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Abstract

Steam turbines are used to generate thermal power in electric power plants. They are important industrial equipment that support societal infrastructure. The stable operation of steam turbines is necessary to maintain long-term electrical power supply. However, low-frequency vibration, which is referred to as steam-whirl-induced vibration, is a self-excited vibration that can damage turbines and hinders stable operation. Therefore, a prediction model and stable margin for steam-whirl-induced vibration in steam turbines should be developed. In this study, we propose a method for modeling steam-whirl-induced vibration using closed-loop system identification. This method directly creates a vibration model from the rotor displacement data. The gain, damping, and natural frequency of the vibration were calculated using this model. Moreover, an equation for the relationship between the damping and load was derived using the model, and the stable margin for increasing the load was estimated. Steam-whirl-induced vibration was modeled using the proposed method for the displacement data obtained from an actual steam turbine. The characteristics of the model are in good agreement with the experimental results, indicating the feasibility of using the model to predict steam-whirl-induced vibration.

Keywords

steam turbine steam-whirl-induced vibration self-excited vibration closed-loop system identification

Introduction

Electric power generation plants play a vital role in daily life because electric power is indispensable for sustaining existing societal infrastructure. In recent decades, power generation using renewable energy has been promoted considering environmental conservation. However, thermal power plants still generate a major share of global electricity (OpenPR.com., 2018). According to the statistical data for 2021 (Renewable Energy Institute, 2021), approximately 55% of the world’s electricity is generated by thermal power plants. In a thermal power plant, heated fuel releases thermal energy, which is transferred to water to generate steam. A steam turbine converts steam energy into electric power. The rotor of the turbine is rotated by steam and the rotational energy is converted into electric power by a generator connected to the end of the rotor. One of the most important design aspects of steam turbines is the stable operation of their rotors. Long-term stable operation of steam turbines is essential for ensuring stable power supply. The rotor is affected by structural and fluid dynamics. In previous studies, rotor stability was evaluated through rotor dynamics analysis (Childs, 1993, Vance, 1988).

Rotors in steam turbines experience low-frequency vibration due to mechanical forces and steam pressure (Kanki et al., 1998). Low-frequency vibration is a unique problem of steam turbines and is called steam-whirl-induced vibration (Hirano et al., 2005, Hirano et al., 2008, Gao et al., 2009). The amplitude of steam-whirl-induced vibration increases with the load on the rotor. Thus, to maintain stable operation of a steam turbine, the amplitude of the steam-whirl-induced vibration should be estimated to determine the load limit. In previous studies, the characteristics of steam-whirl-induced vibration were analyzed using computational fluid dynamics (CFD). Furthermore, vibrational characteristics were examined using numerical calculations based on CFD (Kramer, 1993, Schettel and Nordmann, 2004, Bachschmid et al., 2008, Pollman et al., 1978) and vibration analysis based on experimental data (Hisa et al., 1985, Bachschmid et al., 2006, Kubiak et al., 2007, Pennacchi and Vania, 2011). However, these methods have several drawbacks for practical applications. The time required to obtain accurate results via numerical calculations is considerably high for complex rotating machinery. The evaluation of experimental data requires less time compared with numerical calculations. However, these results may not provide a systematic solution for rotating machinery.

According to the displacement data obtained from a steam turbine, steam-whirl-induced vibration is the forced vibration that appears when the load is lower than the threshold for self-excited vibration (Kanki et al., 1998). Vibration can be modeled as a transfer function using displacement data. A linear model can be applied to rotor dynamics by using a transfer matrix under certain conditions (Childs, 1993). In previous studies, steam-turbine dynamics were modeled at a constant rotational speed using a linear model (Bachschmid et al., 2006, Bachschmid et al., 2008) and transfer function (Cloud et al., 2005, Hu and Palazzolo, 2017, Yabui et al., 2020). Based on the principles of control engineering, rotating machinery supported by a journal bearing, such as a steam turbine, can be considered a closed-loop system. The journal bearing is an analogue controller for positioning a rotor, and the rotating shaft is the control object (Utsumi, 2020, Yabui et al., 2020).

In this study, we propose a modeling method based on a closed-loop system for steam-whirl-induced vibration for application to rotating machinery. In a closed-loop system, steam-whirl-induced vibration is regarded as a perturbation of the control object, and an additional uncertainty modeling method is applied based on the framework of closed-loop system identification. The modeling method models the steam-whirl-induced vibration as a transfer function, which is directly identified using only the displacement data. The proposed method is based on a direct method of closed-loop system identification. There are other system identification methods: indirect, two-stage, and joint input-output methods (Prasad et al., 1977, Linder and Enqvist, 2017, Ase and Katayama, 2018). These methods use information from the feedback controller or the input data for the closed-loop system. However, we decided to use the direct method for system identification for two reasons:

• The characteristics of journal bearings are difficult to identify accurately. The characteristics depend on the operating condition (rotating speed, load, radial clearance, and Reynolds number of oil film). Under constant conditions, the characteristics can be linearized; however, the correct coefficients in the transfer function model are difficult to identify in the actual system.

• In the general rotating machineries, we can only obtain the displacement data. That is, we can obtain the output data but cannot obtain the input data in the block diagram. Therefore, we try to identify the steam-whirl-induced vibration model by using only the displacement data (output data).

This method does not require complex calculations, and the rotor displacement data can be easily measured. The vibration gain, damping ratio, and natural frequency were expressed as functions of the load. The proposed method uses gray-box modeling to evaluate vibrational characteristics.

Modeling of steam-whirl-induced vibration

Experimental system

The target steam turbine used in this study is illustrated in Figure 1. The steam-whirl-induced vibration were measured using a displacement sensor, as shown in Figure 2. The measurement result depends on the mode shape of the rotating shaft. The sensor installation location is limited in the experimental system. Therefore, the location of the displacement sensors was decided to obtain sufficiently the vibration peak from the location option. The operating conditions (load and rotational speed) are illustrated in Figure 3. When the rotational speed reached 3000 rpm, the following four loading conditions were considered: 25%, 38%, 60%, and 100%. Only the load was varied during the operation of the steam turbine. The displacement data (120 sets) were measured under each loading condition at a sampling frequency of 2048 Hz for 1 s. The amplitude spectrum of the displacement is shown in Figure 4. Unbalanced vibration were observed as an amplitude peak at 50 Hz. In addition, the amplitude spectrum increased by approximately 18 Hz with the load. The fundamental cause of the vibration was determined to be the steam whirl, which increased with the load.

Figure 1.

Image of the steam turbine (OpenPR.com., 2018).

Figure 2.

Schematic of the steam turbine.

Figure 3.

Experimental condition of the steam turbine. The rotating speed was changed from 0 to 3000 rpm, and the percentage of load was changed from 0% to 100% as shown in this figure.

Figure 4.

Amplitude spectrum of displacement signal in x, y direction: The displacement signal is measured 2048 point × 120 sets. The amplitude spectrum calculated from each set is overplotted in this figure.

Block diagram of proposed model for steam-whirl-induced vibration

Closed-loop system identification was applied to the steam-whirl-induced vibration model. Although rotating machinery has nonlinear characteristics owing to its complex dynamics, its characteristics can be approximately linearized under certain conditions (Childs, 1993). Previous studies have proposed a linear model for the dynamics of rotating machinery (Cloud et al., 2005, Yabui et al., 2020). Cloud et al. 2005 modeled the rotating machinery dynamics as a transfer function at a constant rotational speed. The characteristics of the linear model were in good agreement with those of actual systems, and the model was used to predict the stability of rotating machinery. Based on previous studies, the proposed method modeled the steam-whirl-induced vibration as a transfer function.

First, we considered the case of operation without steam-whirl-induced vibration. A schematic of the steam turbine used in this study is shown in Figure 5, where u_x and u_y are the journal bearing forces in the x and y directions, respectively; y_x and y_y are the displacements of the rotor in the x and y directions, respectively; P_x and P_y are the models of the rotors; B_xx, B_xy, B_yx, and B_yy are the models of the journal bearings; and the subscripts xx, xy, yx, and yy correspond to the directions of the forces as x → x, x → y, y → x, and y → y, respectively. The rotational speed was maintained at 3000 rpm. Therefore, linear models were used for P_x , P_y , B_xx, B_xy, B_yx, and B_yy. The bearing was the rotor’s analogue positioning controller.

Figure 5.

Block diagram of the steam turbine without steam-whirl-induced vibration.

Second, we considered the case of operation with steam-whirl-induced vibration. A schematic of the steam turbine used in this study is shown in Figure 6, where ΔP_x and ΔP_y are the models of steam-whirl-induced vibration in the x and y directions, respectively. As the vibration were triggered by the eccentricity of the rotor shaft, the displacement of the shaft was considered as the input for ΔP_x and ΔP_y [Hisa et al., 1985, Pennacchi and Vania, 2011]. The characteristics of ΔP_x and ΔP_y varied with the load W. The displacement, including the steam-whirl-induced vibration, is defined as ${\hat{y}}_{x} = y_{x} + Δ y_{x}, {\hat{y}}_{y} = y_{y} + Δ y_{y}$ . In this model, steam-whirl-induced vibration was caused by ΔP and ΔPY, which were varied to increase vibration with W.

Figure 6.

Block diagram of the steam turbine with steam-whirl-induced vibration.

Modeling strategy for steam-whirl-induced vibration

Data processing for the proposed model

The objective of this study was to model ΔP_x and ΔP_y as transfer functions to analyze steam-whirl-induced vibration. The displacement signals without vibrations y_x and y_y are expressed as

\begin{array}{c} y_{x} (t) & = & P_{x} (s) u_{x} (t) \end{array}

(1)

\begin{array}{c} y_{y} (t) & = & P_{y} (s) u_{y} (t) \end{array}

(2)

where t is the time and s is a complex frequency parameter defined in the Laplace transform. P_x(s) and P_y(s) are the transfer-function models of the rotor. The displacement signals with vibration

{\hat{y}}_{x}

and

{\hat{y}}_{y}

are expressed as

\begin{array}{c} {\hat{y}}_{x} (t) & = & P_{x} (s) u_{x} + Δ P_{x} (s) y_{x} (t) \end{array}

(3)

\begin{array}{c} {\hat{y}}_{y} (t) & = & P_{y} (s) u_{y} + Δ P_{y} (s) y_{y} (t) \end{array}

(4)

We obtain the following equations by substituting equations (1) and (2) into equations (3) and (4), respectively

\begin{array}{c} {\hat{y}}_{x} (t) & = & y_{x} (t) + Δ P_{x} (s) y_{x} (t) \end{array}

(5)

\begin{array}{c} {\hat{y}}_{y} (t) & = & y_{y} (t) + Δ P_{y} (s) y_{y} (t) \end{array}

(6)

Equation (5) can be transformed as follows

\begin{array}{c} {\hat{y}}_{x} (t) & = & y_{x} (t) + Δ P_{x} (s) y_{x} (t) \\ Δ y_{x} (t) & = & Δ P_{x} (s) y_{x} (t) \\ (* Δ y_{x} (t) & = & {\hat{y}}_{x} (t) - y_{x} (t)) . \end{array}

(7)

When Δy_x(t) and ${\hat{y}}_{x} (t)$ are Fourier transformed, ΔP_x (s) can be calculated as

Δ P_{x} (s) = \frac{F (Δ y_{x} (t))}{F (y_{x} (t))} .

(8)

F (\cdot)

indicates that the signal within the parentheses is Fourier transformed. The transfer function of the steam-whirl-induced vibration can be derived from Δy_x(t) and Δy_x(t). ΔP_y(s) can be calculated similarly.

Originally, output (displacement) and input signals of ΔP_x(s) and ΔP_y(s) were required to obtain the transfer function. However, the transfer functions ΔP_x(s) and ΔP_y(s) were derived only from the displacement signal in the proposed system, as shown in Figure 6. Therefore, actuators that generate input signals are not required for system identification

Δ P_{y} (s) = \frac{F (Δ y_{y} (t))}{F y_{y} (t)}

(9)

Identification of transfer function model

We estimated the transfer function of the steam-whirl-induced vibration from the actual displacement data. Steam-whirl-induced vibration did not occur immediately after the activation of the steam turbine. The displacement signals measured at this time are denoted by y_x and y_y. As shown in Figure 4, steam-whirl-induced vibration is observed at approximately 18 Hz by increasing W. In this section, we focus on the displacement signals ${\hat{y}}_{x}$ and ${\hat{y}}_{y}$ measured at W = 100%. The time responses of y_x, y_y, $y {\hat{y}}_{x}$ , and ${\hat{y}}_{y}$ are shown in Figure 7. The differences in the signals $Δ y_{x} (t) = {\hat{y}}_{x} (t) - y_{x} (t)$ and $Δ y_{y} (t) = {\hat{y}}_{y} (t) - y_{y} (t)$ are shown in Figure 8. The time responses can be considered as steam-whirl-induced vibration. In the following, we focus on Δy_x(t) values that are larger than those of Δy_y(t).

Figure 7.

Time response of displacement signal of the rotating shaft $y_{x} (t), y_{y} (t) . {\hat{y}}_{x} (t), {\hat{y}}_{y} (t)$ . The displacement signal is measured in 2048 point × 120 sets. The all data set is overplotted in this figure.

Figure 8.

Time response of displacement signal Δyx(t)andΔyy(t) are calculated from data set (2048 point × 120 sets), and all results are overplotted in this figure.

The frequency responses of $F Δ y_{x} (t) / F y_{x} (t)$ were calculated using Fourier transformed data, as shown in Figure 9. The 120 measurements were divided into 6 × 20 datasets, and each dataset was averaged to eliminate noise. In Figure 9, there is a resonance peak at 18 Hz, and the phase is delayed by 180° at approximately 18 Hz. The constant gain observed above 30 Hz was considered to be caused by noise. The frequency response of the steam-whirl-induced vibration was modeled as a second-order transfer function. This was the vibration mode caused by the steam excitation force in the modal coordinates. The second-order transfer function was estimated using the curve-fitting method (Shieh and Cohen, 1978, Ljung, 2003). The estimated transfer function is given by

Δ P_{x} (s) = \frac{22500}{s^{2} + 35.81 s + 14250} .

(10)

Figure 9.

Frequency responses of $F Δ y_{x} (t) / F y_{x} (t)$ and ΔP_x(s). Black lines $F Δ y_{x} (t) / F y_{x} (t)$ , red broken line ΔP_x(s). $F Δ y_{x} (t) / F y_{x} (t)$ is calculated from the summarized data sets 2048 point × 20 sets × 6patterns (made from 2048 point × 120 sets). The red broken line ΔP_x(s) is estimated using the curve-fitting method.

The frequency response of the transfer function is indicated by a dashed line in Figure 9. The frequency response strongly agreed with the experimental results. Additionally, Δy_x(t) was back-calculated using the measured signal y_x as the input for ΔP_x(s). Figure 10 compares the results obtained for y_x and Δy_x(t). The time responses of Δy_x(t) strongly agreed with those of y_x.

Figure 10.

Time response of Δy_x(t) estimated by the transfer function model. Δy_x(t) is calculated from ΔPx(s), and the original data is calculated from averaging the all measurement data (2048 point × 120 sets).

ΔP_x(s) was determined from the displacement data measured at W = 25%, 38%, and 60%. The frequency responses of $F Δ y_{x} (t) / F y_{x} (t)$ and ΔP_x(s) are shown in Figure 11. The transfer functions are presented in Table1. The proposed method was used to model the vibration as a transfer function. It is difficult to use it directly for nonlinear system identification. However, the dynamics of nonlinear systems can be linearized around the equilibrium point. For a rotating machine, the nonlinearity originates from the dynamics of the journal bearing, and the dynamics can be linearized under certain conditions (constant rotating speed, load, etc.). Therefore, the proposed method can be applied to the displacement data measured using a nonlinear rotor under constant conditions.

Figure 11.

Frequency responses of $F Δ y_{x} (t) / F y_{x} (t)$ and ΔP_x(s). Top: W = 25%, middle: W = 38%, bottom: W = 60%. Black lines: $F Δ y_{x} (t) / F y_{x} (t)$ , red broken line: ΔP_x(s). At each load condition, $F Δ y_{x} (t) / F y_{x} (t)$ is calculated from the summarized data sets 2048 point × 20 sets × 6 patterns (made from 2048 point × 120 sets). The red broken line ΔP_x(s). is estimated using the curve-fitting method.

Table 1.

ΔP_x(s) for each load.

Load W	ΔP_x(s)
25%	$\frac{22500}{s^{2} + 115.6 s + 20880}$
38%	$\frac{22500}{s^{2} + 96.8 s + 19110}$
60%	$\frac{22500}{s^{2} + 78.0 s + 16920}$
100%	$\frac{22500}{s^{2} + 35.8 s + 14250}$

Proposed modeling method and stability analysis for steam-whirl-induced vibration

Stability analysis based on the determined model

The transfer function was estimated from the measured data using a black-box modeling method. However, the parameters of the transfer function were transformed into residue, damping, and stiffness in the modal coordinates. Therefore, the characteristics of the transfer function were evaluated with respect to the mechanical parameters, which were calculated using gray-box modeling. Based on this knowledge, we propose a method for estimating the stable margin of steam-whirl-induced vibration. When the system is considered a lumped mass system, the transfer function is described based on the equation of motion as follows

N (s) = \frac{α}{s^{2} + \frac{C_{x} (W)}{M_{x} (W)} s + \frac{K_{x} (W)}{M_{x} (W)}} .

(11)

where α is the residue; M_x, C_x, and K_x are the mass, damping, and stiffness, respectively; and W denotes the load percentage. The model normalized by dividing both sides by the mass M_x is as follows

N^{*} (s) = \frac{α^{*} (W)}{s^{2} + C_{x}^{*} (W) s + K_{x}^{*} (W)}

(12)

At W = 100%, the values of $α^{*} (W), C_{x}^{*} (W)$ , and $K_{x}^{*}$ were obtained as 22,500, 35.81, and 14,250, respectively, by comparing the results of ΔP_x(s) and N*(s). Although the values were calculated in modal coordinates, physically meaningful values, such as the residue, damping, and stiffness, were obtained.

Figure 12 shows the values of $α^{*} (W), C_{x}^{*} (W)$ , and $K_{x}^{*}$ at different values of W. The relationships between the load and each coefficient were obtained using the polyfit function of MATLAB [Mathworks, 2021], as follows

\begin{array}{c} α^{*} (W) & = & 22500 \end{array}

(13)

\begin{array}{c} C_{x}^{*} (W) & = & - 1.0 W + 139.4 \end{array}

(14)

\begin{array}{c} K_{x}^{*} (W) & = & - 85.0 W + 22530 \end{array}

(15)

Figure 12.

Relationship between the load and the identified parameters, $α^{*} (W), C_{x}^{*} (W)$ , and $K_{x}^{*}$ .

The following observations were made from this result:

• α*(W) did not depend on W.

• $C_{x}^{*} (W)$ decreased as W increased.

• $K_{x}^{*} (W)$ increased as W decreased.

Although the residue (mode gain) was constant, the natural frequency and damping depended on the load. In this model, steam-whirl-induced vibration is considered to occur when the damping becomes negative. As shown in Figure 12, self-excited vibration occurs when the load is approximately 1.3 times the maximum load.

Validation of the proposed method

To validate the proposed method, the displacement data of the steam turbine were measured on different days. The operating conditions (rotational speed and load on the steam turbine) are illustrated in Figure 13. When the rotational speed reached 3000 rpm, the displacement data were measured for each loading condition (25%, 38%, 60%, and 75%). Additionally, the displacement data were measured for two loading patterns: gradually increasing and decreasing the load. The frequency responses of $F Δ y_{x} (t) / F y_{x} (t)$ were calculated from the measured displacement data.

Figure 13.

Experimental condition of the turbine for validation check. The rotating speed was changed from 0 to 3000 rpm, and the percentage of load was changed from 0% to 75% as shown in this figure.

The transfer functions obtained via curve fitting are presented in Table2. The value of α*(W) was the same as that obtained in Stability analysis based on the determined model. A comparison between the results for

C_{x}^{*} (W)

and

K_{x}^{*}

is presented in Figure 14. For W = 25%, 38%, and 60%, the values of

C_{x}^{*} (W)

and

K_{x}^{*}

were similar to those obtained in Stability analysis based on the determined model. Furthermore, the values of

C_{x}^{*} (W)

and

K_{x}^{*}

measured at W = 75% lie on the prediction line derived from the results presented in Stability analysis based on the determined model. Thus, the transfer function model obtained using the proposed method was successfully validated. The transfer functions of the steam-whirl-induced vibration were also identified under each condition. The frequency responses are shown in Figure 15.

Table 2.

ΔP_x(s) for validation.

Load	ΔP_x(s) load increasing	ΔP_x(s) load decreasing
25%	$\frac{22500}{s^{2} + 113.6 s + 20160}$	$\frac{22500}{s^{2} + 115.9 s + 19990}$
38%	$\frac{22500}{s^{2} + 100.4 s + 19460}$	$\frac{22500}{s^{2} + 99.5 s + 19110}$
60%	$\frac{22500}{s^{2} + 77.3 s + 16590}$	$\frac{22500}{s^{2} + 79.5 s + 16430}$
75%	$\frac{22500}{s^{2} + 62.5 s + 15630}$	$\frac{22500}{s^{2} + 64.4 s + 15320}$

Figure 14.

Relationship between the load and the identified parameters with the validation data, $α^{*} (W), C_{x}^{*} (W)$ , and $K_{x}^{*}$ .

Figure 15.

Frequency responses of $F Δ y_{x} (t) / F y_{x} (t)$ and ΔP_x(s) for the validation check. Black lines: $F Δ y_{x} (t) / F y_{x} (t)$ , red broken line: ΔP_x(s). At each load condition in validation check, $F Δ y_{x} (t) / F y_{x} (t)$ is calculated from the summarized data sets 2048 point × 20 sets × 6 patterns (made from 2048 point × 120 sets). The red broken line ΔP_x(s) is estimated using the curve-fitting method.

Conclusion

We proposed a system identification method to analyze steam-whirl-induced vibration. The following conclusions were drawn:

• The steam-whirl-induced vibration can be identified using the direct closed-loop identification system. The difference in displacement signals between the cases with and without steam-whirl-induced vibration is used in this modeling method.

• The characteristics of the steam-whirl-induced vibration can be modeled as a second-order transfer function. The frequency and time response of the vibration are in good agreement with the experimental data.

• The fluctuation of each coefficient with respect to the load was estimated by calculating the stiffness and damping in the modal coordinates from the transfer function.

• We confirmed the validity of the model by applying the proposed method to the operation data for a steam turbine from different days. The model parameters were almost the same.

• The proposed method requires displacement data only. The displacement sensor is used for most rotating machineries, and it is easy to incorporate in cases where it has not been adopted. Therefore, the proposed method can be applied to rotating machineries.

The following scope for future research is drawn based on the conclusions of this study:

• The instability condition due to the steam-whirl-induced vibration can be predicted using the damping parameter in the proposed model. When the damping parameter becomes negative, the rotating machinery becomes unstable. A feasibility study for the instability prediction will be conducted on a test bench.

• The effectiveness of the proposed method has been verified for a steam turbine. To check the generality of the model, it will be used to model the vibration of other steam turbines. Moreover, we will attempt to identify other types of rotating machineries.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Shota Yabui

Tsuyoshi Inoue

References

Ase

Katayama

. A joint input-output approach to closed-loop identification of hammerstein-wiener systems. Proc ISCIE Int Symp Stochastic Syst Theor Its Appl 2018; 2019: 121–129.

Bachschmid

Vania

Pennacchi

. Modelling of steam whirl excitation and evaluation of stability margin. In: Proceedings of the international conference on tribology, 20–22 September 2006, Parma, Italy.

Bachschmid

Pennacchi

Vania

. Steam-whirl analysis in a high pressure cylinder of a turbo generator. Mech Syst Signal Process 2008; 22(1): 121–132.

Childs

. Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis. New York, NY: John Wiley & Sons, 1993.

Cloud

Maslen

, et al. Practical applications of singular value decomposition in rotordynamics. Aust J Mech Eng 2005; 2(1): 21–32.

Gao

Dai

Wang

, et al. Rotordynamic stability under partial admission conditions in a large power steam turbine. In: Proceedings of the ASME turbo expo 2009: Power for land, sea, and air volume 6 structures and dynamics, Orlando, FL, USA, June 8–12 2009, pp. 795–802.

Hirano

Guo

Kirk

. Application of computational fluid dynamics analysis for rotating machinery-part II: labyrinth seal analysis, ASME. J Eng Gas Turbines Power 2005; 127(4): 820–826.

Hirano

Sasaki

Sakakida

, et al. Evaluation of rotordynamic stability of a steam turbine due to labyrinth seal force. J Power Energy Syst 2008; 2(3): 945–955.

Hisa

Kitamura

Sakakida

. Steam excited vibrations in rotor-bearing systems. Turbomachinery 1985; 13(6): 329–334.

10.

Palazzolo

. Solid element rotordynamic modeling of a rotor on a flexible support structure utilizing multiple-input and multiple-output support transfer functions. ASME J Eng Gas Turbines Power 2017; 139(1): 012503.

11.

Kanki

Kaneko

Kurosawa

, et al. Prevention of low-frequency vibration of high-capacity steam turbine units by squeeze-film damper. ASME J Eng Gas Turbines Power 1998; 120(2): 391–396.

12.

Kramer

. Steam whirl. In: Dynamics of Rotors and Foundations. Berlin, Heidelberg: Springer, 1993.

13.

Kubiak

Childs

Rodriguez

, et al. Investigation into a “steam whirl” which affected HP rotors of 300 MWsteam turbines. In: Proceedings of the ASME power conference, San Antonio, TX, USA, 17–19 July 2007, pp. 291–298.

14.

Linder

Enqvist

. Identification of systems with unknown inputs using indirect input measurements. Int J Control 2017; 90(4): 729–745.

15.

Ljung

. Linear system identification as curve fitting. In: A

Rantzer

Byrnes

(eds). Directions in Mathematical Systems Theory and Optimization. Lecture Notes in Control and Information Sciences, vol 286. Berlin, Heidelberg: Springer, 2003.

16.

Mathworks . Polyfit: Polynomial Curve Fitting. 2021. https://www.mathworks.com/help/matlab/ref/polyfit.html

17.

OpenPR.com . Steam Turbine Market 2024: Global Industry Top Key Players. 2018, https://www.openpr.com/news/1181104/steam-turbine-market-2024-global-industry-top-key-players-siemens-mitsui-fuji-electric-toshiba-kawasaki-heavy-industries-ge-bhel-eliott-group-ansaldo-energia-harbine-turbine.html

18.

Pennacchi

Vania

. Analysis of the instability phenomena caused by steam in high-pressure turbines. Shock and Vibration 2011; 18: 20.

19.

Pollman

Schwerdtfeger

Termuehlen

. Flow excited vibrations in high-pressure turbines (steam whirl). ASME J Eng Power 1978; 100(2): 219–228.

20.

Prasad

Sinha

Mahalanabis

. Two-stage identification of closed-loop systems. IEEE Trans Automatic Control 1977; 22(6): 987–988.

21.

Renewable Energy Institute . [Global] World and Countries (Quartely). 2021. https://www.renewableei.org/en/statistics/international/?cat=quarterly

22.

Schettel

Nordmann

. Rotordynamics of Turbine Labyrinth Seals - a Comparison of CFD Models to Experiments, IMechE paper C623/018. 2004, pp. 13–22.

23.

Shieh

Cohen

. Transfer function fitting from experimental frequency-response data. Comput Electr Eng 1978; 5(3): 205–212.

24.

Utsumi

. Vibration analysis of a cylindrical rotor partially filled with liquid considering nonlinearity of liquid motion. Mech Eng J 2020; 7(4): 19.

25.

Vance

. Rotordynamics of Turbomachinery. Wiley-Interscience, 1988.

26.

Yabui

Chiba

Suzuki

, et al. Analysis of whirling vibration due to morton effect by using frequency transfer function model. J Bearings ASME J Dyn Sys Meas Control 2020; 142(4): 041010.

Analysis of low-frequency vibration in a steam turbine based on closed-loop system identification

Abstract

Keywords

Introduction

Modeling of steam-whirl-induced vibration

Experimental system

Block diagram of proposed model for steam-whirl-induced vibration

Modeling strategy for steam-whirl-induced vibration

Data processing for the proposed model

Identification of transfer function model

Proposed modeling method and stability analysis for steam-whirl-induced vibration

Stability analysis based on the determined model

Validation of the proposed method

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References