Reduced-order models for large-amplitude vibrations of shells including in-plane inertia

Abstract

. Reduced-order models for large-amplitude vibrations of shells including in-plane inertia. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2008, 197 (21-24) AbstractNon-linear normal modes (NNMs) are used in order to derive reduced-order models for large amplitude, geometrically non-linear vibrations of thin shells. The main objective of the paper is to compare the accuracy of different truncations, using linear and non-linear modes, in order to predict the response of shells structures subjected to harmonic excitation. For an exhaustive comparison, three different shell problems have been selected: (i) a doubly curved shallow shell, simply supported on a rectangular base; (ii) a circular cylindrical panel with simply supported, in-plane free edges; and (iii) a simply supported, closed circular cylindrical shell. In each case, the models are derived by using refined shell theories for expressing the strain-displacement relationship. As a consequence, in-plane inertia is retained in the formulation. Reduction to one or two NNMs shows perfect results for vibration amplitude lower or equal to the thickness of the shell in the three cases, and this limitation is extended to two times the thickness for two of the selected models.

Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

“…These algorithms have been extensively used for computing the forced response and limit cycles of nonlinear dynamical systems [6][7][8][9][10][11]. Doedel and co-workers used them for the computation of periodic orbits during the free response of conservative systems [12,13].…”

Nonlinear frequency response analysis of structural vibrations

“…The most common projection method is modal reduction or truncation [14,15], where the transformation matrix Φ is composed from a subset of linear eigenvectors, see (38). It is widely used in linear structural dynamics, but there is only a limited applicability in nonlinear analysis (see also later examples in Section 5.1).…”

“…In [30] the properties of isogeometric finite element discretizations in the context of linear eigenvalue problems such as (38) were examined already. For one-dimensional rods and beams, it has been shown analytically and numerically that spline-based finite elements are more accurate than Lagrangian finite elements.…”

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

“…2 After application of the nonlinear coordinate change (31a), (31b), the dynamics are expressed with the normal variables (R p , S p ) and are substantially simplified. Moreover, it opens the doors to reduced-order modeling since in the normal dynamics, all invariant-breaking terms have been cancelled; see, e.g., [25][26][27][28]. In the case where no internal resonance exists between the eigenfrequencies {ω p } p=1,...,N of the system (28), the normal dynamics can be reduced to a single master coordinate, say (R p , S p ), by simply cancelling all the others normal coordinates: ∀k = p, R k = S k = 0.…”

“…By doing so, V depends only on two variables, which renders the computations much more efficient. In particular, this method could be an alternative if one wants to get an upper bound for a N -dofs problem with N large as, e.g., in [28] where ROMs are derived for shell vibrations problems with routinely N = 20. In that case, computing the full criterion will lead to define a 40-dimensional grid, which will be impossible in a reasonable computation time.…”