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

This paper investigates the free vibration characteristics of truncated conical shells with variable thickness using the Haar wavelet method. Based on the Love first-approximation theory, the governing partial differential equations are formulated, which are transformed into ordinary differential equations using the separation of variables. The Haar wavelet discretization method is introduced to predict the dynamic characteristics of truncated circular conical shells with linearly and parabolically varying thickness. The present analysis is validated by comparing the numerical results with those available in the literature and very good agreement is achieved. The effects of geometrical parameters and boundary conditions on the vibration characteristics of conical shells with variable thickness are presented. The advantages of this method consist in its simplicity, fast convergence and high precision.

Free vibration analysis of cylindrical 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.

Vibration characteristics of rotating functionally graded circular cylindrical shell with variable thickness under thermal environment

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.

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.

To investigate the influence of Coriolis force on vibration characteristics of mistuned bladed disk, a bladed disk with 22 blades is employed and the effects of different rotational speeds and excitation engine orders on the maximum forced response are discussed considering the effects of Coriolis forces. The results show that if there are frequency veering regions, the largest split of double natural frequencies of each modal family considering the effects of Coriolis forces appears at frequency veering region. In addition, the amplitude magnification factor considering the Coriolis effects is increased by 1.02% compared to the system without considering the Coriolis effects as the rotating speed is 3000 rpm, while the amplitude magnification factor is increased by 2.76% as the rotating speed is 10000 rpm. The results indicate that the amplitude magnification factor may be moderately enhanced with the increasing of rotating speed. Moreover, the position of the maximum forced response of bladed disk may shift from one blade to another with the increasing of the rotational speed, when the effects of Coriolis forces are considered.

A numerical solution for vibration analysis of composite laminated conical, cylindrical shell and annular plate structures

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.

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.

Effects of rotation on heat transfer on leading and trailing sides of gas turbine blades has been extensively studied in the past. It has been established for typical two-pass channel that radially outward flow (first pass) has higher heat transfer on trailing side and lower heat transfer on leading side and vice versa for radially inward flow (second pass). Rotation induces three forces on the coolant flow — Coriolis, Centrifugal and Buoyancy forces. The direction of Coriolis force depends on the relative angle between the coolant flow and the rotation direction, because of which the direction of Coriolis force is different in individual passes — which in turn results in non-uniform distribution of high heat transfer regions on leading and trailing walls. The present study is focused on utilizing the Coriolis force favorably in both the passes by rotating the typical arrangement of two-pass channels by 90°. Firstly, smooth two pass duct (Model A-smooth) having typical arrangement of coolant flow and rotation direction is studied. The second configuration is the corresponding ribbed channel (Model A-ribbed) featuring V-shaped ribs on both leading and trailing walls. The rib-height-to-channel hydraulic diameter ratio was 0.125, rib pitch-to-rib height ratio was 8, and channel aspect ratio was unity. Model B was obtained by rotating the Model A by 90° and changing the coolant inlet port as well. Model B had three configurations — (a) smooth duct, (b) single sided ribbed duct, and (c) double sided ribbed duct. Detailed heat transfer coefficients were measured by transient liquid crystal thermography under rotating conditions. In order to match the direction of Buoyancy force as it exists in actual engines, colder air was passed during the transient experiment. The heat transfer experiments were carried out at a Reynolds number of 20000 and Rotation numbers of 0, 0.05 and 0.1. The Nusselt numbers have been reported in two forms, (a) normalized with respect to Dittus-Boelter correlation for developed turbulent flow in circular duct, (b) normalized with corresponding Nusselt number obtained from smooth channel experiments. The effects of Coriolis force and centrifugal force on heat transfer has been discussed in detail. A new model has been proposed based on the understanding and findings of the present study, which has positive effects of rotation on both leading and trailing walls.

Flow transitions and combined free and forced convective heat transfer in a rotating curved circular tube

In order to clarify the turbulence behavior in impeller channels of turbomachinery, effects of the Coriolis and centrifugal forces on the turbulent boundary layer on the concave and convex surfaces were studied experimentally, using a rotating channel of rectangular cross section whose aspect ratio was 2 : 1. Velocity profiles and turbulence intensities were also compared with those for a channel having an aspect ratio of 4 : 1. In the present rotating channel it was found that secondary flows generated by the centrifugal and Coriolis forces have a large effect on the time-averaged and turbulence components in the boundary layer along the concave and convex surfaces, and that the suppression of turbulence expected on the lower-pressure side of a two-dimensional flow is not observed because of the accmulation of lower energy fluids and the interference of the fluids with the primary flow.

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.