# Discussion of Stick-Slip Dynamics of 2DOF Sliding Systems Based on Dynamic Vibration Absorbers Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling

#### 2.1. Analytical Model

_{a}is connected to the rigid base by a linear spring with a stiffness k

_{a}and a dashpot with a damping coefficient c

_{a}. In the primary vibration system, a primary mass m is then connected in a series to the additional mass m

_{a}by a linear spring with a stiffness k and a dashpot with a damping coefficient c. The mass in the primary system is in contact with a moving plate that moves horizontally with a constant driving speed V under a normal load W. The friction force F acts on the contact interface between the mass and the moving plate.

#### 2.2. Governing Equations

_{a}denote the displacement from the natural length of the springs of the primary and additional masses, respectively, and (

^{•}) denotes the derivative with respect to time t.

_{s}acts on the contact surface, it acts along the positive direction of x. In contrast, for the slip-I state ($\dot{x}<V$) and slip-II state ($\dot{x}>V$), kinetic friction F

_{k}acts along the positive and negative directions, respectively. Therefore, F in Equation (1) can be expressed as follows:

_{s}is lower than the maximum static friction force F

_{smax}. Thus, the following equation can be written with the static friction coefficient μ

_{s}:

_{s}approaches μ

_{s}W, the friction state changes from the stick state to the slip-I state. During the slip-I and slip-II states, F

_{k}can be expressed using the kinetic friction coefficient μ

_{k}as follows:

#### 2.3. Dimensionless Description of Governing Equations

_{a}and dimensionless time τ, as follows:

_{n}is the natural frequency of the primary vibration system (rad/s), defined as:

_{a}, and τ, the governing equations, Equations (6)–(14), can be rewritten as follows:

_{a}, ζ, λ, and γ are defined as follows:

_{a}, ζ, λ, and γ.

_{ext}was applied to the primary mass, as shown in Figure 3. The equation of motion for the analysis model shown in Figure 3 is the same as when the friction force F in Equation (1) is replaced with the periodic excitation force f

_{ext}(=fsin(ωt)), where f and ω are the amplitude and angular frequency, respectively, of the periodic external force.

_{opt}(=m

_{a_opt}/m), optimally tuned stiffness ratio K

_{opt}(=k

_{a_opt}/k), and optimally tuned damping ratio ζ

_{a}

_{opt}(=c

_{a_opt}/(mk)

^{0.5}) under different damping ratio ζ (=c/(mk)

^{0.5}) values. As shown in Figure 4, we calculated a best fit curve for the results of Harik and Issa [20] to determine the ζ dependence for each optimally tuned parameter. The resulting formulae are as follows:

_{n}) are the dimensionless displacement and the forced frequency ratio, respectively; H

_{max}is the maximum dimensionless displacement under each condition. As seen in Figure 5, H

_{max}is greatly reduced when the optimally tuned DVA parameters are applied.

_{max}, as shown in Figure 6. We obtained the following formula:

## 3. Numerical Results

_{opt}, K = K

_{opt}, ζ

_{a}= ζ

_{a}

_{opt}, ζ = 0.1, and γ = 2, and (b) M = M

_{opt}, K = 2K

_{opt}, ζ

_{a}= ζ

_{a}

_{opt}, ζ = 0.1, and γ = 2. Thus, the results in Figure 7a,b correspond to the use of the optimally tuned DVA and the non-optimally tuned DVA systems, respectively. Previous research shows that stick-slip occurs under a large λ condition [11]. Considering that an increase in λ corresponds to an increase in the normal load W, a decrease in velocity V, and a decrease in support stiffness k, the observed stick-slip characteristics exhibit typical stick-slip behavior. Figure 7 clearly shows that setting the value for the DVA parameter strongly affects the occurrence and non-occurrence of stick-slip.

_{a}, ζ, λ, and γ. Because the DVA parameters are optimally set based on Equations (28)–(30), that is, M

_{aopt}, K

_{aopt}, and ζ

_{aopt}are the dependent variables of ζ, the dynamic behavior of the present system is characterized only by the three parameters λ, ζ, and γ. Because γ does not affect the occurrence and non-occurrence of stick-slip [11], it is possible to completely determine when stick-slip occurs by investigating the occurrence and non-occurrence of stick-slip when λ and ζ are changed.

_{eq}= 0. Thus, this configuration corresponds to the stick-slip occurrence condition when no DVA is used. The solid line drawn in Figure 8a is the discriminant equation of the boundary conditions for the occurrence and non-occurrence of stick-slip when not using the DVA system, based on Equation (37).

## 4. Discussion

_{ext}(=fsin(ωt)) was applied to the primary mass, as shown in Figure 3. If the frequency of the cyclic external force is in the region close to the natural frequency of the system (i.e., ω = ω

_{n}) under the condition that the dynamic absorber parameters are optimized, the behavior of the system can be approximated by an equivalent 1DOF vibration system. Figure 9 shows the equivalent 1DOF system that can replace the 2DOF DVA system shown in Figure 1. In Figure 9, the magnitude of c

_{eq}represents the stick-slip suppression effect of the DVA.

_{max}and the damping ratio of the equivalent model ζ

_{eq}= (c + c

_{eq})/(2(mk)

^{0.5}) is given by the following equation:

_{eq}. Here, ζ

_{eq}is the equivalent damping ratio including the damping effect of the optimally applied DVA, and ζ is the damping ratio of the primary vibration system. Therefore, Equation (33) holds only when the DVA is optimally adjusted.

_{eq}. From Figure 5, it is found that the behavior of the main vibration system in the 2DOF system and the equivalent 1-DOF system are in good agreement in the range close to ω = ω

_{n}.

_{ss}and amplitude A

_{ss}is formulated as follows

_{n}= ω

_{n}/2π and A

_{n}= V/ω

_{n}.

_{eq}, as follows:

_{eq}= 0 in the equivalent model shown in Figure 9. Thus, the results in Figure 8a corresponds to the results of stick-slip occurrence conditions when c

_{eq}= 0 in Figure 9. Therefore, by replacing ζ

_{eq}with ζ in Equation (37), it is possible to obtain the boundary equation for the occurrence and non-occurrence of stick-slip when the DVA is not attached, as shown below:

_{opt}, K

_{opt}, ζ

_{aopt}) were determined using Equations (28)–(30). At the same time, the optimum value of the DVA parameters (m

_{a_opt}, k

_{a_opt}, c

_{a_opt}) were determined using Equations (28)–(30). Using the results, we were able to design a DVA for the specific situation. Based on Equation (33), we quantified the equivalent damping ratio ζ

_{eq}when the optimum vibration absorber was installed. Finally, using Equation (38), we were able to quantify the values of λ at which stick-slip occurs. Because λ includes the velocity V and load W, it was possible to quantify the velocity range or normal load range in which stick-slip was expected to occur.

_{opt}, K = K

_{opt}/2, ζ

_{a}= ζ

_{a}

_{opt}, and γ = 2; and in Figure 11b the parameters are set to M = M

_{opt}, K = 2K

_{opt}, ζ

_{a}= ζ

_{a}

_{opt}, and γ = 2. When the parameters of the DVA diverge from those defined by Equations (28)–(30), the stick-slip occurrence region may be wider or narrower than the region predicted by Equation (37). Importantly, the discriminant of Equation (37) does not necessarily show the maximum stick-slip suppression ability of the DVA. The importance of Equation (37) is that if the parameters of the DVA are set to the values defined by Equations (28)–(30), then Equation (37) represents the region where stick-slip reliably does not occur. Thus, the theoretical prediction tool derived in this study can be utilized effectively when making a quantitative design based on theory.

_{a}and dimensionless velocities ${\xi}^{\prime}$ and ${{\xi}^{\prime}}_{a}$ under λ = 1, ζ = 0.01. Under these conditions, stick-slip does not occur in either case and the initial fluctuations decrease and decay with time. A closer look reveals that for K = K

_{opt}/2, i.e., Figure 12a, the oscillations of the main vibration system (black line) and the secondary vibration (DVA) system (gray line) are nearly in opposite phases. On the other hand, in case K = k

_{opt}× 2, i.e., Figure 12b, each mass point is oscillating in perfectly in-phase. Figure 13 shows the time variations of dimensionless displacements ξ and ξ

_{a}and dimensionless velocities ${\xi}^{\prime}$ and ${{\xi}^{\prime}}_{a}$ under λ = 10, ζ= 0.01. In Figure 13a, stick-slip does not occur, but in Figure 13b, stick-slip is observed. This corresponds to the fact that in Figure 11, the stick-slip region is wide for K = k

_{opt}× 2 and extremely narrow for K = K

_{opt}/2. From the above, it can be seen that the vibration modes have a significant effect on the stick-slip suppression observed in Figure 11a. In the case of K = K

_{opt}/2, the two masses oscillate in opposite phases as a second-order mode, and their motions cancel each other out to produce the stick-slip suppression effect. In this case, stick-slip can be suppressed even at a higher λ than the condition predicted by the stick-slip occurrence discriminant formula shown in Equation (37). In other words, by setting the phases of the two masses well, it is possible to obtain a large vibration suppression effect beyond the vibration suppression effect of a general dynamic absorber. Although this effect is not discussed in depth in this study, it is considered to be an important finding for stick-slip suppression in 2DOF vibration systems.

_{eq}. Next, λ and ζ

_{eq}are substituted into Equation (38). If the same equation holds, stick-slip suppression using an elastic foundation is possible. By substituting ζ into Equations (28)–(30), we can then obtain M

_{opt}, K

_{opt}, and ζ

_{aopt}. From these, the optimal parameters of the elastic foundation, m

_{a_opt}, k

_{a_opt}, and c

_{a_opt}, are calculated. Finally, we redesign the targeted sliding system based on the obtained design parameters of the sliding system with elastic foundation (DVA).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

List of Symbols (units) | |

c | damping coefficient of the primary vibration system (N s/m) |

c_{a} | damping coefficient of the DVA system (N s/m) |

c_{a_opt} | optimally tuned damping coefficient of the DVA system (N s/m) |

c_{eq} | equivalent damping coefficient |

f | amplitude of the periodic external force (N) |

f_{ext} | periodic external force (N) |

F | friction force (N) |

F_{k} | kinetic friction force (N) |

F_{s} | static friction force (N) |

F_{smax} | maximum static friction force (N) |

H | dimensionless displacement under without DVA system |

H_{max} | maximum value of H |

k | stiffness of the primary vibration system (N/m) |

k_{a} | stiffness of the DVA system (N/m) |

k_{a_opt} | optimally tuned stiffness of the DVA system (N/m) |

K | stiffness ratio |

K_{opt} | optimally tuned stiffness ratio |

m | mass of the primary vibration system (kg) |

m_{a} | additional mass of the DVA system (kg) |

m_{a_opt} | optimally tuned additional mass of the DVA system (kg) |

M | mass ratio |

M_{opt} | optimally tuned mass ratio |

t | time (s) |

V | driving speed (m/s) |

W | normal load (N) |

x | displacement of the primary mass (m) |

$\dot{x}$ | velocity of the primary mass (m/s) |

$\ddot{x}$ | acceleration of the primary mass (m/s^{2}) |

x_{a} | displacement of the additional mass (m) |

${\dot{x}}_{a}$ | velocity of the additional mass (m/s) |

${\ddot{x}}_{a}$ | acceleration of the additional mass (m/s^{2}) |

γ | dimensionless parameter |

λ | dimensionless parameter |

µ_{k} | kinetic friction coefficient |

µ_{s} | static friction coefficient |

τ | dimensionless time |

ω | angular frequency of the periodic external force |

ω_{n} | natural frequency of the primary vibration system (rad/s) |

Ω | forced frequency ratio |

ξ | dimensionless displacement of the primary mass |

${\xi}^{\prime}$ | dimensionless velocity of the primary mass |

${\xi}^{\u2033}$ | dimensionless acceleration of the primary mass |

ξ_{a} | dimensionless displacement of the additional mass |

${{\xi}^{\prime}}_{a}$ | dimensionless velocity of the additional mass |

${{\xi}^{\u2033}}_{a}$ | dimensionless acceleration of the additional mass |

ζ | damping ratio |

ζ_{a} | damping ratio of the DVA system |

ζ_{aopt} | optimally tuned damping ratio of the DVA system |

ζ_{eq} | equivalent damping coefficient |

## References

- Koenen, A.; Sanon, A. Tribological and vibroacoustic behavior of a contact between rubber and glass (application to wiper blade). Tribol. Int.
**2007**, 40, 1484–1491. [Google Scholar] [CrossRef] - Le Rouzic, J.; Le Bot, A.; Perret-Liaudet, J.; Guibert, M.; Rusanov, A.; Douminge, L.; Bretagnol, F.; Mazuyer, D. Friction-induced vibration by Stribeck’s law: Application to wiper blade squeal noise. Tribol. Lett.
**2013**, 49, 563–572. [Google Scholar] [CrossRef] - Khattab, M.; Wasfy, T. Prediction of Dynamic Stick-Slip Events in Belt-Drives Using a High-Fidelity Finite Element Model. J. Comput. Nolinear Dynam.
**2022**, 17, 064501. [Google Scholar] [CrossRef] - Wu, Y.; Leamy, M.J.; Varenberg, M. Minimizing self-oscillation in belt drives: Surface texturing. Tribol. Int.
**2020**, 145, 106157. [Google Scholar] [CrossRef] - Lee, S.M.; Shin, M.W.; Lee, W.K.; Jang, H. The correlation between contact stiffness and stick–slip of brake friction materials. Wear
**2013**, 302, 1414–1420. [Google Scholar] [CrossRef] - Lazzari, A.; Tonazzi, D.; Conidi, G.; Malmassari, C.; Cerutti, A.; Massi, F. Experimental evaluation of brake pad material propensity to stick-slip and groan noise emission. Lubricants
**2018**, 6, 107. [Google Scholar] [CrossRef] [Green Version] - Dolce, M.; Cardone, D.; Croatto, F. Frictional behavior of steel-PTFE interfaces for seismic isolation. Bull. Earthq. Eng.
**2005**, 3, 75–99. [Google Scholar] [CrossRef] - Gandelli, E.; De Domenico, D.; Dubini, P.; Besio, M.; Bruschi, E.; Quaglini, V. Influence of the breakaway friction on the seismic response of buildings isolated with curved surface sliders: Parametric study and design recommendations. Structures
**2020**, 27, 788–812. [Google Scholar] [CrossRef] - Quaglini, V.; Dubini, P.; Furinghetti, M.; Pavese, A. Assessment of Scale Effects in the Experimental Evaluation of the Coefficient of Friction of Sliding Isolators. J. Earthq. Eng.
**2019**, 26, 525–545. [Google Scholar] [CrossRef] - Feeny, B.; Guran, A.; Hinrichs, N.; Popp, K. Historical review on dry friction and stick–slip phenomena. Appl. Mech. Rev.
**1998**, 51, 321–341. [Google Scholar] [CrossRef] - Nakano, K. Two dimensionless parameters controlling the occurrence of stick-slip motion in a 1-DOF system with Coulomb friction. Tribol. Lett.
**2006**, 24, 91–98. [Google Scholar] [CrossRef] - Nakano, K.; Maegawa, S. Occurrence limit of stick-slip: Dimensionless analysis for fundamental design of robust-stable systems. Lubr. Sci.
**2010**, 22, 1–18. [Google Scholar] [CrossRef] - Kado, N.; Sato, N.; Tadokoro, C.; Skarokek, A.; Nakano, K. Effect of yaw angle misalignment on brake noise and brake time in a pad-on-disc-type apparatus with unidirectional compliance for pad support. Tribol. Int.
**2014**, 78, 41–46. [Google Scholar] [CrossRef] - Nakano, K.; Maegawa, S. Stick-slip in sliding systems with tangential contact compliance. Tribol. Int.
**2009**, 42, 1771–1780. [Google Scholar] [CrossRef] - Popp, K.; Rudolph, M. Vibration control to avoid stick-slip motion. J. Vib. Control
**2004**, 10, 1585–1600. [Google Scholar] [CrossRef] - Kruse, S.; Tiedemann, M.; Zeumer, B.; Reuss, P.; Hetzler, H.; Hoffmann, N. The influence of joints on friction induced vibration in brake squeal. J. Sound Vib.
**2015**, 340, 239–252. [Google Scholar] [CrossRef] [Green Version] - Yang, L.; Li, H.; Ahmadian, M.; Ma, B. Analysis of the influence of engine torque excitation on clutch judder. J. Vib. Control
**2015**, 23, 645–655. [Google Scholar] [CrossRef] - Maegawa, S.; Itoigawa, F. Design method for suppressing stick-slip using dynamic vibration absorber. Tribol. Int.
**2019**, 140, 105866. [Google Scholar] [CrossRef] - Den Hartog, J.P. Mechanical Vibrations; McGraw-Hill: New York, NY, USA, 1956. [Google Scholar]
- Harik, R.S.; Issa, J.S. Design of a vibration absorber for harmonically forced damped systems. J. Vib. Control
**2015**, 21, 1810–1820. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**Analytical model in this study without the sliding surface and with a periodic external force f

_{ext}applied at the primary mass.

**Figure 4.**Optimally tuned DVA parameters depending on ζ in the analytical model shown in Figure 3; the open circles show the numerical results derived in [20] and the solid curves show the approximate formula shown in Equations (28)–(30). (

**a**) Effect of ζ on M

_{opt}, (

**b**) Effect of ζ on K

_{opt}, (

**c**) Effect of ζ on ζ

_{a_opt}.

**Figure 5.**Vibration response curve under the different DVA parameter conditions; dashed line shows vibration response curve with equivalent 1DOF vibration system with ζ

_{eq}.

**Figure 7.**Time variations of dimensionless displacement ξ under different λ with (

**a**) optimally tuned DVA (M = M

_{opt}, K = K

_{opt}, ζ

_{a}= ζ

_{a}

_{opt}, ζ = 0.1 and γ = 2) and (

**b**) non-optimally tuned DVA (M = M

_{opt}, K = 2K

_{opt}, ζ

_{a}= ζ

_{a}

_{opt}, ζ = 0.1 and γ = 2).

**Figure 8.**The occurrence and non-occurrence map of stick-slip obtained by numerical simulation under γ = 2; the solid circles show the occurrence conditions for stick-slip; the open circles show the non-occurrence conditions of stick-slip (i.e., damped vibration conditions); and the solid curves show the boundary between the occurrence and non-occurrence of stick-slip derived by (

**a**) Equation (39), without DVA and (

**b**) Equation (37), with optimally tuned DVA.

**Figure 11.**The occurrence and non-occurrence map of stick-slip obtained by numerical simulation under γ = 2 for investigating the stick-slip suppression effect of non-optimally tuned DVA systems. (

**a**) M = M

_{opt}, K = K

_{opt}/2, ζ

_{a}= ζ

_{a}

_{opt;}(

**b**) M = M

_{opt}, K = K

_{opt}× 2, ζ

_{a}= ζ

_{a}

_{opt}.

**Figure 12.**Time variations of dimensionless displacements ξ and ξ

_{a}and dimensionless velocities ${\xi}^{\prime}$ and ${{\xi}^{\prime}}_{a}$ under λ = 1, ζ = 0.01. (

**a**) M = M

_{opt}, K = K

_{opt}/2, ζ

_{a}= ζ

_{a}

_{opt;}(

**b**) M = M

_{opt}, K = K

_{opt}× 2, ζ

_{a}= ζ

_{a}

_{opt}.

**Figure 13.**Time variations of dimensionless displacements ξ and ξ

_{a}and dimensionless velocities ${\xi}^{\prime}$ and ${{\xi}^{\prime}}_{a}$ under λ = 10, ζ = 0.01. (

**a**) M = M

_{opt}, K = K

_{opt}/2, ζ

_{a}= ζ

_{a}

_{opt;}(

**b**) M = M

_{opt}, K = K

_{opt}× 2, ζ

_{a}= ζ

_{a}

_{opt}.

**Figure 14.**Trajectories in a phase plane with ξ and ${\xi}^{\prime}$ an under λ = 1, ζ = 0.01. (

**a**) M = M

_{opt}, K = K

_{opt}/2, ζ

_{a}= ζ

_{a}

_{opt;}(

**b**) M = M

_{opt}, K = K

_{opt}× 2, ζ

_{a}= ζ

_{a}

_{opt}.

**Figure 15.**Trajectories in a phase plane with ξ and ${\xi}^{\prime}$ an under λ = 10, ζ = 0.01. (

**a**) M = M

_{opt}, K = K

_{opt}/2, ζ

_{a}= ζ

_{a}

_{opt;}(

**b**) M = M

_{opt}, K = K

_{opt}× 2, ζ

_{a}= ζ

_{a}

_{opt}.

**Figure 16.**Schematics of the design strategy for suppressing stick-slip using optimally tuned elastic foundation.

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**MDPI and ACS Style**

Maegawa, S.; Liu, X.; Itoigawa, F.
Discussion of Stick-Slip Dynamics of 2DOF Sliding Systems Based on Dynamic Vibration Absorbers Analysis. *Lubricants* **2022**, *10*, 113.
https://doi.org/10.3390/lubricants10060113

**AMA Style**

Maegawa S, Liu X, Itoigawa F.
Discussion of Stick-Slip Dynamics of 2DOF Sliding Systems Based on Dynamic Vibration Absorbers Analysis. *Lubricants*. 2022; 10(6):113.
https://doi.org/10.3390/lubricants10060113

**Chicago/Turabian Style**

Maegawa, Satoru, Xiaoxu Liu, and Fumihiro Itoigawa.
2022. "Discussion of Stick-Slip Dynamics of 2DOF Sliding Systems Based on Dynamic Vibration Absorbers Analysis" *Lubricants* 10, no. 6: 113.
https://doi.org/10.3390/lubricants10060113