CN108427857B - Rotor variable stiffness determination method and system based on any spatial oblique crack - Google Patents

Rotor variable stiffness determination method and system based on any spatial oblique crack Download PDF

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CN108427857B
CN108427857B CN201810319424.2A CN201810319424A CN108427857B CN 108427857 B CN108427857 B CN 108427857B CN 201810319424 A CN201810319424 A CN 201810319424A CN 108427857 B CN108427857 B CN 108427857B
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焦卫东
常永萍
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Zhejiang Normal University CJNU
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Abstract

The invention discloses a variable stiffness determination method and system based on a space arbitrary oblique crack rotor, wherein the method comprises the following steps: determining a total strain energy model of a spatial arbitrary oblique crack rotor; determining the flexibility parameter of the rotor with any oblique cracks in space according to the total strain energy model; determining a variable stiffness model of the rotor with any spatial oblique cracks according to the flexibility parameter; and inputting the parameter values of the space arbitrary oblique crack rotor to be detected into the variable stiffness model to determine the variable stiffness of the space arbitrary oblique crack rotor so as to determine the variable stiffness of the space arbitrary oblique crack rotor, and thus, the rotor strength can be conveniently designed according to the variable stiffness in the later period and the fault diagnosis can be conveniently carried out on the rotor.

Description

Rotor variable stiffness determination method and system based on any spatial oblique crack
Technical Field
The invention relates to the technical field of determining variable stiffness of a rotor, in particular to a method and a system for determining variable stiffness of the rotor based on any spatial oblique crack.
Background
Modern rotor systems work in harsh mechanical and thermal stress environments for a long time, fatigue cracks occur occasionally, and the operation safety of the system is seriously threatened if the fatigue cracks are not monitored in time. The crack dynamics is the basis of rotor crack monitoring, and relates to the contents of dynamic modeling and analysis of cracks and the like.
The prior art studies are mostly directed to transverse cracks (crack planes perpendicular to both the rotor axis and the base plane) as shown in fig. 3. In recent years, the research direction has started to focus on oblique cracks. Oblique cracks result in more coupling of different parameters in the stiffness matrix than transverse cracks, which in turn results in coupling of rotor bending, torsion and even longitudinal vibrations. Bachschmid et al teach that the rotor crack plane is normally perpendicular to the shaft axis direction (i.e., transverse), but that cracks propagate in the helical direction under high torque and high bending moment loads, thereby forming helical or skewed cracks. Based on a fine three-dimensional nonlinear model, a simpler one-dimensional finite element model of the crack rotor is established, and numerical simulation and experimental verification are carried out on the crack propagation behavior under the combined action of the transverse crack breathing mechanism and the inclined crack bending torsion. Han et al have conducted numerical simulation studies on the dynamic characteristics of a gear-rotor-bearing system containing oblique cracks, considering that large torque due to gear meshing forces tends to induce oblique crack propagation. The variable stiffness characteristics of the transverse crack rotor and the inclined crack rotor are compared by Liangliu in China, and the typical nonlinear dynamic characteristics of a double-disc double-breathing type crack rotor system are further researched. Summer constant and the like analyze the influence factors of the rigidity of the inclined crack rotor in the state that the cracks are completely opened. However, most of these characteristics that determine the variable stiffness of the rotor are for transverse-oblique or straight-oblique cracks in which the crack plane is oblique to the axis of the shaft and perpendicular to the base plane of the shaft, as shown in fig. 4. For any spatial oblique crack shown in fig. 5-6 (i.e. the crack plane is not perpendicular to the axis of the rotating shaft and the base plane of the rotating shaft), the geometric model of the crack has substantial difference compared with the transverse and transverse-oblique geometric models, so that the methods for determining the variable stiffness characteristics of the rotor are also greatly different, and in addition, the derivation process is extremely complex, so that the variable stiffness characteristics of the rotor for any spatial oblique crack have not been researched so far. Based on the above problems, how to determine the variable stiffness characteristics of the rotor based on the spatial arbitrary oblique cracks becomes a technical problem to be solved urgently in the field.
Disclosure of Invention
The invention aims to provide a variable stiffness determining method and system based on a space arbitrary oblique crack rotor, so as to determine the variable stiffness of the space arbitrary oblique crack rotor, and facilitate designing the rotor strength according to the variable stiffness and performing fault diagnosis on the rotor at the later stage.
In order to achieve the above object, the present invention provides a variable stiffness determination method for a rotor based on a spatially arbitrary oblique crack, the method comprising:
determining a total strain energy model of a spatial arbitrary oblique crack rotor;
determining the flexibility parameter of the rotor with any oblique cracks in space according to the total strain energy model;
determining a variable stiffness model of the rotor with any spatial oblique cracks according to the flexibility parameter;
inputting the parameter value of the space arbitrary oblique crack rotor to be detected into the variable stiffness model to determine the variable stiffness of the space arbitrary oblique crack rotor; the parameter values of the space random inclined crack rotor to be detected are the length l, the diameter D, the crack depth a and the crack surface space azimuth angle theta of the space random inclined crack rotor1And theta2At least one of them.
Optionally, the determining a total strain energy model of the spatial arbitrary oblique crack rotor specifically includes:
acquiring strain energy of a crack-free rotor;
determining stress density factors of three displacement modes of crack surface opening, slipping and shearing;
determining external strain energy according to stress density factors of three displacement modes of crack surface opening, slipping and shearing;
and determining a total strain energy model of the oblique crack rotor in any space according to the strain energy and the external strain energy of the crack-free rotor.
Optionally, the external strain energy is determined according to stress density factors of three displacement modes of crack face opening, slipping and shearing, and the specific formula is as follows:
Figure BDA0001624858630000021
wherein E ═ E/(1-v), m s1+ v, v is Poisson's ratio, Ki Ⅰ、Ki ⅡAnd Ki ⅢStress density factors U describing three displacement modes of crack surface opening, sliding and shearingcFor the applied strain energy, E is the Young's modulus, A is the cross-sectional area of the rotor shaft, and dA is the derivative of A in the integral operation.
Optionally, the flexibility parameter of any spatial oblique crack rotor is determined according to the total strain energy model, and the specific formula is as follows:
Figure BDA0001624858630000031
wherein, U0+UcAs a model of total strain energy, U0Strain energy of flawless rotor, UcFor the application of strain energy uiFor nodal displacement and nodal force along the ith coordinate direction, PiIs the nodal force along the ith coordinate direction, PjIs the nodal force along the jth coordinate direction.
Optionally, the variable stiffness model of the rotor with any spatial oblique cracks is determined according to the flexibility parameter, and the specific formula is as follows:
Kc＝TG-1TT；
wherein G ═ { G ═ Gij},i＝1、2、…、6，j＝1、2、…、6，gijThe flexibility parameter of the rotor with any oblique cracks in space is shown, and T is a transformation matrix.
The invention also provides a variable stiffness determination system based on the space arbitrary oblique crack rotor, which comprises the following components:
the total strain energy model determining module is used for determining a total strain energy model of the rotor with any oblique cracks in space;
the compliance parameter determining module is used for determining compliance parameters of the rotor with any oblique cracks in space according to the total strain energy model;
the variable stiffness model determining module is used for determining a variable stiffness model of the rotor with any oblique cracks in space according to the flexibility parameter;
the variable stiffness determining module is used for inputting the parameter values of the any oblique crack rotor in the space to be detected into the variable stiffness model to determine the variable stiffness of the any oblique crack rotor in the space; the parameter values of the space random inclined crack rotor to be detected are the length l, the diameter D, the crack depth a and the crack surface space azimuth angle theta of the space random inclined crack rotor1And theta2At least one of them.
Optionally, the total strain energy model determining module specifically includes:
an acquisition unit for acquiring strain energy of a crack-free rotor;
the stress density factor determining unit is used for determining the stress density factors of three displacement modes of crack surface opening, slippage and shearing;
the external strain energy determining unit is used for determining external strain energy according to stress density factors of three displacement modes of crack surface opening, slippage and shearing;
and the total strain energy model determining unit is used for determining a total strain energy model of the oblique crack rotor in any space according to the strain energy and the external strain energy of the crack-free rotor.
Optionally, the external strain energy is determined according to stress density factors of three displacement modes of crack face opening, slipping and shearing, and the specific formula is as follows:
Figure BDA0001624858630000041
wherein E ═ E/(1-v), m s1+ v, v is Poisson's ratio, Ki Ⅰ、Ki ⅡAnd Ki ⅢStress density factors U describing three displacement modes of crack surface opening, sliding and shearingcFor the applied strain energy, E is the Young's modulus, A is the cross-sectional area of the rotor shaft, and dA is the derivative of A in the integral operation.
Optionally, the flexibility parameter of any spatial oblique crack rotor is determined according to the total strain energy model, and the specific formula is as follows:
Figure BDA0001624858630000042
wherein, U0+UcAs a model of total strain energy, U0Strain energy of flawless rotor, UcFor the application of strain energy uiFor nodal displacement and nodal force along the ith coordinate direction, PiIs the nodal force along the ith coordinate direction, PjIs the nodal force along the jth coordinate direction.
Optionally, the variable stiffness model of the rotor with any spatial oblique cracks is determined according to the flexibility parameter, and the specific formula is as follows:
Kc＝TG-1TT；
wherein G ═ { G ═ Gij},i＝1、2、…、6，j＝1、2、…、6，gijThe flexibility parameter of the rotor with any oblique cracks in space is shown, and T is a transformation matrix.
According to the specific embodiment provided by the invention, the invention discloses the following technical effects:
firstly, determining a total strain energy model of a rotor with any spatial oblique cracks; then determining the flexibility parameter of the rotor with any oblique cracks in space according to the total strain energy model; determining a variable stiffness model of the rotor with any spatial oblique cracks according to the flexibility parameter; finally, inputting the parameter value of the space arbitrary oblique crack rotor to be detected into the variable stiffness model to determine the variable stiffness of the space arbitrary oblique crack rotor; the variable stiffness of the rotor with any oblique cracks in the space is determined, and the design of the rotor strength and the fault diagnosis of the rotor according to the variable stiffness in the later period are facilitated.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
FIG. 1 is a flow chart of a variable stiffness determination method of a rotor based on any spatial oblique crack according to an embodiment of the invention;
FIG. 2 is a structural diagram of a variable stiffness determination system of a rotor based on any oblique crack in space according to an embodiment of the invention;
FIG. 3 is a transverse crack rotor diagram according to an embodiment of the present invention;
FIG. 4 is a cross-skewed crack rotor diagram according to an embodiment of the present invention;
FIG. 5 is a diagram of a spatially arbitrary skewed crack rotor of an embodiment of the present invention;
FIG. 6 is a comprehensive comparison graph of the stress of three types of cracked rotors in the embodiment of the invention;
FIG. 7 is a graph comparing the stiffness of three types of cracked rotors according to the example of the invention;
FIG. 8 is a graph of the variation of stiffness characteristics of a spatially random skewed crack rotor with the variation of the attitude of the crack face in accordance with an embodiment of the present invention;
FIG. 9 is a graph of the variation of stiffness characteristics of a spatially random skewed crack rotor with crack depth according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention aims to provide a variable stiffness determining method and system based on a space arbitrary oblique crack rotor, so as to determine the variable stiffness of the space arbitrary oblique crack rotor, and facilitate designing the rotor strength according to the variable stiffness and performing fault diagnosis on the rotor at the later stage.
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
FIG. 1 is a flow chart of a variable stiffness determination method of a rotor based on any spatial oblique crack according to an embodiment of the invention; as shown in fig. 1, the present invention provides a variable stiffness determination method for a rotor based on any spatial oblique crack, the method comprising:
step 10: and determining a total strain energy model of the spatial arbitrary oblique crack rotor.
Step 20: and determining the flexibility parameter of the rotor with any spatial oblique cracks according to the total strain energy model.
Step 30: and determining a variable stiffness model of the rotor with any spatial oblique cracks according to the flexibility parameter.
Step 40: and inputting the parameter values of the space arbitrary oblique crack rotor to be detected into the variable stiffness model to determine the variable stiffness of the space arbitrary oblique crack rotor. The parameter values of the space random inclined crack rotor to be detected are the length l, the diameter D, the crack depth a and the crack surface space azimuth angle theta of the space random inclined crack rotor1And theta2At least one of them.
The following detailed analysis of each step was performed:
step 10: determining a total strain energy model of a spatial arbitrary oblique crack rotor, specifically comprising:
step 101: obtaining strain energy U of crack-free rotor0。
Step 102: determining stress density factors of three displacement modes of crack surface opening, slipping and shearing;
(1) the stress density factor for the open mode is formulated as:
Figure BDA0001624858630000061
Figure BDA0001624858630000062
Figure BDA0001624858630000063
Figure BDA0001624858630000071
Figure BDA0001624858630000072
Figure BDA0001624858630000073
(2) the stress density factor formula of the slip mode is: :
Figure BDA0001624858630000074
Figure BDA0001624858630000075
Figure BDA0001624858630000076
Figure BDA0001624858630000077
Figure BDA0001624858630000078
Figure BDA0001624858630000079
(3) the shear mode stress density factor formula is:
Figure BDA00016248586300000710
Figure BDA0001624858630000081
Figure BDA0001624858630000082
Figure BDA0001624858630000083
Figure BDA0001624858630000084
Figure BDA0001624858630000085
wherein, F1、F2Generalized force, F, of the opening (type I) mode, respectivelyIIGeneralized force in slip (type II) mode, FIIIGeneralized force, P, for shear-off (type III) mode1～P6Respectively, the node forces in the directions of six degrees of freedom, R is the radius of the cross section of the rotor shaft, a is the crack depth, and theta1And theta2Respectively, the angle of the crack plane in space, RbBeta is an integral variable in the crack width direction, k is a shear correction coefficient, and k is 10/9, x is a projection radius on the oblique crack surfaces＝x+βcosθ1X is the distance variable from the crack surface to the left end surface of the rotor shaft, Rb 2＝(R2-β2sin2θ1)/sin2θ2。oi θ1θ2、si θ1θ2And ti θ1θ2The stress corresponding to three displacement modes of crack surface opening (type I), slippage (type II) and shearing (type III) is marked on the stressθ1θ2Indicating the azimuth angle theta of the crack plane1And theta2In the subscript (.)iCorresponding to the force serial numbers of the unit nodes so as to distinguish the stress action on different types of crack surfaces.
Step 103: determining external strain energy according to stress density factors of three displacement modes of crack surface opening, slipping and shearing; the concrete formula is as follows:
Figure BDA0001624858630000086
wherein i is 1,2,3,4,5,6, E' is E/(1-v), m s1+ v, v is Poisson's ratio, Ki Ⅰ、Ki ⅡAnd Ki ⅢStress density describing three displacement modes of crack surface opening, sliding and shearingDegree factor, UcFor the applied strain energy, E is the Young's modulus, A is the cross-sectional area of the rotor shaft, and dA is the derivative of A in the integral operation.
Step 104: determining a total strain energy model of the rotor with any oblique cracks in space according to the strain energy and the external strain energy of the rotor without cracks, wherein the specific formula is as follows:
U＝U0+Uc；
wherein, U0Strain energy of flawless rotor, UcIs the applied strain.
Step 20: determining the flexibility parameter of the rotor with any oblique cracks in space according to the total strain energy model, wherein the specific formula is as follows:
Figure BDA0001624858630000091
wherein, U0+UcAs a model of total strain energy, U0Strain energy of flawless rotor, UcFor the application of strain energy uiFor nodal displacement and nodal force along the ith coordinate direction, PiIs the nodal force along the ith coordinate direction, PjIs the nodal force along the jth coordinate direction.
Figure BDA0001624858630000092
Figure BDA0001624858630000093
Figure BDA0001624858630000101
Figure BDA0001624858630000102
Figure BDA0001624858630000103
Figure BDA0001624858630000104
Figure BDA0001624858630000105
Figure BDA0001624858630000111
Figure BDA0001624858630000112
Figure BDA0001624858630000113
Figure BDA0001624858630000114
Figure BDA0001624858630000121
Figure BDA0001624858630000122
Figure BDA0001624858630000123
Figure BDA0001624858630000124
Figure BDA0001624858630000131
Figure BDA0001624858630000132
Figure BDA0001624858630000133
Figure BDA0001624858630000141
Figure BDA0001624858630000142
Figure BDA0001624858630000143
Wherein, F1、F2Generalized force, F, of the opening (type I) mode, respectivelyIIGeneralized force in slip (type II) mode, FIIIFor shear (type III) mode generalized forces, R is the rotor shaft cross-sectional radius, a is the crack depth, θ1And theta2Respectively, the angle of the crack plane in space, ms1+ v is a new variable defined based on poisson's ratio v, RbBeta is an integral variable in the crack width direction, k is a shear correction coefficient, and k is 10/9, x is a projection radius on the oblique crack surfaces＝x+βcosθ1X is the distance from the crack surface to the left end surface of the rotor shaft, l is the length of the rotor shaft, alphasFor shear deformation parameters, G is the modulus of stiffness, I is the cross-sectional area moment of inertia, E' ═ E/(1-v), ms1+ v, v is Poisson's ratio, Ki Ⅰ、Ki ⅡAnd Ki ⅢStress density factors U describing three displacement modes of crack surface opening, sliding and shearingcFor external strain energy, E is Young's modulus, A is the cross-sectional area of the rotor shaft, dA is the derivative of A in integral operation, Rb 2＝(R2-β2sin2θ1)/sin2θ2，I0Is the cross section active moment of inertia.
Step 30: determining a variable stiffness model of the rotor with any oblique cracks in space according to the flexibility parameter, wherein the specific formula is as follows:
Kc＝TG-1TT；
wherein G ═ { G ═ Gij},i＝1、2、…、6，j＝1、2、…、6，gijThe flexibility parameter of the rotor with any oblique cracks in space is shown, and T is a transformation matrix.
Fig. 2 is a structural diagram of a variable stiffness determining system based on a space random oblique crack rotor according to an embodiment of the present invention, and as shown in fig. 2, the present invention further provides a variable stiffness determining system based on a space random oblique crack rotor, where the system includes:
the total strain energy model determining module 1 is used for determining a total strain energy model of a spatial arbitrary oblique crack rotor;
the compliance parameter determining module 2 is used for determining compliance parameters of the rotor with any spatial oblique cracks according to the total strain energy model;
the variable stiffness model determining module 3 is used for determining a variable stiffness model of the rotor with any spatial oblique cracks according to the flexibility parameter;
the variable stiffness determining module 4 is used for inputting the parameter values of the any oblique crack rotor in the space to be detected into the variable stiffness model to determine the variable stiffness of the any oblique crack rotor in the space; the parameter values of the space random inclined crack rotor to be detected are the length l, the diameter D, the crack depth a and the crack surface space azimuth angle theta of the space random inclined crack rotor1And theta2At least one of them.
The various modules are discussed in detail below:
the total strain energy model determination module 1 specifically includes:
an acquisition unit for acquiring strain energy of a crack-free rotor;
the stress density factor determining unit is used for determining the stress density factors of three displacement modes of crack surface opening, slippage and shearing;
the external strain energy determining unit is used for determining external strain energy according to stress density factors of three displacement modes of crack surface opening, slippage and shearing;
and the total strain energy model determining unit is used for determining a total strain energy model of the oblique crack rotor in any space according to the strain energy and the external strain energy of the crack-free rotor.
The external strain energy determining unit is used for determining external strain energy according to stress density factors of three displacement modes of crack surface opening, sliding and shearing, and the specific formula is as follows:
Figure BDA0001624858630000151
wherein E ═ E/(1-v), m s1+ v, v is Poisson's ratio, Ki Ⅰ、Ki ⅡAnd Ki ⅢStress density factors U describing three displacement modes of crack surface opening, sliding and shearingcFor the applied strain energy, E is the Young's modulus, A is the cross-sectional area of the rotor shaft, and dA is the derivative of A in the integral operation.
The flexibility parameter determining module 2 is used for determining the flexibility parameter of any spatial oblique crack rotor according to the total strain energy model, and the specific formula is as follows:
Figure BDA0001624858630000161
wherein, U0+UcAs a model of total strain energy, U0Strain energy of flawless rotor, UcFor the application of strain energy uiFor nodal displacement and nodal force along the ith coordinate direction, PiIs the nodal force along the ith coordinate direction, PjIs the nodal force along the jth coordinate direction.
The variable stiffness determining module 4 is used for determining a variable stiffness model of any spatial oblique crack rotor according to the flexibility parameter, and the specific formula is as follows:
Kc＝TG-1TT；
wherein G ═ { G ═ Gij},i＝1、2、…、6，j＝1、2、…、6，gijIs the flexibility parameter of the rotor with any oblique cracks in space, T is a transformation matrix,
Figure BDA0001624858630000162
after the variable stiffness model of the rotor with the random oblique cracks in space is obtained, the stiffness change characteristics of different types of cracks can be compared, and the influence rule of the crack types on the stiffness characteristics of the rotor with the cracks is found. In addition, the rigidity change mechanism of the special crack rotor can be determined by researching the variable rigidity characteristic of any spatial oblique crack with different spatial azimuth angles and different crack depths, the multi-freedom-degree coupled vibration characteristic of the rotor with any spatial oblique crack can be further analyzed through dynamic modeling, and the obtained series of analysis results can lay a foundation for the application of rotor strength design, fault diagnosis of a rotor system and the like.
Specific examples thereof
The rotor shaft is respectively modeled by using a Muscovitum beam unit, and the degrees of freedom in the longitudinal direction, the bending direction and the torsion direction are considered. FIG. 6 shows a rotor shaft with arbitrary oblique cracks in space, mass of the rotating disk is m, diameter of the rotating shaft is D, and length of the rotating shaft is L. The crack is located at the left end x and the crack depth is alpha. The unit is stressed as follows: shear force P2、P3And P8、P9Bending moment P5、P6And P11、P12Axial force P1And P7Torque P4And P10。
As the rotor shaft rotates, the cracks open/close alternately, resulting in a periodic variation in the stiffness of the cracked rotor. In order to simulate the variable stiffness characteristic of a crack rotor, crack edges are equally divided into 100 points in one rotation period (360 degrees), and each point determines one CCL crack surface closed line position, so that different integration limits are determined. Thus, the stiffness parameter variation can be considered as a function of CCL position. The invention uses the unit construction of the iron-wood sinco beamThe process of solving the variable stiffness characteristic of the rotor with any oblique cracks in space is explained by taking a rotor shaft as an example, and the determined variable stiffness model of the rotor with any oblique cracks in space not only depends on the types of the cracks, but also depends on the length l, the diameter D, the crack depth a and the space azimuth angle theta of a crack surface of the rotor with any oblique cracks in space1And theta2Is related to (1).
Example one
The invention makes finite element length of arbitrary oblique crack rotor in space 0.7/14-0.05 m, crack depth ratio ā -a/D-0.3, a is crack depth, D is diameter, azimuth angle theta of crack surface1＝45°，θ2And (3) carrying out variable stiffness characteristic analysis by changing the types of the cracks, namely, carrying out transverse, transverse-oblique and spatial arbitrary oblique cracks on the T-type crack, the TS-type crack and the AS-type crack respectively, specifically AS shown in FIG. 7, wherein the stiffness parameters of the T-type crack, the TS-type crack and the AS-type crack are respectively represented by "- - - - - - -", "- - - -" and "-", in FIG. 7.
As can be seen in fig. 7, k11Is a longitudinal stiffness value, k22Is a stiffness value, k, in the horizontal shearing direction33Is a stiffness value, k, in the vertical shearing direction44Is a torsional stiffness value, k55Is a value of stiffness, k, in the vertical bending direction66The horizontal bending direction stiffness value. Spatially arbitrary oblique-crack (AS) rotors have maximum axial and vertical shear stiffness (k) over the entire rotor rotation period compared to transverse-crack and transverse-oblique-crack (T and TS) rotors11And k is33). Torsional stiffness (k) of AS-type cracked rotor at initial rotation44) Slightly smaller than TS type crackle rotor; flexural stiffness (k) of AS-type cracked rotor at end of rotation55And k is66) Slightly smaller than the T-shaped crack rotor. However, the rigidity is almost the greatest as a whole. However, the horizontal shear stiffness (k) of AS-type cracked rotors during most of the rotor rotation22) Is significantly less than the other two types of cracked rotors, although its stiffness is significantly improved at the end of rotation. In addition, torsional stiffness (k) of AS-and TS-type cracked rotors44) Very close, bending stiffness (k)55And k is66) The change trends are similar, which shows thatAzimuth theta of crack plane in torsion and bending directions1Is a major factor affecting the stiffness characteristics of the rotor. And in other directions, the azimuth angle theta2Plays an important role in causing the AS type crack rotor to have obviously different and more complex variable stiffness characteristics from the other two types of crack rotors, particularly in the horizontal shearing direction (k)22) The performance is more obvious, the symmetry of the stiffness curve is seriously damaged, the stiffness value is obviously reduced, and the whole performance is smaller than that of other two types of cracked rotors.
Example two
Maintaining crack plane azimuth θ1Theta is constant at 45 DEG2Varying between 30 deg. and 90 deg.. The stiffness characteristic curve of the AS-type cracked rotor under the combined action of different azimuth angles is shown in FIG. 8. In FIG. 8, the lines "- - - - -", "- + -", and "- - -", are shown as lines in terms of θ2The order of the increments is described for comparison. It should be noted that when θ2At 90 ° the AS cracks regress to TS cracks.
As can be seen in FIG. 8, with θ2Increase of (2), longitudinal stiffness value (k)11) Vertical shear direction stiffness value (k)33) Torsional stiffness value (k)44) And a vertical bending stiffness value (k)55) All exhibit a monotonically decreasing trend. However, the stiffness value (k) in the horizontal bending direction66) And a horizontal shear direction stiffness value (k)22) The monotonous descending trend does not exist, the rigidity change curves are obviously crossed with each other, and the azimuth angle theta of the crack surface of the AS-type crack rotor is shown1And theta2The interaction effect between them. It is clear that this effect has a different effect on the rotor stiffness at different angular positions throughout the rotor's rotational cycle and is therefore dynamic and non-linear.
EXAMPLE III
Further, the crack face azimuth angle theta of the AS-type cracked rotor is maintained1＝45°，θ2The crack depth ratio ā varied from 0.1 to 0.5 at 60 ° left unchanged. The variation curves of the stiffness characteristics corresponding to different crack depths are shown in FIG. 9, in which the lines "" are "- - - -", "-" and- "respectively+ - "and" -o- "are depicted in increasing order of ā.
As can be seen from FIG. 9, AS ā increases, the stiffness parameter k of the AS-type cracked rotor in all directionsiiI is monotonically decreased between 1 and 6. The stiffness value is continuously changed at different rotation angle positions of the whole rotation period of the rotor, but the variable stiffness characteristic curves of the rotor corresponding to different crack depths almost have the same trend and change trend. This shows that the effect of the crack depth parameter on the rotor stiffness is also dynamic, but has a pronounced linear character.
For the system disclosed by the embodiment, the description is relatively simple because the system corresponds to the method disclosed by the embodiment, and the relevant points can be referred to the method part for description.
The principles and embodiments of the present invention have been described herein using specific examples, which are provided only to help understand the method and the core concept of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, the present disclosure should not be construed as limiting the invention.

Claims (8)

1. A variable stiffness determination method based on a space arbitrary oblique crack rotor is characterized by comprising the following steps:
determining a total strain energy model of a spatial arbitrary oblique crack rotor, specifically comprising: acquiring strain energy of a crack-free rotor; determining stress density factors of three displacement modes of crack surface opening, slipping and shearing; determining external strain energy according to stress density factors of three displacement modes of crack surface opening, slipping and shearing; determining a total strain energy model of the oblique crack rotor in any space according to the strain energy and the external strain energy of the crack-free rotor;
the stress density factor formula of the opening mode is specifically as follows:
Figure FDA0003125827360000011
Figure FDA0003125827360000012
Figure FDA0003125827360000013
Figure FDA0003125827360000014
Figure FDA0003125827360000015
Figure FDA0003125827360000016
the formula of the stress density factor of the slip mode is specifically as follows:
Figure FDA0003125827360000017
Figure FDA0003125827360000021
Figure FDA0003125827360000022
Figure FDA0003125827360000023
Figure FDA0003125827360000024
Figure FDA0003125827360000025
the formula of the stress density factor of the shear mode is specifically as follows:
Figure FDA0003125827360000026
Figure FDA0003125827360000027
Figure FDA0003125827360000028
Figure FDA0003125827360000029
Figure FDA00031258273600000210
Figure FDA00031258273600000211
wherein, F1、F2Generalized force in the opening mode, I-type, FIIGeneralized force in slip mode, type II, FIIIGeneralized force for shear mode, type III, P1～P6Respectively, the node forces in the directions of six degrees of freedom, R is the radius of the cross section of the rotor shaft, a is the crack depth, and theta1And theta2Respectively, the angle of the crack plane in space, RbBeta is an integral variable in the crack width direction, k is a shear correction coefficient, and k is 10/9, x is taken as the projection radius on the inclined crack surfaces＝x+βcosθ1X is the distance variable from the crack surface to the left end surface of the rotor shaft, Rb 2＝(R2-β2sin2θ1)/sin2θ2；oi θ1θ2、si θ1θ2And ti θ1θ2Stress corresponding to three displacement modes of crack surface opening, sliding and shearing respectively, marked thereon (·)θ1θ2Indicating the azimuth angle theta of the crack plane1And theta2In the subscript (.)iCorresponding to the force serial numbers of the unit nodes to distinguish the stress action on different types of crack surfaces;
determining the flexibility parameter of the rotor with any oblique cracks in space according to the total strain energy model;
determining a variable stiffness model of the rotor with any spatial oblique cracks according to the flexibility parameter;
inputting the parameter value of the space arbitrary oblique crack rotor to be detected into the variable stiffness model to determine the variable stiffness of the space arbitrary oblique crack rotor; the parameter values of the space random inclined crack rotor to be detected are the length l, the diameter D, the crack depth a and the crack surface space azimuth angle theta of the space random inclined crack rotor1And theta2At least one of them.
2. The variable stiffness determination method based on the space arbitrary oblique crack rotor according to claim 1, wherein the applied strain energy is determined according to stress density factors of three displacement modes of crack face opening, slip and shearing, and the specific formula is as follows:
Figure FDA0003125827360000031
wherein E ═ E/(1-v), ms1+ v, v is Poisson's ratio, Ki Ⅰ、Ki ⅡAnd Ki ⅢStress density factors U describing three displacement modes of crack surface opening, sliding and shearingcFor the applied strain energy, E is the Young's modulus, A is the cross-sectional area of the rotor shaft, and dA is the derivative of A in the integral operation.
3. The variable stiffness determination method based on the space arbitrary oblique crack rotor according to claim 1, wherein the flexibility parameter of the space arbitrary oblique crack rotor is determined according to the total strain energy model, and the specific formula is as follows:
Figure FDA0003125827360000041
wherein, U0+UcAs a model of total strain energy, U0Strain energy of flawless rotor, UcFor the application of strain energy uiFor nodal displacement and nodal force along the ith coordinate direction, PiIs the nodal force along the ith coordinate direction, PjIs the nodal force along the jth coordinate direction.
4. The variable stiffness determination method based on the space arbitrary oblique crack rotor according to claim 1, wherein the variable stiffness model of the space arbitrary oblique crack rotor is determined according to the flexibility parameter, and the specific formula is as follows:
Kc＝TG-1TT；
wherein G ═ { G ═ Gij},i＝1、2、…、6，j＝1、2、…、6，gijThe flexibility parameter of the rotor with any oblique cracks in space is shown, and T is a transformation matrix.
5. A system for determining variable stiffness of a spatially arbitrary skewed crack-based rotor, the system comprising:
the total strain energy model determining module is used for determining a total strain energy model of any spatial oblique crack rotor, and specifically comprises the following steps: an acquisition unit for acquiring strain energy of a crack-free rotor; the stress density factor determining unit is used for determining the stress density factors of three displacement modes of crack surface opening, slippage and shearing; the external strain energy determining unit is used for determining external strain energy according to stress density factors of three displacement modes of crack surface opening, slippage and shearing; the total strain energy model determining unit is used for determining a total strain energy model of the oblique crack rotor in any space according to the strain energy and the external strain energy of the crack-free rotor;
the stress density factor formula of the opening mode is specifically as follows:
Figure FDA0003125827360000042
Figure FDA0003125827360000051
Figure FDA0003125827360000052
Figure FDA0003125827360000053
Figure FDA0003125827360000054
Figure FDA0003125827360000055
the formula of the stress density factor of the slip mode is specifically as follows:
Figure FDA0003125827360000056
Figure FDA0003125827360000057
Figure FDA0003125827360000058
Figure FDA0003125827360000059
Figure FDA00031258273600000510
Figure FDA00031258273600000511
the formula of the stress density factor of the shear mode is specifically as follows:
Figure FDA0003125827360000061
Figure FDA0003125827360000062
Figure FDA0003125827360000063
Figure FDA0003125827360000064
Figure FDA0003125827360000065
Figure FDA0003125827360000066
wherein, F1、F2Generalized force in the opening mode, I-type, FIIGeneralized force in slip mode, type II, FIIIGeneralized force for shear mode, type III, P1～P6Respectively, the node forces in the directions of six degrees of freedom, R is the radius of the cross section of the rotor shaft, a is the crack depth, and theta1And theta2Respectively, the angle of the crack plane in space, RbBeta is an integral variable in the crack width direction, k is a shear correction coefficient, and k is 10/9, x is taken as the projection radius on the inclined crack surfaces＝x+βcosθ1X is the distance variable from the crack surface to the left end surface of the rotor shaft, Rb 2＝(R2-β2sin2θ1)/sin2θ2；oi θ1θ2、si θ1θ2And ti θ1θ2Stress corresponding to three displacement modes of crack surface opening, sliding and shearing respectively, marked thereon (·)θ1θ2Indicating the azimuth angle theta of the crack plane1And theta2In the subscript (.)iCorresponding to the force serial numbers of the unit nodes to distinguish the stress action on different types of crack surfaces;
the compliance parameter determining module is used for determining compliance parameters of the rotor with any oblique cracks in space according to the total strain energy model;
the variable stiffness model determining module is used for determining a variable stiffness model of the rotor with any oblique cracks in space according to the flexibility parameter;
the variable stiffness determining module is used for inputting the parameter values of the any oblique crack rotor in the space to be detected into the variable stiffness model to determine the variable stiffness of the any oblique crack rotor in the space; the space to be detected is arbitrarily oblique crackThe parameter values of the textured rotor are the length l, the diameter D, the crack depth a and the crack surface space azimuth angle theta of the spatially arbitrary oblique crack rotor1And theta2At least one of them.
6. The variable stiffness determination system based on the space arbitrary oblique crack rotor as claimed in claim 5, wherein the applied strain energy is determined according to stress density factors of three displacement modes of crack face opening, slip and shearing, and the specific formula is as follows:
Figure FDA0003125827360000071
wherein E ═ E/(1-v), ms1+ v, v is Poisson's ratio, Ki Ⅰ、Ki ⅡAnd Ki ⅢStress density factors U describing three displacement modes of crack surface opening, sliding and shearingcFor the applied strain energy, E is the Young's modulus, A is the cross-sectional area of the rotor shaft, and dA is the derivative of A in the integral operation.
7. The variable stiffness determination system based on the space arbitrary oblique crack rotor according to claim 5, wherein the flexibility parameter of the space arbitrary oblique crack rotor is determined according to the total strain energy model, and the specific formula is as follows:
Figure FDA0003125827360000072
wherein, U0+UcAs a model of total strain energy, U0Strain energy of flawless rotor, UcFor the application of strain energy uiFor nodal displacement and nodal force along the ith coordinate direction, PiIs the nodal force along the ith coordinate direction, PjIs the nodal force along the jth coordinate direction.
8. The variable stiffness determination system based on the space random oblique crack rotor as claimed in claim 5, wherein the variable stiffness model of the space random oblique crack rotor is determined according to the flexibility parameter, and the specific formula is as follows:
Kc＝TG-1TT；
wherein G ═ { G ═ Gij},i＝1、2、…、6，j＝1、2、…、6，gijThe flexibility parameter of the rotor with any oblique cracks in space is shown, and T is a transformation matrix.
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