Vibration Analysis of Rotary Tapered Axially Functionally Graded Timoshenko Nanobeam in Thermal Environment

Document Type: Research Paper

Authors

1 Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, Iran

2 Mechanical Engineering, Shahid Beheshti University, Tehran, Iran

10.22034/jsm.2019.563759.1273

Abstract

In this paper, vibration analysis of rotary tapered axially functionally graded (AFG) Timoshenko nanobeam is investigated in a thermal environment based on nonlocal theory. The governing equations of motion and the related boundary conditions are derived by means of Hamilton’s principle based on the first order shear deformation theory of beams. The solution method is considered using generalized differential quadrature element (GDQE) method. The accuracy of results are validated by other results reported in other references. The effect of various parameters such as AFG index, rate of cross section change, angular velocity, size effect and boundary conditions on natural frequencies are discussed comprehensively. The results show that with increasing angular velocity, non-dimensional frequency is increased and it depends on size effect parameter. Also, in the zero angular velocity, it can be seen with increasing AFG index, the frequencies are reducing, but in non-zero angular velocity, AFG index shows complex behavior on frequency.

Keywords


[1] Rasheedat M., 2012, Functionally graded material: An overview, Proceedings of the World Congress on Engineering.
[2] Wang S.S., 1983, Fracture mechanics for delamination problems in composite materials, Journal of Composite Materials 17: 210-223.
[3] Vytautas Ostasevicius R.D., 2011, Microsystems dynamics, International Series on Intelligent Systems, Control, and Automation: Science and Engineering.
[4] Li X.B.B., Takashima K., Baek C.W., Kim Y.K., 2003, Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques, Ultramicroscopy 97: 481-494.
[5] Pei J.T.F., Thundat T., 2004, Glucose biosensor based on the microcantilever, Analytical Chemistry 76: 292-297.
[6] Kaya M.O., 2006, Free vibration analysis of a rotating Timoshenko beam by differential transform method, Aircraft Engineering and Aerospace Technology 78:194-203.
[7] Dewey H., Hodges M.J.R., 1981, Free-vibration analysis of rotating beams by a variable-order finite-element method, AIAA Journal 19: 1459-1466.
[8] Chen W.L.L., Ma X., 2011, A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation, Composite Structures 93: 2723-2732.
[9] Akgöz B.C.Ö., 2012, Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory, Archive of Applied Mechanics 82: 423-443.
[10] Asghari M. Kahrobaiyan M.H., Ahmadian M.T., 2010, A nonlinear Timoshenko beam formulation based on the modified couple stress theory, Engineering Science 48: 1749-1761.
[11] Challamel N.,Wang.C.M., 2008, The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnology 19: 345703.
[12] Chen Y.Z.J., Zhang H., 2015, Free vibration analysis of rotating tapered Timoshenko beams via variational iteration method, Vibration and Control 2015: 1-15.
[13] Dehrouyeh-Semnani A., 2015, The influence of size effect on flapwise vibration of rotating microbeams, International Journal of Engineering Science 94: 150-163.
[14] Yong H., Xian-Fang L., 2010, A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, Journal of Sound and Vibration 329: 2291-2303.
[15] Shojaeefard M. H., 2018, Magnetic field effect on free vibration of smart rotary functionally graded nano/microplates: A comparative study on modified couple stress theory and nonlocal elasticity theory, Journal of Intelligent Material Systems and Structures 29(11): 2492-2507.
[16] Shojaeefard M.H., 2018, Vibration and buckling analysis of a rotary functionally graded piezomagnetic nanoshell embedded in viscoelastic media, Journal of Intelligent Material Systems and Structures 29(11): 2344-2361.
[17] Shojaeefard M.H., 2018, Free vibration of an ultra-fast-rotating-induced cylindrical nano-shell resting on a Winkler foundation under thermo-electro-magneto-elastic condition, Applied Mathematical Modelling 61: 255-279.
[18] Korak S., Ranjan G., 2014, Closed-form solutions for axially functionally gradedTimoshenko beams having uniform cross-section and fixed–fixed boundary condition,Composites: Part B 58: 361-370.
[19] Shahba A., Attarnejad R., Tavanaie Marvi M., Hajilar S., 2011, Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites: Part B 42: 801-808.
[20] Shahba A., Sundaramoorthy R., 2012, Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials, Applied Mathematical Modelling 36: 3094-3111.
[21] Yong H., Ling-E Y., Qi-Zhi L., 2013, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Composites: Part B 45: 1493-1498.
[22] Sundaramoorthy R., 2013, Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods, Applied Mathematical Modelling 37: 4440-4463.
[23] Swaminathan K., Naveenkuma D.T., Zenkour A. M., Carrera E., 2014, Stress, vibration and buckling analyses of FGM plates—A stateof-the-art review, Composite Structures 120: 10-31.
[24] Shojaeefard M.H. , Saeidi Googarchin H., Mahinzare M., Ghadiri M. 2018, Free vibration and critical angular velocity of a rotating variable thickness two-directional FG circular microplate, Microsystem Technologies 24: 1525-1543.
[25] Shojaeefard M.H. , Saeidi Googarchin H., Ghadiri M., Mahinzare M., 2017, Micro temperature-dependent FG porous plate: Free vibration and thermal buckling analysis using modified couple stress theory with CPT and FSDT, Applied Mathematical Modelling 50: 633-655.
[26] Bellman R., Casti J., 1971, Differential quadrature and long-term integration, Journal of Mathematical Analysis and Applications 34: 235-238.
[27] Bellman R., Kashef B., Casti J., 1972, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics 10: 40-52.
[28] Shu C., 2000, Differential Quadrature and its Application in Engineering, Springer.
[29] Shu C., Richards B.E., 1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids 15: 791-798.
[30] Viola E., Miniaci M., Fantuzzi N., Marzani A., 2015, Vibration analysis of multi-stepped and multi-damaged parabolic arches using GDQ, Curved and Layered Structures 2: 28-49.
[31] Wang X., GU H., 1997, Static analysis of frame structures by the differential quadrature element method, International Journal for Numerical Methods in Engineering 40: 759-772.
[32] Chen C.-N., 1998, Solution of beam on elastic foundation by DQEM, Journal of Engineering Mechanics 124: 3509-3526.
[33] Shahba A., Attarnejad R., Hajilar S., 2013, A mechanical-based solution for axially functionally graded tapered Euler-Bernoulli beams, Mechanics of Advanced Materials and Structures 20: 696-707.
[34] Yang J., Shen H.-S., 2002, Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments, Journal of Sound and Vibration 255: 579-602.
[35] Shafiei N., Kazemi M., Ghadiri M., 2015, On size-dependent vibration of rotary axially functionally graded microbeam, International Journal of Engineering Science 101: 29-44.
[36] Wang C., Zhang Y., He X., 2007, Vibration of nonlocal Timoshenko beams, Nanotechnology 18: 105401.