Static Response and Buckling Loads of Multilayered Composite Beams Using the Refined Zigzag Theory and Higher-Order Haar Wavelet Method

Abstract

The paper presents a review of Haar wavelet methods and an application of the higher-order Haar wavelet method to study the behavior of multilayered composite beams under static and buckling loads. The Refined Zigzag Theory (RZT) is used to formulate the corresponding governing differential equations (equilibrium/ stability equations and boundary conditions). To solve these equations numerically, the recently developed Higher-Order Haar Wavelet Method (HOHWM) is used. The results found are compared with those obtained by the widely used Haar Wavelet Method (HWM) and the Generalized Differential Quadrature Method (GDQM). The relative numerical performances of these numerical methods are assessed and validated by comparing them with exact analytical solutions. Furthermore, a detailed convergence study is conducted to analyze the convergence characteristics (absolute errors and the order of convergence) of the method presented. It is concluded that the HOHWM, when applied to RZT beam equilibrium equations in static and linear buckling problems, is capable of predicting, with a good accuracy, the unknown kinematic variables and their derivatives. The HOHWM is also found to be computationally competitive with the other numerical methods considered.

Free vibration analysis of tapered Timoshenko beam with higher order Haar wavelet method

“…( 6). Based on the Haar wavelet, the expansion is presented as in which ℎ 𝑖 (𝑥) is the Haar function [18] where 𝑖 = 𝑚 + 𝑘 + 1, 𝑚 = 2 𝑗 is a maximum number of square waves arranged in the interval [𝐴,𝐵] and the parameter 𝑘 indicates the location of the particular square wave [18] The integrals of the Haar functions (7) of order n can be expressed as [13] The differential equation can be satisfied in selected uniform grid points…”

“…Recently, the higher order Haar wavelet method (HOHWM) was introduced in [13] in order to improve the accuracy and convergence of the previously proposed Haar wavelet method. The HOHWM has been applied with success to solving differential equations, vibration, and buckling response of beams [14][15][16][17][18]. Theoretical and numerical analyses of the free and forced vibration of homogeneous and functionally graded Timoshenko beams have been performed [19][20][21][22].…”

A higher-order collocation method based on Haar wavelets for integro-differential equations with two-point integral condition

“…In 2022, M. Ahsan et al extend the HCMHW presented in [44] for the solution of singularly perturbed nonlinear ODEs by utilizing the iterative quasi-linearization technique [47]. The HHWCM is also used to study the static response and buckling loads of multilayered composite beams [48] and vibration analysis of different types of beams [49][50][51]. Furthermore, the HHWCM is recently implemented to solve nonlinear problems with two point-integral boundary conditions [52].…”

Количественная Оценка Расслоения С Помощью Вейвлетов Хаара И Машинного Обучения

“…Вейвлеты Хаара имеют аналитическое решение и могут успешно аппроксимировать производные функций при решении дифференциальных уравнений [22][23][24], таких как уравнение движения для поперечных колебаний балок из слоистых композитов. В [25,26] процедуры, описанной в [29]. Расчетные значения первой собственной частоты (СЧ) сравнили с имеющимися в литературе (табл.…”