CN112364463B - Squirrel-cage elastic support rigidity and stress analysis method - Google Patents
Squirrel-cage elastic support rigidity and stress analysis method Download PDFInfo
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Abstract
The application provides a method for analyzing rigidity and stress of a squirrel-cage elastic support, which is applied to a squirrel-cage elastic support structure and comprises the following steps: calculating the preset displacement delta generated on the squirrel-cage ring according to the characteristics of the squirrel-cage elastic supporting structure ring At j cage bar position angle theta j Is correspondingly positioned at the cantilever end displacement delta of the cage bar sj J is more than or equal to 1 and less than or equal to N, and N is the number of cage bars; according to the elastic deformation theory, cage bars are scattered by beam units, and a cage bar strain matrix B, a stress matrix kappa and a rigidity matrix K are calculated j (ii) a According to the matrix B, kappa, K j Calculating the displacement delta of the cantilever end of the jth cage bar sj While, the jth cage bar stress sigma ji And strain epsilon ji Distribution and acting force of cage bars on the cage ring; calculating a squirrel-cage elastic support rigidity matrix K according to the action resultant force of all cage bars on the squirrel-cage circular ring s . The method can give consideration to both the analysis precision and the calculation efficiency, quickly obtain the rigidity and the stress distribution of the squirrel-cage elastic support, and provide a method with more comprehensive consideration factors for the optimization design of the squirrel-cage elastic support.
Description
Technical Field
The invention belongs to the technical field of mechanical engineering, and particularly relates to a squirrel-cage elastic support rigidity and stress analysis method.
Background
The squirrel-cage elastic support is widely applied to various rotating machines such as an aircraft engine, a gas turbine, a rocket engine and the like, is used for cooperating with a rolling bearing to jointly support a rotor system to work and adjust the critical rotating speed area of the rotor system, so that the working rotating speed area of the rotating machine avoids the resonance area of the rotor system as far as possible, and a large number of design and test processes find that the rigidity of the squirrel-cage elastic support greatly influences the dynamic characteristic of the rotor system, and the stress of the squirrel-cage elastic support can also restrict the working reliability of the whole rotor system.
How to calculate and evaluate the rigidity and the stress mechanics characteristic of the squirrel-cage elastic support quickly, comprehensively, reasonably and accurately, and further understand the strain and stress change rule of the squirrel-cage elastic support cage bars under the multidirectional composite load, is greatly beneficial to optimizing the design and stress distribution of the rigidity of the squirrel-cage elastic support, can further reduce the risks that the rigidity of the squirrel-cage elastic support is not completely matched with the actual requirements of a rotor system and the cage bars are fractured in the working process, and the rotor dynamics design and the rolling bearing dynamics design of the rotary machine become more refined and accurate from this, and finally improve the working reliability of the rotor-support system.
In recent years, some domestic scholars have developed a great deal of theoretical analysis work for squirrel-cage elastic supports, such as the papers: the dynamic characteristics of the gas turbine rotor-squirrel-cage-squeeze film damper are researched (creep equation), but most of the dynamic characteristics are concentrated on the research on the radial rigidity and the maximum stress of the squirrel-cage elastic support by radial load, the axial rigidity, the radial rigidity, the angular rigidity and the torsional rigidity of the squirrel-cage elastic support in the actual working process can influence the dynamic characteristics of a rolling bearing-rotor system, and only a few scholars analyze the relation between the load characteristics and the stress distribution, the radial rigidity and the angular rigidity of the squirrel-cage elastic support based on commercial finite element analysis software.
Generally speaking, the present research method for mechanical characteristics of the squirrel-cage elastic support mainly analyzes the rigidity and stress of the squirrel-cage elastic support through a simplified mechanical analysis calculation formula or commercial finite element analysis software, on one hand, the rigidity and stress calculation analysis or empirical formula of the squirrel-cage elastic support is too much simplified, only the radial rigidity and the maximum stress of cage bars can be calculated, and the rigidity in other directions, the coupling influence of the rigidity in each direction and the stress distribution of the cage bars cannot be calculated, on the other hand, if the rigidity and stress of the squirrel-cage elastic support are analyzed by using the commercial finite element software, although the stress distribution of the cage bars and the rigidity of the squirrel-cage elastic support in each direction can be analyzed in detail, three problems still exist: 1) The accuracy and the reliability of a finite element analysis result of commercial software have a larger relation with the cognition of a designer on mechanical boundary conditions in the squirrel cage elastic supporting use process, the reliability of the calculation result is determined according to the software use experience of the designer and the cognition depth of the designer on the mechanical characteristics, and the instability is larger; 2) The modeling and calculating time of the commercial finite element software analysis process is long, the coupling operation with the rotor-supporting system dynamic characteristic analysis process is difficult to carry out, and a comprehensive and accurate squirrel-cage elastic supporting mechanical model is not considered when the rotor dynamics or the rolling bearing dynamics are calculated; 3) The modeling of commercial finite element analysis software is complicated, the grid division quality can also influence the squirrel-cage elastic support rigidity and stress analysis result to a large extent, and the iterative process of designing, analyzing and feeding back and adjusting the design each time is extremely time-consuming and labor-consuming.
Therefore, it is necessary to establish a squirrel-cage elastic support stiffness and stress analysis method which is efficient, accurate and convenient to transplant into other system-level simulation models.
Disclosure of Invention
The purpose of the invention is as follows: in order to provide a method for analyzing the rigidity and stress of the squirrel-cage elastic support, the main mechanical characteristics of the squirrel-cage elastic support are comprehensively considered based on the method, the rigidity design and the stress optimization design of the squirrel-cage elastic support can be normalized and streamlined, and the squirrel-cage elastic support is designed in the early stage of the squirrel-cage elastic support, so that the method can realize the quick optimization iteration of the structural design and the convenient coupling operation of a squirrel-cage elastic support mechanical model and other support and rotor structural mechanical characteristics.
The application provides a squirrel-cage elastic support rigidity and stress analysis method, the method is applied to a squirrel-cage elastic support structure, the squirrel-cage elastic support structure comprises a squirrel-cage ring and cage bar cantilever ends, the squirrel-cage ring is a non-fixed end used for mounting a bearing in the squirrel-cage elastic support, the cage bar cantilever ends are movable ends used for connecting the squirrel-cage ring on cage bars, and the method comprises the following steps:
calculating the pre-generation of squirrel cage ring according to the characteristics of squirrel cage elastic supporting structureLet the displacement delta ring ＝[x y z θ x θ y θ z ] T When the j cage bar corresponds to the cage bar, the cantilever end of the cage bar shifts delta sj J is more than or equal to 1 and less than or equal to Z, and Z is the number of cage bars;
according to the elastic deformation theory, dispersing each cage bar by using beam units, dividing the j cage bar into N beam units, wherein the j cage bar has N +1 nodes in total, and calculating a strain matrix B, a stress matrix kappa and a rigidity matrix K of the j cage bar j ；
According to the cage bar strain matrix B, the stress matrix kappa and the single cage bar rigidity matrix K j Calculating the fixation of the jth cage bar at one end and the displacement delta of the cantilever end of the cage bar at the other cantilever end sj In the meantime, the ith beam unit stress sigma of the jth cage bar ji And strain epsilon ji I is more than or equal to 1 and less than or equal to N, and the acting force f of the j cage bar cantilever end to the cage ring sj ；
According to the acting force f of the j cage bar cantilever end to the squirrel cage ring sj Calculating the resultant force of the acting forces of all the cage bar cantilever ends to the cage ring, and calculating the displacement delta of the cage elastic support at the preset displacement ring Stiffness matrix K under conditions s 。
Specifically, the mouse cage ring is calculated to generate the preset displacement delta according to the characteristics of the mouse cage elastic supporting structure ring Position angle theta j The cantilever end displacement delta of the cage bar corresponding to the jth cage bar sj The method specifically comprises the following steps:
establishing an inertial coordinate system O by taking the center of the cage ring which does not displace as the origin 1 X 1 Y 1 Z 1 ；
Establishing a squirrel-cage ring fixed coordinate system O by taking the center of the squirrel-cage ring with displacement as the origin 2 X 2 Y 2 Z 2 ；
Establishing cage bar azimuth coordinate system O by taking center of cantilever end of cage bar as origin rj X a Y a Z a Y of the cage bar orientation coordinate system a X through and perpendicular to the inertial system 1 Axis, X of the azimuthal coordinate system a And the system of inertia X 1 Axis parallel, Z of azimuth system a The direction is determined by adopting a right-hand rule;
determining a transformation matrix T from the inertial coordinate system to the jth cage bar azimuth coordinate system according to the spatial position relation between the jth cage bar azimuth coordinate system and the inertial coordinate system ia And calculating the preset displacement delta of the squirrel cage ring ring The vector O from the center of the inertial system to the center of the cantilever end of the j cage bar 1 O’ rj ＝O 1 O 2 +T ia O 2 O’ rj Wherein O is 1 O 2 For a predetermined displacement delta from the center of the inertial system to take place ring Vector of the center of the squirrel cage ring, O 2 O’ rj For generating a predetermined displacement delta ring The vector from the center of the squirrel cage circular ring to the center of the jth cage bar azimuth coordinate system;
according to the jth cage bar cantilever end center position vector O 1 O’ rj And calculating the cantilever end displacement delta of the cage bar corresponding to the jth cage bar sj ＝O 1 O’ rj -O 2 O’ rj 。
Specifically, the j-th cage bar is calculated to be fixed at one end, and the cantilever end displacement delta of the cage bar occurs at the other cantilever end sj In the meantime, the ith beam unit stress sigma of the jth cage bar ji And strain epsilon ji I is more than or equal to 1 and less than or equal to N, and the acting force f of the j cage bar cantilever end to the cage ring sj The method specifically comprises the following steps:
according to the cage bar rigidity matrix obtained by calculation, and by combining whether each node of the j-th cage bar is subjected to displacement constraint or not, rearranging the obtained j-th cage bar rigidity matrix to obtain the following mechanical analysis formula:
wherein, the nodes constrained by the displacement on the jth cage bar comprise nodes at two ends of the cage bar, the nodes not constrained by the displacement on the jth cage bar comprise nodes on the cage bar except the two ends, and d Ej A set of displacements corresponding to nodes on the j-th cage bar subject to displacement constraints, d Fj The j-th cage bar is not suffered fromSet of displacements corresponding to displacement-constrained nodes, f Ej The interaction force between the cantilever end of the cage bar and the squirrel cage ring is applied to the jth cage bar, f Fj The external force, K, on the j-th cage bar which is not subject to the displacement constraint Ej 、K EFj 、K Fj 、K FEj Respectively is a j cage bar rigidity matrix K j Rearranging the internal block matrix;
according to the displacement delta of the cantilever end of the cage bar sj Obtaining the displacement set d corresponding to the nodes constrained by the displacement on the cage bar Ej And the external force f on the node which is not constrained by the displacement on the jth cage bar Fj All elements in (A) are zero according to d in the mechanical analysis formula Ej And f Fj Calculating d Fj And f Ej According to f Ej The interaction force f between the cantilever end of each cage bar and the cage ring can be obtained sj According to d Fj The end point displacement d of the ith beam unit of the jth cage bar can be obtained ji ；
According to the strain matrix B of the beam elements, calculating the strain distribution epsilon in the ith beam element in the jth cage bar ji ＝Bd ji ；
Calculating the stress distribution sigma inside the ith beam unit in the jth cage bar according to the strain matrix kappa of the beam unit ji ＝κd ji 。
Specifically, the acting force f of the cantilever end of the jth cage bar on the squirrel cage ring is used sj Calculating the resultant force of the acting forces of all the cage bar cantilever ends to the cage ring, and calculating the displacement delta of the cage elastic support at the preset displacement ring Stiffness matrix K under conditions s The method specifically comprises the following steps:
according to the obtained interaction force f between the cantilever end of the j cage bar and the squirrel cage ring sj By the formula
And formula
Calculating resultant moment M of mouse cage ring s And resultant force F s ；
By giving a predetermined displacement delta ring Setting a small increment for each element, and utilizing resultant moment M s And the resultant force F s Calculating to obtain a rigidity matrix K by using the idea of differential approximation differentiation s The method comprises the following steps:
wherein,
is represented by F s The derivatives of x, y and z are respectively obtained,
represents M s The derivatives of x, y and z are respectively obtained,
is shown as F s Respectively to theta x 、θ y 、θ z The derivation is carried out by the derivation,
represents M s Respectively to theta x 、θ y 、θ z And (6) derivation.
Specifically, the beam unit comprises an Euler-Bernoulli beam, a Timoshenko beam, a Rayleigh beam and a Shear beam.
Specifically, the method further comprises:
calculating a beam unit rigidity matrix K according to the mechanical characteristics of the selected beam unit and the cage bar structure e ；
According to the beam unit rigidity matrix K e And the displacement coordination and force balance relation among all units is realized through a beam unit rigidity matrix K e Grouping and calculating a j-th cage bar rigidity matrix K j 。
Specifically, a j cage bar rigidity matrix K is set j After rearrangement, an internal block matrix K is formed Ej 、K EFj 、K Fj 、K FEj The method specifically comprises the following steps:
carrying out force balance analysis on the j cage bar to obtain a force balance equation K j d s ＝f s Wherein d is s For the displacement, f, corresponding to each node of the j-th cage bar s Dividing nodes of the cage bars into nodes which are restricted by displacement and nodes which are not restricted by displacement for the force applied to each node of the jth cage bar, wherein the nodes which are restricted by displacement comprise nodes at two ends of the cage bar, the nodes which are not restricted by displacement comprise nodes except the nodes at two ends of the cage bar, and the nodes which are restricted by displacement and the nodes which are not restricted by displacement correspond to d s Is obtained by separately arranging the elements in
At K, while ensuring the force balance equation is constant j In a matrix according to
Node in which displacement is restricted and node which is not restricted by displacement and K j Corresponding relation, will K j The arrangement order of elements is obtained
Further, a block matrix K can be obtained Ej 、K EFj 、K Fj 、K FEj 。
Specifically, calculating the jth cage bar strain matrix B and the stress matrix k specifically includes:
acquiring structural parameters and material parameters of the squirrel-cage elastic support, wherein the structural parameters comprise cage bar thickness, cage bar width, cage bar length, cage bar number, radius of a squirrel-cage circular ring and axial distance between a cage bar cantilever end and the center of the squirrel-cage circular ring, and the material parameters comprise elastic modulus, poisson's ratio and shear modulus of the squirrel-cage elastic support material;
calculating a main inertia moment and a polar inertia moment of the cross section of the cage bar according to the structural parameters and the material parameters;
and calculating a strain matrix B and a stress matrix kappa according to the main inertia moment and the polar inertia moment of the cross section of the cage bar.
The invention has the beneficial effects that: the squirrel-cage elastic support rigidity and stress analysis method provided by the invention has the following specific beneficial effects:
1) The method is easy to realize in a flow manner, is compiled into a modular calculation program, and can quickly and accurately obtain the rigidity of the squirrel-cage elastic support in each direction and the strain and stress distribution of cage bars when the squirrel-cage elastic support is designed and analyzed;
2) The method is convenient to be called by an optimization algorithm of the squirrel-cage elastic support structure, and a more convenient theoretical analysis method is provided for optimizing stress distribution and rigidity design of the squirrel-cage elastic support on the basis of considering both calculation precision and calculation efficiency;
3) The method facilitates the dynamic coupling operation of the squirrel-cage elastic support and the rotor system or the rolling bearing, and provides a more convenient and more comprehensive theoretical analysis method considering factors for the integrated design of the squirrel-cage elastic support and other support structures and the rotor system.
Drawings
FIG. 1 is a flow chart of a squirrel-cage elastic support stiffness and stress analysis provided by an embodiment of the present application;
FIG. 2 is a schematic diagram of a squirrel-cage elastic support coordinate system provided in an embodiment of the present application;
FIG. 3 is a schematic diagram illustrating the relationship between the displacement of cage bars and the displacement of cage rings of the squirrel cage elastic support according to the embodiment of the present invention;
FIG. 4 is a schematic diagram of a squirrel-cage resilient support stiffness matrix provided in accordance with an embodiment of the present application;
FIG. 5 is a partial schematic view of a squirrel cage resilient support stiffness matrix provided in accordance with an embodiment of the present application;
FIG. 6 is a schematic view of the inner edge X of the 2 nd beam unit in the 1 st cage bar elastically supported by the squirrel cage type a Axial stress distribution.
Detailed Description
Example one
The invention is described in detail below with reference to the drawings. Please refer to fig. 1-6 of the specification.
FIG. 1 shows a flow chart of the method of the present invention, which comprises the following main steps:
1) Input squirrel cage elastic support structure and material parameters
Acquiring structural parameters and material parameters of the squirrel-cage elastic support, wherein the structural parameters comprise the specific size of the squirrel-cage elastic support, including cage bar thickness, cage bar width, cage bar length, cage bar number, cage bar circular ring radius and axial distance from the center of a cantilever end of each cage bar to the center of the cage bar circular ring, and the material parameters comprise elastic modulus, poisson ratio and shear modulus of the squirrel-cage elastic support material, and accordingly, calculating main inertia moment and polar inertia moment of the section of each cage bar;
2) Establishing a coordinate system
The established coordinate system is shown in fig. 2, the squirrel-cage elastic support structure comprises a squirrel-cage ring with an annular structure and a cage bar cantilever end, the squirrel-cage ring is a non-fixed end used for mounting a bearing in the squirrel-cage elastic support, and the cage bar cantilever end is a movable end used for connecting the squirrel-cage ring on the cage bar. Establishing an inertial coordinate system O by taking the center of the squirrel-cage ring at the initial position as the origin 1 X 1 Y 1 Z 1 Establishing a squirrel-cage ring fixed coordinate system O by taking the center of the squirrel-cage ring with displacement as the origin 2 X 2 Y 2 Z 2 Coordinate system O in the course of moving cage ring 2 X 2 Y 2 Z 2 Keeping relative static with the squirrel cage ring, and establishing a cage bar azimuth coordinate system O by taking the center of the cantilever end of the cage bar as an origin rj X a Y a Z a Azimuthal coordinate system O rj X a Y a Z a Y of (A) is a X with axis passing through and perpendicular to the inertial system 1 Axis, X of the azimuthal coordinate system a Axis and system of inertia X 1 Axis parallel, Z of azimuth system a The axial direction is determined by adopting a right-hand rule;
3) Analysis of displacement relationship between squirrel cage ring and cantilever end of each cage bar
As shown in FIG. 3, the predetermined displacement delta is given to the squirrel cage ring ring ＝[x y z θ x θ y θ z ] T According to the ith rootDetermining the spatial position relation between the orientation coordinate system and the inertial coordinate system at the cage bars, and determining the transformation matrix T from the inertial coordinate system to the orientation coordinate system ia At this time, the origin O of the inertia system is away from the cantilever end center O 'of the j-th cage bar' rj Has a vector of O 1 O’ rj ＝O 1 O 2 +T ia O 2 O’ rj J is more than or equal to 1 and less than or equal to Z, and Z is the number of cage bars, wherein O 1 O 2 Is the vector from the center of the inertial system to the center of the squirrel cage ring, specifically O 1 O 2 ＝[x y z] T ，O 2 O’ rj A vector from the center of the squirrel cage ring to the center of the jth cage bar formula position coordinate system, specifically O 2 O’ rj ＝[X m R m cos(θ j )R m sin(θ j )] T Wherein R is m Radius of squirrel cage ring, X m The axial distance theta from the center of the cantilever end of the jth cage bar to the center of the squirrel cage ring j Is O 2 O’ rj The vector is in the jth squirrel cage circular ring fixed coordinate system and coordinate axis O 2 The included angle is that after the cantilever end of the jth cage bar is displaced, the center O of the cantilever end of the jth cage bar is not deformed before the jth cage bar is displaced rj To the cantilever end center O 'of the j cage bar after undeformed' rj Has a vector of O rj O’ rj ＝O 1 O’ rj -O 2 O’ rj And calculating the displacement delta of the cantilever end of the jth cage bar sj 。
4) Rigidity and stress analysis is carried out on single cage bar based on elastic deformation theory
Considering the j cage bar as a cantilever beam with one end freedom degree completely restricted and cantilever end displacement influenced by a squirrel cage ring, dividing the j cage bar into N beam units according to a finite element method, wherein the N beam units have N +1 nodes, each beam unit considers torsion, tension and compression and bending deformation of materials, and calculating a rigidity matrix K of the beam unit according to the node e Combining beam unit rigidity matrix K through the relationship of displacement coordination and force balance among the beam units e And further obtaining a j-th cage bar rigidity matrix K j 。
On the basis, according to whether each node of the j-th cage bar is subjected to displacement constraint or not, the method can be obtainedRearranging the rigidity matrix of the jth cage bar, wherein the specific process of the rearrangement is as follows: carrying out force balance analysis on the j cage bar to obtain a force balance equation K j d s ＝f s Wherein d is s For the displacement, f, corresponding to each node of the j-th cage bar s Dividing nodes of the cage bar into nodes which are constrained by the displacement and nodes which are not constrained by the displacement for the force applied to each node of the jth cage bar, wherein the nodes which are constrained by the displacement comprise nodes at two ends of the cage bar, the nodes which are not constrained by the displacement comprise nodes except the nodes at two ends of the cage bar, and the nodes which are constrained by the displacement and the nodes which are not constrained by the displacement correspond to d s Is obtained by separately arranging the elements in
At K, while ensuring the force balance equation is constant j In a matrix according to
Node subjected to displacement constraint and node not subjected to displacement constraint and K j Corresponding relation, will K j The arrangement order of elements is obtained
Further, a block matrix K can be obtained Ej 、K EFj 、K Fj 、K FEj Further, the following mechanical analysis formula is obtained:
wherein d is Fj A displacement set f corresponding to a node which is not subject to displacement constraint on the j-th cage bar Ej The interaction force between the cantilever end of the jth cage bar and the cage ring, K Ej 、K EFj 、K Fj 、K FEj Respectively is a j cage bar rigidity matrix K j Rearranging the internal block matrix;
according to d in the mechanical analysis formula Ej And f Fj Calculating d Fj And f Ej According to f Ej The interaction force f between the cantilever end of each cage bar and the cage ring can be obtained sj According to d Fj The end point displacement d of the ith beam unit of the jth cage bar can be obtained ji ，1≤i≤N；
According to the strain matrix B corresponding to the beam elements, calculating the strain distribution epsilon in the ith beam element in the jth cage bar ji ＝Bd ji ；
Calculating the stress distribution sigma inside the ith beam unit in the jth cage bar according to the strain matrix kappa of the beam unit ji ＝κd ji ；
5) Stress analysis of squirrel cage ring
Obtaining the interaction force f of the cantilever end of the jth cage bar and the cage ring according to the step 2) sj Using the formula
And formula
Calculating the resultant moment M of the squirrel cage ring s Sum and total force F s ；
6) Squirrel-cage elastic support stiffness characteristic analysis
Presetting displacement delta according to squirrel cage ring ring Resultant moment M applied to squirrel cage ring s Sum and total force F s By sequentially adding a small increment to each component of the squirrel-cage ring and utilizing the numerical operation of differential approximation differentiation, the squirrel-cage elastic support rigidity matrix K with 6 rows and 6 columns can be obtained s The method comprises the following steps:
wherein,
is represented by F s The derivatives of x, y and z are respectively obtained,
represents M s Respectively making derivatives of x, y and z,
is represented by F s Respectively to theta x 、θ y 、θ z The derivation is carried out by the derivation,
represents M s Respectively to theta x 、θ y 、θ z And (6) derivation.
The specific embodiment is as follows:
the effectiveness of the method of the present invention is further illustrated by taking as an example a squirrel cage elastomeric support structure. In this configuration, the bearing forces act directly on the cage ring, and the configuration parameters are listed in table 1.
TABLE 1 main parameters of certain type squirrel-cage elastic supporting structure
| (symbol) | Description of the invention | Value taking | Unit |
| R m | Radius of squirrel cage ring | 100 | mm |
| b | Width of cage bar | 2.5 | mm |
| h | Thickness of cage bar | 2.5 | mm |
| Z | Number of cage bars | 76 | |
| L | Length of cage bar | 30 | mm |
| X m | Axial distance of cantilever end of cage bar from center of squirrel cage ring | 10 | mm |
| E | Modulus of elasticity of squirrel cage material | 179 | GPa |
| G | Shear modulus of squirrel cage material | 69 | GPa |
| μ | Poisson's ratio of squirrel cage material | 0.3 |
Based on the parameters given in the table 1, inputting the squirrel-cage elastic supporting structure and the material parameters according to the step 1), and calculating the main inertia moment and the polar inertia moment of the section of the cage bar to be 3.25e-12m respectively 4 And 6.51e-12m 4 ；
According to step 2), an inertial coordinate system O is established, as shown in FIG. 2 1 X 1 Y 1 Z 1 Squirrel-cage circular ring fixed coordinate system O 2 X 2 Y 2 Z 2 Cage bar cantilever end orientation coordinate system O rj X a Y a Z a ；
According to step 3), setting the cage ring to a predetermined displacement of
δ ring ＝[1e-4 1e-4 1e-4 3e-5 3e-5 3e-5] T (m)
Determining a transformation matrix T from the inertial coordinate system to the azimuth coordinate system according to the spatial position relation between the azimuth coordinate system and the inertial coordinate system at the jth cage bar ia At this time, the origin O of the inertia system is separated from the cantilever end center O of the jth cage bar' rj Vector of
O 1 O’ rj ＝O 1 O 2 +T ia O 2 O’ rj
Wherein, O 1 O 2 Is the vector from the center of the inertial system to the center of the squirrel cage ring, specifically O 1 O 2 ＝[1e-4 1e-4 1e-4] T (m)，O 2 O’ rj The vector from the center of the squirrel cage ring to the center of the jth cage bar prescription position coordinate system is
O 2 O’ rj ＝{0.01 0.1cos[2(j-1)π/76]0.1sin[2(j-1)π/76]} T (m)
Calculating the displacement delta of the cantilever end of the jth cage bar after the displacement of the cantilever end of the jth cage bar occurs sj ＝O 1 O’ rj -O 2 O’ rj 。
According to the step 4), carrying out rigidity and stress analysis on the single cage bar
As shown in FIG. 3, consider the j-th cage as a cantilever with one-end freedom fully constrainedDividing the j-th cage bar into 2 Timoshenko beam units by a cantilever beam with end displacement influenced by a cage ring, wherein the j-th cage bar has 3 nodes, and calculating a rigidity matrix K of the Timoshenko beam units by considering torsion, tension and compression and bending deformation of each beam unit and considering the torsion, tension and compression and bending deformation of materials based on the size, the elastic modulus, the shear modulus, the main inertia moment of the section of the cage bar and the polar inertia moment of the squirrel-cage elastic support in the step 1) e Combining beam unit rigidity matrix K through the relationship of displacement coordination and force balance among the beam units e And further obtaining a j-th cage bar rigidity matrix K j 。
On the basis, rearranging the obtained rigidity matrix of the j-th cage bar according to whether each node of the j-th cage bar is subjected to displacement constraint or not to obtain the following mechanical analysis formula:
wherein, the node that receives displacement constraint on the cage strip contains the node at cage strip both ends, and the node that does not receive displacement constraint on the cage strip includes the node except both ends on the cage strip, d Fj A displacement set f corresponding to a node which is not subject to displacement constraint on the j-th cage bar Ej The interaction force between the cantilever end of the jth cage bar and the cage ring, K Ej 、K EFj 、K Fj 、K FEj Respectively is a j cage bar rigidity matrix K j Rearranging the internal block matrix;
according to d in the mechanical analysis formula Ej And f Fj Calculating d Fj And f Ej According to f Ej The mutual acting force f of the cantilever end of the jth cage bar and the cage ring can be obtained sj According to d Fj The end point displacement d of the ith beam unit of the jth cage bar can be obtained ji ，1≤i≤2，1≤j≤67；
According to the strain matrix B and the strain matrix kappa corresponding to the beam unit, calculating the strain distribution epsilon in the ith beam unit in the jth cage bar ji ＝Bd ji And stress distribution σ ji ＝κd ji Drawing according to the calculation resultMaking Y in the azimuth coordinate system of the 1 st cage bar a And Z a Stress of 2 nd beam unit in 1 st cage bar is elastically supported along an azimuth coordinate system X at positions of 1.25 (mm) and 1.25 (mm), respectively a The distribution of axes is shown in FIG. 6, σ in FIG. 6 x 、σ y 、σ z 、σ γ Respectively represent cage bar edges X a Axial tensile and compressive stress, along Y a Axial bending stress, along Z a Axial bending stress, about X a Shear stress due to shaft torsion;
according to the step 5), calculating the interaction force f of the cantilever end of the jth cage bar and the cage ring sj Using the formula
And formulas
Calculating the resultant moment M of the cage ring s ＝[1.98 1.70e2 1.14e2] T (N m) and a resultant force F s ＝[2.83e5 1.97e3 1.97e3] T (N·m)；
According to step 6), presetting displacement delta according to the squirrel cage circular ring ring Resultant moment M applied to squirrel cage ring s Sum and total force F s By sequentially adding a small increment to each component of the squirrel-cage ring and utilizing the numerical operation of differential approximation differentiation, the squirrel-cage elastic support rigidity matrix K with 6 rows and 6 columns can be obtained s The resulting stiffness matrix K s As shown in fig. 4 and 5, fig. 5 is a partial schematic diagram of elements of the matrix shown in fig. 4.
The method can consider the coupling influence among the rigidity of the squirrel-cage elastic support in each direction, quickly calculate the rigidity of the squirrel-cage elastic support in each direction and the strain and stress distribution of each cage bar, provide a more convenient theoretical method for the rigidity design and stress distribution optimization of the squirrel-cage elastic support, and provide a quick, accurate and comprehensive numerical analysis method for the rigidity, strain and stress distribution of the squirrel-cage elastic support and the coupling operation of other structures.
In summary, the invention belongs to the technical field of mechanical engineering, and particularly provides a squirrel-cage elastic support rigidity and stress analysis method, which comprises the following steps: the method comprises the following steps: as shown in fig. 2, a squirrel cage inertial coordinate system, a cage bar fixed body coordinate system and a squirrel cage circular ring fixed body coordinate system are established; step two: as shown in fig. 3, the displacement relationship between the cage ring displacement and the cantilever end of each cage bar is analyzed; step three: according to the elastic deformation theory, strain and stress analysis is carried out on a single cage bar at any position; step four: analyzing and synthesizing the acting force and the acting moment of each cage bar on the squirrel cage circular ring; step five: and analyzing the rigidity characteristic of the squirrel-cage elastic support according to the cage bar acting force applied when the obtained squirrel-cage circular ring generates the preset displacement. The method can give consideration to analysis precision and calculation efficiency on the basis of keeping the main mechanical characteristics of the squirrel cage elastic support, quickly obtain the rigidity and stress distribution of the squirrel cage elastic support, and provide a theoretical method which takes more comprehensive consideration factors for the optimization design of the squirrel cage elastic support and the coupling operation of the squirrel cage elastic support and other structures.
Claims (6)
1. The method for analyzing the rigidity and the stress of the squirrel-cage elastic support is characterized by being applied to a squirrel-cage elastic support structure, the squirrel-cage elastic support structure comprises a squirrel-cage ring and cage bar cantilever ends, the squirrel-cage ring is a non-fixed end of the squirrel-cage elastic support for mounting a bearing, the cage bar cantilever ends are movable ends of cage bars for connecting the squirrel-cage ring, and the method comprises the following steps:
calculating the preset displacement delta generated in the squirrel-cage ring according to the characteristics of the squirrel-cage elastic supporting structure ring ＝[x y z θ x θ y θ z ] T When the j cage bar corresponds to the cage bar, the cantilever end of the cage bar shifts delta sj J is more than or equal to 1 and less than or equal to Z, and Z is the number of cage bars;
according to the elastic deformation theory, dispersing each cage bar by adopting a beam unit, dividing the jth cage bar into N beam units, wherein the jth cage bar has N +1 nodes in total, and calculating a strain matrix B, a stress matrix kappa and a stiffness matrix K of the jth cage bar j ；
According to the cage barStrain matrix B, stress matrix kappa and single cage bar rigidity matrix K j Calculating the fixation of the jth cage bar at one end and the displacement delta of the cantilever end of the cage bar at the other cantilever end sj In the meantime, the stress distribution σ of the ith beam unit of the jth cage bar ji And strain distribution ε ji I is more than or equal to 1 and less than or equal to N, and the acting force f of the j cage bar cantilever end to the cage ring sj ；
According to the acting force f of the j cage bar cantilever end to the squirrel cage ring sj Calculating the resultant force of the acting forces of all the cage bar cantilever ends to the cage ring, and calculating the displacement delta of the cage elastic support at the preset displacement ring Stiffness matrix K under conditions s ；
Calculating the jth cage bar, wherein the jth cage bar is fixed at one end, and the cantilever end displacement delta of the cage bar occurs at the other cantilever end sj In the meantime, stress distribution σ of ith beam unit of jth cage bar ji And strain distribution ε ji I is more than or equal to 1 and less than or equal to N, and the acting force f of the j cage bar cantilever end to the cage ring sj The method specifically comprises the following steps:
according to the calculated cage bar rigidity matrix, in combination with whether each node of the jth cage bar is subjected to displacement constraint or not, rearranging the obtained jth cage bar rigidity matrix to obtain the following mechanical analysis formula:
wherein, the nodes constrained by the displacement on the jth cage bar comprise nodes at two ends of the cage bar, the nodes not constrained by the displacement on the jth cage bar comprise nodes on the cage bar except the two ends, and d Ej Is a set of displacements corresponding to nodes on the jth cage bar subject to displacement constraints, d Fj Is a displacement set f corresponding to the node which is not constrained by the displacement on the jth cage bar Ej The interaction force between the cantilever end of the cage bar on the jth cage bar and the cage ring is f Fj The external force, K, on the j-th cage bar which is not subject to the displacement constraint Ej 、K EFj 、K Fj 、K FEj Respectively is a j cage bar rigidity matrix K j Rearranged inner partA block matrix;
according to the displacement delta of the cantilever end of the cage bar sj Obtaining the displacement set d corresponding to the nodes constrained by the displacement on the cage bar Ej And the external force f borne by the node which is not constrained by the displacement on the jth cage bar Fj All elements in (A) are zero according to d in the mechanical analysis formula Ej And f Fj Calculating d Fj And f Ej According to f Ej The interaction force f between the cantilever end of each cage bar and the cage ring can be obtained sj According to d Fj The end point displacement d of the ith beam unit of the jth cage bar can be obtained ji ；
According to the strain matrix B of the beam unit strains, calculating the strain distribution epsilon in the ith beam unit in the jth cage bar ji ＝Bd ji ；
Calculating the stress distribution sigma inside the ith beam unit in the jth cage bar according to the strain matrix kappa of the beam unit ji ＝κd ji ；
The acting force f of the cantilever end of the jth cage bar on the squirrel cage ring sj Calculating the resultant force of the acting forces of all the cage bar cantilever ends to the cage ring, and calculating the displacement delta of the cage elastic support at the preset displacement ring Stiffness matrix K under conditions s The method specifically comprises the following steps:
according to the obtained interaction force f between the cantilever end of the jth cage bar and the cage ring sj By the formula
And formula
Calculating resultant moment M of mouse cage ring s Sum and total force F s ；
By giving a predetermined displacement delta ring Setting a small increment for each element, and utilizing resultant moment M s And said resultant force F s Calculating to obtain a stiffness matrix K by utilizing the idea of differential approximation differentiation s The method comprises the following steps:
wherein,
is represented by F s Respectively making derivatives of x, y and z,
represents M s The derivatives of x, y and z are respectively obtained,
is represented by F s Respectively to theta x 、θ y 、θ z The derivation is carried out by taking the derivative,
represents M s Respectively to theta x 、θ y 、θ z And (6) derivation.
2. The method for analyzing rigidity and stress of squirrel-cage elastic support according to claim 1, characterized in that the calculation of the preset displacement δ generated in the squirrel-cage ring is based on the characteristics of the squirrel-cage elastic support structure ring Position angle theta j The cantilever end displacement delta of the cage bar corresponding to the jth cage bar sj The method specifically comprises the following steps:
establishing an inertial coordinate system O by taking the center of the cage ring without displacement as the origin 1 X 1 Y 1 Z 1 ；
Establishing a squirrel-cage ring fixed coordinate system O by taking the center of the displaced squirrel-cage ring as an original point 2 X 2 Y 2 Z 2 ；
Establishing cage bar azimuth coordinate system O by taking center of cantilever end of cage bar as origin rj X a Y a Z a Y of the cage bar orientation coordinate system a X through and perpendicular to the inertial system 1 Axis, X of the azimuthal coordinate system a And the inertia system X 1 The axes are parallel to each other and the axis is parallel,z of the azimuthal system a The direction is determined by adopting a right-hand rule;
determining a transformation matrix T from the inertial coordinate system to the jth cage bar azimuth coordinate system according to the spatial position relation between the jth cage bar azimuth coordinate system and the inertial coordinate system ia And calculating the preset displacement delta of the squirrel cage ring ring The vector O from the center of the inertial system to the center of the cantilever end of the j cage bar 1 O′ rj ＝O 1 O 2 +T ia O 2 O′ rj Wherein O is 1 O 2 For a predetermined displacement delta from the center of the inertial system to take place ring Of the center of the squirrel cage ring, O 2 O′ rj For generating a predetermined displacement delta ring The vector from the center of the squirrel cage circular ring to the center of the jth cage bar azimuth coordinate system;
according to the jth cage bar cantilever end center position vector O 1 O′ rj Calculating the displacement delta of the cantilever end of the cage bar corresponding to the jth cage bar sj ＝O 1 O′ rj -O 2 O′ rj 。
3. The method for analyzing rigidity and stress of squirrel-cage elastic support according to claim 1, wherein the beam unit comprises Euler-Bernoulli beam, timoshenko beam, rayleigh beam and Shear beam.
4. The squirrel-cage elastic support stiffness and stress analysis method of claim 1, further comprising:
calculating a beam unit rigidity matrix K according to the mechanical characteristics of the selected beam unit and the cage bar structure e ；
According to beam unit rigidity matrix K e And the displacement coordination and force balance relation among all units is realized through a beam unit rigidity matrix K e Grouping and calculating a j-th cage bar rigidity matrix K j 。
5. The squirrel-cage elastic support stiffness and stress analysis method of claim 1, characterized in that a jth cage bar stiffness matrix K is set j To resumeAfter alignment, an internal block matrix K is formed Ej 、K EFj 、K Fj 、K FEj The method specifically comprises the following steps:
carrying out force balance analysis on the j cage bar to obtain a force balance equation K j d s ＝f s Wherein d is s For the displacement, f, corresponding to each node of the j-th cage bar s Dividing nodes of the cage bars into nodes which are restricted by displacement and nodes which are not restricted by displacement for the force applied to each node of the jth cage bar, wherein the nodes which are restricted by displacement comprise nodes at two ends of the cage bar, the nodes which are not restricted by displacement comprise nodes except the nodes at two ends of the cage bar, and the nodes which are restricted by displacement and the nodes which are not restricted by displacement correspond to d s Is obtained by separately arranging the elements in
At K, while ensuring the force balance equation is constant j In a matrix according to
Node subjected to displacement constraint and node not subjected to displacement constraint and K j Corresponding relation, will K j The element arrangement order is obtained
So as to obtain the block matrix K Ej 、K EFj 、K Fj 、K FEj 。
6. The method for analyzing the rigidity and the stress of the squirrel-cage elastic support according to claim 1, wherein the calculation of the strain matrix B and the stress matrix k of the j cage bar specifically comprises the following steps:
acquiring structural parameters and material parameters of the squirrel-cage elastic support, wherein the structural parameters comprise cage bar thickness, cage bar width, cage bar length, cage bar number, radius of a squirrel-cage circular ring and axial distance between a cage bar cantilever end and the center of the squirrel-cage circular ring, and the material parameters comprise elastic modulus, poisson's ratio and shear modulus of the squirrel-cage elastic support material;
calculating a main inertia moment and a polar inertia moment of the cross section of the cage bar according to the structural parameters and the material parameters;
and calculating a strain matrix B and a stress matrix kappa according to the main inertia moment and the polar inertia moment of the cross section of the cage bar.
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