CN117807723A - Method for rapidly deviating resonance rotating speed of flexible bevel gear transmission system - Google Patents

Abstract

The present invention discloses a method for quickly deviating the resonant speed of a flexible bevel gear transmission system, comprising the steps of: S1, modeling the bevel gear transmission system; S2, calculating dangerous modes: calculating the modal frequency of the bevel gear transmission system, if there is a modal order within 15% of the working speed, then calculating the modal strain energy under each modal order within 15% of the working speed and comparing the total modal strain energy, analyzing whether it is a dangerous modal frequency according to the proportion result, and drawing the vibration mode of the dangerous modal frequency; S3, structural parameter optimization: parametrically studying the structural parameters, bearing positions and stiffness of the bevel gear and the transmission shaft, so that the dangerous modal frequency is far away from the working speed. The present invention can quickly identify the dangerous modal frequency and draw the system modal vibration mode. When there is a dangerous modal frequency near the working speed, the resonant speed is quickly deviated from the working speed by modifying the structural parameters, bearing positions and stiffness of the bevel gear and the transmission shaft.

Description

A method for rapid deviation of resonant speed of flexible bevel gear transmission system

Technical Field

The invention relates to the technical field of structural design of gear transmission systems, and in particular to a method for quickly deviating the resonant speed of a flexible bevel gear transmission system.

Background Art

Bevel gears are used in aviation transmission systems to transmit non-parallel torque or speed. In recent decades, the aerospace field has increasingly demanded lightweight bevel gears. Therefore, in order to reduce mass, bevel gears generally use thin-walled structures. However, this structure is prone to web pitch-diameter vibration problems, which causes the web of the gear to break and affect the long-life operation of the engine. In past studies, there are usually two modeling methods. One is to use a rigid disk for dynamic modeling of bevel gears (rigid model). This method cannot accurately model complex bevel gears and cannot obtain the modal vibration shapes related to the bevel gear body, which brings great errors in the model establishment stage. The other is to use finite element software to establish a model, but finite element software cannot calculate gear dynamics. It is necessary to derive the condensed matrix and then program for dynamic analysis. Although the stiffness matrix and mass matrix derived by the finite element method can ensure that the model has a high-precision modal frequency, the structural parameters of the model are lost and the modal vibration shapes cannot be calculated. The modal frequencies obtained cannot determine which frequencies are prone to excite resonance, that is, the dangerous modal frequencies that are prone to excite resonance cannot be identified, and thus the design cannot be guided to deviate from the resonant speed.

Summary of the invention

The present invention aims to solve at least one of the technical problems existing in the prior art. To this end, the present invention proposes a method for quickly deviating from the resonant speed of a flexible bevel gear transmission system, which can quickly identify dangerous modal frequencies that are easy to excite resonance and draw the system modal vibration shape. When there are dangerous modal frequencies near the working speed, the structural parameters, bearing positions and stiffness of the bevel gear and the transmission shaft are modified to make the resonant speed quickly deviate from the working speed.

The method for quickly deviating from the resonant speed of a flexible bevel gear transmission system according to an embodiment of the present invention comprises the following steps:

S1. Modeling of bevel gear transmission system, including the following steps:

S100, bevel gear and transmission shaft modeling: 20-node hexahedron units are used to model the bevel gear, and the modal synthesis method is used to reduce the dimension of the established bevel gear to reduce the size of the model matrix, and Timoshenko beam units are used to model the transmission shaft;

S110, assembling the driving and driven bevel gear shaft system: coupling the bevel gear and the transmission shaft through a coupling unit, and applying the stiffness of the bearing to the bearing node on the transmission shaft to simulate the bearing;

S120, meshing pair modeling: in the dynamic model, a meshing unit is established between the two bevel gears, and the meshing process is simplified into a spring;

S2. Calculation of dangerous modes: Calculate the modal frequencies of each order of the bevel gear transmission system. If there is a modal order within 15% of the working speed, calculate the modal strain energy of each modal order within 15% of the working speed and compare it with the total modal strain energy to obtain the modal strain energy ratio of each order. According to the modal strain energy ratio result, analyze whether the modal frequency of the corresponding order is a dangerous modal frequency that is easy to excite resonance, and draw the vibration mode of the dangerous modal frequency;

S3. Structural parameter optimization: According to the vibration mode of the dangerous modal frequency, the structural parameters, bearing position and stiffness of the bevel gear and the transmission shaft are parametrically studied to make the dangerous modal frequency away from the operating speed or to convert the dangerous modal frequency into a modal frequency that is not easy to excite resonance.

The method for quickly deviating from the resonant speed of a flexible bevel gear transmission system according to an embodiment of the present invention has at least the following beneficial effects:

The bevel gear, a complex structure, is modeled flexibly based on 20-node hexahedral units, and the transmission shaft is modeled based on Timoshenko beam units. The modeling is accurate and can retain the inherent properties of the model (modal frequency and vibration mode), so that the system modal vibration mode can be drawn to provide guidance for structural optimization. After the bevel gear is modeled, the modal synthesis method is used to shrink the model, which has a fast calculation speed and can greatly reduce the time of the entire process. By calculating the modal frequencies of each order of the bevel gear transmission system, it is analyzed whether there is a modal order within 15% of the working speed, and the modal strain energy ratio of each order mode within 15% of the working speed is calculated, so that the dangerous modal frequency that is easy to excite resonance can be quickly identified according to the modal strain energy ratio result, and then the vibration mode under the dangerous modal frequency is drawn. It can be analyzed according to the vibration mode to find out what the component that causes resonance is, and then the structural parameters of the component that causes resonance are optimized to make the dangerous modal frequency away from the working speed or to convert the dangerous modal frequency into a modal frequency that is not easy to excite resonance, so as to achieve the purpose of deviating the resonant speed from the working speed.

According to some embodiments of the present invention, in step S2, the modal strain energy under the j-th order mode is expressed as:

Wherein, _φmj is the eigenvector related to meshing in the j-th order eigenvector, and _Km is the stiffness matrix of the meshing unit.

According to some embodiments of the present invention, in step S2, the total modal strain energy is expressed as:

Where _Kz is the overall stiffness matrix of the bevel gear transmission system, and _φj represents the j-th order eigenvector.

According to some embodiments of the present invention, step S100 includes the following steps:

S101, using the 20-node hexahedral unit to model the bevel gear, dividing a plurality of the 20-node hexahedral units according to the structure of the bevel gear, and assembling the units according to the unit numbers of the 20-node hexahedral units to obtain the mass matrix and stiffness matrix of the overall bevel gear, wherein the mass matrix of one of the 20-node hexahedral units is:

The stiffness matrix of the 20-node hexahedral element is:

Where N represents the shape function, _ρ1 is the density of the bevel gear, B and B ^* are the strain matrices in the global coordinate system and the isoparametric coordinate system, respectively, D is the elasticity matrix, _Wi , _Wj , _Wk are Gaussian weights, τ is the number of Gaussian points in each direction, J is the Jacobian matrix, and ξ, η, ζ are the coordinates of the integration points.

According to some embodiments of the present invention, the shape function N is expressed as:

Among them, for the corner node, its shape function is expressed as:

For the middle node, its shape function is expressed as:

Where, the values of ξ _a , η _a , and ζ _a are selected according to the positions of the nodes in the isoparametric coordinate system, and the subscript a represents the nodes on the 20-node hexahedral element.

According to some embodiments of the present invention, step S100 includes the following steps:

S102, using the modal synthesis method to obtain a conversion matrix T _cms , and using the conversion matrix T _cms to perform the following transformation:

M _cms = T _cms ^T MT _cms ,

K _cms = T _cms ^T KT _cms ,

Thus, the mass matrix _Mcms and stiffness matrix K _cms of the bevel gear after condensation are obtained, and the dimension reduction condensation of the established bevel gear is realized; wherein, M and K are the mass matrix and stiffness matrix of the overall bevel gear.

According to some embodiments of the present invention, in step S102, the step of obtaining the conversion matrix T _cms by using the modal synthesis method comprises the following steps:

First, according to the characteristics of the bevel gear, each bevel gear has two interfaces, one is the meshing tooth surface of the bevel gear, and the other is the inner ring surface of the bevel gear connected to the transmission shaft. The motion equation of the substructure of the bevel gear is established in the physical coordinates with the node displacement:

Where C is the damping matrix, Q represents the external load vector, R represents the force vector on the interface, and a represents the node displacement vector. a is divided into two parts: internal displacement a _i and interface displacement a _j . The corresponding Q and R can also be divided into two parts. The motion equation of the substructure can be written as:

The free motion equation without considering damping can be written as:

Wherein, _Mii , _Kii are the internal mass matrix and stiffness matrix, _Mjj , _Kjj are the mass matrix and stiffness matrix on the interface, _Mij and _Mji are the coupled mass matrices, _Kij and _Kji are the coupled stiffness matrices, and _Rj is the force vector on the interface;

The free motion equation without damping is then calculated in modal coordinates:

First, fix the interface, that is, set a _j = 0, and calculate the natural vibration mode of the substructure system under the fixed interface, that is, solve the eigenvalue of the following free vibration equation:

By solving the eigenvalues of the free vibration equation, i natural vibration modes can be obtained, which are combined into a matrix represented by Φ _n ;

Then release each degree of freedom on the interface in turn and calculate the static displacement:

From the above formula we can get:

a _i = -K _ii ^-1 K _ij a _j ,

Let the j elements in a _j take unit values in turn and the rest be 0, and obtain the corresponding j groups of static displacement vectors, that is, the constraint modes. The matrix form of a _i is expressed as Φ _j :

At the same time, in order to achieve the purpose of reducing the degree of freedom, the high-order main modes in Φ _n are ignored, and only the k columns of low-order main modes are retained to form Φ _k . The following operation is performed to obtain the transformation matrix T _cms :

Where _Ij is the identity matrix with dimension j*j.

According to some embodiments of the present invention, in step S110, the stiffness matrix of the coupling unit is expressed as:

Where K _C is a 6*6 diagonal matrix whose diagonal values are the same large number.

According to some embodiments of the present invention, in step S120, the stiffness matrix of the meshing unit is expressed as:

K _m = k _m (V) ^T V,

Where _km is the average value of the time-varying meshing stiffness, which is obtained by loading contact analysis, and V represents the meshing vector.

According to some embodiments of the present invention, the engagement vector V is expressed as:

V＝[n _px n _py n _pz 0 0 0 n _gx n _gy n _gz 0 0 0],

In the formula, subscripts p and g represent the driving gear and the driven gear respectively, n _px , n _py , n _pz are the unit components of the equivalent meshing force of the driving gear in the X, Y, and Z directions, and _ngx , n _gy , n _gz are the unit components of the equivalent meshing force of the driven gear in the X, Y, and Z directions.

Additional aspects and advantages of the present invention will be given in part in the following description and in part will be obvious from the following description, or will be learned through practice of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be further described below with reference to the accompanying drawings and embodiments, wherein:

FIG1 is a schematic diagram of a 20-node hexahedral unit;

FIG2 is a schematic diagram of a Timoshenko beam element;

FIG3 is a schematic diagram of a bevel gear shaft model, including (a) a hexahedral unit, (b) a coupling unit, (c) a beam unit, and (d) a bearing unit;

Fig. 4 is a diagram of the system stiffness matrix assembly;

Figure 5 is a top view of the original model of this paper;

Figure 6 is a side view of the original model of this paper;

Figure 7 shows the modal strain energy proportion of each order of the original scheme model in this paper;

Figure 8 is the 30th-order vibration mode of the original model of this paper;

Figure 9 is a top view of the new solution model of this paper;

Figure 10 is a side view of the new solution model of this paper;

Figure 11 shows the modal strain energy proportion of each order of the new scheme model in this paper;

Figure 12 is the 30th-order vibration mode of the new model in this paper.

DETAILED DESCRIPTION

Embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and cannot be understood as limiting the present invention.

In the description of the present invention, it is necessary to understand that descriptions involving orientation, such as orientation or positional relationship indicated as up, down, etc., are based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present invention.

In the description of the present invention, "a plurality" means more than two. If there is a description of "first" or "second", it is only used for the purpose of distinguishing technical features, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features or implicitly indicating the order of the indicated technical features.

In the description of the present invention, unless otherwise clearly defined, terms such as setting, installing, connecting, etc. should be understood in a broad sense, and technicians in the relevant technical field can reasonably determine the specific meanings of the above terms in the present invention in combination with the specific content of the technical solution.

A method for quickly deviating from the resonant speed of a flexible bevel gear transmission system according to an embodiment of the present invention comprises the following steps:

S1. Bevel gear transmission system modeling. Step S1 specifically includes the following steps:

S100, bevel gear and transmission shaft modeling: 20-node hexahedron units are used for flexible modeling of bevel gears to obtain the mass matrix and stiffness matrix of bevel gears, fully simulating the inherent properties of the complex structure of bevel gears, and then the modal synthesis method is used to reduce the dimension of the established bevel gears to reduce the size of the model matrix. At the same time, Timoshenko beam units are used to model the transmission shaft; it should be noted that the bevel gears establish the driving gears (input gears) and the driven gears (output gears), and the transmission shafts establish the driving shafts (input shafts) and the driven shafts (output shafts);

S110, assembly of the driving and driven bevel gear shaft system: couple the bevel gear and the transmission shaft through the coupling unit, couple the coupling node of the inner ring of the bevel gear and the axis node of the transmission shaft where the bevel gear is located, and apply the stiffness of the bearing to the bearing node on the transmission shaft to simulate the bearing. Obviously, the driving gear is coupled with the driving shaft, and the driven gear is coupled with the driven shaft;

S120, meshing pair modeling: in the dynamic model, a meshing unit is established between two bevel gears (driving gear and driven gear), the meshing process is simplified to a spring, the equivalent meshing points of the two bevel gears interact with each other through the spring, the equivalent meshing points are rigidly coupled with the tooth surfaces meshing at the same time, and corresponding input torques and output loads are applied to the input and output positions of the two transmission shafts (driving shaft and driven shaft), respectively; the bevel gear transmission system is modeled through steps S100, S110, and S120;

S2. Calculation of dangerous modes: Calculate the modal frequencies of each order of the bevel gear transmission system. If there are modal orders within 15% of the working speed (i.e., there are modal frequencies under a certain order or multiple orders of modes within 15% of the frequency corresponding to the working speed), then calculate the modal strain energy under each order of mode within 15% of the working speed and compare it with the total modal strain energy to obtain the modal strain energy ratio of each order. According to the modal strain energy ratio result, analyze whether the modal frequency of the corresponding order is a dangerous modal frequency that is easy to excite resonance, and draw the vibration mode of the dangerous modal frequency. It should be noted that the speed within 15% of the working speed is easy to cause resonance, and the dangerous modal frequency is the frequency that is easy to cause resonance. The speed corresponding to the dangerous modal frequency is the resonant speed. The modal frequency of the order with a high modal strain energy ratio is easy to cause resonance.

S3. Structural parameter optimization: parametric study is conducted on the structural parameters, bearing positions and stiffness of bevel gears and transmission shafts according to the vibration mode of dangerous modal frequencies, so that the dangerous modal frequencies are far away from the operating speed or the dangerous modal frequencies are converted into modal frequencies that are not easy to excite resonance, thereby achieving the purpose of deviating the resonant speed from the operating speed; wherein, parametric study is conducted on the structural parameters, bearing positions and stiffness of bevel gears and transmission shafts according to the vibration mode of dangerous modal frequencies refers to confirming the components that cause resonance according to the vibration mode analysis of dangerous modal frequencies, and then modifying the structural parameters of the components that cause resonance for structural optimization.

The method for quickly deviating from the resonant speed of a flexible bevel gear transmission system in an embodiment of the present invention performs flexible modeling of the complex structure of the bevel gear based on a 20-node hexahedral unit, and models the transmission shaft based on the Timoshenko beam unit. The modeling is accurate and can retain the inherent properties of the model (modal frequency and vibration mode), so that the system modal vibration mode can be drawn to provide guidance for structural optimization. After the bevel gear is modeled, the modal synthesis method is used to condense the model, and the calculation speed is fast, which can greatly reduce the time of the entire process. By calculating the modal frequencies of each order of the bevel gear transmission system, it is analyzed whether there is a modal order within 15% of the working speed, and the modal strain energy ratio under each mode within 15% of the working speed is calculated, so that the dangerous modal frequency that is easy to excite resonance can be quickly identified according to the modal strain energy ratio result, and then the vibration mode under the dangerous modal frequency is drawn, and the component that causes resonance can be analyzed according to the vibration mode, and then the structural parameters of the component that causes resonance are optimized, so that the dangerous modal frequency is far away from the working speed or the dangerous modal frequency is converted into a modal frequency that is not easy to excite resonance, so as to achieve the purpose of deviating the resonant speed from the working speed.

It can be understood that, in some specific embodiments, in step S2, the modal strain energy U _mj under the j-th mode is expressed as:

Where φ _mj is the eigenvector related to meshing in the j-th order eigenvector, and K _m is the stiffness matrix of the meshing unit;

The total modal strain energy _UT is expressed as:

Where, _φj represents the j-th order eigenvector, _Kz is the overall stiffness matrix of the bevel gear transmission system, and the overall stiffness matrix can be obtained by assembling the various components of the bevel gear transmission system. Figure 4 is the stiffness matrix assembly diagram of the bevel gear transmission system model established in this paper;

The j-th order modal strain energy ratio is expressed as U _mj /U _T .

In some specific embodiments of the present invention, step S100 includes the following steps:

S101, using 20-node hexahedral units to model the bevel gear, dividing the bevel gear into multiple 20-node hexahedral units according to the structure of the bevel gear, and assembling the units according to the unit numbers of the 20-node hexahedral units to obtain the overall mass matrix and stiffness matrix of the bevel gear;

The mass matrix of a 20-node hexahedral element is:

The stiffness matrix of a 20-node hexahedral element is:

Where N represents the shape function, ρ ₁ is the density of the bevel gear, B and B ^* are the strain matrices in the global coordinate system and the isoparametric coordinate system, respectively, D is the elastic matrix, W _i , W _j , W _k are Gaussian weights, τ is the number of Gaussian points in each direction, J is the Jacobian matrix, ξ, η, ζ are the coordinates of the integration points;

Among them, the shape function N is expressed as:

As shown in Figure 1, for the corner nodes of the hexahedral element, its shape function is expressed as:

For the middle node of the hexahedral element, its shape function is expressed as:

Wherein, the values of ξ _a , η _{a ,} and ζ _a are selected according to the positions of each node in the isoparametric coordinate system, and the subscript a represents the node on the 20-node hexahedral element; it should be noted that the 20-node hexahedral element refers to a hexahedral element with 20 nodes, and each node has three degrees of freedom.

S102, using the modal synthesis method to obtain a conversion matrix T _cms , and using the conversion matrix T _cms to perform the following transformation:

M _cms = T _cms ^T MT _cms ,

K _cms = T _cms ^T KT _cms ,

According to the above formula, the mass matrix M _cms and stiffness matrix K _cms of the bevel gear after condensation are obtained to realize the dimensionality reduction condensation of the established bevel gear; where M and K are the mass matrix and stiffness matrix of the bevel gear as a whole; specifically, the conversion matrix T _cms obtained by the modal synthesis method includes the following steps:

First, according to the characteristics of bevel gears, each bevel gear has two interfaces, one is the meshing tooth surface of the bevel gear, and the other is the inner ring surface of the bevel gear connected to the transmission shaft. The motion equation of the substructure of the bevel gear is established in the physical coordinates with the node displacement:

Where C is the damping matrix, Q represents the external load vector, R represents the force vector on the interface, and a represents the node displacement vector. a is divided into two parts: internal displacement a _i and interface displacement a _j . The corresponding Q and R can also be divided into two parts. The motion equation of the substructure can be written as:

The free motion equation without considering damping can be written as:

Wherein, _Mii , _Kii are the internal mass matrix and stiffness matrix, _Mjj , _Kjj are the mass matrix and stiffness matrix on the interface, _Mij and _Mji are the coupled mass matrices, _Kij and _Kji are the coupled stiffness matrices, and _Rj is the force vector on the interface;

The free motion equations without damping are then calculated in modal coordinates:

First, fix the interface, that is, set a _j = 0, and calculate the natural vibration mode of the substructure system under the fixed interface, that is, solve the eigenvalue of the following free vibration equation:

By solving the eigenvalues of the free vibration equation, we can get i natural vibration modes, which are combined into a matrix represented by Φ _n ; it should be noted that Φ _n can be operated as follows:

Where I _i is the identity matrix with dimension i*i, Ω _i is the matrix with eigenvalues on the main diagonal;

The interface has been fixed above, and then each degree of freedom on the interface is released in turn and the static displacement is calculated:

From the above formula we can get:

a _i = -K _ii ^-1 K _ij a _j ,

Let the j elements in a _j take unit values in turn and the rest be 0, and obtain the corresponding j groups of static displacement vectors, that is, the constraint modes. The matrix form of a _i is expressed as Φ _j :

Φ _j ＝-K _ii ^-1 K _ij I _j ＝-K _ii ^-1 K _ij ,

At the same time, in order to achieve the purpose of reducing the degree of freedom, the high-order main modes in Φ _n are ignored, and only the k columns of low-order main modes are retained to form Φ _k . The following operation is performed to obtain the transformation matrix T _cms :

Where _Ij is the identity matrix with dimension j*j.

S103, using Timoshenko beam units to perform flexible modeling of the transmission shaft, dividing the transmission shaft into multiple Timoshenko beam units according to the structure, and assembling the units according to the unit numbers of the Timoshenko beam units to obtain the overall mass matrix and stiffness matrix of the transmission shaft; as shown in FIG2 , a beam unit consists of two nodes, each node has 6 degrees of freedom, and the displacement array of the unit is:

q ^s =[x _i ,y _i ,z _i ,θ _xi ,θ _yi ,θ _zi ,x _(i+1) ,y _(i+1) ,z _(i+1) ,θ _x(i+1) ,θ _y(i+1) ,θ _z(i+1) ] ^T ,

Where x _i , y _i , _zi are the translational degrees of freedom, θ _xi , θ _yi , θ _zi are the rotational degrees of freedom;

The displacement expression of any point in the unit in all directions is:

z＝N _d q ^s ，

θ _z = N _r q ^s ,

in,

N _d = [0 0 N _d1 0 0 0 0 0 N _d1 0 0 0],

N _r = [0 0 0 0 0 N _r1 0 0 0 0 0 N _r2 ],

Through the above formula, the mass matrix of the Timoshenko beam unit is obtained by combining the Lagrange equation with the kinetic energy formula, and the stiffness matrix of the Timoshenko beam unit is derived by combining the Lagrange equation with the potential energy formula. The mass matrix of a Timoshenko beam unit is expressed as:

The stiffness matrix of a Timoshenko beam element is expressed as:

in,

In the above formula, ρ ₂ is the density of the transmission shaft, υ is Poisson's ratio, E is the elastic model, m is the ratio of the inner diameter to the outer diameter of the beam element, μ is the hollow correction coefficient, G is the shear modulus of the beam element, A is the cross-sectional area of the beam element, l is the length of the beam element, _Is and _Ip are the diameter moment of inertia and polar moment of inertia of the beam element respectively. is the simplifying factor.

In some specific embodiments of the present invention, in step S110, referring to FIG. 3 , the bevel gear and the transmission shaft are coupled by a coupling unit, and the coupling node of the inner ring of the bevel gear and the axis node of the transmission shaft where the bevel gear is located are coupled. The stiffness matrix of the coupling unit of the bevel gear and the transmission shaft is expressed as:

Where K _C is a 6*6 diagonal matrix whose diagonal values are the same large number.

In some specific embodiments of the present invention, in step S120, in the dynamic model, a meshing unit is established between the equivalent meshing points of the two bevel gears (the driving gear and the driven gear). Since the torque caused by the gear meshing has been considered at the equivalent meshing point, the meshing vector V only involves the translational degree of freedom. The meshing vector V is expressed as:

V＝[n _px n _py n _pz 0 0 0 n _gx n _gy n _gz 0 0 0],

In the formula, subscripts p and g represent the driving gear and the driven gear respectively, n _px , n _py , n _pz are the unit components of the equivalent meshing force of the driving gear in the X, Y, and Z directions, and _ngx , n _gy , n _gz are the unit components of the equivalent meshing force of the driven gear in the X, Y, and Z directions; the meshing process of the gears is simplified into a spring, and the stiffness matrix of the meshing unit can be expressed as:

K _m = k _m (V) ^T V,

Where _km is the average value of the time-varying meshing stiffness, which is obtained through loading contact analysis.

Implementation Cases:

The method proposed in this paper is used to establish a model of the traditional bevel gear system, in which the transmission shaft is first established as a solid shaft. The bevel gear parameter table is shown in Table 1 (to distinguish it from the subsequent scheme, the scheme in which the transmission shaft is established as a solid shaft is called the original scheme). The original scheme model is shown in Figures 5 and 6. The modal frequencies of the original scheme model under each mode are calculated. In order to make the data more intuitive, the modal strain energy ratio of the original scheme model under the 0-50th mode is calculated, and the bar chart of the strain energy ratio of each mode is shown in Figure 7.

Table 1 Bevel gear parameters

The input working speed is 6700rpm, then the modal frequency that will excite resonance near this speed needs to be deviated to make the system run stably. The number of teeth of the driving gear is 55, and the system modal frequency corresponding to 6700rpm is 6700/60*55=6142Hz. Therefore, it is necessary to deviate the modal frequency by 15% before and after 6142Hz. Referring to Figure 7, the 30th-order modal strain energy accounts for a high proportion and the corresponding system modal frequency is within 15% of 6142Hz. Therefore, the 30th-order modal frequency is a dangerous modal frequency, and it is necessary to deviate from the 30th-order system modal frequency to meet the conditions for stable operation. The 30th-order vibration mode of the original scheme is shown in Figure 8. The left side of the figure is the driven gear and the shaft, and the right side is the driving gear and the shaft. It can be seen from the figure that it is the pitch diameter vibration mode of the driving and driven gears and the bending coupling vibration mode of the driven shaft.

By adjusting the hollowness of the shaft, the original solid shaft is changed to a hollow shaft, and the position of the bearing is adjusted and the bearing model is replaced, that is, the stiffness of the bearing is changed, and a new scheme is obtained. The parameters of the transmission shaft after adjustment are shown in Table 2 (in Table 2, the length column indicates that a transmission shaft is divided into multiple shaft sections of 9mm, 51mm, 134mm, 15mm, 15mm, and 35mm in length along the axial direction). The new scheme model is shown in Figures 9 and 10. The proportion of strain energy of each order of the new scheme model is shown in Figure 11. It can be found that the proportion of meshing strain energy of the 30th order mode is very low and is no longer a dangerous mode. The frequency of the previous dangerous mode adjacent to it is 4609Hz, and the next order is 7094Hz, which are all outside 15% of 6142Hz, meeting the requirement that the working frequency avoids the resonance frequency range in the project. The vibration mode of the 30th order of the new scheme is drawn as shown in Figure 12. Its vibration mode changes to the bending vibration mode of the driven shaft and the pitch vibration mode of the driven bevel gear. The current vibration mode is not easy to excite resonance.

Table 2 Parameters of each shaft section of the transmission shaft

In addition, for the new scheme, the model was established using the method in this paper and the model was established using the finite element method and compared with the traditional rigid model. The first 20-order system modal frequencies of the three are shown in Table 3, where error 1 is the error between the model in this paper and the finite element model, and error 2 is the error between the model in this paper and the rigid model. It can be seen that error 1 is less than 2% in the first 20 orders, and the error of the first 11 orders of the rigid model is relatively small due to the bending frequency of the shaft, but the errors from the 12th to the 20th order are very large, that is, the rigid model cannot correctly reflect the bevel gear transmission system model.

In addition, it can be seen from Table 3 that there are 20 modal frequencies at 3418 Hz, and the intervals between adjacent modal frequencies are very small. In this way, it is impossible to directly use the modal frequencies of each order to adjust the modal frequencies in the design stage to make the resonant speed away from the working speed, because the modal orders around 15% of the working speed may cause resonance, but if it is not known which modal frequency causes it, then all modal frequencies need to be moved. If the interval is small, then there will be many modal frequencies in this area, and it is impossible to move all modal frequencies in this area. The method in this paper can quickly identify dangerous modal frequencies by calculating the modal strain energy ratio of each mode within 15% before and after the working speed. After that, the vibration mode of the dangerous modal frequency can be drawn for parametric research.

Table 3 System modal frequency comparison

The embodiments of the present invention are described in detail above in conjunction with the accompanying drawings, but the present invention is not limited to the above embodiments, and various changes can be made within the knowledge scope of ordinary technicians in the relevant technical field without departing from the purpose of the present invention.

Claims (10)

1. A method for rapidly deviating the resonance rotating speed of a flexible bevel gear transmission system is characterized by comprising the following steps:

s1, modeling a bevel gear transmission system, which comprises the following steps:

s100, modeling a bevel gear and a transmission shaft: modeling a bevel gear by adopting a 20-node hexahedral unit, performing dimension-reducing polycondensation on the bevel gear by adopting a mode synthesis method so as to reduce the size of a model matrix, and modeling a transmission shaft by adopting a Timoshenko beam unit;

s110, assembling a main driven bevel gear shaft system: coupling the bevel gear and the drive shaft through a coupling unit, and applying rigidity of the bearing to a bearing node on the drive shaft to simulate the bearing;

s120, modeling of meshing pairs: in the dynamics model, a meshing unit is established between the two bevel gears, and the meshing process is simplified into a spring;

s2, dangerous mode calculation: calculating the modal frequency of each step of the bevel gear transmission system, if the modal order exists in the range of 15% of the front and rear working speeds, calculating the modal strain energy of each step in the range of 15% of the front and rear working speeds and comparing the modal strain energy with the total modal strain energy to obtain the modal strain energy duty ratio of each step, analyzing whether the modal frequency of the corresponding step is a dangerous modal frequency which is easy to excite resonance according to the modal strain energy duty ratio result, and drawing the vibration mode of the dangerous modal frequency;

s3, structural parameter optimization: and carrying out parameterization research on structural parameters, bearing positions and rigidity of the bevel gear and the transmission shaft according to the vibration mode of the dangerous mode frequency, so that the dangerous mode frequency is far away from the working rotating speed or is converted into the mode frequency which is not easy to excite resonance.

2. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: in step S2, the modal strain energy in the j-th order mode is expressed as:

in phi _mj Is the feature vector related to engagement in the j th order feature vector, K _m Is the stiffness moment of the engagement unitAn array.

3. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: in step S2, the total modal strain energy is expressed as:

wherein K is _z Is the overall rigidity matrix phi of the bevel gear transmission system _j Representing the feature vector of the j-th order.

4. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: step S100 includes the steps of:

s101, modeling the bevel gear by adopting the 20-node hexahedral unit, dividing a plurality of 20-node hexahedral units according to the structure of the bevel gear, and assembling the units according to the unit numbers of the 20-node hexahedral units to obtain a total mass matrix and a rigidity matrix of the bevel gear, wherein the mass matrix of one 20-node hexahedral unit is as follows:

the stiffness matrix of one 20-node hexahedral unit is:

wherein N represents a shape function, ρ ₁ Is the density of bevel gears, B and B ^* The strain matrix under the global coordinate system and the iso-coordinate system respectively, D is an elastic matrix, W _i 、W _j 、W _k For Gaussian weight, τ is the number of Gaussian points in each direction, J is the Jacobian momentAnd the matrix, xi, eta and zeta are integral point coordinates.

5. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 4, wherein: the shape function N is expressed as:

wherein, for an angular node, its shape function is expressed as:

for intermediate nodes, its shape function is expressed as:

in xi _a 、η _a 、ζ _a The values of (a) are selected according to the position of each node in the iso-reference frame, and the subscript a indicates the node on the 20-node hexahedral cell.

6. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 4, wherein: step S100 includes the steps of:

s102, obtaining a conversion matrix T by adopting a mode synthesis method _cms By means of the transformation matrix T _cms The following transformations were performed:

M _cms ＝T _cms ^T MT _cms ，

K _cms ＝T _cms ^T KT _cms ，

thereby obtaining a mass matrix M after polycondensation of the bevel gears _cms And a stiffness matrix K _cms Realizing the established dimension-reducing polycondensation of the bevel gear; wherein M and K are the bevel gearsAn overall mass matrix and a stiffness matrix.

7. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 6, wherein: in step S102, the conversion matrix T is obtained by using a mode synthesis method _cms The method comprises the following steps:

firstly, according to the characteristics of the bevel gears, each bevel gear is provided with two interfaces, one is an engagement tooth surface of the bevel gear, the other is an inner ring surface of the bevel gear connected with the transmission shaft, and a motion equation of a substructure of the bevel gear is established under physical coordinates by node displacement:

wherein C is a damping matrix, Q represents an external load vector, R represents a force vector at the interface, a represents a node displacement vector, and a is divided into internal displacements a _i And interface displacement a _j The two parts, Q and R can be divided into two parts, and the motion equation of the substructure can be written as:

the free equation of motion for which damping is not a consideration is written as:

wherein M is _ii 、K _ii Is an internal mass matrix and a stiffness matrix, M _jj 、K _jj Is the mass matrix and stiffness matrix at the interface, M _ij And M _ji Is a coupling quality matrix, K _ij And K _ji Is a coupling stiffness matrix, R _j Is the force vector at the interface;

the free motion equation without damping is then calculated in modal coordinates:

the interface is fixed firstly, namely, the a is made _j =0, calculating the natural mode of the subsystem at the fixed interface, i.e. solving the eigenvalues of the following free vibration equation:

by solving the eigenvalue of the free vibration equation, i natural vibration modes can be obtained, and the combination is expressed as phi by a matrix _n ；

Each degree of freedom at the interface is then released in turn and the static displacement is then calculated:

from the above formula:

a _i ＝-K _ii ^-1 K _ij a _j ，

let a _j The j elements in the model are sequentially taken as unit values, the rest are 0, and the corresponding j groups of static displacement vectors, namely constraint modes, a, are obtained _i The form combined into matrix is expressed as phi _j ：

Φ _j ＝-K _ii ^-1 K _ij I _j ＝-K _ii ^-1 K _ij ，

Meanwhile, to achieve the purpose of reducing the degree of freedom, phi is omitted _n Only the k rows of low-order main modes are reserved to form phi _k The following operation is performed to obtain a transformation matrix T _cms ：

Wherein I is _j Is an identity matrix with dimension j x j.

8. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: in step S110, the stiffness matrix of the coupling unit is expressed as:

wherein K is _C A diagonal matrix of 6*6, the diagonal value of which is an equally large number.

9. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 1, wherein: in step S120, the stiffness matrix of the engagement unit is expressed as:

K _m ＝k _m (V) ^T V，

wherein k is _m As an average value of time-varying engagement stiffness, obtained by load contact analysis, V represents an engagement vector.

10. The method for rapidly deviating the resonance rotational speed of a flexible bevel gear transmission according to claim 9, wherein: the meshing vector V is expressed as:

V＝[n _px n _py n _pz 0 0 0 n _gx n _gy n _gz 0 0 0]，

wherein subscripts p and g respectively represent a driving gear and a driven gear, n _px 、n _py 、n _pz Is the unit component of the equivalent meshing force of the driving gear in the X, Y, Z direction, n _gx 、n _gy 、n _gz Is a unit component of the equivalent meshing force of the driven gear in the direction X, Y, Z.