Dynamic Property Regulation of Axially Functional Graded Beams for Airfoil Structure

2016 ◽  
Author(s):  
Bo Xin ◽  
Zhaolong Zhang ◽  
Jie Zhang ◽  
Yiming Rong
Keyword(s):  
Mode Shape ◽  
Dynamic Property ◽  
Material Properties ◽  
Fundamental Mode ◽  
Complex Geometry ◽  
Mass Density ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Axial Direction ◽  
Cantilever Beams

The development of additive manufacturing technology makes it possible to realize a designed distribution of material properties in complex geometry. The distribution can be made not only in layers, but also in the axial direction of a beam. Axially functional graded (AFG) beams may play an important role in light-weight design for airfoil structure in aerospace industry. In this research, the free vibration of AFG cantilever beams is studied by using Euler-Bernoulli beam theory. It is taken into consideration that material properties, including elasticity modulus and mass density vary continuously along the axial direction of AFG beam according to an elliptic load distribution based on the life curve of airfoil. The governing differential equations of motion has been analyzed and solved to calculate the natural frequency and fundamental mode shape accurately. In order to verify the accuracy of the present formulation and results, the natural frequencies and fundamental mode shape of several AFG beams are obtained, and compared with the dynamic simulation results from ABAQUS. Good agreement is observed. The results indicate that the proposed method is effective at forecasting the dynamic property of AFG cantilever beams for airfoil structure.

Analysis of sandwich Timoshenko porous beam with temperature-dependent material properties: Magneto-electro-elastic vibration and buckling solution

2019 ◽  
Vol 25 (23-24) ◽  
pp. 2875-2893 ◽  
Author(s):  
M. Bamdad ◽  
M. Mohammadimehr ◽  
K. Alambeigi
Keyword(s):  
Timoshenko Beam ◽  
Material Properties ◽  
Vinylidene Fluoride ◽  
Equations Of Motion ◽  
Sandwich Beam ◽  
Beam Theory ◽  
Distribution Patterns ◽  
Functionally Graded ◽  
Geometrical Parameters ◽  
Temperature Dependent

Vibration and buckling analysis of a magneto-electro-elastic sandwich Timoshenko beam with a porous core and poly-vinylidene fluoride (PVDF) matrix reinforced by carbon nanotubes (CNTs) is considered as face layers and material properties of CNTs and PVDF are assumed to be temperature-dependent. Different CNT distribution patterns including uniform distribution, AV (which top and bottom face sheets have functionally graded-A (FG-A) and functionally graded-V (FG-V) CNT distribution patterns, respectively) and VA patterns are employed. The governing equations of motion are derived based on Timoshenko beam theory, and Navier's solution is used to solve these equations. The sandwich beam resting on a Pasternak foundation and face layers are subjected to electric and magnetic potentials. The effect of different parameters such as porosity coefficient, electric and magnetic potential, parameters of foundation, and geometrical parameters are investigated on vibration and buckling behavior of the sandwich beam. Numerical results of this paper show that porosity distribution has a significant effect on the stiffness of the sandwich beam. The results can be used for future analysis of magneto-electro-mechanical sandwich systems as actuators and sensors.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

Probabilistic Oblique Impact Analysis of Functionally Graded Plates – A Multivariate Adaptive Regression Splines Approach

2021 ◽  
Author(s):  
P. K. Karsh ◽  
Bindi Thakkar ◽  
R. R. Kumar ◽  
Vaishali ◽  
Sudip Dey
Keyword(s):  
Material Properties ◽  
Mass Density ◽  
Plate Thickness ◽  
Functionally Graded ◽  
Multivariate Adaptive Regression Splines ◽  
Low Velocity Impact ◽  
Fe Model ◽  
Low Velocity ◽  
Velocity Impact ◽  
Mars Model

Purpose: To investigate the probabilistic low-velocity impact of functionally graded (FG) plate using the MARS model, considering uncertain system parameters. Design/methodology/application: The distribution of various material properties throughout FG plate thickness is calculated using power law. For finite element (FE) formulation, isoparametric elements with eight nodes are considered, each component has five degrees of freedom. The combined effect of variability in material properties such as elastic modulus, modulus of rigidity, Poisson’s ratio, and mass density are considered. The surrogate model is validated with the FE model represented by the scatter plot and the probability density function (PDF) plot based on Monte Carlo simulation (MCS). Findings: The outcome of the degree of stochasticity, impact angle, impactor’s velocity, impactor’s mass density, and point of impact on the maximum value of contact force (CFmax ), plate deformation (PDmax), and impactor deformation (IDmax ) are determined. A convergence study is also performed to determine the optimal number of the constructed MARS model’s sample size. Originality/value: The results illustrate the significant effects of uncertain input parameters on FGM plates’ low-velocity impact responses by employing a surrogate-based MARS model.

Free vibration analysis and post-critical free vibrations of nanocomposite rotating beams reinforced with graphene platelet

2021 ◽  
pp. 107754632110511
Author(s):  
Arameh Eyvazian ◽  
Chunwei Zhang ◽  
Farayi Musharavati ◽  
Afrasyab Khan ◽  
Mohammad Alkhedher
Keyword(s):  
Natural Frequency ◽  
Thermal Environment ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Weight Fraction ◽  
Free Vibrations ◽  
Graphene Platelet ◽  
The Impact

Treatment of the first natural frequency of a rotating nanocomposite beam reinforced with graphene platelet is discussed here. In regard of the Timoshenko beam theory hypothesis, the motion equations are acquired. The effective elasticity modulus of the rotating nanocomposite beam is specified resorting to the Halpin–Tsai micro mechanical model. The Ritz technique is utilized for the sake of discretization of the nonlinear equations of motion. The first natural frequency of the rotating nanocomposite beam prior to the buckling instability and the associated post-critical natural frequency is computed by means of a powerful iteration scheme in reliance on the Newton–Raphson method alongside the iteration strategy. The impact of adding the graphene platelet to a rotating isotropic beam in thermal ambient is discussed in detail. The impression of support conditions, and the weight fraction and the dispersion type of the graphene platelet on the acquired outcomes are studied. It is elucidated that when a beam has not undergone a temperature increment, by reinforcing the beam with graphene platelet, the natural frequency is enhanced. However, when the beam is in a thermal environment, at low-to-medium range of rotational velocity, adding the graphene platelet diminishes the first natural frequency of a rotating O-GPL nanocomposite beam. Depending on the temperature, the post-critical natural frequency of a rotating X-GPL nanocomposite beam may be enhanced or reduced by the growth of the graphene platelet weight fraction.

Bending Vibrations of Variable Section Beams

1956 ◽  
Vol 23 (1) ◽  
pp. 103-108
Author(s):  
E. T. Cranch ◽  
Alfred A. Adler
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Cantilever Beams ◽  
Constant Depth ◽  
Bending Vibrations ◽  
Variable Section

Abstract Using simple beam theory, solutions are given for the vibration of beams having rectangular cross section with (a) linear depth and any power width variation, (b) quadratic depth and any power width variation, (c) cubic depth and any power width variation, and (d) constant depth and exponential width variation. Beams of elliptical and circular cross section are also investigated. Several cases of cantilever beams are given in detail. The vibration of compound beams is investigated. Several cases of free double wedges with various width variations are discussed.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

Design with Nonhomogeneous Materials—Part I: Pure Bending of Prismatic Bars

1987 ◽  
Vol 109 (1) ◽  
pp. 82-86 ◽  
Author(s):  
V. K. Stokes
Keyword(s):  
Tensile Test ◽  
Material Properties ◽  
Elastic Moduli ◽  
Mechanical Response ◽  
Mechanical Design ◽  
Beam Theory ◽  
Pure Bending ◽  
Strongly Coupled ◽  
Point To Point ◽  
Nonhomogeneous Materials

Because material properties vary from point to point in nonhomogeneous materials, there is some question as to what “properties” are measured in tests such as the tensile test, and how such “properties” can be used in the mechanical design process. In this paper, the mechanical response of nonhomogeneous prismatic bars in pure bending has been shown to depend on parameters that are strongly coupled combinations of geometry and material properties. The purely geometry based inertia tensor in homogeneous beam theory is replaced in the nonhomogeneous case by the rigidity tensor, which combines geometry and material properties. Interpretations for the average elastic moduli, which would be determined by tests on nonhomogeneous materials, have been explored. Also discussed is the usefulness of such average moduli for predicting the mechanical response of nonhomogeneous bars.

scholarly journals Vibration and Instability of Rotating Composite Thin-Walled Shafts with Internal Damping

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Ren Yongsheng ◽  
Zhang Xingqi ◽  
Liu Yanghang ◽  
Chen Xiulong
Keyword(s):  
Virtual Work ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Dynamical Analysis ◽  
Design Parameters ◽  
Internal Damping ◽  
Analysis Method ◽  
Governing Equations ◽  
Thin Walled

The dynamical analysis of a rotating thin-walled composite shaft with internal damping is carried out analytically. The equations of motion are derived using the thin-walled composite beam theory and the principle of virtual work. The internal damping of shafts is introduced by adopting the multiscale damping analysis method. Galerkin’s method is used to discretize and solve the governing equations. Numerical study shows the effect of design parameters on the natural frequencies, critical rotating speeds, and instability thresholds of shafts.

scholarly journals FORCED RESPONSE VIBRATION OF SIMPLY SUPPORTED BEAMS WITH AN ELASTIC PASTERNAK FOUNDATION UNDER A DISTRIBUTED MOVING LOAD

2020 ◽  
Vol 4 (2) ◽  
pp. 1-7
Author(s):  
Fatai Hammed ◽  
M. A. Usman ◽  
S. A. Onitilo ◽  
F. A. Alade ◽  
K. A. Omoteso
Keyword(s):  
Shear Modulus ◽  
Moving Load ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Forced Response ◽  
Dynamic Responses ◽  
Moving Force ◽  
Pasternak Elastic Foundation

In this study, the response of two homogeneous parallel beams with two-parameter Pasternak elastic foundation subjected to a constant uniform partially distributed moving force is considered. On the basis of Euler-Bernoulli beam theory, the fourth order partial differential equations of motion describing the behavior of the beams when subjected to a moving force were formulated. In order to solve the resulting initial-boundary value problem, finite Fourier sine integral technique and differential transform scheme were employed to obtain the analytical solution. The dynamic responses of the two beams obtained was investigated under moving force conditions using MATLAB. The effects of speed of the moving force, layer parameters such as stiffness (K_0) and shear modulus (G_0 ) have been conducted for the moving force. Various values of speed of the moving load, stiffness parameters and shear modulus were considered. The results obtained indicates that response amplitudes of both the upper and lower beams increases with increase in the speed of the moving load. Increasing the stiffness parameter is observed to cause a decrease in the response amplitudes of the beams. The response amplitudes decreases with increase in the shear modulus of the linear elastic layer.

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