Jawaid I. Inayat-Hussain

On the bifurcations of a rigid rotor response in squeeze-film dampers

Abstract The effectiveness of squeeze-film dampers in controlling vibrations in rotating machinery may be limited by the nonlinear interactions between large rotor imbalance forces with fluid-film forces induced by dampers operating in cavitated conditions. From a practical point of view, the occurrence of nonsynchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The bifurcations in the response of a rigid rotor supported by cavitated squeeze-film dampers resulting from such interactions are presented in this paper. The effects of design and operating parameters, namely the bearing parameter (B), gravity parameter (W), spring parameter (S) and unbalance parameter (U), on the bifurcations of the rotor response are investigated. Spring parameter (S) values between 0 and 1 are considered. A spring parameter value of S=0 represents the special case of dampers without centering springs. With the exception of the case S=1, jump phenomena appeared to be a common bifurcation that occurred at certain combinations of B, W and U irrespective of the value of S. Period-doubling and secondary Hopf bifurcations which occurred for low values of S (⩽0.3) were not observed for the higher values S⩾0.5. For very low stiffness values (S

Control of dual acting magnetic bearing actuator system using fuzzy logic

In this paper, a fuzzy logic scheme is developed to improve the performance of a magnetic bearing control system. The developed fuzzy logic strategy is utilized in the non-linear control of a dual-acting magnetic bearing actuator system. The mathematical model of a dual acting magnetic bearing axis has been developed. The non-linearity in this system is due to the relationship between the forces generated in the electromagnetic actuator and the coil current and the air gap between the rotor and the stator. Non-linearities in magnetic bearing systems limit the control effectiveness and the region of stable performance. The model is used for dynamic simulation purposes and it is not directly used in the fuzzy controller design. Active magnetic bearing with adaptive fuzzy control showed a good improvement in step response compared to a linear PID or a steady state controller. With the presented fuzzy logic based strategy, nonlinear controllers for nonlinear applications can be designed with little effort.

Stability and bifurcation of a rigid rotor in cavitated squeeze-film dampers without centering springs

Abstract The continuation method is employed to investigate the non-linear response of a rigid rotor supported by squeeze-film dampers without centering springs. The Floquet transition matrix method, which was used to examine the stability of the periodic response, indicated that the rotor may lose stability via period-doubling and saddle-node bifurcations at certain parameter values. The continuation method, which allows for switching from one solution branch to another, enabled the solution branches of period-two, four, eight and sixteen to be mapped out. By simultaneously varying the unbalance and bearing parameters, the combination of these two parameters that avoids the occurrence of non-synchronous vibration or jump phenomena in this system was determined.

Fuzzy bang-bang relay control of a single-axis active magnetic bearing system

Abstract Active magnetic bearings (AMB) are presently being utilized in various classes of rotating machinery. Although the rotor-AMB systems are open loop unstable, they are easily stabilized using feedback control schemes of which the PID controller is the most commonly used. The PID controller is however only effective at the vicinity of the rotor’s equilibrium position where the dynamics of the rotor-AMB system is linearized. Significant deviation of the rotor’s motion from this equilibrium position may occur due to large imbalance forces. In this situation, the nonlinearity in AMBs, which arises from the relationship between the electromagnetic force, coil current and air gap, may render the PID controller ineffective. For the control of nonlinear systems, artificial intelligence techniques such as fuzzy and hybrid techniques are effective. In this paper, a new fuzzy controller is proposed for the control of a single-axis AMB system. This controller is based on the bang–bang scheme, which is an old but effective technique to control nonlinear systems in optimal time. The performance of the proposed integrated fuzzy bang–bang relay controller (FBBRC) was found to be superior to that of the optimized PD controller and the conventional fuzzy logic controller. Comparison of the FBBRC with the fuzzy logic controller cascaded with a hard limiter (FBBC) relay revealed almost equal performance. High frequency chattering was however observed in the steady-state response of the FBBC. Such chattering is known to cause instability and distortion in the amplifiers that are used to supply current to the magnetic bearing actuators.

Chaos in the unbalance response of a rigid rotor in cavitated squeeze-film dampers without centering springs

Abstract Numerical investigation on the unbalance response of a rigid rotor supported by squeeze-film dampers without centering springs revealed some complex bifurcation features that have not been previously reported in the literature. With the variation of the unbalance parameter (U), the period-1 solution was found to undergo a sequence of period-doubling bifurcations that eventually resulted in chaotic motion. The existence of a period-3 solution, which formed a closed bifurcation curve consisting of a pair of saddle nodes, was for the first time observed in such a system. The chaotic attractor arising from the period-doubling cascade of the period-1 solution, which was observed to co-exist with the period-3 attractor in a narrow range of U values, was eventually annihilated in a collision with the unstable period-3 orbit in a boundary crisis. Similar to the bifurcations of the period-1 solution, the period-3 solution was also found to bifurcate into solutions of period-6 and period-12, which eventually led to chaotic motion. A chaotic attractor was also observed to co-exist with a period-4 orbit. The period-4 orbit was found to undergo a sequence of reverse period-doubling bifurcations resulting in a large amplitude period-1 orbit. The occurrence of non-synchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The magnitude of rotor unbalance where non-synchronous and chaotic motion were observed in this study, although higher than the permissible unbalance level for rigid rotating machinery, may nevertheless occur with in-service erosion of the rotor or in the event of a partial or an entire blade failure.

Fuzzy control scheme for dual-acting magnetic bearing actuator system

Active magnetic bearings are increasingly being utilized in rotating machinery applications as an alternative to the conventional rolling-clement and fluid-film bearing types. Magnetic bearing is an open-loop unstable system, and in most practical applications, a PID controller is utilized to ensure stable operation of the rotating machinery. The PID controller, however, becomes ineffective when the machine operates in highly nonlinear regimes. This paper develops a fuzzy logic control scheme to improve the performance of a dual-acting magnetic bearing actuator system operating in nonlinear regimes. The nonlinearity in this system is due to the relationship between the forces generated in the electromagnetic actuator and the coil current and the air gap between the rotor and the stator. The dynamic response of the magnetic bearing actuator system based on the fuzzy logic scheme proposed in this work was found to be much better as compared to the response of the system based on the conventional PD controller. The fuzzy logic control scheme presented in this work may be used for the nonlinear control of systems that operates in nonlinear regimes.

Fuzzy logic based control of rotor motion in active magnetic bearings

Active magnetic bearings (AMB) are increasingly being used as an alternative to rolling element and fluid-film bearings in rotating machinery application. Stable operation of AMB can only be achieved via feedback control of which the most widely used controllers are of the linear PID type. Under extreme conditions, however, the dynamic response of the rotor in AMB becomes highly nonlinear. As a result, the linear controllers are no longer capable of suppressing or controlling the bifurcations of the rotor response. In order to suppress the non-linearity in AMB, a nonlinear control strategy is required. One of such strategies is the use of fuzzy logic based control approach. This paper aims to investigate and analyze the dynamical response of rotors in active magnetic bearings in AMB and presents the development and implementation of a fuzzy logic control strategy for suppressing the non-synchronous response in AMB. This control strategy is expected to stabilize the rotor-bearing system or to delay its onset of instability. Through modeling and simulation of the non-linear rotor response, it is found that for certain operating parameters, the rotor in AMB exhibits non-synchronous response (quasi-periodic or chaotic). However, the use of fuzzy logic control has been able to eliminate the undesirable non-synchronous response and improve the rotor stability performance

Bifurcations in the response of a Jeffcott rotor with rotor-to-stator rub

In the design of rotating machines the clearances between the rotating and stationary parts are often made smaller in order to increase their efficiency. Although the smaller clearance provides an advantage from the performance point of view, it is however not the case when viewed from the dynamics perspective as the possibility of rotor-to-stator rub to occur increases significantly. Rubbing in rotating machines is a highly nonlinear phenomenon and therefore it is very important to understand its dynamics. In this work the bifurcations in the response of a Jeffcott rotor subjected to rotor-to-stator rub is numerically investigated. In particular the influence of the rotor imbalance, contact stiffness and friction coefficient on the response of the rotor is examined for a range of operating speed. For the range of parameters investigated in this work, numerical results revealed the occurrence of sub-synchronous, quasi-periodic and chaotic vibrations in the response of the rotor. The range of operating speed where sub-synchronous and non-synchronous vibrations were found in the rotor’s response was seen to increase as a result of increasing the magnitudes of the rotor imbalance and contact stiffness. The response of the rotor was generally found to be invariant to the variation of the friction coefficient magnitude except for the combination of the largest values of rotor eccentricity and contact stiffness, whereby the range of sub-synchronous and non-synchronous vibrations was observed to increase as the magnitude of the friction coefficient became larger. For these particular magnitudes of rotor eccentricity and contact stiffness, the rotor response exhibited hysteresis phenomena resulting in the presence of multiple attractors for the same set of parameter values.Copyright

A comparative study of policies in Q-learning for foraging tasks

Q-learning is a machine learning technique that learns what to do and how to map states to actions to maximize rewards. Q-learning has been applied to various tasks such as foraging, soccer and prey-pursuing robots. In this paper, a simple foraging task has been considered to study the influences of the policies reported in the open literatures. A mobile robot is used to search and retrieve pucks back to a home location. The goal of this study is to identify an efficient policy for q-learning which maximizes the number of pucks collected and minimizes the number of collisions in the environment. Policies namely greedy, epsilon-greedy, Boltzmann distribution and random search are used to study their performances in the foraging task and the results are presented.

Multiple attractors in the response of a flexible rotor in active magnetic bearings with geometric coupling

Numerical results on the response of a flexible rotor supported by nonlinear active magnetic bearings are presented. Nonlinearity arising from the magnetic actuator forces that are nonlinear functions of the coil current and the air gap between the rotor and the stator, and from the geometric coupling of the magnetic actuators is incorporated into the mathematical model of the flexible rotor - active magnetic bearing system. For relatively large values of the geometric coupling parameter, the response of the rotor with the variation of the speed parameter within the range 0.05 ⋚ Ω ⋚ 5.0 displayed a rich variety of nonlinear dynamical phenomena including sub-synchronous vibrations of periods -2, -3, -6, -9, and -17, quasi-periodicity and chaos. Numerical results also reveal the occurrence of bi-stable operation within certain ranges of the speed parameter where multiple attractors may co-exist at the same speed parameter value depending on the operating speed of the rotor.