Nonlinear modelling and parameter influence of supercritical transmission shaft with dry friction damper

Abstract

Supercritical transmission shafts, which have one or more critical speeds below their working speeds, are becoming more popular in new rotorcraft designs. To attenuate the excessive transcritical vibration, dry friction damper is a prevailing choice. In this paper, we focus on the basic working mechanism and parameter influence of the dry friction damper for supercritical transmission shaft. Mathematical model of the dry friction damper, which fully considers the nonlinear rub-impact and side-dry-friction effects, is proposed and integrated with finite element model of the transmission shaft to investigate nonlinear interactions between the shaft and damper. It is demonstrated through systematic numerical simulations that a typical transcritical response with dry friction damper can be divided into 4 sub-regions and the dry friction damper takes effect only within region II and III respectively through hard-stopping and side-dry-friction effects. In addition, effects of nonlinear bearing force, transcritical acceleration and initial location of the damper are discussed in detail. Moreover, influences of 3 key damper parameters, that is the rub-impact clearance, the critical slip force and the circumferential friction coefficient, are further investigated, which provides a guidance for designs of the dry friction damper. Finally, prototypes of the dry friction damper are designed, manufactured and tested on a rotor dynamics test rig. For the first time, the theoretical analysis and numerical simulation results are quantitatively verified by an experiment.

Appendices

Appendix

Nonlinear bearing force model

A ball bearing before and after deformation is schematically plotted in Fig. 

Fig. 18

Schematic illustration of the nonlinear ball bearing force model

18, where $\Delta u$ and $\Delta v$ respectively denote the relative displacements of the inner ring with respect to the outer ring along $Y$ and $Z$ directions. The contact deformation of the $j$th roller can be written as:

$$ \delta_{j} = \Delta u{\text{cos}}\phi_{j} + \Delta v{\text{sin}}\phi_{j} - \delta_{b0} $$

(9)

where ${\delta }_{b0}$ is the backlash of the ball bearing, ${\phi }_{j}$ represents the angular position of the $j$th roller relative to the positive $Y$ direction, which can be written as:

$$ \phi_{j} = \frac{{2\pi \left( {j - 1} \right)}}{{N_{b} }} + \frac{{R_{bi} \Omega t}}{{R_{bi} + R_{bo} }} $$

(10)

where ${N}_{b}$ denotes the number of the rollers, $\Omega$ depicts the rotational speed, and ${R}_{bi}$ and ${R}_{bo}$ respectively describe the radius of the inner and outer rings. According to the Hertzian contact theory, the reaction force between the roller and rings is

$$ F_{j} = k_{b} \Theta \delta_{j}^{3/2} $$

(11)

where ${k}_{b}$ depicts the contact stiffness, $\Theta$ is defined as in the main body of the paper and retains a value of 1 only when ${\delta }_{j}\ge 0$, otherwise is equal to 0. The total reaction forces between the rollers and rings along $Y$ and $Z$ directions can be written as:

$$ \left[ {\begin{array}{*{20}c} {f_{by} } \\ {f_{bz} } \\ \end{array} } \right] = \mathop \sum \limits_{j = 1}^{{N_{b} }} \left[ {\begin{array}{*{20}c} {k_{b} \Theta \delta_{j}^{3/2} {\text{cos}}\phi_{j} } \\ {k_{b} \Theta \delta_{j}^{3/2} {\text{sin}}\phi_{j} } \\ \end{array} } \right]. $$

(12)

Rub-impact force model

There exists an initial clearance ${\delta }_{0}$ between the shaft and damper ring. Rub-impact only takes place when the relative displacement of the shaft related to the damper ring exceeds the clearance. As schematically illustrated in Fig. 

Fig. 19

Schematic illustration of the rub-impact force model

19, the rub-impact position is defined by an angle $\varphi$, and the following expressions hold:

$$ \cos \varphi = \frac{{u_{s} - u_{d} }}{\delta },\;\sin \varphi = \frac{{v_{s} - v_{d} }}{\delta } $$

(13)

where ${\mathbf{q}}_{d}={\left[{u}_{d},{v}_{d}\right]}^{T}$ and ${\mathbf{q}}_{s}={\left[{u}_{s},{v}_{s}\right]}^{T}$ are respectively the translational displacements of the damper ring and corresponding rub-impact node of the transmission shaft, and $\delta$ depicts the relative displacement of the transmission shaft related to the damper ring and can be expressed as:

$$ \delta = \sqrt {\left( {u_{s} - u_{d} } \right)^{2} + \left( {v_{s} - v_{d} } \right)^{2} } . $$

(14)

The rub-impact force between the shaft/damper includes two components, that is the radial impact component ${f}_{rr}$ and tangential rub component ${f}_{rt}$. The radial impact component is assumed to be elastic and linearly proportional to the interference between the shaft and damper ring, which can be expressed as:

$$ f_{rr} = \left\{ \begin{gathered} 0,\quad \quad \quad \quad \delta < \delta_{0} \hfill \\ k_{r} \left( {\delta - \delta_{0} } \right),\;\delta \ge \delta_{0} \hfill \\ \end{gathered} \right. $$

(15)

where ${k}_{r}$ is the radial impact stiffness. The tangential rub component is assumed to be a Coulomb type and always retains a direction opposite to the relative velocity at the contact point and can be expressed as:

$${f}_{rt}={\mu }_{r}{f}_{rr}\mathrm{sign}\left({v}_{rel}\right)$$

(16)

where ${\mu }_{r}$ is the tangential friction coefficient and ${v}_{rel}$ denotes the relative velocity at the contact point, which can be expressed as

$${v}_{rel}=\frac{{D}_{so}}{2}\Omega +\left({\dot{v}}_{s}\mathrm{cos}\varphi -{\dot{u}}_{s}\mathrm{sin}\varphi \right)-\left({\dot{v}}_{d}\mathrm{cos}\varphi -{\dot{u}}_{d}\mathrm{sin}\varphi \right)$$

(17)

where ${D}_{so}$ depicts the outer diameter of the shaft and $\Omega$ represents the rotational speed of the shaft. The radial and tangential components of the rub-impact force can be further written in the $Y$ and $Z$ coordinates by taking the following transformation:

$$ \left[ {\begin{array}{*{20}c} {f_{ry} } \\ {f_{rz} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - \cos \varphi } & {\sin \varphi } \\ { - \sin \varphi } & { - \cos \varphi } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {f_{rr} } \\ {f_{rt} } \\ \end{array} } \right]. $$

(18)

Further introduce the function $\Theta$, which is defined as: $\Theta =1$, if $\delta \ge {\delta }_{0}$ and $\Theta =0$, if $\delta <{\delta }_{0}$, the rub-impact force can be expressed as

$$\left[\begin{array}{c}{f}_{ry}\\ {f}_{rz}\end{array}\right]=\Theta {k}_{r}\left(\delta -{\delta }_{0}\right)\left[\begin{array}{c}-\mathrm{cos}\varphi +{\mu }_{r}\mathrm{sign}\left({v}_{rel}\right)\mathrm{sin}\varphi \\ -\mathrm{sin}\varphi -{\mu }_{r}\mathrm{sign}\left({v}_{rel}\right)\mathrm{cos}\varphi \end{array}\right]$$

(19)