WO2023035582A1 - 一种液体火箭发动机高速重载球轴承接触动态特性仿真方法 - Google Patents

一种液体火箭发动机高速重载球轴承接触动态特性仿真方法 Download PDF

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WO2023035582A1
WO2023035582A1 PCT/CN2022/081259 CN2022081259W WO2023035582A1 WO 2023035582 A1 WO2023035582 A1 WO 2023035582A1 CN 2022081259 W CN2022081259 W CN 2022081259W WO 2023035582 A1 WO2023035582 A1 WO 2023035582A1
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ball bearing
contact
heavy
speed
duty
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French (fr)
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杜飞平
谭永华
王春民
陈晖�
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西安航天动力研究所
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/15—Vehicle, aircraft or watercraft design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

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  • the invention belongs to the field of dynamic characteristics of liquid rocket engine ball bearings, and mainly relates to a method for simulating the contact dynamic characteristics of liquid rocket engine high-speed and heavy-duty ball bearings.
  • Ball bearings are important supporting and rotating parts that transmit motion and bear loads, and are widely used in major equipment technical fields such as aerospace propulsion systems, ship engineering, and precision instruments. Ball bearings are mainly used in the liquid rocket engine turbopump rotor system because they have the characteristics of high manufacturing precision, high limit speed, strong bearing capacity, and can bear radial load and axial load at the same time.
  • Ball bearings are composed of structural elements such as balls, inner rings, outer rings, and cages. Their dynamic characteristics include dynamic contact angles between balls and inner and outer rings, contact load, contact deformation, contact stiffness, and roll-to-roll ratio. The most concerned is the dynamic contact angle between the ball and the inner and outer rings.
  • Ball bearings are key components of liquid rocket engine turbopump rotor systems. very harsh. It is generally believed that the DN (bearing inner diameter (mm) ⁇ speed (r/min)) value exceeds 0.6 ⁇ 10 6mm r/min is a high-speed ball bearing, and in the liquid rocket engine, the DN value of the ball bearing reaches 2.5 ⁇ 10 6mm r/min, and the ratio of the radial equivalent dynamic load Pr to the rated dynamic load Cr is often greater than 0.15, so it can be regarded as a high-speed heavy-duty ball bearing.
  • DN bearing inner diameter (mm) ⁇ speed (r/min)
  • the object of the present invention is to provide a kind of liquid rocket engine high-speed heavy-duty ball bearing contact dynamic characteristic simulation method, can obtain the liquid rocket engine ball bearing from static to dynamic, single load to combined load and low speed to The contact dynamics characteristic parameters and changing law at very high speed.
  • a method for simulating the contact dynamic characteristics of a high-speed, heavy-duty ball bearing of a liquid rocket engine including the following steps:
  • each ball in the ball bearing is obtained
  • the contact dynamic characteristics include the contact angle between each ball and the inner ring, and the contact angle between each ball and the outer ring.
  • the method for simulating the contact dynamic characteristics of the high-speed and heavy-duty ball bearing of the liquid rocket engine further includes:
  • the least square method of linear regression is used to simplify the normal contact problem equation to obtain the contact normal stress of the heavy-duty ball bearing contact.
  • a and B both represent the contact surface particles
  • a is the semi-major axis of the contact ellipse
  • b is the semi-minor axis of the contact ellipse
  • e is the parameter of the contact ellipse
  • N(e) is the complete elliptic integral of the first kind
  • M(e ) is the complete elliptic integral of the second kind
  • E 1 is the elastic modulus of the ball
  • E 2 is the elastic modulus of the raceway
  • ⁇ 1 is the Poisson’s ratio of the ball
  • ⁇ 2 is the Poisson’s ratio of the raceway
  • p 0 is the contact The maximum compressive stress at the center of the ellipse
  • the contact ellipse is the ellipse produced when each ball is in contact with the raceway
  • R x and R y respectively represent the equal distances in the direction of the semi-major axis and semi-minor axis of the contact ellipse effective
  • the static contact mechanical characteristics of the low-speed heavy-duty ball bearing include: dimensionless radial deformation of the heavy-duty ball bearing, dimensionless axial deformation of the heavy-duty ball bearing, contact contact of the heavy-duty ball bearing angle as well as the contact equivalent stiffness of the heavy-duty ball bearing.
  • the establishment of the low-speed heavy-duty ball bearing model of the liquid rocket engine according to the contact normal stress of the ball bearing includes:
  • F rx is the force projected to the X direction by the radial load applied to the ball bearing
  • F ry is the force projected to the Y direction by the radial load applied to the ball bearing
  • F a is the axial load of the ball bearing
  • Z is the number of balls
  • ⁇ j is the position angle of the ball bearing and j is the jth ball in the bearing
  • ⁇ ij is the contact angle between the jth ball and the inner ring
  • M rx is the external torque in the X direction on the ball bearing
  • M ry is The torque in the Y direction applied to the ball bearing
  • M gj is the gyro torque of the jth ball
  • ⁇ j is the distribution coefficient of the frictional moment of the jth ball
  • Q ij is the contact force between the jth ball and the inner ring
  • d m is the pitch circle diameter of the ball bearing
  • f i is the coefficient of curvature radius of the bearing inner ring groove
  • F r is the radial load of the ball bearing
  • F a is the axial load of the ball bearing
  • u r and u a are respectively when the ball bearing bears the radial load F r and the axial load F a at the same time
  • the ball The radial deformation and axial deformation of the bearing
  • K n is the nth iteration
  • S G is the distance between the center of curvature of the inner and outer ring grooves of the angular contact ball bearing
  • ⁇ is the initial contact angle of the ball bearing
  • Mr is the external torque on the ball bearing
  • ⁇ j is the position angle of the heavy-duty ball bearing
  • j is the jth ball in the bearing.
  • the iterative operation using the co-evolutionary particle swarm optimization algorithm of the two populations includes the following steps:
  • Step 1 setting the parameters of the population s1 and the population s2, wherein the parameters include the size of the population, the dimension of the particles, the maximum number of iterations, and acceleration constants c1, c2, inertia weights w1 and w2, the populations s1 and said population s2 forms said dual population;
  • Step 2 randomly initializing the initial position and initial velocity of each particle in the population s1 and the population s2, the individual optimal value pbest id (k) and the global optimal value gbest id (k);
  • Step 3 calculating the fitness value of each particle in the double population, and synchronously updating the position and velocity of each particle, that is, updating the parameter space of the population s1 and the population s; the population s1 and the calculation formula of the inertia weight fitness value Fitness of the population s is:
  • k is the current number of iterations
  • Y 2j S G cos ⁇ + ⁇ r cos ⁇ j ;
  • X1j, Y1j, X2j, Y2j are the geometric quantities of the center of curvature of the ball bearing before and after the load;
  • ⁇ a is the axial deformation of the ball bearing under the action of the axial force Fa;
  • ⁇ r is the axial deformation of the ball bearing at Radial deformation under the action of radial force Fr;
  • ⁇ is the rotation angle of the ball bearing under the action of moment M;
  • f i is the curvature radius coefficient of the inner ring groove of the ball bearing;
  • f 0 is the curvature of the outer ring groove of the ball bearing Radius coefficient;
  • ⁇ ij , ⁇ oj are the contact deformation between the jth ball and the inner and outer rings of the ball bearing;
  • S G is the distance between the centers of curvature of the inner and outer ring grooves of the ball bearing;
  • Expr2, Expr3, Expr4, Expr5 satisfy the following formula:
  • v id (k+1) w ⁇ v id (k)+c 1 ⁇ r 1 ⁇ [pbest id (k)-x id (k)]+c 2 ⁇ r 2 ⁇ [gbest id (k)-x id (k)];
  • x id (k+1) x id (k) + v id (k+1);
  • D is the number of unknowns of the fitness equation
  • vid(k), vid(k+1 ) are the current speed and update speed of the i-th and i+1 generation particle swarms respectively
  • xid(k) and xid(k+1) are the current position and update position of the i-th and i+1 generation particle swarms respectively
  • pbest id (k) is the individual extremum searched by the particle
  • gbest id (k) is the global extremum searched by the population
  • w is the inertia weight, which indicates the influence of the particle on the current velocity, and has the ability to balance global convergence and local convergence
  • c 1 and c 2 are acceleration constants, which represent the acceleration weights that push the particles to the individual extremum and global extremum positions
  • r 1 and r 2 are random numbers in [0,1];
  • Step 4 Update the individual extremum and the global extremum of the dual population, compare each particle in the population s1 and the population s2 with the current individual extremum according to the fitness value, and select the optimal individual extremum value pbest id (k) and the global optimal value gbest id (k) with the smallest fitness value;
  • Step 5 the dynamic cooperation strategy of the two populations, a cooperative relationship occurs through the neighborhood model, and the individual extremum and the global extremum searched by each population in the two populations are shared;
  • Step 6 Judging the termination condition of the algorithm, if the current number of iterations reaches the maximum number of iterations, the loop is terminated, and it is considered that the optimization of the contact angle between each ball and the inner ring in the ball bearing is completed, otherwise return to step 3 and continue the iteration ;
  • Step 7 through the contact angle between each of the balls and the inner ring optimized in step 6, according to the mathematical relationship between the contact angle of the inner ring and the contact angle of the outer ring, the contact between each of the balls and the outer ring can be obtained Angle, so as to complete the simulation of the contact dynamic characteristics of the high-speed heavy-duty ball bearing.
  • the iterative method adopted for the iterative calculation of the low-speed heavy-duty ball bearing model of the liquid rocket engine is the Newton-Raphson iterative method.
  • the DN value of the ball bearing is more than or equal to 2.5 ⁇ 10 6mm r/min; the ratio of the radial equivalent dynamic load Pr of the ball bearing to the rated dynamic load Cr of the ball bearing>0.15.
  • Fig. 1 is the realization flowchart of the inventive method
  • Figure 2 is a graph showing the variation of the contact angle of the inner ring with the axial load
  • Figure 3 is a graph showing the variation of the contact angle of the outer ring with the axial load.
  • the core of the present invention is to provide a method for simulating the contact dynamic characteristics of a high-speed, heavy-duty ball bearing of a liquid rocket engine.
  • the simulating process is executed on matlab, and specifically includes the following steps:
  • Step 1 Establish the equation of the normal contact problem based on the Hertz theory of semi-infinite length space, and then combine the least square method of linear regression to perform simplified calculations to obtain the contact normal stress of the ball bearing;
  • a and B both represent the contact surface particles
  • a is the semi-major axis of the contact ellipse
  • b is the semi-minor axis of the contact ellipse
  • e is the parameter of the contact ellipse
  • N(e) is the complete elliptic integral of the first kind
  • M(e ) is the complete elliptic integral of the second kind
  • E 1 is the elastic modulus of the ball
  • E 2 is the elastic modulus of the raceway
  • ⁇ 1 is the Poisson’s ratio of the ball
  • ⁇ 2 is the Poisson’s ratio of the raceway
  • p 0 is the contact The maximum compressive stress at the center of the ellipse
  • the contact ellipse is the ellipse produced when each ball is in contact with the raceway
  • R x and R y respectively represent the equal distances in the direction of the semi-major axis and semi-minor axis of the contact ellipse effective
  • Step 2 Regardless of the influence of centrifugal force and gyro moment at high speed, a low-speed, heavy-duty ball bearing model of a liquid rocket engine is established based on the contact normal stress.
  • the Newton-Raphson iterative method is used to calculate the static state of the ball bearing.
  • Contact mechanical properties; the static contact mechanical properties of the ball bearing include dimensionless radial deformation of the ball bearing, dimensionless axial deformation of the ball bearing, contact angle ⁇ of the ball bearing, and equivalent contact stiffness Kn of the ball bearing;
  • F r is the radial load of the ball bearing
  • F a is the axial load of the ball bearing
  • u r and u a are respectively when the ball bearing bears the radial load F r and the axial load F a at the same time
  • the ball The radial deformation and axial deformation of the bearing
  • K n is the nth iteration
  • S G is the distance between the center of curvature of the inner and outer ring grooves of the angular contact ball bearing
  • ⁇ is the initial contact angle of the ball bearing
  • M r is the applied torque on the ball bearing
  • ⁇ j is the position angle of the heavy-duty ball bearing
  • j is the jth ball in the bearing.
  • Step 3 Under the same working conditions as Step 2, considering the influence of the centrifugal force and gyro torque at high speed, based on the pseudo-static analysis theory, a high-speed and heavy-duty ball bearing model of the liquid rocket engine is established;
  • the high-speed heavy-duty ball bearing model of the liquid rocket engine satisfies the following expression:
  • F rx is the force projected to the X direction by the radial load applied to the ball bearing
  • F ry is the force projected to the Y direction by the radial load applied to the ball bearing
  • F a is the axial load of the ball bearing
  • Z is the number of balls
  • ⁇ j is the position angle of the ball bearing and j is the jth ball in the bearing
  • ⁇ ij is the contact angle between the jth ball and the inner ring
  • M rx is the external torque in the X direction on the ball bearing
  • M ry is The torque in the Y direction applied to the ball bearing
  • M gj is the gyro torque of the jth ball
  • ⁇ j is the distribution coefficient of the frictional moment of the jth ball
  • Q ij is the contact force between the jth ball and the inner ring
  • d m is the pitch circle diameter of the ball bearing
  • f i is the coefficient of curvature radius of the bearing inner ring groove
  • Step 4 Taking the calculated value of the static contact mechanical properties of the low-speed heavy-duty ball bearing as the initial value, it is substituted into the model of the high-speed heavy-duty ball bearing, and iterated through the co-evolutionary particle swarm optimization algorithm of the double population to calculate the contact dynamic characteristics of each ball;
  • the above contact dynamic characteristics include the contact angle between each ball and the inner ring, and the contact angle between each ball and the outer ring;
  • the co-evolutionary algorithm uses multiple populations to search in parallel. In each iteration process, different evolutionary mechanisms are used, which is not only conducive to the global search, but also helps to quickly converge to the optimal value in the later stage of the search;
  • Step 4.1 Set populations s 1 and s 2, including population size, particle dimension, maximum number of iterations, acceleration constants c 1, c 2, inertia weights w 1 and w 2;
  • Step 4.2 Randomly initialize the initial position and initial velocity of each particle in the population, the individual optimal value pbest id(k) and the global optimal value gbest id(k);
  • Step 4.3 Calculate the fitness value of each particle in the dual population, and update the position and velocity of each particle synchronously, that is, update the parameter space of each population;
  • k is the current number of iterations
  • Y 2j S G cos ⁇ + ⁇ r cos ⁇ j ;
  • X 1j , Y 1j , X 2j , Y 2j are the geometric quantities of the center of curvature of the ball bearing before and after the load;
  • ⁇ a is the axial deformation of the ball bearing under the action of the axial force F a ;
  • ⁇ r is The radial deformation of the ball bearing under the action of radial force F r ;
  • ⁇ is the rotation angle of the ball bearing under the action of moment M;
  • f i is the curvature radius coefficient of the inner ring groove of the ball bearing;
  • f 0 is the The curvature radius coefficient of the ball bearing outer ring groove;
  • ⁇ ij and ⁇ oj are the contact deformation between the jth ball and the inner and outer rings of the ball bearing respectively;
  • S G is the distance between the center of curvature of the inner and outer ring grooves of the ball bearing Distance;
  • Expr2, Expr3, Expr4, Expr5 satisfy the following
  • v id (k+1) w ⁇ v id (k)+c 1 ⁇ r 1 ⁇ [pbest id (k)-x id (k)]+c 2 ⁇ r 2 ⁇ [gbest id (k)-x id (k)];
  • x id (k+1) x id (k) + v id (k+1);
  • D is the number of unknowns in the fitness equation
  • v id (k), v id (k +1) are the current speed and update speed of the i-th and i+1 generation particle swarms
  • x id (k), x id (k+1) are the current position and update speed of the i-th and i+1 generation particle swarms respectively position
  • pbest id (k) is the individual extremum searched by the particle
  • gbest id (k) is the global extremum searched by the population
  • w is the inertia weight, which indicates the influence of the particle on the current velocity, with balanced global convergence and local convergence ability
  • c 1 and c 2 are acceleration constants, which represent the acceleration weights that push particles to the individual extremum and global extremum positions
  • r 1 and r 2 are random numbers in [0,1].
  • Step 4.4 Update the individual extremum and global extremum of the dual population, compare each particle in the dual population with the current individual extremum according to the fitness value, and select the optimal individual extremum pbest id (k) and fitness The global optimal value gbest id (k) with the smallest value;
  • Step 4.5 Dual-population dynamic cooperation strategy, a cooperative relationship occurs through the neighborhood model, and the individual extremum and global extremum found by each are shared;
  • Step 4.6 Determine the termination condition of the algorithm. If the current number of iterations reaches the maximum number of iterations, the loop is terminated, and it is considered that the optimization of the contact angle between each ball and the inner ring is completed, otherwise return to step 4.3 to continue the iteration;
  • Step 4.7 Through the contact angle between each ball and the inner ring optimized in step 4.6, according to the mathematical relationship between the contact angle of the inner ring and the outer ring, the contact angle between each ball and the outer ring can be obtained, thus completing the high-speed Simulation of the contact dynamics of a heavily loaded ball bearing.
  • the angular contact ball bearing B218 was selected.
  • the materials of the inner and outer rings and the balls are all GCr15 steel, and the main parameters of the structure are shown in Table 1. shown.
  • the nonlinear equations of the high-speed heavy-duty ball bearing model can be calculated by using the method simulation method of the present invention, and the inner and outer contact angles of the angular contact ball bearing B218 can be obtained, which is consistent with the test
  • the comparison of values is shown in the table. It can be seen from Table 2 that the maximum relative error between the simulation value of the contact angle of the inner and outer rings of the angular contact ball bearing B218 and the test value is 5.680%. Analysis requirements for bearing dynamic characteristics.

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Abstract

本申请公开了一种液体火箭发动机高速重载球轴承接触动态特性仿真方法，其主要实现步骤是：根据所述球轴承的接触法向应力，建立液体火箭发动机低速重载球轴承模型，通过对所述液体火箭发动机低速重载球轴承模型进行迭代计算，获得重载球轴承静态接触力学特性；基于拟静力学分析理论，建立液体火箭发动机高速重载球轴承模型；以所述重载球轴承静态接触力学特性的计算值作为初值，代入所述高速重载球轴承模型中，通过双种群的协同进化粒子群算法进行迭代，得到所述球轴承中每个滚珠的接触动态特性。通过该方法可获得液体火箭发动机球轴承从静态到动态、单载荷到联合载荷以及低转速到极高转速下的接触动力学特性参数以及变化规律。

Description

一种液体火箭发动机高速重载球轴承接触动态特性仿真方法
本申请要求于2021年09月07日提交中国专利局、申请号为202111041792.3、发明名称为“一种液体火箭发动机高速重载球轴承接触动态特性仿真方法”的中国专利优先权，其全部内容通过引用结合在本申请中。
技术领域
本发明属于液体火箭发动机球轴承动力学特性领域，主要涉及一种液体火箭发动机高速重载球轴承接触动态特性仿真方法。
背景技术
轴承是传递运动和承受载荷的重要支承转动部件，广泛应用于航空宇航推进系统、船舶工程以及精密仪器等重大装备技术领域。在液体火箭发动机涡轮泵转子系统中主要采用球轴承，是因为其具有制造精度高、极限转速高、承载能力强且能同时承受径向载荷和轴向载荷等特点。
球轴承是由滚珠、内圈、外圈和保持架等结构要素构成的，其动态特性包括滚珠与内、外圈的动态接触角、接触载荷、接触变形和接触刚度以及旋滚比等，但最为关注就是滚珠与内、外圈的动态接触角。
传统设计方法通常将球轴承视为标准零件，这对于一般的运动可以满足要求，但是对于现代高速度、高精度和高承载的重大装备系统，球轴承运行工况的复杂性及服役性能的高要求远不同于一般轴承，其动态特性是影响装备系统工作性能和产品质量的关键指标，如随转速变化的球轴承接触刚度。
随着航天液体推进技术的进步，液体火箭发动机日益朝着大推力、高转速、高低温环境以及高可靠性等方向发展，球轴承作为液体火箭发动机涡轮泵转子系统的关键组件，其工作环境变得非常苛刻。一般认为DN(轴承内径(mm)×转速(r/min))值超过0.6×10 6mm·r/min为高速球轴承，而在液体火箭发动机中，球轴承的DN值达到了2.5×10 6mm·r/min，且径向当量动载荷Pr与额定动载荷Cr的比值往往大于0.15，因此可视为高速重载球轴承。在高速重载球轴承中，由于滚珠打滑、疲劳、磨损等引起的轴承失效经常发生，直接导致转子系统精度降低，振动急剧加大，严重时会导致球轴承表面金属脱落，进入 液体火箭发动机泵腔中，而发动机涡轮泵处于高压(大于10MPa)、低温液氧环境，极易导致发动机起火爆炸。但是由于球轴承动力学模型的非线性方程组众多，且对迭代初值敏感，求解不易收敛。因此，目前急需一种能够获取高速重载球轴承动态参数方法。
发明内容
有鉴于此，本发明的目的在于提供一种液体火箭发动机高速重载球轴承接触动态特性仿真方法，通过该方法可获得液体火箭发动机球轴承从静态到动态、单载荷到联合载荷以及低转速到极高转速下的接触动力学特性参数以及变化规律。
本发明的基本思路是：
提供一种液体火箭发动机高速重载球轴承接触动态特性仿真方法，包括以下步骤：
根据所述球轴承的接触法向应力，建立液体火箭发动机低速重载球轴承模型，通过对所述液体火箭发动机低速重载球轴承模型进行迭代计算，获得重载球轴承静态接触力学特性；
基于拟静力学分析理论，建立液体火箭发动机高速重载球轴承模型；
以所述重载球轴承静态接触力学特性的计算值作为初值，代入所述高速重载球轴承模型中，通过双种群的协同进化粒子群算法进行迭代，得到所述球轴承中每个滚珠的接触动态特性；其中，所述接触动态特性包括每个滚珠与内圈的接触角，以及每个滚珠与外圈的接触角。
进一步的，在根据所述球轴承的接触法向应力，建立液体火箭发动机低速重载球轴承模型之前，所述液体火箭发动机高速重载球轴承接触动态特性仿真方法还包括：
采用基于半无限长空间Hertz弹性理论，建立法向接触问题方程；
利用线性回归的最小二乘法对所述法向接触问题方程进行简化运算，得到所述重载球轴承接的触法向应力。
进一步的，利用线性回归的最小二乘法对所述法向接触问题方程进行简化运算后的法向接触问题方程如下式：
式中：
Figure PCTCN2022081259-appb-000001
其中：A和B均表示接触面质点，a为接触椭圆的长半轴，b为接触椭圆的短半轴，e为接触椭圆参数，N(e)为第一类完全椭圆积分，M(e)为第二类完全椭圆积分，E 1为滚珠弹性模量，E 2为滚道的弹性模量，υ 1为滚珠的泊松比，υ 2为滚道的泊松比，p 0为接触椭圆中心处的最大压应力；所述接触椭圆为每个滚珠与滚道接触时产生的椭圆；其中：R x和R y分别表示所述接触椭圆的长半轴和短半轴方向上的等效半径。
进一步的，所述低速重载球轴承静态接触力学特性包括：所述重载球轴承的无量纲径向变形、所述重载球轴承的无量纲轴向变形、所述重载球轴承的接触角以及所述重载球轴承的接触等效刚度。
进一步的，所述根据所述球轴承的接触法向应力，建立液体火箭发动机低速重载球轴承模型包括：
在不考虑所述球轴承在高速转动下的离心力和陀螺力矩的影响下，根据所述球轴承的接触法向应力，建立液体火箭发动机低速重载球轴承模型；
所述基于拟静力学分析理论，建立液体火箭发动机高速重载球轴承模型包括：
在考虑所述球轴承在高速转动下的离心力和陀螺力矩的影响的情况下，基于拟静力学分析理论，建立液体火箭发动机高速重载球轴承模型。
进一步的，所述液体火箭发动机高速重载球轴承模型的表达式为：
Figure PCTCN2022081259-appb-000002
Figure PCTCN2022081259-appb-000003
Figure PCTCN2022081259-appb-000004
Figure PCTCN2022081259-appb-000005
Figure PCTCN2022081259-appb-000006
其中，F rx为球轴承外加的径向载荷投影到X方向的力、F ry为球轴承外加的径向载荷投影到Y方向的力，F a为球轴承的轴向载荷，Z为滚珠个数，ψ j为球轴承的位置角j为轴承内第j个滚珠；α ij为第j个滚珠与内圈的接触角，M rx为球轴承所受外加的X方向的力矩、M ry为球轴承所受外加的Y方向的力矩，M gj为第j个滚珠的陀螺力矩，λ j为第j个滚珠摩擦力矩的分配系数；Q ij为第j个滚珠与内圈的接触力；
Figure PCTCN2022081259-appb-000007
d m为球轴承的节圆直径；f i为轴承内圈沟曲率半径系数；D w为滚珠直径；r i为内圈沟道曲率半径。
进一步的，所述液体火箭发动机低速重载球轴承模型的表达式为：
Figure PCTCN2022081259-appb-000008
Figure PCTCN2022081259-appb-000009
Figure PCTCN2022081259-appb-000010
其中：F r为球轴承的径向载荷，F a为为球轴承的轴向载荷，u r和u a分别为球轴承同时承受径向载荷F r和轴向载荷F a时，所述球轴承产生的径向变形量和轴向变形量，K n为第n次迭代，S G为角接触球轴承内、外圈沟曲率中心之间的距离，α为球轴承的初始接触角，Mr为球轴承所受的外加力矩，ψj为重载球轴承的位置角j为轴承内第j个滚珠。
进一步的，采用所述双种群的协同进化粒子群算法进行迭代运算包括以下步骤：
步骤1，设置种群s1和种群s2的参数，其中，所述参数包括种群的规模、粒子的维数、最大迭代次数，以及加速常数c1、c2，惯性权值w1和w2，所述种群s1和所述种群s2形成了所述双种群；
步骤2，随机初始化所述种群s1和所述种群s2中每个粒子的初始位置和初始速度，个体最优值pbest id(k)和全局最优值gbest id(k)；
步骤3，计算所述双种群中每个粒子的适应度值，并分别同步更新所述每个粒子的位置和速度，即更新所述种群s1和所述种群s的参数空间；所述种群s1和所述种群s的惯性权值适应度值Fitness的计算公式为：
Figure PCTCN2022081259-appb-000011
其中：k为当前的迭代次数；
Expr1满足以下公式：
Expr1＝(X 2j-X 1j) 2+(Y 2j-Y 1j) 2-[(f i-0.5)D w+δ ij] 2；
式中：
Figure PCTCN2022081259-appb-000012
Y 2j＝S Gcosα+δ rcosψ j；
X1j、Y1j根据以下两个公式求解：
(X 2j-X 1j) 2+(Y 2j-Y 1j) 2-[(f i-0.5)D w+δ ij] 2＝0；
Figure PCTCN2022081259-appb-000013
X1j，Y1j，X2j，Y2j为所述球轴承的曲率中心承载前后位置变化的几何量；δ a为所述球轴承在轴向力Fa作用下的轴向变形；δ r为所述球轴承在径向力Fr作用下的径向变形；θ为所述球轴承在力矩M作用下的转角；f i为所述球轴承内圈沟曲率半径系数；f 0为所述球轴承外圈沟曲率半径系数；δ ij、δ oj分别为第j个滚珠与所述球轴承内、外圈的接触变形；S G为所述球轴承内、外圈沟曲率中心之间的距离；Expr2、Expr3、Expr4、Expr5满足以下公式:
Figure PCTCN2022081259-appb-000014
Figure PCTCN2022081259-appb-000015
Figure PCTCN2022081259-appb-000016
Figure PCTCN2022081259-appb-000017
所述粒子的速度和位置更新公式为
v id(k+1)＝w·v id(k)+c 1·r 1·[pbest id(k)-x id(k)]+c 2·r 2·[gbest id(k)-x id(k)]；
x id(k+1)＝x id(k)+v id(k+1)；
其中，i＝1,2,…,m为更新的代数；d＝1,2,…,D为搜索空间的维度，D为适应度方程未知数个数；vid(k)、vid(k+1)分别为第i、i+1代粒子群的当前速度和更新速度；xid(k)、xid(k+1)分别为第i、i+1代粒子群的当前位置和更新位置；pbest id(k)为粒子搜寻到的个体极值；gbest id(k)为种群搜寻到全局极值；w为惯性权值，表示粒子对当前速度的影响，具有平衡全局收敛和局部收敛的能力；c 1和c 2为加速常数，表示将粒子推向个体极值和全局极值位置的加速权值；r 1和r 2为[0,1]的随机数；
步骤4，进行所述双种群的个体极值和全局极值的更新，根据适应度值将所述种群s1和所述种群s2中每个粒子与当前个体极值比较，选择最优的个体极值pbest id(k)以及适应度值最小的全局最优值gbest id(k)；
步骤5，所述双种群动态合作策略，通过邻域模型发生合作关系，共享双种群中各个种群搜索到的个体极值和全局极值；
步骤6，判断算法终止条件，如果当前的迭代次数达到了最大迭代次数，则循环终止，则认为完成了所述球轴承中每个滚珠与内圈的接触角的优化，否则返回步骤3继续迭代；
步骤7，通过步骤6优化后的每个所述滚珠与内圈的接触角，根据所述内圈接触角和外圈接触角的数学关系，即可得到每个所述滚珠与外圈的接触角，从而完成所述高速重载球轴承接触动态特性的仿真。
进一步的，对所述液体火箭发动机低速重载球轴承模型进行迭代计算时采用的迭代法为Newton-Raphson迭代法。
进一步的，所述球轴承的DN值≥2.5×10 6mm·r/min；所述球轴承的径 向当量动载荷Pr与所述球轴承的额定动载荷Cr的比值＞0.15。
本发明与现有技术相比的有益效果是：
(1)本发明在球轴承接触法向应力计算时，当接触椭圆参数e接近1时，椭圆积分的收敛非常慢，借助线性回归的最小二乘法，可以获得关于e、N(e)、M(e)的近似方程，能大幅简化超越方程的计算。
(2)由于模型涉及的非线性方程个数较多，传统的数值求解方法较难收敛到准确解。为解决非线性、不可微和多峰值等复杂的优化问题，首次将双种群协同进化粒子群算法引入到球轴承的分析模型中，采用多个种群并行搜索解空间，有效地解决了传统方法由于缺乏种群多样而容易陷入局部极值的问题，同时由于可以共享种群之间搜寻到的最优值，有效地提高了种群的搜索速度。
附图说明
为了更清楚地说明本发明实施例或现有技术中的技术方案，下面将对实施例或现有技术描述中所需要使用的附图进行说明：
图1为本发明方法的实现流程图；
图2为内圈接触角随轴向载荷变化曲线图；
图3为外圈接触角随轴向载荷变化曲线图。
具体实施方式
本发明的核心是提供了一种液体火箭发动机高速重载球轴承接触动态特性仿真方法，该仿真过程在matlab上执行，具体包括以下步骤：
步骤1：采用基于半无限长空间Hertz弹性理论，建立法向接触问题的方程，再结合线性回归的最小二乘法进行简化运算，求出球轴承接触法向应力；
其中，简化后的法向接触问题方程如下式：
式中：
Figure PCTCN2022081259-appb-000018
Figure PCTCN2022081259-appb-000019
其中：A和B均表示接触面质点，a为接触椭圆的长半轴，b为接触椭圆的短半轴，e为接触椭圆参数，N(e)为第一类完全椭圆积分，M(e)为第二类完全椭圆积分，E 1为滚珠弹性模量，E 2为滚道的弹性模量，υ 1为滚珠的泊松比，υ 2为滚道的泊松比，p 0为接触椭圆中心处的最大压应力；所述接触椭圆为每个滚珠与滚道接触时产生的椭圆；其中：R x和R y分别表示所述接触椭圆的长半轴和短半轴方向上的等效半径。
步骤2：不考虑高速下离心力和陀螺力矩的影响，以接触法向应力为基础，建立液体火箭发动机低速重载球轴承模型，本实施例中采用Newton-Raphson迭代法，计算获得球轴承的静态接触力学特性；所述球轴承的静态接触力学特性包括球轴承无量纲径向变形、球轴承无量纲轴向变形、球轴承的接触角α以及球轴承接触等效刚度Kn；
所述液体火箭发动机低速重载球轴承模型的表达式为：
Figure PCTCN2022081259-appb-000020
Figure PCTCN2022081259-appb-000021
Figure PCTCN2022081259-appb-000022
其中：F r为球轴承的径向载荷，F a为为球轴承的轴向载荷，u r和u a分别为球轴承同时承受径向载荷F r和轴向载荷F a时，所述球轴承产生的径向变形量和轴向变形量，K n为第n次迭代，S G为角接触球轴承内、外圈沟曲率中心之间的距离，α为球轴承的初始接触角，M r为球轴承所受的外加力矩，ψ j为重载球轴承的位置角j为轴承内第j个滚珠。
步骤3：在与步骤2同样的工况下，考虑在高转速下所承受的离心力和陀螺力矩的影响，基于拟静力学分析理论，建立液体火箭发动机高速重载球轴承模型；
其中，液体火箭发动机高速重载球轴承模型的满足以下表达式：
Figure PCTCN2022081259-appb-000023
Figure PCTCN2022081259-appb-000024
Figure PCTCN2022081259-appb-000025
Figure PCTCN2022081259-appb-000026
Figure PCTCN2022081259-appb-000027
其中，F rx为球轴承外加的径向载荷投影到X方向的力、F ry为球轴承外加的径向载荷投影到Y方向的力，F a为球轴承的轴向载荷，Z为滚珠个数，ψ j 为球轴承的位置角j为轴承内第j个滚珠；α ij为第j个滚珠与内圈的接触角，M rx为球轴承所受外加的X方向的力矩、M ry为球轴承所受外加的Y方向的力矩，M gj为第j个滚珠的陀螺力矩，λ j为第j个滚珠摩擦力矩的分配系数；Q ij为第j个滚珠与内圈的接触力；
Figure PCTCN2022081259-appb-000028
d m为球轴承的节圆直径；f i为轴承内圈沟曲率半径系数；D w为滚珠直径；r i为内圈沟道曲率半径。
步骤4：以低速重载球轴承静态接触力学特性计算值作为初值，代入高速重载球轴承模型，通过双种群的协同进化粒子群算法进行迭代，计算出每个滚珠的接触动态特性；所述接触动态特性包括每个滚珠与内圈的接触角，以及每个滚珠与外圈的接触角；
协同进化算法是采用多个种群进行并行搜索，在每次迭代过程中，采用不同的进化机制，既有利于对全局的搜索，也有助于在搜索后期对最优值的快速收敛；
双种群的协同进化粒子群进行迭代运算的具体步骤如下：
步骤4.1：设置种群s 1和s 2，包括种群的规模、粒子的维数、最大迭代次数，以及加速常数c 1、c 2，惯性权值w 1和w 2；
步骤4.2：随机初始化种群中每个粒子的初始位置和初始速度，个体最优值pbest id(k)和全局最优值gbest id(k)；
步骤4.3：计算双种群中每个粒子的适应度值，并分别同步更新每个粒子的位置和速度，即更新各个种群的参数空间；
两个种群的惯性权值适应度值Fitness的计算公式为；
Figure PCTCN2022081259-appb-000029
其中：k为当前的迭代次数；
Expr1满足以下公式：
Expr1＝(X 2j-X 1j) 2+(Y 2j-Y 1j) 2-[(f i-0.5)D w+δ ij] 2；
式中：
Figure PCTCN2022081259-appb-000030
Y 2j＝S Gcosα+δ rcosψ j；
X1j、Y1j根据以下两个公式求解：
(X 2j-X 1j) 2+(Y 2j-Y 1j) 2-[(f i-0.5)D w+δ ij] 2＝0；
Figure PCTCN2022081259-appb-000031
X 1j，Y 1j，X 2j，Y 2j为所述球轴承的曲率中心承载前后位置变化的几何量；δ a为所述球轴承在轴向力F a作用下的轴向变形；δ r为所述球轴承在径向力F r作用下的径向变形；θ为所述球轴承在力矩M作用下的转角；f i为所述球轴承内圈沟曲率半径系数；f 0为所述球轴承外圈沟曲率半径系数；δ ij、δ oj分别为第j个滚珠与所述球轴承内、外圈的接触变形；S G为所述球轴承内、外圈沟曲率中心之间的距离；Expr2、Expr3、Expr4、Expr5满足以下公式:
Figure PCTCN2022081259-appb-000032
Figure PCTCN2022081259-appb-000033
Figure PCTCN2022081259-appb-000034
Figure PCTCN2022081259-appb-000035
所述粒子的速度和位置更新公式为
v id(k+1)＝w·v id(k)+c 1·r 1·[pbest id(k)-x id(k)]+c 2·r 2·[gbest id(k)-x id(k)]；
x id(k+1)＝x id(k)+v id(k+1)；
其中，i＝1,2,…,m为更新的代数；d＝1,2,…,D为搜索空间的维度，D为适应度方程未知数个数；v id(k)、v id(k+1)分别为第i、i+1代粒子群的当前速度和更新速度；x id(k)、x id(k+1)分别为第i、i+1代粒子群的当前位置和更新位置；pbest id(k)为粒子搜寻到的个体极值；gbest id(k)为种群搜寻到全局极值；w为惯性权值，表示粒子对当前速度的影响，具有平衡全局收敛和局部收敛的能力； c 1和c 2为加速常数，表示将粒子推向个体极值和全局极值位置的加速权值；r 1和r 2为[0,1]的随机数。
步骤4.4：进行双种群的个体极值和全局极值的更新，根据适应度值将双种群中每个粒子与当前个体极值比较，选择最优的个体极值pbest id(k)以及适应度值最小的全局最优值gbest id(k)；
步骤4.5：双种群动态合作策略，通过邻域模型发生合作关系，共享各自搜索到的个体极值和全局极值；
步骤4.6：判断算法终止条件，如果当前的迭代次数达到了最大迭代次数，则循环终止，则认为完成了每个滚珠与内圈的接触角的优化，否则返回步骤4.3继续迭代；
步骤4.7：通过步骤4.6优化后的每个滚珠与内圈的接触角，根据内圈接触角和外圈接触角的数学关系，即可得到每个滚珠与外圈的接触角，从而完成了高速重载球轴承接触动态特性的仿真。
对比验证：
为了验证本发明的方法对高速重载球轴承动态特性仿真的可行性与准确性，选择了角接触球轴承B218，其内、外圈与滚珠的材料都是GCr15钢，结构主要参数如表1所示。
表1角接触球轴承B218的结构参数
参数 单位 数值
轴承内径D i mm 90.00
轴承外径D 0 mm 160.00
内圈滚道直径d i mm 102.80
外圈滚道直径d 0 mm 147.40
内圈沟道曲率半径r i mm 11.63
外圈沟道曲率半径r 0 mm 11.63
滚珠直径D w mm 22.23
初始接触角 度 40
滚珠数目Z 个 16
当轴向载荷取17.8kN时，转速取不同数值时，利用本发明的方法仿真方法计算高速重载球轴承模型的非线性方程组，可获得角接触球轴承B218的内、外接触角，与试验值的对比如表所示。从表2中可以看出，角接触球轴承B218的内、外圈接触角仿真值与试验值之间的最大相对误差为5.680％，因此可知该仿真方法精度较高，可满足高速重载球轴承动态特性的分析要求。
表2轴向力17.8kN时球轴承B218接触角的计算值与文献值对比
Figure PCTCN2022081259-appb-000036
当轴承转速n＝15000r/min，轴向载荷取不同值时，角接触球轴承B218内、外接触角与计算值的变化曲线如图2和图3所示。从图中可以得到，利用本发明的方法，球轴承B218内、外圈接触角的仿真值与试验值的趋势十分吻合，仿真相对误差最大为5.128％，且随着轴向力的增大，内、外圈接触角的仿真相对误差减小，可达到2.376％，验证了算法的寻优能力能够降低仿真误差。

Claims (10)

  1. 一种液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于，包括以下步骤：
    根据所述球轴承的接触法向应力，建立液体火箭发动机低速重载球轴承模型，通过对所述液体火箭发动机低速重载球轴承模型进行迭代计算，获得重载球轴承静态接触力学特性；
    基于拟静力学分析理论，建立液体火箭发动机高速重载球轴承模型；
    以所述重载球轴承静态接触力学特性的计算值作为初值，代入所述高速重载球轴承模型中，通过双种群的协同进化粒子群算法进行迭代，得到所述球轴承中每个滚珠的接触动态特性；其中，所述接触动态特性包括每个滚珠与内圈的接触角，以及每个滚珠与外圈的接触角。
  2. 根据权利要求1所述的液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于，在根据所述球轴承的接触法向应力，建立液体火箭发动机低速重载球轴承模型之前，所述液体火箭发动机高速重载球轴承接触动态特性仿真方法还包括：
    采用基于半无限长空间Hertz弹性理论，建立法向接触问题方程；
    利用线性回归的最小二乘法对所述法向接触问题方程进行简化运算，得到所述重载球轴承接的触法向应力。
  3. 根据权利要求2所述的液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于，
    利用线性回归的最小二乘法对所述法向接触问题方程进行简化运算后的法向接触问题方程如下式：
    Figure PCTCN2022081259-appb-100001
    式中：
    Figure PCTCN2022081259-appb-100002
    Figure PCTCN2022081259-appb-100003
    e≈1.0339(R x/R y) 0.636
    N(e)≈1.5277+0.60231n(R y/R x)
    M(e)≈1.0003+0.5968R x/R y
    其中：A和B均表示接触面质点，a为接触椭圆的长半轴，b为接触椭圆的短半轴，e为接触椭圆参数，N(e)为第一类完全椭圆积分，M(e)为第二类完全椭圆积分，E 1为滚珠弹性模量，E 2为滚道的弹性模量，υ 1为滚珠的泊松比，υ 2为滚道的泊松比，p 0为接触椭圆中心处的最大压应力；所述接触椭圆为每个滚珠与滚道接触时产生的椭圆；其中：R x和R y分别表示所述接触椭圆的长半轴和短半轴方向上的等效半径。
  4. 根据权利要求1所述的液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于，所述低速重载球轴承静态接触力学特性包括：所述重载球轴承的无量纲径向变形、所述重载球轴承的无量纲轴向变形、所述重载球轴承的接触角以及所述重载球轴承的接触等效刚度。
  5. 根据权利要求1所述的液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于，所述根据所述球轴承的接触法向应力，建立液体火箭发动机低速重载球轴承模型包括：
    在不考虑所述球轴承在高速转动下的离心力和陀螺力矩的影响下，根据所述球轴承的接触法向应力，建立液体火箭发动机低速重载球轴承模型；
    所述基于拟静力学分析理论，建立液体火箭发动机高速重载球轴承模型包括：
    在考虑所述球轴承在高速转动下的离心力和陀螺力矩的影响的情况下，基于拟静力学分析理论，建立液体火箭发动机高速重载球轴承模型。
  6. 根据权利要求1-5任一项所述的液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于，
    所述液体火箭发动机高速重载球轴承模型的表达式为：
    Figure PCTCN2022081259-appb-100004
    Figure PCTCN2022081259-appb-100005
    Figure PCTCN2022081259-appb-100006
    Figure PCTCN2022081259-appb-100007
    Figure PCTCN2022081259-appb-100008
    其中，F rx为球轴承外加的径向载荷投影到X方向的力、F ry为球轴承外加的径向载荷投影到Y方向的力，F a为球轴承的轴向载荷，Z为滚珠个数，ψ j为球轴承的位置角j为轴承内第j个滚珠；α ij为第j个滚珠与内圈的接触角，M rx为球轴承所受外加的X方向的力矩、M ry为球轴承所受外加的Y方向的力矩，M gj为第j个滚珠的陀螺力矩，λ j为第j个滚珠摩擦力矩的分配系数；Q ij为第j个滚珠与内圈的接触力；
    Figure PCTCN2022081259-appb-100009
    d m为球轴承的节圆直径；f i为轴承内圈沟曲率半径系数；D w为滚珠直径；r i为内圈沟道曲率半径。
  7. 根据权利要求6所述的液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于，
    所述液体火箭发动机低速重载球轴承模型的表达式为：
    Figure PCTCN2022081259-appb-100010
    Figure PCTCN2022081259-appb-100011
    Figure PCTCN2022081259-appb-100012
    其中：F r为球轴承的径向载荷，F a为为球轴承的轴向载荷，u r和u a分别为球轴承同时承受径向载荷F r和轴向载荷F a时，所述球轴承产生的径向变形量和 轴向变形量，K n为第n次迭代，S G为角接触球轴承内、外圈沟曲率中心之间的距离，α为球轴承的初始接触角，Mr为球轴承所受的外加力矩，ψj为重载球轴承的位置角j为轴承内第j个滚珠。
  8. 根据权利要求1-5任一项所述的液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于，采用所述双种群的协同进化粒子群算法进行迭代运算包括以下步骤：
    步骤1，设置种群s1和种群s2的参数，其中，所述参数包括种群的规模、粒子的维数、最大迭代次数，以及加速常数c1、c2，惯性权值w1和w2，所述种群s1和所述种群s2形成了所述双种群；
    步骤2，随机初始化所述种群s1和所述种群s2中每个粒子的初始位置和初始速度，个体最优值pbest id(k)和全局最优值gbest id(k)；
    步骤3，计算所述双种群中每个粒子的适应度值，并分别同步更新所述每个粒子的位置和速度，即更新所述种群s1和所述种群s的参数空间；所述种群s1和所述种群s的惯性权值适应度值Fitness的计算公式为：
    Figure PCTCN2022081259-appb-100013
    其中：k为当前的迭代次数；
    Expr1满足以下公式：
    Expr1＝(X 2j-X 1j) 2+(Y 2j-Y 1j) 2-[(f i-0.5)D w+δ ij] 2；
    式中：
    Figure PCTCN2022081259-appb-100014
    Y 2j＝S Gcosα+δ rcosψ j；
    X1j、Y1j根据以下两个公式求解：
    (X 2j-X 1j) 2+(Y 2j-Y 1j) 2-[(f i-0.5)D w+δ ij] 2＝0；
    Figure PCTCN2022081259-appb-100015
    X 1j，Y 1j，X 2j，Y 2j为所述球轴承的曲率中心承载前后位置变化的几何量；δ a为所述球轴承在轴向力F a作用下的轴向变形；δ r为所述球轴承在径向力F r作用下的径向变形；θ为所述球轴承在力矩M作用下的转角；f i为所述球轴承内圈沟曲率半径系数；f 0为所述球轴承外圈沟曲率半径系数；δ ij、δ oj分别为第j个滚珠与所述球轴承内、外圈的接触变形；S G为所述球轴承内、外圈沟曲率中心之间的距离；Expr2、Expr3、Expr4、Expr5满足以下公式:
    Figure PCTCN2022081259-appb-100016
    Figure PCTCN2022081259-appb-100017
    Figure PCTCN2022081259-appb-100018
    Figure PCTCN2022081259-appb-100019
    所述粒子的速度和位置更新公式为
    v id(k+1)＝w·v id(k)+c 1·r 1·[pbest id(k)-x id(k)]+c 2·r 2·[gbest id(k)-x id(k)]；
    x id(k+1)＝x id(k)+v id(k+1)；
    其中，i＝1,2,…,m为更新的代数；d＝1,2,…,D为搜索空间的维度，D为适应度方程未知数个数；v id(k)、v id(k+1)分别为第i、i+1代粒子群的当前速度和更新速度；x id(k)、x id(k+1)分别为第i、i+1代粒子群的当前位置和更新位置；pbest id(k)为粒子搜寻到的个体极值；gbest id(k)为种群搜寻到全局极值；w为惯 性权值，表示粒子对当前速度的影响，具有平衡全局收敛和局部收敛的能力；c 1和c 2为加速常数，表示将粒子推向个体极值和全局极值位置的加速权值；r 1和r 2为[0,1]的随机数；
    步骤4，进行所述双种群的个体极值和全局极值的更新，根据适应度值将所述种群s1和所述种群s2中每个粒子与当前个体极值比较，选择最优的个体极值pbest id(k)以及适应度值最小的全局最优值gbest id(k)；
    步骤5，所述双种群动态合作策略，通过邻域模型发生合作关系，共享双种群中各个种群搜索到的个体极值和全局极值；
    步骤6，判断算法终止条件，如果当前的迭代次数达到了最大迭代次数，则循环终止，则认为完成了所述球轴承中每个滚珠与内圈的接触角的优化，否则返回步骤3继续迭代；
    步骤7，通过步骤6优化后的每个所述滚珠与内圈的接触角，根据所述内圈接触角和外圈接触角的数学关系，即可得到每个所述滚珠与外圈的接触角，从而完成所述高速重载球轴承接触动态特性的仿真。
  9. 根据权利要求1-5任一项所述的液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于：对所述液体火箭发动机低速重载球轴承模型进行迭代计算时采用的迭代法为Newton-Raphson迭代法。
  10. 根据权利要求1-5任一项所述的液体火箭发动机高速重载球轴承接触动态特性仿真方法，其特征在于：
    所述球轴承的DN值≥2.5×10 6mm·r/min；所述球轴承的径向当量动载荷Pr与所述球轴承的额定动载荷Cr的比值＞0.15。
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Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115994476A (zh) * 2023-03-23 2023-04-21 中国人民解放军63921部队 新型运载火箭发射零秒脱落的仿真验证方法、装置及介质
CN117540579A (zh) * 2023-12-14 2024-02-09 哈尔滨工程大学 一种涉及摩擦热及粗糙度的燃机球轴承动力学分析方法
CN117892559A (zh) * 2024-03-14 2024-04-16 西安现代控制技术研究所 一种超远程制导火箭总体协调多学科分级优化方法
CN117892558A (zh) * 2024-03-14 2024-04-16 西安现代控制技术研究所 一种超远程制导火箭多学科动态优化模型构建方法

Families Citing this family (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113496091B (zh) * 2021-09-07 2021-12-24 西安航天动力研究所 液体火箭发动机高速重载球轴承接触动态特性仿真方法

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2014157047A (ja) * 2013-02-15 2014-08-28 Ntn Corp 転がり軸受の寿命評価方法および寿命評価装置
CN108228991A (zh) * 2017-12-26 2018-06-29 三门峡职业技术学院 一种求解高速角接触球轴承数值的方法
CN110674577A (zh) * 2019-09-23 2020-01-10 武汉理工大学 新能源汽车高转速轴承动力学参数获取方法
CN113496091A (zh) * 2021-09-07 2021-10-12 西安航天动力研究所 液体火箭发动机高速重载球轴承接触动态特性仿真方法

Family Cites Families (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112861271B (zh) * 2019-11-27 2024-04-16 中车株洲电力机车研究所有限公司 风电机组主轴承力学特性的计算方法、系统及介质
CN111895899B (zh) * 2020-07-21 2022-03-25 刘钙 一种三自由度混合磁轴承转子位移自检测方法
CN111985141B (zh) * 2020-09-18 2024-03-22 河南科技大学 一种转盘轴承表面硬化滚道许用接触应力的确定方法
CN112284575B (zh) * 2020-09-18 2021-12-28 西安交通大学 一种基于电涡流位移传感器的轴承载荷在线监测方法
CN112347854B (zh) * 2020-10-12 2024-03-29 西安电子科技大学 滚动轴承故障诊断方法、系统、存储介质、设备及应用
CN112395713A (zh) * 2020-12-23 2021-02-23 南京铁道职业技术学院 一种基于改进粒子群算法的分动箱齿轮优化设计方法

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2014157047A (ja) * 2013-02-15 2014-08-28 Ntn Corp 転がり軸受の寿命評価方法および寿命評価装置
CN108228991A (zh) * 2017-12-26 2018-06-29 三门峡职业技术学院 一种求解高速角接触球轴承数值的方法
CN110674577A (zh) * 2019-09-23 2020-01-10 武汉理工大学 新能源汽车高转速轴承动力学参数获取方法
CN113496091A (zh) * 2021-09-07 2021-10-12 西安航天动力研究所 液体火箭发动机高速重载球轴承接触动态特性仿真方法

Non-Patent Citations (3)

* Cited by examiner, † Cited by third party
Title
PENG, BO: "Simulation Analysis on Working Characteristics of High-Speed Rolling Bearings under High Pulsating Load", CHINA DOCTORAL DISSERTATIONS/MASTER'S THESES FULL-TEXT DATABASE, ENGINEERING SCIENCE II, no. 03, 1 July 2007 (2007-07-01), CN, pages 1 - 69, XP009544386 *
PENG, BO: "Simulation Analysis on Working Characteristics of High-Speed Rolling Bearings under High Pulsating Load", CHINA DOCTORAL DISSERTATIONS/MASTER'S THESES FULL-TEXT DATABASE, ENGINEERING SCIENCE II, no. 03, 15 March 2009 (2009-03-15), ISSN: 1674-0246 *
WANG LIANBAO |, HU XIAO-QIU; RUI HONG-FENG: "Solution of Dynamic Contact Angles for Angular Contact Ball Bearings Based on PSO Algorithm", BEARING, no. 11, 5 November 2013 (2013-11-05), pages 6 - 9, XP093045801, ISSN: 1000-3762, DOI: 10.19533/j.issn1000-3762.2013.11.002 *

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115994476A (zh) * 2023-03-23 2023-04-21 中国人民解放军63921部队 新型运载火箭发射零秒脱落的仿真验证方法、装置及介质
CN117540579A (zh) * 2023-12-14 2024-02-09 哈尔滨工程大学 一种涉及摩擦热及粗糙度的燃机球轴承动力学分析方法
CN117892559A (zh) * 2024-03-14 2024-04-16 西安现代控制技术研究所 一种超远程制导火箭总体协调多学科分级优化方法
CN117892558A (zh) * 2024-03-14 2024-04-16 西安现代控制技术研究所 一种超远程制导火箭多学科动态优化模型构建方法

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