A New Form of Frequency Equation for Functionally Graded Timoshenko Beams with Arbitrary Number of Open Transverse Cracks

Lien, Tran Van; Duc, Ngo Trong; Khiem, Nguyen Tien

doi:10.1007/s40997-018-0152-2

Cited by 7 publications

(7 citation statements)

References 21 publications

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“…By coupling the boundary in equations ( 24) & ( 25) and compatibility conditions in eq. ( 39) and equation (38), we get By substituting equation (37) into equations ( 35) and (36), integration this equations and multiplying by s  , the equation of motion in matrix form is given as…”

Section: Solution Methodsmentioning

confidence: 99%

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Transverse Vibration of Cracked Graded Rayleigh Beam with Axial Motion

Bachaya¹,

Elaikh²

2021

UTJES

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The paper presents the transverse vibration of an open edge crack graded Rayleigh beam with axial motion. The vibration equation was obtained relying on Hamilton's precept and resolved by the approach of Galerkin's. The power-law is adopted to represent the gradient of properties of the material composing the beam along the direction of the thickness. Cracks were modeled as rotational massless spring. The effects of axial velocity, material property change index, location, and depth of cracking on the vibration characteristics are observed. Also, the corresponding mode shapes are found for cracked moving graded Rayleigh beam with simply supported, and fixed-fixed end supports. The results clear up that the natural frequencies drop due to the rise in the axial velocity, the crack depth, and the material property index

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Section: Solution Methodsmentioning

confidence: 99%

“…After integral G and substituting it into equation ( 26), the compliance of the crack of a homogenous beam can be written as [38]:…”

Section: Crack Modelingmentioning

confidence: 99%

Transverse Vibration of Cracked Graded Rayleigh Beam with Axial Motion

Bachaya¹,

Elaikh²

2021

UTJES

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“…Consider a crack at the distance x i from the left side of the beam as shown in Figure 2, The axial displacement, the transverse displacement, force perpendicular to the cross section, bending moment, shear force, and slope of the beam in element i and i þ 1, which is the place of discontinuity segments, must satisfy the following conditions (Aydin, 2013;Van Lien et al, 2019)…”

Section: Properties Of Exponentially Fgmsmentioning

confidence: 99%

Nonlinear frequency behavior of cracked functionally graded porous beams resting on elastic foundation using Reddy shear deformation theory

Forghani

Bazargan-Lari

Zahedinejad

et al. 2022

Journal of Vibration and Control

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This paper presents the nonlinear frequency behavior of cracked functionally graded porous beams subjected to various boundary conditions using Reddy shear deformation theory and Green’s tensor together with the Von Karman geometric nonlinearity. The material properties of the beam vary exponentially in thickness direction. A generalized differential quadrature method (GDQM) in conjunction with a direct numerical iteration method is developed to solve the system of equations derived by means of Hamilton’s principle. Demonstrating the convergence of this method, the verification is performed by using extracted results from a previous study based on the Euler and Timoshenko beam theory. The results for extensive studies are provided to understand the influences of the different gradient indexes, vibration amplitude ratios, porosity indexes, shear and elastic foundation parameters, and boundary conditions on the nonlinear frequency behavior. The location of crack plays the significant role on the nonlinear frequency ratios. The frequency ratios hit the minimum when the crack is located in mid-span of the beam. In this regard, the effect of shear stiffness of foundation is much more than that of Winkler one in the increasing mid-span value of nonlinear frequency ratio. It is also shown that the crack location results in decline of frequency ratios in the mid-span of beam specially for the cracks located closer to surface, the deeper ones lead to get nonlinear frequency ratios rise much more due to the fact that the deeper the crack is, the weaker the crack section becomes. The results of this paper and the effects of these parameters can be used in the optimal design of functionally graded beams and in crack prediction, detection, and monitoring techniques.

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“…The governing equations were solved using frequency contours and response surface models with a genetic algorithm. Lien et al [28] addressed free vibration of a graded Timoshenko beam with multiple cracks. They adopted Hamilton's extended as the basis for deriving the kinematic equations for that system.…”

Section: Introductionmentioning

confidence: 99%

Investigation of Transverse Vibration Characteristics of Cracked Axially Moving Functionally Graded Beam Under Thermal Load

Elaikh

Agboola²

2022

Trends Sci

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This article presents a new analysis of the vibration characteristics of open-edge cracks for graded moving beams under thermal load. The material property gradient is based on the distribution of the power law in the direction of the beam thickness. The vibration equation is obtained depending on the precept of Hamilton and resolved by the extension of Galerkin’s approach. A rotational spring is used to represent the cracking in the beam. The effects of the axial velocity, gradient index, thermal load, and cracking parameters on vibration characteristics are observed. Furthermore, the mode shapes of the simple support cracked moving graded beam are determined. The results show that the rise in the axial velocity, the crack depth, and the material property index led to a decrease in the natural frequencies, which indicates the results obtained. HIGHLIGHTS Stability and vibrations analysis of the cracked gradient beams with an axial motion under a uniform temperature rise The effect of axial velocity, gradient index, thermal load, and cracks parameter was studied Galerkin’s technique is developed for vibrational analysis of cracked graded beams with longitudinal motion The Euler-Bernoulli beam model was used taking into account the effect of rotatory inertia GRAPHICAL ABSTRACT

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A New Form of Frequency Equation for Functionally Graded Timoshenko Beams with Arbitrary Number of Open Transverse Cracks

Cited by 7 publications

References 21 publications

Transverse Vibration of Cracked Graded Rayleigh Beam with Axial Motion

Transverse Vibration of Cracked Graded Rayleigh Beam with Axial Motion

Nonlinear frequency behavior of cracked functionally graded porous beams resting on elastic foundation using Reddy shear deformation theory

Investigation of Transverse Vibration Characteristics of Cracked Axially Moving Functionally Graded Beam Under Thermal Load

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