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Vibration analysis of cylindrical shells by local adaptive differential quadrature method
Abstract
This paper presents the formulation and numerical analysis of circular cylindrical shells by the local adaptive differential quadrature method (LaDQM), which employs both localized interpolating basis functions and exterior grid points for boundary treatments. The governing equations of motion are formulated using the Goldenveizer–Novozhilov shell theory. Appropriate management of exterior grid points is presented to couple the discretized boundary conditions with the governing differential equations instead of using the interior points. The use of compactly supported interpolating basis functions leads to banded and well-conditioned matrices, and thus, enables large-scale computations. The treatment of boundary conditions with exterior grid points avoids spurious eigenvalues. Detailed formulations are presented for the treatment of various shell boundary conditions. Convergence and comparison studies against existing solutions in the literature are carried out to examine the efficiency and reliability of the present approach. It is found that accurate natural frequencies can be obtained by using a small number of grid points with exterior points to accommodate the boundary conditions.
... e nonlinear deformation of elastic beams has applications in the nuclear industry, aviation, biology, and medicine. ere have been many related studies in the field of solid mechanics [7][8][9][10][11][12][13]. Since 1970, some numerical and analytical methods have been proposed to solve these problems of elastic beams, such as the finite element method, spectral methods, and DQ method [7]. ...
... From this work, we can see that the difficulties arise when considering the boundary conditions especially for free edges. Several new methods have been proposed for free boundary conditions, such as local adaptive DQ methods [11], matched interface and boundary method [12], and fictitious domain approach [13]. ese methods have great application impact. ...
... Equations (2)- (6) can be solved when the load distribution is known. Since the solution is the deflection between the location z * � 0 and z * � 1 (where z * � z/h), in order to facilitate the application of the boundary conditions, we use the water flow depth and other parameters to nondimensionalize equations (2)- (6): (11) can be rearranged into 3 dimensionless nonlinear differential equations: ...
According to the characteristics of the reactor internal structure of nuclear power plants, the vibration of the secondary core support pillar in water can be modeled as the vibration of the cantilever beam structure under the action of transverse flow, and its first beam mode is highly likely to be activated. It is thus necessary to dedicate a separate study on the first-order beam mode. In this work, we study the secondary core support pillar in nuclear reactor AP1000 under the action of transverse flow and focus on the derivation of its static cantilever deflection mode shape function in order to lay a foundation for the calculation of hydrodynamic added mass and frequency for the nuclear reactor internal components and their structural integrity evaluation. First, we proposed a set of nonlinear differential equations for the analysis of the single cantilever beam. Second, to solve the nonlinear differential equations, we used a boundary shooting framework in combination with the Runge–Kutta method. The results of the numerical simulation agree with the analytical solution to a very high degree, which demonstrates the effectiveness of the simulation method. Finally, we solved the static deflection mode shape function of the secondary core support pillar under the normal operating conditions. The nonlinear differential model and simulation method proposed in this paper can be used to solve the static cantilever deflection mode shape function of the equipment support tube.
... The results show that the frequency characteristics of FG cylindrical shells are similar to those of isotropic materials. Based on Flu¨gge theory, Zhang et al. [4] studied the free vibration of cylindrical shells under different boundary conditions by using the local adaptive differential quadrature method, the results in this literature are compared with some of the results in this paper. Ghamkhar et al. [5] studied the free vibration of three-layer cylindrical shells with the effect of FGM central layer, and the influence of the thickness and geometry of the intermediate layer on the natural frequency under different classical boundaries was demonstrated. ...
... To verify the method presented in this paper, the method in Section "Theoretical formulation" was used to calculate the natural frequency of a single-layer isotropic shell, FGM cylindrical shells and three-layer FGM cylindrical shells. For the FGMs, when the volume fraction indexes nz and nx are zero, the FGMs becomes an isotropic material composed of one material, which properties are presented in Zhang et al. [4] In this paper, the clamped-clamped (C-C) and the simply supported-simply supported (S-S) boundary conditions are simulated by springs assigned at the ends of the shell. As mentioned in Section "Theoretical formulation", the spring with the stiffness of 10 15 N/m is sufficient to indicate the constraint on degrees of freedom. ...
... The frequency of an isotropic cylindrical shell is shown in Table 4 and compared with the results in the literature of Zhang et al. [4]. Table 5 show the natural frequencies of the FG cylindrical shell with S-S boundary conditions and the data is compared with the results in the literature [1]. ...
In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.
... The common investigated shape functions are Lagrange interpolation polynomials, 38,39 Cardinal sine, 40 Delta Lagrange, and Regularized Shannon kernels (RSKs). [41][42][43][44][45][46] Undoubtedly, numerical DQ methods (polynomialbased differential quadrature (PDQ) or classical DQsinc differential quadrature (SDQ)-discrete singular convolution (DSCDQM)) are not used for solving power conversion of PSCs problems. The main purpose of the present study is to formulate a nonlinear differential equation system for PSC to achieve a higher PCE by these numerical methods. ...
... Also, iterative quadrature technique was used to solve this system. [37][38][39][40][41][42][43][44][45][46] For each method, we ensure the efficiency and convergence by design a MATLAB code to get a numerical solution for this problem. Then, comparing previous experiment, exact, finite difference, SCAPS 1-D simulation software, and finite element techniques with computed results. ...
... 2. Regularized Shannon kernel [44][45][46] : ...
This work presents different numerical methods that are used for the first time in solving Perovskite solar cells (PSCs). Classical differential quadrature, sinc, and discrete singular convolution (Regularized Shannon and Delta Lagrange kernels) methods are employed for studying this problem. The governing equations are derived based on Poisson's and continuity equations. The different quadrature techniques are introduced to convert the system of nonlinear partial differential equations to nonlinear algebraic system. Then, an iterative method is used to solve this system. Convergence and efficiency of the obtained results with error ≤10⁻⁸ depend on various computational characteristics for each technique. The computed results match with previous experiment, exact, finite difference, SCAPS 1-D simulation software, and finite element scheme. Then, the comprehensive parametric study is explored to show the effects of density states, gap energy, thickness, temperatures, lifetimes, wavelength, absorption coefficient, recombination prefactor, and recombination mechanisms whether direct or indirect on power conversion efficiency (PCE) and charge transport of solar cells with and without interface material. After all that have been studied on PSCs, it was found that the best value of PCEs was 32%. Thus, the computed results of the present schemes may be useful for improving the performance level of PSCs.
... This new method has been applied to many mathematical physics and engineering problems by Wei et al. [54,55] and Ng et al. [56]. It was completely shown and proven by many scientists in different areas via different examples [57][58][59][60][61][62][63][64][65][66][67][68][69][70][71] that the method of DSC has good accuracy, efficiency, and rapid convergence. At the beginning of 2000s, the DSC method was introduced via computer realization of some singular convolutions [45,52]. ...
... By the way, the mathematical foundation of the DSC method is older and based on the theory of distributions and the theory of wavelets. In different DSC applications, many DSC kernels such as regularized Shannon's delta (RSD), regularized Dirichlet, regularized Lagrange, and regularized de la Vallée Poussin kernels were used in [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71]. Such a singular convolution is defined as [52] ...
A geometric transformation method based on discrete singular convolution (DSC) is firstly applied to solve the buckling problem of a functionally graded carbon nanotube (FG-CNT)-reinforced composite skew plate. The straight-sided quadrilateral plate geometry is mapped into a square domain in the computational space using a four-node DSC transformation method. Hence, the related governing equations of plate buckling and boundary conditions of the problem are transformed from the physical domain into a square computational domain by using the geometric transformation-based singular convolution. The discretization process is achieved via the DSC method together with numerical differential and two different regularized kernels such as regularized Shannon's delta and Lagrange-delta sequence kernels. The accuracy of the present DSC results is first verified, and then, a detailed parametric study is presented to show the impacts of CNT volume fraction, CNT distribution pattern, geometry of the skew plate and skew angle on the axial and biaxial buckling responses of FG-CNTR composite skew plates with different boundary conditions. Some new results related to critical buckling of an FG-CNT-reinforced composite skew plate are also presented, which can serve as benchmark solutions for future investigations.
... They studied the effect on vibrations of double layered FG shell for various shell constraints, edge conditions and exchange the essential materials making FGMs. Zhang et al. [18] investigated the shell free vibrations for a number of edge conditions by using a differential quadrature type procedure. Naeem et al. [19] analyzed vibration behavior of the tri-layered FGM circular cylinders. ...
... A good agreement is found among the present results and those obtained by other techniques. In Table 1, a comparison of frequency parameters ∆ = ωR √︁ (︀ 1 − µ 2 )︀ ρ/E for simply supported isotropic CS is presented with those in Zhang et al. [18]. Table 3 represents the NFs (Hz) for a simply supported FGM cylindrical shell of type 1 compared with those obtained by Loy et al. [1] and for power law exponents N = 0.5, 1, 2. The frequency at n = 3 which is about 0.9%, 0.5%, and 0.1% minimum than those available in [1]. ...
In this work, we study vibrations of three-layered cylindrical shells with one ring support along its length. Nature of material of the central layer is a functionally graded material (FGM) type. The considered FGM is of stainless steel and nickel. The internal and external layers are presumed to be made of isotropic material i.e., aluminum. The functionally graded material composition of the center layer is assorted by three volume fraction laws (VFL) which are represented by mathematical expressions of polynomial, exponential and trigonometric functions. The implementation of Rayleigh-Ritz method has been done under the Sanders’ shell theory to obtain the shell frequency equation. Natural frequencies (NFs) are attained for the present model problem under six boundary conditions. Use of characteristic beam functions is made for the estimation of the dependence of axial modals. The impact of layer material variations with ring support is considered for many ring positions. Also the effect of volume fraction laws is investigated upon vibration characteristics. This investigation is performed for various physical parameters. Numerous comparisons of values of shell frequencies have been done with available models of such types of results to verify accuracy of the present formulation and demonstrate its numerical efficiency.
... In the DQ method, a partial derivative of a function with respect to a space variable at a discrete point is approximated as a weighted linear sum of the function values at all discrete points in the region of that variable [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]. For simplicity, we consider a one-dimensional function u(x) in the [-1,1] domain, and N discrete points. ...
... Shu and Xue proposed an explicit means of obtaining the weighting coefficients for the HDQ [53]. When the ) (x f is approximated by a Fourier series expansion in the form [42][43][44][45][46][47][48][49][50] ) sin ...
The aim of the present paper is to investigate effect of thickness on frequency. For this, free vibration analysis of circular shells is made via ANSYS and numerical method. Discrete singular convolution (DSC) and differential quadrature methods have been proposed for numerical solution of vibration problem. The formulations are based on the Love's first approximation shell. The performance of the present methodology is also discussed.
... Detailed presentation of the shell theories and some analytical approaches are available in Refs. 1, 2. There exist a number of reports 3-6 using the conventional methods for analyzing the free vibration of cylindrical shells. Zhang et al. 7 presented the local adaptive di®erential quadrature (LaDQM) for the vibration analysis of cylindrical shells. The LaDQM uses both the localized interpolating basic functions and exterior grid points for the boundary treatment. ...
... The stress resultants in terms of forces and moments can be de¯ned by using Love's¯rst approximation theory as follows 7,10 : ...
This paper applies the Chebyshev collocation method to finding accurate solutions of natural frequencies for circular cylindrical shells. The shells with different boundary conditions are considered in the parametric study. By using the method to solve the coupled differential equations of motion governing the vibration of the shell, numerical results are obtained from the algebraic eigenvalue equation using the Chebyshev differentiation matrices. And the results satisfy both the geometric and force boundary conditions. Based on the numerical examples, the proposed method shows its capacity and reliability in predicting accurate frequency results for circular cylindrical shells with various boundary conditions as compared to some exact solutions available in the literature.
... The incremental differential quadrature method (IDQM) has been developed by Hashemi et al. [339][340][341][342], and it has been basically applied for solving shallow water and water flow equations. Another method is the local adaptive differential quadrature method (LaDQM), which has been used by Zhang et al. [343], Ng et al. [344], and Wu et al. [345][346][347] for solving several engineering problems related to structural components, heat and waves. LaDQM has banded weighting coefficient matrices instead of sparse ones, and it is similar to the local generalized differential quadrature (LGDQ), which is a localized version of the GDQ method [348][349][350][351][352][353][354][355][356][357][358][359]. ...
... Local generalized differential quadrature (LGDQ) method [348] 2001 Domain-free discretization (DFD) method [407] 2003 Moving least squares differential quadrature (MLSDQ) method [418] 2003 State-space-based differential quadrature method (SSDQM) [411] 2004 Spline-based differential quadrature method (SDQM) [461] 2006 Incremental differential quadrature method (IDQM) [339] 2006 Local adaptive differential quadrature method (LaDQM) [343] 2007 Differential quadrature based on the interpolation of the highest derivatives (DQIHQ) method [395] 2008 Differential quadrature Trefftz method (DQTM) [393] 2009 Random differential quadrature (RDQ) method [399] 2010 ...
A survey of several methods under the heading of SFEM (Strong Formulation Finite Element Method) is presented. These approaches are distinguished from classical termed Weak Formulation Finite Element Method (WFEM). The main advantage of the SFEM is that it uses differential quadrature method (DQM) for the discretization of the equations and the mapping technique for the coordinate transformation from the Cartesian to the computational domain. Moreover the element connectivity is performed by using kinematic and static conditions so that displacements and stresses are continuous across the element boundaries. Numerical investigations integrate this survey by giving graphical details on the subject.
... From our articles [48][49][50][51][52], we introduced many various test functions. But, within this study, two different test functions are used to compute ℝ k ij [56][57][58][59]: First test function is Delta Lagrange Kernel (DLK), which can be written as: ...
In the present study, the free vibration analysis of piezoelectric composite nanobeams is performed with various boundary conditions. Novel numerical techniques based on quadrature schemes with various test functions are applied with the generalized Caputo kind to solve the boundary value fractional problems. The accuracy and validity of the present analysis are demonstrated by comparing the present results with the previously published ones. Moreover, a parametric analysis is carried out to examine the effect of some material parameters, fractional order derivatives, and geometric dimensions on the vibration attitude of the nanobeam. This parametric study proven the ability of our techniques to provide a solution to most complex fractional differential equations. It is revealed that the fractional differential quadrature methods decrease the time and computational costs.
... Still, in both cases, a convergence analysis is necessary. Zhang et al. [28] used the local adaptive differential quadrature method, which employs localised interpolating basis functions and exterior grid points for boundary treatments. Pellicano [29] proposed a method for analysing linear and nonlinear vibrations of circular cylindrical shells with different boundary constraints, using harmonic functions and Chebyshev polynomials for displacements and a numerical technique for the resolution. ...
Although free vibrations of thin-walled cylinders have been extensively addressed in the relevant literature, finding a good balance between accuracy and simplicity of the procedures used for natural frequency assessment is still an open issue. This paper proposes a novel approach with a high potential for practical application for rapid esteem of natural frequencies of thin-walled cylinders under different boundary conditions. Starting from Donnell–Mushtari’s shell theory, the differential problem is simplified by using the principle of virtual work and introducing the flexural waveforms of a beam as constrained as the cylinder. Hence, the formulation is reduced to the eigenvalue problem of an equivalent 3 × 3 dynamic matrix, which depends on the cylinder geometry, material, and boundary conditions. Several comparisons with experimental, numerical, and analytical approaches are presented to prove model reliability and practical interest. An excellent balance between fast usability and accuracy is achieved. The user-friendliness of the model makes it suitable to be implemented during the design stage without requiring any deep knowledge of the topic.
... Consequently, ℜ ( ) ℜ ( ) can be determined by differentiating Equation (15), as follows [24][25][26][27]: ...
In recent times, the global community has been faced with the unprecedented challenge of the coronavirus disease (COVID-19) pandemic, which has had a profound and enduring impact on both global health and the global economy. The utilization of mathematical modeling has become an essential instrument in the characterization and understanding of the dynamics associated with infectious illnesses. In this study, the utilization of the differential quadrature method (DQM) was employed in order to anticipate the characterization of the dynamics of COVID-19 through a fractional mathematical model. Uniform and non-uniform polynomial differential quadrature methods (PDQMs) and a discrete singular convolution method (DSCDQM) were employed in the examination of the dynamics of COVID-19 in vulnerable, exposed, deceased, asymptomatic, and recovered persons. An analysis was conducted to compare the methodologies used in this study, as well as the modified Euler method, in order to highlight the superior efficiency of the DQM approach in terms of code-execution times. The results demonstrated that the fractional order significantly influenced the outcomes. As the fractional order tended towards unity, the anticipated numbers of vulnerable, exposed, deceased, asymptomatic, and recovered individuals increased. During the initial week of the inquiry, there was a substantial rise in the number of individuals who contracted COVID-19, which was primarily attributed to the disease’s high transmission rate. As a result, there was an increase in the number of individuals who recovered, in tandem with the rise in the number of infected individuals. These results highlight the importance of the fractional order in influencing the dynamics of COVID-19. The utilization of the DQM approach, characterized by its proficient code-execution durations, provided significant insights into the dynamics of COVID-19 among diverse population cohorts and enhanced our comprehension of the evolution of the pandemic. The proposed method was efficient in dealing with ordinary differential equations (ODEs), partial differential equations (PDEs), and fractional differential equations (FDEs), in either linear or nonlinear forms. In addition, the stability of the DQM and its validity were verified during the present study. Moreover, the error analysis showed that DQM has better error percentages in many applications than other relevant techniques.
... Consequently, ℜ (1) and ℜ (2) can be given by differentiating Equation (18) as follows [39,[52][53][54]: ...
The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for convection and diffusion depends on both spatial and transient factors. By using the block marching technique, we transform these equations into an algebraic system using differential quadrature methods and the Caputo-type fractional operator. Next, we develop a MATLAB program that generates code capable of solving the fractional convection–diffusion equation in (1+2) dimensions for each shape function. Our goal is to ensure that our methods are reliable, accurate, efficient, and capable of convergence. To achieve this, we conduct two experiments, comparing the numerical and graphical results with both analytical and numerical solutions. Additionally, we evaluate the accuracy of our findings using the L∞ error. Our tests show that the differential quadrature method, which relies mainly on the discrete singular convolution shape function, is a highly effective numerical approach for fractional convective diffusion problems. It offers superior accuracy, faster convergence, and greater reliability than other techniques. Furthermore, we study the impact of fractional order derivatives, velocity, and positive diffusion parameters on the results.
... In our model, we use a differential quadrature method (DQM) with drastically different shape functions to solve the problem of drug diffusion through a thin membrane. These shape functions are the Lagrange interpolation function [21,31], the Delta Lagrange kernel, and the Regularized Shannon kernel [32][33][34][35], and they have been successfully applied to the problem of drug diffusion through a thin membrane. Furthermore, the partial differential equation is transformed into an ordinary differential equation (ODE) using a one-parameter group transformation [4] to reduce the two independent variables by one. ...
The primary goal of this work is to solve the problem of drug diffusion through a thin membrane using a differential quadrature approach with drastically different shape functions, such as Lagrange interpolation and discrete singular convolution (the delta Lagrange kernel and the regularized Shannon kernel). A nonlinear partial differential equation with two time- and space-dependent variables governs the system. To reduce the two independent variables by one, the partial differential equation is transformed into an ordinary differential equation using a one-parameter group transformation. With the aid of the iterative technique, the differential quadrature methods change this equation into an algebraic equation. Then, using a MATLAB program, a code is created that solves this equation for each shape function. To ensure the validity, efficiency, and accuracy of the developed techniques, the computed results are compared to previous numerical and analytical solutions. In addition, the L∞ error is applied. As a consequence of the numerical outcomes, the differential quadrature method, which is primarily based on a discrete singular convolution shape function, is an effective numerical method that can be used to solve the problem of drug diffusion through a thin membrane, guaranteeing a higher accuracy, faster convergence, and greater reliability than other techniques.
... In all these studies, due to the axial symmetry of the shell, the discretization of the equations has been done only in the longitudinal direction. Zhang et al., 47 used a version of the differential quadrature method called the Local Adaptive Differential Quadrature Method (LaDQM) to study the vibrations of the cylindrical shell. Relationships are written based on the Goldenveizer-Novozhilov shell theory and different boundary conditions were studied. ...
In this examination, the free vibrations of complete composite shells with rectangular openings based on first-order shear deformation theory have been studied. The equations are generally written in such a way that they can be converted to any of Donnell, Love, or Sanders theories. To study the shell with the opening of the problem-solving space, it is elementalized in such a way that the boundary conditions and loading are uniform at the edges of each element. For each element, the governing equations, the boundary conditions of the edges, and the compatibility conditions at the common boundary of the adjacent elements are discretized by the generalized differential quadrature method in the longitudinal and peripheral directions, and by assembling them, a system of algebraic equations is formed. Finally, the natural frequency of the structure is calculated using the solution of the eigenvalue. To validate this method, the results are compared with the results of some articles as well as the results of Abaqus finite element software. After ensuring the efficiency of the present method, it has been used to study the effect of different parameters on the vibrational behavior of shells with and without apertures. These studies show that relatively small openings (c/L <0.3) have little effect on the natural frequency of the shell, regardless of the material and the porcelain layer of the shell. While reducing the ratio of length to radius or increasing the thickness of the shell is also effective in reducing the effects of opening. In addition, the effect of peripheral openings is far less than longitudinal openings.
... In recent years, numerous studies have been conducted on the free and steady vibration of cylindrical shell, including Ritz method, 1-4¯n ite element method (FEM), 5 transfer matrix method, 6,7 dynamic sti®ness method. [8][9][10] and di®erential quadrature method (DQM), 11,12 etc. Considering the in°uence of aspect ratio, number of modes and thermal environment, Ebrahimi et al. 13 investigated the thermal buckling and forced vibration behaviors of the cylindrical shell by using Hamilton's principle. Based on the Flügge shell theory, Zhang et al. 14,15 converted the variable coe±cient partial di®erential equations into mutually coupled linear equations, the forced vibration response of the elliptical cylindrical shell was presented by solving the coupled equations, and the calculation results were in good agreement with those obtained by FEM. ...
In this paper, the experimental and Jacobi–Ritz method (JRM) have been adopted to analyze the forced vibration analysis of uniform and stepped circular cylindrical shells with general boundary conditions. The simply supported cylindrical shell at both ends is taken as an experimental model, and the free, steady and transient vibration characteristics of structures under hammer and fixed exciter are recorded. The results show that the results of JRM are in sensible agreement with those in experiment. In addition, the results for various boundary conditions, structural parameter are also presented. On this basis, the Newmark-β integration method is adopted to realize the time domain solutions for transient vibration response, and the frequency domain results can be obtained by using Fourier transformations from time domain results. Finally, the line spectrum vibration response results of the structure are presented under the random excitation load, and the research can supply technical support for the vibration control of cylindrical shell structure.
... Table 1 demonstrates values of frequency comparison of simply supported cylindrical shell. A comparison of the frequency parameters is made with those results observed by Loy et al. (1999) and Naeem and Sharma (2000) in Tables 1, 2 and 3. Table 4 shows a judgment with those frequencies obtained by Zhang et al. (2006). The shell has been supported on ends with simply supported end conditions. ...
This paper deals with the specific influence of three different fraction laws for vibrational analysis of rotating cylindrical shells. The rotating cylindrical shells are stabilized by ring-stiffeners to increase the stiffness and strength. Isotopic materials are the constituents of these rings. The frequencies are investigated versus circumferential wave number, length- and height-to- radius ratios using three volume fraction laws. Moreover, the effect of rotation speed is investigated. It is examined that the backward and forward frequencies increase and decrease on increasing the ratio of height- and length-to-radius ratio. When the position of ring supports increases, the backward and forward frequency first increases and obtains its maximum value at the shell mid length position and then decreases and get a bell shape with clamped-clamped and clamped-free conditions. The assessment of present model is judged with the comparison of non-rotating and rotating results with former exploration.
... Table 1 demonstrates values of frequency comparison of simply supported cylindrical shell. A comparison of the frequency parameters is made with those results observed by Loy et al. (1999) and Naeem and Sharma (2000) in Tables 1, 2 and 3. Table 4 shows a judgment with those frequencies obtained by Zhang et al. (2006). The shell has been supported on ends with simply supported end conditions. ...
... The method of discrete singular convolution first used at the end of the 90 s by Wei and his coauthors [150][151][152][153], in which they have proposed some singular kernels, namely, Hilbert, Abel, and delta types, in some mathematical physics and computational mechanics problems. Then, the method of DSC has been utilized in different problems in the area of mathematical physics and computational solid and fluid mechanics [154][155][156][157][158][159][160][161][162][163][164][165][166][167][168][169][170]. It was completely shown and proven by many scientists in different areas via different examples that the method of discrete singular convolution (DSC) has good accuracy, easy for applications, efficiency, and rapid convergence. ...
This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.
... Therefore, vibration analysis is important in order to improve the behavior of structures such as cylindrical and conical shells. Several researchers have studied vibration of cylindrical [9][10][11][12][13][14][15][16][17][18][19] and conical shells [20][21][22][23][24][25][26][27]. For large amplitudes, the nonlinear model of the system can better describe the structure's vibration behavior. ...
This paper presents the combination of multiple scale method and modal analysis in order to investigate nonlinear vibration of laminated composite angle-ply cylindrical and conical shells. The shells are modeled considering the shear deformation and rotary inertia while the geometrical nonlinearity is modeled using von Karman approach. Hamilton principle is used for obtaining the basic equations of the system. These equations are converted to nonlinear ordinary differential equations depending on time variable using Ritz method. The results of this study are validated against the results of open literature and good agreement is observed. The effects of several parameters including the layers' angle, the number of the layers, semi-vertex angle, length, radius and also each layer's thickness on nonlinear frequency ratio, fundamental linear frequency and nonlinear frequency are illustrated in details.
... For free edge, Taylor series expansion [90][91][92][93][94] is used for discretization. Based on the first-order shear deformation theory, the governing equations for buckling of thick plates are also given D 11 ...
... Results for an isotropic cylindrical shell with following edge conditions, simply supported-simply supported (ş-ş), clamped-clamped (ς-ς) and clamped-free (ς-f ), are compared with the results available in open literature to ensure the validity, authenticity and robustness of the current technique. Tables 1 and 2 show the comparisons of frequency parameters with those in the Zhang et al. 7 for ş-ş and ς-ς isotropic cylindrical shells. Comparison of natural frequencies (Hz) with those available in Loy & Lam 4 for ς-f isotropic cylindrical shell is presented in the Table 3. ...
In this research, vibration frequency analysis of three layered functionally graded material (FGM) cylinder-shaped shell is studied with FGM central layer and the internal and external layers are of homogenous material. Strain and curvature-displacement relations are taken from Sander’s shell theory. The shell frequency equation is obtained by employing the Rayleigh Ritz method. Influence on natural frequencies (NFs) is observed for various thickness of the middle layer. The characteristics beam functions are used to estimate the dependence of axial modal. Results are obtained for thickness to radius ratios and length to radius ratios for different edge conditions. The validity of this method is checked for numerous results in the open literature.
... Na slici (6) prikazani su oblici oscilovanja dobijeni primenom MDK i programa u Matlab-u, koji odgovaraju sopstvenim frekvencijama određenim u Tabeli 2. ) with F-F boundary conditions and given m= 1,2,3,4,5,6. The results obtained by the DSM based on the Flügge thin shell theory are compared with the exact results of Zhang [18], as well as those obtained by Abaqus. Xhang used the Goldenveizer-Novozhilov thin shell theory ( [19], [20]) and the state-space method to obtain the homogenous differential equations. ...
In this paper the dynamic stiffness method is used for free vibration analysis of a circular cylindrical shell. The dynamic stiffness matrix is formulated on the base of the exact solution for free vibration of a circular cylindrical shell according to the Flügge thin shell theory. The matrix is frequency dependent and, besides the stiffness, includes inertia and damping effects. The derived dynamic stiffness matrix is implemented in the code developed in a Matlab program for computing natural frequencies and mode shapes of a circular cylindrical shell. Several numerical examples are carried out. The obtained results are validated against the results obtained by using the commercial finite element program Abaqus as well as the available analytical solutions from the literature.
... Numerical methods have been widely used in modeling and analysis of buckling of cylindrical shells. In addition to the traditional methods, like finite differences, finite elements method (FEM), boundary element method (BEM), etc., differential quadrature (DQ) [16], discrete singular convolution (DSC) [17,18] and meshless method [19][20][21] have also gradually risen. However, the complicated modeling, tedious mathematical formulas, and/ or programming applicability existed in the above methods, cause inconvenience when used in practice. ...
In practice, the steel pipe-jacking can be regarded as a thin-walled cylindrical shell mainly subjected to jacking force in the axial direction and surrounded by the soil which is usually simplified and modeled as an elastic foundation. In this paper, the elastic buckling behavior of steel jacking pipes primarily under axial compression and with the Pasternak foundation is analyzed by the finite strip method (FSM). The elastic foundation is considered in the stiffness matrix through the strain energy, and the deformation in the longitudinal direction is simulated by the series functions in FSM. A parametric study is conducted to analyze buckling of cylindrical shells embedded in different elastic foundations. It indicates that the Pasternak foundation is more conducive to prevent buckling of cylindrical shells under axial compression. The critical length and the lower bound of buckling loads are obtained, and they offer the basis for optimal design of steel pipe-jacking. Finally, the case study combined with the buckling accident in the steel pipe-jacking event is presented. The present buckling analysis of soil-embedded cylindrical shells under axial compression provides design guidance for steel pipe-jacking construction.
... Zhang et al. [36] used the local adaptive differential quadrature method for the nonlinear vibrations study of simply supported, clamped and free cylindrical shells. The nonlinear equations of motion were formulated by means of the Goldenveizer-Novozhilov shell theory. ...
... Pellicano [11] conducted both theoretical and experimental analyses on linear and nonlinear vibration based on the Sanders-Koiter theory [22][23] for different boundary conditions; in this case, the analysis was also performed using numerical resolution techniques. Recently, further approaches to the problem were developed: Xing et al. [12], working from the Donnell-Mushtari theory [24], resolved the problem for different boundary conditions via the variables separation method associated with the Newton iterative method; moreover, both Xie et al. [13] and Zhang et al. [25] analysed different boundary conditions using the Goldenveizer-Novozhilov theory [26] but with different numerical approaches, the former used the Haar wavelet numerical method, while the latter used the local adaptive differential quadrature method. Khalili et al. [27] presented a formulation of 3D refined higher-order shear deformation theory for the free vibration analysis of simply supported-simply supported and clamped-clamped cylindrical shells and the solutions are obtained using the Galerkin numerical method. ...
... Many numerical methods have also been used for free vibration analysis of thin circular cylindrical shells in recent years. Zhang et al. [17] analyzed free vibration of circular cylindrical shells through a local adaptive differential quadrature method (LaDQM). ...
... Amabili [4] investigated largeamplitude vibrations of circular cylindrical shells with different boundary conditions and subjected to radial harmonic excitation in the spectral neighborhood of the lowest resonances. Zhang et al. [5] presented the numerical analysis of circular cylindrical shell by the local adaptive differential quadrature method which employed both localized interpolating basis functions and exterior grid points for boundary treatments. The governing equations of motion were formulated using the Goldenveizer-Novozhilov shell theory. ...
In this study a theoretical analysis for non-linear vibrations of buried gas-filled pipes caused by leakage has been investigated. We considered the pipe as a cylindrical shell buried in an isotropic, homogeneous elastic medium. The standard form of the Donnell’s non-linear shallow shell equation is used to model the free vibration of the pipe and the effects of surrounding medium on the pipe radial displacement is modeled by the potential function. The gas-structure interaction is analyzed by the Weaver-Unny model. By combination of these models and using the Galerkin method, the pipe wall radial displacement was obtained. A series of experiments were carried out on a steel pipe. The pipe was fitted with a continuous leak source positioned on the top of it and two ends of pipe were capped by plates. Two R15a sensors were mounted on the pipe. The pipe assembly was buried in a channel and sand was poured on it. The AE signals caused by leakage were recorded in large frequencies about 0–400 kHz. To study the accuracy of the present model, noises and unwanted signals were removed through the wavelet transform and then Fast Fourier Transform (FFT) was taken from theoretical and experimental results. The FFT results of theoretical model are compared with exciting experimental data and in all cases a good agreement is observed so that a percentage error less than 7% is resulted.
... Two types of the circumferential stiffeners were taken: outer ring and inner ring. (Zhang et al. 2006) presented the formulation and numerical analysis of circular cylindrical shells by the local adaptive differential quardrature method, which was applied to both localized interpolating basis functions and exterior grid points for boundary treatments. (Pellicano, 2007) presented a method for analyzing linear and nonlinear vibrations of circular cylindrical shells based on different end conditions. ...
In the current analysis vibration characteristics of a cylindrical shell composed of three layers are examined. This configuration is framed by three layers of different materials in thickness direction such that the inner and outer layers are of isotropic nature and functionally graded material is used for the middle layer. The shell is supported on Winkler and Pasternak foundations. Love shell equations are considered to study the vibration problem. The Winkler and Pasternak foundations are combined with the shell dynamical equations in the transverse direction. The present shell problem is solved by using wave propagation approach. A few comparisons of shell frequencies are done to verify the validity and accuracy of the present technique.
... In Section 6, the symbol N indicates the number of points per edge used for the current mesh. The global algebraic system can be computed by solving the eigenvalue problem (44) for the multi-domain case. It is worth noting that, the GDQ scheme is applied to each regular sub-domain in the same manner as in the single domain. ...
This paper provides a new technique for solving free vibration problems of composite arbitrarily shaped membranes by using Generalized Differential Quadrature Finite Element Method (GDQFEM). The proposed technique, also known as Multi-Domain Differential Quadrature (MDQ), is an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The multi-domain method can be directly applied to regular sub-domains of rectangular shape, as well as to elements of general shape when a coordinate transformation is considered. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the parent space, called computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is simple and straightforward. Computer investigations concerning a large number of membrane geometries have been carried out. GDQFEM results are compared with those presented in literature and a perfect agreement is observed. Membranes of complex geometry with a material inhomogeneity are also carefully examined. Numerical results referring to some new unpublished geometric shapes are reported to let comparisons with further research on this subject.
The main aim of this book is to analyze the mathematical
fundamentals and the main features of the Generalized
Differential Quadrature (GDQ) and Generalized Integral
Quadrature (GIQ) techniques. Furthermore, another interesting
aim of the present book is to shown that from the two numerical
techniques mentioned above it is possible to derive two different
approaches such as the Strong and Weak Finite Element
Methods (SFEM and WFEM), that will be used to solve various
structural problems and arbitrarily shaped structures.
A general approach to the Differential Quadrature is
proposed. The weighting coefficients for different basis
functions and grid distributions are determined. Furthermore,
the expressions of the principal approximating polynomials
and grid distributions, available in the literature, are shown.
Besides the classic orthogonal polynomials, a new class of basis
functions, which depend on the radial distance between the
discretization points, is presented. They are known as Radial
Basis Functions (or RBFs). The general expressions for the
derivative evaluation can be utilized in the local form to reduce
the computational cost. From this concept the Local Generalized
Differential Quadrature (LGDQ) method is derived.
The Generalized Integral Quadrature (GIQ) technique can be
used employing several basis functions, without any restriction
on the point distributions for the given definition domain.
To better underline these concepts some classical numerical
integration schemes are reported, such as the trapezoidal rule or
the Simpson method. An alternative approach based on Taylor
series is also illustrated to approximate integrals. This technique
is named as Generalized Taylor-based Integral Quadrature (GTIQ)
method.
The major structural theories for the analysis of the
mechanical behavior of various structures are presented in depth
in the book. In particular, the strong and weak formulations
of the corresponding governing equations are discussed and
illustrated. Generally speaking, two formulations of the same
system of governing equations can be developed, which are
respectively the strong and weak (or variational) formulations.
Once the governing equations that rule a generic structural
problem are obtained, together with the corresponding boundary
conditions, a differential system is written. In particular, the
Strong Formulation (SF) of the governing equations is obtained.
The differentiability requirement, instead, is reduced through
a weighted integral statement if the corresponding Weak
Formulation (WF) of the governing equations is developed. Thus,
an equivalent integral formulation is derived, starting directly
from the previous one. In particular, the formulation in hand is
obtained by introducing a Lagrangian approximation of the
degrees of freedom of the problem.
The need of studying arbitrarily shaped domains or
characterized by mechanical and geometrical discontinuities
leads to the development of new numerical approaches that
divide the structure in finite elements. Then, the strong form or
the weak form of the fundamental equations are solved inside
each element. The fundamental aspects of this technique,
which the author defined respectively Strong Formulation Finite
Element Method (SFEM) and Weak Formulation Finite Element
Method (WFEM), are presented in the book.
A numerical approach in applying the differential quadrature method (DQM) to the stability of cylindrical shells subjected to axial flow is presented. Donnell–Mushtari shell equation and unsteady Bernoulli’s equation are respectively applied to model the shell motion and solve the fluid force. DQM discretizes the shell equation and its boundary conditions. Lagrange interpolation and trigonometric series are applied to solve the fluid force at the non-uniform grid points. Then the fluid–structure interaction equation based on DQM is established. The modal and fluid-induced instability analysis is carried out by eigenvalue analysis. The accuracy of the present method is verified via comparison with other theories and commercial software. The relationship between the shell frequencies and critical flow speeds is found, then a formula for predicting the critical flow speed with comprehensive accuracy validation is first derived. It can predict the critical flow speed using the shell’s frequencies in air and fluid but not through the traditional eigenvalue-based analysis. A brief theoretical explanation of this formula is given by the analysis of the principal mass and stiffness of the system. It may have great potential application for a vibrating structure with closed boundaries in axial flow.
In pipe jacking projects, there are several factors that could influence construction efficiency. Pipe dimensions are one of the main ones that affect the cost and safety of pipe jacking operations. However, the main factors influencing pipeline stability and the maximum allowable jacking length, which usually is called the critical jacking length, are rarely explored. In this paper, three-dimensional finite-element models were conducted to investigate the factors influencing the critical jacking length during pipe jacking, including pipe dimensions, soil specifications, buried depth, and overcut values using commercially available software. The critical jacking length of a steel pipe in an alluvial soil was estimated using the displacement control method, and the probable failure modes of steel pipes were determined. Two cases were selected to compare the pipeline stresses with field-measured data to validate the numerical results. The results suggested the developed model could estimate the critical jacking length based on the site characteristics and pipeline specifications, and a critical jacking length surface could be presented for each site based on the site properties, which would be beneficial to select required pipe segment dimensions at the preliminary design steps
A dual-functional gradient carbon nanotube-reinforced composite(DFG-CNTRC) in which both the matrix and the carbon nanotubes(CNTs) are assumed to be functionally graded is proposed. The matrix is the metal–ceramic functionally graded material(FGM) whose properties change gradually along the thickness direction, aiming to further improve the mechanical properties and natural frequencies of the functionally graded structures. By adopting orthogonal polynomials to expand the shell displacement fields and the Rayleigh–Ritz method is used to derive the vibration differential equation. Then, based on the first-order shear deformation theory(FSDT), Donnell-type kinematic assumptions, and artificial spring technique, a general model for analyzing the free vibration of rotating DFG-CNTRC laminated shells with arbitrary boundary conditions is provided. The effects of rotating speed, CNTs’ volume fraction, matrix’ change and the middle layer’s thickness on the traveling waves are discussed, and the modal characteristics of DFG-CNTRC laminated shells are analyzed.
Free vibration behavior of a composite shell/panel with and without a central square cutout has been studied using Multi-domain Generalized Differential Quadrature (GDQ) method. A physical domain is decomposed into several elements in such a way that all the elements have uniform thickness and material properties, as well as continuous loading and boundary conditions at their edges. The governing equations are derived based on the first-order shear deformation theory. They are formulated in a general form and can be converted to Donnell’s, Love’s, and Sanders’ theories. In addition, compatibility conditions are considered at the interface boundaries of adjacent elements as well as proper boundary conditions on other elements‘ edges. The GDQ method is employed to discretize these equations in both longitudinal and circumferential directions. By assembling these discretized relations, a system of algebraic equations will be generated. These equations can be solved through an eigenvalue solution to compute the natural frequencies of the whole shell/panel. Numerical results obtained by the presented method are compared with ABAQUS results and those available in the literature. After verifying the accuracy and precision of the proposed method, it is employed to study the effect of an opening on the vibrational behavior of composite shells and panels through a parametric study. The influence of the presence of a cutout and its size is investigated on shells/panels with various material properties, layup orientations, thicknesses, dimensions, and boundary conditions. The obtained results can be used as a benchmark for further researches.
Exact analytical solutions for free vibration of isotropic and orthotropic cylindrical shells with uniform and stepped thickness subject to general boundary conditions are presented by means of a symplectic analytical approach. The Reissner shell theory is adopted to formulate a theoretical model. By introducing a Hamiltonian system, the governing higher order partial differential equation is reduced to a set of ordinary differential equations which can be analytically solved by separating the variables. Applying the end boundary and interface continuous conditions, a set of analytical characteristic frequency equations are obtained, and exact solutions can be determined. To ensure accuracy and validity of the symplectic method, the analytical solutions for uniform and stepped cylindrical shells with isotropic or orthotropic material properties and with arbitrary boundary conditions are compared with available published data and FEM solution. A set of comprehensive new result for orthotropic stepped cylindrical shells under classical and elastic restraints are presented. Some typical mode shapes for various examples are illustrated. In addition, the effects of boundary conditions and orthotropic properties on vibration frequency are analyzed. The result shows that the fundamental frequency is higher for a boundary with more constraints.
This work studies the performance of composite solar cell model consisting of perovskite absorber layer (CH3NH3PbI3), electron transport layer (TiO2 or PCBM), and hole transport layer (Spiro-OMeTAD or CuI) via sinc and discrete singular convolution quadrature techniques. The governing equations for hole transport layer, electron transport layer, and absorber layer (perovskite) are derived depending on the equations of continuity and Poisson. Different quadrature and block-marching techniques convert these equations to the nonlinear algebraic system. Then, we apply the iterative method to solve the problem of nonlinearity. For each scheme, a programmed code is designed to get a numerical solution for composite perovskite solar cells by MATLAB. To ensure the efficiency and convergence of these schemes, the obtained results match with previous experiments and other numerical schemes. From these comparisons, the discrete singular convolution quadrature technique gives high accuracy, speed of convergence, and more reliability than other methods with error up to 10⁻⁸. Consequently, the effects of different thicknesses, mobilities, band gaps, temperatures, absorption factors, wavelengths, and doping concentrations on current density, open-circuit voltage, and efficiency of solar cells are investigated in a full parametric study. Hence, the obtained effects of the current schemes increase the efficiency of perovskite solar cells.
Novelty Statement
Sinc and discrete singular convolution quadrature techniques are applied to reach high performance for perovskite absorber layer sandwiched between electron transport layer (ETL) and hole transport layer (HTL). The study demonstrates that the discrete singular convolution quadrature method gives high efficiency and accurate results at different parameters. It is obtained that the power conversion efficiency is approximately 35% at specific parameters.
Convolution and indirect meshless techniques are presented to solve the free vibration problem of irregular composite plates surrounded by a nonlinear elastic matrix with three parameters. Linear and parabolic foundations are examined. The displacement field is derived based on a first order transverse shear theory. In vibration study, the governing equation is reduced for the solution of the nonlinear eigenvalue problem and is solved by a direct iterative method. The influences of variable thickness, foundation stiffness, and supporting conditions on frequency values and mode shapes are investigated. The accuracy and efficiency of the two methods are carried out in two ways, firstly by comparison with exact, classical quadrature, and Rayleigh-Ritz approaches, and secondly by calculated the CPU time. Furthermore, some detailed results are presented to explore the effects of elastic and geometric properties of the vibrated irregular plate. From the results, it is found that the indirect moving least squares quadrature technique is an efficient scheme for vibration plates containing material discontinuity.
The size-dependent sound transmission loss problem of an air-filled functionally graded material cylindrical shell subjected to a plane progressive sound wave with even and uneven porosity distributions was analytically studied using a nonlocal strain gradient and the first-order shear deformation theories. To simulate the heterogeneous material, the impressive material properties were supposed to be associated with the porosity volume fraction model, based on a power-law model. They were suggested to be constantly changeable along the thickness direction. The motion equations were extracted by Hamilton’s principle and then solved using the Fourier-Bessel series. The accuracy of the obtained formulation were strictly proved by comparing the data accessible in the literature. Parameter studies reveal the effects of material gradient index, size scale factors, porosity volume fraction, and incident angles on the variations in the amplitude of sound transmission loss through the nanoshell.
Free vibration characteristics of thick skew plates reinforced by functionally graded carbon nanotubes (CNTs) reinforced composite are presented. Discrete singular convolution (DSC) method is used for the numerical solution of vibration problems via geometric mapping technique. Using the geometric transformation via a four‐node element, the straight‐sided quadrilateral physical domain is mapped into a square domain in the computational space. Then the method of discrete singular convolution with some singular kernels such as Regularized Shannon's delta (RSD) and Lagrange's delta kernels (LDK) have been used for spatial discretizing of the resulting governing equation of motion. Calculated results have been presented in order to show the effects of volume fraction of CNT, skew angles, CNT distribution types, plate aspect ratio and length‐to‐thickness ratio on the frequency of CNT reinforced skew plate. The current results are compared with the related results available in the literature and obtained by different methods. It is shown that reasonable accurate results are obtained for free vibration of nanocomposite plates with less computational effort for higher modes. Several test examples for different plate have been selected to demonstrate the convergence properties, accuracy, and simplicity in numerical implementation of DSC procedures. This approach has verified the accuracy and applicability of DSC method to the class of problem considered in this study. Furthermore, in the numerical examples in the scope of the study, the results obtained with DSC method using a coarser grid are more accurate than the values obtained by finite elements and differential quadrature (DQ) methods. It is also revealed that the method of discrete singular convolution is a promising and potential approach for computational mechanics of non‐rectangular plates with nanocomposite reinforced. This article is protected by copyright. All rights reserved.
A porous material contains a structure in which its density reduces when the volume increases due to voids. The high strength, low weight and absorption of the sound or impact convert the porous materials desirable with vast applications to different fields of science and technology. In this paper, an analytical method is proposed for investigating the vibrations behavior of thick porous cylinders for various boundary conditions. The porosity variation is function of the thickness as symmetric, asymmetric or uniform. For mathematical modeling, the first-order shear deformation theory is used as displacement field, by considering the transverse normal strain effect. Hamilton’s principle in conjunction with the linear kinematic relations and Biot constitutive equations are employed to extract the motion equations. The governing equations contain four coupled partial differential equations. These equations are solved analytically and the natural frequencies and mode shapes are determined. A parametric study is performed and the effect of the materials and mechanical properties is studied for different boundary conditions. The results are compared with the finite element methods and the available results in the literature.
Most of the research on truncated conical shells focuses on the dynamics with classical boundary conditions. However, less attention has been given to the non-classical boundary conditions due to the lack of a unified displacement function, such as a boundary with point constraint or partial constraint, or a boundary with added mass, which exists in engineering practice. In this study, we investigate the free vibration of the truncated conical shell with arbitrary boundary conditions, including elastic and inertia force constraints. The equations of motion with elastic boundary constraints are formulated by employing Hamilton's principle and the thin-walled shallow shell theory of the Donnell type. The solutions of the shells are obtained using Fourier series in circumferential directions and power series in meridional directions, with various boundary conditions achieved through the choice of stiffness, which is a unified solution procedure. The procedure proposed was validated by comparing the results obtained with results available in the literature on classical boundary conditions and with the finite element method results for the non-classical boundary conditions. Numerical simulations were carried out to illustrate the sensitivity of the shell frequency to the stiffness parameters and the moment of inertia effect on frequency, and to present the circumferential modal jumping phenomena with patterns.
In this paper, the free vibration of viscoelastic nanotube under longitudinal magnetic field is investigated. The governing equation is formulated by utilizing Timoshenko beam model and Kelvin-Voigt model based on the nonlocal strain gradient theory. The local adaptive differential quadrature method (LADQM) is applied in the analyzing procedure. We also investigated the influences of the nonlocal parameter, structural damping coefficient, material length scale parameter and the longitudinal magnetic field on the natural frequencies of the system. The results of this research may be helpful for understanding the potential applications of nanotubes in Nano-Electromechanical System.
Combining Goldenveizer-Novozhilov shell theory, thin plate theory and electro-elastic surface theory, the size-dependent vibration of nano-sized piezoelectric double-shell structures under simply supported boundary condition is presented, and the surface energy effect on the natural frequencies is discussed. The displacement components of the cylindrical nano-shells and annular nano-plates are expanded as the superposition of standard Fourier series based on Hamilton's principle. The total stresses with consideration of surface energy effect are derived, and the total energy function is obtained by using Rayleigh-Ritz energy method. The free vibration equation is solved, and the natural frequency is analyzed. In numerical examples, it is found that the surface elastic constant, piezoelectric constant and surface residual stress show different effects on the natural frequencies. The effect of surface piezoelectric constant is the maximum. The effect of dimensions of the double-shell under different surface material properties is also examined.
This paper presents a new technique based on the Ritz method for the damage detection of circular cylindrical shell structures. Sander's thin shell theory together with the Ritz method is used to analyse the dynamic behaviour of circular cylindrical shells. The crack damage on the shell surface is modelled by a rotational line spring along the circumference of the shell. Different damage scenarios are investigated by changing the crack locations and rotational spring stiffness. Modal parameters of shells with different damage patterns are obtained and compared. Wavelet analysis is carried out to detect the discontinuities in the mode shape where the damage is presented. It is found from the numerical results that the natural frequencies of the shell are insensitive to the crack damage. The wavelet analysis is effective to detect the damage in the circular cylindrical shell.
Free vibrations of the laminated composite cylindrical shell with clamped edges are considered in this paper. Equations of the theory of laminated shells taking into account the average transverse shear strains are employed for the vibration analysis. A solution of the equations of motion of the shell is based on the Fourier decomposition and the Galerkin method and yields an analytical formula for the calculation of a fundamental frequency. Applications of this formula to the determination of the fundamental frequencies for the filament-wound composite cylindrical shells are demonstrated using numerical examples. The calculations have been verified by comparison with a finite-element solution. It has been shown that the analytical formula presented in this paper provides an efficient means for rapid and reliable calculation of the fundamental frequency which can be used for the assessment of the structural stiffness of the shells in the design analysis.
Natural frequency characteristics of a thin-walled multiple layered cylindrical shell under lateral pressure are studied. The multiple layered cylindrical shell configuration is formed by three layers of isotropic material where the inner and outer layers are stainless steel and the middle layer is aluminum. The multiple layered shell equations with lateral pressure are established based on Love’s shell theory. The governing equations of motion with lateral pressure are employed by using energy functional and applying the Ritz method. The boundary conditions represented by end conditions of the multiple layered cylindrical shell are simply supported-clamped(SS-C), free-clamped(F-C) and simply supported-free(SS-F). The influence of different lateral pressures, different thickness to radius ratios, different length to radius ratios and effect of the asymmetric boundary conditions on natural frequency characteristics are studied. It is shown that the lateral pressure has effect on the natural frequency of multiple layered cylindrical shell and causes the natural frequency to increase. The natural frequency of the developed multilayered cylindrical shell is validated by comparing with those in the literature. The proposed research provides an effective approach for vibration analysis shell structures subjected to lateral pressure with an energy method.
A general approach is presented for the vibration studies of rotating cylindrical shells having arbitrary edges. The present analysis is based on the Sanders' shell theory and the effects of centrifugal and Coriolis forces as well as initial hoop tension due to rotating are all taken into account. By taking the characteristic orthogonal polynomial series as the admissible functions, the Rayleigh–Ritz method is employed to derive the frequency equations of rotating cylinders with classical homogeneous boundary conditions. Further, utilizing artificial springs to simulate the elastic constraints imposed on the cylinders’ edges, one can derive the frequency equations of rotating cylindrical shells with more general boundary conditions by considering the strain energy of artificial springs during the Rayleigh–Ritz procedure. To validate the approach proposed in this paper, a series of comparison and convergence studies are performed and the investigations demonstrate high accuracy and low computational cost of the present approach. Finally, some further numerical results are given to illustrate the influence of the variations of spring stiffness on the frequencies of rotating cylinders.
This paper introduces a discrete singular convolution algorithm for solving the Fokker–Planck equation. Singular kernels of the Hilbert-type and the delta type are presented for numerical computations. Various sequences of approximations to the singular kernels are discussed. A numerical algorithm is proposed to incorporate the approximation kernels for physical applications. Three standard problems, the Lorentz Fokker–Planck equation, the bistable model and the Henon–Heiles system, are utilized to test the accuracy, reliability, and speed of convergency of the present approach. All results are in excellent agreement with those of previous methods in the field.
Numerical solution of high-order differential equations with multi-boundary conditions is discussed in this paper. Motivated by the discrete singular convolution algorithm, the use of fictitious points as additional unknowns is proposed in the implementation of locally supported Lagrange polynomials. The proposed method can be regarded as a local adaptive differential quadrature method. Two examples, an eigenvalue problem and a boundary-value problem, which are governed by a sixth-order differential equation and an eighth-order differential equation, respectively, are employed to illustrate the proposed method.
In this paper, free vibrations of laminated composite cylindrical shells are investigated by the global method of generalized differential quadrature (GDQ). The GDQ method was developed to improve the differential quadrature (DQ) method for the computation of weighting coefficients. The differential equations of motion are formulated using Love's first approximation classical shell theory. The spatial derivatives in both the governing equations and the boundary conditions are discretized by the GDQ method. The GDQ method is examined by comparing its results with those available in the literature. It is demonstrated that, with the use of the GDQ method, the natural frequencies can be easily and accurately obtained by using a considerably small number of grid points.
Starting from Flügge's three equations of motion for a uniform thin cylindrical shell, the paper gives a general solution, from which the dependence of natural frequencies on shell dimensions and mode number can be investigated for any end conditions. This solution requires the assumption of a natural frequency and the determination of the corresponding shell length for the prescribed end conditions. Numerical results are given for shells with clamped ends and for shells with free ends; the variation of frequency factor and of mode shape with dimensional and mode parameters is shown and the accuracy of approximate theories assessed.
Based on the same concept as generalized differential quadrature (GDQ), the method of Fourier expansion-based differential quadrature (FDQ) was developed and then applied to solve the Helmholtz eigenvalue problems with periodic and non-periodic boundary conditions. In FDQ, the solution of a partial differential equation is approximated by a Fourier series expansion. The details of the FDQ method and its implementation to sample problems are shown in this paper. It was found that the FDQ results are very accurate for the Helmholtz eigenvalue problems even though very few grid points are used. ©1997 John Wiley & Sons, Ltd.
This paper presents new exact solutions for vibration of thin circular cylindrical shells with intermediate ring supports, based on the Goldenveizer–Novozhilov shell theory (Theory of thin shells; The theory of thin elastic shells). An analytical method is proposed to study the vibration behaviour of the ring supported cylindrical shells. In the proposed method, the state-space technique is employed to derive the homogenous differential equation system for a shell segment and a domain decomposition approach is developed to cater for the continuity requirements between shell segments. Exact frequency parameters are presented in tables and design charts for circular cylindrical shells having multiple intermediate ring supports and various combinations of end support conditions. These exact vibration frequencies may serve as important benchmark values for researchers to validate their numerical methods for such circular cylindrical shell problems.
Considerable attention has been devoted in recent years to the use of shell equations for the prediction of the dynamic behavior of thin cylindrical shells as an element in a missile or spacecraft. The complexity involved in the use of the shell equations must be tolerated for problems that require modes having several circumferential waves (e.g., prediction of panel flutter or response to acoustic loading). Moreover the minimum natural frequency usually corresponds to a mode having two or more circumferential waves. For prediction of overall vehicle behavior, however, one is primarily interested in axisymmetric (n = 0) and beamtype (n = 1) modes; in these instances the problem can be considerably simplified by considering the cylinder as a bar for n = 0 modes or a compact beam for n = 1 modes. The present paper examines the accuracy of these engineering approximations as compared with exact solutions from Flugge's shell equations and discusses the error in terms of frequency, mode shape, modal forces, and generalized mass. Consideration is given to the effect of shell bending stiffness and the influence of boundary conditions on these parameters. © 1969 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.
The method of differential quadrature is demonstrated by solving the two-dimensional Poisson equation. The results for three test problems are compared with the exact analytical solutions and the numerical solutions obtained by others for the Galerkin, the control-volume and the five-point finite difference methods. The method of differential quadrature leads to more accurate results for comparable levels of computational effort.
The numerical technique of differential quadrature for the solution of linear and non-linear partial differential equations, first introduced by Bellman and his associates, is applied to the equations governing the deflection and buckling behaviour of one- and two-dimensional structural components. Separate transformations are used for higher-order derivatives, as suggested by Mingle, thus extending the method to treat fourth-order equations and to include multiple, boundary conditions in the respective co-ordinate directions. Results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods. The application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort.
A global method of generalized differential quadrature is applied to solve the two-dimensional incompressible Navier-Stokes equations in the vorticity-stream-function formulation. Numerical results for the flow past a circular cylinder were obtained using just a few grid points. A good agreement is found with the experimental data.
This paper presents a novel computational approach, the discrete singular convolution (DSC) algorithm, for analysing plate structures. The basic philosophy behind the DSC algorithm for the approximation of functions and their derivatives is studied. Approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels. Unified features of the DSC algorithm for solving differential equations are explored. It is demonstrated that different methods of implementation for the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. The use of the algorithm for the vibration analysis of plates with internal supports is discussed. Detailed formulation is given to the treatment of different plate boundary conditions, including simply supported, elastically supported and clamped edges. This work paves the way for applying the DSC approach in the following paper to plates with complex support conditions, which have not been fully addressed in the literature yet. Copyright © 2002 John Wiley & Sons, Ltd.
Based on the work of generalized differential quadrature (GDQ), a global method of generalized integral quadrature (GIQ) is developed in this paper for approximating an integral of a function over a part of the closed domain. GIQ approximates the integral of a function over the part of the whole closed domain by a linear combination of all the functional values in the whole domain with higher order of accuracy. The weighting coefficients of GIQ can be easily determined from those of GDQ. Applications of GDQ and GIQ to solve boundary layer equations demonstrated that accurate numerical results can be obtained using just a few grid points.
The methodology and numerical solution of problems concerning transport processes via the method of differential quadrature are presented. Application of the method is demonstrated by solving a simple one-dimensional, time-dependent (transient) diffusion process involving an irreversible reaction without any flux across the end boundary. In addition, the same technique is used (for the first time to the authors' knowledge) to solve a steady-state problem. For this purpose, a convection-diffusion problem involving an irreversible reaction is considered. The demonstration is carried out in two ways, (1) using the Bellman et al. technique which employs approximation formulas for higher order partial derivatives derived by iterating the linear quadrature approximation for the first order partial derivative, and (2) using individual quadratures to approximate the partial derivatives of first, as well as higher orders, as suggested by Mingle. Both approaches give the same results; however, the latter saves an appreciable amount of iterative computing effort despite the fact that it requires separate weighting coefficients for each individual quadrature. Since the technique of differential quadrature can produce solutions with sufficient accuracy even when using as few as three discrete points, both the programming task and computational effort are alleviated considerably. For these reasons the differential quadrature approach appears to be very practical in solving a variety of problems related to transport phenomena.
This paper deals with the treatment of mixed and nonuniform boundary conditions in the global method of generalized differential quadrature (GDQ) for the vibration analysis of rectangular plates. The discretized mixed and nonuniform boundary conditions are directly substituted into the governing equations in order to derive the eigenvalue equation system. The present approach overcomes the drawbacks of previous approaches such as the δ-technique in treating the boundary conditions. The accuracy and efficiency of the present approach are studied by comparing its solutions with available results. Some new solutions for plates with nonuniform and mixed boundary conditions are presented, as well as details of the GDQ method.
The technique of differential quadrature (DQ) for the solution of a partial differential equation is extended and generalized. The general formulation for determining the weighting coefficients of the first order derivative is obtained, and a recurrence relationship for determining the weighting coefficients of the second and higher order partial derivatives is achieved. For parallel computation, the multi-domain GDQ scheme was also developed, and successfully applied to the solution of the incompressible Navier-Stokes (N-S) equations. Numerical examples include the flow past a backward facing step, the flow past a square step, and driven cavity flow. For the driven cavity flow problem, two formulations of the N-S equations (vorticity-stream function and vorticity-velocity) and three methods for dealing with the interface between subdomains (i.e. patched by enforcing continuity to the function and its normal derivative; patched by using a Lagrange interpolation scheme; and overlapped) were studied comparatively. In addition, an attempt to develop a general code which can be run on any array of processors without modification to the program was discussed, and then successfully applied to the driven cavity flow problem.
A general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type. The solution is obtained through a direct solution procedure in which Sanders' shell equations are used with the axial modal displacements represented as simple Fourier series expressions. Stokes' transformation is exploited to obtain correct series expressions for the derivatives of the Fourier series. An explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions. The accuracy of the method is checked against available data. The method is used to find the modal characteristics of the thermal liner model of the U.S. Fast Test Reactor (FTP). The numerical results obtained are compared with finite element method solutions.
The numerical solution of nonlinear partial differential equations plays a prominent role in numerical weather forecasting, optimal control theory, radiative transfer, and many other areas of physics, engineering, and biology. In many cases all that is desired is a moderately accurate solution at a few points which can be calculated rapidly.In this paper we wish to present a simple direct technique which can be applied in a large number of cases to circumvent the difficulties of programming complex algorithms for the computer, as well as excessive use of storage and computer time. We illustrate this technique with the solution of some partial differential equations arising in various simplified models of fluid flow and turbulence.
Numerical solutions to transient nonlinear diffusion problems are obtained by the method of differential quadrature. The accuracy of the solutions is inferred by comparison with the analytical solution for the linear case. The particular problems associated with general boundary conditions are handled by the use of integral methods. Examples from heat diffusion include composite media and radiation-enhanced conduction.
This communication introduces a novel scheme for the treatment of free edge supports in the analysis of beams by using the discrete singular convolution (DSC) algorithm. Accommodating free edges has been a challenging issue in the DSC analysis of beams, plates, and shells. An iteratively matched boundary (IMB) method is proposed to overcome the difficulty. Numerical experiments are carried out to demonstrate that the proposed IMB method works very well in dealing with arbitrary combinations of beam edge supports.
This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is demonstrated that different implementations of the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. Three benchmark stochastic systems, the repulsive Wong process, the Black-Scholes equation and a genuine nonlinear model, are employed to illustrate the robustness and to test accuracy of the present approach for the solution of the Fokker-Planck equation via a time-dependent method. An additional example, the incompressible Euler equation, is used to further validate the present approach for more difficult problems. Numerical results indicate that the present unified approach is robust and accurate for solving the Fokker-Planck equation. Comment: 19 pages
Application of generalized differential quad-rature to solve two-dimensions incompressible Navier-stress equa-tions
- C Shu
- Richard
- Be
Shu C, Richard BE. Application of generalized differential quad-rature to solve two-dimensions incompressible Navier-stress equa-tions. International Journal of Numerical Methods in Fluids 1992;15: 791–8.
Vibration of shells. NASA SP-288
- A W Leissa
Leissa AW. Vibration of shells. NASA SP-288, 1973.