Numerical Simulation of the Dry Friction Constrained System Based on Coulomb Stick-Slip Motion

Abstract

Due to the non-smooth characteristics of stick-slip friction, analytical solutions for the Dry Friction Constrained System (DFCS) are generally unavailable. Consequently, numerical simulation has become the most widely used approach for analyzing the DFCS. However, the accuracy and efficiency of the numerical algorithm considering the Coulomb stick-slip motion and determining whether stick-slip motion is considered in engineering design to further improve the computational efficiency remain a critical area of study. In this paper, a single-degree-of-freedom DFCS is introduced to address these issues. The Runge-Kutta method, combined with the dichotomy, is employed to accurately capture the stick-slip transition point. The normal load and dry friction are both symmetrically and evenly distributed at contact surfaces. Firstly, stick-slip motion analyses are performed, and response characteristics of the DFCS are discussed. Then, the convergence characteristics of the numerical algorithm are analyzed, and the optimal iteration step size and the zero-velocity interval are determined. Finally, whether stick-slip motion is considered in numerical simulation in the design of the DFCS in engineering practice is analyzed based on the dimensionless external force and frequency ratio. The criteria for determining whether stick-slip motion is considered in engineering design are established, which can improve both computational accuracy and efficiency.

1. Introduction

It is well known that dry friction widely exists in mechanical systems, which is sometimes harmful and sometimes useful. In any case, accurately modeling the dry friction force at the interface is important in the dynamic simulation of the DFCS. Coulomb [1] stated that the dry friction force is independent of the magnitude of velocity, and developed the first mathematical friction model (the Coulomb friction model). As a classical friction model, the Coulomb friction model has been widely used to model the dry friction force in dynamics of various engineering fields, such as automobile brake system [2,3], helicopter tail driveline [4,5], soil–structure interaction [6,7], seismic isolation for civil structures [8,9], friction and wear of materials [10,11], artificial joint wear [12,13], and vibration reduction of turbine blades [14,15]. To introduce the issues in numerical algorithms based on the Coulomb stick-slip motion discussed in this paper, the main contents of the Coulomb friction model are reiterated as follows:

(1)

Dry friction force always impedes relative motion or its tendency, and sliding occurs suddenly regardless of contact deformation.

(2)

Static friction exists before sliding, where the body remains relatively stationary and in static balance. Static friction and dynamic friction are both proportional to the normal load, and the proportionality coefficients are called the static friction coefficient and sliding friction coefficient, respectively.

(3)

The friction coefficients are dependent on the material properties of the friction interface and independent of the relative velocity of motion.

Based on the above three points, the mathematical model of the Coulomb friction model is expressed in Equation (1), in which the difference between the dynamic and static friction coefficients is neglected.

$$f=\left\{\begin{array}{c}\mu N (v>0, \mathrm{slip})\\ f_{\mathrm{s}} (v=0, \mathrm{stick}, -\mu N\le f_{\mathrm{s}}\le \mu N)\\ -\mu N (v<0, \mathrm{slip})\end{array}\right.$$

(1)

In Equation (1), $f$ is the dry friction force, while $f_{\mathrm{s}}$ is the static friction force, which can be obtained through the static equilibrium condition. $N$ is the normal load at the contact interface, $v$ is the relative velocity, and $\mu$ is the friction coefficient. It should be pointed out that the zero velocity cannot be captured in a numerical simulation. When a body keeps moving constantly, the static friction theory will not be involved, and the zero velocity does not need to be captured in a numerical algorithm. Otherwise, when a body’s motion is reciprocating or periodical and the active force acting on the body is less than the maximum static friction force at the zero velocity, stick state (the static friction) occurs during the motion and the stick-slip motion should be considered. With this situation, the zero velocity should be captured in a numerical algorithm. The main research about the dynamics of the DFCS based on Coulomb friction is summarized as follows.

Den Hartog [16] introduced the Coulomb friction model to model the dry friction contact between the damper and the blade, and the vibration reduction characteristics of a single-degree-of-freedom blade damping system were studied. Based on the Coulomb friction model, Earles and William [17] considered the influence of normal load on the dry friction damping characteristics of the shrouded blade. In [18], the forced vibration response of a single-degree-of-freedom torsional system with Coulomb friction under a periodically varying normal load was studied. Allara [19] proposed a model to characterize the friction contact of non-spherical contact geometries obeying the Coulomb friction law with a constant friction coefficient and constant normal load. Xie et al. [20] considered the contact effects between the adjacent shrouds and established the numerical solution method of the twisted shrouded blade with shroud contact using Timoshenko beam theory and Coulomb’s friction law. Gastaldi et al. [21] studied the nonlinear characteristics of the two-blades-plus-damper system based on the Coulomb friction model. In the above studies, the stick-slip motion was not discussed, and Equation (1) is essentially transformed into Equation (2) with this situation.

$$f=\left\{\begin{array}{c}\mu N (v>0, \mathrm{slip})\\ -\mu N (v<0, \mathrm{slip})\end{array}\right.$$

(2)

However, with the Coulomb friction model, the stick state may exist, and there is a jump in the friction force at the stick-slip transition in engineering. Researchers pay more and more attention to the stick-slip motion analysis. Karnopp [22] introduced a zero-velocity interval to capture the stick-slip transition numerically and analyzed the dynamics of a one-dimensional dry friction damping system. When the velocity is within this range, it is considered to be zero. In [23], the forced vibration response of a single-degree-of-freedom torsion system considering the Coulomb stick-slip motion was studied when the normal load varied periodically. Xia [24] proposed a model for investigating the stick-slip motion caused by dry friction of a two-dimensional oscillator under arbitrary excitations and provided a numerical approach to investigate the system with the Coulomb friction law. He et al. [25,26] considered the differences between static and dynamic friction coefficients and improved the Coulomb stick-slip motion analysis by introducing the exponential velocity-dependent friction coefficient model. Guo et al. [27] established a simplified computational model of a twisted shrouded blade with impact and friction. The impact force was modeled by a linear spring model, and the friction force was generated by the Coulomb friction model and stick-slip motion analysis. A one-dimensional model [28] and two-dimensional model [29] of the integrally shrouded group blades were established considering stick-slip motion. A three-dimensional mortar-based frictional contact algorithm was proposed in an explicit scheme which incorporated the LuGre model to account for dynamical frictional behaviours such as stick-slip [30]. It was found that the stick-slip behavior is more likely to occur at low sliding speeds and high normal loads [31]. Though the Coulomb stick-slip motion is considered in these studies, how to set an optimized zero-velocity interval to improve the computational accuracy and efficiency in the numerical algorithm has not been discussed, which is important for numerical simulation of the DFCS.

In summary, there are still two main issues to be further discussed. (1) When the stick state occurs and the stick-slip motion is considered, exploring and confirming an optimal zero-velocity interval needs to be analyzed to improve the accuracy and efficiency of the numerical algorithm, which is also the foundation of the next issue. (2) Compared with not considering stick-slip motion, using the dichotomy to perform stick-slip motion analysis can better ensure the accuracy of the numerical algorithm but may result in a significant decrease in computational efficiency. Thus, for a definite DFCS, a criterion for determining whether stick-slip motion is considered in numerical algorithms of engineering design based on the Coulomb stick-slip motion needs to be established. It is necessary to propose an accurate and efficient numerical algorithm for dynamic analysis based on the Coulomb stick-slip motion.

Based on the analysis above, this paper is outlined as follows. In Section 2, the dynamic model of a single-degree-of-freedom DFCS is introduced, and the Coulomb stick-slip motion analysis is performed first. Then, based on the Coulomb friction model, the response characteristics of the DFCS considering and not considering the stick-slip motion are both analyzed, which is the foundation of the following research. In Section 3, considering the Coulomb stick-slip motion, setting the iteration step size of the numerical algorithm is discussed first, then the influences of the zero-velocity interval on the computational accuracy and efficiency of the numerical algorithm are discussed fully. Finally, the optimal zero-velocity interval and iteration step size for numerical simulation considering and not considering stick-slip motion are determined. In Section 4, external parameters of the DFCS affecting the deviation of numerical results considering and not considering stick-slip motion are determined first. With the iteration step size and the optimal zero-velocity interval determined in Section 3, the influences of dimensionless external force and frequency ratio on the deviation of the two types of numerical results are analyzed in detail and the criterion for determining whether stick-slip motion is considered in engineering design is established. Section 5 concludes the work.

2. A Single-Degree-of-Freedom DFCS and Its Response Characteristics

2.1. Dynamic Model and the Coulomb Stick-Slip Motion Analysis

A typical DFCS, where the first-order bending vibration is considered and is simplified as a single-degree-of-freedom mass-spring model, is presented in this section. Assuming the foundation is fixed, the dynamic model is shown in Figure 1. The normal load and dry friction are both symmetrically and evenly distributed at contact surfaces; thus, the friction can be modeled by a macro-slip model. Combining with the vibration theory, the dynamic equation of the DFCS can be expressed in Equation (3).

$$m\ddot{x}+c\dot{x}+kx+f=F_0\sin (\omega t)$$

(3)

In the typical DFCS, $m$ is the equivalent mass and $c$ is the equivalent viscous damping, while $k$ is the equivalent stiffness and $x$ is the displacement of the mass relative to the fixed foundation. $F_0\sin \omega t$ is the external force and $f$ is the dry friction force at the friction interface. $F_0$ and $\omega$ are, respectively, the amplitude and the frequency of the external force. The fourth-order Runge-Kutta method is introduced to solve Equation (3).

As the design and manufacturing of components of mechanical systems, such as turbine blades, are standardized, the parameters and constraints of the components are both determined when they are installed. Thus, the external parameters affecting the dynamic behavior of the DFCS in this paper mainly include $N$, $F_0$ and $\tilde{\omega }$. Defining the natural frequency ${\omega }_n=\sqrt{k/m}$, the frequency ratio $\tilde{\omega }=\omega /{\omega }_n$, and $\tau ={\omega }_{n}t$, $F_0\sin (\omega t)$ is transformed into $F_0\sin (\tilde{\omega }\tau )$. The friction force $f$ is determined through the following Coulomb stick-slip motion analysis and can be expressed as a function of $\mu N$.

When a stick state occurs, the Coulomb stick-slip motion analysis should be carried out, and the stick-slip transition is captured with the dichotomy to suppress the error accumulation in the numerical algorithm. The zero-velocity interval ${\epsilon }_v$ is decisive to the accuracy and efficiency of the numerical algorithm. When the absolute value of velocity is within the zero-velocity interval, the system may be at a stick state, and the dry friction force can be determined as follows.

When $\left|\dot{x}\right|>{\epsilon }_v$, the lumped mass is at a slip state, the friction force is the sliding friction force; when $\dot{x}>{\epsilon }_v$, $f=\mu N$; when $\dot{x}<-{\epsilon }_v$, $f=-\mu N$;

When $\left|\dot{x}\right|<{\epsilon }_v$, denote $F_0\sin \omega t-c\dot{x}-kx$ as the external non-friction force combining with Equation (3):

When $\left|F_0\sin \omega t-c\dot{x}-kx\right|>\mu N$, the lumped mass is also at slip state; when $F_0\sin \omega t-c\dot{x}-kx>\mu N$, $f=\mu N$; when $F_0\sin \omega t-c\dot{x}-kx<-\mu N$, $f=-\mu N$;

When $\left|F_0\sin \omega t-c\dot{x}-kx\right|\le \mu N$, the lumped mass is at a static balance state, the static dry friction force is equal to the external non-friction force acting on the mass, $f=F_0\sin (\omega t)-c\dot{x}-kx$.

Based on the analysis above, Equation (1) is transformed into Equation (4).

$$f=\left\{\begin{array}{cc}\mu N\mathrm{sgn}(\dot{x}),& \left|\dot{x}\right|\ge {\epsilon }_v\\ \left\{\begin{array}{cc}{F}_0\sin (\omega t)-c\dot{x}-kx,& \left|F_0\sin (\omega t)-c\dot{x}-kx\right|<\mu N\\ \mu N\mathrm{sgn}(F_0\sin (\omega t)-c\dot{x}-kx),& \left|F_0\sin (\omega t)-c\dot{x}-kx\right|\ge \mu N\end{array}\right.& \left|\dot{x}\right|<{\epsilon }_v\end{array}\right.$$

(4)

Thus, the flowchart of the dichotomy based on the fourth-order Runge-Kutta method can be illustrated in Figure 2.

In Figure 2, $T$ is the period of the system’s steady response and it is equal to $2\pi /\omega$, while $\Delta t$ is the iteration step size equaling to $T/h_0$ and $h_0$ is the number of iterations in a period. The $t_n$ and $t_{n+1}$ are the corresponding time points for the $n$-th and $n+1$-th step, while $t_m$ is the time point in dichotomy. Then, $v_n$, $v_m$ and $v_{n+1}$ are the velocities at $t_n$, $t_m$ and $t_{n+1}$, respectively. The specific implementation details of the dichotomy are as follows. The maximum number of iterations is set to 16 to balance accuracy and efficiency. The iteration terminates when the absolute velocity falls within the zero-velocity interval ($v_m\le \left|{\epsilon }_{\mathrm{v}}\right|$) or the iteration count exceeds 16. To prevent skipping zero-crossings or numerical oscillation, the time step is pre-optimized to be sufficiently small, ensuring that sign changes of the velocity typically correspond to a single physical stick-slip transition point, which is then precisely captured by the dichotomy.

2.2. Response Characteristics of the DFCS

In order to deeply investigate the influence of considering stick-slip motion on the system’s response and explore the optimal zero-velocity interval in the next, the response characteristics of the DFCS considering and not considering stick-slip motion are both analyzed first.

Referring to reference [28], the mass $m$, system stiffness $k$, viscous damping $c$, and dry friction coefficient $\mu$ are taken; the main inherent parameters of the DFCS are shown in

Table 1. The main parameters of the system.

Sign	Value
$m$	0.2 kg
$c$	5.65 N·s/m
$k$	1 × 105 N/m
$\mu$	0.5

.

To ensure the accuracy of the numerical simulation, values of zero-velocity interval and iterative step size are strictly selected. With $\tilde{\omega }=0.6$, $N=40 \mathrm{N}$ and $F_0=30 \mathrm{N}$, the typical simulation results are displayed in Figure 3.

From Figure 3, the steady motion of the damping system is periodic and the minimum period of the steady response $T$ is equal to that of the external force. In frequency spectrum, $\omega$ is the frequency of the external force. There are no fractional frequencies, and only odd multiple frequencies can be observed when the friction contact does not separate.

Significant presence of high-order harmonic components indicates that the amplitude and vibration energy should be considered to analyze the convergence of the numerical algorithm and to study the influence of whether stick-slip motion is considered on the system’s vibration behavior.

3. Convergence Analysis of the Numerical Algorithm of the DFCS Considering Stick-Slip Motion

For the DFCS, the zero-velocity interval is very important for computational accuracy and efficiency when the Coulomb stick-slip motion needs to be considered in the numerical algorithm.

In this section, to analyze the influence of the zero-velocity interval on the convergence characteristics of the system response in detail, the setting of the iteration step size of the numerical algorithm is analyzed first. The amplitude and vibration energy of the system’s response are both adopted as the convergence criterion. The vibration energy of a steady cycle of the system is shown in Equation (5) and the convergence criterion is shown in Equation (6). When the two convergence criteria are simultaneously satisfied, the numerical algorithm is considered to converge.

$$E={\displaystyle \underset{T}{\int }{x}^{2}dt}$$

(5)

$$\left\{\begin{array}{l}\left|A_i-A_{i-1}\right|/\left|A_i\right|\le {\delta }_{\mathrm{A}}\\ \left|E_i-E_{i-1}\right|/\left|E_i\right|\le {\delta }_{\mathrm{E}}\end{array}\right.$$

(6)

In Equation (6), $A_i$ and $A_{i-1}$ are the amplitude of the steady response corresponding to varying $h_0$, while $E_i$ and $E_{i-1}$ are the vibration energy of a steady-state cycle. The convergence error limits matching to the two-convergence criterion are ${\delta }_{\mathrm{A}}$ and ${\delta }_{\mathrm{E}}$.

Based on the two convergence criteria, influences of the iteration step size and zero-velocity interval on the convergence characteristics of the numerical algorithm are analyzed with the three external parameters including $N$, $F_0$ and $\tilde{\omega }$.

3.1. Analysis of Setting the Iteration Step Size of the Numerical Algorithm

A sufficiently small ${\epsilon }_v$ is selected to perform the analysis of the setting iteration step size of the numerical algorithm. Based on repeated tests to balance the computational accuracy and computational efficiency, define ${\epsilon }_v={10}^{-6} \mathrm{m}/\mathrm{s}$, ${\delta }_{\mathrm{A}}={\delta }_{\mathrm{E}}=0.1\%$. Figure 4 shows the simulation results.

From Figure 4, the response amplitude and vibration energy of the system gradually stabilize as $h_0$ increases and are within a wide range. Meanwhile, as $h_0$ continues to increase, computational accuracy will not improve and the computational efficiency will decrease. Analyzing three sets of simulation results, an appropriate iteration step size interval about $T/1200$ can be determined based on the two convergence criteria. Meanwhile, simulation results prove that the iteration step size interval about $T/1200$ is also applicable to the numerical algorithm without considering the stick-slip motion.

3.2. Convergence Characteristics Analysis of the Numerical Algorithm Based on the Zero-Velocity Interval

With the iteration step size determined in Section 3.1, the influence of the zero-velocity interval on the convergence characteristics of the system is analyzed in detail, and the optimal zero-velocity interval is explored to obtain good computational accuracy and efficiency. Figure 5 shows the simulation results.

From Figure 5, the response amplitude and vibration energy of the system eventually stabilize with the decrease of the zero-velocity interval ${\epsilon }_v$. With the three external parameters of the system including $\tilde{\omega }$, $N$ and $F_0$, the simulation results show that the optimal zero-velocity interval is about $8\times {10}^{-4} \mathrm{m}/\mathrm{s}$ based on the two convergence criteria. As the zero-velocity interval continues to decrease from $8\times {10}^{-4} \mathrm{m}/\mathrm{s}$, the computational accuracy of the numerical algorithm will not improve while the computational efficiency will significantly decrease.

The simulation results of the decrease in computational efficiency as the zero-velocity interval decreases from $8\times {10}^{-4} \mathrm{m}/\mathrm{s}$ are shown in Figure 6, all of which are conducted in the same computing environment with the same $\omega$ and same number of computation cycles. As shown in Equation (7), $t_{\mathrm{d}}$ is the reduction rate of the computational efficiency of the numerical algorithm.

$$t_{\mathrm{d}}={\displaystyle \frac{t^{\prime }-t}{t}}\times 100\%$$

(7)

In Equation (7), $t^{\prime }$ is the runtime corresponding to varying zero-velocity interval from $8\times {10}^{-4} \mathrm{m}/\mathrm{s}$ while $t$ is the runtime corresponding to the zero-velocity interval equaling to $8\times {10}^{-4} \mathrm{m}/\mathrm{s}$.

From Figure 6, as the zero-velocity interval decreases, there is a rapid decrease in the computational efficiency. When ${\epsilon }_v=1\times {10}^{-7} \mathrm{m}/\mathrm{s}$, the computational efficiency can decrease more than 120%. The $t_{\mathrm{d}}$-${\epsilon }_{\mathrm{v}}$ curves under different external parameters are basically overlapping in Figure 6, as the computational efficiency mainly depends on the zero-velocity interval. This phenomenon also indicates that the influence of external parameters on the computational efficiency is much smaller than that within the zero-velocity interval.

4. Establishing the Criteria for Whether Stick-Slip Motion Is Taken into Account in Engineering Design

Compared with not considering the stick-slip motion, considering it improves the computational accuracy, but may lead to a significant decrease in computational efficiency. Thus, analyzing and revealing the system’s external parameter ranges under which the stick-slip motion should be considered can further improve the computational efficiency of the numerical algorithm of the DFCS design in engineering.

In this section, with the optimal zero-velocity interval and step size determined in Section 3, the influence of whether stick-slip motion is considered on computational efficiency is studied first. On this basis, a criterion for whether stick-slip motion is considered in the numerical algorithm of dynamic analysis of the DFCS is established, which will be of great value for the design of the DFCS in engineering.

4.1. Analysis of the Decrease in Computational Efficiency Caused by Considering Stick-Slip Motion

To compare the computational efficiency of considering and not considering stick-slip motion, $t_{\Delta }$ is defined as shown in Equation (8) to represent the difference in the run time to complete the two types of calculations.

$$t_{\Delta }={\displaystyle \frac{t_2-t_1}{t_1}}\times 100\%$$

(8)

In Equation (8), $t_1$ and $t_2$ are the runtime of not considering stick-slip motion and considering stick-slip motion, respectively, which are taken as the average of five similar calculations. The simulation results are displayed in Figure 7.

In Figure 7, the influences of the three external parameters of the system, $\tilde{\omega }$, $N$ and $F_0$, on $t_{\Delta }$ are displayed in detail. The simulation results show that the difference in run time of the two types of simulation exhibits different variation characteristics with the three parameters varying. What is more, compared with not considering stick-slip motion in the numerical algorithm, considering it increases the computation time to complete the same simulation with the same computer by more than 225%. This further proves the necessity of analyzing whether stick-slip motion is considered in engineering design.

4.2. Comparative Analysis of Simulation Results Considering Stick-Slip Motion and Not Considering It

4.2.1. Determination of External Parameters Affecting the Deviation of the Two Types of Numerical Results

In [29], for a one-dimensional micro-slip model and a finite model of the platform damper, it has been shown that the reduction rate of the system response amplitude is related to the dimensionless normal load ($\mu N/F_0$). For the DFCS in this paper, the relevant conclusions are worth further discussion to reduce the external parameters that affect the following analysis.

When the friction force is zero, $A_0$ is the steady response amplitude and $E_0$ is the vibration energy of the system of a steady-state cycle. When the dry friction force exists, $A_1$ and $A_2$ are the steady response amplitudes corresponding to not considering stick-slip motion and considering stick-slip motion, while $E_1$ and $E_2$ are the corresponding vibration energy of a steady-state cycle. Define ${\tilde{A}}_1=A_1/A_0$, ${\tilde{A}}_2=A_2/A_0$, ${\tilde{E}}_1=E_1/E_0$, ${\tilde{E}}_2=E_2/E_0$. The numerical results considering stick-slip motion are considered to be accurate, $A_{\Delta }$ and $E_{\Delta }$ are defined in Equation (9) to describe the deviation of the numerical results caused by not considering stick-slip motion.

$$\left\{\begin{array}{c}{A}_{\Delta }=\left|{\displaystyle \frac{A_1-A_2}{A_2}}\right|\times 100\%=\left|{\displaystyle \frac{\frac{A_1}{A_0}-\frac{A_2}{A_0}}{\frac{A_2}{A_0}}}\right|\times 100\%=\left|{\displaystyle \frac{{\tilde{A}}_1-{\tilde{A}}_2}{{\tilde{A}}_2}}\right|\times 100\%\\ E_{\Delta }=\left|{\displaystyle \frac{E_1-E_2}{E_2}}\right|\times 100\%=\left|{\displaystyle \frac{\frac{E_1}{E_0}-\frac{E_2}{E_0}}{\frac{E_2}{E_0}}}\right|\times 100\%=\left|{\displaystyle \frac{{\tilde{E}}_1-{\tilde{E}}_2}{{\tilde{E}}_2}}\right|\times 100\%\end{array}\right.$$

(9)

The dimensionless external force is defined as ${\tilde{F}}_0=F_0/N$. Setting ${\tilde{F}}_0\in [0.6,1.5]$, increasing the $F_0$ and keeping ${\tilde{F}}_0$ constant. As simulation results are consistent under different $\tilde{\omega }$, simulation results under different ${\tilde{F}}_0$ with $\tilde{\omega }=0.6$ are displayed in Figure 8 (not considering stick-slip motion) and Figure 9 (considering it).

As shown in Figure 8 and Figure 9, whether considering stick-slip motion or not, for different values of ${\tilde{F}}_0$, ${\tilde{A}}_1$, ${\tilde{A}}_2$ and ${\tilde{E}}_1$, ${\tilde{E}}_2$ also keep constant with $F_0$ varying. This indicates that the external parameters affecting the difference between two types of numerical results can be transformed from three parameters ($\tilde{\omega }$, $N$ and $F_0$) into the two parameters (${\tilde{F}}_0$ and $\tilde{\omega }$), which can simplify the establishment of the criterion for whether stick-slip motion is considered in engineering design.

4.2.2. Deviation Analysis of the Simulation Results for Whether Stick-Slip Motion Is Considered

To enable the DFCS to vibrate, ${\tilde{F}}_0\ge \mu =0.5$ is set. Meanwhile, the excitation frequency ($\tilde{\omega }\in [0.5,1.5]$) near the resonance frequency of the DFCS is selected. Influences of the dimensionless external force ${\tilde{F}}_0$ and the frequency ratio $\tilde{\omega }$ on the $A_{\Delta }$ and $E_{\Delta }$ are fully discussed, and the simulation results are respectively shown in Figure 10 and Figure 11.

From Figure 10 and Figure 11, there is a wide range of ${\tilde{F}}_0$ and $\tilde{\omega }$ under which stick-slip motion does not need to be considered in the numerical algorithm in the engineering design of DFCS, and this external parameter range can be determined by the accuracy requirement of the calculation. Obviously, when $0.54\le {\tilde{F}}_0\le 0.58$ and $0.5\le \tilde{\omega }\le 0.76$, $A_{\Delta }\ge 10\%$; when $0.53\le {\tilde{F}}_0\le 0.6$ and $0.5\le \tilde{\omega }\le 0.84$, $E_{\Delta }\ge 10\%$. When $0.53\le {\tilde{F}}_0\le 0.61$ and $0.5\le \tilde{\omega }\le 0.88$, $A_{\Delta }\ge 5\%$; when $0.52\le {\tilde{F}}_0\le 0.63$ and $0.5\le \tilde{\omega }\le 0.94$, $E_{\Delta }\ge 5\%$. As shown in Figure 10 and Figure 11, when ${\tilde{F}}_0=0.55,\tilde{\omega }=0.64$, $A_{\Delta }$ reaches a maximum value of 18.69% and when ${\tilde{F}}_0=0.54,\tilde{\omega }=0.58$, $E_{\Delta }$ reaches a maximum value of 33.79%, which indicates that the simulation results of considering and not considering stick-slip motion are significantly different under some external parameters of the system. The stick-slip motion must be considered in this case. The mechanism of the deviation is illustrated in Figure 12 (when ${\tilde{F}}_0=0.54$ and $\tilde{\omega }=0.55$, $A_{\Delta }=15\%$) and Figure 13 (when ${\tilde{F}}_0=0.8$ and $\tilde{\omega }=1$, $A_{\Delta }=2\%$).

As shown in Figure 12a and Figure 13a, when stick-slip motion is not considered, the friction force repeatedly jumps between $-\mu N$ and $\mu N$ around the zero velocity; this is not reasonable in engineering practice and will lead to calculating error. Meanwhile, when stick-slip motion is considered, the stick state is captured by the dichotomy, and no jumps exist for the dry friction force, which is consistent with engineering practice, and this precisely indicates that considering stick-slip motion can enhance the calculation accuracy. Comparing Figure 12 with Figure 13, it can be seen that when the time proportion of the stick state in a steady time cycle is higher, the deviation of the simulation results is larger. For the DFCS, the time proportion of the stick state in a steady time cycle can be mainly determined by ${\tilde{F}}_0$ and $\tilde{\omega }$.

5. Discussion

The difference between the static friction coefficient and the kinetic friction coefficient is actually one of the key physical mechanisms for the generation of “stick-slip” oscillation, ignoring the condition ${\mu }_{\mathrm{s}}>{\mu }_{\mathrm{k}}$ leads to the model being unable to capture the self-excited vibration phenomenon caused by the decreasing characteristic of frictional force (negative damping), when the system is mainly in forced vibration and the excitation force frequency is much higher than the friction-induced frequency, the condition ${\mu }_{\mathrm{s}}>{\mu }_{\mathrm{k}}$ can be ignored.

6. Conclusions

In this paper, the numerical algorithm for dynamic analysis of a single-degree-of-freedom DFCS based on the Coulomb stick-slip motion is discussed. When the Coulomb stick-slip motion is considered, the optimization of the zero-velocity interval and step size is analyzed. Then, a criterion for determining whether stick-slip motion is considered in numerical algorithms in engineering design is explored and established. The main conclusions are displayed as follows.

(1)

When stick-slip motion is considered, simulation results show that optimization of the zero-velocity interval and step size can significantly improve the computational efficiency of the numerical algorithm.

(2)

When $0.53\le {\tilde{F}}_0\le 0.61$ and $0.5\le \tilde{\omega }\le 0.88$, $A_{\Delta }\ge 5\%$; when $0.52\le {\tilde{F}}_0\le 0.63$ and $0.5\le \tilde{\omega }\le 0.94$, $E_{\Delta }\ge 5\%$, for this instance, stick-slip motion must be considered. When ${\tilde{F}}_0=0.55, \tilde{\omega }=0.64$, $A_{\Delta }$ reaches a maximum value of 18.69% and when ${\tilde{F}}_0=0.54, \tilde{\omega }=0.58$, $E_{\Delta }$ reaches a maximum value of 33.79%.

(3)

Compared with not considering stick-slip motion, considering it in the numerical algorithm increases the runtime by more than twice. The stick-slip motion does not need to be considered in the numerical algorithm of the engineering design within a wide range of the system’s external parameters, which can be determined by combining the requirements of computation accuracy. Identifying the parameter range that does not need to consider the stick-slip motion can significantly enhance the computational efficiency while maintaining the accuracy. However, this conclusion is based on a simple single-degree-of-freedom model. Further in-depth studies are needed for more complex situations.

(4)

Compared with considering stick-slip motion, not considering it may cause repeated jumps for the friction force around the zero velocity, which is the main reason for the calculation error, and considering stick-slip motion can enhance the calculation accuracy. The time proportion of the stick state in a steady time cycle mainly determines the deviation of the two types of simulation results.