# Experimental Characterization of Friction in a Negative Stiffness Nonlinear Oscillator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Negative Stiffness Oscillator

## 3. Experimental Nonlinear Characterization and Equation of Motion

#### 3.1. Identification of the Restoring Force

#### 3.2. Characterization of the Friction Force

- A dynamical damping friction term ${f}_{d}\left(z\right)$, with a Coulomb-like function having a transition velocity ${v}_{t}$;

- A static friction term ${f}_{s}\left(z\right)$ to account for the stiction force and the Stribeck effect;

- A linear viscous damping term $c\dot{z}$, to account for possible viscous forces generated by the lubricated slider.

## 4. Optimization-Based Identification Strategy

- The external forcing term is periodic, and the response is expected to be periodic as well (excluding the chance of chaotic motion);

- The evolution of the dissipated energy with the input amplitude is directly correlated with the dissipation functional form, i.e., the damping force $\mathcal{D}$;

- It depends both on the response amplitude and the phase with the forcing term.

#### 4.1. Experimental Estimation of the Energy Dissipated per Cycle (EDC)

#### 4.2. Numerical Computation via Harmonic Balance Method with Amplitude Continuation

#### Illustrative Example

## 5. Results and Discussion

#### 5.1. Sensitivity Analysis

#### 5.2. Nonlinear Amplitude Response Curves

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Qualitative representation of the potential $\mathcal{U}\left(z\right)$. Red dots: equilibrium positions.

**Figure 3.**Photos of the experimental setup. (

**a**) Negative equilibrium position ${z}_{-}^{*}$; (

**b**) positive equilibrium position ${z}_{+}^{*}$.

**Figure 4.**Measured cross-well oscillations under random excitation in [27].

**Figure 5.**Experimental restoring surface from [27] with cross-well random excitation in (

**a**); experimental restoring force (blue dots, ${\u03f5}_{s}=0.1\%$ and identified restoring force (red line) in (

**b**); experimental damping forces around ${z}_{-}^{*}$ (green dots, ${\u03f5}_{d}=0.1\%$) and ${z}_{+}^{*}$ (yellow dots, ${\u03f5}_{d}=0.1\%$) in (

**c**).

**Figure 6.**Stepped sine tests. Oscillations around the positive equilibrium in (

**a**) and around the negative equilibrium in (

**b**).

**Figure 7.**Experimental restoring surface with stepped sine excitations in (

**a**); experimental restoring force (blue dots, ϵ

_{s}= 0.1%) and identified restoring force (red line) in (

**b**); experimental damping forces around ${z}_{-}^{*}$(green dots, ${\u03f5}_{d}=0.1\%$) and ${z}_{+}^{*}$ (yellow dots, ${\u03f5}_{d}=0.1\%$) in (

**c**).

**Figure 8.**Variation in the experimental damping force across the position of the moving mass, $\left|z-\tilde{z}\right|<{\u03f5}_{d}$ with ${\u03f5}_{d}=0.1\%$.

**Figure 9.**Energy dissipated per cycle in (

**a**) with excitation amplitude of 1.3 N. Blue: oscillations around the positive equilibrium position; orange: oscillations around the negative equilibrium position. Corresponding response-input plots in (

**b**) and (

**c**) with highlighted areas.

**Figure 10.**Experimental estimation of the dissipated energy per cycle. Blue line: oscillations around ${z}_{+}^{*}$; orange line: oscillations around ${z}_{-}^{*}$. Bars represent the quantity ${S}_{k}^{mes}\pm 3{\sigma}_{k}^{mes}$.

**Figure 11.**Harmonic balance method (HBM) with amplitude continuation, illustrative example. Dashed lines represent unstable paths. Response amplitude in (

**a**); energy dissipated per cycle (EDC) in (

**b**); normalized harmonic coefficients in (

**c**).

**Figure 12.**Comparison between the HBM and numerical integration, illustrative example. Blue line: numerical integration with amplitude sweep down sine excitation; orange line: numerical integration with amplitude sweep up sine excitation; black line: the HBM with amplitude continuation.

**Figure 15.**Energy dissipated per cycle. Continuous lines: experimental estimations ${S}^{mes}\pm 3{\sigma}^{mes}$; dashed-dotted lines: final model predictions. Blue line: oscillations around ${z}_{+}^{*}$; orange line: oscillations around ${z}_{-}^{*}$.

**Figure 16.**Sensitivity analysis on the parameters of the identification. Dashed-dotted red line: reference EDC with the identified values. Blue lines: EDCs corresponding to a 20%, 50%, 70%, 120% and 150% variation of the selected parameter.

**Figure 17.**The HBM with amplitude continuation, final model. Response amplitude in (

**a**); normalized harmonic coefficients in (

**b**).

**Figure 18.**Bifurcation map of the system in the range 0.1–3.7 N. Black dots: experimental observations; red line: nonlinear amplitude response curve with the HBM.

k_{3} (N/m^{3}) | k_{2} (N/m^{2}) | k_{1} (N/m) | k_{0} (N) |
---|---|---|---|

7.35 · 10^{5} | 1.56 · 10^{3} | 550 | 2.4 |

$\mathit{m}\left(\mathbf{k}\mathbf{g}\right)$ | $\mathit{k}\left(\mathbf{N}/\mathbf{m}\right)$ | $\mathit{c}\left(\mathbf{N}\mathbf{s}/\mathbf{m}\right)$ | ${\mathit{f}}_{\mathit{d}}\left(\mathbf{N}\right)$ | ${\mathbf{k}}_{\mathit{s}}$ | ${\mathit{v}}_{\mathit{t}}\left(\mathbf{m}/\mathbf{s}\right)$ |
---|---|---|---|---|---|

1.3 | 800 | 1 | 1 | 1.2 | 10^{−2} |

${\mathit{f}}_{\mathit{d}}\left({\mathit{z}}_{-}^{*}\right)\left(\mathbf{N}\right)\text{}$ | ${\mathit{f}}_{\mathit{d}}\left(0\right)\left(\mathbf{N}\right)$ | ${\mathit{f}}_{\mathit{d}}\left({\mathit{z}}_{+}^{*}\right)\left(\mathbf{N}\right)\text{}$ | ${\mathit{k}}_{\mathit{s}}\text{}$ | $\mathit{c}\left(\mathbf{N}\mathbf{s}/\mathbf{m}\right)\text{}$ | ${\mathit{v}}_{\mathit{t}}\left(\mathbf{m}/\mathbf{s}\right)\text{}$ |
---|---|---|---|---|---|

$0.33$ | $0.67$ | $0.66$ | $1.07$ | $2.02$ | $1.5\cdot {10}^{-2}$ |

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**MDPI and ACS Style**

Anastasio, D.; Marchesiello, S. Experimental Characterization of Friction in a Negative Stiffness Nonlinear Oscillator. *Vibration* **2020**, *3*, 132-148.
https://doi.org/10.3390/vibration3020011

**AMA Style**

Anastasio D, Marchesiello S. Experimental Characterization of Friction in a Negative Stiffness Nonlinear Oscillator. *Vibration*. 2020; 3(2):132-148.
https://doi.org/10.3390/vibration3020011

**Chicago/Turabian Style**

Anastasio, Dario, and Stefano Marchesiello. 2020. "Experimental Characterization of Friction in a Negative Stiffness Nonlinear Oscillator" *Vibration* 3, no. 2: 132-148.
https://doi.org/10.3390/vibration3020011