This work focuses on two main facets of rotor-AMB systems, namely the analysis of the system (i.e. the direct problem) and the error parameter identification (i.e. the inverse problem). The identification algorithm developed on the basis of the reduced EOM is used to estimate the crack, unbalance, viscous damping and AMB stiffness parameters.
Importance of Study
For greater reliability and better performance of rotating machines, in addition to monitoring the health of the rotors, the search continues for the best combination of speed, power, compactness and service life. However, very few articles are available on the identification of cracks with the active control of rotors.
Condition Monitoring of Rotating Machinery
Isermann and Balle (1997) presented a brief overview of the historical development of model-based fault detection, as well as suggestions for standardizing the fault detection terminology. For a good expert system, a detailed design of the system must be taken care of.
Crack Modeling
Mathematical models are particularly needed for the identification of the crack depth and location using inverse methods. Crack models are mostly based on the increase in flexibility at the cross section due to the appearance of the crack.
Crack Identification in Rotors
- Signal Based Approaches
- Model Based Approaches
- Condensation Schemes
Due to the complexity in crack identification, trend analysis for increasing vibration response amplitudes 1×, 2× and 3× is suggested. As the excitation frequency increases, the neglected inertia term in the reduction process becomes more important.
Experimental Studies on Rotor Cracks
We considered the FEM and calculated the bending moments at the nodes of the crack element. The 1× and 3× components were considered for identification only in the case of 47% crack depth.
AMB and their Applications in Rotor Crack Detection
Gap (or displacement) sensors read the instantaneous position of the rotor shaft and pass the signal to the controller. In their work, the rotor was supported on conventional ball bearings with the AMB located in the middle of the rotor space.
Outcome of the Literature Review
Modeling errors and determination of the damage mechanism are the areas that have a study space in model-based condition monitoring. The full spectrum was mainly discussed for the signal-based identification of the crack and other faults.
Objectives and Scope of the Present Study
The effect on the estimates of different levels of measurement noise in responses will also be tested. For the crack model containing rotational coordinates, a dynamic reduction scheme will be used to eliminate rotational displacements from the equations of motion.
Organization of the Thesis
To account for such modeling deficiencies, the effect of modeling error (or bias error) on estimates will be quantified. In this chapter, in addition to the cracks and stiffness of the AMB, the dynamic parameters of the bearing deposit are also defined.
Introduction
The combination of unbalance, crack and AMB forcing acting together produces excitations of several harmonics of the rotor spin frequency. The steady-state response of the system to these excitations can essentially be studied using time histories and frequency responses.
Model of a 2-DOF Cracked Rotor with AMB Support
- Equations of Motion of Cracked Jeffcott Rotor
- Linearization of System Equations of Motion
- Modeling of the Crack
- Transformation Matrix and Equations of Motion in Inertial Coordinates
- Restitution due to AMB Support
Force vectors on the RHS of the above-mentioned EOMs are due to the crack and the unbalance, respectively. The negative sign of the additive crack stiffness implies a reduction in stiffness due to the appearance of the crack.
Analyses of Forced Responses
- Estimation of Displacement and Current Harmonics in Time Domain
- Estimation of Displacement and Current Harmonics from Full Spectrum
- Multi-Harmonic Quadrature Reference Signal and Phase Compensation Algorithm
Here si is the magnitude and φi is the phase of the harmonic of the complex reference signal on FFT. By subtracting this angle from the phase of the harmonic of the displacement and current signals, viz.
Generation of Simulated Responses
The rotor response with and without AMB in the support system is shown in Figure 2-8 and it can be seen that the presence. A comparison of the speed, accuracy and suitability of both methods is presented in the next section.
Harmonic Analysis of Generated Responses
For correct and meaningful use of the entire spectrum, certain precautions must be observed. The full spectrum of the displacement and AMB current signals can be used to select the participating harmonics using the standard peak detection algorithm of MATLABTM.
Development of Identification Algorithm
In such cases, where n spin rates are considered within the range of interest, the regression matrices will take the form, as. The results obtained in the estimation of parameters considered in the identification algorithm are presented in the next section.
Results and Discussions
It is noticeable from Table 2-5 that the estimate in lower speed range is quite robust against instrument errors. Effect of bias error magnitude on results of estimation in lower speed range is summarized in Table 2-7.
Concluding Remarks
For a generalized case of static deflection due to rotor self-weight, (δy = 0 and 0 .. δ = ) the cracking force vector fcr , defined in Eq. The equations obtained for the individual subsystems viz. shaft, disc, AMB and slot are combined together and boundary conditions are applied to obtain the fitted rotor equations of motion.
Introduction
The governing equations of motion of a cracked Jeffcott rotor with the AMB support were developed in the previous chapter, which were solved to obtain the vibration displacement and AMB current responses. In the present chapter, a 4-DOF cracked rotor is modeled taking into account the gyroscopic effect due to the offset disk and a link crack excitation function to introduce the breathing of crack.
System Configurations
System Equations of Motion
- Crack Model
- Transformation of Stiffness Matrices
- Equations of Motion in the Presence of AMB Support
Modeling of the crack effect for a 4-DOF model of the rotor is presented in the following section. T is the force vector corresponding to the combined flexibility matrix (ie of the intact shaft as well as of the cracked section) in the rotating frame of reference.
Time and frequency responses
With SCEF expressed by Eq. 2.11), using Euler's formula, the crack excitation force can be expressed in the sum of different harmonics as The system response in the time domain, v(t), can be obtained by time integration of Eq. 3.27) and the frequency domain response, vi, can be obtained from the full spectrum v(t).
Crack Identification Algorithm
- Application of the Dynamic Reduction Scheme
- Estimation of the Rotor and AMB Parameters
For parameter identification, the frequency domain Eqn. 3.42) is divided into known and unknown parts and rearranged so that the unknown rotor and AMB parameters are grouped in a vector. The harmonics of vibrations and current responses, as required in Eqn. 3.47) will be obtained from FFT-based full-spectrum extraction properly compensated for the phase ambiguity.
Generation of Simulated Responses
The rotor rotation speed and simulation time are inputs and the forces along the coordinate directions are the output of this subsystem block. The importance and amount of participation of the gyroscopic matrix G in the rotor dynamics can be evaluated by comparing the responses with and without the inclusion of the matrix G in the ramp up test.
Results and Discussions
For further analysis, two speed ranges are selected - the lowest speed range below 1580 rpm (vibration displacement below 1.27×10-5 m) and the highest speed range above 5592 rpm (vibration displacement below 1 ,24×10-5 m). From Table 3-3 some observations can be made regarding the impact of signal noise on the performance of the developed identification procedure.
Concluding Remarks
This method has been faithfully applied in this chapter to obtain the displacement and current harmonics used in the identification process. An appropriate reduction scheme was applied to eliminate the need for this measurement and the identification plan was developed based solely on the master (i.e., retained) DOFs.
Introduction
System Configuration
The angular position of the imbalance with respect to the perpendicular to the crack front is quantified by constant angles, β1 to βn, for each slice. The complex vibration displacement and static deflection will be used to define the forcing expressions due to the cracking and AMB effects.
Statement of Excitation Forces
- Crack Forces
- Residual Unbalance Forces
- Restitution force due to AMB
4.7) can be written in terms of the complex vibration displacement vector qcv and the complex current ic as. Once excitation forces are available in terms of the vibration displacement vector, they can be placed into appropriate equations during FE modeling.
Finite Element Modelling of the Rotor System
- Sub-models
- Assembled Equations of Motion of the Cracked-Rotor AMB System
- Time and Frequency Responses
- Reduction of Unwanted DOFs
The components of the vector fd were explained in Section 4.3.2. Matrices Md and Gd are the disc mass and gyroscope matrices, respectively, as detailed in Appendix B. The action of AMB was described in Section 4.3.3. The vectors ηAM B and ic contain the displacement and current at the AMB location, as described in Section 4.3.3; all other entries of both vectors are zero.
Development of Identification Algorithm
Since, the frequency domain motion matrix equation and subsequent identification equations use positive and negative frequency harmonics, ηi, of the time domain response, η( )t. Extracting the full spectrum of the time domain signal and applying phase correction to this process is covered in detail under sections 2.3.2 and 2.3.3 in chapter 2 and the procedures developed there are used in the following sections.
Generation of Simulated Responses
The response and information from the rotor system available at multiple rotational speeds can be accommodated in the matrix equation as described in Section 2.6 of Chapter 2. The rotor block performs the dynamics of the cracked rotor as defined in the system configuration.
Results and Discussions
A representative surge response at node 3 and the Hilbert envelope (Feldman, 2011) of the displacement response is shown in Figure 4-10. The estimation results and the effect of signal noise on estimation quality are shown in Table 4-2.
Comparison with Numerical Models
Overall, the three models and the identification algorithm developed based on them give similar results.
Concluding Remarks
The present chapter serves to establish a procedure for modeling the rotor-AMB systems with greater complexities; the shortcomings of the current model in the form of simplifying assumptions will be addressed in the next chapter. In the next chapter, the model is extended to a very generic situation with multiple disks and multiple AMB support; and the support dynamics were also considered.
- Introduction
- System Configuration
- Derivation of the Excitation Forces
- Support Bearing Forces
- Finite Element Modelling of the Rotor-AMB System
- Sub-models
- Assembled Equations of Motion of the Cracked-Rotor AMB System
- Time and Frequency Responses
- Development of Identification Equations
- Numerical Experiments
- Generation of Simulated Response
- Results and Discussions
- Concluding Remarks
All submodels of the current model, except the support bearing submodel, were developed in Chapter 4. The axle submodel, crack submodel, disc submodel and AMB submodels are described in Section 4.4. 1 in Chapter 4. Equations obtained for individual subsystems, namely the shaft, crack, disc, AMB and support bearings are combined and boundary conditions are applied to obtain the composite equations of motion of the rotor.
Summary of the present work
Major Conclusions and Recommendations from the Present Work
Limitations and Applicability
Scope for Future Work
