# Approach for Calibrated Measurement of the Frequency Response for Characterization of Compliant Interface Elements on Vibration Test Benches

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theory

#### 2.2. Calibration Function of the Frequency Response

#### 2.3. The Unknown Calibration Values

#### 2.4. Dynamic Response Measurement Systems for AIEs with Translatory Motion

^{2}leads to a frequency range from 3 to 23 Hz, since lower frequencies at this acceleration would result in too high displacements.

#### 2.5. Masses and Compliant Elements under Characterization

## 3. Results and Discussion

#### 3.1. Dynamic Characterization of the System

#### 3.2. Calibration of the Measurement System

#### 3.3. The Dynamic Response of the Masses

#### 3.4. Evaluation of the Dynamic Response of the Compliant Elements

#### 3.5. Findings from the Performed Dynamic Calibration

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

abs | absolute value or magnitude in polar coordinate system |

AC | accelerance |

AIE | adjustable impedance element |

AM | apparent mass |

arg | argument in polar coordinate system |

AS | apparent stiffness |

FRF | frequency response function |

MI | mechanical impedance |

MO | mobility |

RE | receptance |

## References

- Heyden, E.; Lindenmann, A.; Oltmann, J.; Bruchmueller, T.; Krause, D.; Matthiesen, S. Adjustable Impedance Elements for Testing and Validation of System Components. In Symposium Lightweight Design in Product Development; Ermanni, P., Meboldt, M., Wartzack, S., Krause, D., Zogg, M., Eds.; CMASLab, ETH: Zurich, Switzerland, 2018; pp. 44–46. [Google Scholar]
- Lalanne, C. Sinusoidal Vibration, 3rd ed.; John Wiley & Sons, Ltd.: Chichester, UK, 2014. [Google Scholar] [CrossRef]
- Moldenhauer, B.; Allen, M.; DeLima, W.J.; Dodgen, E. Using Hybrid Modal Substructuring with a Complex Transmission Simulator to Model an Electrodynamic Shaker. In Dynamic Substructures; Linderholt, A., Allen, M.S., Mayes, R.L., Rixen, D., Eds.; Conference Proceedings of the Society for Experimental Mechanics Series; Springer International Publishing: Cham, Switzerland, 2020; Volume 4, pp. 23–34. [Google Scholar] [CrossRef]
- Karlicek, A.C.; Dilworth, B.J.; McDaniel, J.G. System Characterization and Design Using Mechanical Impedance Representations. In Special Topics in Structural Dynamics & Experimental Techniques; Epp, D.S., Ed.; Conference Proceedings of the Society for Experimental Mechanics Series; Springer International Publishing: Cham, Switzerland, 2021; Volume 5, pp. 121–128. [Google Scholar] [CrossRef]
- Kim, J.Y.; Jeong, W.B.; Lee, S.B.; Lee, B.H. An Experimental Approach for Structural Dynamic Modification of Fixture in Vibration Test Control. JSME Int. J. Ser. C
**2001**, 44, 334–340. [Google Scholar] [CrossRef][Green Version] - Rivin, E. Design Techniques for Reducing Structural Deformations (Stiffness Enhancement Techniques). In Handbook on Stiffness & Damping in Mechanical Design; Rivin, E.I., Ed.; ASME: New York, NY, USA, 2010; pp. 371–453. [Google Scholar] [CrossRef]
- Heyden, E.; Hartwich, T.S.; Schwenke, J.; Krause, D. Transferability of Boundary Conditions in Testing and Validation of Lightweight Structures. In Proceedings of the 30th Symposium Design for X (DFX 2019), Jesteburg, Germany, 18–19 September 2019; pp. 85–96. [Google Scholar] [CrossRef]
- Lindenmann, A.; Heyden, E.; Matthiesen, S.; Krause, D. Adjustable Impedance Elements for Testing and Validation of Aircraft Components and Hand-Held Power Tools. In Stuttgarter Symposium für Produktentwicklung SSP 2019; Binz, H., Bertsche, B., Bauer, W., Riedel, O., Spath, D., Roth, D., Eds.; Fraunhofer-Institut für Arbeitswirtschaft und Organisation IAO: Stuttgart, Germany, 2019; pp. 63–72. [Google Scholar]
- Vanderborght, B.; Albu-Schaeffer, A.; Bicchi, A.; Burdet, E.; Caldwell, D.G.; Carloni, R.; Catalano, M.; Eiberger, O.; Friedl, W.; Ganesh, G.; et al. Variable impedance actuators: A review. Robot. Auton. Syst.
**2013**, 61, 1601–1614. [Google Scholar] [CrossRef][Green Version] - van Ham, R.; Sugar, T.; Vanderborght, B.; Hollander, K.; Lefeber, D. Compliant actuator designs. IEEE Robot. Autom. Mag.
**2009**, 16, 81–94. [Google Scholar] [CrossRef] - Tagliamonte, N.L.; Sergi, F.; Accoto, D.; Carpino, G.; Guglielmelli, E. Double actuation architectures for rendering variable impedance in compliant robots: A review. Mechatronics
**2012**, 22, 1187–1203. [Google Scholar] [CrossRef] - Petit, F.; Friedl, W.; Höppner, H.; Grebenstein, M. Analysis and Synthesis of the Bidirectional Antagonistic Variable Stiffness Mechanism. IEEE/ASME Trans. Mechatron.
**2015**, 20, 684–695. [Google Scholar] [CrossRef] - Stucheli, M.; Foehr, A.; Meboldt, M. Work density analysis of adjustable stiffness mechanisms. In Proceedings of the 2016 IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, 16–21 May 2016; pp. 648–654. [Google Scholar] [CrossRef]
- Awad, M.I.; Gan, D.; Cempini, M.; Cortese, M.; Vitiello, N.; Dias, J.; Dario, P.; Seneviratne, L. Modeling, design & characterization of a novel Passive Variable Stiffness Joint (pVSJ). In Proceedings of the 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, Korea, 9–14 October 2016; pp. 323–329. [Google Scholar] [CrossRef]
- Vuong, N.D.; Li, R.; Chew, C.M.; Jafari, A.; Polden, J. A novel variable stiffness mechanism with linear spring characteristic for machining operations. Robotica
**2017**, 35, 1627–1637. [Google Scholar] [CrossRef] - Hošovský, A.; Pite’, J.; Židek, K.; Tóthová, M.; Sárosi, J.; Cveticanin, L. Dynamic characterization and simulation of two-link soft robot arm with pneumatic muscles. Mech. Mach. Theory
**2016**, 103, 98–116. [Google Scholar] [CrossRef] - Li, Z.Q.; Xu, Y.L.; Zhou, L.M. Adjustable fluid damper with SMA actuators. Smart Mater. Struct.
**2006**, 15, 1483–1492. [Google Scholar] [CrossRef] - Deng, H.; Deng, J.; Yue, R.; Han, G.; Zhang, J.; Ma, M.; Zhong, X. Design and verification of a seat suspension with variable stiffness and damping. Smart Mater. Struct.
**2019**, 28, 065015. [Google Scholar] [CrossRef] - Xing, Z.; Yu, M.; Sun, S.; Fu, J.; Li, W. A hybrid magnetorheological elastomer-fluid (MRE-F) isolation mount: Development and experimental validation. Smart Mater. Struct.
**2016**, 25, 015026. [Google Scholar] [CrossRef] - Sun, S.; Deng, H.; Du, H.; Li, W.; Yang, J.; Liu, G.; Alici, G.; Yan, T. A Compact Variable Stiffness and Damping Shock Absorber for Vehicle Suspension. IEEE/ASME Trans. Mechatron.
**2015**, 20, 2621–2629. [Google Scholar] [CrossRef] - Sun, S.; Tang, X.; Li, W.; Du, H. Advanced vehicle suspension with variable stiffness and damping MR damper. In Proceedings of the 2017 IEEE International Conference on Mechatronics (ICM), Churchill, VIC, Australia, 13–15 February 2017; pp. 444–448. [Google Scholar] [CrossRef]
- Wu, T.H.; Lan, C.C. A wide-range variable stiffness mechanism for semi-active vibration systems. J. Sound Vib.
**2016**, 363, 18–32. [Google Scholar] [CrossRef] - Meng, K.; Sun, Y.; Pu, H.; Luo, J.; Yuan, S.; Zhao, J.; Xie, S.; Peng, Y. Development of Vibration Isolator With Controllable Stiffness Using Permanent Magnets and Coils. J. Vib. Acoust.
**2019**, 141, 954. [Google Scholar] [CrossRef] - Jujjavarapu, S.S.; Memar, A.H.; Karami, M.A.; Esfahani, E.T. Variable Stiffness Mechanism for Suppressing Unintended Forces in Physical Human–Robot Interaction. J. Mech. Robot.
**2019**, 11, 1–7. [Google Scholar] [CrossRef][Green Version] - Dong, R.G.; Welcome, D.E.; McDowell, T.W.; Wu, J.Z. Measurement of biodynamic response of human hand–arm system. J. Sound Vib.
**2006**, 294, 807–827. [Google Scholar] [CrossRef] - Ewins, D.J. Modal Testing: Theory and Practice, 4th ed.; Mechanical engineering research studies; Research Studies Pr: Taunton, UK, 1989; Volume 2. [Google Scholar]
- McConnell, K.G. Vibration Testing: Theory and Practice; Wiley: New York, NY, USA, 1995. [Google Scholar]
- International Organization for Standardization. General Requirements for the Competance of Testing and Calibration Laboratories (ISO/IEC 17025:2017). 03/2018. Available online: https://www.iso.org/standard/66912.html (accessed on 25 July 2021).
- Silva, J.M.M.; Maia, N.M.M.; Ribeiro, A.M.R. Cancelation of Mass-Loading Effects of Transducers and Evaluation of Unmeasured Frequency Response Functions. J. Sound Vib.
**2000**, 236, 761–779. [Google Scholar] [CrossRef] - Silva, J.M.M.; Maia, N.M.M.; Ribeiro, A.M.R. Some Applications of Coupling/Uncoupling Techniques in Structural Dynamics: Part 1: Solving the Mass Cancellation Problem. In Proceedings of the SPIE—The International Society for Optical Engineering, Orlando, FL, USA, 21–22 April 1997; pp. 1431–1439. [Google Scholar]
- Maia, N.M.M.; Silva, J.M.M.; Ribeiro, A.M.R. Some Applications of Coupling/Uncoupling Techniques in Structural Dynamics: Part 2: Generation of the Whole FRF Matrix from Measurements on a Single Column-The Mass Uncoupling Method (MUM). In Proceedings of the SPIE— The International Society for Optical Engineering, Orlando, FL, USA, 21–22 April 1997; pp. 1440–1452. [Google Scholar]
- Laukotka, F.; Hartwich, T.S.; Hauschild, J.; Heyden, E.; Schmidt, J.; Schwenke, J.; Wegner, M.; Wortmann, N.; Krause, D. Entwicklung und Anwendung von Sonderprüfständen. In Produktentwicklung und Konstruktionstechnik; Krause, D., Hartwich, T.S., Rennpferdt, C., Eds.; Produktentwicklung und Konstruktionstechnik; Springer: Heidelberg, Germany, 2020; Volume 19, pp. 177–205. [Google Scholar] [CrossRef]
- Brandt, A. Noise and Vibration Analysis: Signal Analysis and Experimental Procedures; John Wiley and Sons Ltd.: Chichester, UK, 2011. [Google Scholar] [CrossRef]
- Lang, G.F.; Snyder, D. Understanding the Physics of Electrodynamic Shaker Performance. In Sound and Vibration—Dynamic Testing Reference Issue; Data Physics Corporation: San Jose, CAM, USA, 2001; pp. 24–35. [Google Scholar]
- Stone, C.J.; Koo, C.Y. Additive Splines in Statistics. Proc. Am. Stat. Assoc. ASA
**1985**, 45, 48. [Google Scholar] - Harrell, F.E. Regression Modeling Strategies: With Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis; Springer International Publishing: Cham, Switzerland, 2015. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) mechanical model of a mass-damper-spring system; (

**b**) mass separated into ${m}_{sensor}$ and ${m}_{testobj.}$.

**Figure 2.**(

**a**) Hydraulic test bench for low frequencies adapted from [32]; (

**b**) electrodynamic test bench for high frequencies.

**Figure 3.**Test setup for the compliant element at: (

**a**) low frequency; (

**b**) high frequency test bench.

**Figure 4.**(

**a**) Compliant element A with two rubber buffers aligned; (

**b**) compliant element B with one rubber buffer aligned; (

**c**) size of rubber buffer.

**Figure 7.**Average residual e (Equation (19)) of $H{I}_{pp}\left(f\right)$ over degree of fitting polynoma for the low frequency (

**a**) and high frequency (

**b**) test bench. The red circle marks the polynomial degree whose relative change of the residual is less than 1%.

**Figure 9.**FRFs AM, MI, AS and its phase directly measured and the calibrated FRFs of the compliant element A over frequency.

**Figure 10.**Apparent Stiffness directly measured $A{S}_{meas.}$ and calibrated $A{S}_{testobj.}$ of the compliant element A at the low frequency test bench.

**Figure 11.**Calibrated FRFs AM, MI, AS and its phase of the compliant elements A and B with double and single rubber buffer in each stack over frequency.

${\mathit{m}}_{\mathbf{sensor}}$ | ${\mathit{m}}_{1}$ | ${\mathit{m}}_{2}$ | ${\mathit{m}}_{3}$ | ${\mathit{m}}_{4}$ | |
---|---|---|---|---|---|

low freq. test bench | 0.863 kg | 2.482 kg | 4.965 kg | 7.448 kg | 9.9316 kg |

high freq. test bench | 1.133 kg | 0.234 kg | 0.467 kg | 0.7011 kg | 0.9315 kg |

Average Deviation ${\mathit{e}}_{\mathit{AM},\mathit{abs}}$ (${\mathit{e}}_{\mathit{AM},\mathit{rel}}$) | Low Freq. Test Bench | High Freq. Test Bench |
---|---|---|

static calibration | 0.6252 kg (12.0 %) | 1.1583 kg (252.6 %) |

mass cancellation by Ewins [26] | 0.4567 kg (8.50 %) | 0.1125 kg (18.23 %) |

$H{I}_{pp,fit}\left({m}_{0}\right)$ by Dong et al. [25] | 0.0982 kg (1.68 %) | 0.0628 kg (10.46 %) |

$H{I}_{pp,fit}\left({m}_{i}\right)$ over added masses | 0.0433 kg (0.74 %) | 0.0237 kg (4.21 %) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Heyden, E.; Lindenmann, A.; Matthiesen, S.; Krause, D.
Approach for Calibrated Measurement of the Frequency Response for Characterization of Compliant Interface Elements on Vibration Test Benches. *Appl. Sci.* **2021**, *11*, 9604.
https://doi.org/10.3390/app11209604

**AMA Style**

Heyden E, Lindenmann A, Matthiesen S, Krause D.
Approach for Calibrated Measurement of the Frequency Response for Characterization of Compliant Interface Elements on Vibration Test Benches. *Applied Sciences*. 2021; 11(20):9604.
https://doi.org/10.3390/app11209604

**Chicago/Turabian Style**

Heyden, Emil, Andreas Lindenmann, Sven Matthiesen, and Dieter Krause.
2021. "Approach for Calibrated Measurement of the Frequency Response for Characterization of Compliant Interface Elements on Vibration Test Benches" *Applied Sciences* 11, no. 20: 9604.
https://doi.org/10.3390/app11209604