CN114912256B - Spiral bevel gear vibration response analysis method containing crack fault - Google Patents
Spiral bevel gear vibration response analysis method containing crack fault Download PDFInfo
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Abstract
The method comprises the steps of obtaining parameters of the spiral bevel gear and parameters of root crack faults; equidistant slicing is carried out on the spiral bevel gear along the tooth width direction, and parameters of gear slicing are obtained; determining the meshing state of the gear slices under different rotation angles, and calculating the relation between the meshing range and the meshing position of the gear slices and the rotation angles; establishing a single-piece spiral bevel gear crack meshing stiffness calculation model by adopting a potential energy method, and solving to obtain the gear time-varying meshing stiffness under spiral bevel gear crack faults; and establishing a dynamic model under the arc tooth bevel gear crack fault by adopting a centralized parameter method, and solving to obtain the vibration response characteristic under the arc tooth bevel gear crack fault. The invention has the advantages of accurate calculation and high efficiency. The meshing stiffness obtained by the method can be used for detection and diagnosis of arc tooth bevel gear crack faults and research of vibration response characteristics.
Description
Technical Field
The invention belongs to the technical field of health monitoring and fault diagnosis of a transmission system, and particularly relates to a vibration response analysis method for a spiral bevel gear with a crack fault.
Background
The spiral bevel gear has the advantages of high transmission ratio, stable transmission, strong bearing capacity, compact structure, low noise and the like, so that the spiral bevel gear is widely applied to power and motion transmission among staggered shafts and is a key component for power and motion transmission in a transmission system. Gear tooth root cracks are used as a common gear fault form, can cause vibration and noise increase of a transmission system, are key causes of serious faults such as gear breakage and the like, and seriously affect the stability and safety of a spiral bevel gear transmission system. In order to monitor the running state of the spiral bevel gear transmission system by means of vibration signals and the like, dynamic analysis is needed to be carried out on the transmission system after the occurrence of gear tooth root cracks, vibration response characteristics when crack faults occur are researched, and a basis for monitoring and diagnosing the gear tooth root crack faults is provided. The gear meshing stiffness is used as important internal excitation in a gear transmission system, and whether the meshing stiffness after the occurrence of tooth root cracks of a gear can be accurately obtained and a dynamic model under the fault of the spiral bevel gear cracks is established is a key for researching the fault dynamics of the spiral bevel gear cracks.
The above information disclosed in the background section is only for enhancement of understanding of the background of the invention and therefore may contain information that does not form the prior art that is already known to a person of ordinary skill in the art.
Disclosure of Invention
The invention aims to provide a vibration response analysis method for a spiral bevel gear with a crack fault, which is used for accurately calculating the time-varying meshing stiffness under the crack fault of the spiral bevel gear, establishing a dynamic model under the crack fault of the spiral bevel gear and solving the dynamic model.
In order to achieve the above object, the present invention provides the following technical solutions:
the invention relates to a spiral bevel gear vibration response analysis method containing crack faults, which comprises the following steps:
step 1, measuring to obtain parameters of a spiral bevel gear and parameters of a tooth root crack, wherein the parameters of the gear comprise tooth number, large end modulus, tooth width, pressure angle, radial deflection coefficient, top clearance coefficient, tooth top height coefficient and tooth width midpoint spiral angle, and the parameters of the tooth root crack comprise crack depth q k Crack length l k Crack propagation angle v k ；
Step 2, equidistant slicing is carried out on the spiral bevel gear along the tooth width direction to obtain a slicing gear and slice gear parameters are obtained, wherein the slice gear parameters comprise the width of each slice, the distance from the large end of the gear, the radius of a reference circle, the radius of a base circle, the radius of a root circle and the radius of a top circle;
step 3, determining meshing states of the slicing gears under different rotation angles based on the slicing gear parameters, and calculating the relation between the meshing point position of the slicing gears and the rotation angles of the gears;
step 4, based on the relation between the meshing point position and the gear corner, establishing a crack meshing stiffness calculation model of the single-piece spiral bevel gear by adopting a potential energy method and solving the crack meshing stiffness calculation model, so as to obtain the time-varying meshing stiffness of the gear under the crack failure of the spiral bevel gear;
and 5, establishing a dynamic model under the transmission crack fault of the spiral bevel gear based on the time-varying meshing stiffness of the gear by adopting a centralized parameter method, and solving to obtain the vibration response characteristic under the transmission crack fault of the spiral bevel gear.
In the vibration response analysis method for the spiral bevel gear with the crack fault, in the step 2, the spiral bevel gear is equally divided into a plurality of pieces along the tooth width direction, the tooth width of each piece of slicing gear is delta b=b/N, b is the tooth width of the spiral bevel gear, N is the number of gear slices, and the slicing parameters of each piece of slicing gear are obtained through calculation.
In the vibration response analysis method of the spiral bevel gear with the crack fault, in the step 3), the relationship between the meshing point position and the rotation angle of the slicing gear is as follows:
wherein θ k1 、θ k2 Respectively represent the large end face rotation angle alpha of the main and driven wheels of the kth gear k1，p 、α k1，g Respectively represent the angle position alpha of the meshing point on the large end face of the main driven wheel of the kth gear tk Represents the pressure angle of the large end face of the kth gear,the rotation angles of the k-th piece of gears of the driving wheel and the driven wheel relative to the large end faces are respectively shown.
In the spiral bevel gear vibration response analysis method containing the crack fault, in the step 4), the rigidity of the gear comprises: hertz contact stiffness k kh Bending stiffness k kb Shear deformation stiffness k ks Stiffness in axial compression k ka And elastic matrix deformation stiffness k kf The calculation of the crack meshing stiffness of the single piece spiral bevel gear with an equivalent tooth count of less than 42 is performed in two cases:
case 1: h is a ka ≥h ko &α k1 ≥α ka
Case 2: h is a ka ＜h ko or h ka ≥h ko &α k1 ≥α ka
Wherein v is poisson's ratio, E, G is modulus and shear modulus, respectively, α k Is the pressure angle of any point on the tooth surface, alpha ka For the pressure angle, alpha, corresponding to the crack end position kr For the root circle half-tooth angle alpha corresponding to the crack end position k1 、α k2 、α k3 、α k4 The pressure angle of the meshing point on the tooth surface, the base circle half tooth angle, the root circle half tooth angle and the pressure angle when the meshing point is on the root circle are respectively shown as gamma k Is any point on the crackCorresponding root circle half angle, x k D is the distance from any point on the tooth surface to the tooth root k 、d k1 The distance from the contact point on the tooth surface to the tooth root, the distance between the root circle and the base circle along the direction of the involute extension line of the tooth profile, h k 、h ka 、h ko The distance between the meshing point on the tooth surface and the central line of the tooth is respectively the meshing point, the crack end position and the distance between the meshing point on the tooth top circle and the central line of the tooth, r kb 、r kf The base circle radius and the root circle radius of the slice gear are respectively I kx 、I dk1 、I kaγ 、I kxa Respectively the tooth profile at any position x on involute, the tooth profile at the transition curve section of tooth root, the tooth root circle and any point x on tooth profile k Moment of inertia at the center plane of the gear tooth, A kx 、A dk1 、A kaγ 、A kxa Respectively the tooth profile at any position x on involute, the tooth profile at the transition curve section of tooth root, the tooth root circle and any point x on tooth profile k Cross-sectional area of tooth at center of tooth, q k1 U is the crack depth of the upper part of the gear tooth centerline kf S is the distance from the intersection point of the force action line and the gear tooth center line to the tooth root kf Is the arc length of the root of a single tooth, L * 、M * 、P * And Q is equal to * Are all equal to h, theta kf The related polynomial, h is the ratio of the radius of the root circle to the radius of the shaft hole, theta kf The arc of the tooth root corresponds to the central half angle.
In the vibration response analysis method of the spiral bevel gear with the crack fault, the single tooth meshing stiffness and the time-varying meshing stiffness of the spiral bevel gear are as follows:
single tooth engagement stiffness:
time-varying engagement stiffness:
wherein k is kb1 、k ks1 、k ka1 、k kf1 Representing the gear stiffness, k, of the drive wheel kb2 、k ks2 、k ka2 、k kf2 The index i=1, 2,3 indicates the gear stiffness of the driven wheel, and the index i=1, 2,3 indicates the 1 st, 2 nd, and 3 rd pairs of meshing gears at the time of multi-tooth meshing.
In the spiral bevel gear vibration response analysis method containing the crack fault, the spiral bevel gear transmission has 8 degrees of freedom, each gear has 3 translational degrees of freedom and 1 rotational degree of freedom, and the relative displacement of meshing tooth surfaces of two bevel gears along the normal direction of meshing points is as follows:
X n ＝(x 1 -x 2 )a+(y 1 -y 2 )b+(z 1 -z 2 -r m1 θ 1 +r m2 θ 2 )c+e(t)
where a=cos δ 1 cosα n sinβ m +sinδ 1 sinα n ，b＝sinδ 1 cosα n sinβ m -cosδ 1 sinα n ，c＝cosβ m cosα n E (t) represents the static transmission error of bevel gear transmission, r m1 、r m2 Indicating the index circle radius of the midpoint of the main wheel and the driven wheel;
dynamic engagement force F of driving wheel and driven wheel n 、F n The' calculation formula is:
wherein k is m Representing the time-varying meshing stiffness of the meshing of the bevel gear pair with the arc teeth, c m Represents the meshing damping of the meshing of the gear pair;
the kinetic equation of spiral bevel gear transmission is:
wherein m is 1 、m 2 The mass of the main wheel and the driven wheel is I 1 、I 2 The moment of inertia of the driving wheel and the driven wheel around the respective axes r m1 、r m2 Represents the index circle radius, theta of the midpoint of the driving wheel and the driven wheel 1 And theta 2 The rotation degrees of freedom of the main wheel and the driven wheel respectively, T in 、T out To input and load moment, x i ，y i ，z i Representing three degrees of freedom, F, of the driving wheel and the driven wheel, respectively xi ，F yi ，F zi (i=1, 2) are the components of the meshing force of the driving and driven wheels in the three degrees of freedom, x i ，y i ，z i (i=1, 2) three translational degrees of freedom, k, of the driving and driven wheels respectively xi ，k yi ，k zi 、c xi ，c yi ，c zi (i=1, 2) is the equivalent support stiffness and support damping in the respective degrees of freedom on the driven and driven axles, respectively.
In the technical scheme, the spiral bevel gear vibration response analysis method with crack faults has the following beneficial effects: according to the method, the relation between the meshing point position and the rotation angle of the spiral bevel gear slice is calculated, a single-piece spiral bevel gear crack meshing stiffness calculation model is established, time-varying meshing stiffness under spiral bevel gear crack faults is calculated according to a potential energy method, and a dynamics model under the spiral bevel gear crack faults is established according to the stiffness by adopting a centralized parameter method. The invention fully plays the advantages of resolving the time-varying meshing stiffness of the spiral bevel gear by an analytic method, and has the advantages of accurate calculation and high efficiency. The meshing stiffness solved by the method can provide a key basis for dynamics and diagnosis research of arc tooth bevel gear crack faults.
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In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings that are needed in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments described in the present invention, and other drawings may be obtained according to these drawings for a person having ordinary skill in the art.
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a schematic view of the slicing process of the spiral bevel gear of the present invention;
FIG. 3 is a schematic view of the relationship between meshing position and rotation angle of a spiral bevel gear according to the present invention;
FIG. 4 is a schematic view of a calculation model of crack engagement stiffness of a single piece spiral bevel gear according to the present invention;
FIG. 5 is a time-varying mesh stiffness under a spiral bevel gear crack failure of an example of the invention;
FIG. 6 is a schematic representation of a kinetic model of a spiral bevel gear crack failure according to the present invention;
fig. 7 is a time domain plot of a spiral bevel gear crack failure vibration response signal of an example of the invention.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the embodiments of the present invention, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments. All other embodiments, based on the embodiments of the invention, which are apparent to those of ordinary skill in the art without inventive faculty, are intended to be within the scope of the invention.
Accordingly, the following detailed description of the embodiments of the invention provided in figures 1 through 7 of the drawings is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, based on the embodiments of the invention, which are apparent to those of ordinary skill in the art without inventive faculty, are intended to be within the scope of the invention.
It should be noted that: like reference numerals and letters denote like items in the following figures, and thus once an item is defined in one figure, no further definition or explanation thereof is necessary in the following figures.
In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "length", "width", "thickness", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", "clockwise", "counterclockwise", etc. indicate orientations or positional relationships based on the orientations or positional relationships shown in the drawings are merely for convenience in describing the present invention and simplifying the description, and do not indicate or imply that the apparatus or elements referred to must have a specific orientation, be configured and operated in a specific orientation, and thus should not be construed as limiting the present invention.
Furthermore, the terms "first," "second," and the like, are used for descriptive purposes only and are not to be construed as indicating or implying a relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defining "a first" or "a second" may explicitly or implicitly include one or more such feature. In the description of the present invention, the meaning of "a plurality" is two or more, unless explicitly defined otherwise.
In the present invention, unless explicitly specified and limited otherwise, the terms "mounted," "connected," "secured," and the like are to be construed broadly, and may be, for example, fixedly connected, detachably connected, or integrally formed; can be directly connected or indirectly connected through an intermediate medium, and can be communicated with the inside of two elements or the interaction relationship of the two elements. The specific meaning of the above terms in the present invention can be understood by those of ordinary skill in the art according to the specific circumstances.
In the present invention, unless expressly stated or limited otherwise, a first feature "above" or "below" a second feature may include both the first and second features being in direct contact, as well as the first and second features not being in direct contact but being in contact with each other through additional features therebetween. Moreover, a first feature being "above," "over" and "on" a second feature includes the first feature being directly above and obliquely above the second feature, or simply indicating that the first feature is higher in level than the second feature. The first feature being "under", "below" and "beneath" the second feature includes the first feature being directly under and obliquely below the second feature, or simply means that the first feature is less level than the second feature.
In order to make the technical scheme of the present invention better understood by those skilled in the art, the present invention will be further described in detail with reference to the accompanying drawings.
A spiral bevel gear vibration response analysis method with crack faults comprises the following steps:
1) Obtaining parameters of a spiral bevel gear and parameters of root cracks;
2) Equidistant slicing is carried out on the spiral bevel gear along the tooth width direction, and parameters of gear slicing are obtained;
3) Determining meshing states of the gear slices at different corners, and calculating the relation between the meshing point positions of the gear slices and the corners;
4) Establishing a single-piece spiral bevel gear crack meshing stiffness calculation model by adopting a potential energy method, and solving to obtain the gear time-varying meshing stiffness under spiral bevel gear crack faults;
5) And establishing a dynamic model under the transmission crack fault of the spiral bevel gear by adopting a centralized parameter method, and solving to obtain the vibration response characteristic under the transmission crack fault of the spiral bevel gear.
Preferably, in step 1), the gear root crack simplifies the expansion path, ignoring the crack width, considering only the expansion of the root crack in the depth and length directions. Root crack through crack depth q k Crack length l k Crack propagation angle v k Description is made.
Preferably, in step 2), the spiral bevel gear is equally divided into a plurality of pieces along the tooth width direction, the tooth width of each piece of thin-piece gear is Δb=b/N, b is the tooth width of the spiral bevel gear, N is the number of gear slices, and the parameter of each slice is obtained through calculation.
Preferably, in step 3), the relationship between the meshing point position and the rotation angle of each meshing spiral bevel gear slice is:
wherein,θ k1 、θ k2 respectively represent the large end face rotation angle alpha of the main and driven wheels of the kth gear k1，p 、α k1，g Respectively represent the angle position alpha of the meshing point on the large end face of the main driven wheel of the kth gear tk Represents the pressure angle of the large end face of the kth gear,the rotation angles of the k-th piece of gears of the driving wheel and the driven wheel relative to the large end faces are respectively shown.
Through the steps, the corner relation between adjacent spiral bevel gear slices and the relation between the meshing point position of the spiral bevel gear slices and the corner are established.
Preferably, in the step 4), the step of establishing the crack meshing stiffness calculation model of the single-chip spiral bevel gear by using a potential energy method comprises the following steps:
the rigidity of the gear includes: hertz contact stiffness k kh Bending stiffness k kb Shear deformation stiffness k ks Stiffness in axial compression k ka And elastic matrix deformation stiffness k kf . Assuming that the equivalent number of teeth is less than 42 and the crack does not cross the gear tooth center line, the crack fault is now expanding from the large end face to the small end face on the driving wheel, and each gear is a penetrating crack. The calculation of the crack meshing stiffness of the single piece spiral bevel gear can be performed in two cases:
case 1: h is a ka ≥h ko &α k1 ≥α ka
Case 2: h is a ka ＜h ko or h ka ≥h ko &α k1 ≥α ka
Wherein v is poisson's ratio, E, G is modulus and shear modulus, respectively, α k Is the pressure angle of any point on the tooth surface, alpha ka For the pressure angle, alpha, corresponding to the crack end position kr For the root circle half-tooth angle alpha corresponding to the crack end position k1 、α k2 、α k3 、α k4 The pressure angle of the meshing point on the tooth surface, the base circle half tooth angle, the root circle half tooth angle and the pressure angle when the meshing point is on the root circle are respectively shown as gamma k Corresponding root circle half angle, x, for any point on the crack k D is the distance from any point on the tooth surface to the tooth root k 、d k1 The distance from the contact point on the tooth surface to the tooth root, the distance between the root circle and the base circle along the direction of the involute extension line of the tooth profile, h k 、h ka 、h ko The distance between the meshing point on the tooth surface and the central line of the tooth is respectively the meshing point, the crack end position and the distance between the meshing point on the tooth top circle and the central line of the tooth, r kb 、r kf The base circle radius and the root circle radius of the slice gear are respectively I kx 、I dk1 、I kaγ 、I kxa Respectively the tooth profile at any position x on involute, the tooth profile at the transition curve section of tooth root, the tooth root circle and any point x on tooth profile k Moment of inertia at the center plane of the gear tooth, A kx 、A dk1 、A kaγ 、A kxa Respectively the tooth profile at any position x on involute, the tooth profile at the transition curve section of tooth root, the tooth root circle and any point x on tooth profile k Cross-sectional area of tooth at center of tooth, q k1 U is the crack depth of the upper part of the gear tooth centerline kf S is the distance from the intersection point of the force action line and the gear tooth center line to the tooth root kf Is the arc length of the root of a single tooth, L * 、M * 、P * And Q is equal to * Are all equal to h, theta kf The related polynomial, h is the ratio of the radius of the root circle to the radius of the shaft hole, theta kf The arc of the tooth root corresponds to the central half angle.
The single tooth and time-varying meshing stiffness of the spiral bevel gear can be obtained by the method:
single tooth engagement stiffness:
time-varying engagement stiffness:
wherein k is kh1 、k kb1 、k ks1 、k ka1 、k kf1 For the gear rigidity of the driving wheel, k kh2 、k kb2 、k ks2 、k ka2 、k kf2 For the gear stiffness of the driven wheel, subscripts i=1, 2,3 are the 1 st, 2 nd, and 3 rd pairs of meshing gears at the time of multi-tooth meshing.
Preferably, in step 5), the spiral bevel gear system has 8 degrees of freedom, i.e. 3 translational degrees of freedom and 1 rotational degree of freedom per gear. The shaft and bearing supporting the gears act equivalently on the mid-point of the tooth width. The relative displacement of the meshing tooth surfaces of the two bevel gears along the normal direction of the meshing point is as follows:
X n ＝(x 1 -x 2 )a+(y 1 -y 2 )b+(z 1 -z 2 -r m1 θ 1 +r m2 θ 2 )c+e(t)
where a=cos δ 1 cosα n sinβ m +sinδ 1 sinα n ，b＝sinδ 1 cosα n sinβ m -cosδ 1 sinα n ，c＝cosβ m cosα n E (t) is the static transmission error of bevel gear transmission, r m1 、r m2 The index circle radius is the midpoint of the main and driven wheels;
dynamic engagement force F of driving wheel and driven wheel n 、F n The' calculation formula is:
wherein k is m C is the time-varying meshing stiffness of the meshing of the bevel gear pair with the arc teeth m Represents the meshing damping of the meshing of the gear pair;
the kinetic equation of spiral bevel gear transmission is:
wherein m is 1 、m 2 The mass of the main wheel and the driven wheel is I 1 、I 2 The moment of inertia of the driving wheel and the driven wheel around the respective axes r m1 、r m2 Represents the index circle radius, theta of the midpoint of the driving wheel and the driven wheel 1 And theta 2 The rotation degrees of freedom of the main wheel and the driven wheel respectively, T in 、T out To input and load moment, x i ，y i ，z i Representing three degrees of freedom, F, of the driving wheel and the driven wheel, respectively xi ，F yi ，F zi (i=1, 2) are the components of the meshing force of the driving and driven wheels in the three degrees of freedom, x i ，y i ，z i (i=1, 2) three translational degrees of freedom, k, of the driving and driven wheels respectively xi ，k yi ，k zi 、c xi ，c yi ，c zi (i=1, 2) are the directions of the degrees of freedom on the driving and driven axles respectivelyEquivalent support stiffness and support damping.
In one embodiment, as shown in fig. 1, a spiral bevel gear vibration response analysis method including a crack failure includes the steps of:
1) Obtaining parameters of a spiral bevel gear and parameters of root cracks;
in this step, basic parameters of the spiral bevel gear are shown in table 1. The cracks are arranged on the driving wheel, and the root cracks pass through the crack depth q k Crack length l k Crack propagation angle v k Description is made.
TABLE 1 parameters relating to spiral bevel gears
2) Equidistant slicing is carried out on the spiral bevel gear along the tooth width direction, and parameters of gear slicing are obtained;
the spiral bevel gear is equally divided into a plurality of pieces along the tooth width direction, and the specific process is shown in figure 2. The relevant parameters of the spiral bevel gear slicing gear are shown in table 2.
TABLE 2 parameters related to spiral bevel gear slicing gears
3) Calculating the relation between the meshing point position and the rotation angle of the meshing gear slices;
in this step, the relationship between the position of the meshing point and the rotation angle is deduced according to fig. 3, and the relationship between the position of each meshing point of the meshing spiral bevel gear slices and the rotation angle is obtained as follows:
wherein θ k1 、θ k2 Respectively represent the large end face rotation angle alpha of the main and driven wheels of the kth gear k1，p 、α k1，g Respectively represent the angle position alpha of the meshing point on the large end face of the main driven wheel of the kth gear tk Represents the pressure angle of the large end face of the kth gear,the rotation angles of the k-th piece of gears of the driving wheel and the driven wheel relative to the large end faces are respectively shown.
Through the steps, the corner relation between adjacent spiral bevel gear slices and the relation between the meshing point position of the spiral bevel gear slices and the corner are established.
4) Establishing a single-piece spiral bevel gear crack meshing stiffness calculation model by adopting a potential energy method, and solving to obtain gear time-varying meshing stiffness under spiral bevel gear crack failure, wherein the spiral bevel gear crack meshing stiffness calculation model is shown in fig. 4;
the rigidity of the gear includes: hertz contact stiffness k kh Bending stiffness k kb Shear deformation stiffness k ks Stiffness in axial compression k ka And elastic matrix deformation stiffness k kf . Assuming that the equivalent number of teeth is less than 42 and the crack does not cross the gear tooth center line, the crack fault is now expanding from the large end face to the small end face on the driving wheel, and each gear is a penetrating crack. The calculation of the crack meshing stiffness of the single piece spiral bevel gear can be performed in two cases:
case 1: h is a ka ≥h ko &α k1 ≥α ka
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Case 2: h is a ka ＜h ko or h ka ≥h ko &α k1 ≥α ka
Wherein v is poisson's ratio, E, G is modulus and shear modulus, respectively, α k Is the pressure angle of any point on the tooth surface, alpha ka For the pressure angle, alpha, corresponding to the crack end position kr For the root circle half-tooth angle alpha corresponding to the crack end position k1 、α k2 、α k3 、α k4 The pressure angle of the meshing point on the tooth surface, the base circle half tooth angle, the root circle half tooth angle and the pressure angle when the meshing point is on the root circle are respectively shown as gamma k Corresponding root circle half angle, x, for any point on the crack k D is the distance from any point on the tooth surface to the tooth root k 、d k1 The distance from the contact point on the tooth surface to the tooth root, the distance between the root circle and the base circle along the direction of the involute extension line of the tooth profile, h k 、h ka 、h ko The distance between the meshing point on the tooth surface and the central line of the tooth is respectively the meshing point, the crack end position and the distance between the meshing point on the tooth top circle and the central line of the tooth, r kb 、r kf The base circle radius and the root circle radius of the slice gear are respectively I kx 、I dk1 、I kaγ 、I kxa Respectively the tooth profile at any position x on involute, the tooth profile at the transition curve section of tooth root, the tooth root circle and any point x on tooth profile k Moment of inertia at the center plane of the gear tooth, A kx 、A dk1 、A kaγ 、A kxa Respectively the tooth profile at any position x on involute, the tooth profile at the transition curve section of tooth root, the tooth root circle and any point x on tooth profile k Cross-sectional area of tooth at center of tooth, q k1 U is the crack depth of the upper part of the gear tooth centerline kf S is the distance from the intersection point of the force action line and the gear tooth center line to the tooth root kf Is the arc length of the root of a single tooth, L * 、M * 、P * And Q is equal to * Are all equal to h, theta kf The related polynomial, h is the ratio of the radius of the root circle to the radius of the shaft hole, theta kf The arc of the tooth root corresponds to the central half angle.
The single tooth and time-varying meshing stiffness of the spiral bevel gear can be obtained by the method:
single tooth engagement stiffness:
time-varying engagement stiffness:
wherein I is kb1 、k ks1 、k ka1 、k kf1 For the gear rigidity of the driving wheel, k kb2 、k ks2 、k ka2 、k kf2 For the gear stiffness of the driven wheel, subscripts i=1, 2,3 are the 1 st, 2 nd, and 3 rd pairs of meshing gears at the time of multi-tooth meshing.
Using the parameters in Table 1, a spiral bevel gear vibration response analysis method containing a crack failure according to the present invention was performed at a crack depth of q k1 Crack propagation angle v =4mm k When the angle is 45 DEG, the crack l with different lengths k Calculation was performed on the stiffness at =0.5b, 0.75b, 1b, resulting in a curve bevel gear time-varying mesh stiffness comparison graph at different crack lengths as shown in fig. 5.
5) Establishing a dynamic model under the transmission crack fault of the spiral bevel gear by adopting a centralized parameter method and solving to obtain the vibration response characteristic under the transmission crack fault of the spiral bevel gear;
the spiral bevel gear dynamic model is shown in fig. 6, a coordinate system XOY is established, and the spiral bevel gear transmission system has 8 degrees of freedom, namely, a movement degree of freedom along 3 coordinate directions and a rotation degree of freedom along an axial direction: { x 1 ，y 1 ，z 1 ，θ 1 ，x 2 ，y 2 ，z 2 ，θ 2 } T . The shaft and bearing supporting the gears act equivalently on the mid-point of the tooth width. The relative displacement of the meshing tooth surfaces of the two bevel gears along the normal direction of the meshing point is as follows:
X n ＝(x 1 -x 2 )a+(y 1 -y 2 )b+(z 1 -z 2 -r m1 θ 1 +r m2 θ 2 )c+e(t)
where a=cos δ 1 cosα n sinβ m +sinδ 1 sinα n ，b＝sinδ 1 cosα n sinβ m -cosδ 1 sinα n ，c＝cosβ m cosα n E (t) is the static transmission error of bevel gear transmission, r m1 、r m2 The index circle radius is the midpoint of the main and driven wheels;
dynamic engagement force F of driving wheel and driven wheel n 、F n The' calculation formula is:
wherein k is m C is the time-varying meshing stiffness of the meshing of the bevel gear pair with the arc teeth m Represents the meshing damping of the meshing of the gear pair;
the kinetic equation of spiral bevel gear transmission is:
wherein m is 1 、m 2 The mass of the driving wheel and the driven wheel is I 1 、I 2 Is a driving wheel andmoment of inertia, θ, of the driven wheels about their respective axes 1 And theta 2 One degree of freedom of rotation, T, of the driving wheel and the driven wheel respectively in 、T out To input and load moment, F xi ，F yi ，F zi (i=1, 2) is the component force of the meshing force of the driving wheel and the driven wheel along the three degrees of freedom, x i ，y i ，z i (i=1, 2) three translational degrees of freedom, k, of the driving wheel and the driven wheel, respectively xi ，k yi ，k zi 、c xi ，c yi ，c zi (i=1, 2) is equivalent supporting rigidity and supporting damping in the direction of the degree of freedom of each of the driving wheel and the driven wheel shaft respectively.
According to the method for analyzing meshing stiffness and dynamic response under spiral bevel gear crack fault, which is disclosed by the invention, by adopting parameters shown in table 1 and utilizing the stiffness obtained in the step 4), the input shaft rotation frequency is set to be 30Hz, and the crack depth is set to be q k1 Crack propagation angle v =4mm k Simulation of response to crack faults of different lengths is carried out to obtain vibration response results of the spiral bevel gear when crack faults of spiral bevel gear occur, wherein the vibration response time domain diagram is shown in figure 7, and the crack length of the root crack parameter is l k When=0.5 b, 0.75b, 1b, the impact phenomenon occurs once every 16 periods of the relative displacement acceleration of the driving wheel, and the interval time between each impact is 0.033s.
Finally, it should be noted that: the described embodiments are intended to be illustrative of only some, but not all, of the embodiments disclosed herein and, based on the embodiments disclosed herein, all other embodiments that may be made by those skilled in the art without the benefit of the teachings herein are intended to be within the scope of this application.
While certain exemplary embodiments of the present invention have been described above by way of illustration only, it will be apparent to those of ordinary skill in the art that modifications may be made to the described embodiments in various different ways without departing from the spirit and scope of the invention. Accordingly, the drawings and description are to be regarded as illustrative in nature and not as restrictive of the scope of the invention, which is defined by the appended claims.
Claims (6)
1. A spiral bevel gear vibration response analysis method containing crack faults is characterized by comprising the following steps:
step 1, measuring to obtain parameters of a spiral bevel gear and parameters of a tooth root crack, wherein the parameters of the gear comprise tooth number, large end modulus, tooth width, pressure angle, radial deflection coefficient, top clearance coefficient, tooth top height coefficient and tooth width midpoint spiral angle, and the parameters of the tooth root crack comprise crack depth q k Crack length l k Crack propagation angle v k ；
Step 2, equidistant slicing is carried out on the spiral bevel gear along the tooth width direction to obtain a slicing gear and slice gear parameters are obtained, wherein the slice gear parameters comprise the width of each slice, the distance from the large end of the gear, the radius of a reference circle, the radius of a base circle, the radius of a root circle and the radius of a top circle;
step 3, determining meshing states of the slicing gears under different rotation angles based on the slicing gear parameters, and calculating the relation between the meshing point position of the slicing gears and the rotation angles of the gears;
step 4, establishing a crack meshing stiffness calculation model of the single-piece spiral bevel gear by adopting a potential energy method based on the relation between the position of the meshing point and the gear corner, and solving the crack meshing stiffness calculation model to further obtain the time-varying meshing stiffness of the gear under the crack failure of the spiral bevel gear;
and 5, based on the time-varying meshing stiffness of the gear, establishing a dynamic model under the transmission crack fault of the spiral bevel gear by adopting a centralized parameter method, and solving to obtain the vibration response characteristic under the transmission crack fault of the spiral bevel gear.
2. The method for analyzing vibration response of spiral bevel gear with crack failure according to claim 1, wherein in step 2, the spiral bevel gear is preferably cut into a plurality of pieces at equal intervals along the tooth width direction, the tooth width of each piece of slicing gear is Δb=b/N, b is the tooth width of the spiral bevel gear, N is the number of gear slices, and the slicing parameters of each piece of slicing gear are calculated.
3. The method for analyzing vibration response of spiral bevel gear with crack failure according to claim 2, wherein in step 3), the relationship between the meshing point position of the slicing gear and the rotation angle is:
wherein θ k1 、θ k2 Respectively represent the large end face rotation angle alpha of the main and driven wheels of the kth gear k1，p 、α k1，g Respectively represent the angle position alpha of the meshing point on the large end face of the main driven wheel of the kth gear tk Represents the pressure angle of the large end face of the kth gear,the rotation angles of the k-th piece of gears of the driving wheel and the driven wheel relative to the large end faces are respectively shown.
4. A spiral bevel gear vibration response analysis method including a crack failure as claimed in claim 3, wherein in step 4), the rigidity of the gear includes: hertz contact stiffness k kh Bending stiffness k kb Shear deformation stiffness k ks Stiffness in axial compression k ka And elastic matrix deformation stiffness k kf The calculation of the crack meshing stiffness of the single piece spiral bevel gear with an equivalent tooth count of less than 42 is performed in two cases:
case 1: h is a ka ≥h ko &α k1 ≥α ka
Case 2: h is a ka ＜h ko or h ka ≥h ko &α k1 ≥α ka
Wherein v is poisson's ratio, E, G is modulus and shear modulus, respectively, α k Is the pressure angle of any point on the tooth surface, alpha ka For the pressure angle, alpha, corresponding to the crack end position kr For the root circle half-tooth angle alpha corresponding to the crack end position k1 、α k2 、α k3 、α k4 The pressure angle of the meshing point on the tooth surface, the base circle half tooth angle, the root circle half tooth angle and the pressure angle when the meshing point is on the root circle are respectively shown as gamma k Corresponding root circle half angle, x, for any point on the crack k D is the distance from any point on the tooth surface to the tooth root k 、d k1 The distance from the contact point on the tooth surface to the tooth root, the distance between the root circle and the base circle along the direction of the involute extension line of the tooth profile, h k 、h ka 、h ko The distance between the meshing point on the tooth top circle and the center line of the gear tooth is respectively equal to r kb 、r kf Separately of slicing gearsBase radius, root radius, I kx 、I dk1 、I kaγ 、I kxa Respectively the tooth profile at any position x on involute, the tooth profile at the transition curve section of tooth root, the tooth root circle and any point x on tooth profile k Moment of inertia at the center plane of the gear tooth, A kx 、A dk1 、A kaγ 、A kxa Respectively the tooth profile at any position x on involute, the tooth profile at the transition curve section of tooth root, the tooth root circle and any point x on tooth profile k Cross-sectional area of tooth at center of tooth, q k1 U is the crack depth of the upper part of the gear tooth centerline kf S is the distance from the intersection point of the force action line and the gear tooth center line to the tooth root kf Is the arc length of the root of a single tooth, L * 、M * 、P * And Q is equal to * Are all equal to h, theta kf The related polynomial, h is the ratio of the radius of the root circle to the radius of the shaft hole, theta kf The arc of the tooth root corresponds to the central half angle.
5. The method for analyzing vibration response of spiral bevel gear with crack failure according to claim 4, wherein the single tooth meshing stiffness and time-varying meshing stiffness of the spiral bevel gear are:
single tooth engagement stiffness:
time-varying engagement stiffness:
wherein k is kb1 、k ks1 、k ka1 、k kf1 Representing the gear stiffness, k, of the drive wheel kb2 、k ks2 、k ka2 、k kf2 The index i=1, 2,3 indicates the gear stiffness of the driven wheel, and the index i=1, 2,3 indicates the 1 st, 2 nd, and 3 rd pairs of meshing gears at the time of multi-tooth meshing.
6. The method for analyzing vibration response of spiral bevel gear with crack failure according to claim 5, wherein the spiral bevel gear has 8 degrees of freedom, each gear has 3 translational degrees of freedom and 1 rotational degree of freedom, and the relative displacement of the meshing tooth surfaces of the two spiral bevel gears along the normal direction of the meshing point is:
X n ＝(x 1 -x 2 )a+(y 1 -y 2 )b+(z 1 -z 2 -r m1 θ 1 +r m2 θ 2 )c+e(t)，
where a=cos δ 1 cosα n sinβ m +sinδ 1 sinα n ，b＝sinδ 1 cosα n sinβ m -cosδ 1 sinα n ，c＝coSβ m coSα n E (t) represents the static transmission error of bevel gear transmission, r m1 、r m2 Indicating the index circle radius of the midpoint of the main wheel and the driven wheel;
dynamic engagement force F of driving wheel and driven wheel n 、F n The' calculation formula is:
wherein k is m C represents the time-varying meshing stiffness of the bevel gear pair with arc teeth m Represents the meshing damping of the meshing of the gear pair;
the kinetic equation of spiral bevel gear transmission is:
wherein m is 1 、m 2 The mass of the main wheel and the driven wheel is I 1 、I 2 The moment of inertia of the driving wheel and the driven wheel around the respective axes r m1 、r m2 Represents the index circle radius, theta of the midpoint of the driving wheel and the driven wheel 1 And theta 2 The rotation degrees of freedom of the main wheel and the driven wheel respectively, T in 、T out To input and load moment, x i ，y i ，z i Representing three degrees of freedom, F, of the driving wheel and the driven wheel, respectively xi ，F yi ，F zi (i=1, 2) are the components of the meshing force of the driving and driven wheels in the three degrees of freedom, x i ，y i ，z i (i=1, 2) three translational degrees of freedom, k, of the driving and driven wheels respectively xi ，k yi ，k zi 、c xi ，c yi ，c zi (i=1, 2) is the equivalent support stiffness and support Yang Ni in the direction of the white pitch on the driving and driven axles, respectively.
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