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Investigation of Resonance Frequency and Stability of Solutions in a Continuous Rotor System

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Abstract

Purpose

The primary objective of this work is to analyze the effect of different system parameters on the resonance frequency of the linear rotor system. The effect of internal damping on the stability of the system is also investigated and a critical frequency ratio separating the stable and unstable regions is obtained.

Methods

A continuous rotor system is modeled by considering some critical factors, like the gyroscopic and rotary inertia effects of disc and shaft cross-sections, internal damping, large shaft deformation, and restriction to shaft axial motion at the bearings. The bearings are replaced by spring-dashpot systems along both horizontal and vertical directions. The governing partial differential equations (PDE’s) for the vibrations of the disc along the horizontal and vertical directions are derived by employing Hamilton's principle. The governing equations are then reduced to a set of ordinary differential equations (ODEs) using the method of modal projection. The large deformation, restriction to axial motion of the shaft, and the nonlinear stiffness of the end springs yield nonlinearities in the system governing equations. However, only the linear system is considered in this paper.

Results and Conclusion

The parameters in the dimensionless form of the governing equations are functions of some independent variables. These independent variables, on the other hand, are associated with the material and geometrical properties of the rotor system. The effect of the independent parameters on the system dynamics is analysed wherein the variation of the dependent parameters is also monitored. An appropriate design of a rotor system can be achieved through a methodical analysis, like the one this study addresses.

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Data Availability

The datasets generated during and/or analysed during the current study are not publicly available due to ongoing research or future publications but are available from the corresponding author on reasonable request.

Abbreviations

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

Mass of disk

\(L\) :

Length of shaft

\(A\) :

Cross sectional area of shaft

\(\rho \) :

Density of shaft material

\(e\) :

Eccentricity of disk mass center away from the geometric center

\({I}_{dx}\) :

Mass moment of disc about horizontal (\(X\)-) direction (disk radial direction)

\({I}_{dy}\) :

Polar mass moment of inertia of disk (\(Y\Rightarrow \) axial direction of the disk)

\(I=A{R}_{s}^{2}\) :

Area moment of inertia of shaft cross-section

\({R}_{s}\) :

Radius of gyration of shaft cross-section

\({L}_{1}\left({l}_{1}\right)\) :

Dimensional (dimensionless) disk position on shaft from left end bearing

\({\Psi }_{1}(y)\) :

1St orthonormal mode shape

\(t \left(\tau \right)\) :

Dimensional (dimensionless) time

\(U\left(Y,t\right) (u\left(y,\tau \right))\) :

Dimensional (dimensionless) displacement of disk along \(X\)- direction

\(W\left(Y,t\right) (w\left(y,\tau \right))\) :

Dimensional (dimensionless) displacement of disk along vertical (\(Z\)-) direction

\({\theta }_{x}and{\theta }_{y}\) :

Rotational displacements about horizontal and vertical directions, respectively

\(T\) :

Total kinetic energy of the rotor system

\(V\) :

Total strain energy of the shaft

\({r}_{e }\left(=e/L\right)\) :

Dimensionless eccentricity ratio

\({r}_{m }\left(=\rho AL/{M}_{d}\right)\) :

Dimensionless mass ratio

\(r={R}_{s}/L\) :

Dimensionless radius of gyration

\({r}_{g}=I/A{L}^{2}={r}^{2}\) :

Shaft geometric parameter

\({r}_{f }\left(=\Omega /\Upsilon\right)\) :

Dimensionless frequency ratio

\(\Omega \) :

Spin speed of the rotor

\(\Upsilon=\sqrt{EI/\left(\rho A{L}^{4}\right)}\) :

Natural frequency of shaft

\({r}_{dx }\left(={I}_{dx }/\rho A{L}^{3}\right)\) :

Mass moment of inertia of the disc about \(x\)-axis

\({r}_{dy }\left(={I}_{dy }/\rho A{L}^{3}\right)\) :

Mass moment of inertia of the disc about \(y\)-axis

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

Undamped natural frequency (dimensionless) of the rotor system

\(G\) :

Dimensionless gyroscopic coefficient

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

Effective mass

\({q}_{0}\) :

Amplitude of excitation (dimensionless)

\({\varvec{\updelta}}\) :

Dirac delta function

\({C}_{ex}({c}_{ex})\) :

Dimensional (dimensionless) damping constant due to air resistance in horizontal direction

\({C}_{ez}({c}_{ez})\) :

Dimensional (dimensionless) damping constant due to air resistance in vertical direction

\({C}_{1}({c}_{1})\) :

Dimensional (dimensionless) viscous damping constant of left bearing

\({C}_{2}({c}_{2})\) :

Dimensional (dimensionless) viscous damping constant of right bearing

\({C}_{i}({c}_{i})\) :

Dimensional (dimensionless) internal damping coefficient

\({\upmu }_{x} and {\upmu }_{z}\) :

Dimensionless damping factor in horizontal and vertical direction, respectively

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

Internal Damping factor

\({K}_{11}({k}_{11})\) :

Dimensional (dimensionless) linear spring coefficient along \(X\)- direction

\({K}_{21}({k}_{21})\) :

Dimensional (dimensionless) linear spring coefficient along \(Z\)- direction

\({K}_{13}({k}_{13})\) :

Dimensional (dimensionless) non-linear spring stiffness along \(X\)- direction

\({K}_{23}({k}_{23})\) :

Dimensional (dimensionless) non-linear spring stiffness along \(Z\)- direction

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Acknowledgements

This work has been supported by Startup Research Grant (SRG) funded by DST-SERB under Project File no. SRG/2019/001445 and National Institute of Technology Calicut under Faculty Research Grant. We are also grateful to Dr. M. D. Narayanan for his support and advice.

Funding

This work has been supported by DST SERB—SRG under Project File no. SRG/2019/001445 and National Institute of Technology Calicut under Faculty Research Grant. We are also grateful to Dr. M. D. Narayanan for his support and advice.

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Institute of Technology Calicut, Kerala, 673601, India

    Amit Malgol, K. P. Vineesh & Ashesh Saha

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  1. Amit Malgol
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  2. K. P. Vineesh
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  3. Ashesh Saha
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Amit Malgol, Vineesh K P and Ashesh Saha. The first draft of the manuscript was written by Amit Malgol and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Amit Malgol.

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Malgol, A., Vineesh, K.P. & Saha, A. Investigation of Resonance Frequency and Stability of Solutions in a Continuous Rotor System. J. Vib. Eng. Technol. (2024). https://doi.org/10.1007/s42417-024-01347-7

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  • DOI: https://doi.org/10.1007/s42417-024-01347-7

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