Applications of adaptive stiffness suspensions to vibration control of a high-speed stiff rotor with tilting pad bearings

Abstract

The current paper discusses the optimum parameter setting of asymmetric high-static low-dynamic stiffness (HSLDS) suspensions to reduce vibrations of a high-speed symmetric rotary system, excited by an unbalance force. The rotating system consists of a shaft that is supported by tilting pad journal bearings on the asymmetric HSLDS suspensions. The Reynolds equation is solved numerically to obtain the oil pressure distribution for each pad of bearing. With the aim of calculating the hydrodynamic forces applied to each pad, an analytical approach is presented. Its results are validated using a numerical integration approach. Given the considerable difference between the shaft mass and pads moment of inertia, the mathematical equations governing the motions of disk, journal, bearing and pads are solved implementing a routine specified for stiff ordinary differential equations in MATLAB. The optimum parameters of HSLDS suspensions are obtained, using a multi-objective genetic algorithm. Design objectives are considered to minimize the vibrations of rotor, journal, and bearing and bearing force transmission to the external supports. The efficiency of optimum HSLDS suspensions in reducing the vibrations of journal, bearing and rotor within the operating speed range is shown. The high performance of the designed suspensions in decreasing the bearings force transmission is proved as well. In addition, the design robustness to uncertainties in HSLDS suspensions parameters is studied.

Appendix

To find partial derivatives of G at first, following parameters are defined as:

$$\begin{gathered} n = \sqrt {1 - u_{p}^{2} - w_{p}^{2} } b = w_{p} - \left( {1 + u_{p} } \right)\tan \frac{\upsilon }{2}b_{u} = \frac{\partial b}{{\partial u}},b_{w} = \frac{\partial b}{{\partial w}},b_{uu} = \frac{{\partial^{2} b}}{{\partial u^{2} }},b_{ww} = \frac{{\partial^{2} b}}{{\partial w^{2} }}, \hfill \\ n_{u} = \frac{\partial n}{{\partial u}},n_{w} = \frac{\partial n}{{\partial w}},n_{uu} = \frac{{\partial^{2} n}}{{\partial u^{2} }},n_{ww} = \frac{{\partial^{2} n}}{{\partial w^{2} }} \hfill \\ \end{gathered}$$

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The partial derivatives of G function can be written as,

$$\begin{aligned} \frac{{\partial^{2} G}}{{\partial u^{2} }} & = - \frac{{4n_{u} }}{{\left( {1 + \frac{{b^{2} }}{{n^{2} }}} \right)h^{2} }}\left( {\frac{{b_{w} }}{n} - \frac{{bn_{u} }}{{n^{2} }}} \right) + 2\tan^{ - 1} \left( \frac{b}{n} \right)\left( {\frac{{2n_{u}^{2} }}{{n^{3} }} - \frac{{n_{ww} }}{{n^{2} }}} \right) \\ & \quad + \frac{2}{h}\left( { - \frac{{\left( {\frac{{2bb_{u} }}{{n^{2} }} - \frac{{2b^{2} n_{u} }}{{n^{3} }}} \right)\left( {\frac{{b_{u} }}{n} - \frac{{bn_{u} }}{{n^{2} }}} \right)}}{{\left( {1 + \frac{{b^{2} }}{{n^{2} }}} \right)^{2} }} + \frac{{ - \frac{{2b_{u} n_{u} }}{{n^{2} }} + \frac{{2bn_{u}^{2} }}{{n^{3} }} + \frac{{b_{uu} }}{n} - \frac{{bh_{uu} }}{{n^{2} }}}}{{1 + \frac{{b^{2} }}{{n^{2} }}}}} \right) \\ \end{aligned}$$

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$$\begin{aligned} \frac{{\partial^{2} G}}{\partial u\partial w} & = 4\tan \left( \frac{b}{n} \right)\frac{{n_{w} n_{u} }}{{n^{3} }} - \frac{{2\left( {\frac{{b_{w} }}{n} - \frac{{bn_{w} }}{{n^{2} }}} \right)n_{u} }}{{\left( {1 + \frac{{b^{2} }}{{n^{2} }}} \right)^{2} }} - \frac{{2n_{w} \left( {\frac{{b_{u} }}{n} - \frac{{bn_{u} }}{{n^{2} }}} \right)}}{{\left( {1 + \frac{{b^{2} }}{{n^{2} }}} \right)n^{2} }} - \frac{{2\left( {\frac{{2bb_{w} }}{{n^{2} }} - \frac{{2b^{2} n_{w} }}{{n^{3} }}} \right)\left( {\frac{{b_{x} }}{n} - \frac{{bn_{x} }}{{n^{2} }}} \right)}}{{\left( {1 + \frac{{b^{2} }}{{n^{2} }}} \right)n}} \\ & \quad - \frac{{2\tan^{ - 1} \left( \frac{b}{n} \right)n_{uw} }}{{n^{2} }} + \frac{2}{{\left( {1 + \frac{{b^{2} }}{{n^{2} }}} \right)h}}\left( { - \frac{{n_{w} b_{u} }}{{n^{2} }} - \frac{{b_{w} n_{u} }}{{n^{2} }} + \frac{{2bn_{y} n_{x} }}{{n^{3} }} + \frac{{b_{uw} }}{n} - \frac{{bn_{uw} }}{{n^{2} }}} \right) \\ \end{aligned}$$

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$$\begin{aligned} \frac{{\partial^{2} G}}{{\partial w^{2} }} & = - \frac{{4n_{w} }}{{\left( {1 + \frac{{b^{2} }}{{n^{2} }}} \right)h^{2} }}\left( {\frac{{b_{u} }}{n} - \frac{{bn_{w} }}{{n^{2} }}} \right) + 2\tan^{ - 1} \left( \frac{b}{n} \right)\left( {\frac{{2n_{w}^{2} }}{{n^{3} }} - \frac{{n_{uu} }}{{n^{2} }}} \right) \\ & \quad + \frac{2}{h}\left( { - \frac{{\left( {\frac{{2bb_{w} }}{{n^{2} }} - \frac{{2b^{2} n_{w} }}{{n^{3} }}} \right)\left( {\frac{{b_{w} }}{n} - \frac{{bn_{w} }}{{n^{2} }}} \right)}}{{\left( {1 + \frac{{b^{2} }}{{n^{2} }}} \right)^{2} }} + \frac{{ - \frac{{2b_{w} n_{w} }}{{n^{2} }} + \frac{{2bn_{w}^{2} }}{{n^{3} }} + \frac{{b_{ww} }}{n} - \frac{{bh_{ww} }}{{n^{2} }}}}{{1 + \frac{{b^{2} }}{{n^{2} }}}}} \right) \\ \end{aligned}$$

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