Vibrations of rotating cylindrical shells with functionally graded material using wave propagation approach

Abstract

Fundamental natural frequencies of rotating functionally graded cylindrical shells have been calculated through the improved wave propagation approach using three different volume fraction laws. The governing shell equations are obtained from Love’s shell approximations using improved rotating terms and the new equations are obtained in standard eigenvalue problem with wave propagation approach and volume fraction laws. The effects of circumferential wave number, rotating speed, length-to-radius, and thickness-to-radius ratios have been computed with various combinations of axial wave numbers and volume fraction law exponent on the fundamental natural frequencies of nonrotating and rotating functionally graded cylindrical shells using wave propagation approach and volume fraction laws with simply supported edge. In this work, variation of material properties of functionally graded materials is controlled by three volume fraction laws. This process creates a variation in the results of shell frequency. MATLAB programming has been used to determine shell frequencies for traveling mode (backward and forward) rotating motions. New estimations show that the rotating forward and backward simply supported fundamental natural frequencies increases with an increase in circumferential wave number, for Type I and Type II of functionally graded cylindrical shells. The presented results of backward and forward simply supported fundamental natural frequencies corresponding to Law I are higher than Laws II and III for Type I and reverse effects are found for Type II, depending on rotating speed. Our investigations show that the decreasing and increasing behaviors are noted for rotating simply supported fundamental natural frequencies with increasing length-to-radius and thickness-to-radius ratios, respectively. It is found that the fundamental frequencies of the forward waves decrease with the increase in the rotating speed, and the fundamental frequencies of the backward waves increase with the increase in the rotating speed. This investigation has been made with three different volume fraction laws of polynomial (Law I), exponential (Law II), and trigonometric (Law III). The presented numerical results of nonrotating isotropic and rotating functionally graded simply supported are in fair agreement with parts of other earlier numerical results.

Effect of Power Law-Index for Accurate Natural Frequencies of Rotating Isotropic SWCNTs with Ring Supports

“…For backward and forward waves, it can be seen that the frequency amplitudes of both types of SWCNTs remain same at  = 0. The frequency curves of forward waves are slightly less than forward waves validating the earlier computation [12,78] The presented Galerkin's technique for backward and forward waves of these tubes provides higher estimation definitely for whole range of  . It is noted that from the figures on increasing the rotational speed, the backward waves monotonically increases and forward waves decreases, respectively for C-C and C-F.…”

“…Several techniques [78,79,80] have been proposed to analyze the vibration of CNT, but here, Galerkin's method to discretize the governing equation of motion. The coordinate system ,,…”

“…In addition, for the vibration of CNTs, many researchers have used different numerical techniques; for example modified coupled theory [22], 3D-nonlocal theory [36,37,64] differential quadrature method [23,41], wave propagation approach [42,44,45] [13,22,24,25,37,38,53,54,61,[75][76][77][78] In present work, we analyze the vibrations of isotropic rotating single-walled carbon nanotube (SWCNTs) with ring supports using Love shell theory based on Galerkin's method, which is our particular motivation. Since there is no evidence in the literature regarding present model where such problem have been studied.…”

Vibration Characteristics of Single-Walled Carbon Nanotubes Based on Nonlocal Elasticity Theory Using Wave Propagation Approach (WPA) Including Chirality

“…Prior to this, many techniques have been sequentially used to study the vibration of CNTs [41][42][43]. Previously, the current approach was used continuously to study the vibration of carbon nanotubes [27,29,[44][45][46][47].…”

Nonlinear coupled axial–torsional vibration of single‐walled carbon nanotubes using homotopy perturbation method

“…Therefore, it is essential to consider the coupling effect between axial and torsional vibrations of SWCNTs, especially for studying stability conditions of CNTs. It can be seen from the previous works on the vibration of SWCNTs that most of existing SWCNTs systems have focused on the bending [9,10], torsional [11][12][13], radial [14,15], longitudinal [16,17] or rotational [18,19] vibrations behaviour of the shafts, solely and the coupling effect between the vibrational modes were ignored. The coupled vibration of SWCNTs is an interesting subject because of the complexity of the equations and the analytical solutions are difficult to obtain.…”