Browse > Article
http://dx.doi.org/10.12989/csm.2013.2.2.159

Nonlinear stability and bifurcations of an axially accelerating beam with an intermediate spring-support  

Ghayesh, Mergen H. (Department of Mechanical Engineering, McGill University)
Amabili, Marco (Department of Mechanical Engineering, McGill University)
Publication Information
Coupled systems mechanics / v.2, no.2, 2013 , pp. 159-174 More about this Journal
Abstract
The present work aims at investigating the nonlinear dynamics, bifurcations, and stability of an axially accelerating beam with an intermediate spring-support. The problem of a parametrically excited system is addressed for the gyroscopic system. A geometric nonlinearity due to mid-plane stretching is considered and Hamilton's principle is employed to derive the nonlinear equation of motion. The equation is then reduced into a set of nonlinear ordinary differential equations with coupled terms via Galerkin's method. For the system in the sub-critical speed regime, the pseudo-arclength continuation technique is employed to plot the frequency-response curves. The results are presented for the system with and without a three-to-one internal resonance between the first two transverse modes. Also, the global dynamics of the system is investigated using direct time integration of the discretized equations. The mean axial speed and the amplitude of speed variations are varied as the bifurcation parameters and the bifurcation diagrams of Poincare maps are constructed.
Keywords
axially moving beams; nonlinear dynamics; additional spring-support;
Citations & Related Records
Times Cited By KSCI : 5  (Citation Analysis)
연도 인용수 순위
1 Ravindra, B. and Zhu, W.D. (1998), "Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime", Arch. Appl. Mech., 68(3-4), 195-205.   DOI   ScienceOn
2 Riedel, C.H. and Tan, C.A. (2002), "Coupled, forced response of an axially moving strip with internal resonance", Int. J. Nonlinear Mech., 37(1), 101-116.   DOI   ScienceOn
3 Saffari, H., Mohammadnejad, M. and Bagheripour, M.H. (2012), "Free vibration analysis of non-prismatic beams under variable axial forces", Struct. Eng. Mech., 43(50).
4 Sahebkar, S.M., Ghazavi, M.R., Khadem, S.E. and Ghayesh, M.H. (2011), "Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well", Mech. Mach. Theory, 46(5), 743-760.   DOI   ScienceOn
5 Song, Z., Li, W. and Liu, G. (2012), "Stability and non-stationary vibration analysis of beams subjected to periodic axial forces using discrete singular convolution", Struct. Eng. Mech., 44(4), 487-499.   DOI   ScienceOn
6 Suweken, G. and Van Horssen, W.T. (2003), "On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-like case", J. Sound Vib., 264(1), 117-133.   DOI   ScienceOn
7 Suweken, G. and Van Horssen, W.T. (2003), "On the weakly nonlinear, transversal vibrations of a conveyor belt with a low and time-varying velocity", Nonlinear Dynam., 31(2), 197-223.   DOI   ScienceOn
8 Tang, Y.Q., Chen, L.Q. and Yang, X.D. (2008), "Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions", Int. J. Mech. Sci., 50(10-11), 1448-1458.   DOI   ScienceOn
9 Yang, X.D. and Chen, L.Q. (2005), "Bifurcation and chaos of an axially accelerating viscoelastic beam", Chaos Soliton. Fract., 23(1), 249-258.   DOI   ScienceOn
10 Pakdemirli, M., Ulsoy, A.G. and Ceranoglu, A. (1994), "Transverse vibration of an axially accelerating string", J. Sound Vib., 169(2), 179-196.   DOI   ScienceOn
11 Pellicano, F. and Vestroni, F. (2000), "Nonlinear dynamics and bifurcations of an axially moving beam", J. Vib. Acoust., 122(1), 21-30.   DOI   ScienceOn
12 Ghayesh, M.H. and Balar, S. (2010), "Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams", Appl. Math. Model., 34(10), 2850-2859.   DOI   ScienceOn
13 Ghayesh, M.H., Kazemirad, S. and Amabili, M. (2012), "Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance", Mech. Mach. Theory, 52(0), 18-34.   DOI   ScienceOn
14 Ghayesh, M.H. and Khadem, S.E. (2007), "Non-linear vibration and stability analysis of a partially supported conveyor belt by a distributed viscoelastic foundation", Struct. Eng. Mech., 27(1), 17-32.   DOI   ScienceOn
15 Huang, J.L., Su, R.K.L., Li, W.H. and Chen, S.H. (2011), "Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances", J. Sound Vib., 330(3), 471-485.   DOI   ScienceOn
16 Ghayesh, M.H. and Paidoussis, M.P. (2010), "Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array", Int. J. Nonlinear Mech., 45(5), 507-524.   DOI   ScienceOn
17 Ghayesh, M.H., Paidoussis, M.P. and Modarres-Sadeghi, Y. (2011), "Three-dimensional dynamics of a fluid-conveying cantilevered pipe fitted with an additional spring-support and an end-mass", J. Sound Vib., 330(12), 2869-2899.   DOI   ScienceOn
18 Ghayesh, M.H., Yourdkhani, M., Balar, S. and Reid, T. (2010), "Vibrations and stability of axially traveling laminated beams", Appl. Math. Comput., 217(2), 545-556.   DOI   ScienceOn
19 Kural, S. and O zkaya, E. (2012), "Vibrations of an axially accelerating, multiple supported flexible beam", Struct. Eng. Mech., 44(4).
20 Marynowski, K. (2004), "Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension", Chaos Soliton. Fract., 21(2), 481-490.   DOI   ScienceOn
21 Marynowski, K. and Kapitaniak, T. (2002), "Kelvin-Voigt versus Burgers internal damping in modeling of axially moving viscoelastic web", Int. J. Nonlinear Mech., 37(7), 1147-1161.   DOI   ScienceOn
22 Movahedian, B. (2012), "Dynamic stiffness matrix method for axially moving micro-beam", Int. Multiscale Mech., 5(4).
23 O z, H.R. and Pakdemirli, M. (1999), "Vibrations of an axially moving beam with time-dependent velocity", J. Sound Vib., 227(2), 239-257.   DOI   ScienceOn
24 Oz, H.R., Pakdemirli, M. and Boyacı, H. (2001), "Non-linear vibrations and stability of an axially moving beam with time-dependent velocity", Int. J. Nonlinear Mech., 36(1), 107-115.   DOI   ScienceOn
25 Pakdemirli, M. and Ulsoy, A.G. (1997), "Stability analysis of an axially accelerating string", J. Sound Vib., 203(5), 815-832.   DOI   ScienceOn
26 O z, H.R., Pakdemirli, M. and O zkaya, E. (1998), "Transition behaviour from string to beam for an axially accelerating material", J. Sound Vib., 215(3), 571-576.   DOI   ScienceOn
27 O zkaya, E. and Pakdemirli, M. (2000), "Vibrations of an axially accelerating beam with small flexural stiffness", J. Sound Vib., 234(3), 521-535.   DOI   ScienceOn
28 Pakdemirli, M. and O z, H.R. (2008), "Infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations", J. Sound Vib., 311(3-5), 1052-1074.   DOI
29 Chen, L.Q., Ding, H. and Lim, C.W. (2012), "Principal parametric resonance of axially accelerating viscoelastic beams: multi-scale analysis and differential quadrature verification", Shock Vib., 19(4), 527-543.   DOI
30 Chen, L.Q. and Yang, X.D. (2005), "Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models", Int. J. Solids Struct., 42(1), 37-50.   DOI   ScienceOn
31 Ding, H. and Chen, L.Q. (2010), "Galerkin methods for natural frequencies of high-speed axially moving beams", J. Sound Vib., 329(17), 3484-3494.   DOI   ScienceOn
32 Ding, H. and Chen, L. (2009), "Nonlinear dynamics of axially accelerating viscoelastic beams based on differential quadrature", Acta Mech. Solida Sinica, 22(3), 267-275.   DOI   ScienceOn
33 Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B. and Wang, X. (1998), AUTO 97: Continuation and bifurcation software for ordinary differential equations (with homcont), Concordia University, Montreal, Canada.
34 Ghayesh, M. (2012), "Stability and bifurcations of an axially moving beam with an intermediate spring support", Nonlinear Dynam., 69(1), 193-210.   DOI   ScienceOn
35 Ghayesh, M.H. (2010), "Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation", Int. J. Nonlinear Mech., 45(4), 382-394.   DOI   ScienceOn
36 Ghayesh, M., Alijani, F. and Darabi, M. (2011), "An analytical solution for nonlinear dynamics of a viscoelastic beam-heavy mass system", J. Mech. Sci. Tech., 25(8), 1915-1923.   DOI   ScienceOn
37 Ghayesh, M.H. (2008), "Nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide", J. Sound Vib., 314(3-5), 757-774.   DOI   ScienceOn
38 Ghayesh, M.H. (2009), "Stability characteristics of an axially accelerating string supported by an elastic foundation", Mech. Mach. Theory, 44(10), 1964-1979.   DOI   ScienceOn
39 Ghayesh, M.H. (2011), "Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance", Int. J. Mech. Sci., 53(11), 1022-1037.   DOI   ScienceOn
40 Ahmadian, M., Nasrabadi, V. and Mohammadi, H. (2010), "Nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to mono-frequency excitation", Acta Mech., 214(3), 357-373.   DOI
41 Bayat, M., Pakar, I. and Bayat, M. (2013), "On the large amplitude free vibrations of axially loaded Euler-Bernoulli beams", Steel Compos. Struct., 14(1), 73-83.   DOI   ScienceOn
42 Ghayesh, M.H. (2011), "On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: parametric study", Acta Mech. Solida Sinica, 24(4), 373-382.   DOI   ScienceOn
43 Ghayesh, M.H., Amabili, M. and Farokhi, H. (2013), "Coupled global dynamics of an axially moving viscoelastic beam", Int. J. Nonlinear Mech., 51(0), 54-74.   DOI   ScienceOn
44 Ghayesh, M.H. and Balar, S. (2008), "Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams", Int. J. Solids Struct., 45(25-26), 6451-6467.   DOI   ScienceOn