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Modeling and dynamic analysis of a full flexible shaft–disk–blades system with nonlinear energy sinks (NESs) using modified SAFE diagram

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Abstract

In this paper, the application of nonlinear energy sinks (NESs) for the vibration suppression of the blades in a flexible shaft–disk–blades system is studied. The system is a simplified model of an industrial steam turbine rotor, which includes a shaft, a disk, and 37 packets of 7-connected blades. A two-level model order reduction is used to obtain the mathematical model of this industrial system. The frequency diagram of the whole system is obtained through the modal analysis of a cyclic symmetric finite element model of a single sector, which is practically the first model reduction. Some new families of flexible shaft–disk–blades system’s natural modes with only two harmonics appear on this diagram that are mostly related to the dynamic of the shaft. Hence, it is nominated for modified SAFE (Singh’s Advanced Frequency Evaluation) diagram in this study. Then, as a second model reduction, a two-degree of freedom reduced-order model of the sector is identified to model the system behavior around the 11th family of natural modes. A strong coupling between rotor components occurs at the first nodal diameter of this mode. The NESs, which consist of a small mass, an essentially nonlinear stiffness, and a linear damping, are mounted on the disk to indirectly suppress the vibration of the blades. Numerical study of system behavior and tuning of NES parameters show that the strongly modulated response occurs near the resonance, and the optimum NESs could effectively reduce the out-of-plane vibration of the blades.

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Abbreviations

\({c}_{b}\) :

Blade aerodynamic damping (\(2\pi {\zeta }_{b}\)), N s/m

\({c}_{N}\) :

NES linear damping, \(N s/m\)

\([E]\) :

Fourier transformation matrix

\(\{{e}_{i}\}\) :

ith Column of Fourier matrix

\({f}_{b}\) :

External force applied to the blade, N

\([K]\) :

Sector stiffness matrix

\([{K}_{c}]\) :

Coupling stiffness matrix

[\(\widetilde{K}\)], [\({\widetilde{K}}_{D}\)]:

Stiffness matrices of the system

\({k}_{b}\) :

Blade stiffness, N/m

\({k}_{c}\) :

Inter sector coupling stiffness, N/m

\({k}_{\rm ds}\) :

Disk–shaft stiffness, N/m

\({k}_{N}\) :

NES nonlinear stiffness coefficient, N/m3

\([M]\) :

Sector mass matrix

[\(\widetilde{M}\)], [\({\widetilde{M}}_{D}\)]:

System mass matrices

\({m}_{b}\) :

Blade mass, kg

\({m}_{\mathrm{ds}}\) :

Disk–shaft mass, kg

\({m}_{N}\) :

NES mass, kg

\(\overline{m }\) :

\({m}_{\mathrm{ds}}/{m}_{b}\)

N :

Number of sectors

R :

Disk radius, m

\({{\varvec{U}}}_{{\varvec{i}}}\) :

The ith sector’s DOFs in physical coordinates, m

\({u}_{N}\) :

NES displacement, m

\({u}_{k, b}\) :

Displacement of the blade in the kth modal mode, m

\({u}_{k, \mathrm{ds}}\) :

Displacement of the shaft–disk in the kth modal mode, m

ϵ :

NES mass ratio

γ b, γ ds :

Structural damping ratio of blade and disk–shaft

\({\kappa }_{d},{\kappa }_{c}\) :

Disk and coupling relative stiffnesses

\({\omega }_{b}\) :

Blade bending natural frequency, rad/s

\({\omega }_{k, n}\) :

Frequency of the nth family of natural modes at kth modal mode (or (k − 1)th ND), rad/s

σ :

Frequency detuning parameter

σ k :

Inter-blade phase angle at the kth modal mode of the system

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Authors and Affiliations

  1. Department of Mechanical Rotary Equipment, Niroo Research Institute, P.O. Box 14665-517, Tehran, Iran

    Jafar Aghayari & Saeed Bab

  2. Faculty of Mechanical & Energy Engineering, Shahid Beheshti University, A.C., Tehran, Iran

    Pedram Safarpour & Abbas Rahi

Authors
  1. Jafar Aghayari
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  2. Pedram Safarpour
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Correspondence to Jafar Aghayari.

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Aghayari, J., Safarpour, P., Rahi, A. et al. Modeling and dynamic analysis of a full flexible shaft–disk–blades system with nonlinear energy sinks (NESs) using modified SAFE diagram. J Braz. Soc. Mech. Sci. Eng. 44, 472 (2022). https://doi.org/10.1007/s40430-022-03767-6

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  • DOI: https://doi.org/10.1007/s40430-022-03767-6

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