An investigation is carried out on the systematic analysis of the dynamic behavior of the hybrid squeeze-film damper (HSFD) mounted a rotor-bearing system with strongly nonlinear oil-film force and nonlinear rub-impact force in the present study. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless rotating speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, bifurcation diagrams, maximum Lyapunov exponents, and fractal dimension of the rotor-bearing system. The dynamic behaviors are unlike the usual ways into chaos (

Squeeze-film damper (SFD) bearing is actually a special type of journal bearing with its journal mechanically prevented from rotating but free to vibrate within the clearance space. The hybrid squeeze-film damper (HSFD) and the porous squeeze-film damper (PSFD) are the well-known applications of SFD and also useful for industry. Some literatures discussed dynamic behaviors in SFD bearings and also found many interesting and useful results. Holmes et al. [

Although virtually all physical phenomena in the real world can be regarded as nonlinear, most of these phenomena can be simplified to a linear form given a sufficiently precise linearization technique. However, this simplification is inappropriate for high-power, high rotating speed system and its application during the design and analysis stage may result in a flawed or potentially dangerous operation. As a result, nonlinear analysis methods are generally preferred within engineering and academic circles. The current study performs a nonlinear analysis of the dynamic behavior of a rotor-bearing system equipped with hybrid squeeze-film damper under nonlinear rub-impact force effect. The nondimensional equation of the rotor-bearing system is then solved using the fourth-order Runge-Kutta method. The nonperiodic behavior of this system is characterized using phase diagrams, power spectra, Poincaré maps, bifurcation diagrams, Lyapunov exponents, and the fractal dimension of the system.

Figure

Schematic illustration of hybrid squeeze-film damper mounted the rotor-bearing system.

Cross-section of HSFD model and rub-impact rotor-stator model.

To analyze the pressure distribution, the Reynolds equation for constant lubricant properties and noncompressibility should be assumed, then the Reynolds equation is introduced as follows [

According to the above conditions, the instant oil film pressure distribution is as follows. The instant pressure in rotating direction within the range of

The instant pressure in the axis direction within the range of

The instant oil film forces of the different elements are determined by integrating (

Figure

The equations of rotor motion in the Cartesian coordinates can be written as

In the present study, the nonlinear dynamics of the rotor-bearing system equipped with HSFD shown in Figure

The dynamic trajectories of the rotor-bearing system provide a basic indication as to whether the system behavior is periodic or nonperiodic. However, they are unable to identify the onset of chaotic motion. Accordingly, some other form of analytical method is required. In the current study, the dynamics of the rotor-bearing system are analyzed using Poincaré maps derived from the Poincaré section of the rotor system. A Poincaré section is a hypersurface in the state-space transverse to the flow of the system of interest. In nonautonomous systems, points on the Poincaré section represent the return points of a time series corresponding to a constant interval

In this study, the spectrum components of the motion performed by the rotor-bearing system are analyzed by using the Fast Fourier Transformation method to derive the power spectrum of the displacement of the dimensionless dynamic transmission error. In the analysis, the frequency axis of the power spectrum plot is normalized using the rotating speed,

A bifurcation diagram summarizes the essential dynamics of a rotor-train system and is therefore a useful means of observing its nonlinear dynamic response. In the present analysis, the bifurcation diagrams are generated using two different control parameters, namely the dimensionless unbalance coefficient,

The Lyapunov exponent of a dynamic system characterizes the rate of separation of infinitesimally close trajectories and provides a useful test for the presence of chaos. In a chaotic system, the points of nearby trajectories starting initially within a sphere of radius

The presence of chaotic vibration in a system is generally detected using either the Lyapunov exponent or the fractal dimension property. The Lyapunov exponent test can be used for both dissipative systems and nondissipative (i.e. conservative) systems, but is not easily applied to the analysis of experimental data. Conversely, the fractal dimension test can only be used for dissipative systems but is easily applied to experimental data. In contrast to Fourier transform-based techniques and bifurcation diagrams, which provide only a general indication of the change from periodic motion to chaotic behavior, dimensional measures allow chaotic signals to be differentiated from random signals. Although many dimensional measures have been proposed, the most commonly applied measure is the correlation dimension

In the current study, the attractors in the embedding space were constructed using a total of 60000 data points taken from the time series corresponding to the displacement of the system. Via a process of trial and error, the optimum delay time when constructing the time series was found to correspond to one third of a revolution of the system. The reconstructed attractors were placed in embedding spaces with dimensions of

The nonlinear dynamic equations presented in (

The bifurcation diagram in Figure

Bifurcation diagram of

Subharmonic motion at

Pressure distribution in the static pressure chamber at

Figures

Bifurcation diagram of HSFD rotor-bearing system using dimensionless rotating speed coefficient,

Simulation results obtained for rotor-bearing system with

Phase diagram

Power spectrum

Lyapunov exponent

Poincaré map

Fractal dimension

Simulation results obtained for rotor-bearing system with

Phase diagram

Power spectrum

Lyapunov exponent

Poincaré map

Fractal dimension

Simulation results obtained for rotor-bearing system with

Phase diagram

Power spectrum

Lyapunov exponent

Poincaré map

Fractal dimension

Phase diagrams for the route of subharmonic motion into chaos, out of chaos to periodic response at different rotating speed ratios of

Poincaré maps for the route of subharmonic motion into chaos, out of chaos to periodic response at different rotating speed ratios of

A hybrid squeeze-film damper mounted rotor-bearing system with nonlinear oil-film force and nonlinear rub-impact force has been presented and studied by a numerical analysis of the nonlinear dynamic response in this study. The dynamics of the system have been analyzed by reference to its dynamic trajectories, power spectra, Poincaré maps, bifurcation diagrams, maximum Lyapunov exponents, and fractal dimensions. The bifurcation results can be observed that HSFD may be used to improve dynamic irregularity. The system with rub-impact force effect may be a strongly nonlinear effect, and the bifurcation results show that HSFD mounted rotor-bearing system with rub-impact force effect present nonperiodic motions at low rotating speeds and perform periodic motions at high rotating speeds. The results will enable suitable values of the rotating speed ratio to be specified such that chaotic behavior can be avoided, thus reducing the amplitude of the vibration within the system and extending the system life.

Bearing parameter =

Viscous damping of the rotor disk

Damper eccentricity =

Components of the fluid film force in horizontal and vertical coordinates

Components of the fluid film force in radial and tangential directions

Oil film thickness,

Stiffness of the retaining springs

Proportional gain of PD controller

Derivative gain of PD controller

Bearing length

Masses lumped at the rotor mid-point

Center of rotor gravity

Geometric center of the bearing and journal

Pressure distribution in the fluid film

Pressure of supplying oil

Pressure in the static pressure chamber

Inner radius of the bearing housing

Radius of the journal.

Radial and tangent coordinates

Speed parameter =

Horizontal, vertical and axial coordinates

Damper static displacements

Mass eccentricity of the rotor

Rotational angle

Rotational speed of the shaft

Angle displacement of line

Radial clearance =

The angular position along the oil film from line

Oil dynamic viscosity

Distribution angle of static pressure region

Derivatives with respect to