Parameters of convergence analysis simulation.
Abstract
This chapter presents the modeling procedure, numerical application, and experimental validation of uncertain quantification techniques applied to flexible rotor systems. The uncertainty modeling is based both on the stochastic and fuzzy approaches. The stochastic approach creates a representative model for the flexible rotor system by using the stochastic finite element method. In this case, the uncertain parameters of the rotating machine are characterized by homogeneous Gaussian random fields expressed in a spectral form by using the Karhunen-Loève (KL) expansion. The fuzzy approach uses the fuzzy finite element method, which is based on the α-level optimization. A comparative study regarding the numerical and experimental results obtained from a flexible rotor test rig is analyzed for the stochastic and fuzzy approaches.
Keywords
- rotordynamics
- uncertainty
- fuzziness
- randomness
- experimental validation
1. Introduction
Rotating machines are unavoidably subjected to uncertainties that affect their parameters and, consequently, their dynamic behavior. Thus, mathematical models that encompass variability and randomness are required for the analysis and design of rotating machines instead of using deterministic models.
Uncertain dynamic responses of flexible rotors have been analyzed by applying two main approaches, namely, stochastic and fuzzy. Thus, the uncertainty analysis has been applied in flexible rotors by using the polynomial chaos theory [1], as modeled by considering Gaussian homogeneous stochastic fields discretized by Karhunen-Loève expansion [2] or through the fuzzy approach [3, 4]. These methods are well-established tools that may present limitations and drawbacks depending on the application conveyed.
In this context, this chapter presents two different approaches to model uncertain parameters and to simulate the uncertain dynamic responses of rotating machines. In this way, the stochastic and fuzzy approaches are applied to different parameters of a flexible rotor. The procedure used to obtain the stochastic model of the rotor is based on the stochastic finite element method. Moreover, the fuzzy finite element model of the rotor system is formulated according to the fuzzy approach. Then, the corresponding numerical method used to compute the fuzzy dynamic responses of the rotating machine is described. A comparative study between the stochastic and fuzzy approaches along with the validation of the obtained results by using experimental data is presented.
2. Rotor system model
The deterministic model of a flexible rotor based on the finite element method (FE model) is obtained in this section by following the formulation previously presented in [5]. The rotor system is composed of a flexible shaft, rigid discs, and bearings. Figure 1 shows the finite element used to represent the shaft. In this case, the finite element has two nodes and four degrees of freedom (DOFs) per node. The DOFs are associated with the nodal displacements along the
In this contribution, the FE model of the shaft was obtained based on the Euler-Bernoulli and Timoshenko beam theories. The displacement field along the finite element is represented by a cubic interpolation function. Therefore,
The strain and kinetic energies of the shaft finite element are defined according to analytical equations derived from the variational principle. Therefore, the mass and stiffness elementary matrices of the shaft are given by
where
Rigid discs are introduced in the global FE model of the shaft by considering their corresponding kinetic energy. Thus,
Eq. (2) presents the differential equation that characterizes the dynamic behavior of rotating machines (FE model with
where
3. Stochastic modeling
Among the various methods used to model uncertainties, the stochastic finite element method (SFEM) has been widely applied to complex engineering systems of industrial applications. SFEM presents well-established mathematical fundaments and suitable experimental validation [7]. Some details about the formulation of the SFEM are presented next.
3.1 Stochastic modeling of flexible shafts
The Karhunen-Loève (KL) expansion is used to model the random fields as a spectral representation. Consequently, a random field is represented as a spatial expansion of a random variable that fluctuates randomly. For instance, uncertainties affecting Young’s modulus of the shaft can be evaluated by using the KL expansion. A one-dimensional random field
where
In this work, the exponential covariance is adopted, which is defined as
where
in which
3.2 Stochastic modeling of bearings’ parameters
The uncertainties associated with bearings’ stiffness and damping coefficients of rotating machines can be evaluated by using the following relations:
3.3 Numerical results
In this section, SFEM is applied to the FE model as given by Figure 2. The rotating machine is composed of a horizontal flexible shaft discretized into 20 Euler-Bernoulli’s beam elements, three asymmetric bearings (
In this case, the uncertain random fields associated with Young’s modulus of the shaft are modeled as homogeneous Gaussian stochastic fields, which are represented in the spectral form by using the Karhunen-Loève expansion. The uncertainty variables associated with the stiffness and damping coefficients of the bearings are modeled as random variables. This modeling process considers the frequency- and time-domain vibration responses of the rotating machine in terms of their working envelopes (frequency response functions (FRFs) and orbits).
Initially, the convergence of the stochastic model is verified by changing the number of terms used in the KL expansion and the number of samples considered in MCS (
where
The deterministic and stochastic FRFs were obtained by considering the shaft at rest (
Scenario | ||
---|---|---|
(a) | 1 ≤ |
100 |
(b) | 10 | 1 ≤ |
Figures 3a and b present the upper and lower limits of the RMS envelopes obtained by considering the scenarios (a) and (b) of Table 1, respectively. Note that convergence is achieved for
Figure 4a and b show the FRF and orbit, respectively, obtained by using the deterministic (mean) and stochastic FE models of the rotor system. The uncertain envelopes were determined by applying a 5% dispersion level both in Young’s modulus of the shaft (
4. Fuzzy dynamic analysis
The fuzzy dynamic analysis computes the uncertain dynamic responses of rotating machines by modeling the uncertain parameters as fuzzy variables or fuzzy fields. The fuzzy dynamic analysis is based on the
4.1 Fuzzy variables
Figure 5 presents the definition of fuzzy sets. Considering
Fuzzy variables are represented by using intervals weighted by the membership function, namely,
where 0 ≤
Moreover, according to Figure 5b
If the fuzzy set is convex, each
4.2 Fuzzy dynamic analysis
The fuzzy dynamic analysis is a numerical method used to map a fuzzy input
Figure 6 shows that the fuzzy dynamic analysis is composed of two main steps. The first step consists in discretizing the input fuzzy parameter according to the
In the second step, an optimization problem is performed. This optimization process maximizes and minimizes the value of the output for the mapping model
where
The complete set of the intervals
The fuzzy analysis of either a transient time-domain response or a frequency response function demands the solution of a large number of
4.3 Numerical results
The numerical results for the fuzzy analysis are also obtained by using the rotor FE model presented in Figure 2. In this case, Young’s modulus
The fuzzy responses both on the time and frequency domains show that the fuzzy uncertainty parameters produce a significant variation of the lower and upper curves of the fuzzy envelope. Note that the results obtained in the present analysis are similar to the ones presented in Figure 4, for which the stochastic approach was applied.
5. Comparative study of uncertainty quantification techniques
The uncertainty analysis of dynamic systems has been previously studied by applying techniques based both on stochastic and fuzzy approaches. The fuzzy approach has demonstrated to be more appropriate in the cases of applications for which there is no knowledge regarding the stochastic process that governs the uncertainties themselves.
In the present study, the uncertainties that affect the dynamic response of a flexible rotor system are modeled by using both stochastic and fuzzy approaches. These methodologies have been compared by evaluating the dynamic responses obtained by numerical simulations regarding the frequency responses and time-domain responses. The numerical and experimental results of this section have been obtained from the flexible rotor test rig depicted in Figure 8.
The corresponding FE model was discretized in 33 finite elements, as given by Figure 8b. This rotating machine is composed of a flexible steel shaft of 860 length and 17 mm diameter (
A representative FE model of the rotating machine was obtained by applying a model updating procedure. The differential evolution optimization approach was used to identify the unknown parameters of the FE model, namely, coefficients
Figure 9 shows the simulated Bode diagram obtained by using the parameters identified by the considered optimization procedure. The experimental diagram is added to the figure for comparison purposes. The similarity between the numerical and experimental Bode diagrams demonstrates the representativeness of the obtained FE model.
5.1 Frequency-domain analysis
In the present analysis, the uncertain envelope of the FRF was obtained by considering Young’s modulus of the shaft as uncertain information. Regarding the stochastic approach, uncertain Young’s modulus is modeled as a Gaussian random field with nominal value
For the fuzzy approach, a fuzzy triangular number with the same nominal value and dispersion considered for the stochastic approach (
In this contribution, the performed uncertainty analysis aims at obtaining the minimum and maximum responses of the rotor system, i.e., the bounds of the uncertain dynamic responses. Therefore, the fuzzy uncertainty analysis was devoted to the
Figure 11 presents a comparative evaluation of the FRFs’ uncertain envelopes obtained by applying the stochastic and fuzzy approaches. In this case, the obtained FRFs were determined by considering the force applied along the
5.2 Time-domain analysis
The time-domain analysis was performed based on the orbits of the flexible shaft. This analysis considers uncertainties affecting the stiffness coefficients
The convergence analysis was performed to determine
Considering the fuzzy approach, the uncertain parameter is defined as a fuzzy triangular number with the same nominal value and deviation of the stochastic modeling. The objective function of the
6. Conclusions
This chapter is dedicated to the modeling, numerical methods, and simulations for the uncertainty analysis of flexible rotors. The stochastic and fuzzy approaches showed to be suitable methods to quantify the effect of uncertain parameters on the dynamic responses of rotating machines. The comparative study permitted to evaluate the two studied approaches is based on numerical simulations. Although the numerical results obtained by applying both approaches were similar, the fuzzy approach demands a greater computational effort than the stochastic method. Nevertheless, the stochastic approach requires an extensive mathematical background and an insight knowledge on the uncertain parameters. In this case, the stochastic distribution should be known or assumed. However, both approaches can be applied to the design of rotating machines.
Acknowledgments
The authors are thankful for the financial support provided to the present research effort by CNPq (574001/2008-5, 304546/2018-8, and 431337/2018-7), FAPEMIG (TEC-APQ-3076-09, TEC-APQ-02284-15, TEC-APQ-00464-16, and PPM-00187-18), and CAPES through the INCT-EIE. The authors are also thankful to the companies CERAN, BAESA, ENERCAN, and Foz do Chapecó for the financial support through the R&D project Robust Modeling for the Diagnosis of Defects in Generating Units (02476-3108/2016).
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