Free vibration analysis of Euler–Bernoulli curved beams using two-phase nonlocal integral models

2021 ◽  
pp. 107754632110224
Author(s):  
Pei Zhang ◽  
Hai Qing
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Integral Relation ◽  
Dynamics Simulation ◽  
Curvature Radius ◽  
Elastic Model ◽  
Curved Beams ◽  
Two Phase

Eringen’s nonlocal elastic model has been widely applied to address the size-dependent response of micro-/nanostructures, which is observed in experimental tests and molecular dynamics simulation. However, several recent studies have pointed out that some inconsistent results appear while applying it in the analysis of bounded structures, which indicates that it is necessary to adopt other suitable models. In this work, both the well-posed strain-driven and stress-driven two-phase local/nonlocal integral models are used to study the size effect in the free vibration of Euler–Bernoulli curved beams. The governing equations of motion and the associated boundary conditions are derived on the basis of Hamilton’s principle. The two-phase nonlocal integral relation is transformed into an equivalent differential law with two constitutive boundary conditions. Using the generalized differential quadrature method, the governing equation in terms of displacements is solved numerically. The vibration frequencies of the beam under different boundary conditions are obtained and validated by comparing with those existing results. For all boundary conditions, the nonlocal related parameters of the two types of two-phase nonlocal strategies show consistent softening and stiffening effects on vibration response, respectively. Moreover, the effect of the curvature radius of the beam is also investigated.

Well-posed two-phase nonlocal integral models for free vibration of nanobeams in context with higher-order refined shear deformation theory

2021 ◽  
pp. 107754632110399
Author(s):  
Pei Zhang ◽  
Hai Qing
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Well Posedness ◽  
Governing Equations ◽  
Two Phase

In this article, the well-posedness of several common nonlocal models for higher-order refined shear deformation beams is studied. Unlike the case of classic beams models, both strain-driven and stress-driven purely nonlocal theories lead to an ill-posed issue (i.e., there are excessive mandatory boundary conditions) when considering higher-order shear deformation assumption. As an effective remedy, the well-posedness of strain-driven and stress-driven two-phase nonlocal (StrainDTPN and StressDTPN) models is pertinently evidenced by studying the free vibration problem of nanobeams. The governing equations of motion and standard boundary conditions are derived from Hamilton’s principle. The integral constitutive relation is transformed equivalently to a differential form equipped with two constitutive boundary conditions. Using the generalized differential quadrature method (GDQM), the governing equations in terms of displacements are solved numerically. Numerical results show that both the StrainDTPN and StressDTPN models can predict consistent size-effects of beams with different boundary conditions.

Free Vibration Analysis of Functionally Graded Moderately Thick Annular Sector Plates Using Differential Quadrature Method

2011 ◽  
Vol 110-116 ◽  
pp. 2990-2998 ◽  
Author(s):  
S.H. Mirtalaie ◽  
M.A. Hajabasi ◽  
F. Hejripour
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Differential Quadrature ◽  
Radius Ratio ◽  
Quadrature Method ◽  
Annular Sector

In this paper, the free vibration of moderately thick annular sector plates made of functionally graded materials is studied using the Differential Quadrature Method (DQM). The material properties of the functionally graded plate are assumed to vary continuously through the thickness, according to a power-law distribution. The governing differential equations of motion are derived based on the First order Shear Deformation plate Theory (FSDT) and then solved numerically using DQM under different boundary conditions. The results for the isotropic plates which are derivable with this approach are presented and compared with the literature and they are in good agreement. The natural frequencies of the functionally graded moderately thick annular sector plates under various combinations of clamped, simple supported and free boundary conditions are presented for the first time. The effects of boundary conditions, sector angle, radius ratio, thickness to outer radius ratio, volume fraction exponent and variation of the Poisson’s ratio on the free vibration behavior of the plate are studied

scholarly journals Free Vibration Analysis of Triclinic Nanobeams Based on the Differential Quadrature Method

2019 ◽  
Vol 9 (17) ◽  
pp. 3517 ◽  
Author(s):  
Behrouz Karami ◽  
Maziar Janghorban ◽  
Rossana Dimitri ◽  
Francesco Tornabene
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Strain Gradient ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Differential Quadrature ◽  
Vibration Response ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Beam Model

In this work, the nonlocal strain gradient theory is applied to study the free vibration response of a Timoshenko beam made of triclinic material. The governing equations of the problem and the associated boundary conditions are obtained by means of the Hamiltonian principle, whereby the generalized differential quadrature (GDQ) method is implemented as numerical tool to solve the eigenvalue problem in a discrete form. Different combinations of boundary conditions are also considered, which include simply-supports, clamped supports and free edges. Starting with some pioneering works from the literature about isotropic nanobeams, a convergence analysis is first performed, and the accuracy of the proposed size-dependent anisotropic beam model is checked. A large parametric investigation studies the effect of the nonlocal, geometry, and strain gradient parameters, together with the boundary conditions, on the vibration response of the anisotropic nanobeams, as useful for practical engineering applications.

A modified state space differential quadrature method for free vibration analysis of soft-core sandwich panels

2017 ◽  
Vol 21 (6) ◽  
pp. 1843-1879 ◽  
Author(s):  
Balavishnu Udayakumar ◽  
KV Nagendra Gopal
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
State Space ◽  
Vibration Analysis ◽  
Differential Quadrature Method ◽  
Sandwich Panels ◽  
Free Vibration Analysis ◽  
Differential Quadrature ◽  
Soft Core ◽  
Quadrature Method

Modifications and improvements to conventional state space differential quadrature method are proposed for free vibration analysis of thick, soft-core sandwich panels with arbitrary edge boundary conditions, using an exact two-dimensional elasticity model. The modifications are based on a systematic procedure to implement all possible combinations of edge boundary conditions including simply supported, clamped, free and guided edges. Natural frequencies and mode shapes are obtained and compared with exact elasticity solutions from state space method and approximate solution from finite element simulations; demonstrating the high numerical accuracy and rapid convergence of the modified method. Further, the proposed framework is compared to the conventional implementation of the state space differential quadrature method for thick cantilever sandwich panels and is shown to give results with better accuracy and faster convergence.

Asymmetric free vibration analysis of first-order shear deformable functionally graded magneto-electro-thermo-elastic circular plates

2019 ◽  
Vol 233 (16) ◽  
pp. 5659-5675 ◽  
Author(s):  
Ehsan Arshid ◽  
Ali Kiani ◽  
Saeed Amir ◽  
Mustafa Zarghami Dehaghani
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Circular Plates ◽  
First Order ◽  
Shear Deformable

The current study aims to analyze the asymmetric free vibration behavior of shear deformable functionally graded magneto-electro-thermo-elastic circular plates. The plate’s displacements are described by employing the first-order shear deformation theory and based on the von Karman assumptions, the strains and displacements are related together. Using Hamilton’s principle and variational formulation, the governing motion equations and also the associated boundary conditions have been derived. The generalized differential quadrature method is applied to discretize and solve them. The effects of the most important parameters such as material gradient index, electromagnetic loads, boundary conditions, and also aspect ratio of the plate on the natural frequencies and mode shapes of the plate are considered and discussed in details. The results show that the effect of electric potential on the natural frequency is the opposite of the magnetic one. In other words as the magnetic potential increases, the rigidity of the plate increases too and the frequency enhances. The results are compared and verified with the simpler states in literature. The findings of this study are useful for designing more efficient sensors and actuators used in smart or intelligent structures.

scholarly journals Nonlocal Vibration Analysis of a Nonuniform Carbon Nanotube with Elastic Constraints and an Attached Mass

Materials ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 3445
Author(s):  
Maria Anna De Rosa ◽  
Maria Lippiello ◽  
Enrico Babilio ◽  
Carla Ceraldi
Keyword(s):  
Exact Solution ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Numerical Results ◽  
Small Scale ◽  
Concentrated Mass ◽  
Closed Form Solutions

Here, we consider the free vibration of a tapered beam modeling nonuniform single-walled carbon nanotubes, i.e., nanocones. The beam is clamped at one end and elastically restrained at the other, where a concentrated mass is also located. The equation of motion and relevant boundary conditions are written considering nonlocal effects. To compute the natural frequencies, the differential quadrature method (DQM) is applied. The influence of the small-scale parameter, taper ratio coefficient, and added mass on the first natural frequency is investigated and discussed. Some numerical examples are provided to verify the accuracy and validity of the proposed method, and numerical results are compared to those obtained from exact solution. Since the numerical results are in excellent agreement with the exact solution, we argue that DQM provides a simple and powerful tool that can also be used for the free vibration analysis of carbon nanocones with general boundary conditions for which closed-form solutions are not available in the literature.

OUT-OF-PLANE FREE VIBRATION ANALYSIS OF FUNCTIONALLY GRADED CIRCULAR CURVED BEAMS SUPPORTED ON ELASTIC FOUNDATION

2010 ◽  
Vol 02 (03) ◽  
pp. 635-652 ◽  
Author(s):  
P. MALEKZADEH ◽  
M. R. GOLBAHAR HAGHIGHI ◽  
M. M. ATASHI
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Elastic Foundation ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Curved Beams ◽  
Out Of Plane ◽  
Benchmark Solutions

As a first endeavor, the out-of-plane free vibration analysis of thin-to-moderately thick functionally graded (FG) circular curved beams supported on two-parameter elastic foundation is presented. The formulation is derived based on the first-order shear deformation theory (FSDT), which includes the effects of shear deformation and rotary inertia due to both torsional and flexural vibrations. The material properties are assumed to be graded in the direction normal to the plane of the beam curvature. The differential quadrature method (DQM), as an efficient and accurate method, is employed to discretize the equations of motion and the related boundary conditions. In order to assure the accuracy of the formulation and the method of solution, convergence behavior of the nondimensional natural frequencies is examined for FG circular curved beams and comparison studies with those of isotropic curved beams, available in the literature, are performed. The effects of the elastic foundation coefficients, boundary conditions, the material graded index and different geometrical parameters on the natural frequency parameters of the FG circular curved beams are investigated. The new results can be used as benchmark solutions for future research works.

Large Amplitude Free Vibration Analysis of Thin Annular Sector Plates Using Differential Quadrature Method

2010 ◽  
Author(s):  
S. H. Mirtalaie ◽  
M. A. Hajabasi
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Large Amplitude ◽  
Plate Theory ◽  
Nonlinear Eigenvalue Problem ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Annular Sector

In this paper, the Differential Quadrature Method (DQM) is used to study the large amplitude free vibration of thin annular sector plates. The geometrical nonlinear governing equations of motion are derived based on the classical plate theory and using the von Karman nonlinear strain-displacement relationships. Following the DQ-procedure and employing the concept of new degrees of freedom a nonlinear eigenvalue problem is obtained which is solved iteratively and nonlinear natural frequencies of the plate are obtained. The results show a very good convergence and they are compared with the available literature for the clamped boundary conditions to demonstrate the validity of the work. The effects of boundary conditions, inner to outer radius ratio and sector angle on the large amplitude free vibration of thin plate are studied.

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels integrated with piezoelectric layers

2014 ◽  
Vol 21 (4) ◽  
pp. 571-587 ◽  
Author(s):  
Hamid Reza Saeidi Marzangoo ◽  
Mostafa Jalal
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Simply Supported ◽  
Piezoelectric Layers ◽  
Curved Panels

AbstractFree vibration analysis of functionally graded (FG) curved panels integrated with piezoelectric layers under various boundary conditions is studied. A panel with two opposite edges is simply supported, and arbitrary boundary conditions at the other edges are considered. Two different models of material property variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential law distribution of the material properties through the thickness are considered. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. For the simply supported boundary conditions, closed-form solution is given by making use of the Fourier series expansion, and applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free-end conditions. Natural frequencies of the hybrid curved panels are presented by solving the eigenfrequency equation, which can be obtained by using edges boundary conditions in this state equation. The results obtained for only FGM shell is verified by comparing the natural frequencies with the results obtained in the literature.

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