Multibody dynamics of healthy and faulty rolling element bearings were modeled using vector bond graphs. A 33 degree of freedom (DOF) model was constructed for a bearing with nine balls and two rings (11 elements). The developed model can be extended to a rolling element bearing with n elements and (3×n) DOF in planar and (6×n) DOF in three dimensional motions. The model incorporates the gyroscopic and centrifugal effects, contact elastic deflections and forces, contact slip, contact separations, and localized faults. Dents and pits on inner race and outer race and balls were modeled through surface profile changes. Bearing load zones under various radial loads and clearances were simulated. The effects of type, size, and shape of faults on the vibration response in rolling element bearings and dynamics of contacts in the presence of localized faults were studied. Experiments with healthy and faulty bearings were conducted to validate the model. The proposed model clearly mimics healthy and faulty rolling element bearings.
Sensors in different types and configurations provide information on the dynamics of a system. For a specific task, the question is whether measurements have enough information or whether the sensor configuration can be changed to improve the performance or to reduce costs. Observability analysis may answer the questions. This paper presents a general algorithm of nonlinear observability analysis with application to model-based diagnostics and sensor selection in three-phase induction motors. A bond graph model of the motor is developed and verified with experiments. A nonlinear observability matrix based on Lie derivatives is obtained from state equations. An observability index based on the singular value decomposition of the observability matrix is obtained. Singular values and singular vectors are used to identify the most and least observable configurations of sensors and parameters. A complex step derivative technique is used in the calculation of Jacobians to improve the computational performance of the observability analysis. The proposed algorithm of observability analysis can be applied to any nonlinear system to select the best configuration of sensors for applications of model-based diagnostics, observer-based controller, or to determine the level of sensor redundancy. Observability analysis on induction motors provides various sensor configurations with corresponding observability indices. Results show the redundancy levels for different sensors, and provide a sensor selection guideline for model-based diagnostics, and for observer-based controllers. The results can also be used for sensor fault detection and to improve the reliability of the system by increasing the redundancy level in measurements.
The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton-Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used T. Sinokrot · M. Nakhaeinejad · A. A. Shabana ( ) to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.
Lumped parameter models can not sufficiently describe dynamics of many distributed systems, such as a continuous rotating shaft with bending. The finite element method is employed to embed distributed dynamics of rotating shafts with bending into constitutive laws of bond graphs resulting a finite element bond graph model. Defining the nodal displacements, the finite element mass and stiffness matrices are obtained from energy equations, and represented as constitutive relations for multi-port I and C's. Centrifugal forces are included through multi-port C, and the Coriolis forces are embedded via gyrators. Finally, the FE bond graph is reconstructed in a simple vector bond graph form, which can be integrated into any bond graph model of rotating shafts.
This study illustrates a data driven system identification method for loudspeaker model estimation using the knowledge of the underlying physics of loudspeakers. In this study, diaphragm displacement is analyzed to estimate the model structure and parameters based on impulse response equivalent sampling and autoregressive moving average model. The estimated loudspeaker models are compared in the frequency response function plot. It is shown that the autoregressive moving average (ARMA) based loudspeaker models are comparable to the model estimated by the conventional method based on electrical impedance. Also ARMA modeling strategies with and without knowledge of the physics-based model are compared. Some issues related to ARMA modeling are addressed.
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