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

Jinlong Liu; Shiyu Wang; Zheren Wang; Nan Gao; Dongsheng Zhang

doi:10.1007/s11012-019-01094-0

Meccanica

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

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Authors and affiliations

Jinlong Liu
Shiyu Wang
Zheren Wang
Nan Gao
Dongsheng Zhang

Article

First Online: 03 December 2019

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Abstract

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.

Keywords

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

Abbreviations

RPS: Ring-shaped periodic structure
BTW: Backward traveling wave
FTW: Forward traveling wave

List of symbols

k_v, k_u: Stiffnesses of uniform radial and tangential supports
o₀-x₀y₀z₀: Inertial coordinates
N: Number of discrete supports
β: Discrete supports’ inclination angle relative to the radial direction
k_s: Stiffness of discrete supports
i, j, k: Unit vectors of inertial coordinates
o-xyz: Body-fixed coordinates
Ω₀: 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
q₀: Arbitrary position on the neutral surface
q₁: 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
h: Radial thickness
b: Axial height
ρ: Density
E: Young’s modulus
R: Neutral circle radius
o-rθ: Polar coordinates
v, u: Radial and tangential displacements of a point on neutral plane in polar coordinates
$ \theta $: Position angle
t: Time
N₁: Number of discrete support groups of rotational symmetric RPS
N₂: Number of discrete supports in each group of rotational symmetric RPS
N₃: Number of discrete support groups of mirror symmetric RPS
N₄: 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
T: Kinetic energy
ε_θ: Tangential strain
ε_θ0: Tangential strain in neutral plane
ε_θ1: Membrane strain
$ \hat{U}_{0} $: Potential energy from bending deformation of circular ring
A: Cross-section area
I: 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
r_n: Eigenvalue of RPS
r_n1: The first-order perturbation of the eigenvalue of RPS
$ A_{n} $: Amplitude
~: Complex conjugate operation
i: Imaginary unit, $ {\text{i}} = \sqrt { - 1} $
n: Wavenumber
$ x_{1} $, $ x_{2} $: Plural variables
k, k₁–k₅: 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
c: Cosine
s: Sine
$ {\text{R}} $: Rotational symmetry
M: Mirror symmetry
P: Pure-imaginary
I: Impure-imaginary
bs: Sine-mode of backward traveling wave
bc: Cosine-mode of backward traveling wave
fs: Sine-mode of forward traveling wave
fc: Cosine-mode of forward traveling wave

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Notes

Acknowledgements

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.

Appendix 1

This section calculates the kinetic energy with the method in Ref. [19]. The point q₁ in Fig. 1 is expressed as

$$ \varvec{U}_{1} = \varvec{r}_{\text{0}} + \varvec{r}_{{\text{q}_{0} }} + \varvec{U}_{0} $$

(39)

where $ \varvec{r}_{\text{0}} $, $ \varvec{r}_{{\text{q}_{0} }} $ and $ \varvec{U}_{0} $ are displacement vectors of the geometric center relative to the origin of inertial coordinates, the arbitrary point with respect to the geometric center of the unformed ring, and the instantaneous position on the deformed ring relative to $ {\text{q}}_{0} $. $ \varvec{r}_{{\text{q}_{0} }} $ is written in o-xyz as

$$ \varvec{r}_{{{\text{q}}_{0} }} = x_{{{\text{q}}_{0} }} {\mathbf{i}} + y_{{{\text{q}}_{0} }} {\mathbf{j}} $$

(40)

where $ x_{{{\text{q}}_{0} }} $ and $ y_{{{\text{q}}_{0} }} $ are the coordinates. $ \varvec{U}_{0} $ is written as

$$ \varvec{U}_{0} = U_{0x} {\mathbf{i}} + U_{0y} {\mathbf{j}} $$

(41)

where $ U_{0x} $ and $ U_{0y} $ are vibration displacements about x and y axes, respectively.

In the inertial coordinates, the revolution speed can be expressed as

$$ \varvec{\varOmega}_{ 0} = \varOmega_{x 0} {\mathbf{i}} + \varOmega_{y 0} {\mathbf{j}} + \varOmega_{z 0} {\mathbf{k}} $$

(42)

Assuming the speed of origin is

$$ \varvec{v}_{o} = \frac{{\text{d}\varvec{r}_{ 0} }}{{\text{d}t}} = v_{0x} {\mathbf{i}} + v_{0y} {\mathbf{j}} + v_{0z} {\mathbf{k}} $$

(43)

where $ v_{0x} $, $ v_{0y} $ and $ v_{0z} $ are components in the three orthogonal directions, respectively. The speed of $ {\text{q}}_{0} $ is

$$ \varvec{v}_{{\text{q}_{0} }} = v_{{\text{q}_{0x} }} {\mathbf{i}} + v_{{\text{q}_{0y} }} {\mathbf{j}} + v_{{\text{q}_{0z} }} {\mathbf{k}} $$

(44)

where

$$ v_{{{\text{q}}_{0x} }} = v_{0x} + \dot{U}_{0x} - y_{{{\text{q}}_{0} }} \varOmega_{z0} - U_{0y} \varOmega_{z0} ,\;v_{{{\text{q}}_{0y} }} = v_{0y} + \dot{U}_{0y} - x_{{{\text{q}}_{0} }} \varOmega_{z0} + U_{0x} \varOmega_{z0} $$

$$ v_{{\text{q}_{0z} }} = v_{0z} + y_{{\text{q}_{0} }} \varOmega_{x 0} + U_{0y} \varOmega_{x 0} - x_{{{\text{q}}_{0} }} \varOmega_{y 0} - U_{0x} \varOmega_{y 0} $$

where “·” means the first derivative of variable versus time. Forming the dot product of velocities yields

$$ \begin{aligned} \varvec{v}_{{{\text{q}}_{0} }} \varvec{.v}_{{{\text{q}}_{0} }} & = \dot{U}_{0x}^{2} + \dot{U}_{0y}^{2} + 2\varOmega_{z0} (U_{0x} \dot{U}_{0y} - \dot{U}_{0x} U_{0y} ) + (\varOmega_{y0}^{2} + \varOmega_{z0}^{2} )U_{0x}^{2} \\ & \quad + \,(\varOmega_{x0}^{2} + \varOmega_{z0}^{2} )U_{0y}^{2} - 2\varOmega_{x0} \varOmega_{y0} y_{{{\text{q}}_{0} }} U_{0x} - 2\varOmega_{x0} \varOmega_{y0} x_{{{\text{q}}_{0} }} U_{0y} + 2(\varOmega_{x0}^{2} + \varOmega_{z0}^{2} )y_{{{\text{q}}_{0} }} U_{0y} \\ & \quad + \,2(v_{0x} - \varOmega_{z0} y_{{{\text{q}}_{0} }} )\dot{U}_{0x} + 2(v_{0y} + \varOmega_{z0} x_{{{\text{q}}_{0} }} )\dot{U}_{0y} + 2(\varOmega_{z0} v_{0y} - \varOmega_{y0} v_{0z} )U_{0x} \\ & \quad - \,2(\varOmega_{z0} v_{0x} - \varOmega_{x0} v_{0z} )U_{0y} - 2\varOmega_{x0} \varOmega_{y0} U_{0x} U_{0y} + 2(\varOmega_{y0}^{2} + \varOmega_{z0}^{2} )x_{{{\text{q}}_{0} }} U_{0x} \\ \end{aligned} $$

(45)

Equation (45) excludes the terms being irrelevant to the governing equation of motion. By introducing polar coordinates, the displacements are rewritten as

$$ \left[ {\begin{array}{*{20}c} {U_{{\text{0}x}} } \\ {U_{{\text{0}y}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\cos \theta } & { - \,\sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} v \\ u \\ \end{array} } \right] $$

(46)

Position of q₀ is defined as

$$ \left[ {\begin{array}{*{20}c} {x_{{\text{q}_{0} }} } \\ {y_{{\text{q}_{0} }} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R\cos \theta } \\ {R\sin \theta } \\ \end{array} } \right] $$

(47)

Substituting Eqs. (46) and (47) into (45) yields

$$ \begin{aligned} \varvec{v}_{{{\text{q}}_{0} }} .\varvec{v}_{{{\text{q}}_{0} }} & = \dot{u}^{2} + \dot{v}^{2} + 2\varOmega_{z0} (\dot{u}v - u\dot{v}) + (\varOmega_{y0}^{2} + \varOmega_{z0}^{2} )(v\cos \theta - u\sin \theta )^{2} \\ & \quad + \,2v_{0x} (\dot{v}\cos \theta - \dot{u}\sin \theta ) + 2v_{0y} (\dot{v}\sin \theta + \dot{u}\sin \theta ) + 2R(\varOmega_{z0} \dot{u} + \varOmega_{z0}^{2} v) \\ & \quad - \,2\varOmega_{x0} \varOmega_{y0} \left[ {R(v\sin 2\theta + u\cos 2\theta ) + (v\cos \theta - u\sin \theta )(v\sin \theta + u\cos \theta )} \right] \\ & \quad + \,2(\varOmega_{z0} v_{0y} - \varOmega_{y0} v_{0z} )(v\cos \theta - u\sin \theta ) + 2(\varOmega_{x0} v_{0z} - \varOmega_{z0} v_{0x} )(v\sin \theta + u\cos \theta ) \\ & \quad + \,(\varOmega_{x0}^{2} + \varOmega_{z0}^{2} )(v\sin \theta + u\cos \theta )^{2} + 2\varOmega_{x0}^{2} R(v\sin \theta + u\cos \theta )\sin \theta \\ & \quad + \,2\varOmega_{y0}^{2} R(v\cos \theta - u\sin \theta )\cos \theta \\ \end{aligned} $$

(48)

Appendix 2

See Tables 4, 5, 6, 7 and 8

Table 4

Eigenvalues of rotational symmetric RPS (pure-imaginary $ r_{n0} $, 2n/N₁ = int)

Directions	Modes	Eigenvalues	Cond 1	Cond 2	Cond 3	Cond 4
Backward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} + a_{n} + Q_{1} $	$ 2n\varOmega_{z0} + a_{n} + Q_{1} - \varepsilon k_{s} N_{1} Q_{3} Q_{4} $	$ 2n\varOmega_{z 0} + a_{n} + Q_{1} + \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta / 4\uppi a_{n} $	$ 2n\varOmega_{z 0} + a_{n} + \varepsilon k_{\text{s}} N\sin^{2} \beta / 2\uppi a_{n} $
	Cosine	Real part	0	$ - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} $	$ - \varepsilon k_{\text{s}} N_{1} Q_{4} Q_{5} $	$ - \varepsilon k_{\text{s}} nN\sin 2\beta / 4\uppi a_{n} $
	Sine	Imaginary part	$ 2n\varOmega_{z 0} + a_{n} { + }Q_{1} $	$ 2n\varOmega_{z 0} + a_{n} { + }Q_{1} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} $	$ 2n\varOmega_{z 0} + a_{n} { + }Q_{1} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} $	$ 2n\varOmega_{z 0} + a_{n} + \varepsilon k_{\text{s}} n^{2} N\cos^{2} \beta / 2\uppi a_{n} $
	Sine	Real part	0	$ \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} $	$ \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} $	$ \varepsilon k_{\text{s}} nN\sin 2\beta / 4\uppi a_{n} $
Forward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} $	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} $	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} - \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta / 4\uppi a_{n} $	$ 2n\varOmega_{z 0} - a_{n} - \varepsilon k_{\text{s}} N\sin^{2} \beta / 2\uppi a_{n} $
	Cosine	Real part	0	$ \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} $	$ \varepsilon k_{\text{s}} N_{1} Q_{4} Q_{5} $	$ \varepsilon k_{\text{s}} nN\sin 2\beta / 4\uppi a_{n} $
	Sine	Imaginary part	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} $	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} - \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} $	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} - \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} $	$ 2n\varOmega_{z 0} - a_{n} - \varepsilon k_{\text{s}} n^{2} N\cos^{2} \beta /2\uppi a_{n} $
	Sine	Real part	0	$ - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} $	$ - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} $	$ - \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi a_{n} $

$ Q_{1} = \varepsilon k_{\text{s}} N\left( {\sin^{2} \beta { + }n^{2} \cos^{2} \beta } \right) / 4\uppi a_{n} $, $ Q_{2} = \varepsilon k_{\text{s}} N\left( {\sin^{2} \beta + n^{2} \cos^{2} \beta } \right) /4\uppi b_{n} $, $ Q_{3} = \cos \left( {N_{2} - 1} \right)n\alpha \sin N_{2} n\alpha /\sin n\alpha $, $ Q_{4} = \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right) / 4\uppi a_{n} $, $ Q_{5} = \sin \left( {N_{2} - 1} \right)n\alpha \sin N_{2} n\alpha /\sin n\alpha $, $ Q_{6} = \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right) /4\uppi b_{n} $

Table 5

Eigenvalues of rotational symmetric RPS (impure-imaginary $ r_{n0} $, 2n/N₁ = int)

Directions	Modes	Eigenvalues	Cond 1	Cond 2	Cond 3	Cond 4
Backward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} $	$ 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta /4\uppi b_{n} $	$ 2n\varOmega_{z 0} { + }\varepsilon k_{\text{s}} N_{1} Q_{5} Q_{6} $	$ 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi b_{n} $
	Cosine	Real part	$ - b_{n} + Q_{2} $	$ - b_{n} + Q_{2} - \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{6} $	$ - b_{n} + Q_{2} + \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta /4\uppi b_{n} $	$ - b_{n} + \varepsilon k_{\text{s}} N\sin^{2} \beta / 2\uppi b_{n} $
	Sine	Imaginary part	$ 2n\varOmega_{z 0} $	$ 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta /4\uppi b_{n} $	$ 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} N_{1} Q_{5} Q_{6} $	$ 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi b_{n} $
	Sine	Real part	$ - b_{n} + Q_{2} $	$ - b_{n} + Q_{2} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{6} $	$ - b_{n} + Q_{2} - \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta /4\uppi b_{n} $	$ - b_{n} + \varepsilon k_{\text{s}} n^{2} N\cos^{2} \beta /2\uppi b_{n} $
Forward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} $	$ 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta /4\uppi b_{n} $	$ 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} N_{1} Q_{5} Q_{6} $	$ 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi b_{n} $
	Cosine	Real part	$ b_{n} - Q_{2} $	$ b_{n} - Q_{2} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{6} $	$ b_{n} - Q_{2} - \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta /4\uppi b_{n} $	$ b_{n} - \varepsilon k_{\text{s}} N\sin^{2} \beta /2\uppi b_{n} $
	Sine	Imaginary part	$ 2n\varOmega_{z 0} $	$ 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta /4\uppi b_{n} $	$ 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} N_{1} Q_{5} Q_{6} $	$ 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi b_{n} $
	Sine	Real part	$ b_{n} - Q_{2} $	$ b_{n} - Q_{2} - \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{6} $	$ b_{n} - Q_{2} + \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta /4\uppi b_{n} $	$ b_{n} - \varepsilon k_{\text{s}} n^{2} N\cos^{2} \beta /2\uppi b_{n} $

Table 6

Eigenvalues of rotational symmetric RPS with 2n/N₁$ { \ne } $ int

Directions	Modes	Eigenvalues	Pure-imaginary $ r_{n0} $		Impure-imaginary $ r_{n0} $
Directions	Modes	Eigenvalues	Cond 5	Cond 6	Cond 5	Cond 6
Backward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} + a_{n} + Q_{1} $	$ 2n\varOmega_{z 0} + a_{n} + Q_{1} $	$ 2n\varOmega_{z 0} $	$ 2n\varOmega_{z 0} $
	Cosine	Real part	0	0	$ - b_{n} + Q_{2} $	$ - b_{n} + Q_{2} $
	Sine	Imaginary part	$ 2n\varOmega_{z 0} + a_{n} { + }Q_{1} $	$ 2n\varOmega_{z 0} + a_{n} { + }Q_{1} $	$ 2n\varOmega_{z 0} $	$ 2n\varOmega_{z 0} $
	Sine	Real part	0	0	$ - b_{n} + Q_{2} $	$ - b_{n} + Q_{2} $
Forward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} $	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} $	$ 2n\varOmega_{z 0} $	$ 2n\varOmega_{z 0} $
	Cosine	Real part	0	0	$ b_{n} - Q_{2} $	$ b_{n} - Q_{2} $
	Sine	Imaginary part	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} $	$ 2n\varOmega_{z 0} - a_{n} - Q_{1} $	$ 2n\varOmega_{z 0} $	$ 2n\varOmega_{z 0} $
	Sine	Real part	0	0	$ b_{n} - Q_{2} $	$ b_{n} - Q_{2} $

Table 7

Eigenvalues of mirror symmetric RPS (pure-imaginary $ r_{n0} $)

Directions	Modes	Eigenvalues
Backward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} + a_{n} + \frac{{\varepsilon k_{\text{s}} N_{4} }}{{ 2\uppi a_{n} }}\left( {\sin^{2} \beta { + }n^{2} \cos^{2} \beta } \right){ + }\frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi a_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta \sin 2n\theta_{1,j} - \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right){ \cos }2n\theta_{1,j} } \right]} $ <1>
	Cosine	Real part	$ - \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi a_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta { \cos }2n\theta_{1,j} + \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right)\sin 2n\theta_{1,j} } \right]} $ <2>
	Sine	Imaginary part	$ 2n\varOmega_{z 0} + a_{n} + \frac{{\varepsilon k_{\text{s}} N_{4} }}{{ 2 {\uppi }a_{n} }}\left( {\sin^{2} \beta { + }n^{2} \cos^{2} \beta } \right) - \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi a_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta \sin 2n\theta_{1,j} - \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right){ \cos }2n\theta_{1,j} } \right]} $ <3>
	Sine	Real part	$ \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi a_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta { \cos }2n\theta_{1,j} + \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right)\sin 2n\theta_{1,j} } \right]} $ <4>
Forward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} - a_{n} - \frac{{\varepsilon k_{\text{s}} N_{4} }}{{ 2\uppi a_{n} }}\left( {\sin^{2} \beta { + }n^{2} \cos^{2} \beta } \right) - \frac{{\varepsilon k_{\text{s}} }}{{2\uppi a_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta \sin 2n\theta_{1,j} - \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right){ \cos }2n\theta_{1,j} } \right]} $ <5>
	Cosine	Real part	$ \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi a_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta { \cos }2n\theta_{1,j} + \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right)\sin 2n\theta_{1,j} } \right]} $ <6>
	Sine	Imaginary part	$ 2n\varOmega_{z 0} - a_{n} - \frac{{\varepsilon k_{\text{s}} N_{4} }}{{ 2\uppi a_{n} }}\left( {\sin^{2} \beta { + }n^{2} \cos^{2} \beta } \right){ + }\frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi a_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta \sin 2n\theta_{1,j} - \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right){ \cos }2n\theta_{1,j} } \right]} $ <7>
	Sine	Real part	$ - \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi a_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta { \cos }2n\theta_{1,j} + \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right)\sin 2n\theta_{1,j} } \right]} $ <8>

Table 8

Eigenvalues of mirror symmetric RPS (impure-imaginary $ r_{n0} $)

Directions	Modes	Eigenvalues
Backward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} { + }\frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi b_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta { \cos }2n\theta_{1,j} { + }\left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right)\sin 2n\theta_{1,j} } \right]} $ <1>
	Cosine	Real part	$ - b_{n} + \frac{{\varepsilon k_{\text{s}} N_{4} }}{{ 2\uppi b_{n} }}\left( {\sin^{2} \beta { + }n^{2} \cos^{2} \beta } \right) + \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi b_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta \sin 2n\theta_{1,j} - \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right){ \cos }2n\theta_{1,j} } \right]} $ <2>
	Sine	Imaginary part	$ 2n\varOmega_{z 0} - \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi b_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta { \cos }2n\theta_{1,j} { + }\left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right)\sin 2n\theta_{1,j} } \right]} $ <3>
	Sine	Real part	$ - b_{n} { + }\frac{{\varepsilon k_{\text{s}} N_{4} }}{{ 2\uppi b_{n} }}\left( {\sin^{2} \beta { + }n^{2} \cos^{2} \beta } \right) - \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi b_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta \sin 2n\theta_{1,j} - \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right){ \cos }2n\theta_{1,j} } \right]} $ <4>
Forward	Cosine	Imaginary part	$ 2n\varOmega_{z 0} - \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi b_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta { \cos }2n\theta_{1,j} { + }\left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right)\sin 2n\theta_{1,j} } \right]} $ <5>
	Cosine	Real part	$ b_{n} - \frac{{\varepsilon k_{\text{s}} N_{4} }}{{ 2\uppi b_{n} }}\left( {\sin^{2} \beta { + }n^{2} \cos^{2} \beta } \right) - \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi b_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta \sin 2n\theta_{1,j} - \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right){ \cos }2n\theta_{1,j} } \right]} $ <6>
	Sine	Imaginary part	$ 2n\varOmega_{z 0} { + }\frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi b_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta { \cos }2n\theta_{1,j} { + }\left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right)\sin 2n\theta_{1,j} } \right]} $ <7>
	Sine	Real part	$ b_{n} - \frac{{\varepsilon k_{\text{s}} N_{4} }}{{ 2\uppi b_{n} }}\left( {\sin^{2} \beta { + }n^{2} \cos^{2} \beta } \right) + \frac{{\varepsilon k_{\text{s}} }}{{ 2\uppi b_{n} }}\sum\limits_{j = 1}^{{N_{4} }} {\left[ {n\sin 2\beta \sin 2n\theta_{1,j} - \left( {n^{2} \cos^{2} \beta - \sin^{2} \beta } \right){ \cos }2n\theta_{1,j} } \right]} $ <8>

.

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Authors and Affiliations

Jinlong Liu
- 1
- 2
- 3
Shiyu Wang
- 1
- 2
- 3
Email authorView author's OrcID profile
Zheren Wang
- 1
- 2
- 3
Nan Gao
- 1
- 2
- 3
Dongsheng Zhang
- 4

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

Directions	Modes	Eigenvalues	Cond 1	Cond 2	Cond 3	Cond 4
Backward	Cosine	Imaginary part	\( 2n\varOmega_{z 0} + a_{n} + Q_{1} \)	\( 2n\varOmega_{z0} + a_{n} + Q_{1} - \varepsilon k_{s} N_{1} Q_{3} Q_{4} \)	\( 2n\varOmega_{z 0} + a_{n} + Q_{1} + \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta / 4\uppi a_{n} \)	\( 2n\varOmega_{z 0} + a_{n} + \varepsilon k_{\text{s}} N\sin^{2} \beta / 2\uppi a_{n} \)
	Cosine	Real part	0	\( - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} \)	\( - \varepsilon k_{\text{s}} N_{1} Q_{4} Q_{5} \)	\( - \varepsilon k_{\text{s}} nN\sin 2\beta / 4\uppi a_{n} \)
	Sine	Imaginary part	\( 2n\varOmega_{z 0} + a_{n} { + }Q_{1} \)	\( 2n\varOmega_{z 0} + a_{n} { + }Q_{1} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} \)	\( 2n\varOmega_{z 0} + a_{n} { + }Q_{1} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} \)	\( 2n\varOmega_{z 0} + a_{n} + \varepsilon k_{\text{s}} n^{2} N\cos^{2} \beta / 2\uppi a_{n} \)
	Sine	Real part	0	\( \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} \)	\( \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} \)	\( \varepsilon k_{\text{s}} nN\sin 2\beta / 4\uppi a_{n} \)
Forward	Cosine	Imaginary part	\( 2n\varOmega_{z 0} - a_{n} - Q_{1} \)	\( 2n\varOmega_{z 0} - a_{n} - Q_{1} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} \)	\( 2n\varOmega_{z 0} - a_{n} - Q_{1} - \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta / 4\uppi a_{n} \)	\( 2n\varOmega_{z 0} - a_{n} - \varepsilon k_{\text{s}} N\sin^{2} \beta / 2\uppi a_{n} \)
	Cosine	Real part	0	\( \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} \)	\( \varepsilon k_{\text{s}} N_{1} Q_{4} Q_{5} \)	\( \varepsilon k_{\text{s}} nN\sin 2\beta / 4\uppi a_{n} \)
	Sine	Imaginary part	\( 2n\varOmega_{z 0} - a_{n} - Q_{1} \)	\( 2n\varOmega_{z 0} - a_{n} - Q_{1} - \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} \)	\( 2n\varOmega_{z 0} - a_{n} - Q_{1} - \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{4} \)	\( 2n\varOmega_{z 0} - a_{n} - \varepsilon k_{\text{s}} n^{2} N\cos^{2} \beta /2\uppi a_{n} \)
	Sine	Real part	0	\( - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} \)	\( - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta / 4\uppi a_{n} \)	\( - \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi a_{n} \)

Directions	Modes	Eigenvalues	Cond 1	Cond 2	Cond 3	Cond 4
Backward	Cosine	Imaginary part	\( 2n\varOmega_{z 0} \)	\( 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta /4\uppi b_{n} \)	\( 2n\varOmega_{z 0} { + }\varepsilon k_{\text{s}} N_{1} Q_{5} Q_{6} \)	\( 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi b_{n} \)
	Cosine	Real part	\( - b_{n} + Q_{2} \)	\( - b_{n} + Q_{2} - \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{6} \)	\( - b_{n} + Q_{2} + \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta /4\uppi b_{n} \)	\( - b_{n} + \varepsilon k_{\text{s}} N\sin^{2} \beta / 2\uppi b_{n} \)
	Sine	Imaginary part	\( 2n\varOmega_{z 0} \)	\( 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta /4\uppi b_{n} \)	\( 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} N_{1} Q_{5} Q_{6} \)	\( 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi b_{n} \)
	Sine	Real part	\( - b_{n} + Q_{2} \)	\( - b_{n} + Q_{2} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{6} \)	\( - b_{n} + Q_{2} - \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta /4\uppi b_{n} \)	\( - b_{n} + \varepsilon k_{\text{s}} n^{2} N\cos^{2} \beta /2\uppi b_{n} \)
Forward	Cosine	Imaginary part	\( 2n\varOmega_{z 0} \)	\( 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta /4\uppi b_{n} \)	\( 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} N_{1} Q_{5} Q_{6} \)	\( 2n\varOmega_{z 0} - \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi b_{n} \)
	Cosine	Real part	\( b_{n} - Q_{2} \)	\( b_{n} - Q_{2} + \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{6} \)	\( b_{n} - Q_{2} - \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta /4\uppi b_{n} \)	\( b_{n} - \varepsilon k_{\text{s}} N\sin^{2} \beta /2\uppi b_{n} \)
	Sine	Imaginary part	\( 2n\varOmega_{z 0} \)	\( 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} nN_{1} Q_{3} \sin 2\beta /4\uppi b_{n} \)	\( 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} N_{1} Q_{5} Q_{6} \)	\( 2n\varOmega_{z 0} + \varepsilon k_{\text{s}} nN\sin 2\beta /4\uppi b_{n} \)
	Sine	Real part	\( b_{n} - Q_{2} \)	\( b_{n} - Q_{2} - \varepsilon k_{\text{s}} N_{1} Q_{3} Q_{6} \)	\( b_{n} - Q_{2} + \varepsilon k_{\text{s}} nN_{1} Q_{5} \sin 2\beta /4\uppi b_{n} \)	\( b_{n} - \varepsilon k_{\text{s}} n^{2} N\cos^{2} \beta /2\uppi b_{n} \)

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

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