An experimental and theoretical approach for stiffness of machine tool spindle with fluid bearings

Abstract

Spindle nose stiffness is a leading technical indicator of machine tool spindle. However, the stiffness of machine tool spindle supported by fluid bearings has not been studied thoroughly, by far; there is not an ideal approach to obtain the spindle nose stiffness due to the nonlinear fluid bearings. This paper proposes a new methodology for the spindle nose stiffness including definition of the spindle nose stiffness with the fluid bearings, experimental and theoretical approach to obtain the spindle nose stiffness. A motorized spindle with hydrodynamic water-lubricated spiral groove bearings developed in our laboratory is selected as an object; a theoretical and experimental investigation on the spindle nose stiffness is conducted. The result shows that the proposed methodology is applicable for obtaining the spindle nose stiffness with the fluid bearings; the experimental device has the advantages of simple structure, low cost, and convenient operation. Meanwhile, the spindle nose stiffness can be calculated by simultaneously solving the dynamic model of rotor-bearing system. We expect an improvement for the experimental device by introducing another load device to apply a perturbed load with an any angle to the external load on the spindle nose.

Appendices

Appendix 1

The derivation of equivalent Reynolds equation for thrust bearing:

The transformational relations of pressure gradient of water film between physical coordinate and calculational coordinate for thrust bearing are

$$\left\{\begin{array}{c}\frac{\partial p}{\partial r}=\frac{\partial p}{\partial \xi }-\frac{\partial p}{\partial \eta }\mathrm{sign}\left(\xi -{\xi }_{s}\right)\frac{\mathrm{cot}\beta }{\xi }\\ \frac{\partial p}{\partial \theta }=\frac{\partial p}{\partial \eta }\end{array}\right.$$

(14)

In physical coordinate, the unit volume flow rates along with r and $\theta$ directions of thrust bearing can be written as

$$\left\{\begin{array}{c}{q}_{r}=-\frac{{G}_{r}{h}^{3}}{\mu }\frac{\partial p}{\partial r}\\ {q}_{\theta }=-\frac{{G}_{\theta }{h}^{3}}{\mu r}\frac{\partial p}{\partial \theta }+\frac{\Omega rh}{2}\end{array}\right.$$

(15)

In Eq. (15),

$$\left\{\begin{array}{cc}{G}_{r}=\frac{1}{12},{G}_{\theta }=\frac{1}{12}& \mathrm{Laminar}\left(\mathrm{Re}<{10}^{3}\right)\\ {G}_{r}=\frac{1}{12+0.0043{\mathrm{Re}}^{0.96}},{G}_{\theta }\frac{1}{12+0.0136{\mathrm{Re}}^{0.9}}& \mathrm{Turbulence}\left(\mathrm{Re}\ge {10}^{3}\right)\end{array}\right.$$

(16)

In calculational coordinate, the unit volume flow rates along with $\xi$ and $\eta$ directions of thrust bearing can be described as

$$\left\{\begin{array}{c}{q}_{\xi }=\frac{1}{\sqrt{\alpha }}\left(\frac{\partial \xi }{\partial r}{q}_{r}+\frac{\partial \xi }{r\partial \theta }{q}_{\theta }\right)\\ {q}_{\eta }=\frac{1}{\sqrt{\beta }}\left(r\frac{\partial \eta }{\partial r}{q}_{r}+\frac{\partial \eta }{\partial \theta }{q}_{\theta }\right)\end{array}\right.$$

(17)

In Eq. (17),

$$\left\{\begin{array}{c}\alpha ={\left(\frac{\partial \xi }{\partial r}\right)}^{2}+{\left(\frac{\partial \xi }{r\partial \theta }\right)}^{2}\\ \beta ={\left(r\frac{\partial \eta }{\partial r}\right)}^{2}+{\left(r\frac{\partial \eta }{r\partial \theta }\right)}^{2}\end{array}\right.$$

(18)

The volume flow rates in Eq. (5) can be expressed as

$$\left\{\begin{array}{c}{Q}_{\xi }={\int }_{\eta }\sqrt{\alpha }{q}_{\xi }\xi d\eta \\ {Q}_{\eta }={\int }_{\xi }\sqrt{\beta }{q}_{\eta }d\xi \\ {Q}_{v}={\iint }_{\xi ,\eta }\frac{\partial h}{\partial t}\left|J\right|d\xi d\eta \end{array}\right.$$

(19)

The derivation of equivalent Reynolds equation for journal bearing:

The transformational relations of pressure gradient of water film between physical coordinate and calculational coordinate for journal bearing are

$$\left\{\begin{array}{c}\frac{\partial p}{\partial {x}^{^{\prime}}}=\frac{\partial p}{\partial \xi }\\ \frac{\partial p}{\partial z}=\frac{\partial p}{\partial \eta }-\frac{\partial p}{\partial \xi }\mathrm{cot}{\beta }_{\kappa }\end{array}\right.$$

(20)

In physical coordinate, the unit volume flow rates along with ${x}^{^{\prime}}$ and z directions of journal bearing are

$$\left\{\begin{array}{c}{q}_{{x}^{^{\prime}}}=-\frac{{G}_{{x}^{^{\prime}}}{h}^{3}}{\mu }\frac{\partial p}{\partial {x}^{^{\prime}}}+\frac{Uh}{2}\\ {q}_{z}=-\frac{{G}_{z}{h}^{3}}{\mu }\frac{\partial p}{\partial z}\end{array}\right.$$

(21)

In Eq. (21),

$$\left\{\begin{array}{cc}{G}_{{x}^{^{\prime}}}=\frac{1}{12},{G}_{z}=\frac{1}{12}& \mathrm{Laminar}\left(\mathrm{Re}<{10}^{3}\right)\\ {G}_{{x}^{^{\prime}}}=\frac{1}{12+0.0136{\mathrm{Re}}^{0.9}},{G}_{z}=\frac{1}{12+0.0043{\mathrm{Re}}^{0.96}}& \mathrm{Turbulence}\left(\mathrm{Re}\ge {10}^{3}\right)\end{array}\right.$$

(22)

In calculational coordinate, the unit volume flow rates along with $\xi$ and $\eta$ directions of journal bearing can be expressed as

$$\left\{\begin{array}{c}{q}_{\xi }=\frac{1}{\sqrt{\alpha }}\left(\frac{\partial \xi }{\partial {x}^{^{\prime}}}{q}_{x}^{^{\prime}}+\frac{\partial \xi }{\partial z}{q}_{z}\right)\\ {q}_{\eta }=\frac{1}{\sqrt{\beta }}\left(\frac{\partial \eta }{\partial {x}^{^{\prime}}}{q}_{x}^{^{\prime}}+\frac{\partial \eta }{\partial z}{q}_{z}\right)\end{array}\right.$$

(23)

In Eq. (23),

$$\left\{\begin{array}{c}\alpha ={\left(\frac{\partial \xi }{\partial {x}^{^{\prime}}}\right)}^{2}+{\left(\frac{\partial \xi }{\partial z}\right)}^{2}\\ \beta ={\left(\frac{\partial \eta }{\partial {x}^{^{\prime}}}\right)}^{2}+{\left(\frac{\partial \eta }{\partial z}\right)}^{2}\end{array}\right.$$

(24)

The volume flow rates in Eq. (10) can be described as

$$\left\{\begin{array}{c}{Q}_{\xi }={\int }_{\eta }\sqrt{\alpha }{q}_{\xi }d\eta \\ {Q}_{\eta }={\int }_{\xi }\sqrt{\beta }{q}_{\eta }d\xi \\ {Q}_{v}=\underset{\xi ,\eta }{\overset{}{\iint }}\frac{\partial h}{\partial t}\left|J\right|d\xi d\eta \end{array}\right.$$

(25)

Appendix 2

Table 4 Structure and operation parameters of rotor

4

Table 5 Structure parameters of bearings

5