Modal analysis-based rubber bearing design method suitable for large-span long shaft

Abstract

The invention relates to the technical field of bearing design, and discloses a modal analysis-based rubber bearing design method suitable for a large-span long shaft, and the method comprises the steps: building an II-order modal shape of a rubber bearing, and calculating the installation position of the rubber bearing and a vibration displacement set of mass nodes where an impeller and the rubber bearing are located through secondary modal analysis; constructing a function of the vibration displacement set about the equivalent stiffness and the static stiffness of the rubber bearing; constructing an evaluation function of the equivalent stiffness and the static stiffness of the rubber bearing, and solving an optimal solution of the equivalent stiffness and an optimal solution of the static stiffness when the minimum value of the evaluation function is obtained; equations of the equivalent stiffness and the static stiffness about the gap, the groove depth and the bearing radius and objective functions of the gap, the groove depth and the bearing radius are constructed respectively, and the gap, the groove depth and the bearing radius are optimal solutions when the objective functions take minimum values. According to the invention, the design optimization of the mounting position and structure of the rubber bearing of the large-span long shaft is realized, the vibration amplitude of the shaft system is effectively reduced, and the system stability is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of bearing design, and in particular relates to a rubber bearing design method based on modal analysis suitable for long shafts with large spans. Background Art

[0002] Bearings are widely used supporting components in machinery, and the most widely used bearings are the various types of plain bearings. According to relevant statistics, approximately one-third of the world's energy consumption is due to various forms of friction, of which plain bearings account for approximately one-tenth. Therefore, modern rotating machinery requires bearings with excellent load-bearing capacity, high reliability, and long life. Lubrication with low-viscosity fluids such as water or seawater also requires high load-bearing capacity and wear resistance. Rubber bearings, due to their pollution-free, material-saving, low-cost, wear-resistant, silt-resistant, and corrosion-resistant properties, have gained widespread use in the marine and water pump industries, making them one of the most suitable bearings for underwater use. They hold broad application prospects in industries such as hydraulic turbines, drilling turbines, ore wet separation equipment, marine pumps, slurry pumps, sewage pumps, ocean current generators, and the petrochemical industry. They offer significant advantages in resolving the frequent seal failure and bearing wear issues associated with metal plain bearings used in aqueous media. They also simplify the structure of mechanical lubrication and sealing systems and conserve precious non-ferrous metal bearing materials. Existing rubber bearings for deep-well pump main shafts, vertical long-shaft pumps, and marine pumping equipment are characterized by large spans and long shaft lengths. Due to the long shafts, far support point spacing, significant deflection, and low resonant frequency, these structures place higher demands on the dynamic response performance and adaptability of the bearing system. In order to improve the overall stiffness, a separate rubber bearing needs to be designed in the middle of the rotor.

[0003] Existing design methods for rubber bearings on long-span shafts primarily include designing for load capacity, friction and lubrication, and vibration and noise control. For example, patent application CN 109236721 A, titled "A Rubber Bearing," comprises an outer sleeve and an inner sleeve. The outer sleeve, through its own tension, secures the shaft to a designated position within a steel pipe, preventing the shaft from deflecting during rotation and contacting the steel pipe, providing support and vibration absorption. However, this design approach focuses solely on the rubber bearing itself, failing to consider the overall design of the shafting system. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a rubber bearing design method based on modal analysis suitable for long-span shafts, which is used to design and optimize the installation position and structure of rubber bearings in long-span shaft scenarios, and can effectively reduce the displacement amplitude of shaft system vibration.

[0005] In order to solve the above technical problems, the present invention provides a rubber bearing design method based on modal analysis suitable for long-span long shafts, the process comprising:

[0006] S1. Establish the second-order modal vibration mode of the rubber bearing, and calculate the installation position of the rubber bearing and the vibration displacement set of the mass nodes where the impeller and the rubber bearing are located through secondary modal analysis;

[0007] S2. Construct the equivalent stiffness K of the rubber bearing based on the vibration displacement set eq and static stiffness K s Function:

[0008] S3. Construct the equivalent stiffness K of the rubber bearing eq and static stiffness K s The evaluation function is used to find the optimal solution of equivalent stiffness when the evaluation function is minimized. Optimal solution for static stiffness

[0009] S4. Construct equivalent stiffness K eq and static stiffness K s Regarding the equations for clearance, groove depth, and bearing radius, as well as the objective functions for clearance, groove depth, and bearing radius, the clearance, groove depth, and bearing radius that provide the minimum value for the objective function are the optimal solutions.

[0010] As an improvement of the rubber bearing design method based on modal analysis applicable to long-span long shafts of the present invention:

[0011] The II-order modal vibration mode includes the mass node where the impeller 1 is located being calibrated as node 1, the mass node where the impeller 2 is located being calibrated as node 2, the mass node where the impeller 3 is located being calibrated as node 3, and the mass node where the rubber bearing is located being calibrated as node 4; Node 1 and Node 2 are located on the left side of the shaft, and Node 3 is located on the right side of the shaft; Take 9 points p for the interval z (z=1,2,3,...,9), L is the axis length, node 4 is one of the 9 points, and the distance set from the left axis end is (P1,P2,...P9); equivalent stiffness K eq Including m groups of different values, (K eq1 ,K eq2 ,…K eqm ); static stiffness K s Including n different values, (K s1 ,K s2 ,…K sn ).

[0012] As a further improvement of the rubber bearing design method based on modal analysis applicable to long-span long shafts of the present invention:

[0013] The calculation process of the installation position and vibration displacement set is as follows: first, the equivalent stiffness value K eq1 , static stiffness value K s1 The first modal analysis is performed on the combination of the distance set (P1, P2, ... P9) to obtain the vibration displacement of the nodes where the three impellers are located. and And the vibration displacement of the node where the rubber bearing is located Then the weighted root mean square response criterion is used to calculate the vibration displacement value of the shaft, and the point p where the minimum vibration displacement value of the shaft is located is selected. z As the installation position point s of the rubber bearing;

[0014] The equivalent stiffness value (K eq1 ,K eq2 ,…K eqm ), static stiffness value (K s1 ,K s2 ,…K sn ) and the distance between the installation position point s and the left shaft end are used as examples to perform modal analysis again to obtain the set of vibration displacements.

[0015] As a further improvement of the rubber bearing design method based on modal analysis applicable to long-span long shafts of the present invention:

[0016] The weighted root mean square response criterion is used to calculate the vibration displacement value of the shaft:

[0017]

[0018] Among them, α1, α2, and α3 are the weights of the three impellers, and β is the weight of the rubber bearing displacement.

[0019] As a further improvement of the rubber bearing design method based on modal analysis applicable to long-span long shafts of the present invention:

[0020] The vibration displacement set is:

[0021]

[0022] Among them, U1, U2, U3 and U4 represent the vibration displacement sets of node 1, node 2, node 3 and node 4 located at the installation position s respectively.

[0023] As a further improvement of the rubber bearing design method based on modal analysis applicable to long-span long shafts of the present invention:

[0024] The vibration displacement set is related to the rubber bearing equivalent stiffness K eq and static stiffness K s Function:

[0025]

[0026] Among them, A 1i 、A 2i 、A 3i There are four sets of constants corresponding to the four nodes, m is the mass of the system, and ω is the rotation speed of the axis.

[0027] As a further improvement of the rubber bearing design method based on modal analysis applicable to long-span long shafts of the present invention:

[0028] The equivalent stiffness K of the rubber bearing eq and static stiffness K s The evaluation function of :

[0029] F=U II1 2 (K eq ,K s )+U II2 2 (K eq ,K s )+U II3 2 (K eq ,K s )+U II4 2 (K eq ,K s ) (7)

[0030] When the evaluation function F is at its minimum, the optimal solution of equivalent stiffness is obtained. Optimal solution for static stiffness

[0031] As a further improvement of the rubber bearing design method based on modal analysis applicable to long-span long shafts of the present invention:

[0032] The equivalent stiffness K of the rubber bearing eq Equations for clearance c, groove depth h, and bearing radius R:

[0033]

[0034] Where E is the Young's modulus of the material, t is the effective thickness of the bearing, G is the shear modulus of the material, α is a correction factor to account for the effect of groove depth, and L0 is the length of the bearing.

[0035] Static stiffness K s Equations for clearance c, groove depth h, and bearing radius R:

[0036]

[0037] where I is the moment of inertia of the bearing cross section.

[0038] As a further improvement of the rubber bearing design method based on modal analysis applicable to long-span long shafts of the present invention:

[0039] The objective function of the clearance, groove depth and bearing radius is:

[0040]

[0041] Among them, λ is the weight coefficient;

[0042] The optimal solution of equivalent stiffness Optimal solution for static stiffness Substituting into equations (8), (9) and (10), when the objective function f(c, h, R) is minimized, the optimal clearance c1, optimal groove depth h1 and optimal bearing radius R1 are obtained.

[0043] As a further improvement of the rubber bearing design method based on modal analysis applicable to long-span long shafts of the present invention:

[0044] The optimal clearance c1 ranges from 0.1 mm to 1.0 mm, the ratio of the optimal groove depth h1 to the optimal bearing radius R1 ranges from 0.05 to 0.1, and the optimal bearing radius R1 ranges from 15.1 to 15.3 mm.

[0045] The beneficial effects of the present invention are mainly reflected in:

[0046] While existing improvements often focus on the rubber bearing itself, such as improving the material, this invention, based on modal analysis of the rotor system, constructs an evaluation function for the annular seal gap, groove depth, and bearing radius. This allows for the design and optimization of the rubber bearing's mounting position and structure, effectively reducing shafting vibration and improving system operational stability and safety. Furthermore, this invention incorporates shafting vibration into the design and layout optimization of the rubber bearing's stiffness, enhancing the system's anti-vibration performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] The specific embodiments of the present invention are further described in detail below with reference to the accompanying drawings.

[0048] Figure 1 This is a schematic diagram of the assembly of the rubber bearing and the shaft;

[0049] Figure 2 Schematic diagram of the flow of the rubber bearing design method based on modal analysis applicable to long shafts with large spans according to the present invention;

[0050] Figure 3 is a schematic diagram of the rotor system motion-force model of the present invention;

[0051] Figure 4The harmonic response vibration displacement curves of the rubber bearing before and after the improvement using the method of the present invention are shown. DETAILED DESCRIPTION

[0052] The present invention is further described below with reference to specific embodiments, but the protection scope of the present invention is not limited thereto:

[0053] Example 1: A rubber bearing design method based on modal analysis for long-span shafts

[0054] How to assemble rubber bearing and shaft Figure 1 As shown, c is the gap between the shaft and the rubber bearing, h is the groove depth of the rubber bearing, and R is the radius of the rubber bearing. The gap c, groove depth h, and bearing radius R are the target parameters for design optimization of rubber bearings suitable for long shafts with large spans according to the present invention.

[0055] In the host computer, the rotor system motion-force model is constructed and the II-order modal calculation is performed to calculate the installation position of the rubber bearing and the vibration displacement set of the mass node where the impeller and the rubber bearing are located. Then, the equivalent stiffness K of the vibration displacement set with respect to the rubber bearing is constructed. eq and static stiffness K s function, and construct the equivalent stiffness K of the rubber bearing eq and static stiffness K s The evaluation function is used to find the optimal solution of equivalent stiffness when the evaluation function is minimized. Optimal solution for static stiffness Then construct the equivalent stiffness K eq and static stiffness K s Regarding the equations for clearance, groove depth, and bearing radius, as well as the objective functions for clearance, groove depth, and bearing radius, the clearance, groove depth, and bearing radius that give the minimum objective function are the optimal solutions. The specific process is as follows: Figure 2 As shown:

[0056] Step 1: Determine the installation position of the rubber bearing and the vibration displacement value of the node where the impeller and rubber bearing are located based on the II-order modal vibration analysis

[0057] Step 1.1: Establish the rotor system motion-force model (i.e., the II-order modal vibration shape)

[0058] According to the geometric parameters of the rotor system and each rotor component, the motion-force model of the rotor system (i.e., the II-order modal vibration shape) is established, including the mass nodes where the three impellers are located and the mass node where the rubber bearing is located. There are two impellers installed on the left side of the shaft and one impeller installed on the right side. The mass node where the impeller 1 is located is calibrated as node 1, the mass node where the impeller 2 is located is calibrated as node 2, the mass node where the impeller 3 is located is calibrated as node 3, and the mass node where the rubber bearing is located is calibrated as node 4, as shown in the following example: Figure 3As shown, let the shaft length be L, and Take 9 different points for the interval, with p z (z=1,2,3,...,9) indicates that one of points 1, 2, 3, ..., and 9 is the installation position of the rubber bearing (i.e., node 4). The distances between points 1, 2, 3, ..., and 9 and the left shaft end are P1, P2...P9, respectively. That is, the set of distances between the installation position of the rubber bearing and the left shaft end is (P1, P2,...P9).

[0059] Step 1.2, equivalent stiffness K of rubber bearing in the II-order modal vibration mode eq Take m groups of different values ​​in turn, that is (K eq1 ,K eq2 ,…K eqm ), static stiffness K s Take n different values, that is (K s1 ,K s2 ,…K sn ).

[0060] Step 1.3: Calculate the installation position of the rubber bearing and the vibration displacement of each node

[0061] The calculation formula for the vibration displacement of each node is:

[0062]

[0063] Where a is the displacement of the axis in the x-direction, and b is the displacement of the axis in the y-direction.

[0064] The vibration displacement set needs to undergo a secondary modal analysis. First, the installation position of the rubber bearing is obtained through the first modal analysis calculation, and then the vibration displacement set of each node is obtained through the second modal analysis. The process is as follows:

[0065] (1) The equivalent stiffness value K of the rubber bearing eq1 , the static stiffness value Ks1 and the distance set (P1, P2, ... P9) are combined into a total of 9 groups of cases, which are substituted into the rotor system vibration equation in turn to perform the first modal analysis:

[0066]

[0067] Where M is the mass matrix; is the stiffness matrix; F(t) is the load function that changes with time; {x}, are the vibration displacement and acceleration vector respectively.

[0068]

[0069] Thus, the vibration displacement of the node where the impeller is located under the II-order modal vibration mode is obtained and And the vibration displacement of the node where the rubber bearing is located Specifically:

[0070] Indicates the equivalent stiffness value K eq1 , static stiffness value K s1 , the vibration displacement of node 1 in the II-order mode when the rubber bearing is installed at points 1 to 9 in sequence;

[0071] Indicates the equivalent stiffness value K eq1 , static stiffness value K s1 , the vibration displacement of node 2 in the II-order mode when the rubber bearing is installed at points 1 to 9 in sequence;

[0072] Indicates the equivalent stiffness value K eq1 , static stiffness value K s1 , the vibration displacement of node 3 in the II-order mode when the rubber bearing is installed at points 1 to 9 in sequence;

[0073] Indicates the equivalent stiffness value K eq1 , static stiffness value K s1 , the vibration displacement of node 4 in the II-order mode when the rubber bearing is installed at points 1 to 9 in sequence;

[0074] In order to comprehensively evaluate the influence of different installation positions of rubber bearings on the overall vibration response of the system, the weighted root mean square response criterion is used to calculate the vibration displacement value of the shaft:

[0075]

[0076] Among them, α1, α2, and α3 are the weights of the three impellers, and β is the weight of the rubber bearing displacement. The values ​​of α1, α2, α3, and β are all 0.25.

[0077] Comparison of 9 groups of data RMS (z) size, select the point pz where the vibration displacement value of the shaft is the minimum as the installation position point s of the rubber bearing (s is one of points 1-9) for modal calculation again:

[0078] (2) Taking the equivalent stiffness value of rubber bearing (K eq1 ,K eq2 ,…K eqm ), static stiffness value (K s1 ,K s2 ,…K sn) and the distance between the installation position point s and the left shaft end are substituted into the rotor system vibration equation (2) and modal analysis is performed again. The equivalent stiffness value K here is eq K in eq1 ~K eqm and static stiffness value K s K in s1 ~K sn The value must be 10 5 ~10 9 , and m and n are greater than or equal to 7, a total of m×n groups of examples, so the set of vibration displacements of all four nodes when obtaining the II-order modal vibration shape of each group of examples is:

[0079]

[0080] Among them, U1, U2, U3 and U4 represent the vibration displacement sets of node 1, node 2, node 3 and node 4 (located at the installation position s) respectively.

[0081] Step 2: Construct the four vibration displacement sets U1, U2, U3, and U4 of the nodes under the II-order modal vibration mode with respect to the bearing equivalent stiffness K. eq and static stiffness K s The function is:

[0082]

[0083] Among them, i=1, 2, 3, 4 are used to represent the node number, A 1i 、A 2i 、A 3i There are four sets of constants corresponding to the four nodes, m is the mass of the system, and ω is the rotation speed of the axis.

[0084] Step 3: Construct the equivalent stiffness K of the rubber bearing eq and static stiffness K s Evaluation function F:

[0085] F=U II1 2 (K eq ,K s )+U II2 2 (K eq ,K s )+U II3 2 (K eq ,K s )+U II4 2 (K eq ,K s ) (7)

[0086] Step 4: Obtain the optimal solution of equivalent stiffness corresponding to the minimum value of evaluation function F by solving formula (6) Optimal solution for static stiffness

[0087] Step 5: Construct the equivalent stiffness K of the rubber bearing respectively eq and static stiffness K s The equations and objective functions of the clearance c, groove depth h and bearing radius R are solved to obtain the optimal solutions of the clearance c, groove depth h and bearing radius R.

[0088] Step 5.1: Construct the equivalent stiffness K of the rubber bearing eq Equations for clearance c, groove depth h, and bearing radius R:

[0089]

[0090] Where E is the Young's modulus of the material, t is the effective thickness of the bearing, G is the shear modulus of the material, α is a correction factor to account for the effect of groove depth, and L0 is the length of the bearing.

[0091] Constructing the static stiffness K of rubber bearings s Equations for clearance c, groove depth h, and bearing radius R:

[0092]

[0093] where I is the moment of inertia of the bearing cross section.

[0094] Step 5.2: Construct the objective function for optimizing the clearance c, groove depth h, and bearing radius R of the rubber bearing:

[0095]

[0096] Among them, λ is the weight coefficient, which is 1.

[0097] Step 5.3: The optimal solution of equivalent stiffness obtained in step 4 is Optimal solution for static stiffness Substituting into equations (8), (9) and (10), when the objective function f(c, h, R) is minimized, the optimal clearance c1, optimal groove depth h1 and optimal bearing radius R1 are obtained. The optimal clearance c1 ranges from 0.1 mm to 1.0 mm, the ratio of the optimal groove depth h1 to the optimal bearing radius R1 ranges from 0.05 to 0.1, and the optimal bearing radius R1 ranges from 15.1 to 15.3 mm.

[0098] Step 6: Simulation calculation

[0099] Step 6.1. Based on the geometric parameters of the rotor components of the rubber bearing, establish the rotor system motion-force model, including the mass nodes where the three impellers are located and the mass node where the rubber bearing is located, such as Figure 3 As shown, the equivalent stiffness K eq and static stiffness K s Take 7 different sets of values ​​in turn, specifically: 10 5 , 10 6 , 10 7 、3*10 7 , 6*10 7 , 10 8 and 10 9 .

[0100] The distances between the mass nodes 1-9 where the rubber bearing is located and the left shaft end are 150, 300, 450, 600, 750, 900, 1050, 1200, and 1350 respectively.

[0101] Calculate the vibration displacement of the node: The rubber bearings installed at both ends of the shaft have little effect on the vibration of the shaft system, so points 1 and 9 are discarded and mm is used as the dimension.

[0102] The rubber bearing is installed at point 2, and K eq =10 5 ,K s =10 5 For the calculation example, the vibration displacement of the four nodes of the rotor system is:

[0103] The rubber bearing is installed at point 3, and K eq =10 5 ,K s =10 5 For the calculation example, the vibration displacement of the four nodes of the rotor system is:

[0104] The rubber bearing is installed at point 4, and K eq =10 5 ,K s =10 5 For the calculation example, the vibration displacement of the four nodes of the rotor system is:

[0105] And so on,

[0106] The rubber bearing is installed at point 5, and K eq =10 5 ,K s =10 5 For the calculation example, the vibration displacement of the four nodes of the rotor system is:

[0107] The rubber bearing is installed at point 6, and K eq =10 5 ,K s =10 5 For the calculation example, the vibration displacement of the four nodes of the rotor system is:

[0108] The rubber bearing is installed at point 7, and K eq =10 5 ,K s =10 5 For the calculation example, the vibration displacement of the four nodes of the rotor system is:

[0109] The rubber bearing is installed at point 8, and K eq =10 5 ,K s =10 5 For the calculation example, the vibration displacement of the four nodes of the rotor system is:

[0110] In order to comprehensively evaluate the influence of different installation positions of rubber bearings on the overall vibration response of the system, the weighted root mean square response criterion is adopted: Among them, α1, α2, α3 are the weights of the three impellers, β is the weight of the bearing displacement, z is the installation position of the rubber bearing, α1=α2=α3=0.25, β=0.25

[0111]

[0112] Comparison of the above 7 groups of vibration displacement J RMS It can be seen that when the rubber bearing is installed at point 6, the vibration displacement value of the shaft is the smallest, and then the distance between point 6 and the left shaft end and the equivalent stiffness K eq and static stiffness K s For the purpose of this example, we substitute the vibration equation (2) of the rotor system and perform modal calculation again. The vibration displacement of the four nodes of the rotor system is:

[0113] x 1,1,1 =7.4756,x 2,1,1 =6.5978,x 3,1,1 =6.1568,x 4,1,1 =10.9249.

[0114] x 1,1,2 =7.5462,x 2,1,2 =6.6187,x 3,1,2 =6.0355,x 4,1,2 =9.5346,x 1,1,3 =7.6469,x 2,1,3=6.7696,x 3,1,3 =5.9681,x 4,1,3 =9.1235

[0115] x 1,1,4 =7.4811,x 2,1,4 =6.4423,x 3,1,4 =5.8587x 4,1,4 =9.4633,x 1,1,5 =7.3379,x 2,1,5 =6.3855,x 3,1,5 =5.3015,x 4,1,5 =9.0103

[0116] x 1,1,6 =7.1366,x 2,1,6 =6.3007,x 3,1,6 =5.3129,x 4,1,6 =8.8996,x 1,1,7 =7.4034,x 2,1,7 =6.3527,x 3,1,7 =5.3966,x 4,1,7 =9.5568

[0117] x 1,2,1 =6.9351,x 2,2,1 =6.9688,x 3,2,1 =6.0357,x 4,2,1 =9.2894,x 1,2,2 =6.9532,x 2,2,2 =6.8455,x 3,2,2 =5.7963,x 4,2,2 =9.1433

[0118] x 1,2,3 =6.8832,x 2,2,3 =6.7185,x 3,2,3 =5.8623,x 4,2,3 =9.1042,x 1,2,4 =6.7985,x 2,2,4 =6.1258,x 3,2,4 =5.7964,x 4,2,4 =9.0321

[0119] x 1,2,5 =6.6257,x 2,2,5 =6.3548,x 3,2,5 =5.7782,x 4,2,5 =8.8961,x 1,2,6 =6.5854,x2,2,6 =5.0311,x 3,2,6 =5.6458,x 4,2,6 =8.7654

[0120] x 1,2,7 =6.4334,x 2,2,7 =5.1317,x 3,2,7 =5.5091,x 4,2,7 =8.6154

[0121] x 1,3,1 =6.5216,x 2,3,1 =6.5473,x 3,3,1 =5.9217,x 4,3,1 =8.9876,x 1,3,2 =6.3216,x 2,3,2 =6.3835,x 3,3,2 =5.8299,x 4,3,2 =8.6276

[0122] x 1,3,3 =6.2574,x 2,3,3 =6.1693,x 3,3,3 =5.5249,x 4,3,3 =8.3385,x 1,3,4 =6.0128,x 2,3,4 =6.2174,x 3,3,4 =5.2723,x 4,3,4 =8.6834

[0123] x 1,3,5 =5.5201,x 2,3,5 =5.8298,x 3,3,5 =5.8349,x 4,3,5 =8.2338,x 1,3,6 =5.3598,x 2,3,6 =5.8724,x 3,3,6 =5.6362,x 4,3,6 =8.2035

[0124] x 1,3,7 =5.5691,x 2,3,7 =5.5681,x 3,3,7 =5.6854,x 4,3,7 =8.0772

[0125] x 1,4,1 =6.1787,x 2,4,1 =6.1257,x 3,4,1 =5.6231,x 4,4,1=8.0263,x 1,4,2 =6.2048,x 2,4,2 =5.9359,x 3,4,2 =5.5905,x 4,4,2 =8.6575

[0126] x 1,4,3 =5.8879,x 2,4,3 =5.7873,x 3,4,3 =5.4982,x 4,4,3 =7.8864,x 1,4,4 =5.3589,x 2,4,4 =5.3805,x 3,4,4 =5.4362,x 4,4,4 =7.6438

[0127] x 1,4,5 =5.6058,x 2,4,5 =5.2199,x 3,4,5 =5.3652,x 4,4,5 =7.4746,x 1,4,6 =5.2173,x 2,4,6 =5.1683,x 3,4,6 =5.0175,x 4,4,6 =7.1554

[0128] x 1,4,7 =5.3598,x 2,4,7 =5.2854,x 3,4,7 =5.1127,x 4,4,7 =7.0255

[0129] x 1,5,1 =5.5432,x 2,5,1 =5.4866,x 3,5,1 =5.3592,x 4,5,1 =7.6239,x 1,5,2 =5.4125,x 2,5,2 =5.3652,x 3,5,2 =5.0280,x 4,5,2 =7.3257

[0130] x 1,5,3 =5.1054,x 2,5,3 =5.2327,x 3,5,3 =4.8723,x 4,5,3 =6.9546,x 1,5,4 =4.3449,x 2,5,4 =4.4527,x 3,5,4 =4.2285,x4,5,4 = 5.5128

[0131] x 1,5,5 = 4.9845, x 2,5,5 = 5.0102, x 3,5,5 = 4.8667, x 4,5,5 = 6.3612, x 1,5,6 = 4.8339, x 2,5,6 = 5.1844, x 3,5,6 = 4.9957, x 4,5,6 = 5.7344

[0132] x 1,5,7 = 4.4681, x 2,5,7 = 5.1265, x 3,5,7 = 4.9486, x 4,5,7 = 6.8126

[0133] x 1,6,1 = 5.6836, x 2,6,1 = 5.8439, x 3,6,1 = 5.2876, x 4,6,1 = 7.4699, x 1,6,2 = 5.5212, x 2,6,2 = 5.2199, x 3,6,2 = 4.9668, x 4,6,2 = 7.2053

[0134] x 1,6,3 = 5.2306, x 2,6,3 = 5.6923, x 3,6,3 = 4.6248, x 4,6,3 = 6.9468, x 1,6,4 = 5.1346, x 2,6,4 = 5.0622, x 3,6,4 = 4.5985, x 4,6,4 = 6.5027

[0135] x 1,6,5 = 5.3033, x 2,6,5 = 4.8462, x 3,6,5 = 4.8126, x 4,6,5 = 7.0217, x 1,6,6 = 5.1061, x 2,6,6 = 4.8459, x 3,6,6 = 4.8587, x 4,6,6 = 6.8925

[0136] x 1,6,7 = 4.7527, x 2,6,7=4.9312,x 3,6,7 =4.9219,x 4,6,7 =6.5956

[0137] x 1,7,1 =5.7955,x 2,7,1 =5.7058,x 3,7,1 =5.3803,x 4,7,1 =7.1323,x 1,7,2 =5.2658,x 2,7,2 =5.3978,x 3,7,2 =5.2637,x 4,7,2 =7.0291

[0138] x 1,7,3 =5.4312,x 2,7,3 =5.2632,x 3,7,3 =4.9785,x 4,7,3 =6.9891,x 1,7,4 =5.1623,x 2,7,4 =5.7816,x 3,7,4 =4.6482,x 4,7,4 =6.5683

[0139] x 1,7,5 =4.9462,x 2,7,5 =5.3155,x 3,7,5 =4.8215,x 4,7,5 =6.6429,x 1,7,6 =5.1329,x 2,7,6 =5.2937,x 3,7,6 =4.9749,x 4,7,6 =6.7803

[0140] x 1,7,7 =5.0723,x 2,7,7 =4.8357,x 3,7,7 =4.7841,x 4,7,7 =6.3124

[0141] The vibration displacement sets of the four nodes U1, U2, U3, and U4 are:

[0142]

[0143] Step 6.2: Based on the equivalent stiffness and static stiffness values ​​established in step 6.1, perform modal analysis of the rotor system for each example. According to formula (5), a set of vibration displacement values ​​of four nodes is constructed with respect to the bearing equivalent stiffness K. eq , static stiffness K s The function is:

[0144]

[0145] Step 6.3 Equivalent stiffness K of rubber bearing eq and static stiffness K s The evaluation function F is:

[0146]

[0147] Step 6.4: Solve for the optimal solution of equivalent stiffness when the evaluation function F is minimized Optimal solution for static stiffness

[0148] Step 6.5: Find the optimal solution for clearance, groove depth, and bearing radius.

[0149] Equations for equivalent stiffness and static stiffness with respect to clearance, groove depth, and bearing radius

[0150]

[0151]

[0152] Objective function of rubber bearing clearance, groove depth and bearing radius:

[0153]

[0154] K eq1 ,K s1 Substituting into the equation, when the objective function f(c,h,R) is minimized, the optimal clearance, groove depth and bearing radius are c1=0.278mm, h1=0.831mm, R1=15.167mm respectively.

[0155] The final optimal clearance is rounded to c1 = 0.3 mm, the optimal groove depth is h1 = 0.8 mm, and the optimal bearing radius is R1 = 15.15 mm.

[0156] Step 6.6: Verify the rationality of design optimization

[0157] In the rotor system, the initial parameters before the bearing structure improvement (R0 = 15.2, c0 = 0.5, h0 = 1.2) and the improved optimized parameters c1 = 0.3mm, h1 = 0.8mm, R1 = 15.15mm were analyzed for harmonic response modal analysis to obtain the harmonic response vibration displacement curve, as shown in Figure 2. Figure 4 As shown by Figure 4It can be seen that the harmonic response vibration displacement amplitude obtained by the improved optimization parameters is significantly lower than the displacement amplitude obtained by setting the initial parameters before the improvement. Before the improvement, the vibration of the shaft system was about 33.589μm, and after the improved design, the vibration was 25.471μm. After the structural parameter optimization, the vibration displacement amplitude of the shaft system can be significantly reduced, ensuring the accuracy of the structural design. Therefore, using the modal characteristics analysis results of the rotor system to design and optimize the structural parameters has certain feasibility and important engineering application value.

[0158] It should be noted that the rubber bearing design proposed in the present invention is suitable for large-span rotating shaft systems that have comprehensive requirements for support flexibility, vibration suppression, and fluid lubrication. However, in short-axis, rigid-axis, or other non-long-axis scenarios, due to the high natural frequency of the shaft system, small support point spacing, and high rigidity, there are no special requirements for the flexible characteristics of the rubber bearing, and there is no obvious risk of low-frequency resonance, and they are not suitable for the solution of the present invention.

[0159] Finally, it should be noted that the above examples are merely specific embodiments of the present invention. Obviously, the present invention is not limited to the above examples and is subject to numerous variations. All variations that can be directly derived or conceived by a person of ordinary skill in the art from the disclosure of the present invention are considered to be within the scope of protection of the present invention.

Claims

1. A rubber bearing design method based on modal analysis suitable for long-span shafts, characterized by: S1. Establish the second-order modal vibration mode of the rubber bearing, and calculate the installation position of the rubber bearing and the vibration displacement set of the mass nodes where the impeller and the rubber bearing are located through secondary modal analysis; S2. Construct the equivalent stiffness K of the rubber bearing based on the vibration displacement set eq and static stiffness K s Function: S3. Construct the equivalent stiffness K of the rubber bearing eq and static stiffness K s The evaluation function is used to find the optimal solution of equivalent stiffness when the evaluation function is minimized. Optimal solution for static stiffness S4. Construct equivalent stiffness K eq and static stiffness K s Regarding the equations for clearance, groove depth, and bearing radius, as well as the objective functions for clearance, groove depth, and bearing radius, the clearance, groove depth, and bearing radius that provide the minimum value for the objective function are the optimal solutions.

2. The method for designing a rubber bearing based on modal analysis for a long shaft with a large span according to claim 1, characterized in that: The II-order modal vibration mode includes the mass node where the impeller 1 is located being calibrated as node 1, the mass node where the impeller 2 is located being calibrated as node 2, the mass node where the impeller 3 is located being calibrated as node 3, and the mass node where the rubber bearing is located being calibrated as node 4; Node 1 and Node 2 are located on the left side of the shaft, and Node 3 is located on the right side of the shaft; Take 9 points p for the interval z (z=1,2,3,...,9), L is the axis length, node 4 is one of the 9 points, and the distance set from the left axis end is (P1,P2,...P9); equivalent stiffness K eq Including m groups of different values, (K eq1 ,K eq2 ,…K eqm ); static stiffness K s Including n different values, (K s1 ,K s2 ,…K sn ).

3. The method for designing a rubber bearing based on modal analysis for a long shaft with a large span according to claim 2, characterized in that: The calculation process of the installation position and vibration displacement set is as follows: first, the equivalent stiffness value K eq1 , static stiffness value K s1 The first modal analysis is performed on the combination of the distance set (P1, P2, ... P9) to obtain the vibration displacement of the nodes where the three impellers are located. and And the vibration displacement of the node where the rubber bearing is located Then the weighted root mean square response criterion is used to calculate the vibration displacement value of the shaft, and the point p where the minimum vibration displacement value of the shaft is located is selected. z As the installation position point s of the rubber bearing; The equivalent stiffness value (K eq1 ,K eq2 ,…K eqm ), static stiffness value (K s1 ,K s2 ,…K sn ) and the distance between the installation position point s and the left shaft end are used as examples to perform modal analysis again to obtain the set of vibration displacements.

4. The method for designing a rubber bearing based on modal analysis for a long shaft with a large span according to claim 3, characterized in that: The weighted root mean square response criterion is used to calculate the vibration displacement value of the shaft: Among them, α1, α2, and α3 are the weights of the three impellers, and β is the weight of the rubber bearing displacement.

5. The method for designing a rubber bearing based on modal analysis for a long shaft with a large span according to claim 4, characterized in that: The vibration displacement set is: Among them, U1, U2, U3 and U4 represent the vibration displacement sets of node 1, node 2, node 3 and node 4 located at the installation position s, respectively.

6. The method for designing a rubber bearing based on modal analysis for a long shaft with a large span according to claim 5, characterized in that: The vibration displacement set is related to the rubber bearing equivalent stiffness K eq and static stiffness K s Function: Among them, A 1i 、A 2i 、A 3i There are four sets of constants corresponding to the four nodes, m is the mass of the system, and ω is the rotation speed of the axis.

7. The method for designing a rubber bearing based on modal analysis for a long shaft with a large span according to claim 6, characterized in that: The equivalent stiffness K of the rubber bearing eq and static stiffness K s The evaluation function of : F=U II1 2 (K eq ,K s )+U II2 2 (K eq ,K s )+U II3 2 (K eq ,K s )+U II4 2 (K eq ,K s ) (7) When the evaluation function F is at its minimum, the optimal solution of equivalent stiffness is obtained. Optimal solution for static stiffness 8. The method for designing a rubber bearing based on modal analysis for a long shaft with a large span according to claim 7, characterized in that: The equivalent stiffness K of the rubber bearing eq Equations for clearance c, groove depth h, and bearing radius R: Where E is the Young's modulus of the material, t is the effective thickness of the bearing, G is the shear modulus of the material, α is a correction factor to account for the effect of groove depth, and L0 is the length of the bearing. Static stiffness K s Equations for clearance c, groove depth h, and bearing radius R: where I is the moment of inertia of the bearing cross section.

9. The method for designing a rubber bearing based on modal analysis for a long shaft with a large span according to claim 8, characterized in that: The objective function of the clearance, groove depth and bearing radius is: Among them, λ is the weight coefficient; The optimal solution of equivalent stiffness Optimal solution for static stiffness Substituting into equations (8), (9) and (10), when the objective function f(c, h, R) is minimized, the optimal clearance c1, optimal groove depth h1 and optimal bearing radius R1 are obtained.

10. The method for designing a rubber bearing based on modal analysis for a long shaft with a large span according to claim 9, characterized in that: The optimal clearance c1 ranges from 0.1 mm to 1.0 mm, the ratio of the optimal groove depth h1 to the optimal bearing radius R1 ranges from 0.05 to 0.1, and the optimal bearing radius R1 ranges from 15.1 to 15.3 mm.

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