Industrial Applications of Nonlinear Modal Modeling: Two Case Studies 53 Eqs (8) through (10) are substituted into Eq. (4). The result is rearranged and cast to the frequency domain to yield h F(q|q|) . . . F q|q|Ns−1 F( ˙q| ˙q|) . . . F ˙q| ˙q|Nd−1 i k1 .. . kNs c1 .. . cNd = [F(fq −¨q −c0 ˙q −k0q)] (11) where F() is the Fourier Transform and represent columns of data in the above matrices. The polynomial coefficients for the nonlinear restoring forces can be solved for in a least-squares since using: k1 .. . kNs c1 .. . cNd =h F(q|q|) . . . F q|q|Ns−1 F( ˙q| ˙q|) . . . F ˙q| ˙q|Nd−1 i + [F(fq −¨q −c0 ˙q −k0q)] (12) where the “+” is the Moore-Penrose Pseudoinverse. The parameterized fr,nl (q(t), ˙q(t)) is now identified for the mode of interest. Note that data from multiple tests can be stacked within each matrix on the right-hand-side of Eq. (12). For both of the case studies in this work, sine beats at multiple levels were conducted and the corresponding data were all used to fit the parameters. This entire process is repeated for each mode of interest that is deemed nonlinear. All modes (linear and nonlinear) are then concatenated into a nonlinear modal model of the entire structure: I{¨q(t)}+C0{˙q(t)}+K0{q(t)}+nfr,nl {q(t)}, n ˙ q(t)o o={fq(t)} (13) whereIis the identity matrix, C0 is a diagonal linear damping matrix with entries of individual c0,j for each modej, K0 is a diagonal linear stiffness matrix with entries of individual k0,j for each mode j, andnfr,nl {q(t)}, n ˙ q(t)o ois the vector of nonlinear restoring forces defined by Eq. (8) for each nonlinear mode. In this formulation, rowj of Eq. (13) corresponds to Eq (4) for each mode j. The response of the entire structure, {x(t)}, to an arbitrary force {f (t)}can be simulated using Eq. (13) by first casting the force in physical coordinates to modal coordinates using Eq. (3). With assumed modal displacement and velocity initial conditions and {fq(t)}, Eq. (13) is used to obtain {q(t)} via time simulation. These modal responses are then cast back to physical coordinates, {x(t)}, using Eq. (1). These {x(t)} are the simulated response of the structure using the identified NLMM and incorporate the amplitude-dependent stiffness and damping due to the nonlinear modes. Generating an explicit nonlinear model of an industrial structure where the physics are exactly captured is difficult to update via system identification experiments. Moreover, the computational cost to simulate the response of a structure to arbitrary inputs with this model is prohibitively expensive. Conversely, the NLMM is highly computationally efficient and, while not exactly modelling the underlying physics, is able to capture the amplitude dependence of the stiffness and damping of each nonlinear mode. Therefore, the NLMM approach is attractive for industrial applications. Case Studies: Industrial Applications of the NLMM In Sections 4.1 and 4.2, the NLMM identification process described in Section 3.3 was implemented on two industrial structures. After conducting linear modal analysis, sine beats were used to excite each structure to high levels. The NLMMs were then identified, and truth tests conducted. The accuracy of the NLMM was evaluated by comparing its predicted response to the truth test forcing to that measured from the experiment. Additionally, for Case Study 2, NFA testing was conducted to evaluate a potential modal interaction. This latter result was not used explicitly in the NLMM identification for this case but is shown to demonstrate that modal interactions occur in industry. It also motivates future research to incorporate this nonlinear behavior into the NLMM process.
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