Effect of axially functionally graded material on whirling frequencies and critical speeds of a spinning Timoshenko beam

2018 ◽  
Vol 192 ◽  
pp. 355-367 ◽  
Author(s):  
Yixin Huang ◽  
Tianshu Wang ◽  
Yang Zhao ◽  
Pingping Wang
Keyword(s):  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Functionally Graded ◽  
Critical Speeds ◽  
Graded Material ◽  
Axially Functionally Graded Material

Stability of the Size-Dependent and Functionally Graded Curvilinear Timoshenko Beams

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
J. Awrejcewicz ◽  
A. V. Krysko ◽  
S. P. Pavlov ◽  
M. V. Zhigalov ◽  
V. A. Krysko
Keyword(s):  
Differential Equations ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Stability Loss ◽  
Functionally Graded ◽  
Beam Deflection ◽  
Timoshenko Beams ◽  
Size Dependent ◽  
Rigid Layer ◽  
Graded Material

The size-dependent model is studied based on the modified couple stress theory for the geometrically nonlinear curvilinear Timoshenko beam made from a functionally graded material having its properties changed along the beam thickness. The influence of the size-dependent coefficient and the material grading on the stability of the curvilinear beams is investigated with the use of the setup method. The second-order accuracy finite difference method is used to solve the problem of nonlinear partial differential equations (PDEs) by reducing it to the Cauchy problem. The obtained set of nonlinear ordinary differential equations (ODEs) is then solved by the fourth-order Runge–Kutta method. The relaxation method is employed to solve numerous static problems based on the dynamic approach. Eight different combinations of size-dependent coefficients and the functionally graded material coefficient are used to study the stress-strain responses of Timoshenko beams. Stability loss of the curvilinear Timoshenko beams is investigated using the Lyapunov criterion based on the estimation of the Lyapunov exponents. Beams with/without the size-dependent behavior, homogeneous beams, and functionally graded beams having the same stiffness are investigated. It is shown that in straight-line beams, the size-dependent effect decreases the beam deflection. The same is observed if the most rigid layer is located on the top of the beam. In the curvilinear Timoshenko beam, such a location of the most rigid layer essentially improves the beam strength against stability loss. The observed transition of the largest Lyapunov exponent from a negative to positive value corresponds to the transition from a precritical to postcritical beam state.

On modal analysis of axially functionally graded material beam under hygrothermal effect

2019 ◽  
Vol 234 (5) ◽  
pp. 1085-1101 ◽  
Author(s):  
Pankaj Sharma ◽  
Rahul Singh ◽  
Muzamal Hussain
Keyword(s):  
Modal Analysis ◽  
Functionally Graded Material ◽  
Volume Fraction ◽  
Axial Direction ◽  
Functionally Graded ◽  
Element Analysis ◽  
Hygrothermal Effect ◽  
Graded Material ◽  
Eigen Frequencies ◽  
Axially Functionally Graded Material

This investigation focuses on the modal analysis of an axially functionally graded material beam under hygrothermal effect. The material constants of the beam are supposed to be graded smoothly along the axial direction under both power law and sigmoid law distribution. A finite element analysis with COMSOL Multiphysics® (version 5.2) package is used to find the Eigen frequencies of the beam. The accuracy of the technique is authenticated by relating the results with the prior investigation for reduced case. The effects of moisture changes, temperature, and volume fraction index, length-to-thickness ratio on the Eigen frequencies are investigated in detail. It is believed that the present investigation may be useful in the design of highly efficient environmental sensors for structural health monitoring perspective.

Vibration analysis of an axially functionally graded material non-prismatic beam under axial thermal variation in humid environment

2021 ◽  
pp. 107754632110371
Author(s):  
Rahul Singh ◽  
Pankaj Sharma
Keyword(s):  
Functionally Graded Material ◽  
Temperature Rise ◽  
Differential Quadrature Method ◽  
Functionally Graded ◽  
Quadrature Method ◽  
Thermal Variation ◽  
Harmonic Differential Quadrature Method ◽  
Prismatic Beam ◽  
Graded Material ◽  
Axially Functionally Graded Material

The vibration analysis of an axially functionally graded material non-prismatic Timoshenko beam under axial thermal variation in humid environment is carried out using the harmonic differential quadrature method. In this modeling, the length and width of the beam remains constant whereas thickness of the beam is linearly varied along the axis of the beam. The material properties are temperature dependent and are assumed to be varied continuously along the axial direction according to power law distribution. Three types of temperature variations are considered in this study, that is, uniform temperature rise, linear temperature rise, and non-linear temperature rise. The temperature of the beam remains constant under uniform temperature rise condition and it is varied linearly and nonlinearly along the length of beam for rest of the conditions. The beam is subjected to uniform moisture concentration to impose humidity. Hamiltonian’s approach is used to derive the governing equations of motion. The resultant governing equations are then solved using the harmonic differential quadrature method to obtain the natural frequencies of the axially functionally graded material non-prismatic beam. The results obtained using the harmonic differential quadrature method are compared with results obtained for special cases. The effects of thermal variation, humidity, non-homogeneity parameter, and end conditions on natural frequencies of the non-prismatic beam are reported.

Effect of cracks on nonlinear flexural vibration of rotating Timoshenko functionally graded material beam having large amplitude motion

2017 ◽  
Vol 232 (6) ◽  
pp. 930-940 ◽  
Author(s):  
B Panigrahi ◽  
G Pohit
Keyword(s):  
Centrifugal Force ◽  
Functionally Graded Material ◽  
Timoshenko Beam ◽  
Ritz Method ◽  
Functionally Graded ◽  
Neutral Axis ◽  
Spanwise Direction ◽  
Governing Equations ◽  
Graded Material ◽  
Fixed End

This study investigates the stiffening effect due to rotation on the nonlinear vibrational characteristics for cracked Timoshenko beam for the first time. Fixed end of the beam is attached to a rotating hub. Functionally graded material is taken into consideration, in which the properties vary as a continuous function along the depth of the beam. An elastic mass-less rotational spring is assumed in the place of crack, which splits the beam into two different parts. The point on the neutral axis at the fixed end is assumed to be the center of rotation of the beam. Centrifugal force is considered to act towards the spanwise direction and along the neutral axis. An additional displacement due to rotation of the beam along with the centrifugal force is incorporated with the energy formulation. Timoshenko beam theory and classical Ritz method is employed to derive the governing equations. In order to solve the nonlinear governing equations, direct substitution iterative technique is used. Effects of various parameters such as rotating speeds, radius of hub, depth of crack, location of crack, and different functionally graded material properties on linear and nonlinear vibration characteristics are studied. Validity of the present methodology is assured by comparing the results with some of the results from the existing literatures.

Effects of Support Conditions to the Post-Buckling Behaviors of Axially Functionally Graded Material Rods

2017 ◽  
Vol 730 ◽  
pp. 502-509 ◽  
Author(s):  
Buntara Sthenly Gan ◽  
Thanh Huong Trinh ◽  
Takahiro Hara ◽  
Dinh Kien Nguyen ◽  
Thi Thom Tran
Keyword(s):  
Finite Element ◽  
Material Property ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Support Condition ◽  
Non Linear ◽  
Graded Material ◽  
Post Buckling ◽  
Support Conditions ◽  
Axially Functionally Graded Material

The effects of support conditions to the post-buckling behaviors of rod structures made of Axially Functionally Graded Material (AFGM) are presented. The material property of the rod member is assumed to vary linearly in the axis direction of the member. The non-linear material property of the rod element is formulated in the Finite Element context. The consistent shape functions for the rod element were developed to take into account the varying material property in the finite element formulation. The geometrically non-linear behavior of the rod element is formulated in the context of the updated co-rotational formulation. The non-linear equilibrium equations are solved by using the incremental and iterative procedures in combination with the arc-length control method. The influences of the material distribution on the post-buckling behaviors of the AFGM Williams’ toggle frames under various support conditions are highlighted. As a result, the graded between two materials can increase the post-buckling behaviors of the AFGM rod element regardless of the types of support conditions. The orientation of material distributions combined with the type of support condition can increase the performance of the rod element. The fixed-fixed support condition showed the highest performance of the AFGM rod element.

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