Free vibration of axially functionally graded beams using the asymptotic development method

Cao, Dongxing; Gao, Yuan; Yao, Minghui; Zhang, Wei

doi:10.1016/j.engstruct.2018.06.111

Cited by 52 publications

(26 citation statements)

References 30 publications

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“…According to the Poincaré-Lindstedt method [78][79][80][81][82], we assume the expansion for ω and W ξ ðÞas…”

Section: Asymptotic Perturbation Methods 31 Equation Derivingmentioning

confidence: 99%

“…The asymptotic development method is applied to obtain an approximate analytical expression of the natural frequencies of nonuniform cables and beams [79,80]. Cao et al [81,82] applied the asymptotic development method to analyze the free vibration of nonuniform axially functionally graded (AFG) beams. Tarnopolskaya et al [83] gave the first comprehensive study of the mode transition phenomenon in vibration of beams with arbitrarily varying curvature and cross section on the basis of asymptotic analysis.…”

Section: Introductionmentioning

confidence: 99%

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Free Vibration of Axially Functionally Graded Beam

Cao¹,

Wang²,

Hu³

et al. 2020

Mechanics of Functionally Graded Materials and Structures

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Axially functionally graded (AFG) beam is a special kind of nonhomogeneous functionally gradient material structure, whose material properties vary continuously along the axial direction of the beam by a given distribution form. There are several numerical methods that have been used to analyze the vibration characteristics of AFG beams, but it is difficult to obtain precise solutions for AFG beams because of the variable coefficients of the governing equation. In this topic, the free vibration of AFG beam using analytical method based on the perturbation theory and Meijer G-Function are studied, respectively. First, a detailed review of the existing literatures is summarized. Then, based on the governing equation of the AFG Euler-Bernoulli beam, the detailed analytic equations are derived on basis of the perturbation theory and Meijer G-function, where the nature frequencies are demonstrated. Subsequently, the numerical results are calculated and compared, meanwhile, the analytical results are also confirmed by finite element method and the published references. The results show that the proposed two analytical methods are simple and efficient and can be used to conveniently analyze free vibration of AFG beam.

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“…According to the Poincaré-Lindstedt method [78][79][80][81][82], we assume the expansion for ω and W ξ ðÞas…”

Section: Asymptotic Perturbation Methods 31 Equation Derivingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Free Vibration of Axially Functionally Graded Beam

Cao¹,

Wang²,

Hu³

et al. 2020

Mechanics of Functionally Graded Materials and Structures

Self Cite

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“…Figure 1 shows a typical rectangular cross-sectional AFG tapered beam. In view of the Euler-Bernoulli beam theory, the equation for free transverse vibration of AFG tapered beams is given as presented by (Cao et al, 2018)…”

Section: Governing Equation Based On Euler-bernoulli Beam Theorymentioning

confidence: 99%

“…In a separate investigation, Slaviša et al (2018) proposed the symbolic-numeric method of initial parameters to investigate vibrational characteristics of Euler-Bernoulli AFG tapered beams. Recently, Cao et al (2018Cao et al ( , 2019 used the asymptotic development method to calculate natural frequencies of uniform AFG beams, and the asymptotic perturbation approach was adopted for tapered AFG beams. More recently, Mahmoud (2019) used the Myklestad method (MM) to predict transverse vibration behavior of nonuniform AFG cantilevers loaded at the tips with point masses.…”

Section: Introductionmentioning

confidence: 99%

Free transverse vibrational analysis of axially functionally graded tapered beams via the variational iteration approach

Chen

Dong

Zang

et al. 2020

Journal of Vibration and Control

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Tapered beams constitute functionally graded materials that are widely used in various engineering fields. The vibrational characteristics of tapered beams made of axially varying functionally graded materials are investigated via variational iteration method. Natural frequencies and corresponding mode shapes of axially functionally graded tapered beams are examined, and calculated frequencies are compared with results provided by previous researchers using other approximate methods. The efficiency and accuracy of this technique are demonstrated. The effect of nonhomogeneous material distribution, taper ratio as well as boundary conditions on the dynamical behavior of axially functionally graded tapered beams is thoroughly investigated as well.

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“…Functionally graded materials (FGMs) have proved to be more benefit in comparison with the laminate composites and found widespread application in the high-tech industries such as aerospace, automobile, electronics, optics, chemistry, biomedical engineering etc,... For vibration analysis of FGM structures, a number of methods have been proposed such as analytical (Nguyen, Vo, Nguyen, & Lee, 2016;Sina, Navazi, & Haddadpour, 2009), Rayleigh-Ritz method (Pradhan & Chakraverty, 2013), Galerkin (Sheng & Wang, 2018), combined Fourier series -Galerkin method (Zhu & Sankar, 2004), differential quadrature method (DQM) (Sundaramoorthy Rajasekaran, 2013a;Sınır, Çevik, & Sınır, 2018;Tang, Lv, & Yang, 2019), perturbation method (Sınır et al, 2018), asymptotic development method (Cao, Gao, Yao, & Zhang, 2018), discrete singular convolution and Taylor series expansion method (Wang & Yuan, 2017). While the aforementioned methods are limited to apply for analysis of simple FGM beam structures, the Finite Element Method (FEM) developed in (Alshorbagy, Eltaher, & Mahmoud, 2011;Eltaher, Abdelrahman, Al-Nabawy, Khater, & Mansour, 2014;Mashat, Carrera, Zenkour, Al Khateeb, & Filippi, 2014;Naccache & El Fatmi, 2018) could be used for more complex structures such as that are composed from different beam elements.…”

Section: Introductionmentioning

confidence: 99%

Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method

Lien

Duc²,

Khiem³

2019

Lat. Am. j. solids struct.

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In the present paper, the free and forced vibration of multiple cracked multi span continuous beams made of functionally graded material (FGM) is investigated by the dynamic stiffness method. First, there are constructed dynamic stiffness matrix and nodal load vector of multiple cracked FGM beam element using Timoshenko beam theory and massless double spring model of crack. Then, effect of cracks parameters on free and forced vibration of the FGM continuous beams is examined. The theoretical developments are validated by numerical examples. The obtained results provide an efficient method to analyze free and forced vibration of multiple cracked FGM framed structures and assessment of the behavior of damaged structures.

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Free vibration of axially functionally graded beams using the asymptotic development method

Cited by 52 publications

References 30 publications

Free Vibration of Axially Functionally Graded Beam

Free Vibration of Axially Functionally Graded Beam

Free transverse vibrational analysis of axially functionally graded tapered beams via the variational iteration approach

Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method

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