Free vibration analysis of truncated circular conical shells with variable thickness using the Haar wavelet method

Frequency analysis of rotating truncated conical shells using the Haar wavelet method

In this paper, we present a method using Haar wavelets for solving axially functionally graded (FG) Timoshenko beam equations with non-uniform cross-sections. We compare two different approaches to the solution. The first approach involves approximating the resulting function (i.e., rotation) of the differential equation with a polynomial function using Haar wavelets, which is a classical application in the Haar wavelet method. The second method employs an auxiliary function that uses Haar wavelets, but the rotation or deflection do not directly equal the sought function. The rotation and deflection are derived from this auxiliary function. In both methods, the coupled governing equations are transformed into a single governing equation. Using the Haar wavelet method, this single differential equation transforms into a system of linear algebraic equations. For different boundary conditions, the roots of the characteristic polynomial equation obtained from the system of linear algebraic equations are solved to determine the lowest to highest-order natural frequencies. Our results show that the use of auxiliary function is faster and more consistent with the results available in the literature. We presented several natural frequency predictions of Timoshenko beams with different taper ratios and support conditions that have not been detailed in the literature before.

In this research, the free vibration analysis of lap joint laminated conical shell connected to laminated conical shell with adhesive layer for various boundary conditions are investigated. Based on the first-order shear deformation theory (FSDT), the main field equations of the analytical model are derived and the governing differential equations of laminated conical shell to laminated conical shell adherends lap joint are obtained for homogeneous and elasticity adhesive layers by using Hamilton principle method. Then, based on the generalized differential quadrature method (GDQM), the equilibrium equations of the structure adherends are investigated. The influence of the boundary conditions, the type of the composite materials and adhesive layer, the circumferential wave number, the length to radius ratio of the shell, and the length of overlap to length of shell ratio are studied. Furthermore, the thickness to radius ratio and the thickness of adhesive to thickness of shell ratio, as well as the various cone angles of conical shell on the natural frequency of laminated conical adherends shells connected to the conical shells with adhesive layer are investigated. For validation of the numerical results, the results are compared with the previous research and the comparisons show very good agreement. The numerical results show that with the increase of the internal shell and external shell of semi-angle, the non-dimensional frequency of lap joint connection of two laminated conical shells with adhesive layer decreased and increased, respectively. Furthermore, with the increase of the adhesive thickness to conical shell thickness ratio and lap joint length to length of conical shell ratio, the dimensionless natural frequency of the structure increased.

PurposeThe purpose of this paper is the comparative analysis of Haar Wavelet Method and Optimal Homotopy Asymptotic Method for fractional Fisher type equation. In this paper, two reliable techniques, Haar wavelet method and optimal homotopy asymptotic method (OHAM), have been presented. The Haar wavelet method is an efficient numerical method for the numerical solution of fractional order partial differential equation like the Fisher type. The approximate solutions of the fractional Fisher-type equation are compared with those of OHAM and with the exact solutions. Comparisons between the obtained solutions with the exact solutions exhibit that both the featured methods are effective and efficient in solving nonlinear problems. However, the results indicate that OHAM provides more accurate value than the Haar wavelet method.Design/methodology/approachComparisons between the solutions obtained by the Haar wavelet method and OHAM with the exact solutions exhibit that both featured methods are effective and efficient in solving nonlinear problems.FindingsThe comparative results indicate that OHAM provides a more accurate value than the Haar wavelet method.Originality/valueIn this paper, two reliable techniques, the Haar wavelet method and OHAM, have been proposed for solving nonlinear fractional partial differential equation, i.e. fractional Fisher-type equation. The proposed novel methods are well suited for only nonlinear fractional partial differential equations. It also exhibits that the proposed method is a very efficient and powerful technique in finding the solutions for the nonlinear time fractional differential equations. The main significance of the proposed method is that it requires less amount of computational overhead in comparison to other numerical and analytical approximate methods. The application of the proposed methods for the solutions of time fractional Fisher-type equations satisfactorily justifies its simplicity and efficiency.

The essential objective of this work is to implement Adam Bashforth's Moulton (ABM) and Haar wavelet method (HWM) to solve fractional chemical kinetics and another problem that relates the condensations of carbon dioxide (CO2) and phenyl glycidyl ether (PGE) with two variety of Drichlet and a mixed set of Neumann boundary and Drichlet type conditions respectively. We have been solved the above system of differential equations by Adam Bashforth's Moulton and Haar wavelet operational method where this technique is to convert the system of differential equations into the system of algebraic equation which can be solved easily. This work is expects to contribute the vast advantage of Haar wavelets in chemical science. The Adam Bashforth's Moulton and Haar wavelet method is impressive and convenient for obtaining numerical solutions of chemical engineering type problems. A complete agreement is acheived between Adam Bashforth's Moulton solution and Haar wavelet solution. To manifest about the performance and applicability of the method, two test examples are deliberated.

Free vibration analysis of cylindrical shells using the Haar wavelet method

Flowers are one of the plants that offer a beauty that makes the environment more beautiful and attractive. Not only a decoration, but flowers are also used as cut flowers, sowing flowers and herbal medicine. flowers that grow in Indonesia have two forms, namely the form of a flower and a bunch. In the classification of flower characteristics taken from the flower, images are characteristics of colour, texture and shape. The purpose of this system design is to make a classification of flowers that grow in Indonesia that have different colour, texture and shape characters using the Histogram, Haar Wavelet and Robert Cross methods to get the value of features to be made vector and used on SVM method to determine the types of flowers. SVM is a classifier that has the advantage of being able to process high-dimensional data, without eliminating significant performance degradation. From the experiment of this Indonesian flower classification application, the accuracy of classification success results in the test interest of 81.67% by using the colour, texture and shape features of the extraction results with the Histogram, Haar Wavelet and Robert Cross methods and classified by the SVM method with Linear, Polynomial kernel and RBF.

In this article, a technique called Haar wavelet-Picard technique is proposed to get the numerical solutions of nonlinear differential equations of fractional order. Picard iteration is used to linearize the nonlinear fractional order differential equations and then Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of Picard iteration, solution is updated by the Haar wavelet method. The results are compared with the exact solution.   Key words: Fractional differential equations, Wavelet analysis, Caputo derivative, Haar wavelets, Picard iteration.

Recently, wavelet analysis or wavelet transform has been developed as a mathematical instrument for many problems. In the present paper, the numerical technique is based on Haar wavelet collocation points and operational matrices over a new interval [-ϑ,ϑ]. So, we have established a new operational matrix of integration the Haar wavelets functions for a chosen domain from -ϑ to 0. The operational matrices of integration and product are utilized to convert the linear ordinary differential equations together with variable coefficients into algebraic matrix equation which can be resolved by MATLAB. Three numerical examples have been displayed which involving first order differential equations including variable coefficients. The results illustrate that the suggested approach is completely reasonable in comparison with correct solution. Moreover, the exactness of the variable can be enhanced by increasing the Haar wavelets resolution.

On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation

Effect of boundary condition and variable shell thickness on the vibration behavior of grid-stiffened composite conical shells

In this paper, the axisymmetric free vibrations of truncated conical shells are analyzed by means of an improved shell theory. The equations of vibration and the boundary conditions are obtained from the stational condition of Lagrangian of the shells. The equations of vibration are solved exactly by a solution in series for a conical shell with linearly varied thickness along its axis and the effects of boundary conditions, thickness and semi-vertex angle on the natural frequencies are investigated. As a special case of the conical shell, we compare this theory with Mindlin's circular plate theory and with Mirsky's cylindrical shell theory.

A comparative analysis of the dynamic stability of cylindrical and conical shells with the same geometric and mechanical characteristics under periodic uniformly distributed axial compression was presented. The study of the stability of steady periodic vibrations of thin elastic shells was based on the joint use of the method of curvilinear grids, the projection method and the parameter continuation method combined with the Newton–Kantorovich method. Geometrically nonlinear relations of the thin elastic shells theory are formulated on the basis of the vector approximation of the displacements function in the general curvilinear coordinate system in tensor form and satisfy the Kirchhoff-Love hypothesis. The discretization of the differential equations of the steady forced vibrations in the direction of the generating shells using the method of curvilinear grids was carried out. The components of the elements displacement vectors of the shells middle surface in the circular directionare approximated by trigonometric series. Reduction of the number of generalized coordinates of the discrete dynamic model of shells steady forced vibrations was performed by the Bubnov-Galerkin basis reduction method. A transition from vector ordinary differential equations to a nonlinear system of algebraic equations was made. The construction of a mathematical model of the dynamic stability of steady forced nonlinear vibrations of thin elastic shells was performed according to Floquet's theory using the projection method. The criterion for the loss of stability was the equality to zero of the determinant of the matrix of linearized equations of steady forced nonlinear vibrations of shells according to the Lyapunov theorem. A comparative analysis of frequencies and modes of natural vibrations of cylindrical and conical shells with the same geometric and mechanical characteristics and boundary conditions was performed. Nonlinear steady vibrations of the shells due to periodic axial compression were studied. The critical values of the dynamic load and the corresponding forms of loss of shell stability in the range of lower frequencies of their natural vibrations were obtained.

Augmenting meting and solidification rates of latent heat storage unit (LHSU) is very much essential for its efficient operation. By the effective utilization of natural convection, rate of heat transfer can be enhanced and the conical shell is beneficent in this regard. Using fins further improves the charging and discharging rates. Hence, the current study is focused on analyzing melting and solidification characteristics of a conical shell and tube LHSU along with the effect of fin parameters viz., fin diameter and number of fins. Numerical analysis is chosen for this purpose and the performance is compared via melting/solidification times, energy stored, energy/exergy efficiencies. Initially, the performance of unfinned conical shell is compared with the cylindrical shell without fins and then the effect of fin parameters is presented. For melting process conical shell is found to be superior to cylindrical shell. 34.46% reduction in melting time is noted by using conical shell and rate of energy stored is also higher for conical shell. Increase in fin diameter caused an increase in melting time when 20 number of fins are used, whereas melting time got decreased with the increase in fin diameter when five number of fins are used. Hence, when a greater number of fins are used, lesser diameter is preferred for melting. For discharging process, conical shell took 60% more time than cylindrical shell. Even after using fins, solidification time is not drastically reduced in comparison with cylindrical shell.

为研究超声速轴向流中完全锥壳的气动弹性非线性响应，本文采用活塞理论计算超声速气动力，利用截锥壳的非线性气动弹性运动方程，在截锥壳顶角不为零，小半径无限趋近于零的特定边界条件下，采用一维微分求积法(DQM)离散，数值逼近求解了完全锥壳的颤振临界动压及气动弹性响应。结果表明，不同边界条件对完全锥壳气动颤振临界动压和极限环幅值的影响不大，最小颤振临界对应周向波数相对较小。当动压参数较大时，数值模拟结果显示，顶端自由，底端固支的锥壳还存在一种半稳定的高阶响应极限环，该极限环对应的振幅最大点更靠近锥壳底端。 A study on the nonlinear flutter of circular conical shells in axial supersonic airflow was presented. The piston theory was used to calculate aerodynamic force. Nonlinear aeroelastic equations of circular truncated conical shells were used, when the truncated conical shell angle is not zero and the smaller radius is infinitely close to zero. Under this specific boundary conditions, the flutter critical aerodynamic pressures and nonlinear response of circular conical shells were investigated with one dimensional DQM (differential quadrature method). The results show that, the influence of different boundary conditions on the critical dynamic pressure and the limit cycle amplitude of the circular conical shell is not significant.The wave number corresponding to the minimum critical flutter aerodynamic pressure is relatively small. When the dynamic pressure parameters are large, the numerical simulation results show that there is a semi stable high order limit cycle, which maximum point is closer to the bottom of the cone shell.