# Non-Linear Dynamics of Simple Elastic Systems Undergoing Friction-Ruled Stick–Slip Motions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Mechanical Model

_{A}(t). The rigid block is free to slide with friction over a second rigid support, whose reference point B moves according to a prescribed law, x

_{B}(t). The friction force between this second rigid support and the rigid block is F

_{a}. The position of the block’s center of gravity (G) with respect to the inertial reference frame is x(t). The following equation of motion holds

#### 2.1. The Friction Law: A Modified Version of the Coulomb’s Formulation

#### 2.2. Event-Driven Solution of the Equation of Motion

#### 2.3. End Time of Sticking Phases

#### 2.4. End Time of Sliding Phases

## 3. Remarks on the System Limit Cycles

#### 3.1. Sticking Limit Cycles

#### 3.2. Sliding Limit Cycles

#### 3.3. The Influence of the System Parameters on Its Long-Term Response

#### 3.4. Bifurcation Diagrams

## 4. Numerical Examples

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Gibert, J.M.; Fadel, G.; Daqaq, M.F. On the stick-slip dynamics in ultrasonic additive manufacturing. J. Sound Vib.
**2013**, 332, 4680–4695. [Google Scholar] [CrossRef] - Burridge, R.; Knopov, L. Model and theoretical seismicity. BSSA
**1967**, 57, 341–371. [Google Scholar] [CrossRef] - Chopra, A. Dynamics of Structures: Theory and Applications to Earthquake Engineering, 2nd ed.; Prentice Hall, Inc.: Upper Saddle River, NJ, USA, 2003. [Google Scholar]
- Grigorian, C.; Popov, E. Energy Dissipation with Slotted Bolted Connections, UCB/EERC-94/02; California University, Earthquake Engineering Research Center: Richmond, CA, USA, 1994. [Google Scholar]
- Castaldo, P.; Palazzo, B.; Della Vecchia, P. Seismic reliability of base-isolated structures with friction pendulum bearings. Eng. Struct.
**2015**, 95, 80–93. [Google Scholar] [CrossRef] - Avinash, A.R.; Krishnamoorthy, A.; Kamath, K.; Chaithra, M. Sliding isolation systems: Historical review, modeling techniques, and the contemporary trends. Buildings
**2022**, 12, 1997. [Google Scholar] [CrossRef] - Bhaskararao, A.V.; Jangid, R.S. Seismic analysis of structures connected with friction dampers. Eng. Struct.
**2006**, 28, 690–703. [Google Scholar] [CrossRef] - Ozbulut, O.; Bitaraf, M.; Hurlebaus, S. Adaptive control of base-isolated structures against near-field earthquakes using variable friction dampers. Eng. Struct.
**2011**, 33, 3143–3154. [Google Scholar] [CrossRef] - Yang, T.Y.; Zuo, X.; Rodgers, G.W.; Bagatini-Cachuço, F. Mechanism development and experimental validation of self-centering nonlinear friction damper. Eng. Struct.
**2023**, 287, 116093. [Google Scholar] [CrossRef] - Barsotti, R.; Bennati, S.; Quattrone, F. A simple mechanical model for a wiper blade sliding and sticking over a windscreen. Open Mech. Eng. J.
**2016**, 10, 51–65. [Google Scholar] [CrossRef] - Ferguson, C.D.; Klein, W.; Rundle, J.B. Long-range earthquake fault models. Comput. Phys.
**1998**, 12, 34–40. [Google Scholar] [CrossRef] - Dieterich, J.H. Modeling of rock friction: 1. Experimental results and constitutive equations. J. Geophys.
**1979**, 84, 2161–2168. [Google Scholar] [CrossRef] - Kavvadias, I.; Vasiliadis, L. Finite Element Modeling of Single and Multi-Spherical Friction Pendulum Bearings. In Proceedings of the 6th Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Rhodes Island, Greece, 15–17 June 2017. [Google Scholar]
- Lancioni, G.; Lenci, S.; Galvanetto, U. Non-linear dynamics of a mechanical system with a frictional unilateral constraint. Int. J. Non-Linear Mech.
**2009**, 44, 658–674. [Google Scholar] [CrossRef] - Den Hartog, J. Forced Vibrations with Combined Coulomb and Viscous Friction. Trans. ASME
**1931**, 53, 107–115. [Google Scholar] [CrossRef] - Parnes, R. Response of an Oscillator to a Ground Motion with Coulomb Friction Slippage. J. Sound Vib.
**1984**, 94, 469–482. [Google Scholar] - Marui, E.; Kato, S. Forced Vibration of Base-Excited Single-Degree-of-Freedom System with Coulomb Friction. J. Dyn. Sys Meas. Control
**1984**, 106, 280–285. [Google Scholar] [CrossRef] - Shaw, S.W. On the dynamic response of a system with dry friction. J. Sound Vib.
**1986**, 108, 305–325. [Google Scholar] [CrossRef] - Leine, R.I.; Van Campen, D.H.; De Kraker, A. Stick-slip vibrations induced by alternate friction models. Nonlinear Dyn.
**1998**, 16, 41–54. [Google Scholar] [CrossRef] - Hong, H.K.; Liu, C.S. Non-Sticking Oscillation Formulae for Coulomb Friction Under Harmonic Loading. J. Sound Vib.
**2001**, 244, 883–898. [Google Scholar] [CrossRef] - Popp, K.; Stelter, P. Stick-Slip Vibrations and Chaos. Philos. Trans.
**1990**, 332, 89–105. [Google Scholar] - Andreaus, V.; Casini, P. Dynamics of Friction Oscillators Excited by a Moving Base and/or Driving Force. J. Sound Vib.
**2001**, 245, 685–699. [Google Scholar] [CrossRef] - Licskó, G.; Csernák, G. On the chaotic behaviour of a simple dry-friction oscillator. Math. Comput. Simulat.
**2014**, 95, 55–62. [Google Scholar] [CrossRef] - Csernák, G.; Stépán, G. On the periodic response of a harmonically excited dry friction oscillator. J. Sound Vib.
**2006**, 295, 649–658. [Google Scholar] [CrossRef] - Butikov, E.I. Spring pendulum with dry and viscous damping. Commun. Nonlinear Sci.
**2015**, 20, 298–315. [Google Scholar] [CrossRef] - Wang, X.; Long, X.; Yue, H.; Dai, S. Atluri. Bifurcation analysis of stick-slip vibration in a 2-DOF nonlinear dynamical system with dry friction. Commun. Nonlinear Sci.
**2022**, 111, 106475. [Google Scholar] [CrossRef] - Bennati, S.; Barsotti, R.; Migliaccio, G. A simple model for predicting the nonlinear dynamic behavior of elastic systems subjected to friction. In Proceedings of the XXIV AIMETA Conference 2019, Rome, Italy, 5 May 2020; Lecture Notes in Mechanical Engineering. Carcaterra, A., Paolone, A., Graziani, G., Eds.; Springer: Cham, Switzerland, 2020. [Google Scholar]
- Bennati, S.; Barsotti, R.; Migliaccio, G. A simple model for investigating the non-linear dynamic behavior of elastic systems subjected to stick-slip motion. In Proceedings of the 7th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2019), Crete, Greece, 24–26 June 2019; Volume 3, pp. 4483–4492. [Google Scholar]
- Sampson, J.; Morgan, F.; Reed, D.W.; Muskat, M. Friction behavior during the slip portion of the stick-slip process. J. Appl. Phys.
**1943**, 14, 689–700. [Google Scholar] [CrossRef] - Rabinowicz, E. The Intrinsic Variables affecting the Stick-Slip Process. Proc. Phys. Soc.
**1958**, 71, 668–675. [Google Scholar] [CrossRef] - Pennestrì, E.; Rossi, V.; Salvini, P.; Valentini, P.P. Review and comparison of dry friction force models. Nonlinear Dyn.
**2016**, 83, 1785–1801. [Google Scholar] [CrossRef]

**Figure 1.**Example of structure fitted with a slotted bolted connection (SBC) with steel–brass sliding surfaces.

**Figure 2.**Example of structure fitted with a friction pendulum system (FPS adapted from [6]).

**Figure 3.**Schematic representation of the Burridge–Knopoff model (figure drawn from [1]).

**Figure 4.**Scheme of a single-degree-of-freedom system that is connected elastically to a first rigid support (

**A**) and can slide with friction over a second rigid support (

**B**): m is the mass of the rigid block; k and c are the elastic constant of the spring and the constant coefficient of the linear dashpot between rigid block and rigid support A; ${\mu}_{s}$ and ${\mu}_{d}$ are the static and dynamic friction coefficients that model the tangential interaction between rigid block and rigid support B; N is the normal contact force between rigid block and rigid support B.

**Figure 6.**Partition of the a-$\mathsf{\Omega}$ plane showing possible long-term system responses for the three cases corresponding to $\eta $ = 1/3, $\eta $ = 2/3, and $\eta $ = 1.

**Figure 7.**Bifurcation diagrams in the domain $\mathsf{\Omega}-\stackrel{~}{x}/{x}_{s}$ for a = 10 and $\eta $ = 1/3. Blue points on the diagram correspond to sticking phase, red points are associated with sliding phase.

**Figure 8.**Bifurcation diagrams in the domain $\mathsf{\Omega}-\stackrel{~}{x}/{x}_{s}$ for a = 10 and $\eta $ = 1. Blue points on the diagram correspond to sticking phase, red points are associated with sliding phase.

**Figure 9.**Bifurcation diagrams in the domain $a-\stackrel{~}{x}/{x}_{s}$ for $\mathsf{\Omega}$ = 1/3 and $\eta $ = 1/3. Blue points on the diagram correspond to sticking phase, red points are associated with sliding phase.

**Figure 10.**Bifurcation diagrams in the domain $a-\stackrel{~}{x}/{x}_{s}$ for $\mathsf{\Omega}$ = 1/3 and $\eta $ = 1. Blue points on the diagram correspond to sticking phase, red points are associated with sliding phase.

**Figure 11.**Time history of position and velocity for a = 10, $\eta $ = 1/3, $\mathsf{\Omega}$ = 0.09 (stick–slip limit cycle).

**Figure 12.**Phase portrait of the system for a = 10, $\eta $ = 1/3, $\mathsf{\Omega}$ = 0.09 (stick–slip limit cycle).

**Figure 13.**Time history of position and velocity for a = 10, $\eta $ = 1/3, $\mathsf{\Omega}$ = 0.60 (sliding limit cycle).

**Figure 14.**Phase portrait of the system for a = 10, $\eta $ = 1/3, $\mathsf{\Omega}$ = 0.60 (sliding limit cycle).

**Figure 15.**Time history of position and velocity for a = 10, $\eta $ = 1/3, $\mathsf{\Omega}$ = 0.99 (sticking limit cycle).

**Figure 16.**Phase portrait of the system for a = 10, $\eta $ = 1/3, $\mathsf{\Omega}$ = 0.99 (sticking limit cycle).

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 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**

Barsotti, R.; Bennati, S.; Migliaccio, G.
Non-Linear Dynamics of Simple Elastic Systems Undergoing Friction-Ruled Stick–Slip Motions. *CivilEng* **2024**, *5*, 420-434.
https://doi.org/10.3390/civileng5020021

**AMA Style**

Barsotti R, Bennati S, Migliaccio G.
Non-Linear Dynamics of Simple Elastic Systems Undergoing Friction-Ruled Stick–Slip Motions. *CivilEng*. 2024; 5(2):420-434.
https://doi.org/10.3390/civileng5020021

**Chicago/Turabian Style**

Barsotti, Riccardo, Stefano Bennati, and Giovanni Migliaccio.
2024. "Non-Linear Dynamics of Simple Elastic Systems Undergoing Friction-Ruled Stick–Slip Motions" *CivilEng* 5, no. 2: 420-434.
https://doi.org/10.3390/civileng5020021