A method for predicting the gas hammer vibration of a hydrostatic bearing based on static analysis

Abstract

The present invention discloses a method for predicting the hammer vibration of a gas static pressure bearing based on static analysis. The method includes the following steps: Step 1: Establish a physical model of a small-hole throttling gas static pressure thrust bearing; Step 2: Conduct a static theoretical analysis of the small-hole throttling gas static pressure thrust bearing based on the fluid lubrication theory, and derive the expression of the static stiffness of the thrust bearing; Step 3: Conduct a dynamic theoretical analysis of the small-hole throttling gas static pressure thrust bearing based on the fluid lubrication theory, derive the relationship that needs to be satisfied between the dynamic stiffness and the static stiffness of the bearing when the small-hole throttling gas static pressure thrust bearing is stable, and further derive a simplified stability criterion for the thrust bearing; Step 4: Based on the physical model of the small-hole throttling gas static pressure thrust bearing, give a geometric representation of the ultimate stiffness of the gas static pressure thrust bearing. This method can judge whether the small-hole throttling gas static pressure thrust bearing generates hammer vibration phenomenon only by relying on the static analysis of the thrust bearing.

Description

Technical Field

[0001] The present invention belongs to the field of hydrostatic sliding bearings, and relates to a method for predicting the gas hammer vibration of a gas hydrostatic bearing, in particular to a method for judging the gas hammer vibration stability criterion of a small orifice throttling gas hydrostatic thrust bearing based on static analysis. Background Art

[0002] With the continuous improvement of the moving speed and moving accuracy in ultra-precision equipment, on the basis that the static characteristics meet the performance requirements, the dynamic characteristics of gas hydrostatic bearings have an increasingly serious impact on the overall performance of ultra-precision equipment, and have become a hot and difficult issue in the mechanism research and engineering application of gas hydrostatic bearings. The following two unstable phenomena have first attracted attention to the dynamic performance of gas bearings: the gas hammer vibration phenomenon and the whirl instability phenomenon. The gas hammer vibration (self-excited vibration, whistling) phenomenon mainly appears in thrust bearings with external throttling, and the (half-speed) whirl instability phenomenon appears in high-speed radial bearings. In engineering applications, the main problem faced in designing a small orifice throttling gas hydrostatic thrust bearing is to predict whether the thrust bearing will generate gas hammer vibration under the design parameters obtained through static analysis while obtaining the design parameters of large load capacity and high stiffness. At present, the commonly used stability criterion for judging whether a small orifice throttling gas hydrostatic thrust bearing will generate gas hammer vibration must go through complex calculations and cannot intuitively judge the stability of the thrust bearing. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for predicting the gas hammer vibration of a gas hydrostatic bearing based on static analysis, which can judge whether a small orifice throttling gas hydrostatic thrust bearing generates gas hammer vibration only by relying on the static analysis of the thrust bearing.

[0004] The purpose of the present invention is achieved through the following technical solutions:

[0005] A method for predicting the gas hammer vibration of a gas hydrostatic bearing based on static analysis includes the following steps:

[0006] Step 1: Establish a physical model of a small orifice throttling gas hydrostatic thrust bearing;

[0007] Step 2: Conduct static theoretical analysis on the small orifice throttling gas hydrostatic thrust bearing based on the fluid lubrication theory, and derive the static stiffness expression of the thrust bearing:

[0008]

[0009] where K S is the static stiffness, represents the effective load area, W is the static load capacity, p d is the gas film gap pressure, and h is the gas film thickness. is the mass flowing into the restrictor, is the mass flowing into the atmosphere through the boundary in the air film gap;

[0010] Step 3: Based on the fluid lubrication theory, perform a dynamic theoretical analysis on the orifice-throttled aerostatic thrust bearing, derive the dynamic stiffness expression of the thrust bearing and the relationship that needs to be satisfied between the dynamic stiffness and the static stiffness of the orifice-throttled aerostatic thrust bearing when it is stable, and further derive a simplified stability criterion for the thrust bearing, where:

[0011] The dynamic stiffness expression of the thrust bearing is:

[0012]

[0013] where, K S is the static stiffness, has the dimension of time, s represents the perturbation frequency of the dynamic bearing, m g is the total mass of the gas in the bearing clearance and the pressure equalizing grooves;

[0014] The relationship that needs to be satisfied between the dynamic stiffness and the static stiffness of the orifice-throttled aerostatic thrust bearing when it is stable is:

[0015] τ1 > τ2;

[0016] K ∞ > K S ;

[0017] where, K ∞ is the limit stiffness of the aerostatic thrust bearing;

[0018] Let Then the stability criterion for the thrust bearing is:

[0019] λ > 1 or log 10 λ > 0;

[0020] Step 4: Based on the physical model of the orifice-throttled aerostatic thrust bearing, give a geometric representation of the limit stiffness of the aerostatic thrust bearing:

[0021]

[0022] where, δV represents the volume of the region covered by the pressure after the throttle hole.

[0023] Compared with the prior art, the present invention has the following advantages:

[0024] Based on the fluid lubrication theory, the present invention analyzes the static and dynamic characteristics of a small orifice throttling gas static thrust bearing, and derives the conditions that the dynamic stiffness and static stiffness of the thrust bearing should satisfy when the small orifice throttling gas static thrust bearing is in a stable state. Taking this as a bridge, the static analysis results are used to judge whether the thrust bearing has hammer vibration. In engineering practice, through the stability criterion given by the present invention, it is possible to judge whether the thrust bearing will produce hammer vibration phenomenon only by analyzing the static characteristics of the small orifice throttling gas static thrust bearing, without the need to perform complex calculations to solve dynamic parameters such as α, ξ, θ, r, q as in the traditional stability criterion. Description of the Drawings

[0025] Figure 1 Schematic diagram of the derivation process of the hammer vibration stability criterion for a small orifice throttling gas static thrust bearing based on static analysis;

[0026] Figure 2 Schematic diagram of the physical model of a small orifice throttling gas static thrust bearing;

[0027] Figure 3 Schematic diagram of the static characteristic curve of the thrust bearing;

[0028] Figure 4 Schematic diagram of the equivalent control model of a small orifice throttling type thrust bearing;

[0029] Figure 5 Schematic diagram of the process for designing a small orifice throttling gas static thrust bearing based on the present invention;

[0030] Figure 6 Design model of a thrust bearing for a large-scale measuring turntable;

[0031] Figure 7 For the upper thrust bearing log 10 Relationship between λ and the gas film thickness. Detailed Implementation Modes

[0032] The technical solution of the present invention will be further described below in conjunction with the drawings, but it is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.

[0033] The present invention provides a method for predicting the hammer vibration of a gas static bearing based on static analysis, as Figure 1 shown, the method includes the following steps:

[0034] Step 1: Establish a physical model of a small orifice throttling gas static thrust bearing, as Figure 2As shown, the physical model includes a fluid domain and a solid domain, namely, the air film of the orifice-throttled aerostatic thrust bearing and the thrust plate of the orifice-throttled aerostatic thrust bearing.

[0035] Step 2: Based on the fluid lubrication theory, a static theoretical analysis is carried out on the orifice-throttled aerostatic thrust bearing, and the expression of the static stiffness of the thrust bearing is derived. The specific derivation process is as follows:

[0036] When the thrust bearing is in a static stable state, after the external load changes, the air film thickness h and the air film gap pressure p d change. At this time, the flow rate m1 flowing through the orifice (only affected by the air film gap pressure p d ) and the flow rate m2 flowing through the air film (affected by both the air film thickness h and the air film gap pressure p d ) change accordingly. According to the law of conservation of flow rate, there is a total differential:

[0037]

[0038] Among them, and The calculation method is as follows:

[0039]

[0040]

[0041] Let Combining equations (2) and (3) and substituting them into equation (1) gives:

[0042]

[0043] Among them, represents the change gradient of the pressure p d in the air film gap along the air film thickness h direction.

[0044] The thrust bearing uses its bearing capacity W(h, p d )(note that it is an integral of the relative pressure) and the difference of the air film thickness h to obtain its static stiffness K S as:

[0045]

[0046] Among them, represents the stiffness generated by the change of the air film thickness h in the thrust bearing gap. In the current aerostatic throttling method, only surface throttling will change the air film thickness in the bearing gap. Therefore, can be called surface stiffness. For a planar aerostatic thrust bearing, the air floating surface of the thrust bearing is planar, and the air film thickness h in the bearing gap is a constant. Then Substituting equation (4) into equation (5) gives:

[0047]

[0048] Among them, represents the effective load-bearing area, which, due to its relation with the characteristics of the restrictor, can be called the restrictor stiffness. For a small-hole throttling gas static thrust bearing, when the gas compressibility is not considered, the flow rate m2 through the lubricating film is proportional to the pressure difference p d across the gas film, proportional to the cube of the clearance h, and inversely proportional to the characteristic length L and the fluid viscosity η, that is, it has the following form:

[0049]

[0050] Substituting equation (7) into equation (6) gives:

[0051]

[0052] After dimensionless treatment, it becomes:

[0053]

[0054] Among them, A t is the surface area of the gas film, p s is the supply pressure of the thrust bearing, is the dimensionless stiffness of the thrust bearing. From equation (9), it can be seen that when h → 0, the static stiffness K S is approximately equal to zero; when h → ∞, K S is also approximately equal to zero. Therefore, the stiffness of the small-hole throttling gas static thrust bearing has Figure 3 the characteristics shown by the static characteristic curve of the thrust bearing. As the clearance increases, the stiffness first increases and then decreases. Therefore, one of the keys to the design of the small-hole throttling gas static thrust bearing lies in the selection of the clearance h op , and only when the clearance h op is well matched with the hole diameter can the optimal stiffness be obtained.

[0055] Step 3: Conduct a static theoretical analysis of the small-hole throttling gas static thrust bearing based on the fluid lubrication theory, derive the expression for the dynamic stiffness of the thrust bearing, derive the relationship that the dynamic stiffness and static stiffness of the small-hole throttling gas static thrust bearing need to satisfy when it is stable, and further derive the simplified stability criterion for the thrust bearing. The specific derivation process is as follows:

[0056] For a dynamic bearing, the unit time change rate of the bearing clearance and the total mass m g (simultaneously affected by the gas film thickness h and the gas film clearance pressure p d ) and the mass flowing into the restrictor (only affected by the gas film clearance pressure pd The mass that flows through the air film gap and into the atmosphere through the boundary) and the influence (simultaneously affected by the air film thickness h and the air film gap pressure p d The change rate of the influence) has the following relationship:

[0057]

[0058] Among them, and The calculation methods are shown in equations (2) and (3), The calculation method of is as follows:

[0059]

[0060] Let Combining equations (2), (3) and (11) and substituting them into equation (10) gives:

[0061]

[0062] After performing Laplace transform on equation (12) and organizing, we get:

[0063]

[0064] Among them, s represents the disturbance frequency of the dynamic bearing. For a gas static thrust bearing that only considers the stiffness of the restrictor, its dynamic stiffness K b (s) is:

[0065]

[0066] Among them, K S is the static stiffness of the system, This equation has been obtained from equation (6), has the dimension of time.

[0067] Figure 4 The equivalent control model of the orifice throttle gas static thrust bearing is shown as resulting in a 180° phase delay in the transfer function. Therefore, the bearing stability condition of the Fuller criterion is τ1 > τ2. At the same time, the limit stiffness when the disturbance frequency approaches infinity is defined as:

[0068]

[0069] Then, from equation (15), the equivalent condition for the orifice throttle gas static thrust bearing not to produce water hammer vibration is: when the disturbance frequency approaches infinity, the limit stiffness generated by the air film is greater than the static stiffness of the bearing system when the disturbance frequency approaches zero, that is: K ∞ > K S .

[0070] To more intuitively judge the stability of the orifice-throttled aerostatic thrust bearing, let:

[0071]

[0072] Then the condition for the stability of the orifice-throttled aerostatic thrust bearing is:

[0073] λ > 1 or log 10 λ > 0 (17)

[0074] Step 4: Based on Figure 2 the physical model of the orifice-throttled aerostatic thrust bearing shown, further give the geometric representation of the ultimate stiffness of the above aerostatic thrust bearing.

[0075] As Figure 2 the physical model of the orifice-throttled aerostatic thrust bearing shown, combined with the gas state equation under isothermal conditions the gas mass m stored in its bearing clearance and total pressure equalizing grooves g is:

[0076]

[0077] where, δV represents the volume of the region covered by the pressure p d after the throttle hole, including the orifice cavity, pressure equalizing grooves, etc. Equation (18) uses the isothermal state equation of the lubricating gas to relate the mass of the stored gas to the static bearing capacity W (note that it is an integral of the absolute pressure). Differentiating from Equation (18) gives:

[0078]

[0079] Substituting Equation (19) into Equation (15) gives the ultimate stiffness when the perturbation frequency of the planar aerostatic thrust bearing approaches infinity as:

[0080]

[0081] Since the area d covered by the pressure p after the throttle hole is very small compared to the total area, it can be approximately simplified to Equation (20) after omitting it:

[0082]

[0083] Equation (21) actually gives the geometric representation of the ultimate stiffness of the orifice-throttled aerostatic thrust bearing, as Figure 3 shown.

[0084] According to the theoretical criterion formula (17) for the stability of the orifice-throttled aerostatic thrust bearing derived above, in engineering practice, according to Figure 5 the process of designing the orifice-throttled aerostatic thrust bearing, it is possible to judge whether the thrust bearing will produce hammer vibration phenomenon only by analyzing the static characteristics of the orifice-throttled aerostatic thrust bearing, and give the maximum clearance that can be designed when the thrust bearing does not produce hammer vibration, without having to perform complex calculations to solve values such as α, ξ, θ, r, q as in the traditional stability criterion.

[0085] Figure 6 As shown in the design model of the thrust bearing of a large measuring turntable, the rotor adopts an I-shaped structure, and the thrust bearing is a closed thrust bearing composed of two upper and lower open thrust bearings with different sizes. The upper and lower thrust bearings and the radial bearing are supplied with gas in three separate ways. Table 1 shows the design parameters of the turntable thrust bearing. According to Figure 5 the process shown for analysis of the upper thrust bearing of this bearing, finally Figure 7 the values of log 10 λ at different air film thicknesses of the upper thrust bearing shown in 10 can be obtained. It can be seen that when the air film thickness h of the upper thrust bearing ≤ 13 μm, log 10 λ > 0. Therefore, when designing the air film thickness of the upper thrust bearing, it is necessary to ensure that the air film thickness h during the normal operation of the upper thrust bearing < 13 μm.

[0086] Table 1

[0087]

Claims

1. A method for predicting the gas hammer vibration of a hydrostatic bearing based on static analysis, characterized in that The method includes the following steps: Step 1: Establish a physical model of a hole-throttle aerostatic thrust bearing; Step 2: Conduct a static theoretical analysis of the hole-throttle aerostatic thrust bearing based on the fluid lubrication theory, and derive the expression for the static stiffness of the thrust bearing. The expression for the static stiffness of the thrust bearing is: ； ； Among them, is the static stiffness, represents the effective bearing area, , , , is the air film gap pressure, h is the air film thickness, is the mass flowing into the restrictor, is the mass flowing from the air film gap into the atmosphere through the boundary, W is the static bearing capacity; Step 3: Conduct a dynamic theoretical analysis of the hole-throttle aerostatic thrust bearing based on the fluid lubrication theory, derive the expression for the dynamic stiffness of the thrust bearing and the relationship that needs to be satisfied between the dynamic stiffness and the static stiffness of the hole-throttle aerostatic thrust bearing when it is stable, and further derive a simplified stability criterion for the thrust bearing. The expression for the dynamic stiffness of the thrust bearing is: ； Among them, , has the dimension of time, represents the disturbance frequency of the dynamic bearing, , , is the total mass of gas in the bearing clearance and the pressure equalizing groove; The relationship that needs to be satisfied between the dynamic stiffness and the static stiffness of the hole-throttle aerostatic thrust bearing when it is stable and does not produce hammer vibration is: ； ； ； Among them, is the limiting dynamic stiffness of the aerostatic thrust bearing; Let , then the stability criterion for the thrust bearing is as follows: or ; Step 4: Based on the physical model of the hole-throttle aerostatic thrust bearing, give a geometric representation of the ultimate dynamic stiffness of the aerostatic thrust bearing. The geometric representation of the ultimate dynamic stiffness of the aerostatic thrust bearing is: Among them, represents the volume of the area covered by the pressure after the throttle orifice; Step 5: In engineering practice, conduct a static theoretical analysis of the hole-throttle aerostatic thrust bearing to solve for the relative load-carrying capacity, absolute load-carrying capacity, and static stiffness of the thrust bearing, and substitute the solved static parameters into the stability criterion to obtain the maximum air film thickness at which the thrust bearing does not produce hammer vibration under the current design parameters.

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