Model based identification of internal and external damping in a cracked rotor

74 Figure 2.8 Full spectrum of displacement at 88 rad/s (a) amplitude and (b) phase without compensation (c) phase with compensation. 139 Figure 4.7 Full spectrum response at 20 Hz (a) amplitude and (b) phase without compensation, (c) phase of reference signal and (d) phase with compensation (e) amplitude with 1% noise and (f) phase without compensation with 1 % noise, (g) phase of reference signal with 1% noise and (h) phase with compensation with 1% noise.

Figure 1.1 A shaft-disc systems with an interference fit (a) without shaft deformation and (b) with shaft deformation ...............................................................................................................

Introduction

Other destabilizing mechanisms are oil whip, steam swirl, Morton's effect, asymmetric rotor, etc., which are diagnosed as the other main causes of self-excited vibration in rotor system (Adams, 2009).

Health monitoring of rotating machinery

Vibration based monitoring on rotating machinery

Lee and Joh (1994) illustrated the directional frequency response functions (dFRFs) for the analysis of the support anisotropy and axis asymmetry. Detection based on vibration signal of the most common faults such as unbalance, crack, bow and displacement is discussed in the next section.

Table 1.1 Periodicity of common rotor defects

Internal damping in rotors

Internal damping due to rubbing or friction
Hysteretic and other form of internal damping

It was found that the stability achieved due to the incorporation of flexibility and damping in the rotor system support bearing. They studied the stability of the rotating shaft due to internal damping through numerical simulations.

Crack model in rotor systems

The flexibility of the rotor element containing the crack increases when the crack opens and vice versa (Sekhar, 2004a). Most crack models are based on flexibility, which increases due to crack initiation.

Figure 1.2 Various kinds of cracks (a) transverse crack (b) circumferential crack due to: (i) tensile load (ii) moment (iii) twisting moment (c) longitudinal crack (d) slant crack (e) open

Crack identification in rotor system

Model based approaches
Condensation techniques

The next section provides an overview of the identification of cracks in the rotor system. The second harmonics present in the frequency domain signal indicate the presence of cracks in the rotor system. The most difficult work in experimenting with rotor cracks is the initiation of cracks in the laboratory.

Experimental studies on rotor cracks

The crack was formed in the center of the reduced part in the form of a transverse slit by the sparking process. A similar result was observed in the trispectrum, which can classify cracks and misalignment faults. Application of spectral kurtosis (SK) and fast kurtogram in diagnostics for earlier detection of shaft fatigue cracks based on the use of vibration data.

Shortcomings of the present literature

The challenge would be to test the method in a laboratory test setup and verify it in a real machine, even modeling and identifying the flexibility of the foundation. Methods for accurately predicting machine responses at different residual arc values need to be studied. The case of internal damping and different types of misalignment in the rotor system needs to be studied.

Objectives of the present work

To develop mathematical model of 4-DOF rotor system considering gyroscopic effect, external and internal damping, unbalance and cracking force. To numerically develop and test the model-based identification algorithm for the estimation of rotor fault parameters. To develop and numerically test the general identification algorithm for the estimation of bearing parameters, internal and external damping, unbalance and additive crack stiffness parameter.

The organization of the present work

Later EOMs are used to develop a model-based identification algorithm to estimate distinct internal and external damping of the cracked rotor system along with crack addition stiffness and unbalance. This chapter laid the foundation of proposed identification algorithm for the distinct estimation of internal and external damping. Identification methodology developed after elimination of rotational DOFs using condensation scheme is applied for the experimental estimation of internal and external damping.

Introduction

Forces act on the rotor system due to the crack and the imbalance as the rotor rotates. The system vibrates due to these combined forces and excites it at multiple harmonics of the rotor spin frequency. Due to these forces, the rotor whirls in the same direction of spin (i.e. the forward swing) and in reverse direction of spin (i.e. the backward swing) at multiple harmonics.

System model and mathematical formulations

Development of equations of motion of the rotor system
Modeling of the crack
Response due to crack
Generation of full spectrum responses
FFT based full spectrum estimation and phase correction

The updated function σ(t) denotes the switch crack excitation function (SCEF) and it is considered in the present work. It is close to one, when the crack is open or if the shaft is towards the front of the crack in the tension of the shaft. In the identification procedure, a solution of si(t) for a specific harmonic of cracking power generation is considered as Ri(ω)eijωt.

Figure 2.1 (a) A simple rotor in the presence of a transverse switching crack (b) enlarged view of crack

Identification of internal damping and other rotor parameters

The generated responses are further used for parameter estimation using the least square regression technique (i.e., Equations (2.34) and (2.35)), which is illustrated by numerical simulation in the next section. It is common practice to first test the developed identification algorithm through numerical simulation since we have control over all system parameters and the level of noise to be added. Once it is tested, it is then applied to experimental data, and we often do not have an estimate of the system parameters independently by any other method, so it is difficult to verify them.

Numerical simulation

Time domain and full spectrum response
Phase compensation of full spectrum
Identification of parameters

Full-spectrum amplitude and phase without compensation are shown for 88 rad/s and 308 rad/s in Figure 2.8(a-b) and Figure 2.9(a-b), respectively. Likewise, for the same spin speeds, the full spectrum phase with compensation is shown in Figure 2.8(c) and Figure 2.9(c) respectively. The phase of reference signal at spin speed of 88 rad/s and 308 rad/s by the regression based full spectrum is found to be very small and of the order of 10-16 rad as given in Table 2.4.

Table 2.1Assumed initial parameter of rotor model for numerical solution

Conclusions

Therefore, it can be concluded that the proposed identification algorithm is robust to the measurement noise. The experimental analysis of the current cracked simple rotor model is illustrated in the next chapter. It consists of estimating the model parameter, based on the measured responses of an experimental rig.

Experimental setup of a fatigue crack

Experimental design and analysis
Experimental observations

Fatigue crack on the shaft was generated through INSTRON machine as shown in Figure 3.5, in Strength of Material Laboratory at IIT Guwahati based on three point bending test configuration as given in Table 3.2. It was obtained using peaks observed in a reference signal due to phase marker on the motor shaft, as shown in Figure 3.8(a) at 3 Hz rotor speed for 3 complete cycles. Responses can be seen for 3 complete cycles as shown in Figure 3.8(a-c) both for displacements and the reference signal.

Figure 3.4 The shaft with a fatigue crack

FFT based full spectrum analysis

Comparisons in FFT with and without crack
Comparisons of FFT based and Regression-based method
Phase compensation in full spectrum

Phase angles of the reference signal at different harmonics of the full spectrum for slow roll (3 Hz) are given for both methods in Table 3.4. Table 3.4 Displacement amplitudes and their phase angles, and phase angles of reference signals at different harmonics of the full spectrum for the slow operation of the rotor (3 Hz). Full spectrum displacement amplitudes at different harmonics for higher spin rates of 17 Hz, 20 Hz and 22.

Figure 3.12 Full spectrum of measured responses for slow roll at 3 Hz (a) amplitude of displacement, (b) amplitude of reference signal, (c) phase of displacement and (d) phase of

Removing the effect of shaft bow

Comparison of displacements with and without bow and sensor gap

Figure 3.18 Response obtained at 20 Hz rotor spin speed for 20 complete cycles without arc and without sensor gap in (a) vertical direction (b) horizontal direction. Figure 3.19 Response obtained at 22 Hz rotor spin speed in 22 complete cycles without arc and without sensor gap in (a) vertical direction (b) horizontal direction. Rotor trajectories with and without arc and sensor gap can be seen in figure 3.20 to and including figure 3.22 for rotor spin speed at 17 Hz, 20 Hz and 22 Hz respectively.

Figure 3.17 Response obtained at 17 Hz rotor spin speed for 17 complete cycles without bow and without sensor gap in (a) vertical direction (b) horizontal direction

Identification Algorithm

Estimation of the parameters mentioned in vector x2 can be done experimentally through the above regression equation using full spectrum responses.

Estimation of Rotor System Parameters

However, Figures 3.23-3.26 correspond to the measured signal without the arc effect and sensor gaps. But Figures 3.23(b)-3.25(b) were generated from numerical simulation based on parameters identified and listed in Table 3.9 based on experimental responses and are for validation. These are compared with those obtained by experiments at respective speeds, as shown in Figures 3.23(a)-3.25(a).

Table 3.9 Estimated experimental parameters Parameters 17, 18 and 19 Hz

Conclusions

The measured responses, converted to full spectrum, are used in the developed identification algorithm discussed in Chapter 2 to estimate various error parameters. The estimated error parameters through the developed identification algorithm are used in the numerical simulation and find their viability in the full spectrum response. The analysis of the offset disk rotor system is discussed in the next section with gyroscopic effects included.

Introduction

Cracked rotor system model

Configuration of rotor model and equations of motion
Internal damping model
Model of the Crack

Shaft stiffness in the closed state of the crack is K̍rot (which is the same as intact shaft stiffness) and for the crack open state the stiffness is K̍rot-ΔK̍rot. In equation (4.19), pi denotes the coefficient for the ith term in the harmonic function of the crack force excitation, also called the participation factor for individual harmonics. The forward and backward vorticity of the rotor system, Ri, can be obtained through the entire spectrum of v(t). The full spectrum extraction from time domain signals and its application is presented in the next section.

Figure 4.2 The rotor with unbalance, self-weight, internal and external damping forces and displacement (x=u x +u x0 and y=u y ) with respect to the fixed and rotating frame of references

Generation of Full Spectrum Responses

Identification of internal damping and other rotor parameters

Application of the Dynamic Reduction Scheme

In Equation (4.22), the translational and rotational DOFs are considered as the master and slave DOFs, respectively. The cracking force on the right-hand side of equation (4.22) due to the rotational DOFs can be neglected in developing the transformation. Using this transformation and including the shear force due to the rotational DOFs in equation (4.25), we obtain.

Numerical simulations

Time domain and full spectrum response

The full-spectrum amplitude of the complete and incomplete cycle of time-domain signals using the regression-based full-spectrum and FFT-based full-spectrum are compared in Table 4.2 for a spin rate of 20 Hz and in Table 4.3 for a spin rate of 50 Hz. The comparison of full spectrum phase for the spin rate of 20 Hz and 50 Hz for both complete and incomplete signal cycles using the regression and FFT based method is given in Table 4.4 and Table 4.5 respectively. The full spectrum phase angle of the reference signal is also compared to both complete and incomplete signal cycles using the methods mentioned above in Table 4.6 and Table 4.7 for spin rate 20 and 50 Hz.

Figure 4.4 Simulink block of rotor system 4.5.1 Time domain and full spectrum response

Conclusions

Introduction
Experimental analysis and observation of responses
Response comparison analysis
Estimation of rotor system parameters
Validation through numerical simulations
Conclusions

As shown in Figure 5.1, the experimental test rig was set up in the Vibration and Acoustics Laboratory at IIT Guwahati as shown in Figure 5.2. Similarly, for the higher rotational speeds of 15 and 20 Hz, the full-cycle signal acquisition is shown in Figures 5.5 and 5.6, respectively. For the reference signal, Figures 5.7(c) to 9(c) represent the amplitudes and its phases are presented in Figures 5.7(d) to 5.9(d) for 3 Hz, 15 Hz and 20 Hz motor speed.