## 6 Conclusions and Outlook  
The SINDy framework has rapidly developed in different extensions and has been applied to various dynamic systems in the past few years. In this work, we introduce a new sparse regression process for vibration systems based on the SINDy framework and Least Squares Post LASSO (LSPL) algorithms. In the numerical uncertainty analysis, we compared the two regression methods LSST and LSPL in terms of sparsity, convergence, eigenfrequency of oscillator, and coefficient of determination in Monte Carlo Simulations. Based on a through analysis and comparison with characteristic categories, the method SINDy-LSPL delivered a more sparse model and a more accurate prediction of the dynamic behavior. We designed a complete workflow to identify the governing equation of a single-mass oscillator using SINDy-LSPL in experimental tests. The problem of drift effect in the reconstructed velocity and displacement was accounted for by filtering out the low frequencies of reconstructed signals. In this evaluation, we tested two different types of excitation signals, namely impulse and sweep signals. The results reveal that the form and strength of excitation influence the identified result. A potential explanation of this phenomenon lies in the preprocess of training data. A detailed analysis should be dealt with in further research.  
In future work in relation to SINDy-LSPL, we could further optimize the conditioning of the regression problem by improving the signal processing technique in the pre-processing of measurement data, such as denoising of the signals, integral processes, and normalization of the data. On the other hand, the usage of SINDy-LSPL in continuous vibration systems and discrete systems with multiple DOF is also worth exploring. We will also consider integrating SINDy-LSPL into adaptive control algorithms.  
Conceptualization, Y.R. and C.A.; methodology, Y.R.; validation, Y.R.; resources, Y.R. and C.A.; writing--original draft preparation, Y.R.; writing--review and editing, C.A. and T.M.; supervision, C.A. and T.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) grant number 57157498. The publication was funded by DFG and the Open Access Publishing Fund of Technical University of Darmstadt.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The original SINDy algorithm is provided in the supplemental material of [24].  
The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for funding Y.R. within the Sonderforschungsbereich (SFB, Collaborative Research Center) 805 \"Control of Uncertainties in Load-Carrying Structures in Mechanical Engineering\"--project number: 57157498. The authors also thank the Open Access Publishing Fund  
Figure 18: Linear dependency of the identified coefficients (**a**) \(\xi_{2}\) und (**b**) \(\xi_{3}\) on the excitation force \(F_{max}\).  
of Technical University of Darmstadt for supporting the publication. The authors further thank [PERSON] for the possibility to perform experiments on the test stand.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.