CN113177273A - 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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CN113177273A
CN113177273A CN202110416350.6A CN202110416350A CN113177273A CN 113177273 A CN113177273 A CN 113177273A CN 202110416350 A CN202110416350 A CN 202110416350A CN 113177273 A CN113177273 A CN 113177273A
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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 discloses a method for analyzing and solving the contact rigidity and damping of a gas compressor and a turbine blade of a gas turbine, the method is based on the analysis of contact strength under different operating loads, and establishes a point-surface contact, line-surface contact or surface-surface contact friction damping model respectively, and combines the linearization equivalent principle and the energy dissipation principle to obtain contact rigidity and damping agent models of different friction damper structures, the limitation of the traditional method is broken through, the influence of high-order harmonic components is considered, equivalent components of contact rigidity and damping to different harmonic terms are calculated, a multi-working-condition dimensionless equivalent coefficient database is constructed, and calling a database in the numerical model to obtain corresponding equivalent coefficients, analyzing and calculating the contact stiffness and the damping, and providing a proxy model database and a quick equivalent solving method for the industrial vibration reduction analysis of the rotary mechanical blade friction damper.

Description

Gas turbine compressor and turbine blade contact rigidity and damping analysis 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, damping surrounding bands, boss drawing metal, loose drawing metal, damping blocks and the like are common forms, vibration energy is dissipated through friction, the vibration level of parts can be effectively restrained, 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 a method for analyzing and solving the contact rigidity and damping of a gas compressor and a turbine blade of a gas turbine, based on the analysis of the contact strength under different operating loads, a frictional damping model of point-surface contact, line-surface contact or surface-surface contact is respectively established, and combines the linearization equivalent principle and the energy dissipation principle to obtain contact rigidity and damping agent models of different friction damper structures, the method breaks through the limitation of the traditional method, considers the influence of high-order harmonic components, calculates equivalent components of contact rigidity and damping to different harmonic terms, enhances the robustness of the equivalent method to different working conditions, and provides a proxy model database and a rapid equivalent solving method for the industrial vibration reduction analysis of the rotating machinery blade-friction damper (such as a gas turbine, a gas compressor and a turbine blade friction damper, 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:
a method for analyzing and solving contact rigidity and damping of a gas compressor and a turbine blade of a gas turbine comprises the following steps:
step 1, obtaining relevant parameters according to friction damper materials: coefficient of friction between contact surfaces mu0The 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 surfacesc；
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 characteristics between the contact surfaces, obtaining hysteresis loops under the micro-sliding and macro-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 principleeAnd equivalent damping coefficient Ce；
Step 4, repeating the steps 1-3 for multiple working conditionsThe contact rigidity and the damping of the friction damper with different materials, contact characteristics and pressure loads are converted, dimensionless treatment is carried out, and dimensionless equivalent rigidity coefficients of different friction damper structures are constructed
Figure BDA0003026022840000031
And dimensionless equivalent damping coefficient
Figure BDA0003026022840000032
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 2eAnd equivalent damping coefficient Ce；
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 BDA0003026022840000033
And dimensionless equivalent damping coefficient
Figure BDA0003026022840000034
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: a point-surface frictional contact model, a line-surface frictional contact model, and a surface-surface frictional contact model.
The invention is further improved in that in step 2, the friction contact surfaces are just contacted in the rapid lifting load, and when the stress distribution of the contact surfaces is concentrated and distributed in a point shape, the establishment is carried outA point-surface frictional contact model, the describing function of which depends on the magnitude of the relative displacement amplitude: when the relative displacement amplitude D between the contact surfaces is less than or equal to 1.5D0And 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 BDA0003026022840000035
curve CEA:
Figure BDA0003026022840000036
when D > 1.5D0During 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 BDA0003026022840000037
curve C' C ":
Figure BDA0003026022840000038
curve C "E" a ":
Figure BDA0003026022840000041
curve a "a': f is 1 (pi + theta)0＜ωt≤2π)
Wherein
Figure BDA0003026022840000042
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, establishing a surface-surface friction contact model, wherein a description function of the model depends on the magnitude of the relative displacement amplitude: d is less than or equal to D0The dimensionless friction hysteresis loop is:
curve ABCDE:
Figure BDA0003026022840000043
curve EFGA:
Figure BDA0003026022840000044
D＞D0the dimensionless friction hysteresis loop is:
curve DEF:
Figure BDA0003026022840000045
curve FG:
Figure BDA0003026022840000046
curve GABC:
Figure BDA0003026022840000047
curve CD:
Figure BDA0003026022840000048
when the contact surface stress is linearly distributed, a line-surface friction contact model is established, and an expression of the change of the friction force f and the displacement D of the rightmost end of the damper 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-to-surface frictional contact model and surface-to-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＝D cosωt (1)
f＝fk(D)cosωt+fc(D)sinωt (2)
step 3.2, according to the operation of Fourier series coefficient, f is obtainedk(D) And fc(D)：
Figure BDA0003026022840000049
Step 3.3, obtaining by linearization equivalence principle：
Figure BDA0003026022840000051
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 BDA0003026022840000052
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 BDA0003026022840000053
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, wherein the expression is as follows:
Figure BDA0003026022840000054
for the line-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 BDA0003026022840000055
step 3.2, analyzing the mechanical process to obtain an equivalent stiffness coefficient and an equivalent damping coefficient which are respectively as follows:
Figure BDA0003026022840000056
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.
A 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,
dimensionless equivalent stiffness coefficient of point-surface friction contact model
Figure BDA0003026022840000057
And dimensionless equivalent damping coefficient
Figure BDA0003026022840000058
Figure BDA0003026022840000059
Dimensionless equivalent stiffness coefficient of surface-surface frictional contact model
Figure BDA00030260228400000510
And dimensionless equivalent damping coefficient
Figure BDA00030260228400000511
Figure BDA0003026022840000061
Figure BDA0003026022840000062
Dimensionless equivalent stiffness coefficient of line-surface friction contact model
Figure BDA0003026022840000063
And dimensionless equivalent damping coefficient
Figure BDA0003026022840000064
Figure BDA0003026022840000065
Figure BDA0003026022840000066
A further development of the invention is that step 5 comprises:
step 5.1, give harmonic number NhRespectively carrying out N on relative displacement s and friction force f(s) by adopting a multi-harmonic balance methodhExpansion of Fourier series:
Figure BDA0003026022840000067
Figure BDA0003026022840000068
step 5.2, f(s) is obtained according to the friction damping function model, and each right-end coefficient of the formula (9) is obtained according to Fourier transform:
Figure BDA0003026022840000069
Figure BDA00030260228400000610
Figure BDA00030260228400000611
and 5.3, substituting the formulas (8) and (9) into the formula (4) by a linearization equivalent principle, comparing corresponding terms with the same coefficient, and calculating to obtain each order of equivalent stiffness coefficient kenAnd equivalent damping coefficient cen：
Figure BDA0003026022840000071
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 BDA0003026022840000072
in formula (11), k represents the current iteration step number; r ═ ME (ME)2+ CE + KI) s + F-F is an equation residual term, which converges when R is 0;
Figure BDA0003026022840000073
and 5.5, writing the formula (4) into a matrix form to obtain:
f＝(Ke+CeE)s (12)
the equivalent stiffness coefficient matrix obtained from equation (10) is:
Figure BDA0003026022840000074
the equivalent damping coefficient matrix is:
Figure BDA0003026022840000075
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 BDA0003026022840000076
And dimensionless equivalent damping coefficient
Figure BDA0003026022840000077
The expression of (a) is:
Figure BDA0003026022840000081
the coefficients in equation (13) are:
Figure BDA0003026022840000082
Figure BDA0003026022840000083
Figure BDA0003026022840000084
the dimensionless equivalent stiffness coefficient matrix is represented as:
Figure BDA0003026022840000085
the dimensionless equivalent damping coefficient matrix is expressed as:
Figure BDA0003026022840000086
the invention has at least the following beneficial technical effects:
the invention provides a method for analyzing and solving the contact rigidity and damping of a gas compressor and a turbine blade of a gas turbine, which replaces engineering empirical design with parametric theoretical analysis and solution to obtain equivalent rigidity and equivalent damping of a 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.
Further, a linear equivalent 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 linear superposition of elastic force and damping force through stress balance analysis, so that the complicated nonlinear mechanical problem is converted into linear solution, the equivalent stiffness and 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.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings 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 the stiffness and damping of contact between a compressor and a turbine blade of a gas turbine according to an embodiment of the present 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) a tangential friction force time domain curve is obtained; (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 the 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 a following-surface frictional contact model of 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) a tangential friction time domain curve is obtained; (c) the friction force changes with the displacement in the micro-motion sliding stage 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) a tangential friction time domain curve is obtained; (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 in 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 that the tangential force is increased from the minimum value; (e) the hysteresis loop of the line-surface friction contact model is shown.
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 only some of the embodiments 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 friction damper materials: coefficient of friction between contact surfaces mu0The Poisson 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 surfacesc；
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 characteristics between the contact surfaces, obtaining hysteresis loops under the micro-sliding and macro-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 principleeAnd equivalent damping coefficient Ce；
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 BDA0003026022840000111
And nothingEquivalent damping coefficient of line
Figure BDA0003026022840000112
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 2eAnd equivalent damping coefficient Ce；
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 BDA0003026022840000113
And dimensionless equivalent damping coefficient
Figure BDA0003026022840000114
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.
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 times0When 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 1.5D0In the meantime, 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>1.5D0At 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 BDA0003026022840000121
point-surface frictional contact versus tangential contact stiffness under tangential force:
Figure BDA0003026022840000122
(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 BDA0003026022840000123
defining the critical relative displacement amplitude of the point-surface contact model as D0＝μ0Q/kcAmplitude of dimensionless relative motion displacement
Figure BDA0003026022840000124
Dimensionless friction force
Figure BDA0003026022840000125
Non-dimensional friction force amplitude in micro-motion slip stage
Figure BDA0003026022840000126
When the relative displacement amplitude D between the contact surfaces is less than or equal to 1.5D0During 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 BDA0003026022840000127
curve CEA:
Figure BDA0003026022840000128
when D > 1.5D0And then, the damping structure is in a micro-motion slippage macro-slippage coexistence stage, and the corresponding hysteresis curve expression in a dimensionless form under the condition of macro-slippage is as follows:
curve a ' B ' C ':
Figure BDA0003026022840000131
curve C' C ":
Figure BDA0003026022840000132
curve C "E" a ":
Figure BDA0003026022840000133
curve a "a':
Figure BDA0003026022840000134
wherein
Figure BDA0003026022840000135
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:
Figure BDA0003026022840000136
then, the friction force f is subjected to first-order fourier expansion to obtain:
f＝fk(D)cosωt+fc(D)sinωt (2-9)
f can be obtained by operation of Fourier series coefficientk(D) And fc(D)：
Figure BDA0003026022840000137
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 BDA0003026022840000138
wherein: keIs the equivalent stiffness coefficient/N.m between the friction contact surfaces-1，CeIs the equivalent damping coefficient/Ns.m between the friction contact surfaces-1；。
The expression for the friction is found to be:
Figure BDA0003026022840000139
comparing the formula (2-9) with the formula (2-11), the equivalent stiffness coefficient and the equivalent damping coefficient are respectively:
Figure BDA00030260228400001310
the formula (2-12) is subjected to dimensionless processing to respectively obtain:
Figure BDA0003026022840000141
the friction hysteresis curves (2-3) - (2-8) are substituted into the formula (2-13) to carry out integral operation, and the dimensionless equivalent stiffness coefficient can be obtained
Figure BDA0003026022840000142
And dimensionless equivalent damping coefficient
Figure BDA0003026022840000143
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 the micro-motion only slip state, the slip region distribution diagram and the tangential friction force variation curve of the surface-surface friction contact model are respectively shown in fig. 3(a) and 3(b), and the hysteresis curve between the 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 BDA0003026022840000144
wherein L is the length of the cylinder, Q0Is a positive pressure at the center of the contact. Defining the critical relative displacement amplitude of the surface-to-surface contact model as
Figure BDA0003026022840000145
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 D0In 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 when the width of the contact area is 2a0The corresponding dimensionless hysteresis loop expression is shown in fig. 3 (c):
curve ABCDE:
Figure BDA0003026022840000146
curve EFGA:
Figure BDA0003026022840000147
by first-order harmonic balanceCalculating to obtain dimensionless equivalent damping coefficient and equivalent stiffness coefficient
Figure BDA0003026022840000148
Respectively as follows:
Figure BDA0003026022840000149
Figure BDA00030260228400001410
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>D0When 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>D0The corresponding dimensionless hysteresis loop expression is shown in fig. 4 (c):
curve DEF:
Figure BDA0003026022840000151
curve FG:
Figure BDA0003026022840000152
curve GABC:
Figure BDA0003026022840000153
curve CD:
Figure BDA0003026022840000154
wherein:
Figure BDA0003026022840000155
from one toThe equivalent damping coefficient and the equivalent stiffness coefficient without dimension are obtained by the calculation of the order harmonic balance method
Figure BDA0003026022840000156
Respectively as follows:
Figure BDA0003026022840000157
Figure BDA0003026022840000158
in combination with the above
Figure BDA0003026022840000159
And
Figure BDA00030260228400001510
the dimensionless equivalent stiffness coefficient and equivalent damping coefficient curves of the surface-surface contact model obtained in the two slip situations are shown in fig. 4(d) and fig. 4(e), and the expressions are respectively as follows:
Figure BDA00030260228400001511
Figure BDA00030260228400001512
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 BDA0003026022840000161
at this time, the positive pressure Q acting on the damper can be expressed as:
Figure BDA0003026022840000162
assuming that the tangential force acting on the rectangular plate is FaThe expression is as follows:
Fa=F sinωt
wherein FaFor 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 valuea(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 deltad(ii) a As shown in FIG. 5(d), when the force increases from the minimum value, zone C is stretched by the increase of the external force, the direction of the frictional force is consistent with the x-direction, and zone B maintains the negative strain state at the previous moment; the region A is in a viscous, non-slip, zero-strain state and has no frictionA force exists. The rightmost end of the rectangular plate is changed into a stretching state from a compression state, the stretching area is gradually expanded along with the increase of the tangential force until the tangential force reaches the maximum, the whole sliding area is in the stretching state, and the length of the sliding area is deltal. At this point, the tangential force completes one cycle.
Because the energy consumed by the damping structure in one cycle period is equal to the area surrounded by the hysteresis loop, 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 stateiThe length of the slip region in compression being deltad. 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 BDA0003026022840000171
when F is presenta<μ0And 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 BDA0003026022840000172
the equivalent stiffness coefficient can be expressed as:
Figure BDA0003026022840000173
at this time, the dimensionless equivalent damping coefficient is as follows:
Figure BDA0003026022840000174
the dimensionless equivalent stiffness coefficient is shown as follows:
Figure BDA0003026022840000175
when F is presenta>μ0When 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-sliding and macro-sliding, and is represented by the following formula:
W=W1+W2
the energy dissipated by the damping structure due to the pure micro-motion slip is calculated by the following formula:
W1=W(δa)
the energy consumed by the damping structure due to pure macroscopic slippage is calculated by the following formula:
W2=2(B-D)Fa
at this time, in the micro-motion slip macro-slip coexistence stage, the dimensionless equivalent damping coefficient of the line-surface contact model is expressed as:
Figure BDA0003026022840000181
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 BDA0003026022840000182
wherein
Figure BDA0003026022840000183
In combination with the above
Figure BDA0003026022840000184
And
Figure BDA0003026022840000185
obtaining dimensionless equivalent stiffness coefficient and equivalent damping coefficient of the line-surface contact model under two slippage conditionsThe following are respectively:
Figure BDA0003026022840000186
Figure BDA0003026022840000187
as shown in fig. 6, when slip and viscous states coexist in a motion cycle and the weight of the viscous state is large, the influence of the higher harmonic component on the vibration response of the system needs to be considered. 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 BDA0003026022840000188
expanding the independent variable s (t) to N by Fourier serieshThe order is as follows:
Figure BDA0003026022840000189
the first and second derivatives of the displacement s (t) can be expressed as follows:
Figure BDA00030260228400001810
Figure BDA0003026022840000191
then, the friction force f(s) is subjected to Fourier expansion to obtain:
Figure BDA0003026022840000192
the coefficients on the right hand side of equation (5-5) can be obtained by fourier transform of f(s):
Figure BDA0003026022840000193
Figure BDA0003026022840000194
Figure BDA0003026022840000195
exciting F by displacement s (t), friction force F(s)aWritten as a matrix product is of the form:
s(t)＝ΦTs，f(s)＝ΦTf，Fa＝ΦTF (6-6)
wherein,
Φ＝[1 cosωt sinωt…cos(Nhωt)sin(Nhωt)]T
Figure BDA0003026022840000196
Figure BDA0003026022840000197
F＝[0 0 F…0 0]T
the first and second derivatives of the displacement s (t) can be expressed as follows:
Figure BDA0003026022840000198
wherein:
Figure BDA0003026022840000199
then, formula (5-6) is substituted into formula (5-1), and written in matrix form as follows:
MΦTE2s+CΦTEs+KΦTs+ΦTf＝ΦTF (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:
(ME2+CE+KI)s+f＝F
defining equation residual terms as:
R＝(ME2+CE+KI)s+f-F
applying Newton-Raphson to solve the formula iteratively:
Figure BDA0003026022840000201
where 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 linear principle, and the nonlinear frictional force between the contact surfaces can be calculated by linearly adding the elastic force and the damping force, namely:
Figure BDA0003026022840000202
substituting formula (5-8) with formula (5-2), formula (5-3) and formula (5-5) to obtain
Figure BDA0003026022840000203
Calculating to obtain equivalent stiffness coefficient k of each order by comparing corresponding term coefficients to be equalenAnd equivalent damping coefficient cen：
Figure BDA0003026022840000204
Writing equations (5-8) in matrix form:
ΦTf＝KeΦTs+CeΦTEs (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＝(Ke+CeE)s (6-10)
in the equation (5-10), the equivalent stiffness coefficient matrix can be expressed as:
Figure BDA0003026022840000211
the equivalent damping coefficient matrix is:
Figure BDA0003026022840000212
likewise, dimensionless equivalent stiffness coefficients of each order can be obtained
Figure BDA0003026022840000213
And dimensionless equivalent damping coefficient
Figure BDA0003026022840000214
The expression of (a) is:
Figure BDA0003026022840000215
the coefficients in equations (5-11) are:
Figure BDA0003026022840000216
Figure BDA0003026022840000217
Figure BDA0003026022840000218
the dimensionless equivalent stiffness coefficient matrix can be expressed as:
Figure BDA0003026022840000219
the dimensionless equivalent damping coefficient matrix can be expressed as:
Figure BDA0003026022840000221

Claims (9)

1. a method for analyzing and solving contact rigidity and damping of a gas compressor and a turbine blade of a gas turbine is characterized by comprising the following steps:
step 1, obtaining relevant parameters according to friction damper materials: coefficient of friction between contact surfaces mu0The 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 surfacesc；
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 characteristics between the contact surfaces, obtaining hysteresis loops under the micro-sliding and macro-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 principleeAnd equivalent damping coefficient Ce；
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 FDA0003026022830000011
And dimensionless equivalent damping coefficient
Figure FDA0003026022830000012
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 2eAnd equivalent damping coefficient Ce；
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 FDA0003026022830000013
And dimensionless equivalent damping coefficient
Figure FDA0003026022830000014
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.
2. The method for analyzing and solving the contact stiffness and the 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: point-surface frictional contact model, line-surface frictional contact model and surface-surface frictional contact model.
3. 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 2, wherein the steps are as follows2, when the friction contact surfaces just contact in the rapid lifting load and the stress of the contact surfaces is distributed into concentrated point-like distribution, establishing a point-surface friction contact model, wherein a description function of the point-surface friction contact model depends on the magnitude of the amplitude of the relative displacement: when the relative displacement amplitude D between the contact surfaces is less than or equal to 1.5D0And 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 FDA0003026022830000021
curve CEA:
Figure FDA0003026022830000022
when D > 1.5D0During 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 FDA0003026022830000023
curve C' C ":
Figure FDA0003026022830000024
curve C "E" a ":
Figure FDA0003026022830000025
curve a "a':
Figure FDA0003026022830000026
wherein
Figure FDA0003026022830000027
When the contact surface reaches the locking state under the design working condition and the stress distribution of the contact surface does not change any more, a surface-to-surface friction contact model is built,its describing function depends on the magnitude of the relative displacement amplitude: d is less than or equal to D0The dimensionless friction hysteresis loop is:
curve ABCDE:
Figure FDA0003026022830000028
curve EFGA:
Figure FDA0003026022830000029
D>D0the dimensionless friction hysteresis loop is:
curve DEF:
Figure FDA0003026022830000031
curve FG:
Figure FDA0003026022830000032
curve GABC:
Figure FDA0003026022830000033
curve CD:
Figure FDA0003026022830000034
when the contact surface stress is linearly distributed, a line-surface friction contact model is established, and an expression of the change of the friction force f and the displacement D of the rightmost end of the damper 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 the damping of the compressor and the turbine blade of the gas turbine as claimed in claim 3, wherein the step 3 comprises the following steps:
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＝D cosωt (1)
f＝fk(D)cosωt+fc(D)sinωt (2)
step 3.2, according to the operation of Fourier series coefficient, f is obtainedk(D) And fc(D)：
Figure FDA0003026022830000035
And 3.3, obtaining the following components according to a linearization equivalent principle:
Figure FDA0003026022830000036
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 FDA0003026022830000037
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 FDA0003026022830000038
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, wherein the expression is as follows:
Figure FDA0003026022830000039
for a 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 FDA0003026022830000041
step 3.2, analyzing the mechanical process to obtain an equivalent stiffness coefficient and an equivalent damping coefficient which are respectively as follows:
Figure FDA0003026022830000042
5. the method for analyzing and solving the contact stiffness and the damping of the gas turbine compressor and turbine blade according to claim 4, wherein a linear equivalent principle is adopted in the step 3.3, the friction contact surface is simplified into a damping system without a mass spring, 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 the energy consumed by the damping model in one vibration cycle is obtained by calculating the area surrounded by the hysteresis loop by using the energy dissipation principle in step 3.1.
7. The method for analyzing and solving the contact rigidity and the damping of the gas turbine compressor and 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,
dimensionless equivalent stiffness coefficient of point-surface friction contact model
Figure FDA0003026022830000043
And dimensionless equivalent damping coefficient
Figure FDA0003026022830000044
Figure FDA0003026022830000045
Dimensionless equivalent stiffness coefficient of surface-surface friction contact model
Figure FDA0003026022830000046
And dimensionless equivalent damping coefficient
Figure FDA0003026022830000047
Figure FDA0003026022830000048
Figure FDA0003026022830000049
Dimensionless equivalent stiffness coefficient of line-surface friction contact model
Figure FDA0003026022830000051
And dimensionless equivalent damping coefficient
Figure FDA0003026022830000052
Figure FDA0003026022830000053
Figure FDA0003026022830000054
8. The method for analyzing and solving the contact stiffness and the damping of the compressor and the turbine blade of the gas turbine as claimed in claim 7, wherein the step 5 comprises the following steps:
step 5.1, give harmonic number NhRespectively carrying out N on relative displacement s and friction force f(s) by adopting a multi-harmonic balance methodhExpansion of Fourier series:
Figure FDA0003026022830000055
Figure FDA0003026022830000056
step 5.2, f(s) is obtained according to the friction damping function model, and each right-end coefficient of the formula (9) is obtained according to Fourier transform:
Figure FDA0003026022830000057
Figure FDA0003026022830000058
Figure FDA0003026022830000059
and 5.3, substituting the formulas (8) and (9) into the formula (4) by a linearization equivalent principle, comparing corresponding terms with the same coefficient, and calculating to obtain each order of equivalent stiffness coefficient kenAnd equivalent damping coefficient cen：
Figure FDA00030260228300000510
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 FDA0003026022830000061
in formula (11), k represents the current iteration step number; r ═ ME (ME)2+ CE + KI) s + F-F is an equation residual term, which converges when R is 0;
Figure FDA0003026022830000062
and 5.5, writing the formula (4) into a matrix form to obtain:
f＝(Ke+CeE)s (12)
the equivalent stiffness coefficient matrix obtained from equation (10) is:
Figure FDA0003026022830000063
the equivalent damping coefficient matrix is:
Figure FDA0003026022830000064
9. the method for analyzing and solving the contact stiffness and damping of the compressor and the turbine blade of the gas turbine according to claim 8, wherein the equivalent coefficients obtained by the multi-harmonic balancing method in the step 6 are dimensionless to obtain the dimensionless equivalent stiffness coefficients of each order
Figure FDA0003026022830000065
And dimensionless equivalent damping coefficient
Figure FDA0003026022830000066
The expression of (a) is:
Figure FDA0003026022830000067
the coefficients in equation (13) are:
Figure FDA0003026022830000071
Figure FDA0003026022830000072
Figure FDA0003026022830000073
the dimensionless equivalent stiffness coefficient matrix is represented as:
Figure FDA0003026022830000074
the dimensionless equivalent damping coefficient matrix is expressed as:
Figure FDA0003026022830000075
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