Abstract
Purpose
The present study aims to obtain the exact solutions of the free transverse vibration of non-uniform axially functionally graded (NAFG) beams with end point masses and general boundary conditions. Also, the effects of the attached end point masses, rotational and translational elastic supports, and NAFG parameters on the natural frequencies of the power-law NAFG beams are investigated.
Methods
Based on the Euler–Bernoulli beam theory, the governing differential equation of motion was solved accurately using the Bessel functions. Then, the constant coefficients matrices of the power-law NAFG beams with the end point masses and general elastic supports were derived by applying the boundary conditions. The general elastic boundary conditions are modeled with the linear rotational and lateral translational springs. Furthermore, the material and geometrical properties of the NAFG beams are assumed to change continuously and together in the axial direction according to the power-law forms. By taking the constant coefficients matrix determinant equal to zero and calculating the positive real roots, the natural frequencies were obtained. By comparing the responses of the numerical examples with the available solutions, the accuracy and ability of the proposed formulations are demonstrated.
Results and Conclusion
Obtained results show the natural frequencies of the power-law NAFG beam decrease with the increase of the mass ratio and increase with the increase of the stiffness ratios of the supports. Moreover, the natural frequencies of the power-law NAFG beam increase with the increase of NAFG parameters. Depending on the boundary conditions, the mass sensitivity differs from one power-law NAFG beam to another, and from one mode of vibration to another. The exact analytical solutions are listed in tabular and graphical forms and can be used as the benchmark solutions. Moreover, the results presented here can be used for the proper design of composite beams carrying end point masses with different elastic boundary conditions.
Similar content being viewed by others
References
Bambaeechee M (2019) Free vibration of AFG beams with elastic end restraints. Steel Compos Struct 33:403–432
Mabie HH, Rogers CB (1964) Transverse vibrations of tapered cantilever beams with end loads. J Acoust Soc Am 36:463–469
Mabie HH, Rogers CB (1974) Transverse vibrations of double-tapered cantilever beams with end support and with end mass. J Acoust Soc Am 55:986–991
Sankaran GV, Kanaka Raju K, Venkateswara Rao G (1975) Vibration frequencies of a tapered beam with one end spring-hinged and carrying a mass at the other free end. J Appl Mech 42:740–741
Goel RP (1976) Transverse vibrations of tapered beams. J Sound Vib 47:1–7
Lee TW (1976) Transverse vibrations of a tapered beam carrying a concentrated mass. J Appl Mech 43:366–367
Lau JH (1984) Vibration frequencies of tapered bars with end mass. J Appl Mech 51:179–181
Lau JH (1984) Vibration frequencies for a non-uniform beam with end mass. J Sound Vib 97:513–521
Laura PAA, Gutierrez RH (1986) Vibrations of an elastically restrained cantilever beam of varying cross section with tip mass of finite length. J Sound Vib 108:123–131
Alvarez SI, Ficcadenti de Iglesias GM, Laura PAA (1988) Vibrations of an elastically restrained, non-uniform beam with translational and rotational springs, and with a tip mass. J Sound Vib 120:465–471
Yang KY (1990) The natural frequencies of a non-uniform beam with a tip mass and with translational and rotational springs. J Sound Vib 137:339–341
Rossi RE, Laura PAA, Gutierrez RH (1990) A note on transverse vibrations of a Timoshenko beam of non-uniform thickness clamped at one end and carrying a concentrated mass at the other. J Sound Vib 143:491–502
Lee SY, Lin SM (1992) Exact vibration solutions for nonuniform Timoshenko beams with attachments. AIAA J 30:2930–2934
Matsuda H, Morita C, Sakiyama T (1992) A method for vibration analysis of a tapered timoshenko beam with constraint at any points and carrying a heavy tip body. J Sound Vib 158:331–339
Grossi RO, Aranda A, Bhat RB (1993) Vibration of tapered beams with one end spring hinged and the other end with tip mass. J Sound Vib 160:175–178
Auciello NM (1996) LETTER TO THE EDITOR: Free vibrations of a linearly tapered cantilever beam with constraining springs and tip mass. J Sound Vib 192:905–911
Auciello NM (1996) Transverse vibrations of a linearly tapered cantilever beam with tip mass of rotary inertia and eccentricity. J Sound Vib 194:25–34
Auciello NM, Maurizi MJ (1997) On the natural vibrations of tapered beams with attached inertia elements. J Sound Vib 199:522–530
Auciello NM, Nolè G (1998) Vibrations of a cantilever tapered beam with varying section properties and carrying a mass at the free end. J Sound Vib 214:105–119
Wu J, Hsieh M (2000) Free vibration analysis of a non-uniform beam with multiple point masses. Struct Eng Mech 9:449–467
Li QS (2000) An exact approach for free flexural vibrations of multistep nonuniform beams. J Vib Control 6:963–983
Li QS (2002) Free vibration analysis of non-uniform beams with an arbitrary number of cracks and concentrated masses. J Sound Vib 252:509–525
Chen D-W, Wu J-S (2002) The exact solutions for the natural frequencies and mode shapes of non-uniform beams with multiple spring-mass systems. J Sound Vib 255:299–322
Karami G, Malekzadeh P, Shahpari SA (2003) A DQEM for vibration of shear deformable nonuniform beams with general boundary conditions. Eng Struct 25:1169–1178
Wu J-S, Chen D-W (2003) Bending vibrations of wedge beams with any number of point masses. J Sound Vib 262:1073–1090
Wu J-S, Chiang L-K (2004) Free vibrations of solid and hollow wedge beams with rectangular or circular cross-sections and carrying any number of point masses. Int J Numer Methods Eng 60:695–718
De Rosa MA, Maurizi MJ (2005) Dam** in exact analysis of tapered beams. J Sound Vib 286:1041–1047
Wu J-S, Chen C-T (2005) An exact solution for the natural frequencies and mode shapes of an immersed elastically restrained wedge beam carrying an eccentric tip mass with mass moment of inertia. J Sound Vib 286:549–568
Chen D-W, Liu T-L (2006) Free and forced vibrations of a tapered cantilever beam carrying multiple point masses. Struct Eng Mech 23:209–216
Lai H-Y, Chen C-K, Hsu J-C (2008) Free vibration of non-uniform Euler-Bernoulli beams by the Adomian modified decomposition method. CMES - Comput Model Eng Sci 34:87–115
Lin H-Y (2010) An exact solution for free vibrations of a non-uniform beam carrying multiple elastic-supported rigid bars. Struct Eng Mech 34:399–416
Attarnejad R, Shahba A, Eslaminia M (2011) Dynamic basic displacement functions for free vibration analysis of tapered beams. J Vib Control 17:2222–2238
Firouz-Abadi RD, Rahmanian M, Amabili M (2013) Exact solutions for free vibrations and buckling of double tapered columns with elastic foundation and tip mass. J Vib Acoust 135:051017-1–51110
Wang CY (2013) Vibration of a tapered cantilever of constant thickness and linearly tapered width. Arch Appl Mech 83:171–176
Malaeke H, Moeenfard H (2016) Analytical modeling of large amplitude free vibration of non-uniform beams carrying a both transversely and axially eccentric tip mass. J Sound Vib 366:211–229
Sagar Singh S, Pal P, Kumar Pandey A (2016) Mass sensitivity of nonuniform microcantilever beams. J Vib Acoust 138
Nikolić A, Šalinić S (2017) A rigid multibody method for free vibration analysis of beams with variable axial parameters. J Vib Control 23:131–146
Torabi K, Afshari H, Sadeghi M et al (2017) Exact closed-form solution for vibration analysis of truncated conical and tapered beams carrying multiple concentrated masses. J Solid Mech 9:760–782
Huang CA, Wu JS, Shaw H-J (2018) Free vibration analysis of a nonlinearly tapered beam carrying arbitrary concentrated elements by using the continuous-mass transfer matrix method. J Mar Sci Technol Taiwan 26:28–49
Hsu CP, Hung CF, Liao JY (2018) Shock and Vibration, A Chebyshev spectral method with null space approach for boundary-value problems of Euler-Bernoulli beam, 2018. Available from: https://www.hindawi.com/journals/sv/2018/2487697/.
Elishakoff I, Johnson V (2005) Apparently the first closed-form solution of vibrating inhomogeneous beam with a tip mass. J Sound Vib 286:1057–1066
Elishakoff I, Perez A (2005) Design of a polynomially inhomogeneous bar with a tip mass for specified mode shape and natural frequency. J Sound Vib 287:1004–1012
Huang Y, Li X-F (2010) A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. J Sound Vib 329:2291–2303
De Rosa MA, Lippiello M, Maurizi MJ et al (2010) Free vibration of elastically restrained cantilever tapered beams with concentrated viscous dam** and mass. Mech Res Commun 37:261–264
Shahba A, Attarnejad R, Marvi MT et al (2011) Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Compos Part B Eng 42:801–808
Wang CY, Wang CM (2012) Exact vibration solution for exponentially tapered cantilever with tip mass. J Vib Acoust 134:041012-1–41014
Li X-F, Kang Y-A, Wu J-X (2013) Exact frequency equations of free vibration of exponentially functionally graded beams. Appl Acoust 74:413–420
Li XF (2013) Free vibration of axially loaded shear beams carrying elastically restrained lumped-tip masses via asymptotic Timoshenko beam theory. J Eng Mech 139:418–428
Zhang H, Kang YA, Li X-F (2013) Stability and vibration analysis of axially-loaded shear beam-columns carrying elastically restrained mass. Appl Math Model 37:8237–8250
Tang A-Y, Wu J-X, Li X-F et al (2014) Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams. Int J Mech Sci 89:1–11
Tang H-L, Shen Z-B, Li D-K (2014) Vibration of nonuniform carbon nanotube with attached mass via nonlocal Timoshenko beam theory. J Mech Sci Technol 28:3741–3747
Yuan J, Pao Y-H, Chen W (2016) Exact solutions for free vibrations of axially inhomogeneous Timoshenko beams with variable cross section. Acta Mech 227:2625–2643
Chen DQ, Sun DL, Li XF (2017) Surface effects on resonance frequencies of axially functionally graded Timoshenko nanocantilevers with attached nanoparticle. Compos Struct 173:116–126
Rahmani O, Mohammadi Niaei A, Hosseini SAH et al (2017) In-plane vibration of FG micro/nano-mass sensor based on nonlocal theory under various thermal loading via differential transformation method. Superlattices Microstruct 101:23–39
Nikolić A (2017) Free vibration analysis of a non-uniform axially functionally graded cantilever beam with a tip body. Arch Appl Mech 87:1227–1241
Rossit CA, Bambill DV, Gilardi GJ (2017) Free vibrations of AFG cantilever tapered beams carrying attached masses. Struct Eng Mech 61:685–691
Ghadiri M, Jafari A (2018) A nonlocal first order shear deformation theory for vibration analysis of size dependent functionally graded nano beam with attached tip mass: an exact solution. J Solid Mech 10:23–37
Šalinić S, Obradović A, Tomović A (2018) Free vibration analysis of axially functionally graded tapered, stepped, and continuously segmented rods and beams. Compos Part B Eng 150:135–143
Rossit CA, Bambill DV, Gilardi GJ (2018) Timoshenko theory effect on the vibration of axially functionally graded cantilever beams carrying concentrated masses. Struct Eng Mech 66:703–711
Mahmoud MA (2019) Natural frequency of axially functionally graded, tapered cantilever beams with tip masses. Eng Struct 187:34–42
Sun D-L, Li X-F (2019) Initial value method for free vibration of axially loaded functionally graded Timoshenko beams with nonuniform cross section. Mech Based Des Struct Mach 47:102–120
Nguyen KV, Dao TTB, Van Cao M (2020) Comparison studies of the receptance matrices of the isotropic homogeneous beam and the axially functionally graded beam carrying concentrated masses. Appl Acoust 160:107160
Li Z, Xu Y, Huang D (2021) Analytical solution for vibration of functionally graded beams with variable cross-sections resting on Pasternak elastic foundations. Int J Mech Sci 191:106084
Sahu RP, Sutar MK, Pattnaik S (2022) A generalized finite element approach to thefree vibration analysis of non-uniform axially functionally graded beam Scientia Iranica B 29(2):556–571
Liu X, Chang L, Banerjee JR et al (2022) Closed-form dynamic stiffness formulation for exact modal analysis of tapered and functionally graded beams and their assemblies. Int J Mech Sci 214:106887
Rao SS (2019) Vibration of Continuous Systems. John Wiley & Sons Inc
Wang CY, Wang CM (2013) Structural Vibration: Exact Solutions for Strings, Membranes, Beams, and Plates. Florida, CRC Press, Boca Raton
Watson GN (1995) A Treatise on the Theory of Bessel Functions. Cambridge University Press
Çelik İ (2018) Free vibration of non-uniform Euler-Bernoulli beam under various supporting conditions using Chebyshev wavelet collocation method. Appl Math Model 54:268–280
Mao Q (2011) Free vibration analysis of multiple-stepped beams by using Adomian decomposition method. Math Comput Model 54:756–764
Hsu J-C, Lai H-Y, Chen CK (2008) Free vibration of non-uniform Euler-Bernoulli beams with general elastically end constraints using Adomian modified decomposition method. J Sound Vib 318:965–981
De Rosa MA, Auciello NM (1996) Free vibrations of tapered beams with flexible ends. Comput Struct 60:197–202
Funding
The author received no financial support for the research, authorship, and/or publication of this article.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A
For the derivation of the general solution Eq. (9), the compact form of the differential equation obtained in Eq. (8), according to Eq. (4), can be expressed as follows [67]:
Eq. (28) can be factored into:
Each one of the brackets in Eq. (29) is a Bessel operator. One can find the general solution of Wi(X) as:
where
The Eq. (31) is the regular Bessel differential equation, and the solution can be written as:
where C1 and C2 are unknown constants, and Jp and Yp are, respectively, the Bessel functions of first and second kinds of order p. The Eq. (32) is known as the modified Bessel differential equation, and the solution can be expressed as:
where C3 and C4 are unknown constants, and Ip and Kp are, respectively, the modified Bessel functions of first and second kinds of order p. Therefore, the general solution of Wi(X) according to Eq. (30) is:
Appendix B
The elements of the constant coefficients matrix, A for the NAFG beams with the positive gradient coefficient (i.e., c > 0), carrying tip masses and various elastic boundary conditions are as follows:
For the uniform beam, i.e., c = 0, the entries of the unknown constants' matrix, A are as below:
The elements of the constant coefficients matrix, A for the NAFG beams with the negative gradient coefficient (i.e., -1 < c < 0), carrying tip masses and general elastic boundary conditions are as follows:
Rights and permissions
About this article
Cite this article
Bambaeechee, M. Free Transverse Vibration of General Power-Law NAFG Beams with Tip Masses. J. Vib. Eng. Technol. 10, 2765–2797 (2022). https://doi.org/10.1007/s42417-022-00519-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42417-022-00519-7