266 E. Bonisoli et al. 38.3 Multi-Phi Method Linear modal analysis allows to rewrite the equations of a linear dynamic system by means of diagonal matrices only, so uncoupling the problem. The proposal of Multi-Phi is to describe the nonlinearities through a limited number of parameters and to decompose the nonlinear system into a series of linear systems, each one characterised by a reference value of each parameter. In the following, the assumption of a single parameter α is made. It is considered a set of lv linearised models characterised by a reference values αl. Linear modal analysis is performed on each linear systeml, obtaining rl modes, used to build a collection of lv linear models. Each linear model can represent with a good approximation the system behaviour for α ≈αl. Considering a generic l linear system, the equations of motion are described by Eq. (38.1). M(l)¨x+C(l)˙x+K(l)x =f(t) (38.1) where x, ˙x, ¨x ∈ n×1 are respectively displacements, velocities and accelerations of the system, M(l), C(l), K(l) are respectively mass, damping and stiffness matrices of the l linear system and f(t) is the time depending external forcing function applied to the system. Assuming a proportional damping matrix, the modal superposition results: x = (l)η(l) (38.2) where (l) ∈ n×rl and η(l) ∈ rl×1. In Eqs. (38.1) and (38.2) the superscript (l) is referred to the modal coordinates of the lth linear model. If the eigenvectors are unitary modal mass normalised, Eq. (38.1) can be then expressed by means of modal coordinates as Eq. (38.3). I ¨η(l) +diag 2ζlωl ˙η (l) +diag ω 2 l η (l) = (l)Tf(t) (38.3) where ωl and ζl are the vector of natural frequencies and corresponding damping ratios vectors of the l th linear model. The state in terms of physical coordinates can be obtained using Eq. (38.4): ⎧ ⎨ ⎩ x(t) = (t) η(t) +x∞(t) ˙x(t) = (t) ˙η(t) ¨x(t) = (t) ¨η(t) (38.4) In Eq. (38.4) the addition of x∞, called asymptotic configuration, is necessary whenever the set of mode shapes used to simulate the system evolution cannot describe the system state because of non-null boundary conditions. It is considered function of time because it depends on the linearised system considered at each time instant. To determine x(l) ∞, the Guyan reduction is applied, by means of Eq. (38.5). In such equation, xk represents the boundary conditions (known DoFs) while xu represents the not constrained DoFs (unknown). ⎧ ⎪⎨ ⎪⎩ x∞= (− K(l) uu −1 K(l) uk I 3 xk ˙x∞=0 (38.5) Thematrix (t) is varying in time as a consequence of the transition between the linearised system: it is composed by the succession in time of the matrices l, withl =1, . . . , Lif a number of linearised systems equal toLis considered. In the following it will be referred to each linearised system as to a “level”, so that at the level l it is associated the linearised system characterised byα=αl, the set of reduced matrices diag(2ζω)l, diag(ω 2 l ), l and the asymptotic configurationx∞, l. When the nonlinearity is discrete, it is supposed that the parameter α can assume only discrete values. Therefore, during the simulation only one level at the time is considered, depending on the value of α. The evolution of the full system, during the span of time in which α =αl, depends only by the linearised systeml and the transition to another level m occurs whenever α =αm.
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