A Design Method for the Clearance of the Inner Ring Hemispherical Bearing of a Turbopump Based on the Response of the Rotor System

Abstract

The present invention proposes a method for designing the clearance of the inner ring split hemisphere bearings of a turbopump based on the response of the rotor system. First, the radial clearance range corresponding to the axial clearance range is calculated; then, a coupled model of the ball bearing-rotor system is established, and the response of the disk in the vertical direction under different radial clearances is calculated to draw the bifurcation diagram of the system; finally, it is observed whether there is chaotic motion in the bifurcation diagram. If not, the step size is sequentially increased from the minimum value of the radial clearance to the maximum clearance, and the time-domain response of the disk in the horizontal direction under each clearance is calculated; if so, the bearing clearances other than the chaotic motion are selected, and the time-domain response of the disk in the horizontal direction under each clearance is calculated; the clearance values corresponding to the responses not exceeding the limit values are retained to obtain the bearing radial clearance interval for the stable operation of the rotor system. The method proposed by the present invention can obtain the bearing clearance interval for the safe and stable operation of the system according to the rotor response, providing reference and support for the design of the bearing clearance.

Description

Technical Field

[0001] The invention belongs to the technical field related to the design parameters of mechanical structures, and particularly relates to a method for designing the clearance of the inner ring split hemisphere bearing of a turbo pump based on the response of a rotor system. Background Art

[0002] With the rapid development of the aerospace field, the inner ring split hemisphere bearing, as a key basic component for supporting the main shaft of an aeroengine and the rotor of a rocket engine turbo pump, has axial clearance and radial clearance to meet the load-bearing requirements. The special structure of the bearing results in different relationships between its axial clearance and radial clearance compared with ordinary bearings. When the radial clearance of the bearing is too small, the bearing may operate at high temperature and high speed, resulting in problems such as overheating and seizure of the bearing. Excessive radial clearance will cause significant rotor vibration. The axial clearance directly determines the load-bearing capacity and service life of the bearing. When the axial clearance is too large, the rotor will move axially during operation; when the axial clearance is too small, the load-bearing capacity of the bearing will decrease, the service life will be significantly reduced, and even the stability of the entire equipment operation will be lowered. Therefore, it is very important to determine the initial clearance according to the response of the rotor system to ensure the stable and safe operation of the rotor system. At present, the selection of the clearance of the inner ring split hemisphere bearing is more based on the load requirements, and little attention is paid to the system stability for selection. In addition, the special structure of this bearing also makes the relationship between its axial clearance and radial clearance different from that of ordinary bearings.

[0003] In the actual machining process, according to the bearing use, standard bearings are designed with a reference range of axial clearance. For the convenience of measurement, generally, the radial clearance is obtained by measuring the axial clearance. However, the structure of the inner ring split hemisphere bearing dedicated to the rocket engine turbo pump is different from that of ordinary bearings. Therefore, it is necessary to first calculate the radial clearance according to its axial clearance. Selecting the clearance of the bearing according to the response of the rotor is also very important for the bearing life and rotor stability. Summary of the Invention

[0004] To overcome the above-mentioned disadvantages of the prior art, the invention provides a method for designing the clearance of the inner ring split hemisphere bearing of a turbo pump based on the response of a rotor system, aiming to solve the problems of large vibration magnitude and unstable response caused by unreasonable selection of the clearance of the ball bearing-rotor system of the turbo pump.

[0005] To achieve the above object, the technical solution adopted by the invention is:

[0006] A method for designing the clearance of the inner ring split hemisphere bearing of a turbo pump based on the response of a rotor system, comprising the following steps:

[0007] Step 1), determine and input the inner ring groove curvature radius, outer ring groove curvature radius, ball diameter, shim angle and axial clearance reference range of the inner ring split hemispherical bearing. According to the geometric relationship, calculate the corresponding radial clearance range for this axial clearance range, and establish a non-linear force model of the bearing considering the radial clearance;

[0008] Step 2), input the rotor structure parameters, mass, damping and other parameters, establish a coupled model of the ball bearing-rotor system, and write the differential equation of motion;

[0009] Step 3), use the Runge-Kutta method to solve the differential equation of motion, obtain the change of the response of the ball bearing-rotor system under different clearances, and observe whether the results bifurcate or become chaotic;

[0010] Step 4), take the clearance interval where the system does not become chaotic. Starting from the minimum value of the radial clearance, calculate the time-domain response diagram of the disk at the operating speed, and judge whether the peak-to-peak value of the response in the horizontal direction exceeds the limit value. If it does not exceed the limit value, this radial clearance is the radial clearance for the system to operate stably; if it exceeds the limit value, increase the step size on the basis of this radial clearance, calculate the horizontal response again and judge whether it exceeds the limit value until the clearance reaches the maximum value of this radial clearance interval. Finally, obtain the clearance interval that does not exceed the limit value as the range to ensure the stable and safe operation of the system.

[0011] In the said Step 1), the structure of the inner ring split hemispherical bearing is different from that of the ordinary angular contact ball bearing and deep groove ball bearing. To calculate the radial clearance corresponding to the axial clearance, it is necessary to first determine the size of the contact angle α according to the axial clearance:

[0012]

[0013] In the formula, G a is the axial clearance; r i is the inner raceway groove curvature radius; r o is the outer raceway groove curvature radius; D is the ball diameter; α s is the shim angle.

[0014]

[0015] In the formula, w s is the shim width.

[0016] After obtaining the size of the contact angle, the corresponding radial clearance G r can be obtained:

[0017] G r = 2(1 - cosα)(r i + r o - D)-(2r i - D)(1 - cosα s)

[0018] The calculation methods for the non - linear force Q of the bearing in the horizontal direction xi and the non - linear force Q in the vertical direction yi are as follows:

[0019]

[0020]

[0021] Where x and y are the displacements of the shaft center in the horizontal and vertical directions, and K b is the load - deformation constant between the rolling ball and the raceway. C b is the contact stiffness, θ i represents the position angle of the i - th rolling ball, i = 1, 2, …, N, and N is the number of rolling balls.

[0022]

[0023] Where: ω b is the common angular velocity of the rolling balls, and t is the time.

[0024]

[0025] Where: ω i is the angular velocity of the inner ring of the rolling bearing, R i is the radius of the inner - ring raceway, ω o is the angular velocity of the outer ring of the rolling bearing, R o is the radius of the outer - ring raceway.

[0026] In step 2), considering the translational displacements of the ball - bearing - rotor system in the horizontal and vertical directions, the system is discretized, and the mass of the rotating shaft is discretized onto the nodes, obtaining the motion differential equation of the ball - bearing - rotor system as:

[0027]

[0028] Where: M is the system mass matrix, C is the system damping matrix, K is the system stiffness matrix, Q is the bearing non - linear force, F is the system unbalanced force, and U is the system displacement response;

[0029] In step 3), the Runge - Kutta method is used to solve the system equation, obtaining the response diagram of the system with the change of bearing clearance, and observing whether the system bifurcates or becomes chaotic.

[0030] When solving the system equation, attention should be paid to selecting appropriate solution time and step size. If the solution time is too large, the calculation efficiency will be reduced; if the solution time is too small, the system cannot enter the steady state. Therefore, the calculation time should be gradually increased at the beginning, or the efficiency can be improved by increasing the time step size. Generally, for a 6 - degree - of - freedom system, select 1×10 -4Taking s as the time step, it is appropriate to set the total calculation time to 1 s. In the bifurcation diagram, attention should be paid to distinguishing whether the system is in quasi-periodic motion or chaotic motion.

[0031] In step 4), the step size is generally taken as 1 / 20 of the interval length.

[0032] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0033] 1) The relationship between the axial clearance and the radial clearance of the inner ring split hemispherical bearing is clarified, providing a solution for the inconvenience in measuring the radial clearance of the bearing.

[0034] 2) The radial clearance range of the bearing that enables the system to operate stably and safely is obtained, ensuring the reliability of the rotor operation and preventing vibration instability to guarantee the bearing life. Description of the Drawings

[0035] Figure 1 FIG. is a flow chart of a method for designing the clearance of an inner ring split hemispherical bearing of a turbo pump based on the response of a rotor system proposed by the present invention.

[0036] Figure 2 FIG. is an inner ring split hemispherical bearing of a turbo pump according to an embodiment of the method of the present invention.

[0037] Figure 3 FIG. is a simplified model of a ball bearing-rotor system of a turbo pump according to an embodiment of the method of the present invention.

[0038] Figure 4 FIG. is a bifurcation diagram obtained by simulation.

[0039] Figure 5 FIG. is a time-domain response diagram obtained by simulation. Among them, a) is the time-domain response diagram with a clearance of 44 μm; b) is the time-domain response diagram with a clearance of 100 μm. Detailed Embodiments

[0040] The embodiments of the present invention will be described in detail below with reference to the drawings and examples. The embodiment of the present invention is a simplified model of an inner ring split hemispherical bearing-rotor system of a turbo pump, and the clearance range of the bearing that enables the system to operate stably and safely is solved.

[0041] As Figure 1 shown, a method for designing the clearance of an inner ring split hemispherical bearing of a turbo pump based on the response of a rotor system includes the following steps:

[0042] 1) Determine and input the inner ring groove curvature radius, outer ring groove curvature radius, ball diameter, spacer angle, and axial clearance reference range of the inner ring split hemispherical bearing. As shown in Figure 2 the figure, according to the geometric relationship, calculate the radial clearance range corresponding to the axial clearance range, and establish a non-linear force model of the bearing considering the radial clearance;

[0043] To calculate the radial clearance corresponding to the axial clearance, the contact angle α needs to be determined first according to the axial clearance:

[0044]

[0045] In the formula, G a is the axial clearance; r i is the curvature radius of the inner raceway groove; r o is the curvature radius of the outer raceway groove; D is the ball diameter; α s is the shim angle.

[0046]

[0047] In the formula, w s is the shim width;

[0048] After obtaining the contact angle, the corresponding radial clearance G r can be obtained:

[0049] G r = 2(1 - cosα)(r i + r o - D) - (2r i - D)(1 - cosα s )

[0050] The calculation methods for the non - linear force Q xi in the horizontal direction and the non - linear force Q yi in the vertical direction of the bearing are as follows:

[0051]

[0052]

[0053] In the formula, x and y are the displacements of the axis center in the horizontal and vertical directions, K b is the load - deformation constant between the ball and the raceway, C b is the contact stiffness, θ i represents the position angle of the i - th ball, i = 1, 2, …, N, and N is the number of balls;

[0054] 2) Input the rotor structure parameters, mass, damping and other parameters, consider the bearing radial clearance, calculate the bearing non - linear force, and consider the rotor structure and the unbalanced force with periodic changes, establish the coupled model of the ball bearing - rotor system. The model is as Figure 3 shown, and write out the differential equation of motion.

[0055] Considering the translational displacements of the rotor system in the horizontal and vertical directions, the system is discretized into multiple shaft segments. Each shaft segment contains two nodes, and the mass and stiffness of the shaft segment are discretized onto the nodes. The parameters are as follows: m1 = 12 kg, m2 = 11 kg, m rL = m rR = 5.5 kg, c1 = c2 = 2100 N·s / m, c b = 1000 N·s / m, k = 1×10 8 N / m, e = 15 μm, k r = 6×10 7 N / m, d = 0.02 mm, μ = 0.1. The motion equations of the system are:

[0056]

[0057] 3) Solve the motion differential equation using the Runge - Kutta method to obtain the variation of the response of the ball bearing - rotor system under different clearances. From this, obtain the response diagram of the ball bearing - rotor system with the change of bearing clearance, and observe whether the system bifurcates or becomes chaotic. For this example, the rotor response with the change of clearance is as shown Figure 4 .

[0058] In the embodiment of the present invention, first calculate the response of the disk in the vertical direction under different radial clearances, and then draw the bifurcation diagram of the system with the radial clearance as the abscissa and the response value as the ordinate, that is, the so - called response diagram. The radial clearance, as a parameter affecting the non - linear force of the bearing, different values of it are very likely to cause the system response to bifurcate and enter chaotic motion.

[0059] The Runge - Kutta - Fehlberg method is based on the idea of the predictor - corrector algorithm. Starting from (t i , y i ), gradually obtain the information of several points in the interval [t i , t i+1 , and then through appropriate combination, calculate the approximate values y i+1 and i+1 of y(t i+1 ) with 4 - order and 5 - order accuracies respectively. The specific calculation formulas are: Specific calculation formulas are:

[0060]

[0061] In the formula: h - step size.

[0062] According to the calculated approximate results with different accuracies, calculate the local truncation error E, that is:

[0063]

[0064] To make the numerical solution converge to the solution of the original equation, the step size and the current error should satisfy the following relationship, that is:

[0065]

[0066] In the formula: ε — step size control parameter.

[0067] When the local truncation error E meets the error requirement and the result converges, the solution for the next time step is carried out; otherwise, the load step size is reduced and recalculated until the local truncation error E meets the requirement and the result converges.

[0068] 4) Take the clearance interval in which the system does not exhibit chaotic motion. Starting from the minimum value of the radial clearance, calculate the time-domain response of the disk in the horizontal direction corresponding to each radial clearance at the operating speed, and determine whether the peak-to-peak value of the horizontal direction response exceeds the limit value. If it does not exceed the limit value, then this radial clearance is the radial clearance for the stable operation of the system; if it exceeds the limit value, then on the basis of this radial clearance, increase the step size, and the step size is taken as 1 / 20 of the interval length. Calculate the shaft center orbit again and determine whether the response exceeds the limit value until the clearance reaches the maximum value of this interval. Finally, take the clearance interval that does not exceed the limit value as the range to ensure the stable and safe operation of the system. The response in the horizontal direction calculated in this example is as Figure 5 shown. Figure a) is the response diagram of the disk in the horizontal direction when the radial clearance of the bearing is 44 μm, and Figure b) is the response diagram of the disk in the horizontal direction when the radial clearance of the bearing is 100 μm. The abscissa of this diagram is time, and the ordinate is the vibration displacement in the horizontal direction. It can be judged whether the system displacement exceeds the standard under this clearance by the magnitude of the displacement.

[0069] In this step, if there is no chaotic motion in the bifurcation diagram, starting from the minimum value of the radial clearance, increase the step size to the maximum clearance in sequence, calculate the time-domain response of the disk in the horizontal direction under each clearance respectively, and judge whether its response exceeds the specified limit value. Retain the clearance values corresponding to the responses that do not exceed the limit value, and thus obtain the radial clearance interval of the bearing for the stable operation of the rotor system.

[0070] If there is chaotic motion in the bifurcation diagram, select the bearing clearances other than the chaotic motion, calculate the time-domain response of the disk in the horizontal direction under each clearance respectively, and judge whether its response exceeds the specified limit value. Retain the clearance values corresponding to the responses that do not exceed the limit value, and obtain the radial clearance interval of the bearing for the stable operation of the rotor system.

[0071] In an embodiment of the present invention, the required axial clearance range of the half-spherical bearing of the inner ring is 0.20 - 0.55 mm. Input the minimum and maximum values of the axial clearance, and the radial clearances are shown in Table 1. Through Figure 4 it can be observed that when the radial clearance is greater than 0.042 mm, the system does not exhibit chaotic motion.

[0072] Values of radial clearance when determining the axial clearance range in Table 1

[0073]

[0074] When calculating the radial clearance between 0.042 - 0.117 mm, the time-domain response diagram of the system is as Figure 5 shown. According to the time-domain response diagram, within the range of 0.042 mm - 0.117 mm, the peak-to-peak value of the response at the operating speed of the turbine disk does not exceed the limit value of 60 μm. Therefore, the radial clearance of the bearing for the stable and safe operation of the rotor system should be designed to be 0.042 - 0.117 mm.

Claims

1. A design method for the clearance of the inner ring split hemispherical bearing of a turbopump based on the response of a rotor system, characterized in that, It includes the following steps: Step 1), determine and input the inner ring groove curvature radius, outer ring groove curvature radius, ball diameter, shim angle and axial clearance reference range of the inner ring split hemispherical bearing. According to the geometric relationship, calculate the corresponding radial clearance range of this axial clearance range, and establish a non-linear force model of the bearing considering the radial clearance. Step 2), input the rotor structure parameters, mass and damping, establish a coupled model of the ball bearing-rotor system, and write the differential equation of motion. Step 3), use the Runge-Kutta method to solve the differential equation of motion, obtain the change of the response of the ball bearing-rotor system under different clearances, and observe whether the results bifurcate or become chaotic. Step 4), take the clearance interval in which the system does not become chaotic. Starting from the minimum value of the radial clearance, calculate the time-domain response diagram of the disk at the operating speed, and judge whether the peak-to-peak value of the response in the horizontal direction exceeds the limit value. If it does not exceed the limit value, this radial clearance is the radial clearance for the system to operate stably. If it exceeds the limit value, increase the step size on the basis of this radial clearance, calculate the shaft center trajectory again and judge whether the response exceeds the limit value until the clearance reaches the maximum value of this radial clearance interval. Finally, take the radial clearance interval that does not exceed the limit value as the range to ensure the stable and safe operation of the system.

2. A method for designing the clearance of the inner ring split hemisphere bearing of a turbopump based on the response of the rotor system according to claim 1, characterized in that In the said Step 1), the method for calculating the radial clearance corresponding to the axial clearance is: Determine the size of the contact angle α according to the axial clearance: where G a is the axial clearance; r i is the curvature radius of the inner raceway groove; r o is the curvature radius of the outer raceway groove; D is the ball diameter; α s is the shim angle; where w s is the gasket width; After obtaining the magnitude of the contact angle, the corresponding radial clearance G is obtained r : G r = 2(1 - cosα)(r i + r o - D) - (2r i - D)(1 - cosα s ) Nonlinear force Q of the bearing in the horizontal direction xi and nonlinear force Q in the vertical direction yi The calculation method is as follows: where x and y are the displacements of the axis center in the horizontal and vertical directions, and K b is the load-deformation constant of the rolling ball and the raceway, and C b is the contact stiffness, and θ i represents the position angle of the i-th rolling ball, i = 1, 2,..., N, where N is the number of rolling balls; Where: ω b is the angular velocity of the rolling ball's common rotation, t is time, and the calculation of ω b is as follows: Where: ω i is the angular velocity of the inner ring of the rolling bearing, R i is the radius of the inner ring raceway, ω o is the angular velocity of the outer ring of the rolling bearing, R o is the radius of the outer ring raceway.

3. A method for designing the clearance of the inner ring split hemisphere bearing of a turbo pump based on the response of the rotor system according to claim 1, characterized in that, In the said Step 2), considering the translational displacements of the ball bearing-rotor system in the horizontal and vertical directions, discretize the system, and discretize the mass of the rotating shaft onto the nodes, the differential equation of motion of the ball bearing-rotor system is: Where: M is the system mass matrix, C is the system damping matrix, K is the system stiffness matrix, Q is the non-linear force of the bearing, F is the unbalanced force of the system, and U is the displacement response of the system.

4. A design method for the clearance of the inner ring split hemispherical bearing of a turbopump based on the response of the rotor system according to claim 1, characterized in that, In the said Step 3), when solving the system equation, gradually increase the calculation time, or improve the efficiency by increasing the time step size.

5. A method for designing the clearance of the inner ring split hemispherical bearing of a turbo pump based on the response of the rotor system according to claim 1, characterized in that, In step 3), when calculating the 6-degree-of-freedom system, 1×10 -4 s is selected as the time step, and the total calculation time is taken as 1 s; in the bifurcation diagram, distinguish whether the ball bearing-rotor system is in quasi-periodic motion or chaotic motion to obtain the bearing clearance interval in which the system does not exhibit chaotic motion.

6. The design method of the clearance of the inner ring split hemispherical bearing of the turbopump based on the response of the rotor system according to claim 1, characterized in that, In the said Step 4), the step size value is 1 / 20 of the interval length.

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