Regular and Chaotic Vibrations of a Nonlinear Rotor-Stator System

Abstract

The paper is concerned with the analysis of a six-degree-of-freedom non-linear model which describes the vibrations of a rotor. The model takes into account the impacts between the rotating element and a limiter of motion. Using numerical integration and spectrum analysis, the influence of the excitation frequency, static loads, and the position of the limiter of motion on the type of vibrations of the system was studied. A multiparametric analysis has been performed to determine the areas of influence of two system parameters on the type of excited vibrations. Different types of vibration are further illustrated by plots of time histories, frequency spectra, phase portraits, stroboscopic portraits and bifurcation diagrams. The quality index of the system has also been determined and defined as the average value of the rotor kinetic energy. Depending on the parameters of the system, periodic, quasi-periodic or chaotic oscillations take place. The article primarily focuses on the risk of chaotic vibrations occurring in the system.

1. Introduction

The complete elimination of transverse vibration of rotors is practically impossible, primarily due to residual static and dynamic unbalancing [1,2,3,4,5]. In the case of rotors operating at supercritical speeds, there is a possibility that some elements will clash during transient processes. Such impacts are possible, e.g., between the rotor and stator (i.e., limiter of motion), the case which has been studied in references [6,7,8]. As a result of these impact forces, it is possible under some conditions that periodic, almost periodic or chaotic vibrations get excited in the system [9,10,11]. Undesirable rotor impacts and the friction forces occurring between the elements in rotating machinery can lead to the destruction of system components, which negatively affects the efficiency of the device’s operation [12,13,14,15,16]. Many studies describe the impact of collisions between rotating elements of various systems on the type of induced vibrations, including chaotic vibrations. For example, in the work [17], the influence of the parameters of a vibration-isolated hand grinder on the nature of the induced vibrations is examined.

In the literature dealing with impacts in systems including a rotating element, a simplified model of a Jeffcott rotor is usually used, which neglects the gyroscopic effects and the influence of internal friction. The analysis of the simplified model is capable of explaining a number of interesting phenomena brought about by impacts between the elements of the system. Using this model, Karpenko et al. [18,19] and Sun, Xu and Zhou [20] have shown that for the static deflection, which is large compared to the clearance between the rotor and the limiter of motion or for a big amplitude of inertial excitation, subharmonic or chaotic vibrations set in. Wang and Ding, in their work [21], developed a new algorithm (NNMs) for rotor balancing. Using a Jeffcott rotor, authors demonstrated that the NNMs method is significantly more effective than the linear modal method. Vlajic et al. [22] analysed the modified Jeffcott rotor, taking into account torsional vibrations and rotor-stator contact. Using numerical research, the authors determined the vibration areas during which the rotor and stator come into contact. A new method to suppress the resonant vibrations of the non-linear Jeffcott-rotor system has been presented by Saeed et al. in their works [23,24]. In order to suppress the vibrations in the paper [23], a radial Proportional Derivative (PD-) controller was used along with the eight-pole electro-magnetic actuator, while in the paper [24] they employed a proportional integral resonant controller (PIRC).

In references [18,19], the influence of the excitation frequency and the stiffness of the limiter of motion is discussed. Chu and Zhang [25] analysed the influence of the gravity force, Edwards et al. [26] studied the effect of the torsional rigidity of the shaft, whereas Feng and Zhang [27] paid attention to the mode shapes of excited vibrations. Patel et al. [3] also analysed the influence of a rotor-to-stator contact on the lateral-torsional coupled vibrations. In some works, attempts were made to study systems using analytical methods [28,29], e.g., the Krylov-Bogolyubov or Van der Pol methods, which allow for a more precise explanation of certain physical phenomena but are less accurate in the case of strongly nonlinear systems. For example, Kydyrbekuly et al. [28] studied the rotor-stator system for an elastically mounted limiter. They compared the results obtained by the elliptic functions method with the results obtained by the numerical Runge–Kutta–Fehlberg’s 4-order method and the approximate analytical Van der Pol method. In [30], Torres-Contreras et al. proposed a new phase-shift empirical mode decomposition integration (PSEMDI) method to fault diagnose an unbalanced rotor subjected to a friction load. Pust [31] analyses the vibrations of rotors supported on magnetic bearings, accounting for the impact on the retainer bearings. Zhao et al. [32] proposed a bifurcation detection method based on generalised energy transfer to investigate the evolution of motion cascades leading to chaos. The influences of the friction at the interface in the rotor-stator system on the stability of work are studied by Mercier Alexy et al. in [33,34]. In the article [35], Mokhtar et al. investigate the impact of rotor-stator collisions on the bending-torsional vibrations of the system. To analyse the rotor-stator contact phenomenon, the authors utilised finite element methods using a Lagrange multiplier-based contact mechanics approach. Xu et al. [36], using the implicit mapping method, predicted the periodic motions in a nonlinear rotor system and then compared the obtained results with numerical simulation outcomes.

A static deflection big compared to the clearance can be caused by a change in the values of system parameters as a result of a long operation time. The results of the models mentioned above are most often used for diagnostic purposes. For sufficiently small or zero static deflection, chaotic vibrations are rare, and as a rule, quasi-periodic and periodic oscillations are excited. This case has been studied much less, especially for models with several degrees of freedom, which would account for the spatial motion of the rotor or the movement of the limiter.

This work is, on the one hand, of a cognitive nature, focused on examining the influence of many parameters of the system on the character of vibrations, while on the other hand, the work deals with the real rotor-motion limiter system for which there is a risk of damage due to collisions between the elements of this system. The basic purpose of the work is to examine, using vibration-type areas, the simultaneous influence of many parameters of the system on the character of excited vibrations. Analyses such as the one presented here can rarely be found in the available literature. In order to reduce the number of parameters relevant to the qualitative behaviour of the system, dimensionless quantities have been introduced. This fact makes quantitative analyses somewhat more difficult, including the estimation of design parameters of the system for which the system behaves correctly (i.e., no collisions occur). It is also more difficult to compare the research results with the results available in the literature. The qualitative analysis of the rotor-motion limiter system defined in such a way has been carried out in this paper using numerical integration procedures and fast Fourier transform (FFT) algorithms [37,38]. The rotor motion has been analysed for an elastically mounted limiter by examining the spatial transverse vibrations of the rotor and motion limiter. The influence of the most important system parameters on the type of vibrations has been investigated. Additionally, the influence of the position of the motion limiter and collisions between the rotor and stator on the occurrence of chaotic vibrations have been investigated. For this purpose, important parameters have been determined in which the system behaviour is similar in the qualitative senses. Additionally, bifurcation diagrams, time histories, frequency spectra, phase portraits and stroboscopic portraits have been analysed. The results of the analysis can be used to estimate the parameter ranges within which the safest operation of the system can be ensured.

2. Model of the System

In the present paper, a model with six degrees of freedom will be used to describe the vibrations of the system rotor—limiter of motion, as shown in Figure 1. The rotor’s motion is described by the coordinates x₁, y₁ of its geometrical centre S₁ (centre of exactly balanced mass m₁) and the rotation angles α₁ and β₁, whereas the coordinates x₂ and y₂ describe the displacements of the limiter of motion. Axial and torsional vibrations of both rotor and limiter are neglected. It has also been assumed that the rotational speed of the rotor Ω is constant, and thus, the motor has unlimited power. The gyroscopic effects are accounted for in the model. It has also been assumed that the resilient elements are described by the Voigt-Kelvin model with the damping coefficients (ε₀, ε₁, ε₂) proportional to the respective stiffnesses (c₀, c₁₁, c₁₂, c₂). The location of the limiter of motion is specified by the distance d₀ and that of the shaft supports by l₁ and l₂. For the case d₀ = 0, the limiter of motion is called a stator. There is a gap in the clearance δ between the rotor and the limiter of motion. The model takes into account the influence of the inertial forces resulting from the rotor unbalance (through eccentricity e₀) as well as the static forces Q₁ and Q₂.

Using the laws of change of momentum and angular momentum about the rotor centre of mass, the motion of the system is described by the following set of six second-order ordinary differential equations:

$$m_1{\ddot{x}}_1+c_1({\epsilon }_1{\dot{x}}_1+x_1)-c_1{d}_1({\epsilon }_1{\dot{\alpha }}_1+{\alpha }_1)+F_x=Q_1+m_1{e}_0{\Omega }^2\cos \Omega t$$

(1)

$$m_1{\ddot{y}}_1+c_1({\epsilon }_1{\dot{y}}_1+y_1)+c_1{d}_1({\epsilon }_1{\dot{\beta }}_1+{\beta }_1)+F_y=m_1{e}_0{\Omega }^2\sin \Omega t$$

(2)

$$I_1{\ddot{\alpha }}_1-I_0\Omega {\dot{\beta }}_1+(k_1+c_1{d}_1^2)({\epsilon }_1{\dot{\alpha }}_1+{\alpha }_1)-c_1{d}_1({\epsilon }_1{\dot{x}}_1+x_1)+F_x{d}_0=0$$

(3)

$$I_1{\ddot{\beta }}_1+I_0\Omega {\dot{\alpha }}_1+(k_1+c_1{d}_1^2)({\epsilon }_1{\dot{\beta }}_1+{\beta }_1)+c_1{d}_1({\epsilon }_1{\dot{y}}_1+y_1)-F_y{d}_0=0$$

(4)

$$m_2{\ddot{x}}_2+c_2({\epsilon }_2{\dot{x}}_2+x_2)-F_x=Q_2$$

(5)

$$m_2{\ddot{y}}_2+c_2({\epsilon }_2{\dot{y}}_2+y_2)-F_y=0$$

(6)

Here m₁ and m₂ are, respectively, the masses of the rotor and the limiter of motion, and I₁ and I₀ are the moments of inertia of the rotor about the x-axis (or about the y-axis thanks to axial symmetry) and the z-axis. The following symbols are used in Equations (1)–(6):

$$c_1=c_{11}+c_{12}$$

(7)

$$d_1=(c_{11}{l}_1-c_{12}{l}_2)/(c_{11}+c_{12})$$

(8)

$$k_1=c_{11}{c}_{12}{(l_1+l_2)}^2/(c_{11}+c_{12})$$

(9)

The components F_x and F_y of the impact forces take on non-zero values for r > δ, where the distance $r=\sqrt{x_0^2+y_0^2}$ is calculated using the coordinates $x_0=x_1+d_0{\propto }_1-x_2$, $y_0=y_1+d_0{\beta }_1-y_2$ of the vector of the displacement of the rotor axis relative to the limiter of motion (Figure 2). The following relationship is used for the tangential component of the reaction force:

$$T=fN$$

(10)

$$f=\left\{\begin{array}{l}{f}_S\mathrm{sgn}{V}_S\\ f_S{V}_S/V_{S\min }\end{array}\right.\begin{array}{r}if\left|V_S\right|>V_{S\min }\\ if\left|V_S\right|\le V_{S\min }\end{array}$$

(11)

where:

$$V_S=\Omega R+{\displaystyle \frac{(x_0{\dot{y}}_0-y_0{\dot{x}}_0)}{\sqrt{x_0^2+y_0^2}}}$$

(12)

In Equation (11), the modified friction coefficient f is a function of the tangential component of the relative velocity (slip velocity) $V_S$ at the point of contact of the rotor and limiter, which is calculated using Equation (12). In function (12), R is the rotor radius. In some research [7], it is assumed that the tangential component of the impact force is the sum of the dry and viscous friction forces. In this work, due to the very short time of contact between the rotor and the limiter, the simplest hypothesis of dry friction has been adopted [27]. More accurate models of friction forces may be useful when investigating local phenomena caused by collisions. This phenomenon is not analysed in this work. To avoid discontinuity of the friction force, linear approximation has been used after reference [17,27] for $\left|V_S\right|\le V_{S_{min}}$, where $V_{S_{min}}$ is some small value of velocity (0.001 m/s according to [17,27]). The use of the friction law in the form (1–6) reduces the time of numerical calculations in the case when $V_S$ is close to zero without practically influencing the accuracy of analysis.

The normal component of the limiter reaction is taken in the following form:

$$N=c_0[(r-\delta )+{\epsilon }_0\dot{r}]H(r-\delta )H(r-\delta +{\epsilon }_0\dot{r})$$

(13)

where H is the Heaviside (step) function. From Equation (13), it follows that the force N cannot take on negative values (unilateral constraints). The Cartesian components of the impact force are calculated from the following equations:

$$F_x=N\cos \varphi -T\sin \varphi$$

(14)

$$F_y=N\sin \varphi +T\cos \varphi$$

(15)

where the angle ϕ defines the location of the contact point between the rotor and the limiter of motion and is defined as follows: equations:

$$\sin \varphi =y_0/r$$

(16)

$$\cos \varphi =x_0/r$$

(17)

Upon introducing the dimensionless time:

$$\tau ={\omega }_{0}t$$

(18)

where ${\omega }_0=\sqrt{c_1/m_1}$ is the rotor natural frequency, and the dimensionless state vector:

$$\mathsf{u}=\begin{array}{cccccc}[x_1/\delta ,& y_1/\delta ,& l{\alpha }_1/\delta ,& l{\beta }_1/\delta ,& x_2/\delta ,& y_2/\delta {]}^T\end{array}$$

(19)

where l = (l₁ + l₂)/2, the system of Equations (1)–(6) can be written in the following non-dimensional form:

$$u_1^{″}+2{\zeta }_1{u}_1^{\prime }+u_1-{\lambda }_1(2{\zeta }_1{u}_3^{\prime }+u_3)+f_x=e{\omega }^2\cos \omega \tau +q$$

(20)

$$u_2^{″}+2{\zeta }_1{u}_2^{\prime }+u_2+{\lambda }_1(2{\zeta }_1{u}_4^{\prime }+u_4)+f_y=e{\omega }^2\sin \omega \tau$$

(21)

$${\rho }_1{u}_3^{″}-{\rho }_0\omega u_4^{\prime }-{\lambda }_1(2{\zeta }_1{u}_1^{\prime }+u_1)+({\kappa }_1+{\lambda }_1^2)(2{\zeta }_1{u}_3^{\prime }+u_3)+{\lambda }_0{f}_x=0$$

(22)

$${\rho }_1{u}_4^{″}+{\rho }_0\omega u_3^{\prime }+{\lambda }_1(2{\zeta }_1{u}_2^{\prime }+u_2)+({\kappa }_1+{\lambda }_1^2)(2{\zeta }_1{u}_4^{\prime }+u_4)-{\lambda }_0{f}_y=0$$

(23)

$${\mu }_2{u}_5^{″}+{\gamma }_2(2{\zeta }_2{u}_5^{\prime }+u_5)-f_x={\mu }_{2}q$$

(24)

$${\mu }_2{u}_6^{″}+{\gamma }_2(2{\zeta }_2{u}_6^{\prime }+u_6)-f_y=0$$

(25)

The dimensionless quantities appearing in Equations (20)–(25) are defined as follows:

$$\begin{array}{l}\omega =\Omega /{\omega }_0\\ {\gamma }_0=c_0/c_1\\ {\kappa }_1=k_1/c_1{l}^2\\ {\gamma }_2=c_2/c_1\\ {\rho }_0=I_0/m_1{l}^2\end{array}\begin{array}{l}e=e_0/\delta \\ {\lambda }_0=d_0/l\\ {\lambda }_1=d_1/l\\ {\zeta }_2={\epsilon }_2{\omega }_0/2\\ {\rho }_1=I_1/m_1{l}^2\end{array}\begin{array}{l}q=Q_1/c_1\delta \\ {\zeta }_0={\epsilon }_0{\omega }_0/2\\ {\zeta }_1={\epsilon }_1{\omega }_0/2\\ {\mu }_2=m_2/m_1\\ \end{array}$$

(26)

The non-dimensional components of the impact forces f_x and f_y can be calculated from Equations (10)–(17), using in place of N the non-dimensional force:

$$n={\gamma }_0(\rho -1+2{\zeta }_0{\rho }^{\prime })H(\rho -1)H(\rho -1+2{\zeta }_0{\rho }^{\prime })$$

(27)

where ρ = r/δ.

3. Results

The system of non-linear differential Equations (20)–(25), which describe the model, depends on more than ten parameters, some of which essentially influence the solution. The full qualitative analysis of such a model requires the study of the influence of the Parameters (26) on the character of motion.

Figure 3 shows a simplified schematic diagram of the calculations used to investigate the nature of vibrations for selected values of the system parameters. For the purpose of solving the differential Equations (20)–(25), Runge-Kutta algorithms with variable integration steps were used: 4th order one and a more accurate 7th order one. The equations have been integrated over a given time interval (0, NT), where T is the period of the exciting force. The study of vibration type has been carried out in a steady state, analysing the solutions in the interval (KT, NT) for this purpose. The vibration analysis was primarily based on the results of the spectral analysis obtained using the FFT algorithm. In order to improve the assessment of the studied courses, the periodicity conditions of the solutions have been checked in parallel. Additionally, it has been verified whether there are collisions present in the rotor-stator system by noting their average number during the excitation period. The algorithm presented in Figure 3 has been used, among others, to make bifurcation diagrams by finding the components of the state vector at prescribed time instants (the stroboscopic method). In the case of models described by many parameters, it is more useful to determine the zones of the parameters in which the system displays the same qualitative behaviour. It is a more involved problem since it calls for the determination of the vibration type (sub-harmonic, quasi-periodic or chaotic) and the mode of vibration (forward or backward whirl). The algorithm presented in Figure 3 has been used to determine the areas of various types of vibrations by changing the values of two selected system parameters during the calculations. The number of impacts of the rotating element with the limiter during the forcing period and whether there is full or partial rubbing between the rotor and the limiter during the impact are also of importance. Additionally, an index J = E/m₁ω₀² related to the kinetic energy of rotor vibrations has been determined. With the introduction of dimensionless quantities, this index takes the following form:

$$J={\displaystyle \frac{1}{2\left(N-K\right)T}}{\displaystyle \underset{KT}{\overset{NT}{\int }}[u_1^{\prime 2}+u_2^{\prime 2}+{\rho }_1(u_3^{\prime 2}+u_4^{\prime 2})]d\tau }$$

(28)

Selected results of the qualitative analysis are discussed below, where the analysis is limited to the influence of the parameters ω, e, q and λ₀. A small difference between the rigidities of the rotor supports has been assumed (c₁₁ = 0.475, c₁₂ = 0.525, l₁ = l₂ = l), which results in couplings (λ₁ = −0.05) which have an important effect on the mode shapes. Thanks to the assumed asymmetry, spatial vibrations are excited in the analysed system. The following set values of the parameters have been used in the numerical calculations: ρ₀ = 0.01, ρ₁ = 0.05, μ₂ = 2, γ₀ = 1000, γ₂ = 100, κ₁ = 0.9975, Ϛ₀ = 0.001, Ϛ₁ = 0.1, Ϛ₂ = 0.01, ∆ = 1, λ₁ = −0.05, f_S = 0.1. Additionally, the values $R/\delta =25$, $V_{S_{min}}/\mathsf{\Omega }\delta =0.1$ have been used in modelling the friction force.

Figure 4 shows the influence of the excitation frequency (parameter ω) and static force (parameter q) on the vibration type (Figure 4a) and index J (Figure 4b) for the case of the rotor-stator system λ₀ = 0 and e = 0.3. In order to determine the vibration type, the spectra of the state vector coordinates have been analysed. In the case of periodic system responses, the conditions of periodicity of solutions have been additionally examined, which allows us to determine the order of subharmonic vibrations (i.e., nT-periodic). The number n is the ratio of the system response period to the excitation period. All analyses have been carried out in a steady state, assuming K = 100 and N = 150 (where KT is the time without the analysis of the solutions). In calculating the zones, the initial conditions that correspond to the rotor equilibrium conditions were used. Also shown in Figure 4a, the k-curve calculated from the analysis of a linear system (with no contact of the rotor with the limiter of motion), which is defined as loci of static force values q for which the maximum displacement ${\rho }_{max}=r_{max}/\delta$ during steady-state vibrations is equal to unity (${\rho }_{max}=1$). Below this curve, the linear analysis suggests no rotor-limiter collisions. However, nonlinear system analysis reveals that in these zones, steady-state vibrations are possible, during which the rotor and the limiter of motion impact the system in a regular or chaotic way. This phenomenon results from the transient phase of motion during which initial sporadic impacts of the rotor against the limiter of motion can change into systematically repeated clashes, resulting in steady-state vibrations of a complex character. These vibrations have been found to only slightly depend on the initial conditions. In the resonance range (for ω ≈ 1), 1T-periodic vibrations dominate, during which rotor-limiter of motion collisions occur.

Figure 4b shows the influence of parameters ω and q on the quality index J (28). The rotor vibration energy increases with the increase of rotational speed (frequency ω), with significantly higher vibration levels being noted in the chaotic vibration ranges.

Figure 5 shows the areas of periodic, quasi-periodic and chaotic vibrations in the (ω, q) plane for a motion limiter shifted relative to the rotor by a value of λ₀ = 0.3. Similarly to Figure 4a, the k curve is also plotted here. Comparing Figure 4a and Figure 5, one can see the influence of the parameter λ₀ in the higher frequency range. For the λ₀ = 0 (case of the system: rotor-stator) shown in Figure 4a, chaotic vibrations predominate, whereas for λ₀ = 0.3 (Figure 5), mostly sub-harmonic oscillations are excited in the system. With the increase of the excitation frequency, the vibration regions appear in the following order: 1T-periodic, 2T-periodic, 3T-periodic and higher order. Chaotic vibrations and quasi-periodic oscillations occur only for high values of ω and q.

The different character of vibrations for different positions of the limiter of motion is also confirmed by the bifurcation diagrams shown in Figure 6a,b, which correspond to the sections of Figure 4a and Figure 5 for q = 0.7. The frequency intervals with a set of continuous lines are the regions of sub-harmonic oscillations of type 1:n (nT–periodic vibrations), where the number of lines provides information about the order. The mechanism works correctly if the vibration period is equal to the excitation period (single line on the diagram) and, additionally, if no collisions occur. On the bifurcation diagram determined for the rotor-stator system (λ₀ = 0, Figure 6a), the ranges of chaotic vibrations are present only in the low-frequency range. These are the ranges of 1T-, 2T- and less visible 3T-periodic vibrations, separated from each other by the ranges of chaotic vibrations. In the higher frequency range (for ω > 5.8), chaotic or quasi-periodic vibrations are dominant. Analysis of the bifurcation diagram alone does not allow for an unambiguous distinction between these types of vibrations.

With the increase of the distance between the limiter and the rotor (parameter λ₀), the ranges of subsequent nT-periodic vibrations become more distinct, and they shift towards lower frequencies (λ₀ = 0.3, Figure 6b). The subsequent regions of sub-harmonic vibrations of increasing order are separated by narrow zones of quasi-periodic and chaotic oscillations.

Bifurcation diagrams are often used to analyse rotor vibrations because they provide a lot of interesting information regarding the character of vibrations [10,11,15].

It should be pointed out that in the parameter range for which quasi-periodic or chaotic vibrations take place, the solutions of Equations (20)–(25) are sensitive to even very small changes of parameters; the chaotic solutions change easily into quasi-periodic solutions or sub-harmonic solutions of a high order. This is confirmed by the stroboscopic portraits (Figure 7) obtained respectively for chaotic, sub-harmonic and quasi-periodic oscillations. In the case of nT-periodic vibrations, the stroboscopic portrait is a set of n points (Figure 7b); for quasi-periodic vibrations, this portrait is a closed curve (Figure 7a), and for chaotic vibrations, it is a fractal (Figure 7c,d). The shape of the fractal in Figure 7c suggests that the chaotic vibrations are probably the result of successive bifurcations of 4T-periodic vibrations. However, whether the vibrations are quasi-periodic or chaotic, in both cases, there always appear to be harmful impacts between the rotor and limiter.

Figure 8a,b illustrate the influence of the parameters ω, e and λ₀ on the character of vibrations, respectively, for q = 0.4, λ₀ = 0.3 and q = 0.4, e = 0.5. The linear analysis of the steady-state vibrations predicts the occurrence of internal impacts only in the area above curve k (Figure 8a) or between curves k₁ and k₂ (Figure 8b). In the resonant regions, mostly T-periodic oscillations are set in with the full rubbing between the rotor and the limiter of motion. In this region, the rotor axis rotates in the same direction as the rotor revolutions (forward whirl). Outside the resonance region, all types of vibrations take place with partial rubbing between the rotor and limiter.

Figure 8b is determined for large values of parameters q and e (i.e., small values of clearance δ). For this reason, in the case of the rotor-stator system (λ₀ = 0), collisions occur practically in the entire frequency range, which results in periodic vibrations for lower frequencies and chaotic vibrations for higher frequencies. For smaller values of parameters q and e, vibrations without collisions are possible.

For higher values of |λ₀|, quasi-periodic oscillations occur, and by further increasing the value of |λ₀| there appear zones of T-periodic vibrations with no internal impacts. For example, for a relatively high excitation amplitude e = 0.5 and for the static load q = 0.4, in the region lying to the right of the curve k₂ in Figure 8b, there are practically no steady-state impacts between the rotor and the limiter of motion when |λ₀|> 0.4 and 3 < ω < 6. This may suggest that in order to protect the stator, one can use a limiter of motion close to the shaft supports.

Figure 9 shows the bifurcation diagram obtained by the stroboscopic method, which corresponds to the section of Figure 8b. This graph illustrates the effect of the limiter of motion position on the type of excited vibrations for ω = 8. For |λ₀|< 0.25, chaotic oscillations predominate (as might be suggested by the more irregular envelope of the strobe points), whereas for 0.25 < |λ₀| < 0.8, there occur mostly quasi-periodic vibrations. Throughout the whole range of the parameter λ₀, there exist narrow regions of sub-harmonic vibrations of different order. The bifurcation diagram shown in Figure 9 confirms the previous conclusions that the position of the limiter at the rotor mounting point (λ₀ = 0) with incorrectly selected system parameters causes the occurrence of chaotic vibrations.

The analysis of the bifurcation diagram alone does not allow a clear distinction between quasi-periodic and chaotic vibrations. In order to determine the character of vibrations in this case, it is necessary to analyse, for a given parameter λ₀, the spectrum of the calculated time signal. Such analysis was performed to determine the zones of vibration types in the earlier-discussed figures. The study of the stroboscopic phase planes (Figure 10) is also very useful. The results of such an analysis for two selected values of the parameter λ₀ are shown in Figure 10. In the case of chaotic vibrations (Figure 10a), the time histories is irregular, the spectrum is diffused, and it is characterised by the lack of clear peaks, well-defined for concrete frequencies. The phase trajectories (u₁, u_1′) and the trajectory graphs (u₁, u₂) form non-closing curves. For chaotic vibrations, their stroboscopic portraits are fractals (Figure 10a), while in the case of quasi-periodic vibrations, the stroboscopic portraits take the form of closed curves (Figure 10b).

4. Conclusions

The paper discussed an approach to the analysis of non-linear vibrations of the system rotor-limiter of motion. The model with six degrees of freedom used in the study allows for the consideration of the spatial motion of the system and accounts for the unilateral character of the forces between the rotor and the limiter of motion.

An important influence of the transient processes on the steady-state movement of the system has been observed. For a small clearance between the rotor and the limiter of motion, steady-state vibrations with impacts between the elements of the system set in.

Analysis of the results presented in the paper allows us to determine the areas of the most important parameters of the system for which the motion is T-periodic and there are no collisions between the rotor and the limiter. When determining their values, it should be taken into account that for safety reasons, an area of vibrations without collisions should be present in the widest possible operating range of parameters.

The conducted numerical simulations permit us to analyse in detail the influence of the design parameters of the model on the character of vibrations, which significantly depends on the amplitude e and the circular frequency of excitation ω as well as on the static load q. For sufficiently large values of e, in the resonant region, a forward whirl sets in with full rubbing between the rotor and the limiter. The static load of the rotor has a similar effect on the system. The higher the value of the parameter q, the greater the risk of collisions. With the increase of the excitation frequency ω, the character of vibrations becomes more complicated, passing from periodic vibrations of an increasingly higher order to quasi-periodic and chaotic vibrations. The introduced quality index informs about the level of rotor vibrations. With the increase of the rotational speed of the rotor, the vibration energy increases, with significantly higher vibration levels being recorded in the ranges of chaotic vibrations. An important role is also played by the position of the limiter of motion, described by the parameter λ₀. By increasing |λ₀|, the risk of steady-state impacts is reduced.

Full analysis of the system would require the study of the influence of the remaining parameters, such as the damping coefficients Ϛ₀, Ϛ₁, Ϛ₂, the stiffnesses γ₀ γ₂ or the ratio μ₂ of the mass of the limiter to the mass of the rotor. Such a study can be conducted in a similar way to the approach discussed in the paper, which has proved to be very efficient.