Elsevier

Applied Mathematical Modelling

Volume 87, November 2020, Pages 351-371
Applied Mathematical Modelling

Application of non-smooth NES in vibration suppression of rotor-blade systems

https://doi.org/10.1016/j.apm.2020.06.014Get rights and content

Highlights

  • A non-smooth nonlinear energy sink is applied for vibration suppression of rotor-blade system.

  • The effectiveness of vibration suppression of the nonlinear energy sink is numerically studied.

  • And a comparative study with a linear dynamic vibration absorber of the same mass is also carried out.

Abstract

The non-smooth nonlinear energy sink (NSNES) is used to suppress the vibration of the rotor-blade system. Firstly, the structure and working principle of the NSNES for rotor-blade system are introduced. Then, the dynamics model of the rotor-blade-NSNES system is established by Lagrangian method. And then, numerical simulations are applied to evaluate the vibration suppression ability of the NSNES on rotor and blade. The results show that the suppression rates of NSNES on the rotor and the blade can reach 81% and 74% in steady state resonance under given parameters, respectively; and for transient vibration of blade, a 1.85 times dissipating speed is obtained in rotor-blade system with NSNES than that without NSNES. In particular, NSNES has better vibration suppression capability than linear dynamic vibration absorber (LDVA) when both have the same vibration absorption mass.

Introduction

Excessive vibration can cause significant damage to the overall rotating machinery. In the past years, researchers have proposed various methods to reduce vibration most of which can be divided into passive type and active type. The passive vibration suppression methods mainly include variable stiffness or damping [1], dynamic balance [2,3], etc.; active approaches [4], [5], [6] mostly directly actuates to force the required degrees of freedom, thereby actively controlling vibration.

Besides the above methods, it is also very common to add dynamic vibration absorber (DVA) to the rotor system. The centrifugal vibration absorbers [7], [8], [9] are widely applied to suppress torsional vibration, such as in helicopters’ rotors or in internal combustion engines. Doubrawa et al. [10] designed a viscoelastic vibration absorber, in which the model of the coupling system is represented in the modal subspace, and the vibration suppression effect of the absorber has been also verified by experiments; Campos et al. [11] designed a rotating DVA that suppresses the vibration of the inner cylinder of the washing machine. In order to expand the vibration suppression frequency bandwidth, researchers have also invented some active DVAs. For example, Hu et al. [12] developed a LDVA whose natural frequency can be controlled by the switch of the electromagnet; Yao et al. [13] designed a LDVA combining negative stiffness with positive stiffness together, whose natural frequency can be changed by the length of the supporting beams.

The effectiveness of the LDVA has been verified by above literatures, but the LDVA has a narrow absorption band and often has a complicated structure, so its application range is limited. For the purpose of overcoming this problem, NESs have entered into the visual field of the researchers. NES is one of the typical representatives of the nonlinear vibration absorbers. Though the structure of NES is similar to LDVA , the vibration suppression principle, performance and effects of the NES are all different from the LDVA. The adding of NES can make a huge difference in the dynamic performance of the entire system [14], including the Targeted Energy Transfer (TET) behavior in transient vibration suppression [15] and the Strongly Modulated Response (SMR) behavior in steady-state vibration suppression [16].

Refs. [17], [18], [19] used NES to suppress the vibration of the rotor bearing system and presented a good vibration suppression effect. Wang et al. [20] considered the working conditions of multi-frequency excitation and studied the vibration suppression performance of NES on aeroengines. Guo et al. [21] used NES to reduce the vibration of the unbalanced hollow rotor system, and the vibration absorption effect of the NES was compared with the LDVA. Ref. [22] developed a NES to mitigate the vibration of the rotor system and also to reduce the rubbing faults between the rotor and the stator. Dasgupta et al. [23] studied the steady-state and transient dynamics of an eccentric flexible rotating shaft with an additional NES. Yao et al. [24,25] designed a variety of NESs for rotor system vibration, such as a negative stiffness parallel NES and a grounded non-smooth NES. In [26], Bab studied the suppression performance of a smooth NES on the vibration of a rotating beam subjected to external forces, and also compared the vibration suppression effect with the LDVA.

In addition to the above application research, NESs have also been applied to suppress the torsional vibration of rotor system of automotive drivetrains. For example, Ref. [27] studied the suppression of transient vibration of crankshaft by NES. In Ref. [28], Haris et al. studied the suppression effect of NES to torsional vibration of the vehicle transmission system. Next, in Ref. [29], the team conducted an experimental study of NES torsional vibration suppression. In [30,31], multi-NES suppression studies on the transmission system torsional vibration were carried out. Compared with bending vibration, torsional vibration NES placement options are more diverse, placed in the coupling or the end of the shaft system and etc. Torsional vibration NES can achieve a good suppression effect, so the application potential in actual engineering is greater.

Researchers also use NESs to suppress the vibration of rotating blades. Nima et al. [32] studied the aeroelastic behavior of rotor blades when installing NES, and also compared the vibration attenuation performance with two linear dampers. Lee et al. [33] studied the suppression of limit cycle oscillations (LCO) in a van der Pol (VDP) oscillator by a passive grounded or ungrounded NES. Following this research, Lee et al. [34] studied the inhibition performance of NES on aeroelastic instability of LCOs in a two-degree-of-freedom wing model. Finally, the experimental verification of these theoretical observations was presented in the accompanying paper [35].

In rotating machinery, there is often a coupling relationship between the rotor and the blade, so it is more practical and realistic to analyze the vibration of rotor-blade coupling systems. And for the sake of convenience, the small-massed absorbers can be installed inside the rotating part, such as blades or shafts. Duffy et al. [36] analyzed the influence of a self-tuning impact damper on the vibration attenuation of rotating components of turbo-machinery, and showed that the damper can increase the fatigue life of the components. Tehrani et al. [37] studied the application of tuned mass damper (TMD) and nonlinear energy sink (NES) on vibration mitigation of a flexible bladed rotor system.

However, the above papers are all theory studies. Few NESs with actual structures are designed in the vibration suppression of the rotor-blade coupling system. Therefore, in this paper, a NSNES with practical structure is used to suppress the vibration of the rotor-blade coupling system. The dynamics of the rotor-blade-NSNES is studied and the suppression effects of NSNES to the rotor and the blades are analyzed. In particular, the vibration suppression capability of the NSNES and the LDVA of the same mass is comparatively studied.

Section snippets

Structure of the NSNES

The specific structure of the NSNES has been introduced in Ref. [38] and shown in Fig. 1(a), which includes NES mass, piecewise stiffness beam model and support. In Ref. [38], the design concept of applying piecewise linear stiffness spring as NES elastic element is proposed and realized by the combination of leaf springs. And, the effectiveness of the NES in vibration suppression of single-degree-of-freedom systems is verified by numerical simulation and testing. The focus of this article is

Model analysis

The Lagrange equations is applied to obtain the dynamics equations of each analytical model in next sections, which isddt(Ttotalq˙)Ttotalq+Utotalq=Ftotalwhere, Ttotal and Utotal are kinetic energy and potential energy of the system, respectively. Ftotal is external force vector that acts on the system. q,q˙ is generalized coordinate vector of the system.

The structural model of the rotor-blade system is shown in Fig. 3. The rotor model is an isotropic single-axis-single-disk-single-span

Rotor system parameters

In the numerical simulation, the simulation parameters of the rotor system are shown in Table 1.

Blade parameters

In the numerical simulation, the simulation parameters of the blades are shown in Table 2.

NSNES parameters

According to the working principle of the NSNES in Fig. 2(b), it is known that the piecewise linear stiffness knj (j = 1, 2, 3, 4) and the gap values e1, e2, and e3 fit into a cubic nonlinear force. If the function relationship of this cubic nonlinear force is y = kcx3, then kc is the cubic nonlinear stiffness

Structural design of the LDVA

This section will design a LDVA for vibration suppression of the rotor-blade system, which is then compared with the NSNESs above. By redesigning the width of the connecting beams of the I structure in Section 3.2.1, a LDVA can be formed, as shown in Fig. 31. The LDVAs and the NSNESs are added to the same position of the blades, and the dynamics modeling process of the rotor-blade-LDVA system is similar to the rotor-blade-NES connecting beam system modeling process.

Parameters setting of the LDVA

For ease of comparison, the

Conclusions

The NSNES is applied for vibration suppression of rotor-blade system. The main conclusions of this paper can be summarized as follows:

  • (1)

    The NSNESs mounted on the tip of the blades can suppress the vibration of the rotor and the blades in the rotor-blade system, respectively.

  • (2)

    As suppressing the vibration of the rotor and the blade, the vibration suppression rates of the NSNESs can reach 81% and 74% in steady state resonance under given parameters, respectively. And the NSNES has the wide-band

Declaration of Competing Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgement

The authors would like to gratefully acknowledge the National Natural Science Foundation of China (Grant No. U1708257) and the Fundamental Research Funds for Central Universities of China (No. N180313009) for the financial support for this study.

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