scholarly journals Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section

2020 ◽  
Vol 41 (10) ◽  
pp. 1497-1516 ◽  
Author(s):  
Yong Huang
Keyword(s):  
Three Dimensional ◽  
Circular Cross Section ◽  
Vibrational Analysis ◽  
Beam Theory ◽  
Single Equation ◽  
Auxiliary Function ◽  
Functionally Graded ◽  
Beam Model ◽  
Various Boundary Conditions ◽  
Differential Motion

Abstract The bending and free vibrational behaviors of functionally graded (FG) cylindrical beams with radially and axially varying material inhomogeneities are investigated. Based on a high-order cylindrical beam model, where the shear deformation and rotary inertia are both considered, the two coupled governing differential motion equations for the deflection and rotation are established. The analytical bending solutions for various boundary conditions are derived. In the vibrational analysis of FG cylindrical beams, the two governing equations are firstly changed to a single equation by means of an auxiliary function, and then the vibration mode is expanded into shifted Chebyshev polynomials. Numerical examples are given to investigate the effects of the material gradient indices on the deflections, the stress distributions, and the eigenfrequencies of the cylindrical beams, respectively. By comparing the obtained numerical results with those obtained by the three-dimensional (3D) elasticity theory and the Timoshenko beam theory, the effectiveness of the present approach is verified.

scholarly journals Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150790 ◽  
Author(s):  
F. dell’Isola ◽  
I. Giorgio ◽  
M. Pawlikowski ◽  
N. L. Rizzi
Keyword(s):  
Large Deformations ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Beam Theory ◽  
Beam Model ◽  
Computationally Efficient ◽  
Nonlinear Beam ◽  
Pantographic Structures ◽  
Diffusion Of Technology ◽  
Second Gradient

The aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call ‘pantographic structures’. The interest in these materials was increased by the possibilities opened by the diffusion of technology of three-dimensional printing. They can be regarded, once choosing a suitable length scale, as families of beams (also called fibres) interconnected to each other by pivots and undergoing large displacements and large deformations. There are, however, relatively few ‘ready-to-use’ results in the literature of nonlinear beam theory. In this paper, we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model. The homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems. Furthermore, we consider pantographic structures, find the corresponding homogenized second gradient deformation energies and study some planar problems. Numerical solutions for these two-dimensional problems are obtained via minimization of energy and are compared with some experimental measurements, in which elongation phenomena cannot be neglected.

Electromechanical impedance response of a cracked functionally graded beam with imperfectly bonded piezoelectric wafers

2011 ◽  
Vol 22 (16) ◽  
pp. 1899-1912 ◽  
Author(s):  
Wei Yan ◽  
J Wang ◽  
WQ Chen ◽  
WC Li
Keyword(s):  
Lead Zirconate Titanate ◽  
Crack Depth ◽  
Beam Theory ◽  
Lead Zirconate ◽  
Functionally Graded ◽  
Beam Model ◽  
Functionally Graded Beam ◽  
Electromechanical Impedance ◽  
Smart Beam ◽  
Piezoelectric Wafers

An analytical model of a cracked functionally graded beam with attached Lead Zirconate Titanate (PZT) actuator/sensors is proposed in the paper for structural health monitoring. In this model, the dynamic behavior of the piezoelectric patches is considered and a viscoelastic law is adopted to describe the bonding imperfection between the piezoelectric patches and the beam. A piecewisely homogeneous beam model is then employed to approximate the original inhomogeneous beam based on the Timoshenko beam theory. The crack in the beam is treated as a massless rotational spring. In order to develop the recursive formulations to reduce the dimension of the final equations in the method of reverberation-ray matrix (MRRM), new local scattering relations are established for this smart beam using a matrix reduction technique. An analytical expression of the electromechanical impedance (EMI) is derived based on the improved MRRM via the recursive formulations. Comparison with existent experimental results and those predicted by other methods, such as the conventional MRRM, the transfer matrix method (TMM), and the finite element method (FEM), is made to validate the proposed analysis. Furthermore, the effects of various parameters including the crack depth on the EMI signatures are highlighted.

Free Vibration Analysis of Functionally Graded Beams Resting on Elastic Foundation in Thermal Environment

2016 ◽  
Vol 16 (07) ◽  
pp. 1550029 ◽  
Author(s):  
P. Zahedinejad
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Volume Fraction ◽  
Thermal Environment ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Functionally Graded ◽  
Power Law Distribution ◽  
Various Boundary Conditions

The free vibration of functionally graded (FG) beams with various boundary conditions resting on a two-parameter elastic foundation in the thermal environment is studied using the third-order shear deformation beam theory. The material properties are temperature-dependent and vary continuously through the thickness direction of the beam, based on a power-law distribution in terms of the volume fraction of the material constituents. In order to discretize the governing equations, the differential quadrature method (DQM) in conjunction with the Hamilton’s principle is adopted. The convergence of the method is demonstrated. In order to validate the results, comparisons are made with solutions available for the isotropic and FG beams. Through a comprehensive parametric study, the effect of various parameters involved on the FG beam was studied. It is concluded that the uniform temperature rise has more significant effect on the frequency parameters than the nonuniform case.

scholarly journals Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

2020 ◽  
Vol 41 (9) ◽  
pp. 1303-1320
Author(s):  
Zhu Su ◽  
Lifeng Wang ◽  
Kaipeng Sun ◽  
Jie Sun
Keyword(s):  
Transverse Shear ◽  
Three Dimensional ◽  
Beam Theory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Penalty Function Method ◽  
Transverse Displacement ◽  
Three Dimensional Model ◽  
Special Cases

Abstract A quasi-three dimensional model is proposed for the vibration analysis of functionally graded (FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higherorder beam theory are regarded as the special cases of this model. The material properties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme. The energy-based formulation is derived by a variational method integrated with the penalty function method, where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables. The formulation is validated by some comparative examples, and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.

On the dynamics of a massless beam with end mass rotating in the three-dimensional space

2012 ◽  
Vol 227 (3) ◽  
pp. 434-448 ◽  
Author(s):  
Giancarlo Genta
Keyword(s):  
Rotation Axis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Beam Theory ◽  
Beam Model ◽  
Inertial Reference Frame ◽  
Centrifugal Stiffening ◽  
Constant Rate ◽  
Fixed Axis ◽  
Three Dimensional Space

Many elements of machines, like shafts, blades, connecting rods, etc., are often modeled using the so-called beam theory and, when these elements rotate with respect to an inertial reference frame, this rotation can deeply affect their dynamic behavior. The position of the beam with respect to the rotation axis and the possibility that the rotation is more complicated than a simple constant-rate rotation about a fixed axis influence this effect, and different models are usually employed. As a result, different phenomena, like gyroscopic effect, centrifugal stiffening or softening, and instability due to rotation are often mentioned in reference to the different cases. The aim of this article is that of building a much simplified beam model, and to subject it to a compound, nonconstant rate rotation. Since the model can be solved in closed form, at least in several cases, a general discussion of the revant phenomena can be done to shed some light on some aspects, like the instability ranges due to rotation and to the damping of the system.

Free Vibration of Edge Cracked Functionally Graded Microscale Beams Based on the Modified Couple Stress Theory

2017 ◽  
Vol 17 (03) ◽  
pp. 1750033 ◽  
Author(s):  
Şeref Doğuşcan Akbaş
Keyword(s):  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Beam Model ◽  
Cracked Beam ◽  
Stress Theory ◽  
Modified Couple Stress

In this study, the free vibration analysis of edge cracked cantilever microscale beams composed of functionally graded material (FGM) is investigated based on the modified couple stress theory (MCST). The material properties of the beam are assumed to change in the height direction according to the exponential distribution. The cracked beam is modeled as a modification of the classical cracked-beam theory consisting of two sub-beams connected by a massless elastic rotational spring. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new nonclassical beam model reduces to the classical one when the length scale parameter is zero. The problem considered is investigated using the Euler–Bernoulli beam theory by the finite element method. The system of equations of motion is derived by Lagrange’s equations. To verify the accuracy of the present formulation and results, the frequencies obtained are compared with the results available in the literature, for which good agreement is observed. Numerical results are presented to investigate the effect of crack position, beam length, length scale parameter, crack depth, and material distribution on the natural frequencies of the edge cracked FG microbeam. Also, the difference between the classical beam theory (CBT) and MCST is investigated for the vibration characteristics of the beam of concern. It is believed that the results obtained herein serve as a useful reference for research of similar nature.

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