A method for evaluating dynamic characteristics of a single-disk cracked rotor based on similarity theory

Abstract

The present invention provides a method for evaluating the dynamic characteristics of a single-disk cracked rotor based on the similarity theory. First, the dimensions of each part are determined according to the prototype of the single-disk cracked rotor; then, the similarity ratio κ of the shaft diameter is determined using the similarity theory d , and then the similarity ratio κ of the shaft length is obtained successively according to the formula derived from the similarity theory x , the similarity ratio κ of the single-disk diameter D , the similarity ratio κ of the single-disk thickness H , the similarity ratio κ of the bearing stiffness K , and the similarity ratio κ of the crack depth a1 is the same as the similarity ratio κ of the shaft diameter d ; the dimensions of the similarity model are determined according to the similarity ratio, and rotor dynamics analysis is performed on the prototype and the similarity model to analyze their critical speeds and natural frequencies. The present invention considers the influence of cracks in the similarity calculation, and the dynamic characteristics of the prototype can be evaluated and predicted using the similarity model, which has significant engineering practical value.

Description

Technical Field

[0001] The invention relates to a method for evaluating dynamic characteristics of a single-disk cracked rotor based on similarity theory, and belongs to the field of rotating machinery. Background Art

[0002] Crack failure is one of the common failures of rotating machinery. The main reason for the crack is the defects of the rotor material itself or fatigue cracks caused by long-term operation. As the core component of the rotating machinery, the dynamic characteristics of the rotor-support system are directly related to the working performance and structural safety of the entire rotating machinery. Therefore, the study of the dynamic characteristics of the cracked rotor-support system has important academic value and engineering practical value. In real life, for large and complex rotor systems, if the prototype is directly tested, it will bring huge experimental costs and increase experimental risks. If the rotor system is appropriately scaled using similarity theory, it can greatly save costs and reduce experimental difficulty. There are certain results in the study of similarity theory. However, there are still many gaps in the similarity calculation of cracked rotor-support systems. Therefore, it is very important to develop a single-disk crack rotor dynamic characteristics evaluation method based on similarity theory. Summary of the invention

[0003] This patent addresses the gap in the similarity theory of single-disk cracked rotor-support system. Starting from the similarity theory, combined with rotor dynamics and cracked rotors, a calculation method for similarity transformation of the dynamic characteristics of a single-disk cracked rotor is developed. The present invention takes the similarity conversion of cracks into account and keeps the rotor's natural frequency and other characteristics unchanged during the conversion process. This method can reduce the experimental difficulty of cracked rotors, greatly save experimental costs, and improve the safety and reliability of experiments.

[0004] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is as follows:

[0005] A method for evaluating dynamic characteristics of a single-disk cracked rotor based on similarity theory, characterized in that it comprises the following steps:

[0006] The first step is to divide the single-disc crack rotor prototype into three parts: the shaft, the rotating disk and the crack, and determine the size of each part of the single-disc crack rotor prototype;

[0007] The second step is to determine the similarity ratio κ of the shaft diameter according to the similarity conversion requirements. d , and then the axial length similarity ratio κ is obtained in turn according to the formula derived from the similarity theory x , similarity ratio of the turntable diameter κ D , similarity ratio of turntable thickness κ H Similarity ratio of bearing stiffness κ K ; Crack depth similarity ratio Similarity ratio to shaft diameter κ dsame;

[0008] The third step is to obtain a similar model based on the prototype size of the single-disk cracked rotor and the conversion of the similarity ratios of each part;

[0009] The fourth step is to conduct rotor dynamics analysis on the prototype and similar models to analyze their critical speed and natural frequency.

[0010] Furthermore, the dimensions of the single-disk cracked rotor prototype determined in the first step include the length of the shaft, the diameter of the shaft, the thickness of the rotating disk, the outer diameter and the inner diameter of the rotating disk, and the depth of the crack.

[0011] Furthermore, similarity theory is used to perform similarity theory transformation on the single-disk cracked rotor, and the specific steps are as follows:

[0012] 1) According to the vibration equation of the rotor system:

[0013]

[0014] m is the rotor mass per unit length; E is the elastic modulus of the shaft; I is the cross-sectional distance of the shaft; a is the moment of inertia of the turntable mass per unit length relative to the rotating shaft; p is the rotor unbalance force; p=me 1 f 2 , where f is the rotor natural frequency, e 1 is the unbalance curve; x is the length of the shaft; y is the composite deflection of the shaft; t is the time; i is the imaginary unit;

[0015] The inclination angle α and normal stress σ of the shaft elastic curve are introduced to improve the dynamic state of the rotor system, namely:

[0016]

[0017] Among them, M * is the bending moment, M * =Gx, G is gravity; W is the bending section coefficient, W = πd 3 / 32, d is the shaft diameter;

[0018] According to formula (1) (2), the integral simulation method is used to establish similar relationships and obtain similar π groups. According to the π theorem, in order to satisfy the dynamic similarity between the prototype rotor system and the model rotor system, the corresponding π values ​​must be equal:

[0019]

[0020] Subscript c represents a similar model, and subscript n represents a prototype;

[0021] The following factors are introduced to simplify the above formula (3), replacing the corresponding variables in the π group so that it only consists of ω, d, ρ, g, x

[0022] Univariate representation:

[0023]

[0024] G is the gravity of the rotating shaft; J p is the moment of inertia of the turntable; ρ is the density of the shaft material; d is the shaft diameter; ω is the rotation speed of the system; taking E, d, x, ρ of the shaft as the design parameters, the obtained shaft similarity ratio is:

[0025]

[0026] When the material remains unchanged before and after similar conversion, κ ρ =κ E =κ g =1, we get the formula:

[0027]

[0028] 2) According to the moment of inertia and mass formula of the turntable, the similarity relationship of the turntable can be obtained:

[0029]

[0030]

[0031] Where b is the ratio of the inner diameter to the outer diameter of the single-disk crack rotor prototype, D is the outer diameter of the disk; d is the inner diameter of the disk, which is the same as the shaft diameter; H is the disk thickness; ρ' is the disk density; J p is the moment of inertia; κ Jp is the similarity ratio of the turntable moment of inertia; κ m' is the mass similarity ratio of the turntable; c represents the similar model, n represents the prototype. m' is the mass of the turntable;

[0032] When the rotor-support system is similar, the turntable mass and moment of inertia are similar to the shaft mass and moment of inertia, and the material remains unchanged before and after the transformation, so the turntable thickness κ is obtained. H and turntable diameter κ D Similarity ratio:

[0033]

[0034] 3) Since the dynamics are similar, the ratio of the forces in the rotor system and the bearing system is the same; according to the stiffness formula k = m 1 f 2 And into the bearing mass similarity ratio formula Similarity ratio formula with natural frequency The bearing stiffness similarity ratio can be obtained:

[0035]

[0036] 4) Based on a simple straight crack growing on a rotating shaft, the crack depth a 1 The crack front width b 1 Description, for the crack unit, a block method is adopted, and the crack area is divided into crack module I 8, crack module II 9, crack module III 10, and crack module IV 11 from the crack (7 section to both sides; the interface between crack module I 8 and crack module II 9 fits each other without gap, but displacement can occur, so as to simulate the crack; crack module I 8, crack module II 9 and crack module III 10, crack module IV 11, as well as crack module III 10 and crack module IV 11 are all connected and cannot be displaced; in order to simulate that the rest of the parts except the crack are connected to the rotating shaft; since the four modules are connected to the rest of the shaft segments and there is no gap between the faces, when similarity transformation is performed on the crack, similarity transformation is performed on the module where the crack is located; the depth a of the crack 1 The same similarity transformation is performed as a fraction of the shaft diameter, that is, the similarity ratio of the crack depth Similarity ratio to shaft diameter κ d same:

[0037]

[0038] Furthermore, in formula (6), the axis diameter similarity ratio κ is first determined d According to the invariance of the natural frequency before and after the axis similarity transformation, the axis length similarity ratio κ is obtained x , using the formula:

[0039] The technical solution of the present invention can bring the following beneficial effects and effects:

[0040] The present invention converts the dynamic characteristic calculation of a real large single-disk cracked rotor into an acceptable similar model dynamic characteristic calculation by performing similarity theory conversion on the single-disk cracked rotor system. The technical solution of the present invention can ensure that the dynamic characteristics of the prototype rotor and the model cracked rotor, such as the natural frequency and critical speed, are the same. This greatly reduces the danger and experimental cost of cracked rotor experiments, brings new research methods to the dynamic characteristic calculation of cracked rotor systems in engineering, and broadens the application of similarity theory in the field of rotating machinery.

[0041] The present invention can convert the dynamic characteristic calculation of a real large single-disk crack rotor into an acceptable similar model dynamic characteristic calculation by performing similarity theory conversion on the single-disk crack rotor system. The technical solution of the present invention can ensure that the natural frequencies of the prototype rotor and the model crack rotor are the same and the error of the calculated results can be guaranteed to be within 5%, which is conducive to modal experimental research. Due to the limitations of factors such as geometric dimensions, physical space and experimental costs, there are many difficulties in directly experimenting on the prototype rotor system, so the large single-disk crack rotor can be transformed similarly through the technical solution of the invention to greatly reduce the danger and experimental cost of the crack rotor experiment. In the technical solution of the present invention, the straight crack is simplified and similar conversion is performed, which greatly simplifies the workload in the research process, brings a new research method to the dynamic characteristic calculation of the crack rotor system in engineering, and broadens the application of similarity theory in the field of rotating machinery. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 The schematic diagram of the structure of the single-disc cracked rotor of the present invention is

[0043] Figure 2 It is a schematic diagram of the structure of the section where the crack on the shaft is located.

[0044] Figure 3 Schematic diagram of dividing the area around the crack into blocks.

[0045] Figure 4 This is the prototype modal diagram of a single-disk cracked rotor.

[0046] Figure 5 This is the modal diagram of a similar model of a single-disk cracked rotor.

[0047] In the figure:

[0048] 1. Rotating shaft module I, 2. Rotating shaft module II, 3. Rotating shaft module III, 4. Turntable, 5. Rotating shaft module IV, 6. Rotating shaft module V, 7. Crack, 8. Crack module I, 9. Crack module II, 10. Crack module III, 11. Crack module IV. DETAILED DESCRIPTION

[0049] In order to make the technical means, creative features and effects achieved by the present invention easier to understand, the present invention is further described below in conjunction with embodiments and drawings.

[0050] The present invention is a method for evaluating the dynamic characteristics of a single-disk cracked rotor based on similarity theory. Figure 1 This is a typical single-disc crack rotor structure, which mainly includes a rotating shaft and a rotating disk. The rotating disk is located in the middle of the rotating shaft system, and the crack is on the rotating shaft on one side of the rotating disk. The crack only considers the depth. Figure 2 Schematic diagram of crack structure.

[0051] The method for evaluating the dynamic characteristics of a single-disk cracked rotor based on similarity theory of the present invention comprises the following steps:

[0052] The first step is to Figure 1 The structure of the single-disc rotor shown divides the single-disc cracked rotor prototype into three parts: the shaft, the turntable and the crack, and determines the dimensions of each part of the single-disc cracked rotor; including the shaft length, the shaft diameter, the turntable thickness, the outer diameter and inner diameter of the turntable, and the depth of the crack.

[0053] The second step is to determine the similarity ratio κ of the shaft diameter according to the similarity conversion requirements. d , and then the axial length similarity ratio κ is obtained in turn according to the formula derived from the similarity theory x , single disk diameter similarity ratio κ D , single disk thickness similarity ratio κ H Similarity ratio of bearing stiffness κ K ; Crack depth similarity ratio Similarity ratio to shaft diameter κ d same.

[0054] The third step is to obtain a similar model based on the conversion of the prototype size and the similarity ratio of each part;

[0055] The fourth step is to conduct rotor dynamics analysis on the prototype and similar models to analyze their critical speed, natural frequency and modal diagram;

[0056] The specific steps of the second step are as follows:

[0057] 1) According to the vibration equation of the rotor system:

[0058]

[0059] m is the rotor mass per unit length; E is the elastic modulus of the shaft; I is the cross-sectional distance of the shaft; a is the moment of inertia of the turntable mass per unit length relative to the rotating shaft; p is the rotor unbalance force; p=me 1 f 2 , where f is the rotor natural frequency, e 1 is the unbalance curve; x is the length of the shaft; y is the composite deflection of the shaft; t is the time; i is the imaginary unit;

[0060] The inclination angle α and normal stress σ of the shaft elastic curve are introduced to improve the dynamic state of the rotor system, namely:

[0061]

[0062] Among them, M * is the bending moment, M * =Gx, G is gravity; W is the bending section coefficient, W = πd 3 / 32, d is the shaft diameter;

[0063] According to formula (1) (2), the integral simulation method is used to establish similar relationships and obtain similar π groups. According to the π theorem, in order to satisfy the dynamic similarity between the prototype rotor system and the model rotor system, the corresponding π values ​​must be equal:

[0064]

[0065] Subscript c represents a similar model, and subscript n represents a prototype;

[0066] The following factors are introduced to simplify the above formula (3), replacing the corresponding variables in the π group so that it can be represented by only simple variables ω, d, ρ, g, and x:

[0067]

[0068] G is the gravity of the rotating shaft; J p is the moment of inertia of the turntable; ρ is the density of the shaft material; d is the shaft diameter; ω is the rotation speed of the system;

[0069] Taking E, d, x, and ρ of the rotating shaft as design parameters, the similarity ratio of the rotating shaft is obtained as follows:

[0070]

[0071] When the material remains unchanged before and after similar conversion, κ ρ =κ E =κ g =1, we get the formula:

[0072]

[0073] 2) According to the moment of inertia and mass formula of the turntable, the similarity relationship of the turntable can be obtained:

[0074]

[0075]

[0076] Where b is the ratio of the inner diameter to the outer diameter of the single-disk crack rotor prototype, D is the outer diameter of the disk; d is the inner diameter of the disk, which is the same as the shaft diameter; H is the disk thickness; ρ' is the disk density; J p is the moment of inertia; κ Jp is the similarity ratio of the turntable moment of inertia; κ m' is the mass similarity ratio of the turntable; c represents the similar model, n represents the prototype. m' is the mass of the turntable;

[0077] When the rotor-support system is similar, the turntable mass and moment of inertia are similar to the shaft mass and moment of inertia, and the material remains unchanged before and after the transformation, so the turntable thickness κ is obtained. Hand turntable diameter κ D Similarity ratio:

[0078]

[0079] 3) Since the dynamics are similar, the ratio of the forces in the rotor system and the bearing system is the same; according to the stiffness formula k = m 1 f 2 And into the bearing mass similarity ratio formula κ m1 =κ d 2 κ x Similarity ratio formula with natural frequency The bearing stiffness similarity ratio can be obtained:

[0080]

[0081] 4) Based on a simple straight crack growing on a rotating shaft, the crack depth a 1 The crack front width b 1 Description, for the crack unit, a block method is adopted, and the crack area is divided from the crack section 7 to both sides into crack module I 8, crack module II 9, crack module III 10, and crack module IV 11; the interface between crack module I 8 and crack module II 9 fits each other without gap, but displacement can occur, so as to simulate the crack; crack module I 8, crack module II 9 and crack module III10, crack module IV 11, as well as crack module III 10 and crack module IV 11 are all connected and cannot be displaced; in order to simulate the connection between the rest of the parts except the crack and the rotating shaft; because the four modules are connected to the rest of the shaft segments and there is no gap between the faces, when the crack is similarly transformed, the module where the crack is located is similarly transformed; the depth of the crack a 1 The same similarity transformation is performed as a fraction of the shaft diameter, that is, the similarity ratio of the crack depth Similarity ratio to shaft diameter κ d same:

[0082]

[0083] by Figure 1 The single-disk cracked rotor shown in FIG. 1 is taken as an example for verification, and the parameters of the rotor are shown in Table 1.

[0084] Table 1 Single disc crack rotor parameters

[0085] Table 1: Typical single-disc crack rotor prototype parameters

[0086]

[0087]

[0088] According to formula (6), first determine the shaft diameter similarity ratio κ d , and the axis length similarity ratio κ is obtained based on the unchanged natural frequency before and after the axis similarity transformation x In this embodiment, the shaft diameter similarity ratio κ is determined based on the similarity ratio between the similar model and the prototype. d is 0.5, Then we know that: x =0.707.

[0089] According to the inner diameter and outer diameter of the turntable 4 in Table 1, b = 0.19 is calculated. d =0.5,κ x =0.707 Substituting it into formula (8), we can get the disk thickness similarity ratio κ H Similarity ratio κ D :

[0090] κ H =0.34

[0091] κ D =0.713

[0092] According to the stiffness similarity ratio formula (9), we can get: κ K =0.177.

[0093] Since crack 7 grows on the rotating shaft, the crack depth similarity ratio Similarity ratio to shaft diameter κ d same:

[0094]

[0095] According to the above steps, the similarity model parameters after calculation are as follows:

[0096] Table 2: Similarity model parameters

[0097]

[0098]

[0099] The natural frequency, critical speed and modal diagram of the single-disk cracked rotor before and after similarity transformation can be obtained through rotor dynamics calculation. The critical speed and natural frequency are shown in Table 3 and Table 4. The first three modal diagrams of the single-disk cracked rotor are shown in Figure 4 and Figure 5 shown.

[0100] Table 3: Comparison of critical speed of single-disk cracked rotor

[0101] Modal Single disc cracked rotor prototype Similar models error 1 2045.8 2053.4 0.37％ 2 13926 14366 3.16％ 3 22804 23091 1.26％

[0102] Table 4: Comparison of natural frequencies of single-disk cracked rotors

[0103] Modal Single disc cracked rotor prototype Similar models error 1 34.091 34.22 0.38％ 2 203.16 210.84 3.78％ 3 379.04 384.33 1.40％

[0104] By comparing the results, we can see that the method has a high accuracy for the first three critical speeds and natural frequencies through the error calculation of the prototype and similar models, which can meet the general requirements. Figure 4 and Figure 5 From the first three modal diagrams of the prototype and the similar model, we can see that their changing trends are basically the same, which means that the similar model can better describe the modal vibration shape of the prototype.

[0105] Through the above analysis, it can be seen that the method of the present invention can perform effective similarity conversion on a single-disk cracked rotor.

[0106] The embodiments are preferred implementations of the present invention, but the present invention is not limited to the above-mentioned implementations. Any obvious improvements, substitutions or modifications that can be made by those skilled in the art without departing from the essential content of the present invention belong to the protection scope of the present invention.

Claims

1. A method for evaluating the dynamic characteristics of a single-disk cracked rotor based on similarity theory. It is characterized in that The steps include: The first step is to divide the single-disc crack rotor prototype into three parts: the shaft, the rotating disk and the crack, and determine the size of each part of the single-disc crack rotor prototype; The second step is to determine the similarity ratio κ of the shaft diameter according to the similarity conversion requirements. d , and then the axial length similarity ratio κ is obtained in turn according to the formula derived from the similarity theory x , similarity ratio of the turntable diameter κ D , similarity ratio of the turntable thickness κ H Similarity ratio of bearing stiffness κ K ; Crack depth similarity ratio κ a1 Similarity ratio to shaft diameter κ d same; The third step is to obtain a similar model based on the prototype size of the single-disk cracked rotor and the conversion of the similarity ratios of each part; The fourth step is to conduct rotor dynamics analysis on the prototype and similar models to analyze their critical speed and natural frequency; Among them, in the third step, the specific steps of using similarity theory to perform similarity theory transformation on a single-disk cracked rotor are: 1) According to the vibration equation of the rotor system: m is the rotor mass per unit length; E is the elastic modulus of the shaft; I is the cross-sectional distance of the shaft; a is the moment of inertia of the turntable mass per unit length relative to the rotating shaft; p is the rotor unbalance force; p=me 1 f 2 , where f is the rotor natural frequency, e 1 is the unbalance curve; x is the length of the shaft; y is the composite deflection of the shaft; t is the time; i is the imaginary unit; The inclination angle α and normal stress σ of the shaft elastic curve are introduced to improve the dynamic state of the rotor system, namely: Among them, M * is the bending moment, M * =Gx, G is the gravity of the rotating shaft; W is the bending section coefficient, W = πd 3 / 32, d is the shaft diameter; According to formula (1) (2), the integral simulation method is used to establish similar relationships and obtain similar π groups. According to the π theorem, in order to satisfy the dynamic similarity between the prototype rotor system and the model rotor system, the corresponding π values ​​must be equal: Subscript c represents a similar model, and subscript n represents a prototype; The following factors are introduced to simplify the above formula (3), replacing the corresponding variables in the π group so that it can be represented by only simple variables ω, d, ρ, g, and x: G is the gravity of the rotating shaft; J p is the moment of inertia of the turntable; ρ is the density of the shaft material; d is the shaft diameter; ω is the rotation speed of the system; taking E, d, x, ρ of the shaft as the design parameters, the obtained shaft similarity ratio is: When the material remains unchanged before and after similar conversion, κ ρ =κ E =κ g =1, we get the formula: 2) According to the moment of inertia and mass formula of the turntable, the similarity relationship of the turntable can be obtained: Where b is the ratio of the inner diameter to the outer diameter of the single-disk crack rotor prototype, D is the outer diameter of the disk; d is the inner diameter of the disk, which is the same as the shaft diameter; H is the disk thickness; ρ' is the disk density; J p is the moment of inertia; κ Jp is the similarity ratio of the turntable moment of inertia; κ m' is the turntable mass similarity ratio; c represents the similar model, n represents the prototype, and m' is the turntable mass; When the rotor-support system is similar, the turntable mass and moment of inertia are similar to the shaft mass and moment of inertia, and the material remains unchanged before and after the transformation, so the turntable thickness κ is obtained. H and turntable diameter κ D Similarity ratio: 3) Since the dynamics are similar, the ratio of the forces in the rotor system and the bearing system is the same; according to the stiffness formula k = m 1 f 2 And into the bearing mass similarity ratio formula κ m1 =κ d 2 κ x Similarity ratio formula with natural frequency The bearing stiffness similarity ratio can be obtained: 4) Based on a simple straight crack growing on a rotating shaft, the crack depth a 1 The crack front width b 1 Description, for the crack unit, a block method is adopted, and the crack area is divided from the crack (7) section to both sides into crack module I (8), crack module II (9), crack module III (10), and crack module IV (11); the interface between crack module I (8) and crack module II (9) fits each other without gap, but can be displaced, so as to simulate the crack; crack module I (8), crack module II (9) and crack module III (10), crack module IV (11), as well as crack module III (10) and crack module IV (11) are all in a connected state and cannot be displaced; in order to simulate the connection between the rest of the parts except the crack and the rotating shaft; because the four modules are connected to the rest of the shaft segments and there is no gap between the surfaces, when the crack is similarly transformed, the module where the crack is located is similarly transformed; the depth of the crack a 1 The same similarity transformation is performed as a part of the shaft diameter, that is, the similarity ratio κ of the crack depth a1 Similarity ratio to shaft diameter κ d same: k a1 =k d (10)。 2. According to the method for evaluating dynamic characteristics of a single-disk cracked rotor based on similarity theory in claim 1, It is characterized in that The dimensions of the single-disk cracked rotor prototype determined in the first step include the shaft length, the shaft diameter, the thickness of the disk, the outer and inner diameters of the disk, and the depth of the crack.

3. According to the single-disk crack rotor dynamic characteristics evaluation method based on similarity theory as claimed in claim 1, Features: In formula (6), the shaft diameter similarity ratio κ is first determined d According to the invariance of the natural frequency before and after the axis similarity transformation, the axis length similarity ratio κ is obtained x , using the formula:

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