Sparse stability diagrams of LSCF method via strategic pole destabilization using orthogonal matching pursuit

In various engineering fields including mechanical, aerospace, and civil engineering, the identification of modal parameters, including natural frequencies, damping ratios, and mode shapes, is crucial for determining the vibration characteristics of engineered structures. A common method for identifying the modal parameters of structures involves experimental modal analysis using frequency response functions (FRFs) obtained from forced vibration tests. The least squares complex frequency (LSCF) domain method is a widely-used frequency-domain curve-fitting method for the FRFs using the polynomials of high order, which can extract modal parameters with high accuracy. However, increasing the polynomial order tends to result in the generation of non-physical spurious poles that need to be eliminated from the stability diagrams. To overcome this issue, we propose a method that strategically destabilize the stable yet spurious poles of the characteristic polynomials by making their coefficients as sparse as possible, via orthogonal matching pursuit (OMP). This results in sparse stability diagrams because unstable poles can be eliminated from the diagrams. In this paper, the proposed method is first applied to a numerically-obtained FRFs of a rectangular plate using finite element model, and its validity is discussed. Then, the method is applied to experimentally-obtained FRFs of rectangular plates with low-damping and with high-damping. Furthermore, to confirm its applicability to industrial applications with realistic complexity, it has also been applied to the FRFs of the electric machine's stator core used for electric vehicles. Based on the results, we have confirmed that the spurious roots can be eliminated from the stability diagrams without compromising accuracy for the cases considered.

Damage identification using noisy frequency response functions based on topology optimization

This paper proposes a robust damage identification method using noisy frequency response functions (FRFs) and topology optimization. We formulate the damage identification problem as an inverse problem of generating the damage topology of the structure from measured dynamic responses of the structure to given external dynamic loading. The method is based on the minimization of the objective function representing errors between measured FRFs of the structure obtained by experimental modal analysis, and those obtained by harmonic response analysis using finite element analysis. In the minimization process, material distribution, or the topology of the structure is varied and the optimal damage topology is identified as regions with no material assigned as a result of the minimization using the solid isotropic material with penalization (SIMP). In order to overcome the problems caused by the ill-posedness of the inverse problem, it is proposed that the least absolute shrinkage and selection operator (Lasso) regularization, or the penalization to the L1 norm of the design variable be applied to the original objective function. By applying Lasso regularization, the method is expected not only to eliminate spurious damaged regions but also to minimize the effect of measurement noises. This paper first presents the mathematical background and its numerical implementation of the proposed methodology. The method is then applied to the identification of a damage of cantilevered plates. The FRFs were experimentally obtained and the proposed method is applied. It is shown that the method successfully identifies the damage.

Data-driven model order reduction for structures with piecewise linear nonlinearity using dynamic mode decomposition

Piecewise-linear nonlinear systems appear in many engineering disciplines. Prediction of the dynamic behavior of such systems is of great importance from practical and theoretical viewpoint. In this paper, a data-driven model order reduction method for piecewise-linear systems is proposed, which is based on dynamic mode decomposition (DMD). The overview of the concept of DMD is provided, and its application to model order reduction for nonlinear systems based on Galerkin projection is explained. The proposed approach uses impulse responses of the system to obtain snapshots of the state variables. The snapshots are then used to extract the dynamic modes that are used to form the projection basis vectors. The dynamics described by the equations of motion of the original full-order system are then projected onto the subspace spanned by the basis vectors. This produces a system with much smaller number of degrees of freedom (DOFs). The proposed method is applied to two representative examples of piecewise linear systems: a cantilevered beam subjected to an elastic stop at its end, and a bonded plates assembly with partial debonding. The reduced order models (ROMs) of these systems are constructed by using the Galerkin projection of the equation of motion with DMD modes alone, or DMD modes with a set of classical constraint modes to be able to handle the contact nonlinearity efficiently. The obtained ROMs are used for the nonlinear forced response analysis of the systems under harmonic loading. It is shown that the ROMs constructed by the proposed method produce accurate forced response results.

Data-driven forced response analysis with min-max representations of nonlinear restoring forces

This paper discusses a novel data-driven nonlinearity identification method for mechanical systems with nonlinear restoring forces such as polynomial, piecewise-linear, and general displacement-dependent nonlinearities. The proposed method is built upon the universal approximation theorem that states that a nonlinear function can be approximated by a linear combination of activation functions in artificial neural network framework. The proposed approach utilizes piecewise linear springs with initial gaps to act as the activation functions of the neurons of artificial neural networks. A library of piecewise linear springs with initial gaps are constructed, and the contributions of the springs on the nonlinear restoring force are determined by solving the linear regression problems. The piecewise linear springs are realized by combinations of min and max functions with biases. The proposed method is applied to a Duffing oscillator with cubic stiffness, and a piecewise linear oscillator with a gap and their nonlinearities are successfully determined from their free responses. The obtained models are then used for conducting forced response analysis and the results match well with those of the original system. The method is then applied to experimentally-obtained free response data of a cantilevered plate that is subjected to magnetic restoring force, and successfully finds the piecewise linear representation of the magnetic force. It is also shown that the obtained model is capable of accurately capturing the steady-state response of the system subject to harmonic base excitation.