# Experimental Identification of Backbone Curves of Strongly Nonlinear Systems by Using Response-Controlled Stepped-Sine Testing (RCT)

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## Abstract

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## 1. Introduction

## 2. Experimental Modal Analysis with RCT and HFS

## 3. Numerical Validation

#### 3.1. Determination of Backbone Curves by Using Constant-Force Simulations

#### 3.2. Determination of Backbone Curves by Using RCT with HFS

## 4. Experimental Applications

#### 4.1. T-Beam

#### 4.2. Control Fin Actuation Mechanism

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Frequency response curves measured in constant-force stepped-sine testing compared to the ones measured by RCT with HFS: (

**a**) jump phenomenon; (

**b**) premature jump.

**Figure 3.**The 5 DOF system with cubic stiffness nonlinearity [16].

**Figure 4.**Determination of the backbone curves (blue square markers) of the 5 DOF system corresponding to the 1st and 5th DOFs by using constant-force stepped-sine simulations. (

**a**) The backbone curve of the 1st DOF; (

**b**) the backbone curve of the 5th DOF.

**Figure 5.**Construction of the HFS corresponding to the 1st DOF by combining harmonic force spectra with linear interpolation. (

**a**) Harmonic excitation force spectra; (

**b**) HFS of the 1st DOF [16].

**Figure 6.**(

**a**) HFS of the 1st DOF cut with constant force planes; (

**b**) extraction of constant-force frequency response curves (black—ranging from 10 N to 50 N with 5N increments) and identification of the backbone curve (red) of the 1st DOF from HFS.

**Figure 7.**(

**a**) HFS of the 5th DOF cut with constant force planes; (

**b**) extraction of the constant-force frequency response curves (black—ranging from 10 N to 50 N with 5 N increments) and the backbone curve (red) of the 5th DOF from HFS.

**Figure 8.**Comparison of backbone curves obtained from RCT with HFS simulations with those obtained from constant force simulations: (

**a**) the backbone curve of the 1st DOF; (

**b**) the backbone curve of the 5th DOF.

**Figure 10.**(

**a**) Harmonic force spectra of the T-beam measured by RCT; (

**b**) HFS of the T-beam built up by combining harmonic force spectra with linear interpolation [16].

**Figure 11.**Validation of the frequency response curves of the T-beam extracted from HFS by using constant-force test results.

**Figure 12.**Constant-force frequency response curves (0.1 N–1.4 N) and backbone curve of the T-beam at the T-junction obtained by cutting the HFS with various constant force planes.

**Figure 14.**(

**a**) Harmonic force spectra of the control fin actuation mechanism measured by RCT; (

**b**) HFS of the control fin actuation mechanism at the driving point, constructed by combining harmonic force spectra.

**Figure 15.**Comparison of the frequency response curves of the control fin actuation mechanism at the driving point, extracted from HFS with those obtained by using constant-force test results in the sweep-up direction for: (

**a**) force level F1; (

**b**) force level F2; (

**c**) force level F3.

**Figure 16.**Constant-force frequency response curves and the backbone curve of the real control fin actuation mechanism at the driving point, obtained by cutting the HFS with various constant force planes.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Karaağaçlı, T.; Özgüven, H.N. Experimental Identification of Backbone Curves of Strongly Nonlinear Systems by Using Response-Controlled Stepped-Sine Testing (RCT). *Vibration* **2020**, *3*, 266-280.
https://doi.org/10.3390/vibration3030019

**AMA Style**

Karaağaçlı T, Özgüven HN. Experimental Identification of Backbone Curves of Strongly Nonlinear Systems by Using Response-Controlled Stepped-Sine Testing (RCT). *Vibration*. 2020; 3(3):266-280.
https://doi.org/10.3390/vibration3030019

**Chicago/Turabian Style**

Karaağaçlı, Taylan, and H. Nevzat Özgüven. 2020. "Experimental Identification of Backbone Curves of Strongly Nonlinear Systems by Using Response-Controlled Stepped-Sine Testing (RCT)" *Vibration* 3, no. 3: 266-280.
https://doi.org/10.3390/vibration3030019