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Nonlinear dynamic behaviors of rod fastening rotor-hydrodynamic journal bearing system

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Abstract

The model of a rod fastening rotor-finite length journal bearing system is established in this paper. The contact surfaces between the disks and rods are modeled as bending resistance springs with nonlinear stiffness. The gyroscopic effect is considered in the 12 degrees-of-freedom model of the rod fastening rotor-bearing system. Based on the variational principle, an approximate analytical solution of nonlinear oil film forces of finite length journal bearing is proposed by the separation of variables method. By taking rotating speed, eccentricity and bending stiffness as control parameters, the nonlinear behaviors of the system are investigated by the improved Newmark method. The numerical results reveal periodic, period-doubling, period-4, quasi-periodic solutions etc. of rich and complex nonlinear behaviors of the system.

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Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant No. 51375380), Open Project of State Key Laboratory for Strength and Vibration of Mechanical Structures of China (Grant No. SV2014-KF-08), Shaanxi Provincial Natural Science Foundation of China (Grant No. 2014JM2-5082).

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Authors and Affiliations

  1. School of Mechanical and Instrumental Engineering, Xi’an University of Technology, Xi’an, 710048, Shaanxi, People’s Republic of China

    Di Hei, Yanjun Lu, Fuxi Liu & Chao Zhou

  2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Di Hei, Yanjun Lu & Yongfang Zhang

  3. Department of Mechanical and Electrical Engineering, Shaanxi Railway Institute, Weinan, 714000, Shaanxi, People’s Republic of China

    Di Hei

  4. College of Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Yanjun Lu, Yongfang Zhang & Norbert Müller

  5. School of Printing and Packaging Engineering, Xi’an University of Technology, Xi’an, 710048, Shaanxi, People’s Republic of China

    Yongfang Zhang

Authors
  1. Di Hei
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  2. Yanjun Lu
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  3. Yongfang Zhang
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  4. Fuxi Liu
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  5. Chao Zhou
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Corresponding authors

Correspondence to Yanjun Lu or Yongfang Zhang.

Appendices

Appendix 1: The parameters of the rod fastening rotor system

The parameters of the rod fastening rotor system are shown in Figs. 16 and 17.

Fig. 16
figure 16

Parameters of the shaft and disk 1

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Fig. 17
figure 17

Parameters of the shaft and disk 2

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In Fig. 16, \(x_{A}\) and \(y_{A}\) are the displacements of the center of the rotor at station \(A\) in the \(x\) and \(y\) directions. \(x_{B}\) and \(y_{B}\) are the displacements of the center of the rotor at station \(B\) in the \(x\) and \(y\) directions. \(x_{O1}\) and \(y_{O1}\) are the displacements of the center of disk 1 in the \(x\) and \(y\) directions. \(\theta _{x1}\) and \(\theta _{y1}\) are the angles that disk 1 rotates around the \(x\) and \(y\) axes. Other parameters are as follows:

$$\begin{aligned}&{ AO}_{1}=a_{1},\quad { AB}=l,\quad O_{1}B=l-a_{1}=b_{1},\quad x_1 =x_A +\frac{x_B -x_A }{l}a_1 =\frac{b_1 }{l}x_A +\frac{a_1 }{l}x_B,\\&{x}'_1 =x_{O1} -x_1,\quad \theta _{dy1} =\frac{x_B -x_A }{l},\quad \theta _{oy1} =\theta _{y1} -\theta _{dy1}, \\&y_1 =y_A +\frac{y_B -y_A }{l}a_1 =\frac{b_1 }{l}y_A +\frac{a_1 }{l}y_B,\quad {y}'_1 =y_{O1} -y_1,\quad \theta _{dx1} =\frac{y_B -y_A }{l},\quad \theta _{ox1} =\theta _{x1} -\theta _{dx1}. \end{aligned}$$

In Fig. 17, \(x_{O2}\) and \(y_{O2}\) are the displacements of the center of disk 2 in \(x\) and \(y\) directions. \(\theta _{x2}\) and \(\theta _{y2}\) are the angles that disk 2 rotates around \(x\) and \(y\) axes. Other parameters are as follows: \({ AO}_{2}=a_{2}, \quad { AB}=l, \quad O_{2}B=l-a_{2}=b_{2}\),

$$\begin{aligned}&x_2 =x_A +\frac{x_B -x_A }{l}a_2 =\frac{b_2 }{l}x_A +\frac{a_2 }{l}x_B, \quad {x}'_2 =x_{O2} -x_2, \quad \theta _{dy2} =\frac{x_B -x_A }{l},\quad \theta _{oy2} =\theta _{y2} -\theta _{dy2},\\&y_2 =y_A +\frac{y_B -y_A }{l}a_2 =\frac{b_2 }{l}y_A +\frac{a_2 }{l}y_B, \quad {y}'_2 =y_{O2} -y_2, \quad \theta _{dx2} =\frac{y_B -y_A }{l},\quad \theta _{ox2} =\theta _{x2} -\theta _{dx2}. \end{aligned}$$

Appendix 2: The mass, damping, and stiffness matrices

The mass, damping and stiffness matrices are as follows:

$$\begin{aligned}&{\mathbf {M}}=\left[ {{\begin{array}{cccccccccccc} {\bar{{m}}_A }&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ &{} {\bar{{m}}_A }&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ &{} &{} {\bar{{m}}_{O1} }&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ &{} &{} &{} {\bar{{m}}_{O1} }&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ &{} &{} &{} &{} {\bar{{J}}_{d1} }&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ &{} &{} &{} &{} &{} {\bar{{J}}_{d1} }&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ &{} &{} &{} &{} &{} &{} {\bar{{m}}_{O2} }&{} 0&{} 0&{} 0&{} 0&{} 0 \\ &{} &{} &{} &{} &{} &{} &{} {\bar{{m}}_{O2} }&{} 0&{} 0&{} 0&{} 0 \\ &{} &{} &{} &{} &{} &{} &{} &{} {\bar{{J}}_{d2} }&{} 0&{} 0&{} 0 \\ &{} &{} &{} &{} &{} &{} &{} &{} &{} {\bar{{J}}_{d2} }&{} 0&{} 0 \\ &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} {\bar{{m}}_B }&{} 0 \\ &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} {\bar{{m}}_B } \\ \end{array} }} \right] \\&{\mathbf {G}}=\left[ {{\begin{array}{cccccccccccc} 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ &{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad &{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad &{}\quad &{}\quad 0&{}\quad {-\bar{{J}}_{z1} \bar{{\omega }}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad &{}\quad &{}\quad {\bar{{J}}_{z1} \bar{{\omega }}}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 0&{}\quad {-\bar{{J}}_{z2} \bar{{\omega }}}&{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad {\bar{{J}}_{z2} \bar{{\omega }}}&{}\quad 0&{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 0&{}\quad 0 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 0 \\ \end{array} }} \right] \\ \end{aligned}$$
$$\begin{aligned}&{\mathbf {K}}=\left[ {{\begin{array}{cccccc} {\begin{array}{l} \left( K_{11} \frac{\bar{{b}}_1^2 }{\bar{{l}}^{2}}+\frac{K_{44} }{\bar{{l}}^{2}}-\frac{2K_{14} \bar{{b}}_1 }{\bar{{l}}^{2}} \right. \\ \left. \,\,\,+\,K_{11} \frac{\bar{{b}}_2^2 }{\bar{{l}}^{2}}+\frac{K_{44} }{\bar{{l}}^{2}}-\frac{2K_{14} \bar{{b}}_2 }{\bar{{l}}^{2}}\right) \\ \end{array}}&{} 0&{} {\left( -\frac{K_{11} \bar{{b}}_1 }{\bar{{l}}}+\frac{K_{14} }{\bar{{l}}}\right) }&{} 0&{} 0&{} {\left( \frac{K_{44} }{\bar{{l}}}-\frac{K_{14} \bar{{b}}_1 }{\bar{{l}}}\right) } \\ 0&{} {\begin{array}{l} \left( K_{22} \frac{\bar{{b}}_1^2 }{\bar{{l}}^{2}}+\frac{K_{33} }{\bar{{l}}^{2}}-\frac{2K_{23} \bar{{b}}_1 }{\bar{{l}}^{2}}\right. \\ \left. \,\,\,+\,K_{22} \frac{\bar{{b}}_2^2 }{\bar{{l}}^{2}}+\frac{K_{33} }{\bar{{l}}^{2}}-\frac{2K_{23} \bar{{b}}_2 }{\bar{{l}}^{2}}\right) \\ \end{array}}&{} 0&{} {\left( -K_{22} \frac{\bar{{b}}_1 }{\bar{{l}}}+\frac{K_{23} }{\bar{{l}}}\right) }&{} {\left( \frac{K_{33} }{\bar{{l}}}-\frac{K_{23} \bar{{b}}_1 }{\bar{{l}}}\right) }&{} 0 \\ {\left( \frac{K_{14} }{\bar{{l}}}-\frac{K_{11} \bar{{b}}_1 }{\bar{{l}}}\right) }&{} 0&{} {(K_{11} +K_b )}&{} 0&{} 0&{} {K_{14} } \\ 0&{} {\left( \frac{K_{23} }{\bar{{l}}}-\frac{K_{22} \bar{{b}}_1 }{\bar{{l}}}\right) }&{} 0&{} {(K_{22} +K_b )}&{} {K_{23} }&{} 0 \\ 0&{} {\left( \frac{K_{33} }{\bar{{l}}}-\frac{K_{23} \bar{{b}}_1 }{\bar{{l}}}\right) }&{} 0&{} {K_{23} }&{} {K_{33} }&{} 0 \\ {\left( \frac{K_{44} }{\bar{{l}}}-\frac{K_{14} \bar{{b}}_1 }{\bar{{l}}}\right) }&{} 0&{} {K_{14} }&{} 0&{} 0&{} {K_{44} } \\ {\left( -\frac{K_{11} \bar{{b}}_2 }{\bar{{l}}}+\frac{K_{14} }{\bar{{l}}}\right) }&{} 0&{} {-K_b }&{} 0&{} 0&{} 0 \\ 0&{} {\left( -\frac{K_{22} \bar{{b}}_2 }{\bar{{l}}}+\frac{K_{23} }{\bar{{l}}}\right) }&{} 0&{} {-K_b }&{} 0&{} 0 \\ 0&{} {\left( \frac{K_{33} }{\bar{{l}}}-\frac{K_{23} \bar{{b}}_2 }{\bar{{l}}}\right) }&{} 0&{} 0&{} 0&{} 0 \\ {\left( \frac{K_{44} }{\bar{{l}}}-\frac{K_{14} \bar{{b}}_2 }{\bar{{l}}}\right) }&{} 0&{} 0&{} 0&{} 0&{} 0 \\ {\begin{array}{l} \left( \frac{K_{11} \bar{{a}}_1 \bar{{b}}_1 }{\bar{{l}}^{2}}-\frac{K_{44} }{\bar{{l}}^{2}}+\frac{K_{14} (\bar{{b}}_1 -\bar{{a}}_1 )}{\bar{{l}}^{2}}\right. \\ \left. \,\,\,+\,\frac{K_{11} \bar{{a}}_2 \bar{{b}}_2 }{\bar{{l}}^{2}}-\frac{K_{44} }{\bar{{l}}^{2}}+\frac{K_{14} (\bar{{b}}_2 -\bar{{a}}_2 )}{\bar{{l}}^{2}}\right) \\ \end{array}}&{} 0&{} {\left( -\frac{K_{11} \bar{{a}}_1 }{\bar{{l}}}-\frac{K_{14} }{\bar{{l}}}\right) }&{} 0&{} 0&{} {\left( -\frac{K_{44} }{\bar{{l}}}-\frac{K_{14} \bar{{a}}_1 }{\bar{{l}}}\right) } \\ 0&{} {\begin{array}{l} \left( \frac{K_{22} \bar{{a}}_1 \bar{{b}}_1 }{\bar{{l}}^{2}}-\frac{K_{33} }{\bar{{l}}^{2}}+\frac{K_{23} (\bar{{b}}_1 -\bar{{a}}_1 )}{\bar{{l}}^{2}} \right. \\ \left. \,\,\,+\,\frac{K_{22} \bar{{a}}_2 \bar{{b}}_2 }{\bar{{l}}^{2}}-\frac{K_{33} }{\bar{{l}}^{2}}+\frac{K_{23} (\bar{{b}}_2 -\bar{{a}}_2 )}{\bar{{l}}^{2}}\right) \\ \end{array}}&{} 0&{} {\left( -\frac{K_{22} \bar{{a}}_1 }{\bar{{l}}}-\frac{K_{23} }{\bar{{l}}}\right) }&{} {\left( -\frac{K_{33} }{\bar{{l}}}-\frac{K_{23} \bar{{a}}_1 }{\bar{{l}}}\right) }&{} 0 \\ \end{array} }} \right. \end{aligned}$$
$$\begin{aligned} \left. {{\begin{array}{cccccc} {\left( -\frac{K_{11} \bar{{b}}_2 }{\bar{{l}}}+\frac{K_{14} }{\bar{{l}}}\right) }&{} 0&{} 0&{} {\left( \frac{K_{44} }{\bar{{l}}}-\frac{K_{14} \bar{{b}}_2 }{\bar{{l}}}\right) }&{} {\begin{array}{l} \left( \frac{K_{11} \bar{{a}}_1 \bar{{b}}_1 }{\bar{{l}}^{2}}-\frac{K_{44} }{\bar{{l}}^{2}}+\frac{K_{14} (\bar{{b}}_1 -\bar{{a}}_1 )}{\bar{{l}}^{2}}\right. \\ \left. \,\,\,+\,\frac{K_{11} \bar{{a}}_2 \bar{{b}}_2 }{\bar{{l}}^{2}}-\frac{K_{44} }{\bar{{l}}^{2}}+\frac{K_{14} (\bar{{b}}_2 -\bar{{a}}_2 )}{\bar{{l}}^{2}}\right) \\ \end{array}}&{} 0 \\ 0&{} {\left( -K_{22} \frac{\bar{{b}}_2 }{\bar{{l}}}+\frac{K_{23} }{\bar{{l}}}\right) }&{} {\left( \frac{K_{33} }{\bar{{l}}}-\frac{K_{23} \bar{{b}}_2 }{\bar{{l}}}\right) }&{} 0&{} 0&{} {\begin{array}{l} \left( \frac{K_{22} \bar{{a}}_1 \bar{{b}}_1 }{\bar{{l}}^{2}}-\frac{K_{33} }{\bar{{l}}^{2}}+\frac{K_{23} (\bar{{b}}_1 -\bar{{a}}_1 )}{\bar{{l}}^{2}}\right. \\ \left. \,\,\,+\,\frac{K_{22} \bar{{a}}_2 \bar{{b}}_2 }{\bar{{l}}^{2}}-\frac{K_{33} }{\bar{{l}}^{2}}+\frac{K_{23} (\bar{{b}}_2 -\bar{{a}}_2 )}{\bar{{l}}^{2}}\right) \\ \end{array}} \\ {-K_b }&{} 0&{} 0&{} 0&{} {\left( -\frac{K_{11} \bar{{a}}_1 }{\bar{{l}}}-\frac{K_{14} }{\bar{{l}}}\right) }&{} 0 \\ 0&{} {-K_b }&{} 0&{} 0&{} 0&{} {\left( -\frac{K_{22} \bar{{a}}_1 }{\bar{{l}}}-\frac{K_{23} }{\bar{{l}}}\right) } \\ 0&{} 0&{} 0&{} 0&{} 0&{} {\left( -\frac{K_{33} }{\bar{{l}}}-\frac{K_{23} \bar{{a}}_1 }{\bar{{l}}}\right) } \\ 0&{} 0&{} 0&{} 0&{} {\left( -\frac{K_{44} }{\bar{{l}}}-\frac{K_{14} a_1 }{\bar{{l}}}\right) }&{} 0 \\ {(K_{11} +K_b )}&{} 0&{} 0&{} {K_{14} }&{} {\left( -\frac{K_{11} a_2 }{\bar{{l}}}-\frac{K_{14} }{\bar{{l}}}\right) }&{} 0 \\ 0&{} {(K_{22} +K_b )}&{} {K_{23} }&{} 0&{} 0&{} {\left( -\frac{K_{22} \bar{{a}}_2 }{\bar{{l}}}-\frac{K_{23} }{\bar{{l}}}\right) } \\ 0&{} {K_{23} }&{} {K_{33} }&{} 0&{} 0&{} {\left( -\frac{K_{33} }{\bar{{l}}}-\frac{K_{23} \bar{{a}}_2 }{\bar{{l}}}\right) } \\ {K_{14} }&{} 0&{} 0&{} {K_{44} }&{} {\left( -\frac{K_{44} }{\bar{{l}}}-\frac{K_{14} \bar{{a}}_2 }{\bar{{l}}}\right) }&{} 0 \\ {\left( -\frac{K_{11} \bar{{a}}_2 }{\bar{{l}}}-\frac{K_{14} }{\bar{{l}}}\right) }&{} 0&{} 0&{} {\left( -\frac{K_{44} }{\bar{{l}}}-\frac{K_{14} \bar{{a}}_2 }{\bar{{l}}}\right) }&{} {\begin{array}{l} \left( \frac{K_{11} \bar{{a}}_1^2 }{\bar{{l}}^{2}}+\frac{K_{44} }{\bar{{l}}^{2}}+\frac{2K_{14} \bar{{a}}_1 }{\bar{{l}}^{2}}\right. \\ \left. \,\,\,+\,\frac{K_{11} \bar{{a}}_2^2 }{\bar{{l}}^{2}}+\frac{K_{44} }{\bar{{l}}^{2}}+\frac{2K_{14} \bar{{a}}_2 }{\bar{{l}}^{2}}\right) \\ \end{array}}&{} 0 \\ 0&{} {\left( -\frac{K_{22} \bar{{a}}_2 }{\bar{{l}}}-\frac{K_{23} }{\bar{{l}}}\right) }&{} {\left( -\frac{K_{33} }{\bar{{l}}}-\frac{K_{23} \bar{{a}}_2 }{\bar{{l}}}\right) }&{} 0&{} 0&{} {\begin{array}{l} \left( \frac{K_{22} \bar{{a}}_1^2 }{\bar{{l}}^{2}}+\frac{K_{33} }{\bar{{l}}^{2}}+\frac{2K_{23} \bar{{a}}_1 }{\bar{{l}}^{2}}\right. \\ \left. \,\,\,+\,\frac{K_{22} \bar{{a}}_2^2 }{\bar{{l}}^{2}}+\frac{K_{33} }{\bar{{l}}^{2}}+\frac{2K_{23} \bar{{a}}_2 }{\bar{{l}}^{2}}\right) \\ \end{array}} \\ \end{array} }} \right] \end{aligned}$$

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Hei, D., Lu, Y., Zhang, Y. et al. Nonlinear dynamic behaviors of rod fastening rotor-hydrodynamic journal bearing system. Arch Appl Mech 85, 855–875 (2015). https://doi.org/10.1007/s00419-015-0996-5

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