Large amplitude free vibration of orthotropic shallow shells of complex shapes with variable thickness

Awrejcewicz, Jan; Kurpa, L. V.; Shmatko, Tatiyana

doi:10.1590/s1679-78252013000100015

Cited by 14 publications

(12 citation statements)

References 9 publications

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“…Examples, how to define the mentioned solution structures are presented, illustrated and widely discussed in Refs. [4,[17][18][19]23], and therefore this description is omitted in this work.…”

Section: Solution To Linear Problemmentioning

confidence: 99%

“…In the case of geometrically nonlinear problem, the approach proposed in earlier works is applied [4,18,19]. Namely, the unknown functions are presented in the following way:…”

Section: Solution To Nonlinear Problemmentioning

confidence: 99%

“…A review of achievements in this field is presented in works [2,22,24,29]. As follows from the review, there is a small number of papers dedicated to the problem of the nonlinear vibrations of multi-layer plates and shells with variable thickness of layers [4,5,8,28]. A more complex dynamic behavior of the plates and shells caused by the interactive influence of layers and variable thickness is expected.…”

Section: Introductionmentioning

confidence: 99%

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Investigating geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness via the R-functions theory

Awrejcewicz

Kurpa

Shmatko

2015

Composite Structures

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A novel numerical/analytical approach to study geometrically nonlinear vibrations of shells with variable thickness of layers is proposed. It enables investigation of shallow shells with complex forms and different boundary conditions. The proposed method combines application of the R-functions theory, variational Ritz's method, as well as hybrid Bubnov-Galerkin method and the fourth-order Runge-Kutta method. Mainly two approaches, classical and first-order shear deformation theories of shells are used. An original scheme of discretization regarding time reduces the initial problem to the solution of a sequence of linear problems including those related to linear vibrations with a special type of elasticity, as well as problems governed by non-linear system of ordinary differential equations. The proposed method is validated by the investigation of test problems for shallow shells with rectangular planform and applied to new vibration problems for shallow shells with complex planforms and variable thickness of layers.

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Section: Solution To Linear Problemmentioning

confidence: 99%

“…In the case of geometrically nonlinear problem, the approach proposed in earlier works is applied [4,18,19]. Namely, the unknown functions are presented in the following way:…”

Section: Solution To Nonlinear Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Investigating geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness via the R-functions theory

Awrejcewicz

Kurpa

Shmatko

2015

Composite Structures

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“…Consequently, it is important to develop universal and effective methods for investigation of nonlinear vibrations of functionally graded shallow shells of complex planforms and different boundary conditions. Earlier, in papers [Kurpa (2009a); Kurpa et al (2007Kurpa et al ( , 2013; Awrejcewicz et al (2010Awrejcewicz et al ( , 2013Awrejcewicz et al ( , 2015a; Kurpa and Mazur (2010); Kurpa and Shmatko (2014)], the original meshless method aimed at an application of the Rfunctions theory, variational Ritz method, Bubnov-Galerkin procedure, and Runge-Kutta method has been proposed. In reference [Awrejcewicz et al (2015b)] this method has been extended to geometrically nonlinear vibration problems of functionally graded shallow shells of arbitrary planforms.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Geometrically Nonlinear Vibrations of Functionally Graded Shallow Shells of a Complex Shape

Awrejcewicz

Kurpa

Shmatko

2017

Lat. Am. j. solids struct.

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Geometrically nonlinear vibrations of functionally graded shallow shells of complex planform are studied. The paper deals with a power-law distribution of the volume fraction of ceramics and metal through the thickness. The analysis is performed with the use of the R-functions theory and variational Ritz method. Moreover, the Bubnov-Galerkin and the Runge-Kutta methods are employed. A novel approach of discretization of the equation of motion with respect to time is proposed. According to the developed approach, the eigenfunctions of the linear vibration problem and some auxiliary functions are appropriately matched to fit unknown functions of the input nonlinear problem. Application of the R-functions theory on every step has allowed the extension of the proposed approach to study shallow shells with an arbitrary shape and different kinds of boundary conditions. Numerical realization of the proposed method is performed only for one-mode approximation with respect to time. Simultaneously, the developed method is validated by investigating test problems for shallow shells with rectangular and elliptical planforms, and then applied to new kinds of dynamic problems for shallow shells having complex planforms.

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“…Their formulation was based on Novozhilov's linear shallow shell theory. Using Donell's nonlinear shallow shell theory and Kirchhoff's hypothesis, Awrejcewicz et al (2013) studied free vibration analysis of doubly curved orthotropic shallow shells. Viola et al (2013) used a 2D higher order shear deformation theory with nine parameters in order to analyze free vibration analysis of the thick laminated doubly curved shells and panels.…”

Section: Introductionmentioning

confidence: 99%

Improved high order free vibration analysis of thick double curved sandwich panels with transversely flexible cores

Fard

Livani

Ghasemi

2014

Lat. Am. j. solids struct.

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This paper dealt with free vibration analysis of thick double curved composite sandwich panels with simply supported or fully clamped boundary conditions based on a new improved higher order sandwich panel theory. The formulation used the first order shear deformation theory for composite face sheets and polynomial description for the displacement field in the core layer which was based on the displacement field of Frostig's second model. The fully dynamic effects of the core layer and face sheets were also considered in this study. Using the Hamilton's principle, the governing equations were derived. Moreover, effects of some important parameters like that of boundary conditions, thickness ratio of the core to panel, radii curvatures and composite lay-up sequences were investigated on free vibration response of the panel. The results were validated by those published in the literature and with the FE results obtained by ABAQUS software. It was shown that thicker panels with a thicker core provided greater resistance to resonant vibrations. Also, effect of increasing the core thickness in general was significant decreased fundamental natural frequency values. KeywordsFree vibration, Double curved sandwich panel, Boundary conditions, Improved higher order sandwich panel theory.Improved high order free vibration analysis of thick double curved sandwich panels with transversely flexible cores INTRODUCTIONStructural efficiency is an important attribute for aircraft structures. A higher order theory approach, used by Kant and Patil (1991), replaced sandwich structure with an equivalent higher order shear deformable structure, which lacked the ability to determine local buckling modes and imperfection effects on the overall behavior. Using the three-dimensional elasticity equations, Bhimaraddi (1993) studied the static response of orthotropic doubly curved shallow shells. He assumed that the ratio of the shell thickness to its middle surface radius is negligible as compared to unity. The high- er order sandwich panel theory was developed by Frostig et al. (1994Frostig et al. ( , 2004, who considered two types of computational models in order to express governing equations of the core layer. The second model assumed a polynomial description of the displacement fields in the core that was based on displacement fields of the first model. Their theory did not impose any restrictions on distribution of the deformation through thickness of the core. Singh (1999) studied free vibration of the open deep sandwich shells made of thin layers and a moderately thick core. Rayleigh-Ritz method was also used to obtain natural frequencies. The improved higher order sandwich plate theory (IHSAPT), applying the first-order shear deformation theory for the face sheets, was introduced by Malekzadeh et al. (2005Malekzadeh et al. ( , 2006.

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Large amplitude free vibration of orthotropic shallow shells of complex shapes with variable thickness

Cited by 14 publications

References 9 publications

Investigating geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness via the R-functions theory

Investigating geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness via the R-functions theory

Analysis of Geometrically Nonlinear Vibrations of Functionally Graded Shallow Shells of a Complex Shape

Improved high order free vibration analysis of thick double curved sandwich panels with transversely flexible cores

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