CN113177273B - Gas turbine compressor and turbine blade contact rigidity and damping analysis solving method - Google Patents

Gas turbine compressor and turbine blade contact rigidity and damping analysis solving method Download PDF

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CN113177273B
CN113177273B CN202110416350.6A CN202110416350A CN113177273B CN 113177273 B CN113177273 B CN 113177273B CN 202110416350 A CN202110416350 A CN 202110416350A CN 113177273 B CN113177273 B CN 113177273B
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谢永慧
刘铸锋
朱光亚
张荻
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Xian Jiaotong University
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Abstract

The invention relates to a method for analyzing and solving the contact rigidity and the damping of a gas turbine compressor and a turbine blade, which is characterized in that a point-surface contact, line-surface contact or surface-surface contact friction damping model is respectively established on the basis of the analysis of the contact strength under different operating loads, a contact rigidity and damping agent model of different friction damper structures is obtained by combining a linearization equivalence principle and an energy dissipation principle, the limitation of the traditional method is broken through, the influence of a high-order harmonic component is considered, the equivalent components of the contact rigidity and the damping to different harmonic terms are calculated, a multi-working-condition dimensionless equivalent coefficient database is established, a database is called in a numerical model to obtain corresponding equivalent coefficients, the contact rigidity and the damping are analyzed and calculated, and an agent model database and a quick equivalent solving method are provided for the industrial vibration reduction analysis of a rotary mechanical blade friction damper.

Description

Gas turbine compressor and turbine blade contact rigidity and damping analysis and solving method
Technical Field
The invention belongs to a method for analyzing vibration characteristics of blades of rotating machinery, and particularly relates to a method for analyzing and solving the contact rigidity and damping of a gas compressor and a turbine blade of a gas turbine.
Background
In high-speed, high-power rotating machines, vibration damping of key components thereof has been a problem of great concern, for example, severe vibration of gas turbines, compressors, and turbine blades at near-resonance points can produce excessive stress and result in structural failure. The friction damper is widely applied to vibration damping design of blades of industrial rotating machinery due to simple structure and good vibration damping effect. The friction damper is essentially composed of a metal block, common forms include a damping shroud band, boss platinum, loose platinum, a damping block and the like, vibration energy is dissipated through friction, the vibration level of parts can be effectively inhibited, and the safety and reliability of the structure are improved.
When the friction damper is in engineering design, the contact surface is generally considered to reach an ideal surface-surface contact state, the contact rigidity and the damping under the design working condition are converted into an equivalent rigidity coefficient and an equivalent damping coefficient of the contact surface based on the positive pressure load received under the basic load, and the equivalent rigidity coefficient and the equivalent damping coefficient are substituted into a strong nonlinear dynamical equation to be solved, so that the friction damping characteristic of the system under the ideal contact state is obtained. However, in order to meet the requirement of flexible peak regulation operation, the unit needs to complete rapid and deep variable load operation, the working condition at the moment is seriously deviated from the designed working condition, and alternating load is generated, so that the contact surface is changed into the states of point-surface contact and line-surface contact from ideal surface-surface contact, and the energy consumed by the damping contact surfaces under different excitation frequencies is different, and the equivalent stiffness and the equivalent damping obtained under the designed working condition cannot be applied to the whole excitation frequency range; under different exciting forces and positive pressures, the rigidity and damping effect of the contact interface among the gas turbine, the gas compressor, the turbine blade friction damper, the blade root and the wheel rim can also change along with the change of the contact state; on the other hand, the existing stiffness damping calculation method mainly considers the first-order harmonic term of excitation, and ignores the ultraharmonic term and the subharmonic term generated by nonlinear vibration, but when slip and viscous state coexist in the motion period and the weight of the viscous state is large, the influence of the higher-order harmonic component on the vibration response of the system needs to be considered, and these terms often have great influence on the vibration response in the resonance frequency band, resulting in a large resonance frequency band and response estimation deviation.
Disclosure of Invention
The invention aims to provide an analysis and solving method for contact rigidity and damping of a gas compressor and a turbine blade of a gas turbine, which is based on contact strength analysis under different operating loads, respectively establishes a point-surface contact, line-surface contact or surface-surface contact friction damping model, combines a linearization equivalence principle and an energy dissipation principle to obtain contact rigidity and damping agent models of different friction damper structures, breaks through the limitation of the traditional method, considers the influence of high-order harmonic components, calculates equivalent components of the contact rigidity and the damping to different harmonic terms, enhances the robustness of the equivalent method to estimate different working conditions, and provides an agent model database and a rapid equivalent solving method for industrial vibration reduction analysis of a rotating mechanical blade-friction damper (such as the contact friction between a blade root and a wheel rim and the like).
In order to achieve the purpose, the invention adopts the following technical scheme:
the method for analyzing and solving the contact rigidity and the damping of the gas compressor and the turbine blade of the gas turbine comprises the following steps of:
step 1, obtaining relevant parameters according to a friction damper material: coefficient of friction between contact surfaces mu 0 The Poisson's ratio v, the elastic modulus E and the shear modulus G, and obtaining the positive pressure load Q between the friction contact surfaces of the blades under different rotating speed working conditions through analysis to obtain the equivalent contact radius R and the tangential contact rigidity k of the contact surfaces c ；
Step 2, analyzing the contact strength according to the operation load, selecting a corresponding friction model according to the stress distribution between the contact surfaces to describe the friction damping characteristic between the contact surfaces, obtaining a hysteresis loop under the micro-motion sliding and macro-motion sliding states, and establishing a friction damping function model f-s of the contact surfaces;
step 3, performing Fourier expansion on the relative displacement and the friction force by adopting a first-order harmonic balancing method, and calculating to obtain an equivalent stiffness coefficient K between the friction contact surfaces according to a linearization equivalent principle and an energy dissipation principle e And equivalent damping coefficient C e ；
Step 4, repeating the steps 1-3 to convert the contact rigidity and the damping of the friction damper with different materials, contact characteristics and pressure loads under various working conditions, carrying out dimensionless treatment, and constructing dimensionless equivalent rigidity coefficients of different friction damper structures
Figure GDA0003846991980000031
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000032
A database;
step 5, taking high-order harmonic factors into consideration, calculating to obtain an equivalent stiffness coefficient K between friction contact surfaces by adopting a multi-harmonic balance method according to a linearization equivalent principle and the friction damping function model in the step 2 e And equivalent resistanceCoefficient of damping C e ；
Step 6, converting the contact rigidity and damping of the friction damper with different materials, contact characteristics and pressure loads under various working conditions, carrying out dimensionless treatment, and constructing dimensionless equivalent rigidity coefficients of different friction damper structures under high-order harmonics
Figure GDA0003846991980000033
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000034
A database;
and 7, establishing a spring damping unit in the numerical analysis model, calling corresponding equivalent stiffness coefficients and equivalent damping coefficients from a database to average the equivalent stiffness coefficients and the equivalent damping coefficients to the spring damping unit in all directions according to different materials, contact characteristics and pressure loads of the model, and carrying out contact stiffness and damping analysis calculation.
The invention is further improved in that the frictional damping function model in step 2 comprises: point-surface frictional contact model, line-surface frictional contact model and surface-surface frictional contact model.
The invention has the further improvement that in the step 2, when the friction contact surfaces just contact in the rapid lifting load and the stress distribution of the contact surfaces is in concentrated point distribution, a point-surface friction contact model is established, and the description function of the point-surface friction contact model depends on the amplitude of the relative displacement: when the amplitude D of the relative displacement between the contact surfaces is less than or equal to 1.5D 0 And when the friction damper is in a micro-motion sliding state, the friction hysteresis loop between the contact surfaces is as follows:
curve ABC:
Figure GDA0003846991980000035
curve CEA:
Figure GDA0003846991980000036
when D > 1.5D 0 And when the damping structure is in the micro-motion slippage macro-slippage coexistence stage, the friction hysteresis loop between the contact surfaces is as follows:
curve a ' B ' C ':
Figure GDA0003846991980000037
curve C' C ":
Figure GDA0003846991980000038
curve C "E" a ":
Figure GDA0003846991980000041
curve a "a':
Figure GDA0003846991980000042
wherein
Figure GDA0003846991980000043
When the contact surface reaches a locking state under a design working condition and the stress distribution of the contact surface does not change any more, a surface-surface friction contact model is built, and a description function of the surface-surface friction contact model depends on the magnitude of the relative displacement amplitude: d is less than or equal to D 0 The dimensionless friction hysteresis loop is:
curve ABCDE:
Figure GDA0003846991980000044
curve EFGA:
Figure GDA0003846991980000045
D>D 0 the dimensionless friction hysteresis loop is:
curve DEF:
Figure GDA0003846991980000046
curve FG:
Figure GDA0003846991980000047
the curve GABC:
Figure GDA0003846991980000048
curve CD:
Figure GDA0003846991980000049
when the contact surface stress is in linear distribution, a line-surface friction contact model is established, and an expression of the friction force f and the displacement D of the rightmost end of the damper changing along with the sliding length delta is obtained according to the mechanical analysis process.
A further improvement of the invention is that step 3 comprises:
for point-surface frictional contact model and surface-surface frictional contact model:
step 3.1, performing Fourier expansion on the relative displacement s and the friction force f by adopting a first-order harmonic balance method respectively:
s＝Dcosωt (1)
f＝f k (D)cosωt+f c (D)sinωt (2)
step 3.2, according to the operation of Fourier series coefficient, f is obtained k (D) And f c (D)：
Figure GDA00038469919800000410
Step 3.3, obtaining by a linearization equivalent principle:
Figure GDA0003846991980000051
step 3.4, substituting the formula (1) into the formula (4) to obtain an expression of the friction force, wherein the expression is as follows:
Figure GDA0003846991980000052
step 3.5, comparing the formula (5) with the formula (2), and obtaining the equivalent stiffness coefficient and the equivalent damping coefficient which are respectively as follows:
Figure GDA0003846991980000053
and 3.6, substituting the formula (3) into the formula (6) to obtain the equivalent stiffness coefficient and the equivalent damping coefficient, wherein the expressions are as follows:
Figure GDA0003846991980000054
for the line-to-plane frictional contact model:
step 3.1, obtaining a calculation formula of the dissipation energy of the damper in a period according to an energy dissipation principle:
Figure GDA0003846991980000055
step 3.2, analyzing the mechanical process to obtain an equivalent stiffness coefficient and an equivalent damping coefficient which are respectively as follows:
Figure GDA0003846991980000056
the invention has the further improvement that the step 3.3 adopts a linearization equivalence principle, the friction contact surface is simplified into a no-mass spring damping system, and the nonlinear friction force between the contact surfaces is equivalent to the linear superposition of the elastic force and the damping force.
The further improvement of the present invention is that in step 3.1, the energy dissipation principle is adopted, and the energy consumed by the damping model in one vibration cycle is obtained by calculating the area surrounded by the hysteresis loop.
The further improvement of the invention is that in step 4, the rigidity and damping characteristics of the contact surface are separated from the geometric parameters of the damper, and the dimensionless forms of equivalent coefficients of different friction damping function models are obtained; wherein the content of the first and second substances,
dimensionless equivalent stiffness coefficient of point-surface friction contact model
Figure GDA0003846991980000057
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000058
Figure GDA0003846991980000059
Dimensionless equivalent stiffness coefficient of surface-surface friction contact model
Figure GDA0003846991980000061
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000062
Figure GDA0003846991980000063
Figure GDA0003846991980000064
Dimensionless equivalent stiffness coefficient of line-surface friction contact model
Figure GDA0003846991980000065
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000066
Figure GDA0003846991980000067
Figure GDA0003846991980000068
A further development of the invention is that step 5 comprises:
step 5.1, give harmonic number N h Respectively carrying out N on relative displacement s and friction force f(s) by adopting a multi-harmonic balance method h Expansion of Fourier series:
Figure GDA0003846991980000069
Figure GDA00038469919800000610
step 5.2, f(s) is obtained according to the friction damping function model, and coefficients of the right end of the formula (9) are obtained according to Fourier transform:
Figure GDA00038469919800000611
Figure GDA00038469919800000612
Figure GDA0003846991980000071
and 5.3, substituting the formulas (8) and (9) into the formula (4) by a linearization equivalent principle, comparing the corresponding terms with the same coefficient, and calculating to obtain each order of equivalent stiffness coefficient k en And equivalent damping coefficient c en ：
Figure GDA0003846991980000072
And 5.4, expressing the relative displacement and the friction force in a matrix form, and applying a Newton-Raphson method to iteratively solve s:
Figure GDA0003846991980000073
in formula (11)K represents the current iteration step number; r = (ME) 2 + C Ε + KI) s + F-F are equation residual terms that converge when R = 0;
Figure GDA0003846991980000074
and 5.5, writing the formula (4) into a matrix form to obtain:
f＝(K e +C e E)s (12)
the equivalent stiffness coefficient matrix is obtained from equation (10):
Figure GDA0003846991980000075
the equivalent damping coefficient matrix is:
Figure GDA0003846991980000081
the further improvement of the invention is that the equivalent coefficient obtained by the multi-harmonic balancing method in step 6 is dimensionless to obtain the dimensionless equivalent stiffness coefficient of each order
Figure GDA0003846991980000082
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000083
The expression of (c) is:
Figure GDA0003846991980000084
the coefficients in equation (13) are:
Figure GDA0003846991980000085
Figure GDA0003846991980000086
Figure GDA0003846991980000087
the dimensionless equivalent stiffness coefficient matrix is expressed as:
Figure GDA0003846991980000088
the dimensionless equivalent damping coefficient matrix is expressed as:
Figure GDA0003846991980000089
the invention has at least the following beneficial technical effects:
the method for analyzing and solving the contact rigidity and the damping of the gas compressor and the turbine blade of the gas turbine uses parameterized theoretical analysis and solution to replace engineering empirical design, obtains the equivalent rigidity and the equivalent damping of the unit under different operating loads, and considers the influence of different materials, different positive pressure loads and contact conditions;
further, three friction damping models of point-surface contact, line-surface contact and surface-surface contact are provided, basic contact characteristics in the unit load change process are covered, and the corresponding friction damping model can be selected according to the actual contact strength analysis result of the blade-friction damper during analysis, so that the friction hysteresis loop and the description function thereof in each state are accurately obtained.
Furthermore, a linearization equivalence principle is adopted in the first-order harmonic balance method analysis process, the friction contact surface is simplified into a no-mass spring damping system, the nonlinear friction force between the contact surfaces is equivalent into the linear superposition of the elastic force and the damping force through the stress balance analysis, so that the complicated nonlinear mechanical problem is converted into the linear solution, the equivalent stiffness and the equivalent coefficient of a point-surface contact and surface-surface contact model are obtained, the calculated amount can be simplified, and the result precision requirement can be met.
Furthermore, an energy dissipation principle is adopted in the first-order harmonic balance method analysis process, the energy consumed by the damping model in a vibration period can be directly obtained by calculating the area surrounded by the hysteresis loop, and the equivalent rigidity and the equivalent coefficient of the line-surface contact model are obtained by mechanical analysis.
Further, the rigidity and damping characteristics of the contact surface are separated from the geometric parameters of the damper, the dimensionless forms of the equivalent coefficients of different friction damping function models are obtained, induction and comparison are facilitated, and the equivalent rigidity coefficients and the equivalent damping coefficients of corresponding materials can be obtained by multiplying different material parameters on the basis of the dimensionless forms.
Furthermore, the influence of a high-order harmonic component is considered in the analysis and solution method of the contact rigidity and the damping of the gas turbine compressor and the turbine blade, the equivalent components of the contact rigidity and the damping to different harmonic terms are calculated, the analysis precision of a friction damping numerical model can be improved, and the robustness of the equivalent method to different working conditions is enhanced.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art are briefly introduced below; it is obvious that the drawings in the following description are some embodiments of the invention, and that for a person skilled in the art, other drawings can be derived from them without inventive effort.
FIG. 1 is a flow chart of a method for analyzing and solving stiffness and damping of contact between a compressor and a turbine blade of a gas turbine according to an embodiment of the invention;
FIG. 2 is a point-surface contact friction model in an embodiment of the invention: (a) The viscous sliding contact schematic diagram of the point-surface contact friction model is shown; (b) is a tangential friction time domain curve; (c) The friction force changes a hysteresis loop along with displacement in the micro-motion sliding stage; (d) A hysteresis loop with the change of friction force along with displacement in a coexisting state; (e) A dimensionless equivalent stiffness coefficient curve of a point-surface contact friction model is obtained; (f) Is a dimensionless equivalent damping coefficient curve of a point-surface contact friction model.
FIG. 3 is the following-surface frictional contact model for the micromotion slip regime in an embodiment of the invention: (a) Is a schematic diagram of the viscous sliding contact of a surface-surface friction model; (b) is a tangential friction time domain curve; (c) The friction force in the micro-motion sliding stage changes along with the displacement to delay the loop.
FIG. 4 is a following-surface frictional contact model of the micro-motion slip macro-slip coexistence state in the embodiment of the present invention: (a) Is a schematic diagram of the viscous sliding contact of a surface-surface friction model; (b) is a tangential friction time domain curve; (c) A hysteresis loop with the change of friction force along with displacement in a coexisting state; (d) Is a dimensionless equivalent stiffness coefficient curve of a surface-surface friction model; (e) Is a dimensionless equivalent damping coefficient curve of a surface-surface friction model.
FIG. 5 is a line-surface frictional contact model in the micro-motion slip macro-slip coexisting state in the embodiment of the present invention: (a) is a schematic diagram of a line-surface friction contact model; (b) The distribution of a sliding area and a stagnation area under an initial loading state is realized; (c) The distribution of the sliding area and the stagnation area in the process of reducing the tangential force from the maximum value; (d) The sliding area and the stagnation area are distributed in the process of increasing the tangential force from the minimum value; and (e) is a line-surface friction contact model hysteresis loop.
FIG. 6 is a schematic flow chart of solving equivalent stiffness and equivalent damping coefficient by a multi-harmonic balancing method.
In the figure: 1-spherical friction pair; 2-plane friction pair; 3-viscous zone; 4-slip zone; 5-distribution of frictional stress.
Detailed Description
In order to make the purpose, technical effect and technical solution of the embodiments of the present invention clearer, the following clearly and completely describes the technical solution of the embodiments of the present invention with reference to the drawings in the embodiments of the present invention; it is to be understood that the described embodiments are part of the present invention. Other embodiments, which can be derived by one of ordinary skill in the art from the disclosed embodiments without inventive faculty, are intended to be within the scope of the invention.
Referring to fig. 1, the method for analyzing and solving the contact stiffness and damping of the compressor and the turbine blade of the gas turbine provided by the invention mainly comprises the following steps:
step 1, obtaining relevant parameters according to a friction damper material: coefficient of friction between contact surfaces mu 0 The Poisson's ratio v, the elastic modulus E and the shear modulus G, and obtaining the positive pressure load Q between the friction contact surfaces under different rotating speed working conditions through analysis to obtain the equivalent contact radius R and the tangential contact rigidity k of the contact surfaces c ；
Step 2, analyzing the contact strength according to the operation load, selecting a corresponding friction model according to the stress distribution between the contact surfaces to describe the friction damping characteristic between the contact surfaces, obtaining a hysteresis loop under the micro-motion sliding and macro-motion sliding states, and establishing a friction damping function model f-s of the contact surfaces;
step 3, performing Fourier expansion on the relative displacement and the friction force by adopting a first-order harmonic balance method, and calculating to obtain an equivalent stiffness coefficient K between friction contact surfaces according to a linearization equivalent principle and an energy dissipation principle e And equivalent damping coefficient C e ；
Step 4, repeating the steps 1-3 to convert the contact rigidity and the damping of the friction damper with different materials, contact characteristics and pressure loads under various working conditions, carrying out dimensionless treatment, and constructing dimensionless equivalent rigidity coefficients of different friction damper structures
Figure GDA0003846991980000111
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000112
A database;
step 5, taking high-order harmonic factors into consideration, calculating to obtain an equivalent stiffness coefficient K between friction contact surfaces by adopting a multi-harmonic balance method according to a linearization equivalent principle and the friction damping function model in the step 2 e And equivalent damping coefficient C e ；
Step 6, converting the contact rigidity and the damping of the friction damper with different materials, contact characteristics and pressure loads under various working conditions, carrying out non-dimensionalization treatment, and constructing high-order harmonicsDimensionless equivalent stiffness coefficient of different friction damper structures under waves
Figure GDA0003846991980000121
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000122
A database;
and 7, establishing a spring damping unit in the numerical analysis model, calling corresponding equivalent stiffness coefficients and equivalent damping coefficients from a database to average to the spring damping unit in all directions according to different materials, contact characteristics and pressure loads of the model, and carrying out contact stiffness and damping analysis calculation.
Referring to fig. 2, when the friction contact surfaces are just in contact with each other and the stress distribution of the contact surfaces is concentrated in a point-like distribution during the rapid lifting load, a point-surface friction contact model is established, the schematic diagram of the model is shown in fig. 2 (a), and fig. 2 (b) is a time-domain curve of the tangential friction force. Its describing function depends on the magnitude of the relative displacement amplitude: specifying a critical motion displacement D when the relative motion displacement is greater than 1.5 times 0 When the friction force reaches the maximum value, the contact surface completely slides. When the relative motion displacement amplitude D is less than or equal to 1.5D 0 When the contact surface does not slide integrally, the initial loading process is curve OAF, and after reaching the steady state, curve CEA represents the loading process of the excitation force, and curve ABC represents the unloading process, as shown in fig. 2 (c). When the relative motion displacement amplitude D>1.5D 0 At this time, the contact surface is subjected to integral sliding, the initial loading process is curve OF, and after reaching a steady state, curve C "E" A "represents the loading process and A ' B ' C ' C" represents the unloading process, as shown in FIG. 2 (d).
From the theory of elastic contact, the contact half width R can be calculated from the formula (2-1) for the same point-to-surface contact pair of two materials:
Figure GDA0003846991980000123
point-surface frictional contact versus tangential contact stiffness under tangential force:
Figure GDA0003846991980000124
(2-2) formula (2-2): r is the spherical radius, N is the positive pressure, v is the poisson's ratio, E is the elastic modulus, G is the shear modulus, and for isotropic materials,
Figure GDA0003846991980000125
defining the critical relative displacement amplitude of the point-surface contact model as D 0 ＝μ 0 Q/k c Amplitude of displacement of dimensionless relative motion
Figure GDA0003846991980000126
Dimensionless friction force
Figure GDA0003846991980000127
Non-dimensional friction force amplitude in micro-motion slip stage
Figure GDA0003846991980000128
When the amplitude D of the relative displacement between the contact surfaces is less than or equal to 1.5D 0 During the process, the friction damper is in a micro-motion sliding state, and at the moment, dimensionless friction force and relative displacement between contact surfaces are expressed as follows:
curve ABC:
Figure GDA0003846991980000131
curve CEA:
Figure GDA0003846991980000132
when D > 1.5D 0 And when the damping structure is in the micro-motion slippage macro-slippage coexistence stage, the corresponding hysteresis curve expression with dimensionless form under the macro-slippage condition is as follows:
curve a ' B ' C ':
Figure GDA0003846991980000133
curve C' C ":
Figure GDA0003846991980000134
curve C "E" a ":
Figure GDA0003846991980000135
curve a "a':
Figure GDA0003846991980000136
wherein
Figure GDA0003846991980000137
According to the first-order harmonic balance method, when the system is excited by simple resonance, the response of the system is changed by simple resonance, the frequency is the same as that of the exciting force, and the relative motion displacement and the dimensionless form thereof are defined as follows:
s＝Dcosωt，
Figure GDA0003846991980000138
then, the friction force f is subjected to first-order fourier expansion to obtain:
f＝f k (D)cosωt+f c (D)sinωt (2-9)
f can be obtained by operation of Fourier series coefficient k (D) And f c (D)：
Figure GDA0003846991980000139
According to the linearization equivalence principle, the friction contact surface is simplified into a no-mass spring damping system, and the nonlinear friction force between the contact surfaces can be calculated by linearly superposing the elastic force and the damping force, namely:
Figure GDA00038469919800001310
wherein: k e Is the equivalent stiffness coefficient/N.m between the friction contact surfaces -1 ，C e Is the equivalent damping coefficient/Ns.m between the friction contact surfaces -1 ；。
The expression for the friction is found to be:
Figure GDA0003846991980000141
comparing the formula (2-9) with the formula (2-11), the equivalent stiffness coefficient and the equivalent damping coefficient are respectively:
Figure GDA0003846991980000142
the formula (2-12) is subjected to dimensionless processing to respectively obtain:
Figure GDA0003846991980000143
substituting the friction hysteresis curves (2-3) - (2-8) into the formula (2-13) to perform integral operation to obtain the dimensionless equivalent stiffness coefficient
Figure GDA0003846991980000144
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000145
As shown in fig. 2 (e) and 2 (f), respectively.
Referring to fig. 3, when the contact surface reaches the locked state under the design condition and the stress distribution of the contact surface does not change any more, a surface-to-surface friction contact model is built: when the damper is in a micro-motion slip state, a slip region distribution diagram and a tangential friction force change curve of a surface-surface friction contact model are respectively shown in fig. 3 (a) and 3 (b), and a hysteresis curve between contact surfaces in a single period is shown in fig. 3 (c).
From the elastic contact theory, the contact half width R is calculated from the formula (3-1):
Figure GDA0003846991980000146
wherein L is the length of the cylinder, Q 0 Is a positive pressure at the center of the contact. Defining the critical relative displacement amplitude of the surface-to-surface contact model as
Figure GDA0003846991980000147
Its describing function depends on the magnitude of the relative displacement amplitude: when the relative motion displacement amplitude D is less than or equal to D 0 In the meantime, the contact surface does not slide integrally, the initial loading process is curve OA, and after reaching the steady state, curve ABCDE represents the loading process of the exciting force, and curve EFGA represents the unloading process. D is less than or equal to D if the width of the contact area is 2a 0 The corresponding dimensionless hysteresis loop expression is shown in fig. 3 (c):
curve ABCDE:
Figure GDA0003846991980000151
curve EFGA:
Figure GDA0003846991980000152
the dimensionless equivalent damping coefficient and equivalent stiffness coefficient are calculated by a first-order harmonic balance method
Figure GDA0003846991980000153
Respectively as follows:
Figure GDA0003846991980000154
Figure GDA0003846991980000155
referring to fig. 4, when the damper is in a state where both micro-slip and macro-slip coexist, a slip region distribution diagram and a tangential friction force variation curve of the plane-plane model are respectively shown in fig. 4 (a) and 4 (b): when the relative motion displacement amplitude D>D 0 When the contact surface is in the whole sliding state, the initial loading process is the curve OH, and after the stable state is reached, the curve GABCD represents the loading process, and the curve DEFG represents the unloading process. Then D is>D 0 The corresponding dimensionless hysteresis loop expression is shown in fig. 4 (c):
curve DEF:
Figure GDA0003846991980000156
curve FG:
Figure GDA0003846991980000157
curve GABC:
Figure GDA0003846991980000158
the curve CD:
Figure GDA0003846991980000159
wherein:
Figure GDA00038469919800001510
the dimensionless equivalent damping coefficient and equivalent stiffness coefficient are calculated by a first-order harmonic balance method
Figure GDA00038469919800001511
Respectively as follows:
Figure GDA00038469919800001512
Figure GDA00038469919800001513
in combination with the above
Figure GDA00038469919800001514
And
Figure GDA00038469919800001515
the dimensionless equivalent stiffness coefficient and the equivalent damping coefficient curve of the obtained surface-surface contact model under two slippage conditions are shown in fig. 4 (d) and fig. 4 (e), and the expressions are respectively as follows:
Figure GDA0003846991980000161
Figure GDA0003846991980000162
referring to fig. 5, when the contact surface stress is linearly distributed, a line-surface frictional contact model is established and a friction hysteresis loop is obtained. In the line-surface contact model, the damping structure is simplified into a rectangular plate pressed on a rigid plane, and the elastic modulus of the rectangular plate is E; the cross-sectional area is A; the length is l; the coefficient of friction between the contact surfaces is μ, which is constant regardless of the viscous state or the slip state. As shown in fig. 5 (a), the specific structure is that a distributed positive pressure q acts on the surface of the rectangular plate perpendicularly, and the positive pressure acting on the surface of the rectangular plate is assumed to be distributed in a quadratic function along the length direction, and the magnitude of the positive pressure is determined by the following formula:
Figure GDA0003846991980000163
at this time, the positive pressure Q acting on the damper can be expressed as:
Figure GDA0003846991980000164
assuming that the tangential force acting on the rectangular plate is F a The expression is as follows:
F a ＝Fsinωt
wherein F a For the tangential force amplitude, ω is the angular velocity and t is the time.
The changes in friction and relative displacement in the line-surface frictional contact model are as follows: in the initial stage, as shown in fig. 5 (b), the rightmost end of the rectangular plate changes from the rest state to the stretching state in the process of increasing the tangential force from 0 to the maximum value, the length of the stretching section increases along with the increase of the tangential force, and the length of the stretching section reaches the maximum value delta when the tangential force reaches the maximum value a (ii) a As shown in fig. 5 (C), when the tangential force decreases from the maximum value, the region C slips in the x direction due to the decrease of the external force, and the direction of the frictional force is opposite to the x direction; the B area keeps a positive strain state at the previous moment, and the direction of the friction force is consistent with the x direction; the A area is in a viscous, non-slip and zero-strain state, and no friction exists. The rightmost end of the rectangular plate is changed into a compression state from a stretching state, the damper is still in the stretching state, the compression area is gradually increased along with the further reduction of the tangential force until the sliding area is changed into the compression state from the stretching state, at the moment, the tangential force is also reduced to the minimum value, and the length of the compression section reaches the maximum value delta d (ii) a As shown in FIG. 5 (d), when the force increases from the minimum value, the C region is stretched by the increase of the external force, the direction of the frictional force is consistent with the x direction, and the B region maintains the negative strain state at the previous moment; the A area is in a viscous, non-slip and zero-strain state, and no friction exists. The rightmost end of the rectangular plate is changed into a stretching state from a compression state, the stretching region gradually expands along with the increase of the tangential force until the tangential force reaches the maximum, the whole sliding region is in the stretching state, and the length of the sliding region is delta l . At this point, the tangential force completes one cycle.
Surrounded by energy dissipated in one cycle and hysteresis loop due to damping structureThe areas are equal, so that the energy consumed by the damping model in one vibration period can be obtained by calculating the area surrounded by the hysteresis loop. Let u be the rightmost displacement of the rectangular plate and delta be the length of the sliding region in the pulling-up state i Length of slip region in compression is δ d . According to the mechanical analysis process, as shown in fig. 5, the dissipation energy of the damper in one cycle is calculated as follows:
Figure GDA0003846991980000171
when F is present a <μ 0 And when Q is reached, the damper is in a micro-motion slippage stage, namely D is less than or equal to 1. In the micromotion slip single-rod model, the equivalent damping coefficient can be expressed as:
Figure GDA0003846991980000172
the equivalent stiffness coefficient can be expressed as:
Figure GDA0003846991980000173
at this time, the dimensionless equivalent damping coefficient is as follows:
Figure GDA0003846991980000174
the dimensionless equivalent stiffness coefficient is shown as follows:
Figure GDA0003846991980000175
when F is a >μ 0 When Q is reached, the damping structure is in a micro-motion slippage macro-slippage coexisting state, namely D>1. At this time, the energy consumed by the damping structure in a single vibration period consists of micro-slip and macro-slip, and is shown as the following formula:
W＝W 1 +W 2
the energy dissipated by the damping structure due to pure micro-motion slip is calculated by the following formula:
W 1 ＝W(δ a )
the energy consumed by the damping structure due to pure macroscopic slippage is calculated by the following formula:
W 2 ＝2(B-D)F a
at the moment, in the micro-motion slip macro-slip coexistence stage, the dimensionless equivalent damping coefficient of the line-surface contact model is expressed as follows:
Figure GDA0003846991980000181
in the stage of micro-motion slippage and macro-slippage coexistence, the equivalent stiffness coefficient of the line-surface contact model is expressed as follows:
Figure GDA0003846991980000182
wherein
Figure GDA0003846991980000183
In combination with the above
Figure GDA0003846991980000184
And
Figure GDA0003846991980000185
obtaining dimensionless equivalent stiffness coefficients and equivalent damping coefficients of the line-surface contact model under two slippage conditions as follows:
Figure GDA0003846991980000186
Figure GDA0003846991980000187
when slip and viscous states coexist in a motion cycle and the weight of the viscous state is large, the influence of higher harmonic components on the vibration response of the system needs to be considered, as shown in fig. 6. The blade friction damper is abstracted into a mechanical nonlinear vibration system which changes in a sine function manner under the action of external force, and the general expression of a nonlinear dynamic equation on the time domain is obtained as follows:
Figure GDA0003846991980000191
expanding the independent variable s (t) to N by Fourier series h The order is as follows:
Figure GDA0003846991980000192
the first and second derivatives of the displacement s (t) can be expressed as follows:
Figure GDA0003846991980000193
Figure GDA0003846991980000194
then, the friction force f(s) is fourier expanded to obtain:
Figure GDA0003846991980000195
the coefficients of the right-hand side terms of equation (5-5) can be obtained by fourier transform of f(s):
Figure GDA0003846991980000196
Figure GDA0003846991980000197
Figure GDA0003846991980000198
exciting F by using the displacement s (t) and the friction force F(s) a Written as a matrix product is of the form:
s(t)＝Φ T s，f(s)＝Φ T f，F a ＝Φ T F (6-6)
wherein the content of the first and second substances,
Φ＝[1 cosωt sinωt…cos(N h ωt) sin(N h ωt)] T
Figure GDA0003846991980000199
Figure GDA00038469919800001910
F＝[0 0 F…0 0] T
the first and second derivatives of the displacement s (t) can be expressed as follows:
Figure GDA0003846991980000201
wherein:
Figure GDA0003846991980000202
then, formula (5-6) is substituted into formula (5-1), and written in matrix form as follows:
MΦ T E 2 s+CΦ T Εs+KΦ T s+Φ T f＝Φ T F (6-7)
since the equation (5-7) holds for any time t, harmonic coefficients of each order are completely equal, and the following nonlinear algebraic equations can be obtained:
(ME 2 +CΕ+KI)s+f＝F
defining the equation residual terms as:
R＝(ME 2 +CΕ+KI)s+f-F
applying Newton-Raphson to solve the formula iteratively:
Figure GDA0003846991980000203
in the formula, k represents the current iteration step number.
Like equation (2-10), the frictional contact surface is simplified into a no-mass spring damping system according to the equivalent principle of linearization, and the nonlinear frictional force between the contact surfaces can be calculated by linearly adding the elastic force and the damping force, namely:
Figure GDA0003846991980000204
substituting formula (5-8) with formula (5-2), formula (5-3) and formula (5-5) to obtain
Figure GDA0003846991980000211
Calculating to obtain equivalent stiffness coefficient k of each order by comparing corresponding term coefficients to be equal en And equivalent damping coefficient c en :
Figure GDA0003846991980000212
Writing equations (5-8) in matrix form:
Φ T f＝K e Φ T s+C e Φ T Es (6-9)
since the equations (5-9) hold for any time t, the harmonic coefficients of the respective orders are completely equal, and the following nonlinear algebraic equations can be obtained:
f＝(K e +C e E)s (6-10)
in equations (5-10), the equivalent stiffness coefficient matrix can be expressed as:
Figure GDA0003846991980000213
the equivalent damping coefficient matrix is:
Figure GDA0003846991980000214
likewise, dimensionless equivalent stiffness coefficients of each order can be obtained
Figure GDA0003846991980000215
And dimensionless equivalent damping coefficient
Figure GDA0003846991980000216
The expression of (c) is:
Figure GDA0003846991980000217
the coefficients of the terms in the formula (5-11) are:
Figure GDA0003846991980000221
Figure GDA0003846991980000222
Figure GDA0003846991980000223
the dimensionless equivalent stiffness coefficient matrix can be expressed as:
Figure GDA0003846991980000224
the dimensionless equivalent damping coefficient matrix can be expressed as:
Figure GDA0003846991980000225

Claims (7)

1. the method for analyzing and solving the contact rigidity and the damping of the gas compressor and the turbine blade of the gas turbine is characterized by comprising the following steps of:
step 1, obtaining relevant parameters according to a friction damper material: coefficient of friction between contact surfaces mu 0 The Poisson ratio v, the elastic modulus E and the shear modulus G, and obtaining the positive pressure load Q between the friction contact surfaces of the blades under different rotating speed working conditions through analysis to obtain the equivalent contact radius R and the tangential contact rigidity k of the contact surfaces c ；
Step 2, analyzing the contact strength according to the operation load, selecting a corresponding friction model according to the stress distribution between the contact surfaces to describe the friction damping characteristic between the contact surfaces, obtaining a hysteresis loop under the micro-motion sliding and macro-motion sliding states, and establishing a friction damping function model f-s of the contact surfaces;
step 3, performing Fourier expansion on the relative displacement and the friction force by adopting a first-order harmonic balancing method, and calculating to obtain an equivalent stiffness coefficient K between the friction contact surfaces according to a linearization equivalent principle and an energy dissipation principle e And equivalent damping coefficient C e ；
Step 4, repeating the steps 1 to 3 to convert the contact rigidity and the damping of the friction damper with different materials, contact characteristics and pressure loads under various working conditions, carrying out dimensionless treatment, and constructing dimensionless equivalent rigidity coefficients of different friction damper structures
Figure FDA0003846991970000011
And dimensionless equivalent damping coefficient
Figure FDA0003846991970000012
A database;
step 5, considering the high-order harmonic factor, adopting multi-harmonicCalculating to obtain an equivalent stiffness coefficient K between the friction contact surfaces according to a linearization equivalent principle and the friction damping function model in the step 2 by a wave balance method e And equivalent damping coefficient C e ；
Step 6, converting the contact rigidity and damping of the friction damper of different materials, contact characteristics and pressure loads under various working conditions, carrying out dimensionless treatment, and constructing dimensionless equivalent rigidity coefficients of different friction damper structures under high-order harmonics
Figure FDA0003846991970000013
And dimensionless equivalent damping coefficient
Figure FDA0003846991970000014
A database;
and 7, establishing a spring damping unit in the numerical analysis model, calling corresponding equivalent stiffness coefficients and equivalent damping coefficients from a database to average to the spring damping unit in all directions according to different materials, contact characteristics and pressure loads of the model, and carrying out contact stiffness and damping analysis calculation.
2. The method for analyzing and solving the contact stiffness and damping of the compressor and the turbine blade of the gas turbine as claimed in claim 1, wherein the frictional damping function model in the step 2 comprises: a point-surface frictional contact model, a line-surface frictional contact model and a surface-surface frictional contact model.
3. The method for analyzing and solving the contact rigidity and the damping of the gas turbine compressor and the turbine blade according to claim 2, wherein in the step 2, the friction contact surfaces just contact in the rapid lifting load, and when the stress distribution of the contact surfaces is in concentrated point-like distribution, a point-surface friction contact model is established, and the description function of the point-surface friction contact model depends on the amplitude of the relative displacement: when the relative displacement amplitude D between the contact surfaces is less than or equal to 1.5D 0 And when the friction damper is in a micro-motion sliding state, the friction hysteresis loop between the contact surfaces is as follows:
curve ABC:
Figure FDA0003846991970000021
curve CEA:
Figure FDA0003846991970000022
when D > 1.5D 0 During the process, the damping structure is in a micro-motion slippage macro-slippage coexistence stage, and at the moment, the friction hysteresis loop between the contact surfaces is as follows:
curve a ' B ' C ':
Figure FDA0003846991970000023
curve C' C ":
Figure FDA0003846991970000024
curve C "E" a ":
Figure FDA0003846991970000025
curve a "a':
Figure FDA0003846991970000026
wherein
Figure FDA0003846991970000027
When the contact surface reaches a locking state under a design working condition and the stress distribution of the contact surface does not change any more, a surface-surface friction contact model is built, and a description function of the surface-surface friction contact model depends on the magnitude of the relative displacement amplitude: d is less than or equal to D 0 The dimensionless friction hysteresis loop is:
curve ABCDE:
Figure FDA0003846991970000028
curve EFGA:
Figure FDA0003846991970000029
D>D 0 the dimensionless friction hysteresis loop is:
curve DEF:
Figure FDA00038469919700000210
curve FG:
Figure FDA0003846991970000031
curve GABC:
Figure FDA0003846991970000032
the curve CD:
Figure FDA0003846991970000033
when the contact surface stress is in linear distribution, a line-surface friction contact model is established, and an expression of the friction force f and the displacement D of the rightmost end of the damper changing along with the sliding length delta is obtained according to the mechanical analysis process.
4. The method for analyzing and solving the contact stiffness and damping of the compressor and the turbine blade of the gas turbine as claimed in claim 3, wherein the step 3 comprises:
for point-surface frictional contact model and surface-surface frictional contact model:
step 3.1, performing Fourier expansion on the relative displacement s and the friction force f by adopting a first-order harmonic balance method respectively:
s＝Dcosωt (1)
f＝f k (D)cosωt+f c (D)sinωt (2)
step 3.2, according to the operation of Fourier series coefficient, f is obtained k (D) And f c (D)：
Figure FDA0003846991970000034
Step 3.3, obtaining by a linearization equivalent principle:
Figure FDA0003846991970000035
and 3.4, substituting the formula (1) into the formula (4) to obtain an expression of the friction force, wherein the expression is as follows:
Figure FDA0003846991970000036
step 3.5, comparing the formula (5) with the formula (2), and obtaining an equivalent stiffness coefficient and an equivalent damping coefficient which are respectively as follows:
Figure FDA0003846991970000037
and 3.6, substituting the formula (3) into the formula (6) to obtain an expression of the equivalent stiffness coefficient and the equivalent damping coefficient as follows:
Figure FDA0003846991970000038
for the line-to-plane frictional contact model:
step 3.1, obtaining a calculation formula of the dissipation energy of the damper in a period according to an energy dissipation principle:
Figure FDA0003846991970000041
step 3.2, analyzing the mechanical process to obtain an equivalent stiffness coefficient and an equivalent damping coefficient which are respectively as follows:
Figure FDA0003846991970000042
5. the method for analyzing and solving the contact stiffness and damping of the gas turbine compressor and turbine blade according to claim 4, wherein a linearization equivalence principle is adopted in the step 3.3, the friction contact surface is simplified into a no-mass spring damping system, and the nonlinear friction force between the contact surfaces is equivalent to the linear superposition of the elastic force and the damping force.
6. The method for analyzing and solving the contact stiffness and damping of the compressor and the turbine blade of the gas turbine as claimed in claim 4, wherein in step 3.1, the energy consumed by the damping model in one vibration cycle is obtained by calculating the area surrounded by the hysteresis loop by using an energy dissipation principle.
7. The method for analyzing and solving the contact rigidity and the damping of the gas turbine compressor and the turbine blade according to claim 4, wherein in the step 4, the rigidity and the damping characteristic of the contact surface are separated from the geometric parameters of the damper, and the dimensionless forms of the equivalent coefficients of different friction damping function models are obtained; wherein, the first and the second end of the pipe are connected with each other,
dimensionless equivalent stiffness coefficient of point-surface friction contact model
Figure FDA0003846991970000043
And dimensionless equivalent damping coefficient
Figure FDA0003846991970000044
Figure FDA0003846991970000045
Dimensionless equivalent stiffness coefficient of surface-surface friction contact model
Figure FDA0003846991970000046
And dimensionless equivalent damping coefficient
Figure FDA0003846991970000047
Figure FDA0003846991970000048
Figure FDA0003846991970000049
Dimensionless equivalent stiffness coefficient of line-surface friction contact model
Figure FDA0003846991970000051
And dimensionless equivalent damping coefficient
Figure FDA0003846991970000052
Figure FDA0003846991970000053
Figure FDA0003846991970000054
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