Numerical and experimental analysis of the effect of eccentric phase difference in a rotor-bearing system with bolted-disk joint

Abstract

Bolted joints are widely used in industrial rotating machines to fasten the adjacent disks together and affect system dynamic properties. Therefore, there is a strong need to study their influence on the response of such systems. This paper investigated the effect of the eccentric phase difference of the disk and bolted-disk joint on rotor dynamics, which takes into account the time-varying bending stiffness of the bolted joint. The bolted-disk joint is modeled as a two-node element based upon the energy theorem and Lagrange’s principle, where the relative lateral displacement stiffness, relative bending stiffness, and coupling stiffness between the adjacent disks are considered. And then the dynamic model of the rotor-bearing system is derived based on the proposed bolted joint element and lumped mass modeling method. Combining with the Newmark-β integration scheme, the established model allows the dynamic response characteristics of the rotor-bearing system with the bolted joint to be predicted, and the impact of eccentric phase difference in the rotor system on the response to be investigated. The validity of the simulation results was confirmed by experiment. Through the modeling method proposed in this paper and obtained results, the bifurcation characteristics of the bolted joint rotor system can be predicted.

Appendices

Appendix A

$$ {\mathbf{A}}_{{\mathbf{0}}} = \left[ {\begin{array}{*{20}c} {\cos \theta_{Z2} } & {\sin \theta_{Z2} } & 0 \\ { - \sin \theta_{Z2} } & {\cos \theta_{Z2} } & 0 \\ 1 & 0 & 0 \\ \end{array} } \right] $$

(A.1)

$$ {\mathbf{A}}_{1} = \left[ {\begin{array}{*{20}c} {1} & {0} & {0} \\ {0} & {\cos \theta_{X1} } & {\sin \theta_{X1} } \\ {0} & { - \sin \theta_{X1} } & {\cos \theta_{X1} } \\ \end{array} } \right] $$

(A.2)

$$ {\mathbf{A}}_{2} = \left[ {\begin{array}{*{20}c} {\cos \theta_{Y} } & 0 & { - \sin \theta_{Y} } \\ 0 & 1 & 0 \\ {\sin \theta_{Y} } & 0 & {\cos \theta_{Y} } \\ \end{array} } \right] $$

(A.3)

Appendix B

$$ {\mathbf{M}}_{{\text{d}}} = \left[ {\begin{array}{*{20}c} {m_{3} } & 0 & 0 & 0 \\ 0 & {m_{3} } & 0 & 0 \\ 0 & 0 & {J_{{{\text{d}}3}} } & 0 \\ 0 & 0 & 0 & {J_{{{\text{d}}3}} } \\ \end{array} } \right] $$

(B.1)

$$ {\mathbf{G}}_{{\text{d}}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - J_{{{\text{p}}3}} } \\ 0 & 0 & {J_{{{\text{p}}3}} } & 0 \\ \end{array} } \right] $$

(B.2)

$$ {\mathbf{Q}}_{{\text{d}}}^{{\text{u}}} = m_{3} \omega^{2} \left\{ {\begin{array}{*{20}c} {e_{1} {\text{cos}}\left( {\omega t + \varphi_{1} } \right)} \\ {e_{1} {\text{sin}}\left( {\omega t + \varphi_{1} } \right)} \\ 0 \\ 0 \\ \end{array} } \right\} $$

(B.3)

$$ {\mathbf{Q}}_{{\text{d}}}^{{\text{g}}} = m_{3} g\left\{ {\begin{array}{*{20}c} 0 \\ { - 1} \\ 0 \\ 0 \\ \end{array} } \right\} $$

(B.4)

Appendix C

$$ {\mathbf{M}}_{{\text{J}}} { = }\left[ {\begin{array}{*{20}c} {m_{4} } & {} & {} & {} & {} & {} & {} & {} \\ 0 & {m_{4} } & {} & {} & {} & {} & {} & {} \\ 0 & 0 & {J_{{{\text{d}}4}} } & {} & {} & {{\text{sym}}} & {} & {} \\ 0 & 0 & 0 & {J_{{{\text{d}}4}} } & {} & {} & {} & {} \\ 0 & 0 & 0 & 0 & {m_{5} } & {} & {} & {} \\ 0 & 0 & 0 & 0 & 0 & {m_{5} } & {} & {} \\ 0 & 0 & 0 & 0 & 0 & 0 & {J_{{{\text{d}}5}} } & {} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {J_{{{\text{d}}5}} } \\ \end{array} } \right] $$

(C.1)

$$ {\mathbf{G}}_{{\text{J}}} { = }\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - J_{{{\text{p}}4}} } & 0 & 0 & 0 & 0 \\ 0 & 0 & {J_{{{\text{p}}4}} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - J_{{{\text{p}}5}} } \\ 0 & 0 & 0 & 0 & 0 & 0 & {J_{{{\text{p}}5}} } & 0 \\ \end{array} } \right] $$

(C.2)

$$ {\mathbf{K}}_{{\text{J}}} = \left[ {\begin{array}{*{20}c} {k_{{\text{s}}} } & 0 & 0 & {k_{1} } & { - k_{{\text{s}}} } & 0 & 0 & {k_{2} } \\ 0 & {k_{{\text{s}}} } & { - k_{1} } & 0 & 0 & { - k_{{\text{s}}} } & { - k_{2} } & 0 \\ 0 & { - k_{1} } & {k_{\theta } } & 0 & 0 & {k_{2} } & {k_{\theta } - k_{\theta }^{{\prime }} } & 0 \\ {k_{1} } & 0 & 0 & {k_{\theta } } & { - k_{2} } & 0 & 0 & {k_{\theta } - k_{\theta }^{\prime } } \\ { - k_{{\text{s}}} } & 0 & 0 & { - k_{2} } & {k_{{\text{s}}} } & 0 & 0 & { - k_{1} } \\ 0 & { - k_{{\text{s}}} } & {k_{2} } & 0 & 0 & {k_{{\text{s}}} } & {k_{1} } & 0 \\ 0 & { - k_{2} } & {k_{\theta } - k_{\theta }^{\prime } } & 0 & 0 & {k_{1} } & {k_{\theta } } & 0 \\ {k_{2} } & 0 & 0 & {k_{\theta } - k_{\theta }^{\prime } } & { - k_{1} } & 0 & 0 & {k_{\theta } } \\ \end{array} } \right] $$

(C.3)

$$ {\mathbf{Q}}_{{\text{J}}}^{{\text{u}}} = \left[ {m_{4} \omega^{2} e_{2} {\text{cos}}\left( {\omega t + \varphi_{2} } \right),m_{4} \omega^{2} e_{2} {\text{sin}}\left( {\omega t + \varphi_{2} } \right),0,0,m_{5} \omega^{2} e_{3} {\text{cos}}\left( {\omega t + \varphi_{3} } \right),m_{5} \omega^{2} e_{3} {\text{sin}}\left( {\omega t + \varphi_{3} } \right),0,0} \right]^{{\text{T}}} $$

(C.4)

$$ {\mathbf{Q}}_{{\text{J}}}^{{\text{g}}} = \left[ {0, - m_{4} g,0,0,0, - m_{5} g,0,0} \right]^{{\text{T}}} $$

(C.5)

Appendix D

$$ {\tilde{\mathbf{q}}} = \left[ {x_{1} ,\theta_{y1} ,x_{2} ,\theta_{y2} ,x_{3} ,\theta_{y3} ,x_{4} ,\theta_{y4} ,y_{1} ,\theta_{x1} ,y_{2} ,\theta_{x2} ,y_{3} ,\theta_{x3} ,y_{4} ,\theta_{x4} } \right]^{{\text{T}}} $$

(D.1)

$$ {\tilde{\mathbf{M}}} = \left[ {\begin{array}{*{20}c} {{\tilde{\mathbf{M}}}_{x} } & 0 \\ 0 & {{\tilde{\mathbf{M}}}_{y} } \\ \end{array} } \right],\;\;\;{\kern 1pt} {\tilde{\mathbf{M}}}_{x} = {\tilde{\mathbf{M}}}_{y} = {\text{diag}} \left[ {m_{1} ,J_{{{\text{d}}1}} ,m_{2} ,J_{{{\text{d}}2}} ,m_{3} ,J_{{{\text{d}}3}} ,m_{4} ,J_{{{\text{d}}4}} } \right] $$

(D.2)

$$ {\tilde{\mathbf{G}}} = \omega {\tilde{\mathbf{J}}} = \omega \left[ {\begin{array}{*{20}c} 0 & {{\tilde{\mathbf{J}}}_{1} } \\ { - {\tilde{\mathbf{J}}}_{1}^{{\text{T}}} } & 0 \\ \end{array} } \right],\;\;\;{\kern 1pt} {\mathbf{J}}_{1} = {\text{diag}} \left[ {0,J_{{{\text{p}}1}} ,0,J_{{{\text{p}}2}} ,0,J_{{{\text{p}}3}} ,0,J_{{{\text{p}}4}} } \right] $$

(D.3)

$$ {\tilde{\mathbf{K}}} = \left[ {\begin{array}{*{20}c} {{\tilde{\mathbf{K}}}_{x} } & 0 \\ 0 & {{\tilde{\mathbf{K}}}_{y} } \\ \end{array} } \right] $$

(D.4)

$$ {\tilde{\mathbf{K}}}_{x} = \left[ {\begin{array}{*{20}l} {k_{11} } \hfill & {k_{12} } \hfill & {k_{13} } \hfill & {k_{14} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {k_{12} } \hfill & {k_{22} } \hfill & {k_{23} } \hfill & {k_{24} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {k_{13} } \hfill & {k_{23} } \hfill & {k_{33} + k_{{\text{s}}} } \hfill & {k_{34} + k_{1} {\kern 1pt} } \hfill & { - k_{{\text{s}}} } \hfill & {k_{2} } \hfill & 0 \hfill & 0 \hfill \\ {k_{14} } \hfill & {k_{24} } \hfill & {k_{34} + k_{1} {\kern 1pt} } \hfill & {k_{44} { + }k_{\theta } } \hfill & { - k_{2} } \hfill & {k_{\theta } - k_{\theta }^{\prime } } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & { - k_{{\text{s}}} } \hfill & { - k_{2} } \hfill & {k_{55} { + }k_{{\text{s}}} } \hfill & {k_{56} - k_{1} } \hfill & {k_{57} } \hfill & {k_{58} } \hfill \\ 0 \hfill & 0 \hfill & {k_{2} } \hfill & {k_{\theta } - k_{\theta }^{\prime } } \hfill & {k_{56} - k_{1} } \hfill & {k_{66} { + }k_{\theta } } \hfill & {k_{67} } \hfill & {k_{68} } \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {k_{57} } \hfill & {k_{67} } \hfill & {k_{77} } \hfill & {k_{78} } \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {k_{58} } \hfill & {k_{68} } \hfill & {k_{78} } \hfill & {k_{88} } \hfill \\ \end{array} } \right] $$

(D.5)

$$ {\tilde{\mathbf{K}}}_{y} = \left[ {\begin{array}{*{20}l} {k_{11} } \hfill & { - k_{12} } \hfill & {k_{13} } \hfill & { - k_{14} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ { - k_{12} } \hfill & {k_{22} } \hfill & { - k_{23} } \hfill & {k_{24} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {k_{13} } \hfill & { - k_{23} } \hfill & {k_{33} + k_{s} } \hfill & {k_{34} + k_{1} } \hfill & { - k_{s} } \hfill & { - k_{2} } \hfill & 0 \hfill & 0 \hfill \\ { - k_{14} } \hfill & {k_{24} } \hfill & {k_{34} - k_{1} {\kern 1pt} } \hfill & {k_{44} + k_{\theta } } \hfill & {k_{2} } \hfill & {k_{\theta } - k_{\theta }^{\prime } } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & { - k_{s} } \hfill & {k_{2} } \hfill & {k_{55} { + }k_{s} } \hfill & { - k_{56} + k_{1} } \hfill & {k_{57} } \hfill & { - k_{58} } \hfill \\ 0 \hfill & 0 \hfill & { - k_{2} } \hfill & {k_{\theta } - k_{\theta }^{\prime } } \hfill & { - k_{56} + k_{1} } \hfill & {k_{66} { + }k_{\theta } } \hfill & { - k_{67} } \hfill & {k_{68} } \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {k_{57} } \hfill & { - k_{67} } \hfill & {k_{77} } \hfill & { - k_{78} } \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & { - k_{58} } \hfill & {k_{68} } \hfill & { - k_{78} } \hfill & {k_{88} } \hfill \\ \end{array} } \right] $$

(D.6)

The elements of the matrix ${\tilde{\mathbf{K}}}_{x}$ and ${\tilde{\mathbf{K}}}_{y}$ are given as follows:

$$ \left\{ {\begin{array}{*{20}l} {k_{11} = a_{11} } \hfill \\ {k_{12} = a_{21} } \hfill \\ {k_{13} = - a_{11} } \hfill \\ {k_{14} = a_{21} } \hfill \\ \end{array} ,\left\{ {\begin{array}{*{20}l} {k_{33} = a_{12} } \hfill \\ {k_{34} = - a_{21} } \hfill \\ {k_{77} = a_{13} } \hfill \\ {k_{78} = - a_{23} } \hfill \\ \end{array} ,} \right.{\kern 1pt} } \right.{\kern 1pt} \left\{ {\begin{array}{*{20}l} {k_{22} = l_{1} a_{21} - a_{31} } \hfill \\ {k_{23} = - a_{21} } \hfill \\ {k_{24} = a_{31} } \hfill \\ {k_{44} = l_{1} a_{21} - a_{31} } \hfill \\ \end{array} } \right.,\left\{ {\begin{array}{*{20}l} {k_{55} = a_{12} } \hfill \\ {k_{56} = a_{22} } \hfill \\ {k_{66} = l_{2} a_{22} - a_{32} } \hfill \\ {k_{88} = l_{2} a_{22} - a_{32} } \hfill \\ \end{array} } \right. $$

(D.7)

where

$$ \left\{ {\begin{array}{*{20}l} {a_{1i} = \frac{{12{\text{EI}}}}{{l_{i}^{3} }}} \hfill \\ {a_{2i} = \frac{1}{2}l_{i} a_{1i} } \hfill \\ {a_{3i} = \frac{1}{6}l_{i}^{2} a_{1i} } \hfill \\ \end{array} ,\quad i = 1,2} \right. $$

(D.8)

The damping matrix ${\tilde{\mathbf{C}}}$ of the rotor system can be calculated by the following:

$$ {\tilde{\mathbf{C}}} = \alpha {\tilde{\mathbf{M}}} + \beta {\tilde{\mathbf{K}}} $$

(D.9)

And the external force vector can be calculated by:

$$ {\tilde{\mathbf{Q}}}_{u} = \left[ {0,0,m_{2} \tilde{e}_{1} \omega^{2} \cos \left( {\omega t} \right),0,m_{3} \tilde{e}_{2} \omega^{2} \cos \left( {\omega t} \right),0,0,0,} \right.\left. {0,0,m_{2} \tilde{e}_{1} \omega^{2} \sin \left( {\omega t} \right),0,m_{3} \tilde{e}_{2} \omega^{2} \sin \left( {\omega t} \right),0,0,0} \right]^{{\text{T}}} $$

(D.10)

$$ {\tilde{\mathbf{Q}}}_{{\text{b}}} = \left[ {f_{x1}^{{\text{b}}} ,0,0,0,0,0,f_{x4}^{{\text{b}}} ,0,f_{y1}^{{\text{b}}} ,0,0,0,0,0,_{y4}^{{\text{b}}} ,0} \right]^{{\text{T}}} $$

(D.11)

$$ {\tilde{\mathbf{Q}}}_{{\text{g}}} = \left[ {0,0,0,0,0,0,0,0,m_{1} g,0,m_{2} g,0,m_{3} g,0,m_{4} g,0} \right]^{{\text{T}}} $$

(D.12)