170 R.J. Kuether and M.S. Allen the CB-NLROM of a single subcomponent is defined by the low order set of equations, b MCBRqC b KCBqCTT CB fNL;i .q/ fNL;b .q/ D TT CB 0 f.t/ (15.6) The linear portion is exactly the CB model for a linear subsystem [1, 11]. When the physical model contains geometric nonlinearity, however, additional nonlinear stiffness terms are introduced by fNL(q). A functional form of the nonlinearity is not readily extracted from models defined directly in commercial FEA packages, so it is difficult to determine this function analytically. Instead, the functional form of the nonlinearity is approximated as TT CB fNL;i .q/ fNL;b .q/ D™.q/ (15.7) Prior works assume a polynomial function to model the geometric nonlinearity in terms of the free-interface modal coordinates [7, 8], and the same approach is used again here with the CB-NLROM. Hence, the nonlinear function ™(q) in Eq. (15.7) is approximated as follows for the rth mode r .q/ D m Xi D1 m Xj Di Br .i;j/qi qj C m Xi D1 m Xj Di m Xk Dj Ar .i;j;k/qi qjqk (15.8) The nonlinear stiffness coefficients Ar and Br are then determined using a series of static forces applied to the full order, geometrically nonlinear structure. This is the approach used by the Implicit Condensation and Expansion method in [15] to fit the nonlinear portion of the equations. A static load is applied to the system in Eq. (15.1), in which the load is in a series of shapes proportional to a combination of the fixed-interface and constraint modes. The multi-modal force is a permutation of the sums and differences of either one, two or three CB modes in the basis set, and is defined as Fc DMfTCB;1f1 CTCB;2f2 C CTCB;mfmg (15.9) The N 1 load vector Fc is a combination of shapes that each have a load scale factor, fr, for the r th basis vector. The selection of each load factor dictates the level of nonlinearity in each of the characteristic shapes. Typically, each fr is selected such that the force of a single shape applied to the linear model results in a maximum displacement on the order of the thickness of the structure for low frequency modes, but higher order modes typically require lower displacements [4]. The resulting coefficients tend to be sensitive to the selection of the scaling factor, so in practice a trial and error approach determines which load factors work the best. Each static load is then applied to the full order model, then the resulting deformation is extracted and the amplitude of each generalized coordinate is computed using the pseudo-inverse of the CB transformation matrix. Using the resulting applied modal forces and displacements, the nonlinear terms Ar andBr are fit to the reduced static equation, b KCBqC™.q/ DTT CBFc (15.10) The fitting procedure is thoroughly discussed in [4]. After determining the nonlinear static coefficients, the CB-NLROM equations are cast into a matrix form prior to coupling, as done in [18]. First, the quadratic and cubic terms in Eq. (15.8) are separated into two nonlinear restoring force vectors, “and’, respectively, as “D ˇ1 ˇ2 : : : ˇm T where ˇr D m Xi D1 m Xj Di Br .i;j/qi qj (15.11) ’D ˛1 ˛2 : : : ˛m T where ˛r D m Xi D1 m Xj Di m Xk Dj Ar .i;j;k/qi qjqk (15.12) These vectors are then differentiated with respect to each generalized coordinate in order to produce the quadratic and cubic stiffness matrices as,
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