Abstract
This paper presents two methods for numerical calculation of nonlinear normal modes (NNMs) in multi-degree-of-freedom, conservative, nonlinear structural dynamics models. The approaches used are briefly described as follows. Method 1: Starting with small amplitude initial conditions determined by a selected mode of the associated linear system, a small amount of negative damping is added in order to “artificially destabilize” the system; numerical integration of the system equations of motion then produces a simulated response in which orbits spiral outward essentially in the nonlinear modal manifold of interest, approximately generating this manifold for moderate to strong nonlinearity. Method 2: Starting with moderate to large amplitude initial conditions proportional to a selected linear mode shape, perform numerical integration with the coefficient ε of the nonlinearity contrived to vary slowly from an initial value of zero; this simulation methodology gradually transforms the initially flat eigenspace for ε = 0 into the manifold existing quasi-statically for instantaneous values of ε. The two methods are efficient and reasonably accurate and are intended for use in finding NNMs, as well as interesting behavior associated with them, for moderately and strongly nonlinear systems with relatively many degrees of freedom (DOFs).
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References
Rosenberg, R.M.: On nonlinear vibration of systems with many degrees of freedom. Adv. Appl. Mech. 9, 155–242 (1966)
Vakakis, A.F.: Analysis and identification of nonlinear normal modes in vibrating systems. Ph.D. Thesis, California Institute of Technology (1990)
Vakakis, A.F.: Non-similar normal oscillations in a strongly nonlinear discrete system. J. Sound Vib. 158(2), 341–361 (1992)
Vakakis, A.F., Caughey, T.K.: A theorem on the exact nonlinear steady state motions of a nonlinear oscillator. J. Appl. Mech. 59, 418–424 (1992)
Shaw, S.W., Pierre, C.: Nonlinear normal modes and invariant manifolds. J. Sound Vib. 150(1), 170–173 (1991)
Shaw, S.W., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vib. 164(1), 85–121 (1993)
Aubrecht, J., Vakakis, A.F.: Localized and non-localized nonlinear normal modes in a multi-span beam with geometric nonlinearities. J. Vib. Acoustics 118, 533–542 (1996)
Caughey, T.K., Vakakis, A.F.: A method for examining steady state solutions of forced discrete systems with strong nonlinearities. Int. J. Non-Linear Mech. 26(1), 89–103 (1991)
King, M.E., Vakakis, A.F.: An energy based formulation for computing nonlinear normal modes in undamped continuous systems. J. Vib. Acoustics 116, 332–340 (1994)
Pellicano, F., Vakakis, A.F.: Normal modes and boundary layers for a slender tensioned beam on a nonlinear foundation. Nonlinear Dyn. 25, 79–93 (2001)
Vakakis, A.F.: Non-linear normal modes (NNMs) and their applications in vibration theory: An overview. Mech. Syst. Signal Process. 11(1), 3–22 (1997)
Ma, X., Azeez, M.F.A., Vakakis, A.F.: Non-linear normal modes and non-parametric system identification of non-linear oscillators. Mech. Syst. Signal Process. 14(1), 37–48 (2000)
Shaw, S.W., Pierre, C.: Normal modes of vibration for non-linear continuous systems. J. Sound Vib. 169, 319–347 (1994)
Shaw, S.W.: An invariant manifold approach to nonlinear normal modes of oscillation. J. Nonlinear Sci. 4, 419–448 (1994)
Boivin, N., Pierre, C., Shaw, S.W.: Non-linear normal modes, invariance, and modal dynamics approximations of non-linear systems. Nonlinear Dyn. 8, 315–346 (1995)
Pesheck, E., Pierre, C., Shaw, S.W.: Accurate reduced order models for a simple rotor blade model using nonlinear normal modes. Math. Comput. Model. 33, 1085–1097 (2001)
Pesheck, E., Pierre, C., Shaw, S.W.: A new Galerkin-based approach for accurate nonlinear normal modes through invariant manifolds. J. Sound Vib. 249(5), 971–99 (2002)
Nayfeh, A.H., Nayfeh, S.A.: On nonlinear modes of continuous systems. J. Vib. Acoust. 116, 129–136 (1994)
Nayfeh, A.H., Chin, C., Nayfeh, S.A.: On nonlinear modes of systems with internal resonances. J. Vib. Acoust. 118, 340–345 (1996)
Nayfeh, A.H., Nayfeh, S.A.: Nonlinear normal modes of continuous systems with quadratic and cubic nonlinearities. J. Vib. Acoust. 117, 199–205 (1995)
Nayfeh, A.H.: On direct methods for constructing nonlinear normal modes of continuous systems. J. Vib. Control 1(4), 389–430 (1995)
Nayfeh, A.H., Chin, C., Nayfeh, S.A.: Nonlinear normal modes of a cantilever beam. J. Vib. Acoust. 117, 477–481 (1995)
Nayfeh, A.H., Lacarbonara, W., Chin, C.-M.: Nonlinear normal modes of buckled beams: three-to-one and one-to-one internal resonances. Nonlinear Dyn. 18, 253–273 (1999)
Lacarbonara, W., Rega, G., Nayfeh, A.H.: Resonant non-linear normal modes: Part I. Analytical treatment for structural one-dimensional systems. Int. J. Non-Linear Mech. 38, 851–872 (2003)
Yabuno, H., Nayfeh, A.H.: Nonlinear normal modes of a parametrically excited cantilever beam. Nonlinear Dyn. 25, 65–77 (2001)
Burton, T.D., Rhee, W.: On the reduction of nonlinear structural dynamics models. J. Vib. Control 6, 531–556 (2000)
Burton, T.D., Hamdan, M.N.: On the calculation of non-linear normal modes in continuous systems. J. Sound Vib. 197(1), 117–130 (1996)
Burton, T.D., Young, M.E.: Model reduction and nonlinear normal modes in structural dynamics. In: Nonlinear and stochastic dynamics, Bajaj, A.K., Namachchivaya, N.S., Ibrahim, R.A. (eds.) AMD, vol. 192, pp. 9–16. ASME International Mechanical Engineering Congress and Exposition, Chicago, IL (1994)
Slater, J.C.: A numerical method for determining nonlinear normal modes. Nonlinear Dyn. 10, 19–30 (1996)
Zhang, X.H.: Effects of base points and normalization schemes on the non-linear normal modes of conservative systems. J. Sound Vib. 256(3), 447–462 (2002)
Falzarano, J.M., Clague, R.E., Kota, R.S.: Application of nonlinear normal mode analysis to the nonlinear and coupled dynamics of a floating offshore platform with damping. Nonlinear Dyn. 25, 255–274 (2001)
Mazzilli, C.E.N., Baracho Neto, O.G.P.: Evaluation of non-linear normal modes for finite-element models. Comput. Struct. 80, 957–965 (2002)
Soares, M.E.S., Mazzilli, C.E.N.: Nonlinear normal modes of planar frames discretised by the finite element method. Comput. Struct. 77, 485–493 (2000)
Rand, R.H., Ramani, D.V.: Nonlinear normal modes in a system with nonholonomic constraints. Nonlinear Dyn. 25, 49–64 (2001)
Georgiou, I.T., Schwartz, I.B.: Invariant manifolds, nonclassical normal modes, and proper orthogonal modes in the dynamics of the flexible spherical pendulum. Nonlinear Dyn. 25, 3–31 (2001)
Chechin, G.M., Sakhnenko, V.P., Stokes, H.T., Smith, A.D., Hatch, D.M.: Non-linear normal modes for systems with discrete symmetry. Int. J. Non-Linear Mech. 35, 497–513 (2000)
Burton, T.D.: On the amplitude decay of strongly nonlinear damped oscillators. J. Sound Vib. 87(4), 535–541 (1983)
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Burton, T.D. Numerical calculation of nonlinear normal modes in structural systems. Nonlinear Dyn 49, 425–441 (2007). https://doi.org/10.1007/s11071-006-9128-7
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DOI: https://doi.org/10.1007/s11071-006-9128-7