Method for calculating static stiffness of four-sliding-block hydrostatic guideway

Abstract

The invention is suitable for the technical field of mechanical design, and provides a method for calculating the static stiffness of a four-slider hydrostatic guideway, which comprises the following steps of: enabling hydrostatic sliders to be equivalent to parallel spring elements, simplifying the four-slider hydrostatic guideway into a spatial single-degree-of-freedom system, and measuring basic parameters and oil cavity parameters of the four-slider hydrostatic guideway; calculating the rectangular plane oil pad flow Q of the static pressure sliding block; calculating the liquid resistance Rc of the small hole and the liquid resistance Rh of the oil sealing surface; calculating an initial liquid resistance ratio lambda0 and an initial oil cavity pressure p0; calculating the pressure p of the plane rectangular oil pad when the thickness of the oil film changes; the bearing capacity Fi and the static rigidity Si of the oil pad are calculated; calculating a four-sliding-block oil film compression amount matrix delta Z; and establishing a static stiffness calculation model of the four-slider hydrostatic guideway. According to the method, the influence of unbalance loading on system deformation and rigidity is analyzed, a more accurate four-sliding-block hydrostatic guide rail static rigidity calculation scheme is provided, and the defect that modeling based on unbalance loading is lacked in a traditional hydrostatic guide rail rigidity calculation method is overcome.

Description

Technical Field

[0001] The invention belongs to the technical field of mechanical design, and in particular relates to a method for calculating the static stiffness of a four-slider hydrostatic guide rail. Background Art

[0002] As a low-friction, high-precision guide rail system, the four-slider hydrostatic guide rail is often used as a transmission component of a precision mechanical system. Its stiffness performance directly determines the positioning accuracy and dynamic response capability of the system. During the operation of the hydrostatic guide rail, the phenomenon of eccentric loading often occurs, which often causes changes in the mechanical properties of the guide rail system and indirectly affects the performance of the entire transmission system. Under the action of eccentric loading, some sliders bear a large load, the oil film will become thinner due to the increase in pressure, and the support stiffness of the slider will decrease accordingly; while other sliders have a relatively high stiffness due to the small load and the increase in oil film thickness. This uneven oil film thickness and support stiffness change makes it difficult for the guide rail system to maintain uniform stiffness characteristics, resulting in aggravated local deformation of the system and an overall decrease in stiffness. The reduction in stiffness will directly affect the deformation resistance and motion accuracy of the guide rail system. Especially when subjected to dynamic loads or precision movements, the stiffness changes caused by eccentric loading will cause the guide rail system to produce slight vibrations or tilts, thereby affecting the processing or measurement accuracy of the equipment.

[0003] In the prior art, the calculation and analysis of the static stiffness of the four-slider hydrostatic guide usually considers the influence of uniform load or no off-center load. For example, the patent application with application number 202010940197.2 discloses a five-degree-of-freedom static stiffness modeling and calculation method of a hydrostatic cylindrical guide. Its static stiffness calculation method distributes the load evenly on the worktable surface and ignores the influence of uneven load or off-center load position. The limitations of this simplified processing lead to deviations in the existing model when predicting performance under actual working conditions, which is insufficient to provide sufficient scientific basis for the accurate calculation of the static stiffness of the four-slider hydrostatic guide. Summary of the invention

[0004] The purpose of the embodiment of the present invention is to provide a method for calculating the static stiffness of a four-slider hydrostatic guide rail, aiming to solve the problems raised in the above-mentioned background technology.

[0005] The embodiment of the present invention is implemented as follows: a method for calculating the static stiffness of a four-slider hydrostatic guide rail comprises the following steps:

[0006] Step 1: The hydrostatic slider is equivalent to a parallel spring element, the four-slider hydrostatic guide is simplified to a spatial single-degree-of-freedom system, and the basic parameters and oil chamber parameters of the four-slider hydrostatic guide are measured;

[0007] Step 2, calculate the flow rate Q of the rectangular plane oil pad of the static pressure slider;

[0008] Step 3: Calculate the pore liquid resistance R c , Sealing oil surface liquid resistance Rh ;

[0009] Step 4, calculating the initial hydraulic resistance ratio λ0 and the initial oil chamber pressure p0;

[0010] Step 5, calculate the plane rectangular oil cushion pressure p when the oil film thickness changes;

[0011] Step 6: Calculate the oil pad bearing capacity F i and static stiffness S i ;

[0012] Step 7, calculating the oil film compression matrix ΔZ of the four sliders;

[0013] Step 8: Establish a static stiffness calculation model for the four-slider hydrostatic guideway.

[0014] Preferably, in step 1, the four-slider hydrostatic guide rail is simplified into a spatial single-degree-of-freedom system, and the hydrostatic slide only has displacement in the z direction;

[0015] The basic parameters of the guide rail that need to be measured are: the total gravity G of the workbench and sliders and other components, the eccentric load F, and the relative arrangement spacing m, n between the sliders;

[0016] The oil chamber parameters that need to be measured are: oil chamber geometry B, L, b, l, h0, d; oil density, viscosity and oil supply pressure ρ, μ, p s .

[0017] Preferably, in step 2, the calculation formula of the flow rate Q of the rectangular plane oil pad of the static pressure slider is:

[0018]

[0019] Where p is the oil chamber pressure; h is the oil film thickness; B, b, L, l are the geometric dimensions of the oil chamber respectively; μ is the oil viscosity.

[0020] Preferably, in step 3, the pore liquid resistance R c And seal oil surface liquid resistance R h The calculation formula is:

[0021]

[0022] In the formula, α is the flow area coefficient, and the value is α=0.5~0.7; ρ is the oil density; d is the oil inlet hole diameter.

[0023] Preferably, in step 4, the calculation formulas of the initial hydraulic resistance ratio λ0 and the initial oil chamber pressure p0 are:

[0024]

[0025] Where h0 is the oil film thickness in the initial state.

[0026] Preferably, in step 5, the calculation formula of the plane rectangular oil cushion pressure p when the oil film thickness changes is:

[0027]

[0028] In the formula, p s Supply oil pressure to the guide rail system.

[0029] Preferably, in step 6, the oil pad bearing capacity F i and static stiffness S i The calculation formula is:

[0030]

[0031] In the formula, A e is the effective bearing area of ​​the oil film, A e =(Bb)×(Ll).

[0032] Preferably, in step 7, the calculation formula of the four-slider oil film compression matrix ΔZ is:

[0033] ΔZ k+1 =ΔZ k -J -1 (ΔZ k )·F(ΔZ k );

[0034] Where F(ΔZ) is the vector function of the oil film compression, which is obtained by the force balance equation, the moment balance equation and the symmetry constraint; J(ΔZ) is the Jacobian matrix of the vector function F(ΔZ).

[0035] Preferably, in step 8, the established four-slider hydrostatic guideway static stiffness calculation model is:

[0036]

[0037] In the formula, K i is the static stiffness of the oil film of the hydrostatic slider, i = 1 ~ 4; h i is the oil film thickness of each of the four hydrostatic sliders.

[0038] A method for calculating the static stiffness of a four-slider hydrostatic guide provided in an embodiment of the present invention can provide a method for calculating the static stiffness of a four-slider hydrostatic guide rail that takes into account the influence of the eccentric load position. By establishing a mathematical model of the four-slider hydrostatic guide rail system, the influence of the eccentric load on the deformation and stiffness of the system is analyzed, aiming to provide a more accurate four-slider hydrostatic guide rail static stiffness calculation scheme, which makes up for the defect of the traditional hydrostatic guide rail stiffness calculation method that lacks modeling based on eccentric load, contributes to the optimized design of the four-slider hydrostatic guide rail, and improves its performance stability and durability in the high-precision transmission system. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 A simplified diagram of a four-slider hydrostatic guide rail system provided in an embodiment of the present invention;

[0040] Figure 2 A schematic diagram of the oil film structure of a static pressure slider provided in an embodiment of the present invention;

[0041] Figure 3 A schematic diagram of the microelement of oil volume provided by an embodiment of the present invention;

[0042] Figure 4 A schematic diagram of the oil flow process provided by an embodiment of the present invention;

[0043] Figure 5 A curve showing the influence of the oil film thickness on the bearing capacity of the static pressure slider provided in an embodiment of the present invention;

[0044] Figure 6 The influence curve of the oil film thickness on the static stiffness of the hydrostatic slider provided in the embodiment of the present invention;

[0045] Figure 7 A diagram showing the influence of the eccentric load position on the static stiffness of the four-slider hydrostatic guide rail provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0046] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0047] The specific implementation of the present invention is described in detail below in conjunction with specific embodiments.

[0048] A method for calculating the static stiffness of a four-slider hydrostatic guideway comprises the following steps:

[0049] Step 1: Simplify the four-slider hydrostatic guide rail as follows Figure 1 In the spatial single-degree-of-freedom system shown in the figure, the static pressure slider only has displacement in the z direction; Figure 1As mentioned above, the basic parameters of the guide rail that need to be measured are: the total gravity G of the workbench and sliders and other components, the eccentric load F, and the relative arrangement spacing m, n between the sliders; Figure 2 As shown, the oil chamber parameters that need to be measured are: oil chamber geometric dimensions B, L, b, l, h0, d; oil density, viscosity and oil supply pressure ρ, μ, p s ;

[0050] Among them, the reference values ​​of the basic parameters of the four-slider hydrostatic guide are shown in Table 1:

[0051] Table 1

[0052]

[0053]

[0054] The reference values ​​of the hydrostatic slider oil chamber parameters are shown in Table 2:

[0055] Table 2

[0056]

[0057] Step 2: Calculate the flow rate Q of the rectangular plane oil pad of the static pressure slider. The specific process is:

[0058] like Figure 3 As shown in the figure, let the length of the plate in the X direction be l' and the width in the Y direction be b'. Take a volume element of the flowing liquid between the plates for force analysis. In the steady state, the volume element is balanced in the flow direction:

[0059] ∑F x =p1·dy·dz-p2·dy·dz-2τ·dy·dx=0 (1);

[0060] Where p1 and p2 are the pressures at both ends of the microelement; τ is the shear force on the surface of the microelement;

[0061] When the fluid between two opposing plates is in a flowing state, the functional relationship between the flow velocity and the normal distance coordinate y between the plates can be obtained from Newton's shear law:

[0062]

[0063] In the formula, u is the oil flow velocity between the plates, and u=0 at the boundary. Therefore, the definite integral calculation of the oil flow velocity and flow rate with respect to the coordinate Y is:

[0064]

[0065] Where l' is the length of the plate in the X direction; b' is the width of the plate in the Y direction;

[0066] like Figure 2As shown, the oil pad type of the hydrostatic guide rail is a commonly used rectangular plane oil pad, with a pressure equalizing cavity in the middle. The oil is squeezed out from the edge of the pressure equalizing cavity, flows out of the oil sealing surface to the four sides, and finally enters the external environment. The oil sealing edge of the rectangular plane oil pad can be regarded as composed of two (Bb)×l and two (Ll)×b flat plates. Therefore, the total flow Q formula of the rectangular plane oil pad is:

[0067]

[0068] Step 3: Calculate the pore liquid resistance R c , Sealing oil surface liquid resistance R h , the specific process is:

[0069] The flow process of oil in the oil pad in the hydrostatic slider is as follows Figure 4 As shown, according to the definition of liquid resistance, we can get:

[0070]

[0071] In the formula, Δp is the pressure drop of the oil after passing through the fluid resistance, that is, the pressure difference before and after the fluid resistance;

[0072] The oil enters the external environment through the oil seal edge. The external environment pressure is 0. Therefore, substituting formula (4) into the oil seal edge liquid resistance R h :

[0073]

[0074] According to the fluid continuity theorem and Bernoulli equation, the actual flow equation flowing through the small hole can be obtained:

[0075]

[0076] Since the oil is incompressible and leakage is not considered, the flow rate through the small hole is the same as the flow rate of the rectangular plane oil pad, so the small hole liquid resistance R is obtained c for:

[0077]

[0078] Step 4: Calculate the initial hydraulic resistance ratio λ0 and the initial oil chamber pressure p0. The specific process is as follows:

[0079] Since the oil is incompressible and leakage is not considered, the total flow rate through the hydrostatic slider is the same as the flow rate of the rectangular plane oil pad. According to formula (5) and formula (6), the total flow rate through the hydrostatic slider is:

[0080]

[0081] Where λ0 is the hydraulic resistance ratio of the hydrostatic slider, p0 is the initial oil chamber pressure;

[0082] Therefore, by combining formula (6), formula (8) and formula (9), we can obtain the calculation formula of initial hydraulic resistance ratio λ0 and initial oil chamber pressure:

[0083]

[0084] Step 5: Calculate the plane rectangular oil cushion pressure p when the oil film thickness changes. The specific process is:

[0085] When the working load deviates from the initial design load, the oil resistance also changes, and the oil chamber pressure changes. When the total flow through the oil pad is the same, the oil film thickness h is automatically adjusted to reach a new balance with the current actual working load, that is, the adaptability of the liquid hydrostatic support. Under the new balance, the relationship between the sealing oil edge resistance and the oil film thickness is:

[0086]

[0087] Then the expression of the pressure p in the pressure equalization chamber is corrected to:

[0088]

[0089] In the actual working condition of the rectangular plane oil pad, the parameters such as oil film thickness h, oil pad pressure p, and fluid resistance all change with the change of external load. The system will eventually be in steady-state equilibrium. According to formula (8), the relationship between oil pad pressure and oil film thickness can be expressed as:

[0090]

[0091] By combining formula (9), formula (11), formula (12) and formula (13), we can get the formula for calculating the pressure p of the plane rectangular oil pad when the oil film thickness changes:

[0092]

[0093] Step 6: Calculate the oil pad bearing capacity F i and static stiffness S i , the specific process is:

[0094] The oil flows in the throttle, the equalizing cavity and the oil sealing surface, and its pressure distribution is not uniform, but there is a certain pressure drop. The pressure distribution at various locations in the equalizing cavity area is almost linear, and the oil sealing surface decreases step by step along the edge direction. Because the depth of the equalizing cavity is much larger than the thickness of the oil film, the pressure on the bottom area of ​​the equalizing cavity is considered to be equal pressure distribution. Therefore, the calculation formula for the bearing capacity of the static pressure slider oil pad can be obtained from the pressure formula:

[0095]

[0096] In the formula, A e A is the effective bearing area of ​​the oil film of the hydrostatic slider,e =(Bb)×(Ll);

[0097] The stiffness of the oil pad can represent the change in external load corresponding to the supported component within the unit oil film thickness. Therefore, the expression of the stiffness of the oil pad can be obtained by taking the derivative of the bearing capacity of the rectangular oil pad with respect to the oil film thickness h:

[0098]

[0099] Step 7: Calculate the four-slider oil film compression matrix ΔZ. The specific process is:

[0100] The force analysis of the four-slider hydrostatic guide rail shows that the resultant force and moment of the hydrostatic guide rail are both 0 when the force is applied at any position, so the force and moment balance equation can be obtained:

[0101]

[0102] In the formula, x and y are the external loads and coordinates of the hydrostatic guide rail; F1, F2, F3, and F4 are the oil film bearing capacities of each hydrostatic slider;

[0103] The four-slider hydrostatic guide is simplified as follows Figure 1 For the single degree of freedom system shown in the figure, assuming that the hydrostatic slider only has displacement in the z direction, and the workbench and the slider have no deformation, the compression amount of the hydrostatic slider has the following relationship equation:

[0104] Δz1+Δz4=Δz2+Δz3(18);

[0105] Where, Δz1, Δz2, Δz3, and Δz4 are the oil film compressions of the four sliders when subjected to eccentric loads;

[0106] It can be seen from formula (15) that when the oil film thickness changes, the change of its bearing capacity is nonlinear. Therefore, this set of equations is a nonlinear set of equations and usually cannot be solved directly by analytical methods. It can be solved by numerical methods. The steps are as follows:

[0107] Set up a nonlinear system of equations:

[0108] F(ΔZ)=0(19);

[0109] Where ΔZ is the variable vector, ΔZ=(Δz1, Δz2, Δz3, Δz4) T ;

[0110] Define the vector function F(ΔZ) according to formula (17) and formula (18):

[0111]

[0112] Set the initial estimate ΔZ0 and use Newton's method to update the solution to get the new iterative value ΔZk+1 :

[0113] ΔZ k+1 =ΔZ k -J -1 (ΔZ k )·F(ΔZ k )(twenty one)

[0114] Where J(ΔZ) is the Jacobian matrix of F(ΔZ), at point ΔZ k Everywhere

[0115] Step 8: Establish a static stiffness calculation model for the four-slider hydrostatic guideway. The specific process is as follows:

[0116] The four sliders of the hydrostatic guide are in parallel relationship. Combining formula (16) and formula (19), the static stiffness calculation model of the four-slider hydrostatic guide is obtained:

[0117]

[0118] The above calculation process can be implemented using MATLAB programming. To more specifically illustrate the effectiveness of this method, the present invention provides a calculation example result: the basic parameters of the four-slider hydrostatic guide rail and the oil chamber parameters are shown in Table 1 and Table 2. The calculation result curve obtained through steps 1 to 8 is shown in Figures 5 to 7 As shown, when the load F coordinate position is (10, 20), the oil film thickness compression is solved as: ΔZ = (-4.4163, -1.1302, 2.2152, 5.4113) T μm, the total stiffness of the four-slider hydrostatic guide is K=821.1221N / μm.

[0119] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for calculating the static stiffness of a four-slider hydrostatic guideway, characterized in that: The following steps are involved: Step 1: The hydrostatic slider is equivalent to a parallel spring element, the four-slider hydrostatic guide is simplified to a spatial single-degree-of-freedom system, and the basic parameters and oil chamber parameters of the four-slider hydrostatic guide are measured; Step 2, calculate the flow rate Q of the rectangular plane oil pad of the static pressure slider; Step 3: Calculate the pore liquid resistance R c , Oil seal surface liquid resistance R h ; Step 4, calculating the initial fluid resistance ratio λ0 and the initial oil chamber pressure p0; Step 5, calculate the plane rectangular oil cushion pressure p when the oil film thickness changes; Step 6: Calculate the oil pad bearing capacity F i and static stiffness S i ; Step 7, calculating the oil film compression matrix ΔZ of the four sliders; Step 8: Establish a static stiffness calculation model for the four-slider hydrostatic guideway.

2. The method for calculating the static stiffness of a four-slider hydrostatic guideway according to claim 1, characterized in that: In step 1, the four-slider hydrostatic guide is simplified into a spatial single-degree-of-freedom system, and the hydrostatic slider only has displacement in the z direction; The basic parameters of the guide rail that need to be measured are: the total gravity G of the workbench and sliders and other components, the eccentric load F, and the relative arrangement spacing m, n between the sliders; The oil chamber parameters that need to be measured are: oil chamber geometry B, L, b, l, h0, d; oil density, viscosity and oil supply pressure ρ, μ, p s .

3. The method for calculating the static stiffness of a four-slider hydrostatic guideway according to claim 1, characterized in that: In step 2, the calculation formula of the flow rate Q of the rectangular plane oil pad of the static pressure slider is: Where p is the oil chamber pressure; h is the oil film thickness; B, b, L, l are the geometric dimensions of the oil chamber respectively; μ is the oil viscosity.

4. The method for calculating the static stiffness of a four-slider hydrostatic guideway according to claim 3 is characterized in that: In step 3, the pore liquid resistance R c And seal oil surface liquid resistance R h The calculation formula is: In the formula, α is the flow area coefficient, and the value is α=0.5~0.7; ρ is the oil density; d is the oil inlet hole diameter.

5. The method for calculating the static stiffness of a four-slider hydrostatic guideway according to claim 4, characterized in that: In step 4, the calculation formula of the initial hydraulic resistance ratio λ0 and the initial oil chamber pressure p0 is: Where h0 is the oil film thickness in the initial state.

6. The method for calculating the static stiffness of a four-slider hydrostatic guideway according to claim 5, characterized in that: In step 5, the calculation formula of the plane rectangular oil cushion pressure p when the oil film thickness changes is: In the formula, p s Supply oil pressure to the guide rail system.

7. The method for calculating the static stiffness of a four-slider hydrostatic guideway according to claim 6, characterized in that: In step 6, the oil pad bearing capacity F i and static stiffness S i The calculation formula is: In the formula, A e is the effective bearing area of ​​the oil film, A e =(Bb)×(Ll).

8. The method for calculating the static stiffness of a four-slider hydrostatic guideway according to claim 7, characterized in that: In step 7, the calculation formula of the four-slider oil film compression matrix ΔZ is: ΔZ k+1 =ΔZ k -J -1 (ΔZ k )·F(ΔZ k ); Where F(ΔZ) is the vector function of the oil film compression, which is obtained by the force balance equation, the moment balance equation and the symmetry constraint; J(ΔZ) is the Jacobian matrix of the vector function F(ΔZ).

9. The method for calculating the static stiffness of a four-slider hydrostatic guideway according to claim 8, characterized in that: In step 8, the established four-slider hydrostatic guideway static stiffness calculation model is: In the formula, K i is the static stiffness of the oil film of the hydrostatic slider, i = 1 ~ 4; h i is the oil film thickness of each of the four hydrostatic sliders.

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