Abstract
This paper investigates the dynamic response of viscoelastic axially functionally graded (AFG) barrel and hyperboloidal coil springs with variable cross-sectional area. Equations governing the dynamic behaviour of spatial rods are obtained via Timoshenko beam theory. The viscoelastic characteristics of the material are described by Kelvin’s model. The transfer matrix method and stiffness matrix methods are used in combination in the numerical solution of the problem. Stiffness matrices are determined by the transfer matrix method (TMM). Solutions are obtained in the Laplace domain; the results are transformed into the time domain by Durbin’s inverse Laplace transform algorithm. A benchmark solution for verifying non-cylindrical geometry is successfully integrated into the damped forced vibration analysis. A parametric study is conducted in which cylinder radius ratio, damping ratio, material gradient and cross-sectional area are varied for both helical rod geometries mentioned above.
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References
Aribas, U.N., Ermis, M., Eratli, N., Omurtag, M.H.: The static and dynamic analyses of warping included composite exact conical helix by mixed FEM. Composites, Part B, Eng. 160, 285–297 (2019). https://doi.org/10.1016/j.compositesb.2018.10.018
Aribas, U.N., Ermis, M., Kutlu, A., Eratli, N., Omurtag, M.H.: Forced vibration analysis of composite geometrically exact elliptical cone helices via mixed FEM. Mech. Adv. Mat. Struct., 1–19 (2020). https://doi.org/10.1080/15376494.2020.1824048
Bakhshi Khaniki, H., Hosseini-Hashemi, S.: Dynamic transverse vibration characteristics of nonuniform nonlocal strain gradient beams using the generalized differential quadrature method. Eur. Phys. J. Plus 132(11), 1–15 (2017). https://doi.org/10.1140/epjp/i2017-11757-4
Boley, B.A., Weiner, J.H.: Theory of Thermal Stresses. Wiley, New York (1960)
Calim, F.F.: Dynamic analysis of viscoelastic, anisotropic curved spatial rod systems. PhD thesis, Çukurova Univ. Adana (2003)
Calim, F.F.: Forced vibration of helical rods of arbitrary shape. Mech. Res. Commun. 36, 882–891 (2009). https://doi.org/10.1016/j.mechrescom.2009.07.007
Calim, F.F.: Forced vibration of curved beams on two-parameter elastic foundation. Appl. Math. Model. 36, 964–973 (2012). https://doi.org/10.1016/j.apm.2011.07.066
Calim, F.F.: Free and forced vibration analysis of axially functionally graded Timoshenko beams on two-parameter viscoelastic foundation. Composites, Part B, Eng. 103, 98–112 (2016a). https://doi.org/10.1016/j.compositesb.2016.08.008
Calim, F.F.: Dynamic response of curved Timoshenko beams resting on viscoelastic foundation. Struct. Eng. Mech. 59, 761–774 (2016b). https://doi.org/10.12989/sem.2016.59.4.761
Calim, F.F.: Transient analysis of axially functionally graded Timoshenko beams with variable cross-section. Composites, Part B, Eng. 98, 472–483 (2016c). https://doi.org/10.1016/j.compositesb.2016.05.040
Calim, F.F.: Vibration analysis of functionally graded Timoshenko beams on Winkler–Pasternak elastic foundation. Iran. J. Sci. Technol. Trans. Civ. Eng. 44, 901–920 (2020). https://doi.org/10.1007/s40996-019-00283-x
Calim, F.F., Cuma, Y.C.: Vibration analysis of nonuniform hyperboloidal and barrel helices made of functionally graded material. Mech. Based Des. Struct. Mach., 1–15 (2020). https://doi.org/10.1080/15397734.2020.1822181
Calim, F.F., Cuma, Y.C.: Forced vibration analysis of viscoelastic helical rods with varying cross-section and functionally graded material. Mech. Based Des. Struct. Mach., 1–12 (2021). https://doi.org/10.1080/15397734.2021.1931307
Cuma, Y.C., Calim, F.F.: Free vibration analysis of functionally graded cylindrical helices with variable cross-section. J. Sound Vib. 494, 115856 (2021a). https://doi.org/10.1016/j.jsv.2020.115856
Cuma, Y.C., Calim, F.F.: Transient response of functionally graded non-uniform cylindrical helical rods. Steel Compos. Struct. 40, 571–580 (2021b). https://doi.org/10.12989/scs.2021.40.4.571
Eratli, N., Yilmaz, M., Darilmaz, K., Omurtag, M.H.: Dynamic analysis of helicoidal bars with non-circular cross-sections via mixed FEM. Struct. Eng. Mech. 57, 221–238 (2016). https://doi.org/10.12989/sem.2016.57.2.221
Ermis, M., Eratlı, N., Argeso, H., Kutlu, A., Omurtag, M.: Parametric analysis of viscoelastic hyperboloidal helical rod. Adv. Struct. Eng. 19, 1420–1434 (2016). https://doi.org/10.1177/1369433216643584
Girgin, K.: Free vibration analysis of non-cylindrical helices with, variable cross-section by using mixed FEM. J. Sound Vib. 297, 931–945 (2006). https://doi.org/10.1016/j.jsv.2006.05.001
İnan, M.: Initial value problems in elastomechanics and transfer-matrix method (in Turkish). Library of Istanbul Technical University (1964)
Kacar, I., Yildirim, V.: Free vibration/buckling analyses of noncylindrical initially compressed helical composite springs. Mech. Based Des. Struct. Mach. 44, 340–353 (2016). https://doi.org/10.1080/15397734.2015.1066687
Khoram, M.M., Hosseini, M., Hadi, A., Shishehsaz, M.: Bending analysis of bidirectional FGM Timoshenko nanobeam subjected to mechanical and magnetic forces and resting on Winkler–Pasternak foundation. Int. J. Appl. Mech. 12, 2050093 (2020). https://doi.org/10.1142/S1758825120500933
Kiral, E., Ertepinar, A.: Studies on elastic rods subject to diverse external agencies—part III: vibrational analysis of space rods. METU J. Pure Appl. Sci. 7, 55–69 (1974)
Lohar, H., Mitra, A., Sahoo, S.: Nonlinear response of axially functionally graded Timoshenko beams on elastic foundation under harmonic excitation. Curved Layer. Struct. 6, 90–104 (2019). https://doi.org/10.2514/3.6595
Nagaya, K., Takeda, S., Nakata, Y.: Free vibration of coil springs of arbitrary shape. Int. J. Numer. Methods Eng. 23, 1081–1099 (1986). https://doi.org/10.1002/nme.1620230607
Nejad, M.Z., Rastgoo, A., Hadi, A.: Effect of exponentially-varying properties on displacements and stresses in pressurized functionally graded thick spherical shells with using iterative technique. J. Solid Mech. 6, 366–377 (2014a)
Nejad, M.Z., Rastgoo, A., Hadi, A.: Exact elasto-plastic analysis of rotating disks made of functionally graded materials. Int. J. Eng. Sci. 85, 47–57 (2014b). https://doi.org/10.1016/j.ijengsci.2014.07.009
Nejad, M.Z., Jabbari, M., Hadi, A.: A review of functionally graded thick cylindrical and conical shells. J. Comput. Appl. Mech. 48, 357–370 (2017). https://doi.org/10.22059/JCAMECH.2017.247963.220
Nejad, M.Z., Alamzadeh, N., Hadi, A.: Thermoelastoplastic analysis of FGM rotating thick cylindrical pressure vessels in linear elastic-fully plastic condition. Composites, Part B, Eng. 154, 410–422 (2018). https://doi.org/10.1016/j.compositesb.2018.09.022
Noori, A.R., Aslan, T.A., Temel, B.: An efficient approach for in-plane free and forced vibrations of axially functionally graded parabolic arches with nonuniform cross section. Compos. Struct. 200, 701–710 (2018a). https://doi.org/10.1016/j.compstruct.2018.05.077
Noori, A.R., Aslan, T.A., Temel, B.: Damped transient response of in-plane and out-of-plane loaded stepped curved rods. J. Braz. Soc. Mech. Sci. Eng. 40, 28 (2018b). https://doi.org/10.1007/s40430-017-0949-8
Noori, A.R., Aslan, T.A., Temel, B.: Static analysis of FG beams via complementary functions method. Eur. Mech. Sci. 4, 1–6 (2020). https://doi.org/10.26701/ems.590864
Noori, A.R., Aslan, T.A., Temel, B.: Dynamic analysis of functionally graded porous beams using complementary functions method in the Laplace domain. Compos. Struct. 256, 113094 (2021). https://doi.org/10.1016/j.compstruct.2020.113094
Rajasekaran, S., Bakhshi Khaniki, H.: Bending, buckling and vibration of small-scale tapered beams. Int. J. Eng. Sci. 120, 172–188 (2017). https://doi.org/10.1016/j.ijengsci.2017.08.005
Rajasekaran, S., Bakhshi Khaniki, H.: Bending, buckling and vibration analysis of functionally graded non-uniform nanobeams via finite element method. J. Braz. Soc. Mech. Sci. Eng. 40, 549 (2018). https://doi.org/10.1007/s40430-018-1460-6
Rajasekaran, S., Bakhshi Khaniki, H.: Size-dependent forced vibration of non-uniform bi-directional functionally graded beams embedded in variable elastic environment carrying a moving harmonic mass. Appl. Math. Model. 72, 129–154 (2019a). https://doi.org/10.1016/j.apm.2019.03.021
Rajasekaran, S., Bakhshi Khaniki, H.: Finite element static and dynamic analysis of axially functionally graded nonuniform small-scale beams based on nonlocal strain gradient theory. Mech. Adv. Mat. Struct. 26(14), 1245–1259 (2019b). https://doi.org/10.1080/15376494.2018.1432797
Şahan, M.F.: Viscoelastic damped response of cross-ply laminated shallow spherical shells subjected to various impulsive loads. Mech. Time-Depend. Mater. (2017). https://doi.org/10.1007/s11043-017-9339-y
Temel, B., Calim, F.F.: Forced vibration of cylindrical helical rods subjected to impulsive loads. J. Appl. Mech. 70, 281–291 (2003). https://doi.org/10.1115/1.1554413
Temel, B., Calim, F.F., Tütüncü, N.: Quasi-static and dynamic response of viscoelastic helical rods. J. Sound Vib. 271, 921–935 (2004). https://doi.org/10.1016/S0022-460X(03)00760-0
Temel, B., Calim, F.F., Tütüncü, N.: Forced vibration of composite cylindrical helical rods. Int. J. Mech. Sci. 47, 998–1022 (2005). https://doi.org/10.1016/j.ijmecsci.2005.04.003
Wang, G., Zhu, L., Higuchi, K., Fan, W., Li, L.: Solution for free vibration of spatial curved beams. Eng. Comput. 37, 1597–1616 (2019). https://doi.org/10.1108/EC-03-2019-0097
Wang, Y., Xie, K., Fu, T., Zhang, W.: A third order shear deformable model and its applications for nonlinear dynamic response of graphene oxides reinforced curved beams resting on visco-elastic foundation and subjected to moving loads. Eng. Comput. (2021). https://doi.org/10.1007/s00366-020-01238-x
Wu, P., Zhou, D., Liu, W., Lu, W., Wan, L.: Three-dimensional elasticity solution of layered plates with viscoelastic interlayers. Mech. Time-Depend. Mater. 21, 307–329 (2017). https://doi.org/10.1007/s11043-016-9331-y
Wu, P., Zhou, D., Liu, W., Fang, H.: Time-dependent behavior of layered arches with viscoelastic interlayers. Mech. Time-Depend. Mater. (2018). https://doi.org/10.1007/s11043-017-9358-8
Yildirim, V.: Investigation of parameters affecting free vibration frequency of helical springs. Int. J. Numer. Methods Eng. 39, 99–114 (1996). https://doi.org/10.1002/(SICI)1097-0207(19960115)39:1<99::AID-NME850>3.0.CO;2-M
Yildirim, V.: Free vibration analysis of non-cylindrical coil springs by combined use of transfer matrix and the complementary functions methods. Commun. Numer. Methods Eng. 13, 487–494 (1997). https://doi.org/10.1002/(SICI)1099-0887(199706)13:6<487::AID-CNM77>3.0.CO;2-X
Yildirim, V.: A parametric study on the free vibration of non-cylindrical helical springs. J. Appl. Mech. Trans. ASME 65, 157–163 (1998). https://doi.org/10.1115/1.2789019
Yildirim, V.: Expressions for predicting fundamental natural frequencies of non-cylindrical helical springs. J. Sound Vib. 252, 479–491 (2002). https://doi.org/10.1006/jsvi.2001.4005
Yildirim, V., İnce, N.: Natural frequencies of helical springs of arbitrary shape. J. Sound Vib. 204, 311–329 (1997). https://doi.org/10.1006/jsvi.1997.0940
Yu, A.M., Hao, Y.: Effect of warping on natural frequencies of symmetrical cross-ply laminated composite non-cylindrical helical springs. Int. J. Mech. Sci. 74, 65–72 (2013a). https://doi.org/10.1016/j.ijmecsci.2013.04.010
Yu, A.M., Hao, Y.: Warping effect in free vibration analysis of unidirectional composite non-cylindrical helical springs. Meccanica 48, 2453–2465 (2013b). https://doi.org/10.1007/s11012-013-9760-5
Zarezadeh, E., Hosseini, V., Hadi, A.: Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory. Mech. Based Des. Struct. Mach. 48, 480–495 (2020). https://doi.org/10.1080/15397734.2019.1642766
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Cuma, Y.C., Calim, F.F. Dynamic response of viscoelastic functionally graded barrel and hyperboloidal coil springs with variable cross-sectional area. Mech Time-Depend Mater 26, 923–937 (2022). https://doi.org/10.1007/s11043-021-09520-1
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DOI: https://doi.org/10.1007/s11043-021-09520-1