Analytical Analysis for the Nonlinear Phenomena of a Dual-Rotor System at the Case of Primary Resonances

Abstract

Background

This paper proposes an analytical method to study the nonlinear phenomena of a simple dual-rotor system in the case of primary resonances.

Purpose

According to the assumed mode method (AMM), the dynamic equations of the system with four degrees of freedom (4DOF) are established by considering the nonlinearities of the inter-shaft bearing as a nonlinear spring. The 4DOF dynamic equations in real coordinate are transferred into 2DOF dynamic equations in complex coordinate because of the symmetry of the system in two directions.

Methods

Then the amplitude-frequency response equations for both primary resonances are obtained by the multiple scales method and the numerical verification shows that the simplified method of the dynamic equation is correct. Moreover, the primary resonances affected by the typical parameters such as the linear stiffness and the nonlinear stiffness of the inter-shaft bearing, and the rotation speed ratio are discussed in detail afterwards.

Conclusions

The results show that the system will show the jump phenomenon and resonance hysteresis phenomenon when the linear stiffness is rather small and the nonlinear stiffness is rather large. The linear stiffness will suppress these nonlinear phenomena while the nonlinear stiffness will promote them. The results obtained in this paper will contribute to the mechanism analysis of the nonlinear phenomena in a dual-rotor system, which is beyond the reach of the numerical method.

Appendix

The nonlinear contact force of the inter-shaft bearing in the complex coordinate is

$$\begin{aligned} F_{\text{b}} = \frac{{F_{x} }}{{e_{1} }} + i\frac{{F_{y} }}{{e_{1} }} \\ = k_{\text{s}} \left( {X_{\text{i}} - X_{\text{o}} } \right) + e_{1}^{2} k_{\text{n}} \left( {X_{\text{i}} - X_{\text{o}} } \right)\left[ {\left( {X_{\text{i}} - X_{\text{o}} } \right)^{2} - 3\left( {Y_{\text{i}} - Y_{\text{o}} } \right)^{2} } \right] \\ { + }i\left\{ {k_{\text{s}} \left( {Y_{\text{i}} - Y_{\text{o}} } \right) + e_{1}^{2} k_{\text{n}} \left( {Y_{\text{i}} - Y_{\text{o}} } \right)\left[ {\left( {Y_{\text{i}} - Y_{\text{o}} } \right)^{2} - 3\left( {X_{\text{i}} - X_{\text{o}} } \right)^{2} } \right]} \right\} \\ = k_{\text{s}} \left( {r_{\text{i}} - r_{\text{o}} } \right) + e_{1}^{2} k_{\text{n}} \left\{ {\left( {X_{\text{i}} - X_{\text{o}} } \right)\left[ {\left( {X_{\text{i}} - X_{\text{o}} } \right)^{2} - 3\left( {Y_{\text{i}} - Y_{\text{o}} } \right)^{2} } \right] + i\left( {Y_{\text{i}} - Y_{\text{o}} } \right)\left[ {\left( {Y_{\text{i}} - Y_{\text{o}} } \right)^{2} - 3\left( {X_{\text{i}} - X_{\text{o}} } \right)^{2} } \right]} \right\}, \\ \end{aligned}$$

where the term $\left\{ {\left( {X_{\text{i}} - X_{\text{o}} } \right)\left[ {\left( {X_{\text{i}} - X_{\text{o}} } \right)^{2} - 3\left( {Y_{\text{i}} - Y_{\text{o}} } \right)^{2} } \right] + i\left( {Y_{\text{i}} - Y_{\text{o}} } \right)\left[ {\left( {Y_{\text{i}} - Y_{\text{o}} } \right)^{2} - 3\left( {X_{\text{i}} - X_{\text{o}} } \right)^{2} } \right]} \right\}$ is calculated as

$$\begin{aligned} \left\{ {} \right\} & = \left( {X_{\text{i}}^{3} - X_{\text{o}}^{3} - 3X_{\text{i}}^{2} X_{\text{o}} + 3X_{\text{i}} X_{\text{o}}^{2} } \right) - 3\left( {X_{\text{i}} Y_{\text{i}}^{2} - X_{\text{o}} Y_{\text{o}}^{2} - X_{\text{o}} Y_{\text{i}}^{2} - 2X_{\text{i}} Y_{\text{i}} Y_{\text{o}} + X_{\text{i}} Y_{\text{o}}^{2} + 2X_{\text{o}} Y_{\text{i}} Y_{\text{o}} } \right) \\ & \;\;\; + i\left( {Y_{\text{i}}^{3} - Y_{\text{o}}^{3} - 3Y_{\text{i}}^{2} Y_{\text{o}} + 3Y_{\text{i}} Y_{\text{o}}^{2} } \right) - 3i\left( {X_{\text{i}}^{2} Y_{\text{i}} - X_{\text{o}}^{2} Y_{\text{o}} - X_{\text{i}}^{2} Y_{\text{o}} - 2X_{\text{i}} X_{\text{o}} Y_{\text{i}} + X_{\text{o}}^{2} Y_{\text{i}} + 2X_{\text{i}} X_{\text{o}} Y_{\text{o}} } \right) \\ & = \left( {X_{\text{i}}^{3} + iY_{\text{i}}^{3} - 3X_{\text{i}} Y_{\text{i}}^{2} - 3iX_{\text{i}}^{2} Y} \right) - \left( {X_{\text{o}}^{3} + iY_{\text{o}}^{3} - 3X_{\text{o}} Y_{\text{o}}^{2} - 3iX_{\text{o}}^{2} Y_{\text{o}} } \right) \\ & \;\;\; - \;\;3\left( {X_{\text{i}}^{2} X_{\text{o}} + iY_{\text{i}}^{2} Y_{\text{o}} - X_{\text{o}} Y_{\text{i}}^{2} - iX_{\text{i}}^{2} Y_{\text{o}} - 2X_{\text{i}} Y_{\text{i}} Y_{\text{o}} - 2iX_{\text{i}} X_{\text{o}} Y_{\text{i}} } \right) \\ & \;\;\; + \;3\left( {X_{\text{i}} X_{\text{o}}^{2} + iY_{\text{i}} Y_{\text{o}}^{2} - X_{\text{i}} Y_{\text{o}}^{2} - iX_{\text{o}}^{2} Y_{\text{i}} - 2X_{\text{o}} Y_{\text{i}} Y_{\text{o}} - 2iX_{\text{i}} X_{\text{o}} Y_{\text{o}} } \right) \\ & = \left( {X_{\text{i}} - iY_{\text{i}} } \right)^{3} - \left( {X_{\text{o}} - iY_{\text{o}} } \right)^{3} - 3\left( {X_{\text{i}} - iY_{\text{i}} } \right)^{2} \left( {X_{\text{o}} - iY_{\text{o}} } \right) + 3\left( {X_{\text{i}} - iY_{\text{i}} } \right)\left( {X_{\text{o}} - iY_{\text{o}} } \right)^{2} \\ & = \bar{r}_{\text{i}}^{3} - \bar{r}_{\text{o}}^{3} - 3\bar{r}_{\text{i}}^{2} \bar{r}_{\text{o}} + 3\bar{r}_{\text{i}} \bar{r}_{\text{o}}^{2} \\ & = \left( {\bar{r}_{\text{i}} - \bar{r}_{\text{o}} } \right)^{3} . \\ \end{aligned}$$

Therefore, the nonlinear contact force in the complex coordinate can be expressed as

$$F_{\text{b}} = F_{X} + iF_{Y} = k_{\text{s}} \left( {r_{\text{i}} - r_{\text{o}} } \right) + e_{1}^{2} k_{\text{n}} \left( {\bar{r}_{\text{i}} - \bar{r}_{\text{o}} } \right)^{3} = k_{\text{s}} \left( {F_{1} r_{ 1} - F_{2} r_{ 2} } \right) + e_{1}^{2} k_{\text{n}} \left( {F_{1} \bar{r}_{ 1} - F_{2} \bar{r}_{ 2} } \right)^{3} .$$