Analytical Determination of Stick–Slip Whirling Vibrations and Bifurcations in Rotating Machinery

Abstract

In rotating machinery, aerodynamic forces and oil film forces often lead to cross-coupling stiffness. This paper is aimed at studying the stick-slip whirling vibrations induced by the piecewise smooth rotor/stator frictions in a modified flexible rotor subjected to cross-coupling stiffness. Governing equations determining the sliding region and boundaries of piecewise discontinuous friction are defined. This analytical study was conducted to discuss the complex vibrations and bifurcations. Various types of sliding motions (continuous pure rolling, continuous crossing, and grazing–sliding) were observed in this research. Further, as for discussing the impacts of the parameters on nonlinear sliding vibrations, a parametric study was conducted. The obtained results reveal that with an increase in the cross-coupling stiffness coefficient, continuous pure rolling occurs earlier, and the disk vibration time around the contact regime becomes shorter. For studying the self-excited backward whirling vibration of stick–slip nonlinear motions, analytical formulations are established. Detailed vibration amplitude and frequency studies of friction-induced backward whirling vibrations were carried out. Numerical simulations were performed to compare them with the analytical solutions and to validate the results as well. The proposed theory and results provide fresh perspectives for predicting friction-induced whirlings and creating proper designs for turbo machinery.

1. Introduction

In industry, the proper operation of the rotor/stator components in rotating machinery is crucial. Rotor/stator-coupled systems are widely adopted for pumps, motors, compressors, turbines, etc. Friction is common and unavoidable in rotating machinery, which usually causes complex vibrations, wear, and damage. Ensuring the safe and proper operation of the systems is not only vital for longevity but is also important to reduce the cost and damage.

To figure out the impacts of friction on rotating machinery, efforts have been made to model the rotor/stator interaction. The friction from rotor/stator contact is often modeled by discontinuous piecewise functions referred to as Filippov systems. Different motion topologies can occur when trajectories impact different regions on the switching manifold tangentially where sliding or stick–slip bifurcations happen [1,2]. Using Filippov’s method [3] or Utkin’s control method [4], periodic solutions associated with sliding bifurcations are solved. Stability analyses involving sliding bifurcations, the topology of bifurcations, and the corresponding unfoldings are depicted analytically [5]. Michalopoulos et al. [6] conducted an analysis of friction-involved, nonlinear vibrations in gear systems. The nonlinear friction function was defined based on the velocity and forces to yield accurate solutions. They found that the stick–slip vibrations only happened when the velocity was low. Feeny et al. [7] provide a review of the complex phenomena of stick–slip motions in dynamic systems. The existence of solutions, damping properties, friction impacts, chaotic motions, etc., are discussed. Aidanpää [8] made the effort to explore complete stable solutions of a nonlinear, flexible rotor system with friction and stick–slip impacts. Three or four stable solutions were obtained when the oscillation amplitude exceeded the rotor/stator boundaries. In terms of large rotor/stator clearance, stick motions govern the vibration, and damage is caused where the vibration with a large backward whirling frequency and large amplitude happen when the clearance is large. Vlajic et al. [9] focused on the torsional vibration of a rotor/stator system subjected to dry frictions. They found that torsional vibrations could be caused by the stick–slip conditions when forward or backward whirling occurred. The forward whirling triggered torsional vibration when the rotor/stator was in contact, while the backward whirling led to torsional vibration when the rotor/stator was in impact conditions. Yadav and Agnihotri [10] studied the vibration reduction in the friction-involved, nonlinear system. Suitable damping was defined to limit and remove the vibration caused by the sliding motions. The optimal design of the static/kinetic friction ratios was performed to eliminate the friction-induced vibrations of the nonlinear system. Bently and Vlajic [11,12] studied the conditions to trigger the backward whirling in a discontinuous, self-excited, nonlinear system subjected to dry frictions. The critical coefficients of the dry frictions during contact were obtained where the angular velocity of the rotor/stator system was smaller than the critical angular velocity. One focused on finding the intrinsic factors, which caused the occurrence of the backward whirling motions in the dynamic system with dry frictions. The occurrence conditions and damages of rotor/stator-contact dynamic systems were systematically studied [13,14]. The determination criteria of the occurrence of sliding motions are provided by Won and Chung [15]. They define the function of the friction as exponential and polynomial forms with variables of velocities. Wang et al. [16] investigated the sliding motions of a discontinuous, self-excited, nonlinear rotor system with rotor/stator contact. A new approach is proposed in the research, which solves the occurrence conditions of the backward whirling caused by the dry frictions analytically. Various sliding regimes and switching domains have been obtained [17]. Zhou et al. [18] studied a nonlinear rotor system subjected to gas foil forces. The vibration patterns and stability conditions were studied. A new model with LuGre friction was developed to obtain the sliding motions when the system is in contact. Huang et al. [19] investigated the nonlinear vibration patterns of a bearing–rotor system with sliding motions. They established experiments for data to develop theoretical models with rubbing effects. They found that the bearing could be unstable when rubbing happened. Zhang et al. [20] designed a novel piezoelectric actuation device based on sliding motions. A new driving algorithm was developed to increase the stick–slip motion and reduce slipping motions. For the detection of the rotor/stator contact, Pavlenko et al. [21] adopted a clipped sinewave model for the spectral analysis of the vibration state of the rotor/stator interaction. Analytical relationships of the amplitudes for the spectral composition’s components of a signal on the dimensionless radial gap of the rotor/stator system were established. Based on their model, the rotor/stator contact could be diagnosed.

For the analytical determination of stick–slip whirling vibrations and bifurcations, this research first develops the dynamic model of a rotor/stator rubbing system with piecewise, discontinuous friction force and cross-coupling stiffness. Then, the sliding conditions and solutions of the nonlinear rotor/stator system are obtained. The sliding vibrations are analyzed where backward whirling motions are studied. An analytical study of the occurrence of the backward whirling motions is conducted. Finally, a summary of the research is given.

2. Friction-Induced Nonlinear System

In this research, we consider that the rotor/stator system contains a flexible rotor and a frame. The equipment is physically described as a Jeffcott rotor with an interaction with a stator [13,22] in Figure 1.

The rotor is considered a massless shaft which has ideal bearings at two ends, and a disk with mass ($m_r$) mounted at the center of the shaft. The disk is characterized by mass eccentricity (e). $\omega$ stands for the angular speed, $k_s$ stands for the transverse stiffness, and $c_s$ stands for the damping coefficient. For the consideration of the effects around the circumstance, we consider cross-coupling stiffness as $Q_s$ in the rotor/stator system. As a pair with the rotor, we set the stator to be concentric with the rotor and paired with the disk. The stator is modeled as being supported by springs elastically with stiffness ($k_b$) in the radial direction.

The mathematical equation of the rotor/stator system with friction is as follows:

$$\begin{array}{l}m\ddot{r}+c_s\dot{r}+k_{s}r-mr{\dot{\varphi }}^2+\mathsf{\Theta }{k}_b(r-r_0)=me{\omega }^2\cos (\omega t-\varphi )\hfill \\ mr\ddot{\varphi }+c_{s}r\dot{\varphi }+2m\dot{r}\dot{\varphi }-Q_{s}r+\mathsf{\Theta }\mu k_b(r-r_0)·\mathrm{sgn}(v_{rel})=me{\omega }^2\sin (\omega t-\varphi )\hfill \\ v_{rel}=\omega r_d+\dot{\varphi }r\hfill \end{array}$$

(1)

where $r_d$ is the radius of the disk; $r_0$ is the clearance between the disk and stator; r is the vibration radius; $\dot{\varphi }$ is the vibration frequency of the disk. $\mathsf{\Theta }$ is defined as follows:

$$\mathsf{\Theta }=\left\{\begin{array}{l}0. r<r_0\\ 1. r\ge r_0\end{array}\right.$$

$v_{rel}$ stands for the relative velocity between the speed, $\omega r_d$, and $\dot{\varphi }r$ of the disk. We define the direction of friction (F_t) as the variation with the direction of the velocity ($v_{rel}$) (for instance, $\mathrm{sgn}(v_{rel})$). O and O₁ are the geometry centers of the stator and the disk, respectively. $\varphi$ is the whirling motion angle. F_n represents normal force from the stator to the disk. F_t is the Coulomb friction force with a coefficient ($\mu$). Gravity was ignored in the research.

When the rotor/stator is in contact, Equation (1) has the following form:

$$\begin{array}{l}{R}^{″}+2\zeta R^{\prime }+\beta R-R{\varphi }^{\prime 2}+\mathsf{\Theta }(R-R_0)={\mathsf{\Omega }}^2\cos (\mathsf{\Omega }\tau -\varphi )\hfill \\ R{\varphi }^{″}+2\zeta R{\varphi }^{\prime }+2R^{\prime }{\varphi }^{\prime }-\gamma R+\mathsf{\Theta }\mu (R-R_0)·\mathrm{sgn}(V_{rel})={\mathsf{\Omega }}^2\sin (\mathsf{\Omega }\tau -\varphi )\hfill \\ V_{rel}=\mathsf{\Omega }{R}_d+{\varphi }^{\prime }R\hfill \end{array}$$

(2)

where $R^{\prime }=dR/D\tau ,$ ${\varphi }^{\prime }=d\varphi /D\tau ,$ $R=r/e$, $R_0=r_0/e$, $R_d=r_d/e$, $2\zeta =c_s/\sqrt{m{k}_b}$, $\beta =k_s/k_b$, $\gamma =Q_s/k_b$, ${\omega }_2=\sqrt{k_b/m}$, $\mathsf{\Omega }=\omega /{\omega }_2$, $V_{rel}{\omega }_2=v_{rel}/e$, and $\tau ={\omega }_{2}t$. ${\omega }_2$ is considered as the natural frequency of the rotor/stator system when the clearance is zero.

Consider the following representation:

$$x_1=R, x_2=R^{\prime }, x_3=\varphi , x_4={\varphi }^{\prime } \mathrm{and} \theta =\mathrm{mod}(\mathsf{\Omega }\tau ,2\pi )$$

Equation (2) is rewritten as follows:

$$x^{\prime }=\left\{\begin{array}{c}{F}_1(x), H(x)>0\\ F_2(x), H(x)<0\end{array}\right.$$

(3)

where $x={\left(x_1, x_2, x_3, x_4, \theta \right)}^{\mathrm{T}}\in {ℝ}^5$ and $H(x)=\mathsf{\Omega }{R}_d+x_1{x}_4$. $F_1(x)$ and $F_2(x)$ are as follows:

$$\begin{array}{l}{F}_1(x)=\left(\begin{array}{l}{x}_2\hfill \\ {\mathsf{\Omega }}^2\cos (\theta -x_3)-2\zeta x_2-(\beta +1)x_1+x_1{x_4}^2+R_0\hfill \\ x_4\hfill \\ {\displaystyle \frac{{\mathsf{\Omega }}^2\sin (\theta -x_3)}{x_1}}-2\zeta x_4-2{\displaystyle \frac{x_2{x}_4}{x_1}}+\gamma -\mu (1-{\displaystyle \frac{R_0}{x_1}})\hfill \\ \mathsf{\Omega }\hfill \end{array}\right)\\ F_2(x)=\left(\begin{array}{l}{x}_2\hfill \\ {\mathsf{\Omega }}^2\cos (\theta -x_3)-2\zeta x_2-(\beta +1)x_1+x_1{x_4}^2+R_0\hfill \\ x_4\hfill \\ {\displaystyle \frac{{\mathsf{\Omega }}^2\sin (\theta -x_3)}{x_1}}-2\zeta x_4-2{\displaystyle \frac{x_2{x}_4}{x_1}}+\gamma +\mu (1-{\displaystyle \frac{R_0}{x_1}})\hfill \\ \mathsf{\Omega }\hfill \end{array}\right)\end{array}$$

Such a rotor/stator system can be considered as a Filippov system with one degree of smoothness. Such a rotor/stator system may behavior bifurcations. Therefore, complex contact dynamics, in terms of the stick–slip and sliding characteristics induced by dry friction, are studied in this paper.

3. Friction-Induced Vibration

For the research of the piecewise, discontinuous, friction-induced stick–slip whirling vibrations in the rotor/stator system governed in Equation (3), the friction-induced nonlinear motions can be studied with the theories of Filippov and Utkin [3,4].

3.1. Friction-Induced Bifurcations

Consider a very small domain of $S\in {ℝ}^5$ and divide $S$ into two subdomains as $S_1$ and $S_2$. Define two vector fields of $F_1(x)$ and $F_2(x)$, where $F_1$ and $F_2$ are smooth in the rotor/stator nonlinear system.

According to Equation (3), the switching boundary between $S_1$ and $S_2$ is $H(x)=0$. The switching boundary is characterized with discontinuity and refers to curve hypersurfaces ($\mathsf{\Sigma }$). The corresponding subregions and switching manifold are as follows:

$$\left\{\begin{array}{c}{S}_1:=\left\{x\in {ℝ}^5: H(x)=\mathsf{\Omega }{R}_d+x_1{x}_4>0\right\}\\ S_2:=\left\{x\in {ℝ}^5: H(x)=\mathsf{\Omega }{R}_d+x_1{x}_4<0\right\}\\ \sum :=\left\{x\in {ℝ}^5: H(x)=\mathsf{\Omega }{R}_d+x_1{x}_4=0\right\}\end{array}\right.$$

(4)

In the friction-involved rotor/stator system, the oscillation route of the system either runs across the switching manifold ($H(x)=0$) or slides along the boundary. So, we further separate the switching manifold into two domains: the crossing domain with ${\mathsf{\Sigma }}_c$ and the sliding domain with ${\mathsf{\Sigma }}_s$, where ${\mathsf{\Sigma }}_s\cup {\mathsf{\Sigma }}_c=\mathsf{\Sigma }$.

In the case of $F_1(x)$ and $F_2(x)$ pointing to $\mathsf{\Sigma }$, some oscillation routes may be limited by the hypersurface of the switching manifold. So, the boundary of the sliding domains can be identified by the gradient of the vector fields by Filippov’s and Utkin’s methods [3,4]. Here, the sliding domain (${\mathsf{\Sigma }}_s$) and the corresponding boundary ($\partial {\mathsf{\Sigma }}_s$) are defined as follows:

$$\left\{\begin{array}{c}{\mathsf{\Sigma }}_s=\left\{x\in \mathsf{\Sigma }: {\mathcal{L}}_{F_1}H(x)·{\mathcal{L}}_{F_2}H(x)<0\right\}\\ \partial {\mathsf{\Sigma }}_s=\left\{x\in \mathsf{\Sigma }: {\mathcal{L}}_{F_1}H(x)·{\mathcal{L}}_{F_2}H(x)=0\right\}\end{array}\right.$$

(5)

where ${\mathcal{L}}_F$ stands for the Lie derivative along the $F(x)$ flow:

$${\mathcal{L}}_{F}H(x)={\displaystyle \frac{\partial H(x)}{\partial x}}F(x)$$

${\displaystyle \frac{\partial H(x)}{\partial x}}$ represents the gradient of $H(x)$. ${\mathcal{L}}_{F_1}H(x)$ and ${\mathcal{L}}_{F_2}H(x)$ are the normal components of $F_1(x)$ and $F_2(x)$ on the manifold ($\mathsf{\Sigma }$), respectively. ${\mathcal{L}}_{F_1}H(x)$ and ${\mathcal{L}}_{F_2}H(x)$ are given as follows:

$$\left\{\begin{array}{l}{\mathcal{L}}_{F_1}H(x)={\mathsf{\Omega }}^2\sin (\theta -x_3)-2\zeta x_1{x}_4-x_2{x}_4+\gamma x_1-\mu (x_1-R_0)\\ {\mathcal{L}}_{F_2}H(x)={\mathsf{\Omega }}^2\sin (\theta -x_3)-2\zeta x_1{x}_4-x_2{x}_4+\gamma x_1+\mu (x_1-R_0)\end{array}\right.$$

(6)

Based on Utkin’s method, we define the following:

$$\begin{array}{l}{F}_s={\displaystyle \frac{-{\mathcal{L}}_{F_2}H(x)}{{\mathcal{L}}_{F_1}H(x)-{\mathcal{L}}_{F_2}H(x)}}{F}_1(x)+{\displaystyle \frac{{\mathcal{L}}_{F_1}H(x)}{{\mathcal{L}}_{F_1}H(x)-{\mathcal{L}}_{F_2}H(x)}}{F}_2(x)\\ =\left(\begin{array}{l}{x}_2\hfill \\ {\mathsf{\Omega }}^2\cos (\theta -x_3)-2\zeta x_2-(\beta +1)x_1+x_1{x_4}^2+R_0\hfill \\ x_4\hfill \\ -{\displaystyle \frac{x_2{x}_4}{x_1}}\hfill \\ \mathsf{\Omega }\hfill \end{array}\right)\end{array}$$

(7)

as the sliding flow of a such rotor/stator system. $F_s$ points orthogonal to the $H(x)$ and tangential to the $\mathsf{\Sigma }$. Since the deflection is greater than the clearance ($R>R_0$) when backward whirling oscillation from dry friction happens, the condition should be guaranteed:

$${\mathcal{L}}_{F_2}H(x)-{\mathcal{L}}_{F_1}H(x)=2\mu (R-R_0)>0$$

(8)

The condition of (8) guarantees that the sliding domain (${\mathsf{\Sigma }}_s$) is attracting and stable in the normal direction when backward whirling oscillation from dry friction happens.

For sliding bifurcations, the local oscillation moves towards or leaves the boundary, and $\partial {\mathsf{\Sigma }}_s$ should always hold in the second-order Lie derivative, ${{\mathcal{L}}_{F_1}}^{2}H(x)$, which is defined as follows:

$$\begin{array}{lll}{{\mathcal{L}}_{F_1}}^{2}H(x)& =& {\displaystyle \frac{\partial ({\mathcal{L}}_{F_1}H(x))}{\partial x}}{F}_1(x)\\ & =& +2\zeta (\mu -\zeta )x_1+(\mathsf{\Omega }-2x_4){\mathsf{\Omega }}^2\cos (\theta -x_3)-R_0{x}_4+(\beta +1+4{\zeta }^2)x_1{x}_4-x_1{x_4}^3\\ & & +2(3\zeta +{\displaystyle \frac{x_2}{x_1}})x_2{x}_4-\mu {\displaystyle \frac{x_2}{x_1}}{R}_0-2\zeta \mu R_0-(2\zeta +{\displaystyle \frac{x_2}{x_1}}){\mathsf{\Omega }}^2\sin (\theta -x_3)\end{array}$$

(9)

Consider $x=x^{*}$ and $x^{*}\in \partial {{\mathsf{\Sigma }}_s}^{-}$ at the boundary ($\partial {\mathsf{\Sigma }}_s$) of ${\mathsf{\Sigma }}_s$; then we obtain the following:

$$H(x^{*})=0 \mathrm{and} {\mathcal{L}}_{F_1}H(x^{*})=0$$

(10)

Consequently, the crossing–sliding and grazing–sliding bifurcations can be determined [1,5]:

$${{\mathcal{L}}_{F_1}}^{2}H(x^{*})>0$$

(11)

Note that the condition (11) is not sufficient but is necessary for the sliding bifurcations.

3.2. Sliding Vibrations

In this section, we find three categories of sliding oscillations in the friction involved in the rotor/stator system. The parameters considered in the nonlinear rotor system are as follows:

$$\beta =0.04, \gamma =0.05, \zeta =0.05, R_0=1.05, R_d=20R_0$$

(12)

and $\mu \in [0,0.4]$ and $\mathsf{\Omega }\in [0,1]$. For the study of the sliding motions, we characterize the duration of sliding as follows:

$${\tau }_{rel}={\displaystyle \frac{{\tau }_{\mathrm{sliding}}}{T}}$$

(13)

where ${\tau }_{\mathrm{sliding}}$ is the absolute value of the sliding length in one period (T). When ${\tau }_{rel}=1$, the disk performs a pure roll along the stator in one whole period of the oscillation.

Figure 2 and Figure 3 present the results of the pure rolling oscillation. Consider the parameters of $\mathsf{\Omega }=0.2$ and $\mu =0.2$; the rotor/stator system in Equation (3) is characterized with the pure rolling motion of backward whirling oscillation induced by dry friction, as shown in Figure 2a. The green response represents the pure rolling orbit, and the black is the rotor/stator clearance. In Figure 2b, the time history normal components of two vector fields, namely, $F_1(x)$ and $F_2(x)$, and the velocity ($V_{rel}$) are presented. The $V_{rel}$ does not change much during this type of motion and with the value close to zero. It is noted that, in the sliding domain, the condition (5) can always be satisfied in this case, as shown in Figure 2b. Namely, ${\mathcal{L}}_{F_1}H(x)·{\mathcal{L}}_{F_2}H(x)<0$ is always satisfied. So, as for this type of motion, the relative sliding time length can be concluded as ${\tau }_{rel}=1$. Similarly, the pure rolling oscillation in terms of $\mathsf{\Omega }=0.2$ and $\mu =0.2$ is presented in Figure 3.

Figure 4 and Figure 5 present the scenario of continuous crossing oscillation. Consider the parameters of $\mathsf{\Omega }=0.9$ and $\mu =0.16$; the backward whirling oscillation induced by dry friction in the rotor/stator system exhibits the continuous crossing oscillation, as shown in Figure 4a. The blue response represents the continuous crossing oscillation orbits, and the black is the rotor/stator clearance. In Figure 4b, the time history normal components of two vector fields, $F_1(x)$ and $F_2(x)$, and the velocity ($V_{rel}$) are presented. The velocity ($V_{rel}$) remains zero when it goes through the switching manifold. In this case of the sliding domain, the condition (5) in Figure 2b cannot be satisfied. Namely, ${\mathcal{L}}_{F_1}H(x)·{\mathcal{L}}_{F_2}H(x)>0$ is always satisfied. So, for this continuous crossing oscillation, the relative sliding time length can be concluded as ${\tau }_{rel}=1$. Similarly, the continuous crossing oscillation in terms of $\mathsf{\Omega }=0.8$ and $\mu =0.16$ is presented in Figure 5.

When choosing the parameters of $\mathsf{\Omega }=0.4$ and $\mu =0.2$, the rotor/stator system behaviors a backward whirling oscillation induced by dry friction with the characteristics of sliding oscillations and slipping oscillations. Figure 6 and Figure 7 present the scenario of grazing–sliding bifurcation. The green response represents the pure rolling oscillations, and the blue is the slipping motions. In Figure 6b, the velocity ($V_{rel}$) and Lie derivatives are presented. For pure rolling oscillations, the $V_{rel}$ is near zero, while, for slipping oscillations, the $V_{rel}$ is over zero in such a rotor/stator nonlinear system with dry friction. It can be observed that the nonlinear vibration of the $V_{rel}$ consists of two parts: adherence parts with sliding oscillation and non-zero parts with slipping oscillation. In this case with ${\mathcal{L}}_{F_2}H(x)>0$, the sliding motion boundary can be obtained by ${\mathcal{L}}_{F_1}H(x)=0$ of the nonlinear vibration. And the oscillation orbits of the disk go into $S_1$, where the vector field ($F_s$) in Equation (7) equals $F_1$. Meanwhile, because the second-order ${{\mathcal{L}}_{F_1}}^{2}H(x)$ of $F_1(x)$ at the boundary of the sliding domain is larger than zero, conditions (11) and (10) are satisfied, proving the existence of grazing–sliding bifurcations in this rotor/stator system. So, for this kind of motion, the relative sliding time length can be obtained as 0.6478. Similarly, the grazing–sliding bifurcation oscillations in terms of $\mathsf{\Omega }=0.9$ and $\mu =0.2$ are presented in Figure 7.

Figure 8 presents the experimental results of the backward whirling motions of the rotor/stator system. The backward whirling oscillation is very obvious. Figure 8a presents the test platform of the experiment. Figure 8b presents the rotating orbits. The frequency–amplitude characteristics of the backward whirling oscillation are shown in Figure 8c. The negative whirling frequency around −500 Hz appears, which validates the backward whirling oscillations of the rotor/stator rubbing system.

3.3. Cross-Coupling Stiffness Effects

This section studies the effects of the system parameters on friction-induced, nonlinear vibrations and bifurcations. The influence of the cross-coupling stiffness on the sliding time length is presented in the parameter windows of ${\tau }_{\mathrm{rel}}-\mathsf{\Omega }$ and $\mu -\mathsf{\Omega }$.

Consider the parameters based on the experiments in Ref. [11]:

$$\beta =0.04, \zeta =0.05, R_0=1.05, \mathrm{and} R_d=20R_0,$$

The response of ${\tau }_{\mathrm{rel}}-\mathsf{\Omega }$ is calculated during $\mathsf{\Omega }\in [0.03, 1]$ in Figure 9. It is noted that when the backward whirling oscillation induced by dry friction happens, the sliding time length is always less than one. This phenomenon indicates the existence of rotor-rolling and disk-slipping transitions where grazing–sliding bifurcation occurs. Such rotor/stator-coupled oscillation has been recognized as slipping oscillation [11]. The angular velocity ($\mathsf{\Omega }$) is larger at the beginning, where the pure rolling motion part becomes smaller and then increases during one oscillation period. When the cross-coupling stiffness is smaller ($\gamma \le 0.05$), the sliding time length (${\tau }_{\mathrm{rel}}$) reaches the smallest value, around $\mathsf{\Omega }\approx 0.45$. When $\gamma$ is larger ($\gamma \ge 0.12$), the sliding time length (${\tau }_{\mathrm{rel}}$) becomes less than one at $\mathsf{\Omega }>0.33$. This phenomenon indicates that the angular velocity is very high for the chosen parameters of the rotor/stator system on which grazing–sliding bifurcation occurs. When $\gamma$ goes up, the sliding time length (${\tau }_{\mathrm{rel}}$) equal to one happens at a very low angular velocity. Such a phenomenon indicates that the pure rolling oscillation happens at the very low angular velocity, where the backward whirling oscillation induced by dry friction just occurred.

The response of $\mathsf{\Omega }-\mu$ to represent the vibration characteristics of stick–slip whirling is presented in Figure 10. It can be observed that the parameter response of $\mathsf{\Omega }-\mu$ consists of two parts: pure rolling and grazing–sliding bifurcation oscillation. Consider the parameter $\gamma =0$; the black curves of ${\mu }_l$ and ${\mu }_u$ illustrate the coefficients of the dry friction. In such a case, the pure rolling vibration at ${\tau }_{\mathrm{rel}}=1$ governs the motion domain. The stick–slip vibrations that happen are associated with grazing–sliding bifurcations for the range in which the coefficients of the dry friction are smaller than ${\mu }_u$ and larger than ${\mu }_l$. ${\mu }_u$ goes to the minimum value, around $\mathsf{\Omega }\approx 0.45$, which is consistent with the phenomenon in Figure 5. When $\gamma$ increases, the pure rolling vibration at the sliding time length ${\tau }_{\mathrm{rel}}=1$ happens for the larger friction coefficient. This phenomenon indicates that the pure rolling oscillation happens at a very low cross-coupling stiffness. So, the condition of the pure rolling motions can be guaranteed, and the conditions can be applied to predict the nonlinear vibrations and bifurcations of the rotor/stator system with no cross-coupling stiffness [13,14,17].

From the results in Figure 9 and Figure 10, we can conclude the following: (1) In such a rotor/stator-coupled nonlinear system, pure rolling vibrations (${\tau }_{\mathrm{rel}}=1$), continuous crossing vibrations (${\tau }_{\mathrm{rel}}=0$), and grazing–sliding-bifurcation-induced vibrations can occur. (2) If the coefficient ($\mu$) of the dry friction is very large or the cross-coupling stiffness ($\gamma$) is very small, pure rolling vibration may occur. (3) If the angular velocity ($\mathsf{\Omega }$) is very low, the backward whirling oscillation induced by dry friction can be produced in rotor/stator systems, and pure rolling vibration may also happen. Meanwhile, if he cross-coupling stiffness is in the low range, then pure rolling exists for larger ranges.

4. Existence of Backward Whirling

From the above research, we find that at some critical speeds of the backward whirling oscillation induced by dry friction, the nonlinear rotor experiences pure rolling vibration. By a discussion of the relationship between the backward whirling vibration induced by dry friction and the pure rolling vibration, the existence conditions of the backward whirling vibration induced by dry friction can be obtained.

Let us summarize the critical conditions of the sliding vibrations: (1) Sliding vibration occurs on the switching manifold or the relative velocity goes to zero. (2) To keep the sliding vibration on the switching manifold, the product of two vector fields, namely, ${\mathcal{L}}_{F_1}H(x)$ and ${\mathcal{L}}_{F_2}H(x)$, are equal to zero (e.g., ${\mathcal{L}}_{F_1}H(x)·{\mathcal{L}}_{F_2}H(x)=0$). So, the critical condition for the sliding vibration is obtained based on Equations (5) and (6) (e.g., ${\mathcal{L}}_{F_1}H(x)=0$ and ${\mathcal{L}}_{F_2}H(x)=0$). (3) The sliding trajectory of the sliding vibration on the switching manifold are dominated by the dynamic flow ($F_s$). And for this situation, ${x_2}^{\prime }$ (${x_2}^{\prime }=R^{″}$) becomes zero at the critical speed of backward whirling oscillation induced by dry friction.

Therefore, the critical conditions for the sliding vibrations on the switching manifold $H(x)$ are obtained as follows:

$$\begin{array}{l}H=\mathsf{\Omega }{R}_d+R{\varphi }^{\prime }=0\hfill \\ {\mathsf{\Omega }}^2\sin (\theta -\varphi )-2\zeta R{\varphi }^{\prime }-R^{\prime }{\varphi }^{\prime }+\gamma R\pm \mu (R-R_0)=0\hfill \\ {\mathsf{\Omega }}^2\cos (\theta -\varphi )-2\zeta R^{\prime }-(\beta +1)R+R{{\varphi }^{\prime }}^2+R_0=0\hfill \end{array}$$

(14)

The critical condition is represented as follows:

$${\mathsf{\Omega }}^4={[2\zeta R{\varphi }^{\prime }+R^{\prime }{\varphi }^{\prime }-\gamma R\mp \mu (R-R_0)]}^2+{[2\zeta R^{\prime }+(\beta +1)R-R{{\varphi }^{\prime }}^2-R_0]}^2$$

(15)

Since the angular velocity ($\mathsf{\Omega }$) is $\mathsf{\Omega }\ll 1$, ${\mathsf{\Omega }}^4$ can be neglected. By substituting Equation (14) into (15) and keeping the contact condition of $R>R_0$, the governing equation of ${\varphi }^{″}$ is derived as follows:

$$c_3{{\varphi }^{″}}^3+c_2{{\varphi }^{″}}^2+c_1{\varphi }^{″}+c_0=0$$

(16)

where

$$\begin{array}{l}{c}_3=-{\displaystyle \frac{{\mathsf{\Omega }}^2{R_d}^2}{{(2\zeta \mathsf{\Omega }{R}_d+\mu R_0)}^2}}\\ c_2={\displaystyle \frac{R_0}{2\zeta \mathsf{\Omega }{R}_d+\mu R_0}}-3(\gamma -\mu ){\displaystyle \frac{{\mathsf{\Omega }}^2{R_d}^2}{{(2\zeta \mathsf{\Omega }{R}_d+\mu R_0)}^2}}\\ c_1=-3{(\gamma -\mu )}^2{\displaystyle \frac{{\mathsf{\Omega }}^2{R_d}^2}{{(2\zeta \mathsf{\Omega }{R}_d+\mu R_0)}^2}}+2(\gamma -\mu ){\displaystyle \frac{R_0}{2\zeta \mathsf{\Omega }{R}_d+\mu R_0}}-2\zeta {\displaystyle \frac{2\zeta \mathsf{\Omega }{R}_d+\mu R_0}{\mathsf{\Omega }{R}_d}}+(\beta +1)\\ c_0=-{(\gamma -\mu )}^3{\displaystyle \frac{{\mathsf{\Omega }}^2{R_d}^2}{{(2\zeta \mathsf{\Omega }{R}_d+\mu R_0)}^2}}+{(\gamma -\mu )}^2{\displaystyle \frac{R_0}{2\zeta \mathsf{\Omega }{R}_d+\mu R_0}}+(\gamma -\mu )(\beta +1)\end{array}$$

From $H=\mathsf{\Omega }{R}_d+R{\varphi }^{\prime }=0$, the following is obtained:

$$R^{\prime }={\displaystyle \frac{R_{\mathrm{d}}\mathsf{\Omega }{\varphi }^{″}}{{{\varphi }^{\prime }}^2}}$$

(17)

It is easy to note that $R^{\prime }$ has the same orientation as ${\varphi }^{″}$ in Equation (17). So, for ensuring the continuous tangential rolling of the disk, the critical condition can be set to ${\varphi }^{″}=0$ for the existence of a continuous pure rolling vibration in the rotor/stator system.

Based on Equations (16) and (17), the condition for the existence of continuous pure rolling is derived as follows:

$$[4{\zeta }^2(\beta +1)-{(\gamma -\mu )}^2]{R_d}^2{\mathsf{\Omega }}^2+[(\gamma -\mu )+2\mu (\beta +1)]2\zeta R_0{R}_d\mathsf{\Omega }+(\gamma +\mu \beta )\mu {R_0}^2=0$$

(18)

The critical angular velocity (${\mathsf{\Omega }}_c$) for the continuous pure rolling vibration is as follows:

$$\left\{\begin{array}{l}{\mathsf{\Omega }}_c={\displaystyle \frac{-(\gamma +\mu +2\mu \beta )\zeta R_0\pm R_0(\gamma -\mu )\sqrt{{\zeta }^2+\gamma \mu +{\mu }^2\beta }}{[4{\zeta }^2(\beta +1)-{(\gamma -\mu )}^2]R_d}}\hfill \\ \mu \ge 2\zeta \sqrt{\beta +1}+\gamma \hfill \end{array}\right.$$

(19)

Consider the conditions of $\gamma =0$, $\gamma =0.05$, and $\gamma =0.12$ in the rotor/stator system; the existence conditions for the continuous pure rolling are presented as $\mathsf{\Omega }-\mu$ curves in Figure 7.

From the parametric curves in Figure 11, the critical conditions for sliding vibrations in Equation (19) can result in the existence conditions of the backward whirling oscillation induced by dry friction in the rotor/stator system. Moreover, the lower limit of the existence conditions of backward whirling oscillation induced by dry friction in the rotor/stator system can be achieved with $\mu =2\zeta \sqrt{\beta +1}+\gamma$ in Equation (19), which is depicted by the dashed line in Figure 11.

5. Discussion and Conclusions

5.1. Discussion

The continuous pure rolling vibration is characterized with whole sliding domains and the continuous crossing vibration is characterized without sliding domains on the switching manifold. It was also discovered that the proportion of the pure rolling vibration is determined by parameters such as the coefficient of friction, angular velocity of the rotor, and cross-coupling stiffness.

An interesting observation occurs at the stick–slip transitions of the backward whirling vibration induced by dry friction: the augment of the cross-coupling stiffness results in the same effects as those produced by increasing the coefficient of the dry friction, which can also be confirmed [16].

Continuous pure rolling vibrations occur at the existence of backward whirling oscillations induced by dry friction for all cross-coupling stiffnesses. So, through such sliding motions of continuous pure rolling vibrations, the analytical conditions of the critical speed of the rotor/stator system is obtained.

The findings of the study in terms of sliding bifurcations on the backward whirling vibration induced by dry friction are in good agreement with previous discoveries in numerical simulations [12] and experimental testing [11]. Thus, this research provides good perspectives for studying the backward whirling vibration induced by dry friction, and the stick–slip transitions and bifurcations in rotor/stator-coupled systems.

5.2. Conclusions

This study investigates the piecewise, discontinuous, friction-induced stick–slip whirling vibrations and bifurcations in a rotor/stator system. The switching manifold of different motion types is defined as a five-dimension hypersurface. The conclusions can be drawn as follows:

(1)

In such a rotor/stator-coupled nonlinear system, pure rolling vibration (${\tau }_{\mathrm{rel}}=1$), continuous crossing vibration (${\tau }_{\mathrm{rel}}=0$), and grazing–sliding-bifurcation-induced vibration can occur;

(2)

If the coefficient ($\mu$) of the dry friction is very large or the cross-coupling stiffness ($\gamma$) is very small, the pure rolling vibration may occur;

(3)

If the angular velocity ($\mathsf{\Omega }$) is very low, the backward whirling oscillation induced by dry friction can be produced in the rotor/stator system, and the pure rolling vibration may also happen. Meanwhile, if he cross-coupling stiffness is in the low range, pure rolling exists for larger ranges;

(4)

When $\gamma$ increases, the pure rolling vibration at the sliding time length (${\tau }_{\mathrm{rel}}=1$) happens for the larger friction coefficient;

(5)

The stick–slip vibrations that happen are associated with grazing–sliding bifurcations for the range in which the coefficients of the dry friction are smaller than ${\mu }_u$ and larger than ${\mu }_l$;

(6)

As the cross-coupling stiffness coefficient ($\gamma$) increases from 0 to 0.12, the limit friction coefficient of the existence conditions of backward whirling oscillation induced by dry friction in the rotor/stator system is increased from 0.11 to 0.23.