CN103823940B - Static-pressure turntable dynamic response computing method based on overall dynamical model - Google Patents

Abstract

The invention relates to a method for calculating the dynamic response of a static pressure turntable based on an integral dynamic model, comprising: calculating the average oil film thickness of each supporting oil pad and the average oil film thickness of a pre-pressed oil pad, and calculating the thickness of the pre-pressed oil pad and the supporting oil pad Bearing capacity, the overall dynamic model of the static pressure turntable is established, and the numerical solution of the dynamic model of the turntable is calculated by using Matlab to obtain the dynamic response of the turntable. Aiming at the problem that the existing modeling method of the static pressure turntable is not comprehensive due to consideration of factors, thus affecting the calculation accuracy of the turntable response, the present invention provides a modeling and analysis of the dynamic response of the static pressure turntable considering the influence of the surface roughness of the oil pad and the eccentric load. The calculation method improves the accuracy of the model and the calculation precision of the dynamic response, and plays a guiding role in further optimizing the dynamic performance of the static pressure turntable.

Description

A Calculation Method of Dynamic Response of Static Pressure Turntable Based on Integral Dynamic Model

technical field

The invention belongs to the field of static pressure turntable analysis, and relates to a dynamic characteristic modeling and simulation calculation method of a static pressure turntable supported by a single-circle oil pad, in particular to a static pressure turntable considering the influence of oil pad surface roughness and eccentric load Dynamic response modeling and simulation calculation methods.

Background technique

Hydrostatic Rotary Table uses pressurized fluid to separate two surfaces with relative motion and carries them by hydrostatic pressure. Since the kinematic pairs are completely separated by the oil film, the friction between the kinematic pairs is greatly reduced, while its bearing capacity, motion accuracy and life are greatly improved. Because of the many advantages of hydrostatic support, it is widely used in heavy machine tools and becomes one of its key components. The static pressure turntable will inevitably be affected by eccentric dynamic loads and impact loads during use. These problems are particularly obvious in heavy-duty static pressure turntables. To solve these problems, it is first necessary to establish an accurate and comprehensive turntable dynamics At the same time, the dynamic performance of the static pressure turntable is one of the important performances of the turntable. The establishment of the dynamic model of the turntable and the in-depth analysis of the response of the turntable under dynamic load are of great importance for the optimal design of the turntable and the improvement of the comprehensive performance of the turntable.

Scholars at home and abroad have conducted extensive and in-depth research on hydrostatic bearing technology. In March 2012, the paper "Dynamic Modeling and Analysis of Axial Vibration of Turntable with Hydrostatic Guideway" published in "Engineering Mechanics" Volume 29, No. 3, established the axial dynamic model of the turntable, and carried out Analysis and calculation, but did not consider the influence of eccentric load and roughness, nor did it consider the effect of preload, and almost all heavy-duty static pressure turntables are designed with pre-pressure oil pads to provide preload, so this method does not reflect the effect of the turntable actual situation. The invention patent with the publication number CN102980755A "a quantitative static pressure turntable dynamic and static characteristics experimental device" discloses an experimental device for the dynamic and static characteristics of a static pressure turntable, and only considers load balancing for the dynamics modeling of the turntable. The invention patent with the publication number CN103186698A "A Method for Simulation and Optimization of Dynamic and Static Performance of Static Pressure Turntable of Heavy Machine Tool" discloses a method of using ANSYS secondary development to establish the finite element model of the turntable, and then simulate and optimize the turntable. In this method, the oil pad is equivalent to a nonlinear spring, but the nonlinear damping of the oil pad is not considered, and the influence of the roughness of the oil pad is not considered. In short, although the dynamic modeling of the turntable has its own characteristics, the considerations are not comprehensive, and the overall dynamic model of the turntable is not established by comprehensively considering various influencing factors of the turntable, which affects the calculation accuracy of the turntable response.

Contents of the invention

The purpose of the present invention is to provide a calculation method for the dynamic response of the static pressure turntable considering the influence of roughness and eccentric load. The method first calculates the dynamic bearing capacity of the turntable support oil pad and pre-press oil pad, and then establishes the overall dynamics of the turntable model, and finally the response of the turntable is calculated numerically. Applying the invention to the simulation analysis of the dynamic characteristics of the static pressure turntable can accurately calculate the response of the turntable under the action of eccentric dynamic load.

In order to achieve the above object, the present invention adopts the following technical solutions:

In step 1, the deformation of the turntable itself is ignored, that is, the turntable is approximately regarded as a rigid body, and the average oil film thickness of the turntable is considered to be h. The turntable will tilt under the action of eccentric external load, and the oil pads of each support are calculated according to the distribution positions of the oil pads of the turntable. The average oil film thickness and the average oil film thickness of the pre-pressed oil pad.

Step 2, calculate the pressure distribution of the pre-pressed oil pad and the supporting oil pad when considering the extrusion effect and surface roughness according to the relevant theory of fluid mechanics, and then integrate the pressure distribution to obtain the bearing capacity of the pre-pressed oil pad and the supporting oil pad;

Step 3, according to the balance principle of all the forces and moments of the turntable during the period of external load, the dynamic balance equation of the turntable is established. The dynamic equation of the turntable is a second-order nonlinear differential equation.

Step 4, according to the relevant theory of numerical analysis and the Runge-Kutta method, use matlab software to write the solution program of the dynamic equation of the turntable, and finally obtain the numerical solution of the equation, that is, the dynamic response curve of the turntable.

Compared with the prior art, the present invention has the following obvious advantages:

Based on fluid mechanics and numerical analysis theory, the present invention establishes an overall dynamic model of the static pressure turntable, and uses a numerical method to solve the dynamic response of the turntable. Because the influence of the eccentric load on the turntable characteristics and the roughness of the oil pad are considered in the model, the accuracy of the overall dynamic model of the turntable and the response of the turntable is improved, which plays an important guiding role in optimizing the performance of the turntable.

Description of drawings

Figure 1 is a schematic diagram of the structure of the static pressure turntable;

Figure 2 is a schematic diagram of the structure of the turntable support oil pad;

Figure 3 is a schematic diagram of the structure of the pre-pressed oil pad on the turntable;

Figure 4 is a schematic diagram of the force on the turntable;

Fig. 5 is the flowchart of the method involved in the present invention;

Fig. 6 is the response curve of the turntable obtained by the example of the present invention: (a) is the response curve of the turntable when the turntable is subjected to different step load sharing, (b) is the response curve of the turntable when the external load of a constant step is different, (c ), (d) are the response curves of the turntable under constant step load sharing but with different roughness values.

detailed description

The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

Figure 1 is a schematic diagram of the structure of a static pressure turntable supported by a single ring, which is composed of a turntable, a base, a supporting oil pad and a pre-pressed oil pad. The dead weight of the turntable is G, and each oil pad is supplied with oil by a quantitative pump. The oil supply quantity is Q ₀ , and the oil supply quantity of the pre-press oil pad is Q ₁ .

Figure 2 and Figure 3 are schematic diagrams of the structure of the pre-pressing oil pad and the supporting oil pad. The pre-pressing oil pad is an annular oil pad, and the supporting oil pad is a circular oil pad. RMS values of roughness are δ ₁ and δ ₂ respectively.

The flow chart of the calculation method for the dynamic response of the static pressure turntable based on the overall dynamic model is shown in Figure 5, which specifically includes the following steps:

Step 1, calculate the average oil film thickness of each support oil pad and pre-press oil pad.

The structure diagram of the static pressure turntable is shown in Figure 1. The supporting oil pad and the pre-pressing oil pad together form the supporting system. One of the pre-pressing oil pads is installed in the center of the turntable, and several supporting oil pads are evenly distributed along the circumference of the supporting circle. . Neglecting the deformation of the turntable itself, assuming that the turntable is a rigid body, the average oil film thickness of each supporting oil pad and the average oil film thickness of the pre-pressed oil pad are calculated according to the distribution position of each oil pad on the turntable, the formula is as follows:

Wherein, h _i is the average oil film thickness of the i-th supporting oil pad, i=1,2,...,n, n is the number of supporting oil pads; is the angle between the center point of the i-th support oil pad and the center point of the turntable and the x-axis, as shown in Figure 4; h _y is the average oil film thickness of the pre-pressed oil pad; h is the average oil film thickness of all the support oil pads on the turntable Thickness; θ _x , θ _y are the x-direction and y-direction inclinations of the turntable respectively; _RL is the distance between the center of the support oil pad and the center of the turntable, as shown in Figure 1.

Step 2, calculate the bearing capacity of the pre-pressed oil pad and the supporting oil pad.

Step 2.1, find the stochastic Reynolds equation of the oil film.

Assuming that the flow state of the oil pad oil is laminar flow, without considering the influence of temperature, the N-S equation and the continuity equation in the cylindrical coordinate system are simplified according to the fluid mechanics theory:

Among them, p is the pressure of the oil, u _r is the flow velocity of the oil in the r direction, v _z is the flow velocity of the oil in the z direction, and η is the viscosity of the oil.

Solving the above equations yields the one-dimensional stochastic Reynolds equation:

The oil film thickness can be considered to be composed of two parts:

hT = δ(r, θ, ξ) + h( _t ) (6)

Among them, δ = δ ₁ + δ ₂ is the average roughness between the surface of the supporting oil pad and the surface of the guide rail and the surface of the pre-pressed oil pad and the surface of the guide rail, which is the rough part of the thickness of the oil film, and ξ represents the distribution type of the roughness. Since the topography of most surfaces in nature conforms to the Gaussian distribution, the present invention assumes that the roughness of all oil pad surfaces and guide rail surfaces of the turntable conforms to the Gaussian distribution. h represents the smooth part of the oil film thickness.

Step 2.2, calculate the average Reynolds equation.

Divide the surface roughness into two parts, circumferential roughness and radial roughness, and take the expectation of the stochastic Reynolds equation (5) to obtain the average Reynolds equation.

For radial roughness h _T = δ(θ,ξ)+h(t), then the average Reynolds equation is:

For the circumferential roughness h _T =δ(r,ξ)+h(t), the average Reynolds equation is:

Most of the rough surfaces in engineering obey the Gaussian distribution, and the Gaussian distribution approximated by polynomials is:

Among them, σ is the standard deviation; c is half of the random oil film thickness range, and the equation is valid within c=±3σ.

make , when the roughness is the circumferential roughness , when the roughness is the radial roughness Equations (7) and (8) can be uniformly written as:

Similarly, the oil flow rate of the oil pad is obtained as:

Step 2.3, calculate the bearing capacity of the supporting oil pad.

The structure diagram of the supporting oil pad is shown in Figure 2. It is a circular stepped structure, and the main dimensions have been marked in the figure. There are boundary conditions for the supporting oil pad with circular quantitative compensation:

r=R ₁ , p _s =p _s0 ; r=R ₂ , p _s =0; Q(R ₁ )=Q ₀ (12)

Among them, R ₁ and R ₂ are the inner diameter and outer diameter of the supporting oil pad respectively, and p _s0 is the pressure of the oil cavity of the supporting oil pad.

Replacing p in formula (10) with p _s and substituting the boundary conditions of the supporting oil pad, the oil chamber pressure p _s0 of the supporting oil pad and the pressure distribution p _s (r) of the oil sealing edge are obtained as:

The bearing capacity of the circular support oil pad is:

In the formula, F is the bearing capacity of the supporting oil pad.

Step 2.4, calculate the bearing capacity of the pre-pressed oil pad.

The pre-pressure oil pad is an annular oil pad, and its structural diagram is shown in Figure 3. The boundary conditions for the annular oil pad are:

Among them, R _C1 ~ R _C4 are respectively the inner diameter of the oil sealing edge of the pre-pressing oil pad, the inner diameter of the oil cavity of the pre-pressing oil pad, the outer diameter of the oil cavity of the pre-pressing oil pad and the outer diameter of the oil sealing edge of the pre-pressing oil pad.

Replacing p in formula (10) with p _y and bringing it into boundary condition (16), the pressure p _y0 of the oil chamber of the pre-pressed oil pad and the pressure distribution p _y (r) of the oil sealing edge are obtained as follows:

When r∈(R _C1 , R _C2 ), the pressure distribution is:

When r∈(R _C3 , R _C4 ), the pressure distribution is:

The bearing capacity of the annular preload oil pad is:

In the formula, F _y is the bearing capacity of the pre-pressed oil pad.

Step 3, establishing the overall dynamics model of the static pressure turntable.

The force on the turntable is shown in Figure 4. According to the force and moment balance of the turntable when the external load acts, the dynamic model of the turntable is obtained as follows:

in, , is the moment of inertia of the turntable; F _w is the external load, and its offset distance is b; G=Mg is the weight of the turntable, M is the mass of the turntable; _w is the speed of the turntable; F _y is the pre-pressure, and F _i and F _y are calculated by formula (17) and formula (22).

Step 4, find the dynamic response of the turntable.

First, according to (1), (15), (20) and (21), Matlab software is used to write the oil film thickness calculation function, the support oil pad bearing capacity calculation function, the pre-pressed oil pad bearing capacity calculation function and the turntable according to the balance equation Acceleration calculation function in each direction; Then, write the input parameters of the turntable in the main function, including the structural size of the turntable, the structural size of each oil pad, oil parameters, external load and boundary conditions, and then call the bearing capacity calculation The function calculates the bearing capacity of each oil pad, substitutes it into the balance equation to calculate the acceleration in each direction of the turntable, and integrates it to obtain the displacement and velocity of the turntable. Applying the Runge-Kutta method to the above calculation requires four corrections. Repeat the above process to calculate the turntable response at the next time, and the calculation will not be terminated until the time boundary condition is reached.

A calculation example is given below. The parameters of the turntable are shown in Table 1. Apply different external loads to the turntable, and then use Matlab to obtain the numerical solution of the dynamic equation of the turntable. The results are shown in Figure 6 (a), (b), (c), and (d). The response curve of the turntable under step load equalization, (b) is the response curve of the turntable under constant step external load and different offset distances. In the figure, the average oil film thickness of the turntable decreases with the increase of external load. increased by the increase. (c) and (d) are the response curves of the turntable under constant step load sharing but with different roughness values. When the roughness form is circumferential roughness, the average oil film thickness of the turntable is increases, while the law is just the opposite when the roughness form is radial roughness. It can also be seen from the figure that the response curve of the turntable to the step load is relatively smooth, and the stabilization time is short, which reflects the characteristics of high stiffness and large damping of the static pressure support. The obvious influence should be considered and checked when designing the turntable.

Table 1 Static pressure turntable parameters

parameter name value parameter name value Number of bearing pads n 8 Turntable radius R _T /mm 2500 Bearing pad inner diameter R ₁ /mm 160 Turntable supporting ring radius R _L /mm 1500 Bearing pad outer diameter R ₂ /mm 175 Initial oil film thickness of bearing oil pad h ₀ /mm 0.1 Inner diameter of oil sealing side of pre-pressed oil pad R _C1 /mm 185 Initial oil film thickness of pre-pressed oil pad h _y0 /mm 0.6 Inner diameter of pre-pressed oil pad oil chamber R _C2 /mm 220 Oil pad oil supply flow Q ₀ /m ³ /s 1×10 ^-4 Outer diameter of pre-pressed oil pad oil chamber R _C3 /mm 255 Oil viscosity η/Pa.s 0.091 Outer diameter of oil sealing side of pre-pressed oil pad R _C4 /mm 290 Turntable mass M/Kg 30827

Experiments show that the method of the invention can accurately establish the dynamic model of the turntable, quickly calculate the response curve of the turntable, provide theoretical guidance for the optimal design of the turntable, and have certain references for the use of the turntable.

Claims (1)

1. A static pressure turntable dynamic response calculation method based on an overall dynamic model is characterized in that the established model considers the influence of unbalance loading borne by a turntable and the roughness of an oil pad on the turntable dynamic response; the method comprises the following steps:

step 1, calculating the average oil film thickness of each supporting oil pad and each prepressing oil pad;

the supporting oil pad and the prepressing oil pad of the static pressure turntable jointly form a supporting system, one prepressing oil pad is arranged at the central part of the turntable, and a plurality of supporting oil pads are uniformly distributed along the circumference of a supporting circle; neglecting the deformation of the rotary table, assuming that the rotary table is a rigid body, calculating the average oil film thickness of each supporting oil pad and each pre-pressing oil pad according to the distribution position of each oil pad of the rotary table, wherein the formula is as follows:

wherein h is_iThe average oil film thickness of the ith supporting oil pad is shown, i is 1,2, …, n, n is the number of the supporting oil pads;the included angle between the connecting line of the center point of the ith supporting oil pad and the center point of the rotary table and the x axis; h is_yThe average oil film thickness of the pre-pressed oil pad; h is the average oil film thickness of all supporting oil pads of the rotary table; theta_x、θ_yThe inclination angles of the rotary table in the x direction and the y direction are respectively; r_LThe distance between the center of the supporting oil pad and the center of the rotary table;

step 2, calculating the bearing capacity of the pre-pressing oil pad and the supporting oil pad;

step 2.1, solving a random Reynolds equation of the oil film;

assuming that the flow state of oil in the oil pad is laminar flow, and not considering the influence of temperature, simplifying an N-S equation and a continuity equation under a cylindrical coordinate system according to a fluid mechanics theory to obtain the oil pad, wherein the N-S equation comprises the following steps:

$\frac{1}{r}\frac{\∂\left({\mathrm{ru}}_r\right)}{\∂r}+\frac{\∂\left(v_z\right)}{\∂z}=0---\left(2\right)$

$\frac{\∂p}{\∂r}=\mathrm{\η}\frac{{\∂}^2{u}_r}{\∂z^2}---\left(3\right)$

$\frac{\∂p}{\∂r}=0---\left(4\right)$

wherein p is the pressure of the oil, u_rThe flow velocity v of the oil in the r direction_zThe flow velocity of the oil in the z direction is shown, and η is the viscosity of the oil;

solving the above equation can obtain a one-dimensional random reynolds equation:

$\frac{1}{r}\frac{\∂}{\∂r}\left(\frac{{\mathrm{rh}}_T^3}{12\mathrm{\η}}\frac{\∂p}{\∂r}\right)=\frac{\∂h_T}{\∂t}---\left(5\right)$

the oil film thickness can be considered to consist of two parts:

h_T＝(r,θ,ξ)+h(t) (6)

wherein ═₁+₂In order to support the average roughness between the surface of the oil pad and the guide rail surface and between the surface of the pre-pressed oil pad and the guide rail surface, namely the rough part of the oil film thickness, ξ represents the distribution type of the roughness, wherein the shape of most surfaces in nature conforms to Gaussian distribution, so that the roughness of all the surfaces of the oil pad and the guide rail surface of the turntable is assumed to meet the Gaussian distribution, h (t) represents the smooth part of the oil film thickness;

step 2.2, calculating an average Reynolds equation;

dividing the surface roughness into two parts of circumferential roughness and radial roughness, and obtaining an expected average Reynolds equation from the random Reynolds equation (5);

has h for radial roughness_T(θ, ξ) + h (t), the average reynolds equation is:

$\frac{1}{r}\frac{\∂}{\∂r}\left(\frac{rE\left(h_T^3\right)}{12\mathrm{\η}}\frac{\∂p}{\∂r}\right)=\frac{\∂E\left(h_T\right)}{\∂t}---\left(7\right)$

has a circumferential roughness of h_T(r, ξ) + h (t), the average reynolds equation is:

$\frac{1}{r}\frac{\∂}{\∂r}\left(\frac{r}{12\mathrm{\η}E\left(h_T^{-3}\right)}\frac{\∂p}{\∂r}\right)=\frac{\∂E\left(h_T\right)}{\∂t}---\left(8\right)$

rough surfaces in engineering are mostly gaussian-distributed, which is approximated by a polynomial expression:

wherein σ is the standard deviation; c is half of the range of the random oil film thickness, and the equation is effective within +/-3 sigma of c;

order toRoughness is circumferential roughnessWhen the roughness is radial roughnessEquations (7), (8) can be written collectively as:

$\frac{1}{r}\frac{\∂}{\∂r}\left(rE\frac{\∂p}{\∂r}\right)=T---\left(10\right)$

the oil flow of the oil pad obtained by the same method is as follows:

$Q\left(r\right)=-\frac{rE}{12\mathrm{\η}}\frac{\∂p}{\∂r}---\left(11\right)$

step 2.3, calculating the bearing capacity of the supporting oil pad;

the supporting oil pad is of a circular step structure, and has boundary conditions for the circular quantitative compensation:

r_z＝R₁,p＝p₀；r_z＝R₂,p＝0；Q(R₁)＝Q₀； (12)

wherein R is₁、R₂Respectively the inner and outer diameter, p, of the supporting oil pad₀Oil pocket pressure, Q, for supporting oil pads₀Oil supply for supporting the oil pad;

relieving the oil chamber pressure p of the supporting oil pad₀And oil seal edge pressure distribution p (r)_z) Comprises the following steps:

$p_0=\frac{6\mathrm{\η}\ln \left(\frac{R_2}{R_1}\right)}{\mathrm{\π}E}\left(Q_0+\frac{{\mathrm{\πR}}_1^{2}T}{12\mathrm{\η}}-\frac{\mathrm{\π}\left(R_2^2-R_1^2\right)T}{24\mathrm{\η}\ln \left(\frac{R_2}{R_1}\right)}\right)---\left(13\right)$

$p\left(r_z\right)=\frac{{r_z}^{2}T}{4\mathrm{\π}E}-\frac{\left(p_{0}ln\right(\frac{r_z}{R_2})+\frac{T\left(R_2^{2}ln\right(\frac{r_z}{R_1})-R_1^{2}ln(\frac{r_z}{R_2}\left)\right)}{4E})}{ln\left(\frac{R_2}{R_1}\right)}---\left(14\right)$

the bearing capacity of the circular supporting oil pad is as follows:

$F={\mathrm{\πR}}_1^2{p}_0+2\mathrm{\π}\underset{R_1}{\overset{R_2}{\∫}}{r}_{z}p\left(r_z\right){\mathrm{dr}}_z---\left(15\right)$

step 2.4, calculating the bearing capacity of the pre-pressed oil pad;

the pre-pressing oil pad is an annular oil pad, and the annular oil pad has boundary conditions as follows:

$\left\{\begin{array}{c}{r}_y=R_{C1},p=0\\ r_y=R_{C2},p_y=p_{y0}\\ r_y=R_{C3},p_y=p_{y0}\\ r_y=R_{C4},p_y=0\\ Q_1=-Q\left(R_{C2}\right)+Q\left(R_{C3}\right)\end{array}\right.---\left(16\right)$

wherein R is_C1～R_C4The inner diameter of an oil sealing edge of the pre-pressing oil pad, the inner diameter of an oil cavity of the pre-pressing oil pad, the outer diameter of the oil cavity of the pre-pressing oil pad and the outer diameter of the oil sealing edge of the pre-pressing oil pad are respectively;

replacing p in the formula (10) with p_yAnd bringing in boundary conditions (16) to obtain the oil chamber pressure p of the pre-pressed oil pad_y0And the pressure distribution of the oil sealing edge is as follows:

$p_{y0}=\frac{6\mathrm{\η}\ln \left(\frac{R_{C4}}{R_{C3}}\right)\ln \left(\frac{R_{C2}}{R_{C1}}\right)}{\mathrm{\π}E\ln \left(\frac{R_{C4}{R}_{C2}}{R_{C3}{R}_{C1}}\right)}\left(Q_1+\frac{\mathrm{\π}T\left(R_{C3}^2-R_{C2}^2\right)}{12\mathrm{\η}}-\frac{\mathrm{\π}T\left(\ln \left(\frac{R_{C2}}{R_{C1}}\right)\left(R_{C4}^2-R_{C3}^2\right)+\ln \left(\frac{R_{C4}}{R_{C3}}\right)\left(R_{C1}^2-R_{C2}^2\right)\right)}{24\mathrm{\η}\ln \left(\frac{R_{C2}}{R_{C1}}\right)\ln \left(\frac{R_{C4}}{R_{C3}}\right)}\right)---\left(17\right)$

when r is_y∈(R_C1,R_C2) The time pressure distribution is:

$p_1\left(r_y\right)=\frac{{r_y}^{2}T}{4\mathrm{\π}E}-\frac{\left(p_{y0}ln\right(\frac{r_y}{R_{C1}})+\frac{T\left(R_{C1}^{2}ln\right(\frac{r_y}{R_{C2}})-R_{C2}^{2}ln(\frac{r_y}{R_{C1}}\left)\right)}{4E})}{ln\left(\frac{R_{C1}}{R_{C2}}\right)}---\left(18\right)$

when r is_y∈(R_C3,R_C4) The time pressure distribution is:

$p_2\left(r_y\right)=\frac{{r_y}^{2}T}{4\mathrm{\π}E}-\frac{\left(p_{y0}ln\right(\frac{r_y}{R_{C4}})+\frac{T\left(R_{C4}^{2}ln\right(\frac{r_y}{R_{C3}})-R_{C3}^{2}ln(\frac{r_y}{R_{C4}}\left)\right)}{4E})}{ln\left(\frac{R_{C4}}{R_{C3}}\right)}---\left(19\right)$

the bearing capacity of the annular pre-pressing oil pad is as follows:

$F_y=\mathrm{\π}(R_{C3}^2-R_{C2}^2)p_{y0}+2\mathrm{\π}\underset{R_{c1}}{\overset{R_{c2}}{\∫}}{r}_y{p}_1\left(r_y\right){\mathrm{dr}}_y+2\mathrm{\π}\underset{R_{c3}}{\overset{R_{c4}}{\∫}}{r}_y{p}_2\left(r_y\right){\mathrm{dr}}_y---\left(20\right)$

step 3, establishing an integral dynamic model of the static pressure rotary table;

according to the stress and moment balance of the turntable under the external load action, the dynamic model of the turntable is obtained as follows:

wherein,is the moment of inertia of the turntable; f_wIs an external load, and the offset distance is b; g is the weight of the turntable, and M is the mass of the turntable; w is the rotating speed of the rotary table; f_iThe turntable is subjected to supporting forces of the supporting oil pads, F_yFor pre-stressing, F_iAnd F_yCalculated from the formulae (15) and (20);

step 4, solving the dynamic response of the rotary table;

firstly, writing an oil film thickness calculation function, a supporting oil pad bearing capacity calculation function, a pre-pressing oil pad bearing capacity calculation function and acceleration calculation functions of each direction of the rotary table according to formulas (1), (15), (20) and (21) by applying Matlab software; then writing various input parameters of the rotary table in the main function, including the structure size of the rotary table, the structure size of each oil pad, oil parameters, external load and boundary conditions, calling a bearing capacity calculation function to calculate the bearing capacity of each oil pad, substituting a balance equation to calculate the acceleration of the rotary table in each direction, and integrating the acceleration to obtain the displacement and the speed of the rotary table; the calculation is carried out by applying a Longge Kuta method, and four corrections are needed in total; and repeating the above process to calculate the response of the next time turntable, and stopping the calculation until the time boundary condition is reached.