Document Type: Research Paper

**Authors**

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

**Abstract**

In this article, transverse vibration of a cantilever nano- beam with functionally graded materials and carrying a concentrated mass at the free end is studied. Material properties of FG beam are supposed to vary through thickness direction of the constituents according to power-law distribution (P-FGM). The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on Timoshenko beam theory in order to consider the effect of shear deformation and rotary inertia. Hamilton’s principle is applied to obtain the governing differential equation of motion and boundary conditions and they are solved applying analytical solution. The purpose is to study the effects of parameters such as tip mass, small scale, beam thickness, power-law exponent and slenderness on the natural frequencies of FG cantilever nano beam with a point mass at the free end. It is explicitly shown that the vibration behavior of a FG Nano beam is significantly influenced by these effects. The response of Timoshenko Nano beams obtained using an exact solution in a special case is compared with those obtained in the literature and is found to be in good agreement. Numerical results are presented to serve as benchmarks for future analyses of FGM cantilever Nano beams with tip mass.

**Keywords**

[2] Zhang Y. Q., Liu G. R., Wang J. S., 2004, Small-scale effects on buckling of multi walled carbon nanotubes under axial compression, Physical Review B 70(20): 205430.

[3] Eringen A. C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10(1): 1-16.

[4] Eringen A. C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54(9): 4703-4710.

[5] Peddieson J., George R. B., Richard P. M., 2003, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science 41(3): 305-312.

[6] Aydogdu M., 2009, A general nonlocal beam theory: its application to Nano beam bending, buckling and vibration, Physica E: Low-Dimensional Systems and Nanostructures 41(9): 1651-1655.

[7] Phadikar J. K., Pradhan S. C., 2010, Variational formulation and finite element analysis for nonlocal elastic Nano beams and Nano plates, Computational Materials Science 49(3): 492-499.

[8] Pradhan S. C., Murmu T., 2010, Application of nonlocal elasticity and DQM in the flap wise bending vibration of a rotating Nano cantilever, Physica E: Low-Dimensional Systems and Nanostructures 42(7): 1944-1949.

[9] Ghorbanpour Arani A., Kolahchi R., Rahimi pour H., Ghaytani M., Vossough H., 2012, Surface stress effects on the bending wave propagation of Nano beams resting on a pasternak foundation, International Conference on Modern Application of Nanotechnology, Belarus.

[10] Ansari R., Gholami R., Sahmani S., 2011, Free vibration analysis of size-dependent functionally graded

microbeams based on the strain gradient Timoshenko beam theory, Composite Structures 94(1): 221-228.

[11] Ebrahimi F., Salari E., 2015, Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG Nano beams with various boundary conditions, Composites Part B: Engineering 78: 272-290.

[12] Srinath L. S., Das Y. C., 1967, Vibration of beams carrying mass , Journal of Applied Mechanics 34(3): 784-785.

Goel R. P., 1976, Free vibrations of a beam mass system with elastically restrained ends, Journal of Sound and Vibration 47: 9-14.

[13] Saito H., Otomi K., 1979, Vibration and stability of elastically supported beams carrying an attached mass axial and tangential loads, Journal of Sound and Vibration 62: 257-266.

[14] Lau J. H., 1981, Fundamental frequency of a constrained beam , Journal of Sound and Vibration 78: 154-157.

[15] Lauara P. A. A., Filipich C., Cortinez V. H., 1987, Vibration of beams and plates carrying concentrated masses, Journal of Sound and Vibration 117: 459-465.

[16] Liu W. H., Yeh F. H., 1987, Free vibration of a restrained-uniform beam with intermediate masses, Journal of Sound and Vibration 117: 555-570.

[17] Maurizi M. J., Belles P. M., 1991, Natural frequencies of the beam-mass system: comparison of the two fundamental theories of beam vibrations, Journal of Sound and Vibration 150: 330-334.

[18] Maurizi M. J., Belles P. M., 1991, Natural frequencies of the beam-mass system: comparison of the two fundamental theories of beam vibrations, Journal of Sound and Vibration 150: 330-334.

[19] Bapat C.N., Bapat C., 1987, Natural frequencies of a beam with non-classical boundary conditions and concentrated masses, Journal of Sound and Vibration 112: 177-182.

[20] Oz H. R., 2000, Calculation of the natural frequencies of a beam-mass system using finite element method, Mathematical and Computational Applications 5: 67-75.

[21] Low K.H.,1991, A comprehensive approach for the Eigen problem of beams with arbitrary boundary conditions, Computers & Structures 39: 671-678.

[22] Kosmatka J.B., 1995, An improved two-node finite element for stability and natural frequencies of axial-loaded Timoshenko beams, Computers & Structures 57:141-149.

[23] Lin H.P., Chang S.C.,2005, Free vibration analysis of multi-span beams with intermediate flexible constraints, Journal of Sound and Vibration 281: 155-169.

[24] Ferreira A.J.M., Fasshauer G.E., 2006, Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method, Computer Methods in Applied Mechanics and Engineering 196:134-146.

[25] Ruta P., 2006, The application of Chebyshev polynomials to the solution of the nonprismatic Timoshenko beam vibration problem, Journal of Sound and Vibration 296: 243-263.

[26] Laura P.A.A., Pombo J.A., Susemihl E.A., 1974, A note on the vibration of a clamped-free beam with a mass at the free end, Journal of Sound and Vibration 37: 161-168.

[27] Goel R.P., 1976, Free vibrations of a beam-mass system with elastically restrained ends, Journal of Sound and Vibration 47: 9-14.

[28] Parnell L.A., Cobble M.H., 1976, Lateral displacements of a vibrating cantilever beam with a concentrated mass, Journal of Sound and Vibration 44: 499-511.

[29] To C.W.S., 1982, Vibration of a cantilever beam with a base excitation and tip mass, Journal of Sound and Vibration 83: 445-460.

[30] Grant D.A., 1978, The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass, Journal of Sound and Vibration 57: 357-365.

[31] Brunch Jr J.C., Mitchell T.P., 1987, Vibrations of a mass-loaded clamped-free Timoshenko beam, Journal of Sound and Vibration 114: 341-345.

[32] Abramovich H., Hamburger O., 1991, Vibration of a cantilever Timoshenko beam with a tip mass, Journal of Sound and Vibration 148: 162-170.

[33] Abramovich H., Hamburger O., 1992,Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass, Journal of Sound and Vibration 154: 67-80.

[34] Rossi R.E., Laura P.A.A., Avalos D.R., Larrondo H., 1993, Free vibrations of Timoshenko beams carrying elastically mounted concentrated masses, Journal of Sound and Vibration 165: 209-223.

[35] Salarieh H., Ghorashi M., 2006, Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling, International Journal of Mechanical Sciences 48:763-779.

[36] Wu J.S., Hsu S.H., 2007, The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia, Ocean Engineering 34: 54-68.

[37] Lin H.Y., Tsai Y.C., 2007, Free vibration analysis of a uniform multi-span carrying multiple spring-mass systems, Journal of Sound and Vibration 302: 442-456.

[38] Necla T., 2016, Nonlinear vibration of nanobeam with attached mass at the free end via nonlocal elasticity theory, Microsystem Technologies 22(9): 2349-2359.

[39] Simsek M., 2010, Fundamental frequency of functionally graded beams by using different higher order beam theories, Nuclear Engineering and Design 240(4): 697-705.

[40] Pradhan K.k., Chakraverty S., 2014, Effects of different shear deformation theories on free vibration of functionally graded beams, International Journal of Mechanical Science 82:149-160.

Volume 10, Issue 1

Winter 2018

Pages 23-37