pp 1–25 | Cite as

Estimation and elimination of eigenvalue splitting and vibration instability of ring-shaped periodic structure subjected to three-axis angular velocity components

  • Jinlong Liu
  • Shiyu WangEmail author
  • Zheren Wang
  • Nan Gao
  • Dongsheng Zhang


Stable operation is one of the most crucial requirements for resonators in vibratory gyroscopes and ultrasonic motors, but eigenvalue splitting can deteriorate operation stability. This work aims at the estimation and elimination of eigenvalue splitting and vibration instability of resonators arranged in a fashion of ring-shaped periodic structures (RPS). An analytical model is developed by Hamilton’s principle, where in-plane bending displacements, grouped supports and angular velocity components applied about three orthogonal directions are incorporated. Eigensolutions for the proposed rotational and mirror symmetric topologies are formulated by perturbation-superposition method, based on which eigenvalue splitting, vibration instability and their evolution with grouped supports and angular velocity are examined. The results verify the behaviors of splitting and instability share similar rules with those RPS having equally-spaced supports, but they change remarkably with grouping patterns. The dependences of grouping patterns and parameters on vibrations are demonstrated based on sample RPS. The splitting and instability are estimated by eigensolutions, and they can be suppressed or even eliminated by the proposed two types of topologies. Comparisons between the two topologies are made in terms of the requirements from engineering practice. Main results are also compared with those in the open literature.


Ring-shaped periodic structures Rotational symmetry Mirror symmetry Eigenvalue splitting Vibration instability 



Ring-shaped periodic structure


Backward traveling wave


Forward traveling wave

List of symbols

kv, ku

Stiffnesses of uniform radial and tangential supports


Inertial coordinates


Number of discrete supports


Discrete supports’ inclination angle relative to the radial direction


Stiffness of discrete supports

i, j, k

Unit vectors of inertial coordinates


Body-fixed coordinates


Revolution speed of body-fixed coordinates around inertial coordinates

\( \varOmega_{x 0} \), \( \varOmega_{y 0} \), \( \varOmega_{z 0} \)

Speed components around the three axes of inertial coordinates


Arbitrary position on the neutral surface


Instantaneous position of q0 after vibration

\( \varvec{r}_{{0}} \)

Displacement vector of geometric center relative to the origin of inertial coordinates

\( \varvec{r}_{{\text{q}_{0} }} \)

An arbitrary point relative to the geometric center of undeformed circular ring

\( \varvec{U}_{0} \)

The point relative to \( {\text{q}}_{0} \) regarding the deformed circular ring


Radial thickness


Axial height




Young’s modulus


Neutral circle radius


Polar coordinates

v, u

Radial and tangential displacements of a point on neutral plane in polar coordinates

\( \theta \)

Position angle




Number of discrete support groups of rotational symmetric RPS


Number of discrete supports in each group of rotational symmetric RPS


Number of discrete support groups of mirror symmetric RPS


Number of discrete supports in each group of mirror symmetric RPS

\( G_{\text{R}}^{i} \)

The ith (i = 1, 2,… N1) group in rotational symmetric topology in Fig. 2a

\( {\text{L}}_{\text{R}}^{i,j} \)

The jth (j = 1, 2,… N2) support in the ith (i = 1, 2,… N1) group in rotational symmetric topology in Fig. 2a

\( G_{\text{M}}^{i} \)

The ith (i = 1, 2) group in mirror symmetric topology in Fig. 2b

\( {\text{L}}_{\text{R}}^{i,j} \)

The jth (j = 1, 2,… N4) support in the ith (i = 1, 2) group in mirror symmetric topology in Fig. 2b

\( \theta_{i,j} \)

Position angle of the jth (j = 1, 2,… N2 in Fig. 2a or N4 in Fig. 2b) support of the ith (i = 1, 2,… N1 in Fig. 2a or N3 in Fig. 2b) group

\( \alpha \)

Position angle between two adjacent supports in the same group


Kinetic energy


Tangential strain


Tangential strain in neutral plane


Membrane strain

\( \hat{U}_{0} \)

Potential energy from bending deformation of circular ring


Cross-section area


Sectional moment of inertia

\( \hat{U}_{1} \)

Potential energy of discrete supports

\( \varepsilon \)

Non-dimensional small parameter

\( \delta \)

Dirac delta function

\( N_{\text{I}} \)

Group count, which satisfies \( N_{\text{I}} = N_{1} \) for rotational symmetric RPS or \( N_{\text{I}} = N_{3} \) for mirror symmetric one

\( N_{\text{II}} \)

Support count in each group, which satisfies \( N_{\text{II}} = N_{2} \) for rotational symmetric RPS or \( N_{\text{II}} = N_{4} \) for mirror symmetric one

\( \hat{U}_{2} \)

Potential energy of uniform external supports

\( \bar{t} \)

Dimensionless time

\( \bar{k}_{\text{s}} \)

Dimensionless support stiffness

\( \bar{u} \)

Dimensionless tangential displacement

\( \bar{v} \)

Dimensionless radial displacement

\( \bar{k}_{u} \)

Dimensionless tangential support stiffness

\( \bar{k}_{v} \)

Dimensionless radial support stiffness

\( \varvec{v}_{o} \)

Revolution speed of the origin of body-fixed coordinates

\( v_{0x} \), \( v_{0y} \), \( v_{0z} \)

Components of revolution speed in three orthogonal directions in body-fixed coordinates

\( \bar{v}_{0x} \), \( \bar{v}_{0y} \), \( \bar{v}_{0z} \)

Dimensionless components of revolution speed in the three orthogonal directions in body-fixed coordinates

\( r_{n0} \)

Eigenvalue of uniform circular ring


Eigenvalue of RPS


The first-order perturbation of the eigenvalue of RPS

\( A_{n} \)



Complex conjugate operation


Imaginary unit, \( {\text{i}} = \sqrt { - 1} \)



\( x_{1} \), \( x_{2} \)

Plural variables

k, k1k5

Positive integers

\( x_{{{\text{q}}_{0} }} \), \( y_{{{\text{q}}_{0} }} \)

Coordinates of point \( {\text{q}}_{0} \)

\( U_{0x} \), \( U_{0y} \)

Vibration displacements about x and y axes

Superscripts and subscripts

x0, y0, z0

The three axes of the inertial coordinates

v, u

Radial and tangential displacements





\( {\text{R}} \)

Rotational symmetry


Mirror symmetry






Sine-mode of backward traveling wave


Cosine-mode of backward traveling wave


Sine-mode of forward traveling wave


Cosine-mode of forward traveling wave



Authors are grateful to the National Natural Science Foundation of China (Grant Nos. 51675368, 51721003 and 51705519) for supporting this research. Also, authors thank the reviewers and editor for their valuable comments and suggestions on this paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Kim M, Moon J, Wickert JA (2000) Spatial modulation of repeated vibration modes in rotationally periodic structures. ASME J Vib Acoust 122:62–68CrossRefGoogle Scholar
  2. 2.
    Lopez I, Blomb REA, Roozena NB et al (2007) Modelling vibrations on deformed rolling tyres-a modal approach. J Sound Vib 307:481–494ADSCrossRefGoogle Scholar
  3. 3.
    Zhao CS (2007) Ultrasonic motors technologies and applications. Beijing, ChinaGoogle Scholar
  4. 4.
    Xi X, Wu YL, Wu XM et al (2012) Investigation on standing wave vibration of the imperfect resonant shell for cylindrical gyro. Sens Actuators A Phys 179:70–77CrossRefGoogle Scholar
  5. 5.
    Esmaeili M, Durali M, Jalili N (2006) Ring microgyroscope modeling and performance evaluation. J Vib Control 12:537–553CrossRefGoogle Scholar
  6. 6.
    Parker RG, Wu XH (2010) Vibration modes of planetary gears with unequally spaced planets and an elastic ring gear. J Sound Vib 329:2265–2275ADSCrossRefGoogle Scholar
  7. 7.
    McWilliam S, Ong J, Fox CHJ (2005) On the statistics of natural frequency splitting for rings with random mass imperfections. J Sound Vib 279:453–470ADSCrossRefGoogle Scholar
  8. 8.
    Bisegna P, Caruso G (2007) Frequency split and vibration localization in imperfect rings. J Sound Vib 306:691–711ADSCrossRefGoogle Scholar
  9. 9.
    Wu XH, Parker RG (2006) Vibration of rings on a general elastic foundation. J Sound Vib 295:194–213ADSCrossRefGoogle Scholar
  10. 10.
    Parker RG, Mote CD Jr (1996) Exact boundary condition perturbation solutions in eigenvalue problems. J Appl Mech 63:128–135MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wang YY, Wang SY, Zhu DH (2016) Dual-mode frequency splitting elimination of ring periodic structures via feature shifting. Proc IMechE Part C J Mech Eng Sci 230:3347–3357CrossRefGoogle Scholar
  12. 12.
    Fox CHJ (1990) A simple theory for the analysis and correction of frequency splitting in slightly imperfect rings. J Sound Vib 142:227–243ADSCrossRefGoogle Scholar
  13. 13.
    Zhang DS, Wang SY, Liu JP (2014) Analytical prediction for free response of rotationally ring-shaped periodic structures. ASME J Vib Acoust 136:041016CrossRefGoogle Scholar
  14. 14.
    Sun WJ, Wang SY, Xia Y et al (2016) Natural frequency splitting and principal instability of rotating cyclic ring structures. Proc IMechE Part C J Mech Eng Sci 232:66–78CrossRefGoogle Scholar
  15. 15.
    Rourke AK, McWilliam S, Fox CHJ (2002) Multi-mode trimming of imperfect thin rings using masses at preselected locations. J Sound Vib 256:319–345ADSCrossRefGoogle Scholar
  16. 16.
    Gallacher BJ, Hedley J, Burdess JS et al (2005) Electrostatic correction of structural imperfections present in a microring gyroscope. J Microelectromech Syst 14:221–234CrossRefGoogle Scholar
  17. 17.
    Yu RC, Mote CD Jr (1987) Vibration and parametric excitation in asymmetric circular plates under moving loads. J Sound Vib 119:409–427ADSCrossRefGoogle Scholar
  18. 18.
    Wang SY, Sun WJ, Wang YY (2016) Instantaneous mode contamination and parametric combination instability of spinning cyclically symmetric ring structures with expanding application to planetary gear ring. J Sound Vib 375:366–385ADSCrossRefGoogle Scholar
  19. 19.
    Yoon SW, Lee S, Najafi K (2011) Vibration sensitivity analysis of MEMS vibratory ring gyroscopes. Sens Actuators A Phys 171:163–177CrossRefGoogle Scholar
  20. 20.
    Liu YZ, Chen LQ (2010) Nonlinear Vibrations. Beijing, ChinaGoogle Scholar
  21. 21.
    Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Willey, New YorkzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringTianjin UniversityTianjinPeople’s Republic of China
  2. 2.Key Laboratory of Mechanism Theory and Equipment Design of Ministry of EducationTianjin UniversityTianjinPeople’s Republic of China
  3. 3.Tianjin Key Laboratory of Nonlinear Dynamics and ControlTianjinPeople’s Republic of China
  4. 4.College of Aeronautical EngineeringCivil Aviation University of ChinaTianjinPeople’s Republic of China

Personalised recommendations