11 Towards Exact Statistically Independent Nonlinear Normal Modes via the FPK Equation 89 Fig. 11.5 Comparison of the modal mapping.f(y) with varying excitation level (. =100 ) . d2 dt2 g(u) = d dt g (u)˙u =g (u)˙u 2 +g (u)¨u (11.32 ) and substituting into the equation of motion gives: .g (u)˙u 2 + ¨ ug (u) +cg (u)˙u+kg(u) + g(u) 3 =x(t) (11.33 ) which, despite a Gaussian stationary density for x an d u , is clearly a nonlinear ODE. 11.4 Towards Direct MDOF NNMs Although it has been shown that a modal transformation (of sorts) is available in the SISO case, it is of no practical interest for performing nonlinear modal analysis. Unfortunately, the method presented above cannot be trivially extended to the higherdimensional case. The principal issue is the form of the change-of-variables equation for higher-dimensional mappings. For n degrees of freedom, the static forward modal transformation .f(y) = u is a function with n inputs and n outputs. The change-of-variables equation is thus: . Jf = p(u) p(f(y)) (11.34 ) wher e. Jf is the Jacobian of f an d. | · | denotes the determinant operation. The above expression cannot be uniquely solved fo r f and once again a machine-learning approach must be adopted. However, for nonlinear systems where the equations of
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