Rajiv Tiwari

Identification of dynamic bearing parameters: A review

In this paper, we present a review of the experimental identification of dynamic parameters of bearings in a rotating machine. Major emphasis is given to vibration-based identification methods and the review encompasses descriptions of experimental measurement techniques, mathematical modeling, parameter extraction algorithms and uncertainty in the estimates applied to a variety of bearings. The parameter extraction algorithms include the descriptions of governing equations of the rotor-bearing system and identification methods in both time and frequency domains. The identification techniques have been classified based on methods used to excite the system. The review includes a variety of bearings and similar components, which play an active link between the rotating and stationary parts of a machine. Based on the state of the art in bearing identification, conclusions are made and future directions are suggested.

Rolling element bearing design through genetic algorithms

The design of rolling element bearings has been a challenging task in the field of mechanical engineering. While most of the real aspects of the design are never disclosed by bearing manufacturers, the common engineer is left with no other alternative than to refer to standard tables and charts containing the bearing performance characteristics. This paper presents a more viable method to solve this problem using genetic algorithms (GAs). Since the algorithm is basically a guided random search, it weakens the chances of getting trapped in local maxima or minima. The method used has yielded improved performance parameters than those catalogued in standard tables. *Indraneel Chakraborty is currently with Massachusetts Institute of Technology and can be reached at indranil@mit.edu †Vinay Kumar is currently with Mindtree Consulting, India and can be reached at vinayk@mindtree.com

An Optimum Design of Crowned Cylindrical Roller Bearings Using Genetic Algorithms

The long fatigue life is the one of the most important criterion for the design of rolling bearings, however, due to complex and diverse internal geometries, each type of rolling bearings require a different design formulation. To increase the life of cylindrical roller bearings, the profile (or the crowning) of the roller plays an important role. A fiat profile of the rolling element results in the edge stress concentrations at roller ends. A circular crowning of roller eliminates the edge stress concentration at the lower and moderate loads only; however, it develops edge stress concentrations at heavy loads. The logarithmic profile of the roller results in no edge stress concentration at the low, medium, and heavy loads; distribution of contact stresses is also nearly uniform along the length of the roller. A design methodology for the optimum design of cylindrical roller bearings with the logarithmic profile has been outlined. A nonlinear constrained optimization problem has been formulated for the design of cylindrical roller bearings with logarithmic profiles and is optimized by using real-coded genetic algorithms. The change in roller profile has not been accounted for explicitly in the standard definition of the dynamic capacity; hence, for the present case directly the Lundberg―Palmgren life equation has been chosen as an objective function. Design variables include four bearing geometrical parameters and the two logarithmic profile generating parameters are considered. In addition to these, another five design constraint constants are also included, which indirectly affect the fatigue life of cylindrical roller bearings. The five design constraint constants have been given bounds based on the parametric studies through initial optimization runs. The effective length of the roller is taken corresponding to the standard roller diameter, which has standard discrete dimensions. Constraint violation study has been performed to have an assessment of the effectiveness of each of the constraints. A convergence study has been carried out to ensure the global optimum point in the design. A sensitivity analysis of various geometric design parameters has been performed using the Monte Carlo simulation technique, in order to see changes in the fatigue life of the bearing. Illustrations show that the multiplier of the logarithmic profile deviation parameter has more effect on the fatigue life as compared with other geometric parameters.

Development of a Technique to Locate and Quantify a Crack in a Beam Based on Modal Parameters

A model-based algorithm has been developed, in order to locate and quantify the size of a crack, based on free vibration measurements of a cracked beam. Measured natural frequencies (at least two) and corresponding mode shapes of the cracked beam are used in the identification algorithm. The Euler-Bernoulli beam theory is used to model the beam. The crack of the beam is modeled through standard five crack flexibility coefficients, by considering bending effects only. Damping is assumed to be Rayleighs damping. The finite element method is used in the simulation of the cracked beam. The present algorithm is iterative in nature and the iterations are carried out until the estimated and assumed crack depth ratios (as well as its location) become close up to the desired accuracy. The applicability of the algorithm has been tested through numerical examples and is found to be adequate even in the presence of the measurement noise in modes shapes and measurement errors in natural frequencies.

Development of a Novel Algorithm for a Crack Detection, Localization, and Sizing in a Beam Based on Forced Response Measurements

The present work aims at the development of a method for the crack detection, localization and sizing in a beam based on the transverse force and response signals. The Timoshenko beam theory is applied for transverse vibrations of the beam model. The finite element method is used for the cracked beam forced vibration analysis. An open transverse surface crack is considered for the crack model, which contains standard five flexibility coefficients. The effect of the proportionate damping is also included. A harmonic force of known amplitude with sine-sweep frequency is used to dynamically excite the beam, up to few flexible modes, which could be provided with the help of an exciter. In practice, linear degrees of freedom (DOFs) can be measured quite accurately; however, rotational DOFs are difficult to measure accurately. All rotational DOFs, except at crack element, are eliminated by a dynamic condensation scheme; for elimination of rotational DOFs at the crack element, a new condensation scheme is implemented. The algorithm is iterative in nature and starts with a presumption that a crack is present in the beam. For an assumed crack location, flexibility coefficients are estimated with the help of forced responses. The Tikhonov regularization technique is applied in the estimation of bounded crack flexibility coefficients. These crack flexibility coefficients are used to obtain the crack size by minimizing an objective function. With the help of the estimated crack size and measured natural frequency, the crack location is updated. The procedure iterates till the crack size and location get stabilized up to the desired level of accuracy. The algorithm has a potential to detect no crack condition also. The crack flexibility and damping coefficients are estimated as a by-product. Numerical examples, with the simply supported and cantilevered beams, are given to justify the applicability and versatility of the algorithm in practice. With the numerically simulated forced responses, which have the noise contamination and the error in the natural frequency measurements, the estimated crack parameters (i.e., the crack location and size) are in good agreement.

Optimum Design and Analysis of Thrust Magnetic Bearings Using Multi Objective Genetic Algorithms

An optimum design of thrust magnetic bearings has been carried out using multi-objective genetic algorithms (MOGAs). The power-loss and the weight have been selected as the minimization type objective functions for the optimum design. The maximum space available, the maximum current density that can be supplied in the coil, the maximum magnetic flux density that is allowed in the stator-iron (i.e., the magnetic flux density at the saturation), and the load required to be supported have been chosen as constraints. The inner and outer radii of the coil, and the height of the coil have been proposed as design variables. Apart from the comparison of performance parameters in the form of figures and tables, designs are also compared through line diagrams. Post-processing has been done on the final optimized population by studying the variation of different parameters with respect to objective functions. The saturation of magnetic flux density and the saturation of coil current density are observed to be the salient points, where the major changes occur in the behavior of different design parameters. A criterion for the choice of one of the best design based on the minimum normalized distance near the utopia point is used. A sensitivity analysis has been done on a chosen design by giving small perturbations on the design variables. It is observed that the effect of the outer radius of the coil on the objective functions is nearly double as compared to other two design variables.

Development of an Optimum Design Methodology of Cylindrical Roller Bearings Using Genetic Algorithms

In the design of cylindrical roller bearings, the long life is the one of the most important criterion. Bearing standards define the design space available to a designer for deciding internal geometries of the bearing. Based on the rated speed and loading conditions, the design has to satisfy constraints of geometry and strength. An optimum design methodology is needed to achieve this objective. Since the fatigue life is directly proportional to the basic dynamic capacity, for the present case it has been chosen as objective function. It has been optimized by using a constrained non-linear formulation with real-coded genetic algorithms. Design variables include four geometrical parameters: the bearing pitch diameter, the diameter of the roller, the effective length of the roller, and the number of rollers. In addition to these another five design constraint constants are included, which indirectly affect the basic dynamic capacity of cylindrical roller bearings. The five design constraint constants have been given bounds based on the parametric studies through initial optimization runs. The effective length of the roller is taken corresponding to the standard roller diameter, which has standard discrete dimensions. There is good agreement between the optimized and standard bearings in respect to the basic dynamic capacity of the bearing. A constraints violation study has been performed to assess the effectiveness of each of the constraints. A convergence study has been carried out to ensure the global optimum point in the design. A sensitivity analysis of various geometric design parameters has been performed to see changes in the basic dynamic capacity of the bearing, and results show that no geometric parameters have adverse affects.

An Optimal Design Methodology of Tapered Roller Bearings Using Genetic Algorithms

In the design of tapered roller bearings, long life is the one of the most important criterion. The design of bearings has to satisfy constraints of geometry and strength, while operating at its rated speed. An optimal design methodology is needed to achieve this objective (i.e., the maximization of the fatigue life). The fatigue life is directly proportional to the dynamic capacity; hence, for the present case, the latter has been chosen as the objective function. It has been optimized by using a constrained nonlinear formulation with real-coded genetic algorithms. Design variables for the bearing include four geometrical parameters: the bearing pitch diameter, the diameter of the roller, the effective length of the roller, and the number of rollers. These directly affect the dynamic capacity of tapered roller bearings. In addition to these, another five design constraint constants are included, which indirectly affect the basic dynamic capacity of tapered roller bearings. The five design constraint constants have been given bounds based on the parametric studies through initial optimization runs. There is good agreement between the optimized and standard bearings in respect to the basic dynamic capacity. A convergence study has been carried out to ensure the global optimum point in the design. A sensitivity analysis of various design parameters, using the Monte Carlo simulation method, has been performed to see changes in the dynamic capacity. Illustrations show that none of the geometric design parameters have adverse affect on the dynamic capacity.

Optimization of support vector machine based multi-fault classification with evolutionary algorithms from time domain vibration data of gears

In the present work, a multi-fault classification of gears has been attempted by the support vector machine learning technique using the vibration data in time domain. A proper utilization of the support vector machine is based on the selection of support vector machine parameters. The main focus of this article is to examine the performance of the multiclass ability of support vector machine techniques by optimizing its parameters using the grid-search method, genetic algorithm and artificial bee colony algorithm. Four fault conditions were considered. A group of statistical features were extracted from time domain data. The prediction of fault classification is attempted at the same angular speed as the measured data as well as innovatively at the intermediate and extrapolated angular speed conditions. This is due to the fact that it is not feasible to have measurement of vibration data at all continuous speeds of interest. The classification ability is noted and it shows an excellent prediction performance.

Model-Based Switching-Crack Identification in a Jeffcott Rotor With an Offset Disk Integrated With an Active Magnetic Bearing

Vibration characteristics of a cracked Jeffcott rotor with an offset disk under the action of an active magnetic bearing (AMB), implemented to improve the radial positioning of the rotor, has been studied. Presence of the AMB suppresses the vibration induced due to the crack and unbalance; identification of the crack could be made by utilizing the vibration signal in conjunction with the controller current of the AMB. A four degrees-of-freedom (DOF) cracked rotor is modeled considering the gyroscopic effect due to the offset disk and a switching crack excitation function (SCEF) to introduce the breathing of crack. The dynamic condensation is applied to eliminate rotational displacements, which would pose practical difficulty in accurate measurement, from the system equations of motion (EOM) to develop an identification algorithm. Frequency domain transformation of the EOM is made with the help of the full spectrum analysis. An algorithm developed with the purpose of crack identification in the form of additive crack stiffness estimates the viscous damping, disk unbalance, and AMB constants as well. The algorithm has been tested for the measurement noise (in the displacement and the current) and bias errors in system parameters, and found to be robust.