# Simultaneous Regression and Selection in Nonlinear Modal Model Identification

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. Nonlinear Dynamic Equations of Motion

#### 2.2. Generating Static Force-Displacement Data

#### 2.3. Estimating the Nonlinear Stiffness Terms

#### 2.3.1. Least Squares Estimator

#### 2.3.2. Least Absolute Shrinkage and Selection Operator (LASSO)

#### 2.3.3. Repeated K-Fold Cross-Validation and Hyper-Parameter Selection

#### 2.3.4. Discussion on Computational Cost

## 3. Numerical Examples

#### 3.1. Flat Beam

#### 3.1.1. ROM Training and Nonlinear Stiffness Terms

#### 3.1.2. Evaluation of Accuracy

#### 3.2. Curved Panel

#### 3.2.1. Training and Nonlinear Stiffness Identification

#### 3.2.2. Dynamic Accuracy Evaluation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 3.**Mean squared error (MSE) and standard error (SE) of a 1,3-mode reduced order model (ROM) when using k-fold cross validation for each $\lambda $ value. The two subplots on the right are zoomed in portions of the plots on the left and center.

**Figure 4.**Nonlinear stiffness coefficients of the first and third modal equations versus the least absolute selection and shrinkage (LASSO) regularization parameter $\lambda $. The nonlinear stiffness coefficients were normalized with respect to the values estimated using least squares.

**Figure 5.**Frequency energy plot of the first (left) and third (right) nonlinear normal modes (NNMs) of the flat beam for ROMs created using least squares and using LASSO with various penalty terms.

**Figure 8.**Mean squared error (MSE) and standard error (SE) of k-fold cross validation for each $\lambda $ value.

**Figure 9.**Nonlinear stiffness coefficient matrices for three different sets of sparsity approaches: (1) The least squares solution (all terms shown would be retained), (2) the optimal solution identified via cross-validation (terms in white are eliminated), and (3) a solution in which terms with the smallest magnitude are removed (terms in white are eliminated). The colorbar gives the log of the stiffness term; values below ${10}^{-2}$ are zero.

**Figure 11.**Nonlinear normal modes of the curved panel computed from ROM with modes 1, 2, 3, 4, 5, 8, and 10 included within the basis set. Subplot (

**a**), NNM of first mode. Subplot (

**b**), NNM of the second mode. NNMs were computed using the Multi-Harmonic Balance (MHB) method with 5 harmonics included.

**Figure 12.**The first row of contours are the full finite element (FE) model displacements at the NNM solutions designated in Figure 11 for the full ROM model. The second row of contours are the percentage differences in the displacements fields of the optimal model with respect to the full model. The third row of contours are the percentage differences in the displacement fields of the optimal model with respect to the full model.

Point | A | B | C |
---|---|---|---|

LS: All Load Cases | 0.0059 | 0.0070 | 0.0463 |

LS: Single Load 1xThk | 0.0079 | 0.0191 | NA |

LA: All Load Cases-Optimal | 0.0063 | 0.0075 | 0.0513 |

LA: All Load Cases-$1\sigma \phantom{\rule{4pt}{0ex}}error$ | 0.0093 | 0.0108 | 0.1074 |

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**MDPI and ACS Style**

Van Damme, C.; Madrid, A.; Allen, M.; Hollkamp, J.
Simultaneous Regression and Selection in Nonlinear Modal Model Identification. *Vibration* **2021**, *4*, 232-247.
https://doi.org/10.3390/vibration4010016

**AMA Style**

Van Damme C, Madrid A, Allen M, Hollkamp J.
Simultaneous Regression and Selection in Nonlinear Modal Model Identification. *Vibration*. 2021; 4(1):232-247.
https://doi.org/10.3390/vibration4010016

**Chicago/Turabian Style**

Van Damme, Christopher, Alecio Madrid, Matthew Allen, and Joseph Hollkamp.
2021. "Simultaneous Regression and Selection in Nonlinear Modal Model Identification" *Vibration* 4, no. 1: 232-247.
https://doi.org/10.3390/vibration4010016