Haar Wavelet Method for Nonlinear Vibration of Functionally Graded CNT‐Reinforced Composite Beams Resting on Nonlinear Elastic Foundations in Thermal Environment

Abstract

This paper presents a simple and effective approach based on the Haar wavelet discretization method (HWDM) for the nonlinear vibration analysis of carbon nanotube-reinforced composite (CNTRC) beams resting on a nonlinear elastic foundation in a thermal environment. Material properties are assumed to be functionally graded (FG) in the thickness direction and temperature-dependent and are evaluated through the extended rule of mixture. Based on the first-order shear deformation beam theory in conjunction with the von Kármán nonlinearity, the nonlinear governing equations of CNTRC beams on nonlinear elastic foundations are derived as well as the related boundary conditions. Moreover, the initial thermal stress due to uniform temperature rise is considered in this study. To evaluate the nonlinear natural frequencies, the obtained equations are discretized into a set of nonlinear algebraic equations through HWDM and then the direct iteration technique is employed to solve the resulting algebraic equations. The convergence and comparison studies are carried out, and the results indicate that the numerical rate of convergence of the proposed method is in agreement with the convergence theorem, and good accuracy of the present results is observed. Studies on the effects of different parameters such as CNT volume fraction, distribution type of CNTs, foundation stiffness coefficients, boundary condition, slenderness ratio, temperature rise, and initial thermal stress on the linear frequencies and the nonlinear frequency ratios are also reported. This study offers an insight into the vibration behaviors of CNTRC beams resting on nonlinear elastic foundation subjected to the uniform temperature rise.

Higher Order Haar Wavelet Method for Solving Differential Equations

“…where the elements of the matrix P 2 are defined by (7) and the coefficient vector a is determined by substituting the solution (33) and its derivatives in Eq. (31) as…”

“…The HWM is adapted for the analysis of nonlinear integral and integro-differential equations in [24][25][26][27], covering one-and multi-dimensional problems. Solid mechanics, particularly composite structures, are examined using the HWM in [28][29][30][31][32][33]. These studies cover free vibration analysis of orthotropic plates [28], functionally graded composite structures [30][31][32], delamination detection in composite beams [29], and other structures.…”

Analytic Formula for the Vibration and Sound Radiation of a Nonlinear Duct

“…where ρ a is the air density; ω is the excitation frequency; A o and A Q are the displacement amplitudes of the source and panel, respectively; ϕ Q (x, y) is the Q th panel mode shape, which is a double sine function, sin((mπx)/a)sin((nπy)/b) (the panel is simply supported with immovable edges, and hence, the model shape can be written as a double sine function); and m and n are the structural mode numbers. Putting equation (5) into (4) yields the following equation:…”

“…Over the past decades, many studies have considered structural-acoustic problems, plate vibration, and solution methods for nonlinear governing equations (e.g., [1][2][3][4][5][6][7][8][9][10][11][12]). Among various structural-acoustic problems, duct noise has been a particular focus for many years.…”

The recent progress of functionally graded CNT reinforced composites and structures

“…In addition, the nonlinear-to-linear frequency ratio of the sandwich beam with FG-V face sheets was found a little lower than that with the UD face sheets. On the basis of Haar wavelet method, Fan and Huang [163] proposed a numerical model to analyze the nonlinear vibration characteristics of FG-CNTRC beam resting on nonlinear elastic foundations subjected to thermal environment. It was found that the nonlinear frequency ratio increased with increasing temperature.…”