Experimental and numerical analysis of nonlinear dynamics of rotor–bearing–seal system

Nonlinear dynamics and stability of the rotor–bearing–seal system are investigated both theoretically and experimentally. An experimental rotor–bearing–seal device is designed and corresponding tests are carried out. The experimental rotor system is simplified as the Jeffcott rotor. The nonlinear oil–film forces are obtained under the short bearing theory and Muszynska nonlinear seal force model is used. Numerical method is utilized to solve the nonlinear governing equations. Bifurcation diagrams, waterfall plots, Poincaré maps, spectrum plots and rotor orbits are drawn to analyze various nonlinear phenomena and system unstable processes. Theoretical results from numerical analysis are in good agreement with results from experiments. Conclusions are drawn and prove that this study will contribute to the further understanding of nonlinear dynamics and stability of the rotor system with the fluid-induced forces from oil–film bearings and the seals.     

Order reduction of retarded nonlinear systems – the method of multiple scales versus center-manifold reduction

We compare two approaches for determining the normal forms of Hopf bifurcations in retarded nonlinear dynamical systems; namely, the method of multiple scales and a combination of the method of normal forms and the center-manifold theorem. To describe and compare the methods without getting involved in the algebra, we consider three examples: a scalar equation, a single-degree-of-freedom system, and a three-neuron model. The method of multiple scales is directly applied to the retarded differential equations. In contrast, in the second approach, one needs to represent the retarded equations as operator differential equations, decompose the solution space of their linearized form into stable and center subspaces, determine the adjoint of the operator equations, calculate the center manifold, carry out details of the projection using the adjoint of the center subspace, and finally calculate the normal form on the center manifold. We refer to the second approach as center-manifold reduction. Finally, we consider a problem in which the retarded term appears as an acceleration and treat it using the method of multiple scales only. Communicated by G. Rega     

Extension of Lagrangian–Hamiltonian mechanics for continuous systems—investigation of dynamics of a one-dimensional internally damped rotor driven through a dissipative coupling

In this paper, the extended Lagrangian formulation for a one-dimensional continuous system with gyroscopic coupling and non-conservative fields has been developed. Using this formulation, the dynamics of an internally and externally damped rotor driven through a dissipative coupling has been studied. The invariance of the extended or so-called umbra-Lagrangian density is obtained through an extension of Noether’s theorem. The rotor shaft is modeled as a Rayleigh beam. The dynamic behavior of the rotor shaft is obtained and validated through simulation studies. Results show an interesting phenomenon of limiting behavior of the rotor shaft with internal damping beyond certain threshold speeds which are obtained theoretically and affirmed by simulations. It is further observed that there is entrainment of whirling speeds at natural frequencies of the rotor shaft primarily depending on the damping ratio.     

The effect of blade vibration on the nonlinear characteristics of rotor–bearing system supported by nonlinear suspension

In this paper, the complicated nonlinear dynamics of the harmonically forced quasi-zero-stiffness SD (smooth and discontinuous) oscillator is investigated via direct numerical simulations. This oscillator considered that the gravity is composed of a lumped mass connected with a vertical spring of positive stiffness and a pair of horizontally compressed springs providing negative stiffness, which can achieve the quasi-zero stiffness widely used in vibration isolation. The local and global bifurcation analyses are implemented to reveal the complex dynamic phenomena of this system. The double-parameter bifurcation diagrams are constructed to demonstrate the overall topological structures for the distribution of various responses in parameter spaces. Using the Floquet theory and parameter continuation method, the local bifurcation patterns of periodic solutions are obtained. Moreover, the global bifurcation mechanisms for the crises of chaos and metamorphoses of basin boundaries are examined by analysing the attractors and attraction basins, exploring the evolutions of invariant manifolds and constructing the basin cells. Meanwhile, additional nonlinear dynamic phenomena and characteristics closely related to the bifurcations are discussed including the resonant tongues, jump phenomena, amplitude–frequency responses, chaotic seas, transient chaos, chaotic saddles, and also their generation mechanisms are presented.     

Subharmonic resonance of a symmetric ball bearing–rotor system

The performance of a ball bearing–rotor system is often limited by the occurrence of subharmonic resonance with considerable vibration and noise. In order to comprehend the inherent mechanism and the feature of the subharmonic resonance, a symmetrical rotor system supported by ball bearings is studied with numerical analysis and experiment in this paper. A 6DOF rotordynamic model which includes the non-linearity of ball bearings, Hertzian contact forces and bearing internal clearance, and the bending vibration of rotor is presented and an experimental rig is offered for the research of the subharmonic resonance of the ball bearing–rotor system. The dynamic response is investigated with the aid of orbit and amplitude spectrum, and the non-linear system stability is analyzed using the Floquet theory. All of the predicted results coincide well with the experimental data to validate the proposed model. Numerical and experimental results show that the resonance frequency is provoked when the speed is in the vicinity of twice synchroresonance frequency, while the rotor system loses stability through a period-doubling bifurcation and a period-2 motion i.e. subharmonic resonance occurs. It is found that the occurrence of subharmonic resonance is due to the together influence of the non-linear factors, Hertzian contact forces and internal clearance of ball bearings. The effect of unbalance load on subharmonic resonance of the rotor system is minor, which is different from that of the sliding bearing–rotor system. However, the moment of couple has an impact influence on the subharmonic resonances of the ball bearing–rotor system. The numerical and experimental results indicate that the subharmonic resonance caused by ball bearings is a noticeable issue in the optimum design and failure diagnosis of a high-speed rotary machinery.     

Analytical study of nonlinear synchronous full annular rub motion of flexible rotor–stator system and its dynamic stability

The nonlinear synchronous full annular rub motion of a flexible rotor induced by the mass unbalance and the contact-rub force with rigid and flexible stator is studied analytically. The nonlinear property is due to the dry friction force between stator and rotor. The exact solutions of the synchronous full annular rub motion and its run speed regions are obtained. The stability of the synchronous full annular rub motion is discussed analytically. The stability criterion and the stability regions of the synchronous full annular rub motion are obtained. A simplified approximate criterion formula for dynamic stability is also derived under the conditions of large impact stiffness, small damping and small friction. The simplified criterion formula can be used conveniently in engineering and matches the real situations of industry.     

Bifurcation of periodic solutions and invariant tori for a four-dimensional system

In this paper, a four-dimensional system of autonomous ordinary differential equations depending on a small parameter is considered. Suppose that the unperturbed system is composed of two planar systems: one is a Hamiltonian system and another system has a focus. By using the Poincaré map and the integral manifold theory, sufficient conditions for the existence of periodic solutions and invariant tori of the four-dimensional system are obtained. An application of our results to a nonlinearly coupled Van der Pol–Duffing oscillator system is given.     

Simulation and hybrid fuzzy-PID control for positioning of a hydraulic system

Accuracy and precision of position control of hydraulic systems are key parameters for engineering applications in order to set more economical and quality systems. In this context, this paper presents modeling and position control of a hydraulic actuation system consisting of an asymmetric hydraulic cylinder driven by a four way, three position proportional valve. In this system model, the bulk modulus is considered as a variable. In addition, the Hybrid Fuzzy-PID Controller with Coupled Rules (HFPIDCR) is proposed for position control of the hydraulic system and its performance is tested by simulation studies. The novel aspect of this controller is to combine fuzzy logic and PID controllers in terms of a switching condition. Simulation results of the HFPIDCR based controller are compared with the results of classical PID, Fuzzy Logic Controller (FLC), and Hybrid Fuzzy-PID controller (HFPID). As a result, it is demonstrated that Hybrid Fuzzy PID Controller with Coupled Rules is more effective than other controllers.     

Dynamic model based nonlinear tracking control of a planar parallel manipulator

Based on the dynamic model, a novel nonlinear tracking controller is developed to overcome the nonlinear dynamics and friction of a planar parallel manipulator. The dynamic model is formulated in the active joint space, and the active joint friction is described with the Coulomb + viscous friction model. A nonlinear tracking controller is designed to eliminate the tracking error by using the power function. The nonlinear tracking controller is proven to guarantee asymptotic convergence to zero of both the tracking error and error rate with the Barbalat’s lemma. The trajectory tracking experiment of the proposed controller is implemented on an actual five-bar planar parallel manipulator both at the low-speed and high-speed motion. Moreover, the control performances of the proposed controller are compared with the results of the augmented PD (APD) controller.     

Analytical bifurcation analysis of a rotor supported by floating ring bearings

Like with other types of fluid bearings, rotors supported by floating ring bearings may become unstable with increasing speed of rotation due to self-excited vibrations. In order to study the effects of the nonlinear bearing forces, within this contribution a perfectly balanced symmetric rotor is considered which is supported by two identical floating ring bearings. Here, the bearing forces are modeled by applying the short bearing theory for both fluid films. A linear stability analysis about the static equilibrium position of the rotor shows that for a critical revolution speed the real part of an eigenvalue pair changes its sign. By means of a center manifold reduction it is shown that this destabilization of the steady state is due to a Hopf-bifurcation. Furthermore, the type of this bifurcation is determined as well as the existence and stability of limit-cycles. Notably it is found that depending on the parameters of the floating ring bearing subcritical as well as supercritical bifurcations may occur. Additionally, the analytical results obtained from the center manifold reduction are compared to numerical results by a continuation method. In conclusion, the influences of bearing design parameters on the stability and on the limit-cycles are discussed.     

A nonlinear visco-elastoplastic impact model and the coefficient of restitution

A visco-elastoplastic model for the impact between a compact body and a composite target is presented. The model is a combination of a nonlinear contact law that includes energy loss due to plastic deformation and a viscous element that accounts for energy losses due to wave propagation and/or damping. The governing nonlinear equations are solved numerically to obtain the response. A piecewise linear version of the model is also presented, which facilitates analytical solution. The model predictions are compared to those of the well-known and commonly used Hunt–Crossley model. The effects of the various impact parameters, such as impactor mass, velocity, plasticity, and damping, on the impact response and coefficient of restitution are investigated. The model appears to be suitable for a wide range of impact situations, with parameters that are well defined and easily calculated or measured. Furthermore, the resulting coefficient of restitution is shown to be a function of impact velocity and damping, as confirmed by published experimental data.     

Response analysis of a dual-disc rotor system with multi-unbalances–multi-fixed-point rubbing faults

Rotor unbalance and rub-impact are major concerns in rotating machinery. In order to study the dynamic characteristics of these machinery faults, a dual-disc rotor system capable of describing the mechanical vibration resulting from multi-unbalances and multi-fixed-point rub-impact faults is formulated using Euler beam element. The Lankarani–Nikravesh model is used to describe the nonlinear impact forces between discs and casing convex points, and the Coulomb model is applied to simulate the frictional characteristics. To predict the moment of rub-impact happening, a linear interpolation method is carried out in the numerical simulation. The coupling equations are numerically solved using a combination of the linear interpolation method and the Runge–Kutta method. Then, the dynamic behaviours of the rotor system are analysed by the bifurcation diagram, whirl orbit, Poincaré map and spectrum plot. The effects of rotating speed, phase difference of unbalances, convex point of casing and initial clearance on the responses are investigated in detail. The numerical results reveal that a variety of motion types are found, such as periodic, multi-periodic and quasi-periodic motions. Moreover, the energy transfer between the compressor disc and the turbine disc occurs in the multi-fixed-point rubbing faults. Compared with the parameters of the turbine disc, those of the compressor disc can affect the motion of the rotor system more significantly. That is, the responses exhibit simple 1T-periodic motion in the wide range of rotating speed under the conditions of sharp convex point and larger initial clearance. These forms of dynamic characteristics can be effectively used to diagnose the fixed-point rub-impact faults.     

Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems

The problem of reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly a reliable impulsive controller is designed by the impulsive control theory. Then, some sufficient conditions for reliable impulsive lag synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

Projective synchronization of hyperchaotic Lü system and Liu system

This work is concerned with projective synchronization of hyperchaotic Lü system and Liu system by add-order method. Different controllers are designed to projective-synchronize the two nonidentical chaotic systems, active control is used when parameters are known, while the adaptive control law and the parameter update rule are derived via adaptive control when parameters are uncertain. Moreover, the convergence rates of the scheme can be adjusted by changing the control coefficients. Finally, numerical simulations are also shown to verify the results.     

Nonlinear modeling and dynamic analysis of the rotor-bearing system

To overcome the shortcomings of extreme time-consuming in solving the Reynolds equation, two efficient calculation methods, based on the free boundary theory and variational principles for the unsteady nonlinear Reynolds equation in the condition of Reynolds boundary, are presented in the paper. By employing the two mentioned methods, the nonlinear dynamic forces as well as their Jacobians of the journal bearing can be calculated saving time but with the same accuracy. Of these two methods, the one is called a Ritz model which manipulates the cavitation region by simply introducing a parameter to match the free boundary condition and, as a result, a very simple approximate formulae of oil-film pressure is being obtained. The other one is a one-dimensional FEM method which reduces the two-dimensional variational inequality to the one-dimensional algebraic complementary equations, and then a direct method is being used to solve these complementary equations, without the need of iterations, and the free boundary condition can be automatically satisfied. Meanwhile, a new order reduction method is contributed to reduce the degrees of freedom of a complex rotor-bearing system. Thus the nonlinear behavior analysis of the rotor-bearing system can be studied time-sparingly. The results in the paper show the high efficiency of the two methods as well as the abundant nonlinear phenomenon of the system, compared with the results obtained by the usual numerical solution of the Reynolds equation.     

Heat transfer in a rotor–stator system with a radial inflow

The object of the present work is to produce a better understanding of the flow and heat transfer process occurring in a rotor-stator system, with a low aspect ratio and subjected to a superposed radial inflow. The theoretical approach presented in a previous paper (Debuchy et al., Eur. J. Mech. B-Fluids 17 (6) (1998) 791–810) in the framework of laminar, steady, axisymmetric flow is extended to heat transfer effects. The asymptotic model is simplified and new integral relations including temperature are indicated. The experiments, made in a rotor–stator system with a heated stationary disc, are in agreement with the features of the model in the explored range of the gap ratio, Ekman and Rossby numbers. The data include radial and circumferential mean velocity components, air temperature inside the cavity, temperature and temperature-velocity correlations, and also local Nusselt numbers measured on the stationary disc. The flow structure near the axis is found to be strongly affected by the presence of a superposed inflow, as already observed under isothermal conditions. By contrast, the mean temperature, as well as the correlations concerning velocity and temperature are smaller when a radial inflow is assigned.     

Time optimal motions of mechanical system with a prescribed trajectory

The problem of time minimization of a holonomic scleronomic mechanical system on a prescribed trajectory between two specified positions in configuration space is solved. The generalized force with restricted coordinates is taken as the controlling force. The application of the Green theorem (the well-known Miele method in flight mechanics) has shown that at every instant at least one control is at its boundary and possesses controlling functions with interruptions. It is assumed that at least one generalized coordinate exists that is monotonous during the interval of movement. An algorithm for numerical computation is presented for assessing the boundary of the admissible domain in the state space, thus, solving the problem of finding the optimal control as a function of time. Numerical integration is, therefore, carried out forward from the start point and backward from the end point by the use of the Runge-Kutta method. The mentioned procedure is illustrated in the example of time minimization for a manipulator which has its tip moving in a straight line. The application of the presented method simplifies solving of this type of problem compared to other methods, for instance, dynamic programming.     

Bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings

Auxiliary bearings are utilized in practical installations of magnetically suspended rotating machines with the main functions to provide support to the rotating machines during their non-operational period, and to protect the magnetic bearings and the rotating assembly from being damaged due to power loss during operation. The relatively small clearances of these bearings, which are typically half of that of the magnetic bearings, may at time cause contact between the rotor and these bearings to occur even during normal operation of the rotating machines. The work presented herein examines the bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings, which occurs during contact between the rotor and the auxiliary bearings. Numerical results revealed the occurrence of period-doubling bifurcation resulting in vibrations of period-2, -4, -8, -16 and -32, as well as quasi-periodic and chaotic vibrations. The results further showed that for a relatively small rotor imbalance magnitude, which is within the prescribed level of certain classes of practical rotating machinery, such nonlinear dynamical phenomena would not have been discovered had the auxiliary bearings forces been omitted in the model of the rotor-bearing system. As these bearings are essential elements in practical installations of magnetically supported rotating machines, failure to include them in the rotor-bearing model may result in incorrect prediction of the rotor’s vibration response.     

On uniqueness and continuous dependence of solutions in viscoelastic mixtures

This note deals with the isothermal linear theory of porous viscoelastic mixtures. Questions of uniqueness and continuous dependence for solutions of various classes of initial boundary value problems in mixtures consisting of two constituents: a porous elastic solid and a porous Kelvin–Voigt material are studied. The Lagrange identity and Logarithmic convexity methods are used to establish uniqueness and continuous dependence results, with no definiteness assumptions upon the internal energy.     

Nonlinear normal modes of a self-excited system driven by parametric and external excitations

Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical simulations and an essential difference between linear and nonlinear modes is pointed out.