An improved dynamic model of the spline coupling with misalignment and its load distribution analysis

Abstract

Spline couplings allow for a certain amount of misalignment and relative sliding between their internal and external components. However, the misalignment could cause serious uneven load distribution and aggravate the wear of a spline coupling. So far, the effects of misalignment on the load distribution of the spline coupling aren't fully understood. To solve the above problem, an improved dynamic model of the spline coupling is established, which introduces the static misalignment caused by installation and manufacturing errors and the dynamic misalignment introduced by the dynamic vibration displacement between the internal and external splines. The classical potential energy method is adopted to derive the meshing stiffness, and then the equivalent stiffness and meshing excitation force of the spline coupling with misalignment is obtained. The accuracy of the method proposed has been proved by software. The load distribution of the spline coupling with various misalignments is studied. The results show that: the misalignment would cause serious uneven load distribution, especially the static parallel misalignment. Meanwhile, the dynamic misalignment has a small effect on the load distribution, which can be ignored during load distribution analysis. The improved model can be widely applied to rotor systems connected by spline couplings.

Appendix

$$ \left\{ {\begin{array}{*{20}l} {F_{Nx} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \left\{ {\left. \begin{gathered} \lambda_{ji} + [\Delta x + (j{\text{d}}z - B/2)\Delta \theta_{y} ]\sin (\theta_{i} - \alpha_{ji} ) + \hfill \\ \quad \quad \quad \;\;\;[\Delta y - (j{\text{d}}z - B/2)\Delta \theta_{x} ]\cos (\theta_{i} - \alpha_{ji} ) \hfill \\ \end{gathered} \right\}} \right.\sin (\theta_{i} - \alpha_{ji} )} } } \hfill \\ {F_{Ny} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \left\{ {\left. \begin{gathered} \lambda_{ji} + [\Delta x + (j{\text{d}}z - B/2)\Delta \theta_{y} ]\sin (\theta_{i} - \alpha_{ji} ) + \hfill \\ \quad \quad \quad \;\;\;[\Delta y - (j{\text{d}}z - B/2)\Delta \theta_{x} ]\cos (\theta_{i} - \alpha_{ji} ) \hfill \\ \end{gathered} \right\}} \right.\cos (\theta_{i} - \alpha_{ji} )} } } \hfill \\ {M_{Nx} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \left\{ {\left. \begin{gathered} \lambda_{ji} + [\Delta x + (j{\text{d}}z - B/2)\Delta \theta_{y} ]\sin (\theta_{i} - \alpha_{ji} ) + \hfill \\ \quad \quad \quad \;\;\;[\Delta y - (j{\text{d}}z - B/2)\Delta \theta_{x} ]\cos (\theta_{i} - \alpha_{ji} ) \hfill \\ \end{gathered} \right\}} \right.\cos (\theta_{i} - \alpha_{ji} )(j{\text{d}}z - B/2)} } } \hfill \\ {M_{Ny} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \left\{ {\left. \begin{gathered} \lambda_{ji} + [\Delta x + (j{\text{d}}z - B/2)\Delta \theta_{y} ]\sin (\theta_{i} - \alpha_{ji} ) + \hfill \\ \quad \quad \quad \;\;\;[\Delta y - (j{\text{d}}z - B/2)\Delta \theta_{x} ]\cos (\theta_{i} - \alpha_{ji} ) \hfill \\ \end{gathered} \right\}} \right.\sin (\theta_{i} - \alpha_{ji} )(j{\text{d}}z - B/2)} } } \hfill \\ \end{array} } \right. $$

(29)

$$ \left\{ {\begin{array}{*{20}l} {k_{N11} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \sin^{2} (\theta_{i} - \alpha_{ji} ),k_{N22} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \cos^{2} (\theta_{i} - \alpha_{ji} )} } } } } \hfill \\ {k_{N33} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} \begin{gathered} k_{ji}^{T} \cos^{2} (\theta_{i} - \alpha_{ji} )(B/2 - j{\text{d}}z)^{2} - \hfill \\ k_{ji}^{T} \left\{ {\left. \begin{gathered} \lambda_{ji} + [\Delta x + (j{\text{d}}z - B/2)\Delta \theta_{y} ]\sin (\theta_{i} - \alpha_{ji} ) - \hfill \\ [\Delta y - (j{\text{d}}z - B/2)\Delta \theta_{x} ]\cos (\theta_{i} - \alpha_{ji} ) \hfill \\ \end{gathered} \right\}} \right.\sin (\theta_{i} - \alpha_{ji} )(r_{{\text{f}}} + L_{ji} )\sin \theta_{i} \hfill \\ \end{gathered} } } \hfill \\ {k_{N44} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} \begin{gathered} k_{ji}^{T} \sin^{2} (\theta_{i} - \alpha_{ji} )(B/2 - j{\text{d}}z)^{2} - \hfill \\ k_{ji}^{T} \left\{ {\left. \begin{gathered} \lambda_{ji} + [\Delta x + (j{\text{d}}z - B/2)\Delta \theta_{y} ]\sin (\theta_{i} - \alpha_{ji} ) - \hfill \\ [\Delta y - (j{\text{d}}z - B/2)\Delta \theta_{x} ]\cos (\theta_{i} - \alpha_{ji} ) \hfill \\ \end{gathered} \right\}} \right.\cos (\theta_{i} - \alpha_{ji} )(r_{{\text{f}}} + L_{ji} )\cos \theta_{i} \hfill \\ \end{gathered} } } \hfill \\ {k_{N12} = k_{N21} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \sin [2(\theta_{i} - \alpha_{ji} )]/2} } } \hfill \\ {k_{N13} = k_{N31} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \sin [2(\theta_{i} - \alpha_{ji} )](B/2 - j{\text{d}}z)/2} } } \hfill \\ {k_{N14} = k_{N41} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \sin^{2} (\theta_{i} - \alpha_{ji} )(j{\text{d}}z - B/2)} } } \hfill \\ {k_{N23} = k_{N32} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \cos^{2} (\theta_{i} - \alpha_{ji} )(B/2 - j{\text{d}}z)} } } \hfill \\ {k_{N24} = k_{N42} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \sin [2(\theta_{i} - \alpha_{ji} )](j{\text{d}}z - B/2)/2} } } \hfill \\ {k_{N34} = k_{N43} = \sum\limits_{j = 1}^{{n_{Z} }} {\sum\limits_{i = 1}^{z} {k_{ji}^{T} \sin [2(\theta_{i} - \alpha_{ji} )](j{\text{d}}z - B/2)^{2} /2} } } \hfill \\ \end{array} } \right. $$

(30)

$$ {\mathbf{T}}{ = }\left[ {\begin{array}{*{20}c} {\cos \Omega t} & { - \sin \Omega t} & {} & {} \\ {\sin \Omega t} & {\cos \Omega t} & {} & {} \\ {} & {} & {\cos \Omega t} & { - \sin \Omega t} \\ {} & {} & {\sin \Omega t} & {\cos \Omega t} \\ \end{array} } \right] $$

(31)

$$ {\mathbf{K}}_{0} { = }\frac{1}{2}\left[ {\begin{array}{*{20}c} {k_{11} + k_{22} } & {k_{12} - k_{21} } & {k_{13} + k_{24} } & {k_{14} - k_{23} } \\ {k_{21} - k_{12} } & {k_{22} + k_{11} } & {k_{23} - k_{14} } & {k_{24} + k_{13} } \\ {k_{31} + k_{42} } & {k_{32} - k_{41} } & {k_{33} + k_{44} } & {k_{34} - k_{43} } \\ {k_{41} - k_{32} } & {k_{42} + k_{31} } & {k_{43} - k_{34} } & {k_{44} + k_{33} } \\ \end{array} } \right] $$

(32)

$$ {\mathbf{K}}_{1} { = }\frac{1}{2}\left[ {\begin{array}{*{20}c} {k_{11} - k_{22} } & {k_{12} + k_{21} } & {k_{13} - k_{24} } & {k_{14} + k_{23} } \\ {k_{21} + k_{12} } & {k_{22} - k_{11} } & {k_{23} + k_{14} } & {k_{24} - k_{13} } \\ {k_{31} - k_{42} } & {k_{32} + k_{41} } & {k_{33} - k_{44} } & {k_{34} + k_{43} } \\ {k_{41} + k_{32} } & {k_{42} - k_{31} } & {k_{43} + k_{34} } & {k_{44} - k_{33} } \\ \end{array} } \right] $$

(33)

$$ {\mathbf{K}}_{2} { = }\frac{1}{2}\left[ {\begin{array}{*{20}c} { - k_{12} - k_{21} } & {k_{11} - k_{22} } & { - k_{14} - k_{23} } & {k_{13} - k_{24} } \\ {k_{11} - k_{22} } & {k_{21} + k_{12} } & {k_{13} - k_{24} } & {k_{23} + k_{14} } \\ { - k_{32} - k_{41} } & {k_{31} - k_{42} } & { - k_{34} - k_{43} } & {k_{33} - k_{44} } \\ {k_{31} - k_{42} } & {k_{41} + k_{32} } & {k_{33} - k_{44} } & {k_{43} + k_{34} } \\ \end{array} } \right] $$

(34)