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Incorporating the velocity slip effect of the gas flow at the solid boundary, the performance and dynamic response of a micro gas-bearing-rotor system are investigated in this paper. For the characteristic length scale of the micro gas bearing, the gas flow in the bearing resides in the slip regime rather than in the continuum regime. The modified Reynolds equations of different slip models are presented. Gas pressure distribution and load carrying capacity are obtained by solving the Reynolds equations with finite different method (FDM). Comparing results from different models, it is found that the second order slip model agrees reasonably well with the benchmarked solutions obtained from the linearized Boltzmann equation. Therefore, dynamic coefficients derived from the second order slip model are employed to evaluate the linear dynamic stability and vibration characteristics of the system. Compared with the continuum flow model, the slip effect reduces dynamic coefficients of the micro gas bearing, and the threshold speed for stable operation is consequently raised. Also, dynamic analysis shows that the system responses change with variation of the operating parameters including the eccentricity ratio, the rotational speed, and the unbalance ratio.

In an attempt to address the ever rising demand for high performance compact power source, a new branch of micro-electro-mechanical system called power MEMS has recently been defined^{[1, 2]}. Among them, the nascent project to develop micro scale gas turbine generators at MIT is specifically targeted for high power density applications. These machines are supported by gas bearings, as shown in ^{[2]}. The micro rotor and the micro bearing are fabricated by micro fabrication technology. Different from the traditional rotor system, the micro rotor and the bearing are made of Silicon.

Operating at such high speed, an accurate assessment of the vibration characteristics of the micro rotor system must be made so as to design highly reliable rotary machines and avoid occurrence of dangerous sliding wear and rub impact effect. Thus, the high speed rotor dynamic characteristic of the micro rotor-bearing system is a major concern in the mechanical design of micro motors and such kind of power MEMS. There are many dynamic phenomena and nonlinear problems that can be met for MEMS that may challenge successful operation of these machines. Some researches, which aimed at analyzing and simulating the rotor dynamics in micro rotating machinery^{[3, 4]}, have been done during the past two decades. Piekos^{[5]} developed a pseudo-spectral method to facilitate orbit simulation of the rotor. Savoulides^{[6]} constructed a small-scale gas bearing model for the rotor. Wang^{[7]} analyzed the bifurcation of a rotor supported by a self-acting gas journal bearing, and the analysis focused on the dynamic behavior of the rotor-bearing system. In these researches, the lubricated gas film was mostly treated as continuum flow, and effects of the micro fluid mechanics have not been considered.

One of the most interesting characteristics in micro rotor-bearing systems is that the nature of the lubricated gas flow departs from the continuum flow regime. For a micro gas bearing, the thickness of the gas film is comparable with the gas molecular mean free path. At this scale, the gas layer adjacent to the surface does not satisfy the no-slip velocity boundary condition. Some researchers have made contributions and improvements to this field. Burgdofer^{[8]} introduced the first-order slip velocity model and derived the modified Reynolds equation. Hsia^{[9]} and Mitsuya^{[10]} presented the second-order slip model and the 1.5-order modified Reynolds equation. Huang^{[11]} discussed the second-order slip effect on the micro gas bearing steady-state operation performance. Y.H Sun^{[12]} presented analytical investigations of slip flow between the flying head and disk in the hard disk drive. However, these researchers mainly focused on the micro fluid mechanics, and few attentions have been paid to the performance of the micro journal bearing.

In this study, the performance of the micro bearing is evaluated and the stability of the micro rotor system is discussed. The gas velocity slip effect on the solid boundary is taken into account, and the modified Reynolds equation is solved with the finite difference method (FDM). By comparison of performances of the micro gas bearing such as flow rate, gas pressure distribution, load carrying capacity and attitude angle derived from different models, the slip effect on the performances of the bearing is discussed and the second order slip model is chosen to evaluate dynamic characteristics of the system. The linear dynamic coefficients are employed to acquire the linear threshold speed for stable operation, and the dynamic responses of different speed regimes are compared. It is found that, for a micro gas bearing, the velocity slip effect would result in smaller dynamic coefficients. As a result, the stability of the micro rotor system is compromised and the threshold speed for stable operation is pushed upwards.

_{b}_{a}

The mass of the rotor is denoted by _{m}_{L}

In view of the length scale of the micro bearing, the thickness of the film would approach 0.1 ^{[13]}, the gas flow in the bearing resides in the slip regime (10^{−3} < Kn < 0.1), in which the velocity slip occurs at the solid boundary and the traditional Reynolds equation needs to be modified. If we adopt the first-order slip model, the flow slippage on the surface of the journal and bushing is given as^{[8]}
^{[9]}
^{[10]}

Using the non-dimensional parameters expressed in

_{a}

In this section, the steady-state performance of the bearing is studied. Results obtained from the continuum flow, the first-order slip model, the second-order slip model and the 1.5 order slip model are compared with each other and benchmarked against those of the linearized Boltzmann equation[

The non-dimensionalized flow rates are given by Mitsuya^{[10]} and Sun^{[12]} as follows:

Continuum flow:

First-order slip flow:

Second-order slip flow:

1.5-order slip flow:
^{[14]} as follow
^{[14]}

_{n}

Solving the Reynolds equations with finite difference method, the gas pressure of the gas film is obtained. Integrating the dimensionless pressure on the surface of the journal, the dimensionless bearing force components along the radial direction _{r} and tangential direction _{t} as shown in

Then the non-dimensional load-carrying capacity ξ and the attitude angle

For the micro gas bearing presented in this discussion, the non-dimensional flow rate is compared in

In this discussion, it is assumed that the micro rotor-bearing system is comprised of a rigid rotor and rigid mounted bushing. Also, the rotor is perfectly co-axial with the bushing. Thus, the motion equations of the rotor in the Cartesian coordinates can be written as follows:
_{x}_{y}

_{1}=_{2}=_{3}=_{4}=

The gas-film forces are nonlinear function and can be secured by solving the Reynolds equation with FDM. However, in most practical calculations, it is assumed that perturbation of the rotor from its initial position is small enough that the gas-film forces could be considered as linear. Thus the gas-film force increments are modeled by a first-order Taylor expansion^{[15, 16]} as follows
_{x}_{y}_{ij}_{ij}

Subject matrix

Referring to

The numerical simulation of the rotor dynamical response can be achieved by solving the Reynolds equation and the motion equation subsequently.

Given a slight disturbance, a balanced rotor will converge to its static equilibrium position or whirl around the static equilibrium if the rotational speeds is above the threshold speed. And it will crash into the bushing when the rotational speed is less than the threshold speed. With the continuum model assumption, the dynamic responses of the micro gas bearing at different rotational speed are shown in

By comparison between

In this discussion, the whirl is caused by the gas film force. It can be seen from

For a practical rotor, due to the fabrication defect, the mass center would inevitably depart from the geometry center, and the rotor is unbalanced. From the above discussion, it is known that the continuum flow assumption will lead to a mistaken result of the dynamic characteristics of the system. Thus, the second order slip flow is adopted for the dynamic analysis of the unbalanced micro rotor. For difference rotational speed, the dynamical responses of an unbalance motor are acquired, which are shown in

For some unbalanced rotor, the oscillation amplitude will keep growing and the rotor will crash into the bushing (

With the increase of the unbalance ratio, the whirl amplitude increases, as shown in

In this study, the performances of the micro gas bearing are presented. Besides, the stability and the dynamical response of the micro rotor supported by the micro gas bearing are discussed. Considering the micro length scale and the micro fabrication constraints, the driving flow mechanisms in the micro gas bearing and the rotor dynamic characteristics of the bearing system are different from those of conventional gas bearings. From the analysis and the discussion, some conclusions could be drawn:

In most operation conditions of the micro gas bearing presented in the paper, the second order slip model predicts a good approximation to the linearized Boltzmann equation;

The slip effect at the solid boundary increases the flow rate of the gas flow. Hence, the gas pressure of the lubricated gas film is decreased, and the load carrying capacity of the micro gas bearing is reduced.

The dynamic coefficients of a micro rotor-bearing system are decreased due to the slip effect, which means that the system stability is compromised. Therefore, the micro rotor must operate at much higher speed to keep stable.

When subject to a slight disturbance, a balance micro rotor will converge to or whirl around the static equilibrium position when the rotational speed is not less than the threshold speed. Otherwise the disturbance will be enhanced and the rotor will crash into the bushing.

Considering the slip effect, the rotor's whirl frequency is decreased, and the micro rotor needs more time to reach a stable operation or crash into the bushing.

For an unbalanced micro rotor, the rotor would not converge to the static equilibrium position. The stable operation is a slight whirl which is excited by the unbalance force. The whirl frequency is equal to the rotational frequency.

The whirl amplitude increases with the increase of the unbalance ratio. And the rotor would not stay stable when the unbalance ratio is too big.

This work was supported by the the National Outstanding Youth Foundation of China under Grant No.10325209.

Schematic cross-section of the micro-motor integrated with gas bearing.

Schematic of the journal bearing in the micro rotor bearing system (not to scale).

The force component of the micro gas bearing.

Comparison of non-dimensional flow rate

Comparison of the gas pressure.

Comparison of the load carrying capacity

Comparison of the attitude angle.

The comparison of the dynamic coefficients at

The response of the rotor at different bearing number (continuum flow).

The response of the rotor at different bearing number (2^{nd} slip flow).

The whirl frequency spectrum of the micro rotor.

The dynamical response of an unbalanced rotor (^{nd} slip flow).

The dynamical response of an unbalanced rotor (^{nd}slip flow).

The frequency spectrum of an unbalanced rotor whirling motion (^{nd} slip .flow)

The maximum and the minimum eccentricity ratio of the rotor during the whirling motion versus the unbalance ratio (

The geometry and the environment parameters

Descriptions | Parameters | Value |
---|---|---|

Radius of the rotor | 2 mm | |

Length of the rotor and the bearing | 300 μm | |

Average clearance | 12 μm | |

Material of the rotor | Si | |

Density | 2.33g/cm^{3} | |

Material of the gas | ||

Viscosity | 1.8×10^{-5} Pa·s | |

Environment pressure | _{a} |
1.01325 Pa |