modal-based approach. The approach is an enhancement of classical wave transmission formula. Unlike most of the existing numerical or experimental techniques, the approach uses uncoupled plate modal information and treats substructure by means of averaged modal impedances. The procedure introduced here is verified using analytical definitions of infinite orthotropic plates which physically resemble to laminated plates for (under) specific conditions, and is tested by performing experimental power injection method (PIM) for an actual, right-angled composite structure. In the development process, force and moment transmissions are individually considered in order to be consistent with analytical formulations. Modal information of composite plates is statistically evaluated by the discrete singular convolution method with random boundary conditions. Proposed methodology not only provides an efficient use of SEA method in high frequency vibration analysis of composite structures, but also enhances SEA accuracy in mid frequency region in which conventional SEA fails. Furthermore, the effect of orientation angles of laminations on SEA parameters are also discussed in mid and high frequency regions.

Numerical solution of 2d seepage flow problem using discrete singular convolution method

PurposeThe purpose of this paper is to describe the implementation of discrete singular convolution (DSC) method to steady seepage flow while presenting one of the possible uses of DSC method in geotechnical engineering. It also aims to present the implementation of DSC to the problems with mixed boundary conditions.Design/methodology/approachSecond order spatial derivatives of potential and stream functions in Laplace's equation are discretized using the DSC method in which the regularized Shannon's delta kernel is used as an approximation to delta distribution. After implementation of boundary conditions, the system of equations is solved for the unknown terms.FindingsThe results are compared with those obtained from the finite element method and the finite difference method.Originality/valueThe method is applied to the flow problem through porous medium for the first time.

Memory-Efficient Modeling of Reverberation Chambers Using Hybrid Recursive Update Discrete Singular Convolution-Method of Moments

A hybrid method is proposed for memory-efficient analysis of reverberation chambers (RCs). In the hybrid method, the cavity is modeled by the recursive update discrete singular convolution (RUDSC) method, and antennas and stirrers inside the cavity are simulated using the flexible method of moments (MoM). In order to solve DSC and MoM unknowns separately, a layer-based elimination algorithm is utilized to eliminate the DSC unknowns. The MoM unknowns are then solved by a direct solver. Once the solution for the MoM model is obtained, the original RC is equivalent to a cavity excited by known current sources. The equivalent problem is finally solved using the RUDSC method. Taking advantage of the layer-based elimination algorithm and the recursive update technique, the memory requirement of the new hybrid method is much smaller than that of using a direct solver. Numerical simulations are presented to show the efficacy of the proposed method. It is shown that the the proposed method substantially reduces the memory cost of RC modeling, which extends RC analysis to higher frequencies.

Vibration analysis of laminated composite conical shells by the method of discrete singular convolution based on the shear deformation theory

Free vibration analysis of laminated conical shells is presented by using the numerical solution of governing differential equations of motion based on transverse shear deformation theory. Results are presented for isotropic, orthotropic, and laminated cases for conical shells. Free vibrations of circular cylindrical shells and annular plates are treated as special cases. To verify the accuracy of this method, comparisons of the present results are made with results available in the open literature. Numerical results in vibrations of laminated conical shells are presented for different geometric and material parameters.

Extreme-Value-Based Statistical Bounding of Low, Mid, and High Frequency Responses of a Forced Plate With Random Boundary Conditions

The problem of statistically bounding the response of an engineering structure with random boundary conditions is addressed across the entire frequency range: from the low, through the mid, to the high frequency region. Extreme-value-based bounding of both the FRF and the energy density response is examined for a rectangular linear plate with harmonic point forcing. The proposed extreme-value (EV) approach, previously tested only in the low frequency region for uncoupled and acoustically-coupled uncertain structures, is examined here in the mid and high frequency regions in addition to testing at low frequencies. EV-based bounding uses an asymptotic threshold exceedance model of Type-I, to extrapolate the m-observational return period to an arbitrarily-large batch of structures. It does this by repeatedly calibrating the threshold model at discrete frequencies using a small sample of response data generated by Monte Carlo simulation or measurement. Here the discrete singular convolution (DSC) method – a transfrequency computation approach for deterministic vibration - is used to generate Monte Carlo samples. The accuracy of the DSC method is first verified (i) in terms of the spatial distribution of total energy density and (ii) across the frequency range, by comparison with a mode superposition method and Statistical Energy Analysis (SEA). EV-based bound extrapolations of the receptance FRF and total energy density are then compared with: (i) directly-estimated bounds using a full set of Monte Carlo simulations and (ii) with total mean energy levels obtained with SEA. This paper shows that for a rectangular plate structure with random boundary conditions, EV-based statistical bounding of both the FRF and total energy density response is generally applicable across the entire frequency range.

Buckling and Vibration of Elastically Restrained Standing Vertical Plates

This paper investigates the buckling and vibration of heavy standing plates with rotational elastic edge constraints. The discrete singular convolution (DSC) method as a powerful numerical technique is applied to derive the governing eigenvalue equation. Convergence and comparison studies are conducted to authenticate the correctness and accuracy of the DSC approach. Accurate first-known vibration solutions for elastically restrained vertical plates subjecting to body forces/self-weight are presented. Some contour mode shapes for the vibration of elastically restrained vertical plates are also depicted for illustration.

Nonlinear vibration analysis of laminated plates resting on nonlinear two-parameters elastic foundations

In the present manuscript, geometrically nonlinear free vibration analysis of thin laminated plates resting on non-linear elastic foundations is investigated. Winkler-Pasternak type foundation model is used. Governing equations of motions are obtained using the von Karman type nonlinear theory. The method of discrete singular convolution is used to obtain the discretised equations of motion of plates. The effects of plate geometry, boundary conditions, material properties and foundation parameters on nonlinear vibration behavior of plates are presented.

Efficient Modeling of Three-Dimensional Reverberation Chambers Using Hybrid Discrete Singular Convolution-Method of Moments

Efficient modeling of a three-dimensional reverberation chamber (RC) is achieved by combining the discrete singular convolution (DSC) method and the method of moments (MoM). An RC usually consists of a metallic cavity and one or two conducting stirrers, whose size is normally small compared to the chamber size. The large cavity is efficiently modeled by the DSC method, and the stirrer is simulated by the flexible MoM. Exploiting the property of RWG basis, solutions from the two methods are combined together using the equivalence principle. The validity and advantages of the proposed hybrid technique are shown through comparisons with the commercial software FEKO. Employing the high efficiency of the DSC method, the hybrid technique can analyze one stirrer position of a medium-sized RC in a few hundred seconds on a single personal computer, for which FEKO needs thousands of seconds CPU time. The memory requirement of the proposed method is also less than that of FEKO. Furthermore, our hybrid method provides efficient calculation of electric field strength at a large number of field points, which is of great interest in RC analysis. Simulations show that our method only takes 1.7 seconds to compute electric field strength at 4026 field points.