Nonlinear dynamics of a dual-rotor-bearing system with active elastic support dry friction dampers

Abstract

In present study, the modified harmonic balance-alternating frequency/time domain (HB-AFT) method with embedded arc-length continuation method is used to study the nonlinear dynamic characteristics of dual-rotor-bearing system with active elastic support dry friction damper (ESDFD). The friction on the contact interface between ESDFD moving and stationary disks is described by two-dimensional (2D) friction contact model. The dynamic model of dual-rotor is established by conical Timoshenko beam element and rigid disk element, while the inter-shaft bearing force is obtained by Hertz contact model. The reduced order model (ROM) of dual-rotor is constructed by the Craig-Bampton method. Based on the ROM, the modified HB-AFT method with embedded arc-length continuation procedure is used to solve the periodic solutions of dual-rotor system under unbalance excitation. The Floquet theory is employed to determine the stability of periodic solutions. Then the impact of key parameters such as Hertz contact stiffness and radial clearance of inter-shaft bearing, eccentricity of disk, and modal damping ratio on the primary resonance characteristics and inter-shaft bearing dynamic load of the dual-rotor system are revealed without considering ESDFD. Moreover, the influence of ESDFD normal force on the primary resonance peak and inter-shaft bearing dynamic load is investigated with considering ESDFD. The optimal normal force and controllable region for ESDFD to control the vibration under the target mode of dual-rotor system is determined. Furthermore, a control strategy based on altering normal force within the controllable region is designed. Results show that under the proposed control strategy, the damping effect of ESDFD significantly reduces the vibration amplitude of dual-rotor system and mitigates the dynamic load of inter-shaft bearing when passing through the resonance region, completely suppresses the bi-stable phenomenon and vibration jump behavior of dual-rotor system. Accordingly, the structural damage caused by excessive vibration is reduced, which demonstrates promising engineering applications of active ESDFD.

Appendices

Appendix A: Dynamic equation of conical beam element

Due to the cross-section parameters vary along axial position of conical beam element, they can be calculated using mean values by assuming the section is short. As a result, transverse shear deformation coefficient can be expressed as follows

$$ \varphi_{s} = \frac{{12EI_{m} \kappa }}{{GA_{m} l^{2} }} $$

(A90)

where

$$ I_{m} = \frac{{\pi \left( {R_{m}^{4} - r_{m}^{4} } \right)}}{4} $$

(A91)

$$ A_{m} = \pi \left( {R_{m}^{2} - r_{m}^{2} } \right) $$

(A92)

$$ \kappa = \frac{7 + 6\upsilon }{{6\left( {1 + \upsilon } \right)}}\left[ {1 + \frac{20 + 12\upsilon }{{7 + 6\upsilon }}\left( {\frac{{R_{m} r_{m} }}{{R_{m}^{2} + r_{m}^{2} }}} \right)^{2} } \right] $$

(A93)

$$ R_{m} = \sqrt {\frac{1}{2}\left( {R_{i}^{2} + R_{j}^{2} } \right)} $$

(A94)

$$ r_{m} = \sqrt {\frac{1}{2}\left( {r_{i}^{2} + r_{j}^{2} } \right)} $$

(A95)

According to the FEM and Lagrange’s principle, the dynamic matrix for conical beam element can be expressed as follows

$$ \left[ {{\mathbf{M}}_{e}^{t} } \right]_{con} = \frac{{\rho A_{L} l}}{{1260\left( {1 + \varphi_{s} } \right)^{2} }}\left[ {\begin{array}{*{20}c} {m_{1} } & 0 & 0 & {lm_{2} } & {m_{3} } & 0 & 0 & { - lm_{4} } \\ 0 & {m_{1} } & { - lm_{2} } & 0 & 0 & {m_{3} } & {lm_{4} } & 0 \\ 0 & { - lm_{2} } & {l^{2} m_{5} } & 0 & 0 & { - lm_{6} } & { - l^{2} m_{7} } & 0 \\ {lm_{2} } & 0 & 0 & {l^{2} m_{5} } & {lm_{6} } & 0 & 0 & { - l^{2} m_{7} } \\ {m_{3} } & 0 & 0 & {lm_{6} } & {m_{8} } & 0 & 0 & { - lm_{9} } \\ 0 & {m_{3} } & { - lm_{6} } & 0 & 0 & {m_{8} } & {lm_{9} } & 0 \\ 0 & {lm_{4} } & { - l^{2} m_{7} } & 0 & 0 & {lm_{9} } & {l^{2} m_{10} } & 0 \\ { - lm_{4} } & 0 & 0 & { - l^{2} m_{7} } & { - lm_{9} } & 0 & 0 & {l^{2} m_{10} } \\ \end{array} } \right] $$

(A96)

$$ \left[ {{\mathbf{M}}_{e}^{r} } \right]_{con} = \frac{{\rho I_{L} }}{{210l\left( {1 + \varphi_{s} } \right)^{2} }}\left[ {\begin{array}{*{20}c} {m_{11} } & 0 & 0 & {lm_{12} } & { - m_{11} } & 0 & 0 & {lm_{13} } \\ 0 & {m_{11} } & { - lm_{12} } & 0 & 0 & { - m_{11} } & { - lm_{13} } & 0 \\ 0 & { - lm_{12} } & {l^{2} m_{14} } & 0 & 0 & {lm_{12} } & { - l^{2} m_{15} } & 0 \\ {lm_{12} } & 0 & 0 & {l^{2} m_{14} } & { - lm_{12} } & 0 & 0 & { - l^{2} m_{15} } \\ { - m_{11} } & 0 & 0 & { - lm_{12} } & {m_{11} } & 0 & 0 & { - lm_{13} } \\ 0 & { - m_{11} } & {lm_{12} } & 0 & 0 & {m_{11} } & {lm_{13} } & 0 \\ 0 & { - lm_{13} } & { - l^{2} m_{15} } & 0 & 0 & {lm_{13} } & {l^{2} m_{16} } & 0 \\ {lm_{13} } & 0 & 0 & { - l^{2} m_{15} } & { - lm_{13} } & 0 & 0 & {l^{2} m_{16} } \\ \end{array} } \right] $$

(A97)

$$ \left[ {{\mathbf{K}}_{e} } \right]_{con} = {\mathbf{K}}_{e}^{b} + {\mathbf{K}}_{e}^{s} $$

(A98)

$$ {\mathbf{K}}_{e}^{b} = \frac{{{\rm E}I_{L} }}{{105l^{3} \left( {1 + \varphi_{s} } \right)^{2} }}\left[ {\begin{array}{*{20}c} {k_{1} } & 0 & 0 & {lk_{2} } & { - k_{1} } & 0 & 0 & {lk_{3} } \\ 0 & {k_{1} } & { - lk_{2} } & 0 & 0 & { - k_{1} } & { - lk_{3} } & 0 \\ 0 & { - lk_{2} } & {l^{2} k_{4} } & 0 & 0 & {lk_{2} } & {l^{2} k_{5} } & 0 \\ {lk_{2} } & 0 & 0 & {l^{2} k_{4} } & { - lk_{2} } & 0 & 0 & {l^{2} k_{5} } \\ { - k_{1} } & 0 & 0 & { - lk_{2} } & {k_{1} } & 0 & 0 & { - lk_{3} } \\ 0 & { - k_{1} } & {lk_{2} } & 0 & 0 & {k_{1} } & {lk_{3} } & 0 \\ 0 & { - lk_{3} } & {l^{2} k_{5} } & 0 & 0 & {lk_{3} } & {l^{2} k_{6} } & 0 \\ {lk_{3} } & 0 & 0 & {l^{2} k_{5} } & { - lk_{3} } & 0 & 0 & {l^{2} k_{6} } \\ \end{array} } \right] $$

(A99)

$$ {\mathbf{K}}_{e}^{s} = \frac{{GA_{L} \varphi_{s}^{2} }}{{12\kappa_{s} l\left( {1 + \varphi_{s} } \right)^{2} }}\left[ {\begin{array}{*{20}c} {k_{7} } & 0 & 0 & {lk_{8} } & { - k_{7} } & 0 & 0 & {lk_{8} } \\ 0 & {k_{7} } & { - lk_{8} } & 0 & 0 & { - k_{7} } & { - lk_{8} } & 0 \\ 0 & { - lk_{8} } & {l^{2} k_{9} } & 0 & 0 & {lk_{8} } & {l^{2} k_{9} } & 0 \\ {lk_{8} } & 0 & 0 & {l^{2} k_{9} } & { - lk_{8} } & 0 & 0 & {l^{2} k_{9} } \\ { - k_{7} } & 0 & 0 & { - lk_{8} } & {k_{7} } & 0 & 0 & { - lk_{8} } \\ 0 & { - k_{7} } & {lk_{8} } & 0 & 0 & {k_{7} } & {lk_{8} } & 0 \\ 0 & { - lk_{8} } & {l^{2} k_{9} } & 0 & 0 & {lk_{8} } & {l^{2} k_{9} } & 0 \\ {lk_{8} } & 0 & 0 & {l^{2} k_{9} } & { - lk_{8} } & 0 & 0 & {l^{2} k_{9} } \\ \end{array} } \right] $$

(A100)

$$ \left[ {{\mathbf{G}}_{e} } \right]_{con} = \frac{{\rho I_{L} }}{{105l\left( {1 + \varphi_{s} } \right)^{2} }}\left[ {\begin{array}{*{20}c} 0 & {m_{11} } & { - lm_{12} } & 0 & 0 & { - m_{11} } & { - lm_{13} } & 0 \\ { - m_{11} } & 0 & 0 & { - lm_{12} } & {m_{11} } & 0 & 0 & { - lm_{13} } \\ {lm_{12} } & 0 & 0 & {l^{2} m_{14} } & { - lm_{12} } & 0 & 0 & { - l^{2} m_{15} } \\ 0 & {lm_{12} } & { - l^{2} m_{14} } & 0 & 0 & { - lm_{12} } & {l^{2} m_{15} } & 0 \\ 0 & { - m_{11} } & {lm_{12} } & 0 & 0 & {m_{11} } & {lm_{13} } & 0 \\ {m_{11} } & 0 & 0 & {lm_{12} } & { - m_{11} } & 0 & 0 & {lm_{13} } \\ {lm_{13} } & 0 & 0 & { - l^{2} m_{15} } & { - lm_{13} } & 0 & 0 & {l^{2} m_{16} } \\ 0 & {lm_{13} } & {l^{2} m_{15} } & 0 & 0 & { - lm_{13} } & { - l^{2} m_{16} } & 0 \\ \end{array} } \right] $$

(A101)

in which

$$ \begin{array}{l} m_{1} = \left( {105\alpha _{1} + 42\alpha _{2} + 420} \right)\varphi _{s}^{2} + \left( {210\alpha _{1} + 78\alpha _{2} + 882} \right)\varphi _{s} + \left( {108\alpha _{1} + 38\alpha _{2} + 468} \right) \\ m_{2} = \left( {\begin{array}{*{20}r} \hfill {42} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {21} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {105} \\ \end{array} } \right)\varphi _{s}^{2} /2 + \left( {\begin{array}{*{20}r} \hfill {81} \\ \end{array} \alpha _{1} + 36\alpha _{2} + \begin{array}{*{20}r} \hfill {231} \\ \end{array} } \right)\varphi _{s} /2 + \left( {\begin{array}{*{20}r} \hfill {42} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {17} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {132} \\ \end{array} } \right)/2 \\ m_{3} = \left( {\begin{array}{*{20}r} \hfill {105} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {63} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {210} \\ \end{array} } \right)\varphi _{s}^{2} + \left( {\begin{array}{*{20}r} \hfill {189} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {111} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {378} \\ \end{array} } \right)\varphi _{s} + \left( {\begin{array}{*{20}r} \hfill {81} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {46} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {162} \\ \end{array} } \right) \\ m_{4} = \left( {42\alpha _{1} + \begin{array}{*{20}r} \hfill {21} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {105} \\ \end{array} } \right)\varphi _{s}^{2} /2 + \left( {\begin{array}{*{20}r} \hfill {81} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {42} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {189} \\ \end{array} } \right)\varphi _{s} /2 + \left( {\begin{array}{*{20}r} \hfill {36} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {19} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {78} \\ \end{array} } \right)/2 \\ m_{5} = \left( {\begin{array}{*{20}r} \hfill {21} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {12} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {42} \\ \end{array} } \right)\varphi _{s}^{2} /4 + \left( {\begin{array}{*{20}r} \hfill {18} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill 9 \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {42} \\ \end{array} } \right)\varphi _{s} /2 + \left( {\begin{array}{*{20}r} \hfill 9 \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill 4 \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {24} \\ \end{array} } \right)/2 \\ m_{6} = \left( {\begin{array}{*{20}r} \hfill {63} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {42} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {105} \\ \end{array} } \right)\varphi _{s}^{2} /2 + \left( {\begin{array}{*{20}r} \hfill {108} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {69} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {189} \\ \end{array} } \right)\varphi _{s} /2 + \left( {\begin{array}{*{20}r} \hfill {42} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {25} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {78} \\ \end{array} } \right)/2 \\ m_{7} = \left( {\begin{array}{*{20}r} \hfill {21} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {12} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {42} \\ \end{array} } \right)\varphi _{s}^{2} /4 + \left( {\begin{array}{*{20}r} \hfill {21} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {12} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {42} \\ \end{array} } \right)\varphi _{s} /2 + \left( {\begin{array}{*{20}r} \hfill 9 \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill 5 \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {18} \\ \end{array} } \right)/2 \\ m_{8} = \left( {\begin{array}{*{20}r} \hfill {315} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {252} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {420} \\ \end{array} } \right)\varphi _{s}^{2} + \left( {\begin{array}{*{20}r} \hfill {672} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {540} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {882} \\ \end{array} } \right)\varphi _{s} + \left( {\begin{array}{*{20}r} \hfill {360} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {290} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {468} \\ \end{array} } \right) \\ m_{9} = \left( {\begin{array}{*{20}r} \hfill {63} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {42} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {105} \\ \end{array} } \right)\varphi _{s}^{2} /2 + \left( {\begin{array}{*{20}r} \hfill {150} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {105} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {231} \\ \end{array} } \right)\varphi _{s} /2 + \left( {\begin{array}{*{20}r} \hfill {90} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {65} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {132} \\ \end{array} } \right)/2 \\ m_{{10}} = \left( {\begin{array}{*{20}r} \hfill {21} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {12} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {42} \\ \end{array} } \right)\varphi _{s}^{2} /4 + \left( {\begin{array}{*{20}r} \hfill {24} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {15} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {42} \\ \end{array} } \right)\varphi _{s} /2 + \left( {\begin{array}{*{20}r} \hfill {15} \\ \end{array} \alpha _{1} + \begin{array}{*{20}r} \hfill {10} \\ \end{array} \alpha _{2} + \begin{array}{*{20}r} \hfill {24} \\ \end{array} } \right)/2 \\ m_{{11}} = 126\delta _{1} + 72\delta _{2} + 30\delta _{4} + 45\delta _{3} + 252 \\ m_{{12}} = \left( { - 42\delta _{1} - 21\delta _{2} - 7.5\delta _{4} - 12\delta _{3} - 105} \right)\varphi _{s} + 21\delta _{1} + 15\delta _{2} + 7.5\delta _{4} + 10.5\delta _{3} + 21 \\ m_{{13}} = \left( { - 63\delta _{1} - 42\delta _{2} - 22.5\delta _{4} - 30\delta _{3} - 105} \right)\varphi _{s} + 21 - 7.5\delta _{4} - 7.5\delta _{3} - 6\delta _{2} \\ m_{{14}} = \left( {17.5\delta _{1} + 7\delta _{2} + 2\delta _{4} + 3.5\delta _{3} + 70} \right)\varphi _{s}^{2} + \left( {35 - 7\delta _{1} - 3.5\delta _{4} - 5\delta _{3} - 7\delta _{2} } \right)\varphi _{s} + 7\delta _{1} + 4\delta _{2} + 2\delta _{4} + 2.75\delta _{3} + 28 \\ m_{{15}} = \left( { - 17.5\delta _{1} - 10.5\delta _{2} - 5\delta _{4} - 7\delta _{3} - 35} \right)\varphi _{s}^{2} + \left( {17.5\delta _{1} + 10.5\delta _{2} + 5\delta _{4} + 7\delta _{3} + 35} \right)\varphi _{s} + 3.5\delta _{1} + 3\delta _{2} + 2.5\delta _{4} + 2.75\delta _{3} + 7 \\ m_{{16}} = \left( {52.5\delta _{1} + 42\delta _{2} + 30\delta _{4} + 35\delta _{3} + 70} \right)\varphi _{s}^{2} + \left( {42\delta _{1} + 42\delta _{2} + 37.5\delta _{4} + 40\delta _{3} + 35} \right)\varphi _{s} + 21\delta _{1} + 18\delta _{2} + 15\delta _{4} + 16.25\delta _{3} + 28 \\ \end{array} $$

(A102)

$$ \begin{array}{l} k_{1} = 630\delta_{1} + 504\delta_{2} + 396\delta_{4} + 441\delta_{3} + 1260 \\ k_{2} = - \left( {105\delta_{1} + 105\delta_{2} + 84\delta_{4} + 94.5\delta_{3} } \right)\varphi_{s} + 210\delta_{1} + 147\delta_{2} + 114\delta_{4} + 126\delta_{3} + 630 \\ k_{3} = \left( {105\delta_{1} + 105\delta_{2} + 84\delta_{4} + 94.5\delta_{3} } \right)\varphi_{s} + 420\delta_{1} + 357\delta_{2} + 282\delta_{4} + 315\delta_{3} + 630 \\ k_{4} = \left( {52.5\delta_{1} + 35\delta_{2} + 21\delta_{4} + 26.25\delta_{3} + 105} \right)\varphi_{s}^{2} + \left( {210 - 42\delta_{4} - 42\delta_{3} - 35\delta_{2} } \right)\varphi_{s} \\ + 105\delta_{1} + 56\delta_{2} + 36\delta_{4} + 42\delta_{3} + 420 \\ k_{5} = \left( { - 52.5\delta_{1} - 35\delta_{2} - 21\delta_{4} - 26.25\delta_{3} - 105} \right)\varphi_{s}^{2} \\ + \left( { - 105\delta_{1} - 70\delta_{2} - 42\delta_{4} - 52.5\delta_{3} - 210} \right)\varphi_{s} \\ + 105\delta_{1} + 91\delta_{2} + 78\delta_{4} + 84\delta_{3} + 210 \\ k_{6} = \left( {52.5\delta_{1} + 35\delta_{2} + 21\delta_{4} + 26.25\delta_{3} + 105} \right)\varphi_{s}^{2} \\ + \left( {210\delta_{1} + 175\delta_{2} + 126\delta_{4} + 147\delta_{3} + 210} \right)\varphi_{s} \\ + 315\delta_{1} + 266\delta_{2} + 204\delta_{4} + 231\delta_{3} + 420 \\ k_{7} = 6\alpha_{1} + 4\alpha_{2} + 12 \\ k_{8} = 3\alpha_{1} + 2\alpha_{2} + 6 \\ k_{9} = 1.5\alpha_{1} + \alpha_{2} + 3 \\ \end{array} $$

(A103)

The parameters related to geometric dimensions of conical shaft section can be expressed as

$$ \alpha_{1} = {{2\pi \left( {R_{i} \Delta R - r_{i} \Delta r} \right)} \mathord{\left/ {\vphantom {{2\pi \left( {R_{i} \Delta R - r_{i} \Delta r} \right)} {A_{i} }}} \right. \kern-0pt} {A_{i} }} $$

(A104)

$$ \alpha_{2} = {{\pi \left( {\Delta R^{2} - \Delta r_{i}^{2} } \right)} \mathord{\left/ {\vphantom {{\pi \left( {\Delta R^{2} - \Delta r_{i}^{2} } \right)} {A_{i} }}} \right. \kern-0pt} {A_{i} }} $$

(A105)

$$ \delta_{1} = {{\pi \left( {R_{i}^{3} \Delta R - r_{i}^{3} \Delta r} \right)} \mathord{\left/ {\vphantom {{\pi \left( {R_{i}^{3} \Delta R - r_{i}^{3} \Delta r} \right)} {I_{i} }}} \right. \kern-0pt} {I_{i} }} $$

(A106)

$$ \delta_{2} = {{3\pi \left( {R_{i}^{2} \Delta R^{2} - r_{i}^{2} \Delta r_{i}^{2} } \right)} \mathord{\left/ {\vphantom {{3\pi \left( {R_{i}^{2} \Delta R^{2} - r_{i}^{2} \Delta r_{i}^{2} } \right)} {2I_{i} }}} \right. \kern-0pt} {2I_{i} }} $$

(A107)

$$ \delta_{3} = {{\pi \left( {R_{i} \Delta R^{3} - r_{i} \Delta r^{3} } \right)} \mathord{\left/ {\vphantom {{\pi \left( {R_{i} \Delta R^{3} - r_{i} \Delta r^{3} } \right)} {I_{i} }}} \right. \kern-0pt} {I_{i} }} $$

(A108)

$$ \delta_{4} = {{\pi \left( {\Delta R^{4} - \Delta r_{i}^{4} } \right)} \mathord{\left/ {\vphantom {{\pi \left( {\Delta R^{4} - \Delta r_{i}^{4} } \right)} {4I_{i} }}} \right. \kern-0pt} {4I_{i} }} $$

(A109)

in which

$$ \Delta R = R_{j} - R_{i} $$

(A110)

$$ \Delta r = r_{j} - r_{i} $$

(A111)

A_i and I_i represent the cross-section area and moment of inertia of node i

$$ A_{i} = \pi \left( {R_{i}^{2} - r_{i}^{2} } \right) $$

(A112)

$$ I_{i} = {{\pi \left( {R_{i}^{4} - r_{i}^{4} } \right)} \mathord{\left/ {\vphantom {{\pi \left( {R_{i}^{4} - r_{i}^{4} } \right)} 4}} \right. \kern-0pt} 4} $$

(A113)

Appendix B: Geometric and physical parameters of the model

See Table 

Table 4 Geometrical dimensions of shaft elements

4,

Table 5 Physical parameters of disk elements

5,

Table 6 Parameters of inter-shaft bearing

6,

Table 7 Parameters of ESDFD

7 and

Table 8 Parameters of supports

8.