Considering full spectrum analysis of the geared rotor system, information on rotor vibrations and AMB current, estimation of system parameters, i.e. unbalance force vector or force vector due to torque K Stiffness matrix of geared rotor M Mass matrix of geared rotor.
Introduction and Literature Review
Introduction
- Gear Dynamic Models
- Lumped Parameter Models
- Finite Element Models
- Evaluation of Gear Parameters
- Numerical Simulation Results
- Time Domain Solution
- Frequency Domain Solution
The nonlinear elasticity term resulting from the reverse reaction was expressed by a descriptive function and. multi-harmonic responses of multi-degree-of-freedom nonlinear systems were obtained. In the next section, some of the techniques for predicting and diagnosing gear faults are presented. However, it has been recognized that TE can be measured by phase demodulation of the shaft encoder signals hard-coupled to each of the gears in the mesh, i.e., low-speed and low-load GTE, low-speed and higher-load STE high, and DTE at higher speed and higher load.
Since the dynamic transmission error is excited by the static transmission error, it is useful to reduce the noise by addressing the static transmission error. The alternating component of static transmission error is reduced by specifying tooth modifications that compensate for stiffness variation. Another method by Gahler et al. natural frequencies and damping, using a predefined model structure. Unfortunately, there are also few limitations with piezoelectric actuator technology, i.e., structural vibrations intensify at frequencies close to the resonant frequency, so the positioning speed of piezo-based systems is limited by the lowest frequency of structural vibration, causing significant positioning errors.
In an effort to estimate these parameters, an identification algorithm has been developed. The desired goal is achieved by undertaking the following objectives: The area between the stator poles is 'Aa' the cross-sectional area of the flux in air which is equal to the area of the iron core and the holes have 'N' number of coil turns. The structural matrices, i.e., the mass, stiffness and damping matrices, and the state (displacement vector) and force vectors, are divided into sub-vectors and sub-matrices related to the main DOFs to be maintained, and slave DOFs, which must be eliminated.
The frequency domain EOMs, Eq. 3.20), can be broken down as follows, where subscript m and s represent the masters and slaves, respectively, and L = p, g,e, i.e. the three different excitation frequencies. In inertial frame of reference, the energy equation for the gear-rotor AMB system with gyroscopic effect, i.e. the kinetic energy 'T', potential energy 'U' and Rayleigh's dissipation function 'D' and the force vector 'P' in considering planer motion can be expressed as.
Representation of Equation of Motion in Complex Form
The frequency domain EOMs (4.18) can be divided as the subscript m and s representing the master and slave DOFs, respectively, and H = p, g, i.e. the three different excitation frequencies. The identifiable parameters are then solved using the least-squares regression method, which is given by the expression 4.36) are suitably recombined to determine the final sought 20 identifiable system parameters (i.e. the gear mesh stiffness, gear mesh damping. The parameter estimation is done by using multi-harmonic amplitude and phase information of the full spectrum in the identification problem.
Here, the rotor-AMB physical parameters listed in Tables 4.1 and 4.2 are assumed to be known values for the identification problem. As in the previous examples, Gaussian noise is added to the time-domain signals to simulate the actual measurement conditions, and a random solution is obtained from the numerical simulation to check the robustness of the identification algorithm. Therefore, the sensitivity of the identification algorithm is checked by adding bias errors in the density, length, and Young's modulus of the system.
Results of the estimation based on two speed ranges taken for the identification problem with clean and corrupted signals are summarized in Tables 4.3 and 4.4.
Introduction
Design of Geared Rotor (AMB) System
The centerline of the shaft is defined along the Z axis, and the X and Y axes are located at the centers of the gears. The gear is driven by the motor, and the output gear is loaded with the braking load. Due to the flexible coupling, the effect of transverse vibration is almost negligible at the motor and brake end, both of which are mounted on a rigid pedestal/foundation.
Thus, the dynamic transmission error due to eccentricities present in the geared rotor system is discussed by considering the gear runouts (i.e., leakages leading to mass imbalance in the shafts and elastic, damping forces at the opening point) of together with the AMBs.
Development of Laboratory Test Rig…
- Components of Test Rig
- Instrumentation of Test Rig
As known, bellows couplings have high torsional stiffness, which is good for torque transmissions with angle maintenance, they can mitigate all forms of misalignment, so they are used to connect different rotor components, i.e. magnetic brake (load 0~1 Nm) and to the rotary torque converter. Two radial magnetic bearing drives are located on either side of the gear shafts next to the input and output gears and near the ball bearings. The radial magnetic bearing actuator and core are made of laminated silicon steel plates, i.e. of CRGO (Cold Rolled Grain Oriented) steel, built as a stack of perforated round laminated plates so that core losses and eddy current losses are low.
It is then connected to the rotating shafts via bellows couplings using standard splined shafts. Eddy current sensors are placed between the pinion and the AMBs and near the pinions, since the vibrations that occur due to the misalignment of the pinions are transmitted to the flexible shaft and then to the entire system. ControlDesk offered a variety of virtual instruments to build and configure virtual dashboards according to needs and a platform to run a seamless experiment.
The deviation in vibration responses due to coupled effect is not much visible due to the low torque/load variation and is also influenced by the geometric parameters of the rotor.
Experimental Analysis
Therefore, the controller gains mentioned in this work can be used up to 29 Hz of operating speed. The rotor response is checked using digitally tuned KP, KD and KI values (refer to Table 5.1). The identification of gear rotor parameters from the measured signals of the laboratory test equipment is explained for different speed ranges in the present section. A mathematical model is developed for the numerical simulation of the proposed geared rotor AMB system and experimentally verified by a laboratory test rig.
A new identification algorithm was developed using regression equations from the mathematical model, which estimated the system error parameters, i.e. gear mesh stiffness, gear mesh damping, runout of input gear and phase, runout of output gear and phase, AMB displacement stiffness constant, AMB current stiffness constant, mean transmission error and phase, variable transmission error corresponding to different harmonics and respective phase angles. The numerical model of the proposed geared rotor AMB system is experimentally verified by a laboratory test rig. The numerical results of the proposed model of geared rotor AMB system were compared with the results obtained from the experimental test rig.
More advanced modeling of the gear-rotor AMB system would be important in simulating rotor dynamic behavior.
Conclusions and Future Scope
Summary of the Present Work
The estimated parameters were tested with different random noises and modeling errors to check the robustness of the developed algorithm. Next, the gyroscopic effect on the transverse vibrations of the geared rotor AMB system is analyzed if the gears can be moved from the middle of the rotor span. Relative to torsional vibration, the lateral vibration of the gear pair is more easily affected by the gyroscopic effect.
The experimental data helped understand the system dynamics and test the efficiency of the developed algorithm. Measurement noise and modeling errors were added to check the robustness of the algorithm, yielding favorable results. Based on the proposed model, a test rig was fabricated to check the feasibility and practical implementation of the proposed model that can control gear vibration and noise.
Thus, it can be concluded that AMBs can effectively prevent damage to gearbox components with active vibration control.
Main Contribution of the Research Work
Limitations of the Thesis Work
Since a radial magnetic bearing acts in radial direction, there was not much effect on torsional vibration of the system seen numerically. Although good consistency was found between the simulated and experimental responses showing the operational feasibility of the proposed method. An improvement can be made by deploying a more robust active control scheme involving an adaptive controller, which can be more effective in reducing the coupled vibrations regardless of the speed and load variation.
The identification algorithms are model-based and developed using a linear mathematical model with constant mesh stiffness and damping. Due to inadequate sensors, the torsional vibrations could not be measured and the effect of AMBs on the torsional vibrations of the gears could not be understood experimentally. Because of this, the numerical results obtained from the identification algorithm developed in the case of coupled torsional-lateral vibrations could not be experimentally validated.
In reality, the measurement locations on the shaft may not be accessible due to other mountings over the shaft.
Recommendations for Future Work
Under impact load conditions, an adaptive control method for dynamic load suppression based on torque compensation can be proposed. Key design parameters that affect the dynamic performance of the system, such as axle stiffness, inertia, etc., can be identified for further improvements in vibration suppression and shock resistance. Assembly or assembly errors, such as if the axes are not perfectly parallel, distortions arise and the contact areas between the mating teeth can be significantly altered.
Thus, the present analysis can be extended by considering the effect of different types of linkage misalignment on gear transmission error. Appendix A1: Conversion of Complex Regression Equations into Real Regression Equations for 4 DOF Geared Rotor-AMB System. For i=0, regression equations are after separating the real and imaginary parts. excitation frequency is L =p then equations are after the separation of the real and imaginary parts. excitation frequency is L =g then equations after the separation of the. real and imaginary parts are. excitation frequency is L = e then equations to. separation of the real and imaginary parts is.
Appendix A2: Conversion of complex regression equations to real regression equations for the 8 DOF geared rotor-AMB system.
