A Dynamic-Balancing Testing System Designed for Flexible Rotor

Abstract

In this paper, a dynamic-balancing testing system is designed. The innovative feature of the testing system is the dynamic balancing of the rotor system with robustness and high balance efficiency which meets the requirements of engineering application. The transient characteristic-based balancing method (TCBM) interface and the influence coefficient method (ICM) interface are designed in the testing system. The TCBM calculates the unbalance by the transient vibration responses while accelerating rotor operating without trail-weight. The ICM calculates the unbalance by the steady-state vibration responses while the rotor system operates with trail-weight and constant speed. The testing system has the functions of monitoring operations synchronously, measuring and recording the required vibration responses, analyzing the dynamic characteristics, and identifying the unbalance parameters. Experiments of the single disc rotor system are carried out, and the maximum deflection of the measuring point has decreased by 73.11% after balancing by the TCBM interface. The maximum amplitude of the measuring point at 2914 r/min has decreased by 77.74% after balancing by ICM interface, while the maximum deflection during the whole operation has decreased by 70.00%. The experiments prove the effectiveness of the testing system, while the testing system has advantages of convenient and intuitive operation, high balance efficiency, and security.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Rotating machinery is widely used in power generation, petrochemical, metallurgy, aerospace, and other fields for its rotating function required in special working conditions. Lu et al. [1] introduced the actual applications of the order reduction methods in solving rotor speed-varying transient problems. Fu et al. [2] analyzed the transient vibrations of an accelerating rotor system under both random and uncertain-but-bounded parameters. In the field of substructures of the rotor system, Xie and Zhu [3] investigated the lubrication characteristics of floating ring bearing under the circumstance of considering multicoupling factors. With the rapid development of high speed and heavy power rotating machinery, detecting and eliminating faults of the rotor system have always been a hot issue in the field of rotating machinery design, research, and development. In the past decades, the research contents of rotor fault detection and elimination include the identification of initial unbalance [4–6], the resonance, flutter and damage monitoring of the blade [7–9], the looseness [10–12], the rotor/stator rubbing [13–16], crack [17–19], and misalignment effects [20–22].

In engineering, the rotor fault of the initial unbalance caused by the process preparation error must be solved. And the rotor system is required to be prebalanced after design and production to reduce the extra vibration response caused by the initial unbalance of the rotor system, which makes the first step of detecting and eliminating faults is the rotor dynamic balancing. The mature dynamic-balancing methods of the rotor system at present are the influence coefficient method (ICM) [23, 24] and the modal balancing method (MBM) [25–27]. However, the ICM requires balancing with constant rotational speed. Multiple constant rotational speeds are required to solve the contradiction equations for balancing in a certain range of rotating speed which will definitely reduce the efficiency of the balancing process. The MBM identifies the unbalance parameters by modes of each order and the core principle of the MBM is mode superposition.

Both the above two methods require test runs with trail-weight while adding trail-weight to the rotor system may lead to two problems. One is that the position of adding trail-weight may be sensitive to the main structure of the rotor and may lead to the failure of obtaining the response during the test run with trail-weight. The other is that there is no basis to determine the qualities and directions of the trail-weight. Once the acute angle is formed by the direction of adding trail-weight and unbalance azimuth, the vibration response of the rotor system will increase and may cause damage to the main structure of the rotor system.

The mature approach of dynamic balancing without trial weights is to optimize the ICM and the MBM. The parameters that the ICM or the MBM required for calculating unbalance can be obtained through dynamic simulation of the rotor system in this way. And the unbalance of the rotor system can be identified by the ICM or the MBM without trail-weight. For example, Yao et al. [28] proposed a dual-objective optimization method and integrated the proposed method with least-square ICM. Bin et al. [29, 30] investigated the weighted influence coefficient matrix by adding simulated exciting force to the finite element model of the multidisc series shafting rotor system and then identified the unbalance of the rotor system by the ICM without trail-weight. Li et al. [31] calculated the dynamic characteristics of the rotor by finite element analysis and identified the unbalance of the four-disc simply supported structure rotor and turbo-shaft engine power turbine rotor by N and N + 2 plane MBM without trial weights, respectively. Ye et al. [32] introduced dynamic similitude theory and dimensional analysis method for optimizing the ICM and proposed the balancing method without trial weights by analyzing the similitude relationship of the influence coefficient between similarity and prototype rotor system.

Other approaches for balancing the rotor system without trail-weight are based on the dynamic characteristics of the rotor system. Zou et al. [33] estimated unbalance loads of the state space model of the rotor system by using the Kalman filter. Zhao et al. [34] identified the unbalance of the rotor system through transient characteristics of unbalance loads, while the unbalance loads are estimated by vibration response and system characteristics of the rotor system.

Although the methods mentioned above have achieved a good balancing effect in the dynamic-balancing field, the interface of the dynamic-balancing testing system designed and developed in relevant filed is less than other fields such as the interfaces of control systems designed based on excellent control algorithms [35, 36], the testing platforms of robot operation designed based on computer vision, virtual reality, and mechanical learning algorithms [37, 38]. The most beneficial function of the dynamic-balancing testing system is field balancing of the rotor system based on the interface integrated by source programs of the specific dynamic-balancing method. Meanwhile, the functions of displaying the measuring signal during operation, analyzing the measuring data in the process of acquisition, are also designed in the testing system, which will simplify the balancing process and improve the balancing efficiency.

In this paper, we proposed a novel method for balancing the rotor system without trail-weight which is named the transient characteristic-based balancing method (TCBM). The excitation force of the accelerating unbalanced rotor system will be calculated by the transient vibration responses and dynamic load identification technique. And the unbalance parameters are identified by the analysis of the amplitudes and phases of the calculated excitation forces. For simplifying the balancing process and improving the balancing efficiency in experimental research and engineering, the dynamic-balancing testing system is integrated by TCBM. The interface of the ICM is also integrated for demonstrating the accuracy of the unbalance identification by the TCBM and making the testing system more universal.

The basic theories and the balancing processes of the TCBM and ICM are introduced in Sections 2.1 and 2.2, respectively. The composition of the dynamic-balancing testing system, the functions, and the general requirements of the hardware and software modules are introduced, respectively, in Section 3. To verify the balancing effect of the dynamic-balancing testing system, the dynamic-balancing experiments of a single disc rotor system are carried out in Section 4. The balancing results prove that the dynamic-balancing testing system can be effectively applied to the field of dynamic balancing, while the testing system owns the benefits of concise interfaces and high balancing efficiency and security.

2. The Basic Methods of the Dynamic-Balancing Testing System

2.1. The Basic Theory of the TCBM

The TCBM calculates the unbalance parameters of the rotor system by the transient vibration responses and dynamic load identification technique. The TCBM does not require the test run with trail-weight and the TCBM is suitable for balancing procedures with the conditions whatever variable rotational speeds or constant rotational speeds. Through the dynamic load identification technique, there is a relationship between the vibration response and the excitation force of the rotor system which can be expressed as$$\begin{array}{}\text{(1)}& \mathbf{U}\omega =\mathbf{H}\omega \mathbf{F}\omega ⟶\mathbf{F}\omega ={{\mathbf{H}}^T\mathbf{H}}^{-1}{\mathbf{H}}^T\mathbf{U}\omega ,\end{array}$$where U (ω) is the Fourier transform of U (t), U (t) is the dynamic response vector which represents the output of the system, which is a column vector that contains L elements, and L represents the total number of vibration measurement points. F (ω) is the Fourier transform of F (t), F (t) is the external force vector which represents the input of the system, F (t) is a column vector that contains P elements, and P represents the total number of the unbalance excitation force while L ≥ P in general. H (ω) is the FRF matrix of the system.

Considering the condition of L = P, the FRF matrix consists of the modal FRF matrix and the modal shapes matrix which can be described as$$\begin{array}{}\text{(2)}& \mathbf{H}\omega =\mathbf{\Phi }{\mathbf{H}}_q\omega {\mathbf{\Phi }}^T,\end{array}$$where H_q (ω) = diag [H₁ (ω), H₂ (ω), ···, H_r (ω), ···, H_N (ω)] is the modal FRF matrix and H_r (ω) is the rth modal FRF of the system which can be expressed as$$\begin{array}{}\text{(3)}& H_r\omega =\frac{1}{M_r{\omega }_r^2-{\omega }^2+2j\omega {\omega }_r{\xi }_r},\hspace{1em}r=\mathrm{1,2},\dots ,N,\end{array}$$where M_r, ω_r, and ξ_r are the rth modal mass, modal frequency, and modal damping ratio of the system, respectively. ω is time-varying frequency.

For a nonproportional damping system, the unbalance excitation force can be expressed as$$\begin{array}{}\text{(4)}& \mathbf{F}\omega ={{\mathbf{\Phi }}^{\ast }{{\mathbf{\Phi }}^{\ast }}^T}^{-1}{\mathbf{\Phi }}^{\ast }{\mathbf{M}}_r{\mathbf{\omega }}_r^2-{\mathbf{\omega }}^2+2j\mathbf{\omega }{\mathbf{\omega }}_r{\mathbf{\xi }}_r{{\mathbf{\Phi }}^T\mathbf{\Phi }}^{-1}{\mathbf{\Phi }}^T\mathbf{U}\omega ,\end{array}$$where M_r is the diagonal matrix composed of modal mass, Φ = {ϕ₁, ϕ₂, ···, ϕ_r, ···, ϕ_N}, Φ∗ = {ϕ∗₁, ϕ∗₂, ···, ϕ∗_r, ···, ϕ∗_N}, (r = 1, 2, ···, N) are the complex modal matrices of the rotor system, and ϕ_r, ϕ∗_r are the complex modal vectors; the above parameters can be calculated by FEM or modal experiment. ω_r and ξ_r are the modal frequency and modal damping ratio, respectively, ω_r and ξ_r can be obtained by the Bode diagram of the rotor system, and ω is the diagonal matrix composed of the time-varying frequency which can be counted by setting the Key Phase Signal (KPS) to the rotor test rig.

In the premise of the radial displacement of the rotor system which can be expressed by the horizontal and the vertical components, the displacement vector in equation (4) can be divided into x and y directions corresponding to the horizontal and the vertical components. Assuming that the rotor system is isotropic and considering the difficulty of obtaining higher-order modes in simulations or experiments of the rotor system, the calculated excitation force vectors can be rewritten by equation (4) which is based on the front M real modal analysis of the rotor system as shown in the following equation:$$\begin{array}{}\text{(5)}& \begin{array}{c}{\mathbf{F}}_x\omega ={{\mathbf{\Phi }}^{\ast }{\mathbf{\Phi }}^{\ast T}}^{-1}{\mathbf{\Phi }}^{\ast }{\mathbf{M}}^{\ast }{\mathbf{\omega }}^{\ast 2}-{\mathbf{\omega }}^2+2j\mathbf{\omega }{\mathbf{\omega }}^{\ast }{\mathbf{\xi }}^{\ast }{{\mathbf{\Phi }}^T\mathbf{\Phi }}^{-1}{\mathbf{\Phi }}^T{\mathbf{U}}_x\omega ,\\ {\mathbf{F}}_y\omega ={{\mathbf{\Phi }}^{\ast }{\mathbf{\Phi }}^{\ast T}}^{-1}{\mathbf{\Phi }}^{\ast }{\mathbf{M}}^{\ast }{\omega }^{\ast 2}-{\mathbf{\omega }}^2+2j\mathbf{\omega }{\mathbf{\omega }}^{\ast }{\mathbf{\xi }}^{\ast }{{\mathbf{\Phi }}^T\mathbf{\Phi }}^{-1}{\mathbf{\Phi }}^T{\mathbf{U}}_y\omega ,\end{array}\end{array}$$where F_x (ω) and F_y (ω) represent the calculated excitation force vector of the x and y direction, respectively, U_x (ω) and U_y (ω) represent the displacement of the x and y direction, respectively, U_x (ω) = [x₁ (ω), x₂ (ω), ···, x_N (ω)]^T, U_y (ω) = [y₁ (ω), y₂ (ω), ···, y_N (ω)]^T, [Φ]_N × M = [ϕ₁, ϕ₂, ···, ϕ_r, ···, ϕ_M], [Φ^∗]_N×M = [ϕ∗₁, ϕ∗₂, ···, ϕ∗_r, ···, ϕ∗_M], M∗ = diag [M₁, M₂, ···, M_r, ···, M_M], ω∗ = diag [ω₁, ω₂, ···, ω_r, ···, ω_M], ω = diag [ω, ω, ···, ω]_M×M, and ξ∗ = diag [ξ₁, ξ₂, ···, ξ_r, ···, ξ_M].

In this way, we can calculate the unbalance excitation forces of the rotor system by the vibration response and the modal parameters of the rotor system.

The excitation force vectors calculated by equation (5) consist of the excitation force of the discs in the time domain which can be expressed as$$\begin{array}{}\text{(6)}& \begin{array}{c}{F_{x}t}_{Z\times 1}=F_{x1}t,F_{x2}t,\dots ,F_{xi}t,\dots ,F_{xZ}t,\\ {F_{y}t}_{Z\times 1}=F_{y1}t,F_{y2}t,\dots ,F_{yi}t,\dots ,F_{yZ}t,\end{array}\end{array}$$where Z is the number of discs. The elements of the excitation force vectors contain the unbalance parameters of the discs which can be rewritten as$$\begin{array}{}\text{(7)}& \begin{array}{c}{F}_{xi}t=m_i{e}_{i}At\cos B_{x}t+{\phi }_{ei},\\ F_{yi}t=m_i{e}_{i}At\sin B_{y}t+{\phi }_{ei},\end{array}\end{array}$$where m_ie_iA (t) is the term of amplitude and m_i and e_i represent mass and eccentricity of ith disc, respectively. $A=\sqrt{{\dot{\phi }}^{4}t+{\alpha }^2}$, where $\dot{\phi }t$ and α are the angular velocity and acceleration of the disc. $\dot{\phi }t=\dot{\phi }{t}_0+\alpha t$, and $\dot{\phi }{t}_0$ is the initial angular velocity. B_x (t) and B_y (t) are the terms of phases, B_x (t) = φ (t) + ψ_x (t) and B_y (t) = φ (t)+ψ_y (t). φ (t) is the rotational angle of the disc, $\phi t=\phi t_0+\dot{\phi }tt+1/2\alpha t^2$, and φ (t₀) is the initial angle of the disc. Generally, we assume φ (t₀) = 0, and ψ_x (t) and ψ_y (t) can be expressed as$$\begin{array}{}\text{(8)}& \begin{array}{cc}{\psi }_{x}t=\arctan -\frac{\alpha }{{\dot{\phi }}^{2}t},& {\psi }_{x}t\in -\frac{\pi }{2},0,\\ {\psi }_{y}t=\arctan -\frac{\alpha }{{\dot{\phi }}^{2}t},& {\psi }_{y}t\in -\frac{\pi }{2},0.\end{array}\end{array}$$

On the premise of the rotor system operating with constant rotational speed, $\dot{\phi }{t}_0$ is a fixed value, α = 0, $\phi t=\dot{\phi }{t}_{0}t$, ψ_x (t) = ψ_y(t) = 0. For the rotor system operates with variable rotational speeds, $\dot{\phi }{t}_0$ varies with time and α is approximated as a fixed value; $\dot{\phi }{t}_0$ and α can be calculated by the KPS.

To conclude, the data of the unbalance force calculated by equation (4) are introduced to equations (7) and (8), and the unbalance parameters (eccentricities and azimuths) of the rotor system can be calculated by equations (7) and (8).

The balancing process of the TCBM is shown in Figure 1.

[figure omitted; refer to PDF]

Meanwhile, a total number of 13 signal windows are displayed in three columns in the data acquisition interface, including 1 KPS display window, 4 measured signal display windows, and the rotational speed-amplitude, rotational speed-phase display windows of the Bode diagram corresponding to the measured signals. The other channels can be displayed by selecting the “channel options” menu at the left end of the interface as shown in Figure 6. The right end of the data acquisition interface displays the amplitude and phase of the last point in the Bode diagram drawn by the measured signals of each channel. The basic parameters are set below the acquisition interface which includes rotational speed, cycle, and frequency of the rotor test rig, basic setting of sampling. The clear button and the options of adjusting the rotational speed-phase of the Bode diagram are arranged in the lower right corner, which can complete the functions of clearing the plot or phase shift according to the requirements.

3.2.1. The System Operation of the Software

The system operation menu is set in the upper left corner of the software, which has two options in the drop-down menu. One is the acquisition channel setting button and the other is the sensor coefficient setting button. The acquisition channel setting interface is shown in Figure 7. The marks of A and B in the Displacement Sensor part represent the data transmission interface of the ECD sensor and GD sensor, respectively. And a total of 16 displacement channels are set for recording the vibration response. The marks of A and B in the KPS part represent the data transmission interface of the PE sensor and ECD sensor, respectively. The Acceleration Sensor part owns 6 data transmission interfaces for measuring the acceleration of measuring points.

[figure omitted; refer to PDF]

The sensor coefficient setting interface is shown in Figure 8, which represents the relationship of the sensors’ input and output:$$\begin{array}{}\text{(19)}& \begin{array}{c}{x}_{\text{output}}=\Pi x_{\text{input}},\\ x_{\text{plot}}=K{x}_{\text{output}}+b,\end{array}\end{array}$$where Π is the relationship between the input signal and the output signal of the sensor. x_plot is the display signal of the data acquisition interface. K and b are the sensor coefficient, and x_plot should be equal to x_input during operation.

[figure omitted; refer to PDF]

As shown in Figure 15, the maximum deflections of the disc before and after balancing are 4.50 × 10^–4 m and 1.21 × 10^–4 m, respectively. The maximum deflection of the disc has decreased by 73.11% after balancing by the TCBM interface, which suggests the effectiveness of the TCBM interface in balancing the single disc rotor system.

4.2. The Verification of the ICM Interface in Dynamic Balancing

Requiring the rotor system operates with lower angular acceleration at first and observing the response signals of the rotor system on the acquisition interface, we can obtain the maximum vibration response of the rotor system while operating near the critical speed. For the single disc rotor system shown in Figure 12, the rotational speed corresponding to the maximum vibration response of the rotor system is 2914 r/min through observation. On the premise of a constant rotational speed of 2914 r/min, the response data of the measuring point in the situation of the rotor system operating without trail-weight and with trail-weight are measured and recorded by the ICM interface, respectively. The trail-weight is a mass of 0.72 g placed at the threaded hole of 0°. As shown in Figure 16, the “without trail-weight” module in the ICM interface is used for analyzing the response data of the rotor system while operating without trail-weight, and the “with trail-weight” module in the ICM interface is used for inputting the parameters of the trail-weight and analyzing the response data of the rotor system while operating with trail-weight. The counterweight calculated by the ICM interface is 1.32 g∠304.64°, which should be decomposed into counterweights of 0.62 g to the threaded hole of ∠292.5° and 0.73 g to the threaded hole of ∠315°. Operating with the constant rotational speed of 2914 r/min after adding the counterweights to the disc, the comparison of the steady-state vibration responses before and after balancing by the ICM interface is shown in Figure 17. Figure 18 is the comparison of the experimental transient deflections before and after balancing by the ICM interface.

[figure omitted; refer to PDF]

It can be summarized from Figures 17 and 18 that the steady-state vibration amplitudes of 2914 r/min and transient deflections are all significantly decreased after balancing by the ICM interface. The maximum vibration amplitudes of the rotor system at 2914 r/min before and after balancing by ICM interface are 4.47 × 10^–4 m and 9.95 × 10^–5 m, which have decreased by 77.74%. In Figure 18, the maximum deflection has decreased by 70.00% while the maximum deflections of the disc before and after balancing are 4.50 × 10^–4 m and 1.35 × 10^–4 m, respectively. These results prove that the ICM interface has good performance for balancing the single disc rotor system.

In conclusion, the experiments in this section show that the dynamic-balancing testing system succeeds in balancing the single disc rotor system by both the TCBM interface and the ICM interface. The TCBM interface identifies the unbalance parameters by the transient vibration response while the rotor system operates with constant angular acceleration, and the maximum deflection of the measuring points has decreased by 73.11% after balancing by the TCBM interface. Other than the TCBM interface, the ICM interface requires steady-state vibration response data while the rotor system should operate with constant rotational speed. In addition, the ICM interface requires two operations of the rotor system while one measures the signal of the rotor system without trail-weight and the other measures the signal of the rotor system with trail-weight. The steady-state vibration response amplitudes of 2914 r/min are significantly decreased after balancing by the ICM interface, and the maximum deflection of the measuring points has decreased by 70.00% after balancing by the ICM interface. The dynamic-balancing results calculated by the above two interfaces prove that the dynamic-balancing testing system can be effectively applied to the field of dynamic balancing of the flexible rotor system. And the testing system is more convenient than the source program of the dynamic-balancing methods in balancing the rotor system, simplifying the balancing steps and improving the balancing efficiency and safety.

5. Conclusion

In this paper, we designed a dynamic-balancing testing system for the flexible rotor system, whose innovative features are the interfaces according to the principles of TCBM and ICM for dynamic balancing of the rotor system. The functions of the testing system are monitoring the operations of the rotor synchronously, measuring and recording the required vibration response of the rotor, analyzing the dynamic characteristics of the rotor, and finally identifying the unbalance parameters of the rotor. The experiments of the single disc flexible rotor are carried out to detect the functions of the testing system and verify the effectiveness of the dynamic-balancing method. According to the experiment tests, some conclusions are made:

(1) The experimental tests of the dynamic-balancing testing system targeted to the single disc rotor system show that the hardware and the software can work well and realize their own respective functions.

(2) The maximum deflection of the measuring point has decreased by 73.11% after balancing by the TCBM interface. The maximum amplitude of the measuring point at 2914 r/min has decreased by 77.74% after balancing by ICM interface, while the maximum deflection during the whole operation has decreased by 70.00%. The dynamic balancing results demonstrate the application of the above interfaces are both satisfactory for the field of dynamic balancing.

(3) In addition to the above good dynamic-balancing results, the testing system designed in this paper improves safety in balancing due to its function of monitoring the operations of the rotor synchronously. Meanwhile, the abstract source programs are integrated and designed in the form of buttons on the interface of the testing system. The functions of the source programs can be realized by clicking the corresponding buttons on the interface, which will make the balancing steps clearer. The functions of analyzing the measuring signals synchronously and processing the recording data on the interface make the testing system more efficient in signal processing.

In conclusion, the dynamic-balancing testing system could be successfully used in the field balancing of rotor system which has the advantages of simplifying the balancing process and improving the balancing efficiency.

Acknowledgments

This study was funded by the National Natural Science Foundation of China (Grant nos. 12072263, 11972295, and 11802235), Natural Science Foundation of Shaanxi Province (Grant no. 2020JQ-129), and State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures (Grant no. KF2020-26).

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