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Nonlinear normal modes and dynamic balancing for a nonlinear rotor system

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Abstract

The basis of rotor dynamic balancing procedures depends on the linearization hypothesis of the system. However, some dynamical characteristics of mechanical elements of rotor systems are nonlinear. To increase the balancing efficiency, an improved algorithm is proposed by applying the Nonlinear Normal Modes (NNMs) to the modal balancing procedure. To demonstrate the accuracy and effectiveness of the proposed method, a Jeffcott rotor with nonlinear restore force is balanced by both the NNMs method and the linear modal method for comparation. The simulations show that the balancing results by the NNMs method are significantly better than those by the linear modal method, regardless of levels of nonlinearity and eccentricity, by comparing the percentage reduction in vibration amplitude at critical frequencies, spectra of responses and resonance curves.

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Data availability

The data that support the findings of this study are available from the corresponding author, upon reasonable request.

Abbreviations

\({A}_{{i}_{k}}\) :

Complex amplitude of the rotor response

\({a}_{{i}_{k}}\) :

Amplitude spectrum of the rotor response

\({\mathbf{b}}_{n}\) :

Trial balance vector

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

Total unbalance correction

\(\mathbf{C}\) :

Damping matrix

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

Eccentricity

\({\mathbf{F}}_{{\text{nl}}}\) :

Nonlinear function

\({\mathbf{F}}_{{\text{e}}}\) :

Unbalanced excitation

\({F}_{{\text{x}}},{F}_{y}\) :

Elastic restoring force in horizontal and vertical directions, respectively

\({f}_{1},{f}_{2}\) :

Dimensionless mass eccentricities

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

Gravity centers

\(\mathbf{K}\) :

Stiffness matrix

\({k}_{1},{k}_{2}\) :

Linear and nonlinear stiffness coefficients of the shaft

\(\mathbf{M}\) :

Mass matrix

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

Geometric centers

\(\mathbf{r}\) :

Response vector of the rotor system

\({\mathbf{r}}^{\boldsymbol{*}}\) :

Rotor response after attaching the trial masses

\({\mathbf{r}}_{{\text{d}}}\) :

Difference of the vibrations before and after adding the trial balance

\({u}_{{i}_{k}}\) :

Amplitude of the kth harmonic component in the NNM shapes

\({\mathbf{u}}_{i}\) :

NNM shapes in the complex amplitude vector form

\(\mathbf{U}\) :

Nonlinear mode shapes of the rotor system

\(x,\dot{x},\ddot{x}\) :

Displacement, velocity and acceleration in, respectively, of the horizontal oscillation

\(y,\dot{y},\ddot{y}\) :

Displacement, velocity and acceleration in, respectively, of the vertical oscillation

\({x}_{i,st},{y}_{i,st}\) :

Static displacements in horizontal and vertical directions, respectively

\({x}_{i,0}^{*},{y}_{i,0}^{*}\) :

Perturbation around the static equilibrium in horizontal and vertical directions, respectively

\({{\varvec{x}}}_{p}\) :

Periodic solutions of the NNM motion

\(\alpha \) :

Phase angle

\({\alpha }_{{n}_{{i}_{k}}}\) :

Correction factor for each of the frequency components

\({{\varvec{\upalpha}}}_{n}\) :

Correction coefficient vector of the trial balance

\(\mathrm{\angle }{\beta }_{{i}_{k}}\) :

Phase spectrum of the rotor response

\(\delta \) :

Arbitrary positive real scalar

\({\theta }_{{i}_{k}}\) :

Phase of the k th harmonic component in the NNM shapes

\({\lambda }_{1},{\lambda }_{2}\) :

Mechanical stiffness ratios

\(\Omega \) :

Dimensionless frequency ratio

\({\mu }_{1},{\mu }_{2}\) :

Dimensionless damping ratios

\(\omega \) :

Angular speed

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

Angular frequency of the kth harmonic component in the NNM shapes

NNMs:

Nonlinear normal modes

LNMs:

Linear normal modes

mX:

The rotational and the multiple of the rotational frequency components, m = 1, 2, 3,

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Acknowledgements

This work is supported by the National Natural Science Foundation of China through the grant Nos. 12021002 and 12132010.

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Authors and Affiliations

  1. School of Materials Science and Engineering, Tianjin University, Tianjin, 300072, China

    Tianzhu Wang

  2. Tianjin Key Laboratory of Advanced Joining Technology, Tianjin, 300072, China

    Tianzhu Wang

  3. Department of Mechanics, Tianjin University, Tianjin, 300350, China

    Qian Ding

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  1. Tianzhu Wang
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Wang, T., Ding, Q. Nonlinear normal modes and dynamic balancing for a nonlinear rotor system. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09654-0

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