CN112069710A - Prediction method for self-excited vibration of gas static pressure main shaft - Google Patents

Prediction method for self-excited vibration of gas static pressure main shaft Download PDF

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CN112069710A
CN112069710A CN202010927115.0A CN202010927115A CN112069710A CN 112069710 A CN112069710 A CN 112069710A CN 202010927115 A CN202010927115 A CN 202010927115A CN 112069710 A CN112069710 A CN 112069710A
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equation
spindle
gas
static pressure
main shaft
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CN112069710B (en
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王伟
王超
程旭浩
龚维纬
庞飞
王超群
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University of Electronic Science and Technology of China
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    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
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Abstract

The invention discloses a prediction method for self-excited vibration of a gas static pressure main shaft, which comprises the following steps: s1, establishing a physical model of the gas static pressure spindle system; s2, dividing a finite element grid for the gas film of the gas static pressure main shaft by using a finite element method to obtain pressure distribution of a flow field applied to the gas static pressure main shaft; s3, establishing a motion control equation of the gas static pressure main shaft; s4, establishing a finite element model of the gas static pressure main shaft; and S5, calculating to obtain the self-excited vibration characteristic of the aerostatic spindle according to the finite element model of the aerostatic spindle, and obtaining the self-excited vibration curve of the aerostatic spindle. The method realizes the simulation calculation of the self-excited vibration characteristic of the ultra-precise gas static pressure main shaft, and realizes the self-excited vibration prediction of the gas static pressure main shaft system by the assistance of modal analysis.

Description

Prediction method for self-excited vibration of gas static pressure main shaft
Technical Field
The invention belongs to the technical field of dynamics simulation, and particularly relates to a prediction method for self-excited vibration of a gas static pressure spindle.
Background
With the further development of ultra-precision manufacturing industry, people have higher and higher requirements on the precision of the main shaft. The gas lubrication technology gradually replaces the traditional oil film lubrication support and other forms, and becomes an important means for ensuring the ultra-precision machining precision. In engineering practice, the bearing capacity and static rigidity of the bearing are usually increased by measures such as increasing the air supply pressure of an air source, changing the forms and structural parameters of the bearing and the restrictor, and the like. However, under certain conditions, the bearing can cause the main shaft to vibrate in a self-excited manner, the vibration amplitude of the main shaft is large, continuous squeal is accompanied, the main shaft is unstable destructive vibration, the main shaft is just like air hammering a workpiece, the main shaft is also called as air hammer vibration, and the main shaft can be damaged when the main shaft is serious, so that the main shaft cannot work normally. The current research mainly focuses on the static and dynamic characteristics of the aerostatic bearing, the research on self-excited vibration mostly focuses on optimizing the structure, and the traditional simulation method cannot predict the generation of the self-excited vibration of the main shaft.
Disclosure of Invention
The invention aims to solve the problem that the self-excited vibration of a main shaft cannot be predicted in the prior art, and provides a prediction method for the self-excited vibration of a gas static pressure main shaft, which can realize the simulation calculation of the self-excited vibration characteristic of an ultra-precise gas static pressure main shaft and realize the self-excited vibration prediction of a gas static pressure main shaft system through the assistance of modal analysis.
The purpose of the invention is realized by the following technical scheme: the prediction method for the self-excited vibration of the gas static pressure spindle comprises the following steps:
s1, establishing a physical model of the gas static pressure spindle system, wherein the physical model comprises a gas film of the gas static pressure spindle and a rotor of the gas static pressure spindle;
s2, dividing a finite element grid for the gas film of the gas static pressure main shaft by using a finite element method based on a gas lubrication principle to obtain pressure distribution of a flow field applied to the gas static pressure main shaft;
s3, based on a solid control equation, firstly, obtaining the system natural frequency of the gas static pressure spindle system under the fluid-solid coupling effect by using a modal method, and then establishing a motion control equation of the gas static pressure spindle;
s4, based on the fluid-solid coupling control equation, coupling the pressure distribution data obtained in the step S2 and the motion control equation in the step S3 together, and establishing a finite element model of the gas static pressure spindle;
and S5, calculating to obtain the self-excited vibration characteristic of the aerostatic spindle according to the finite element model of the aerostatic spindle, and obtaining the self-excited vibration curve of the aerostatic spindle.
Further, in step S2, the method for obtaining the pressure distribution data W of the aerostatic spindle by numerical calculation by using the gas lubrication principle and the finite element method to simulate the entire solution domain by dividing the finite mesh into the gas film of the aerostatic spindle system is as follows:
and (2) applying a fluid control equation to obtain the pressure distribution condition of the gas film of the gas static pressure main shaft, wherein the fluid control equation is a continuity equation, a momentum equation and an energy equation:
Figure BDA0002668805700000021
where ρ is the fluid density, t is the time,
Figure BDA0002668805700000022
is the speed;
p is the pressure on the fluid micro-elements,
Figure BDA0002668805700000023
is unit mass force, mu is gas viscosity coefficient;
e is the total energy of the fluid micelle, including internal energy, kinetic energy and potential energy; k is a radical ofeffFor effective heat transfer coefficient, T is temperature, JjIs the diffusion flux of component j, τeffFor effective viscous stress, ShIs a volumetric heat source term, hjIs the enthalpy of the fluid portion;
div () is the divergence solution, grad is the gradient solution,
Figure BDA0002668805700000024
Is a gradient operator,
Figure BDA0002668805700000025
Is the temperature difference;
the control equation is rewritten into the form of a generic variable equation:
Figure BDA0002668805700000026
wherein the content of the first and second substances,
Figure BDA0002668805700000027
are general variables, represent
Figure BDA0002668805700000028
Or T;
in the case of a generalized diffusion coefficient,
Figure BDA0002668805700000029
is a generalized source term;
carrying out grid division on a gas film flow field calculation domain by utilizing a finite element method discrete fluid control equation, wherein the discrete equation is as follows:
Figure BDA00026688057000000210
v represents a infinitesimal integration volume;
integrating the differential equations in the control volume, wherein the corresponding steady-state solving equations and the corresponding unsteady-state solving equations are respectively as follows:
Figure BDA00026688057000000211
Figure BDA00026688057000000212
is a normal vector of a infinitesimal body, A is the integral area of the infinitesimal, and delta t is the time variation;
and (3) solving the gas film pressure distribution p by using a solving equation according to the analysis condition, and further solving the pressure distribution W:
Figure BDA0002668805700000031
s is the gas film calculation area, r1Is the inner diameter of the gas film, r2For the gas film outer diameter, r and θ are the conversions of the integration of the area integral over ds in the column coordinates.
Further, the method for describing the physical motion of the spindle by using the modal shape calculation method and the solid control equation in step S3 is as follows:
the differential equation of motion for a mass system is:
Figure BDA0002668805700000032
wherein [ M]Is the quality matrix of the spindle system, [ C ]]Is the damping matrix of the spindle system, [ K ]]Is the stiffness matrix of the spindle system, [ F ]]Is the external excitation matrix to which the principal axis is subjected,
Figure BDA0002668805700000033
is the acceleration of the motion of the spindle system,
Figure BDA0002668805700000034
the movement speed of the main shaft system is shown, and x is the movement displacement of the main shaft system;
when there is no external load and damping is neglected, equation (6) is reduced to the free vibration equation
Figure BDA0002668805700000035
When in use
Figure BDA0002668805700000036
When the system is in a normal state, the equation has a non-zero solution, and the vibration equation at the moment is the mode shape of the system, wherein omeganIs the system natural frequency;
the solids control equation is:
Figure BDA0002668805700000037
where ρ issIs the density of the solid, and is,
Figure BDA0002668805700000038
is the solid-domain local acceleration vector, σsIs the Cauchy stress tensor, fsIs the volume force vector, whose governing equation is derived from newton's second law;
for the energy equation of the solid part, the thermal deformation term caused by the temperature difference is added:
Figure BDA0002668805700000039
wherein alpha isTIs the temperature dependent coefficient of thermal expansion.
Further, the pressure distribution data obtained in step S4 and the motion control equation in step three are coupled together, and the method for establishing the finite element model of the aerostatic spindle is as follows:
Figure BDA00026688057000000310
wherein tau is stress, d is displacement, q is heat flow, T is temperature, subscript f represents fluid, subscript s represents solid, and the vibration characteristic of the main shaft system can be calculated by a fluid-solid coupling control equation.
The invention has the beneficial effects that: the method realizes the simulation calculation of the self-excited vibration characteristic of the ultra-precise gas static pressure main shaft, and realizes the self-excited vibration prediction of the gas static pressure main shaft system by the assistance of modal analysis.
Drawings
FIG. 1 is a flow chart of a method of the present invention for predicting self-excited vibration of a aerostatic spindle;
FIG. 2 is a schematic view of the aerostatic spindle configuration of the present invention;
FIG. 3 is a finite element mesh partition of the gas film in the aerostatic spindle system of the present invention;
FIG. 4 is a radial pressure profile of the aerostatic spindle of the present invention;
FIG. 5 is a finite element model of the aerostatic spindle of the present invention;
FIG. 6 is a modal shape curve of the aerostatic spindle of the present invention;
fig. 7 is a vibration characteristic curve of the aerostatic spindle according to the invention.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
As shown in fig. 1, the method for predicting self-excited vibration of a gas static pressure spindle of the present invention includes the following steps:
s1, establishing a physical model of the gas static pressure spindle system, wherein the physical model comprises a fluid domain and a solid domain, namely a gas static pressure spindle gas film and a gas static pressure spindle rotor, and the gas static pressure spindle is schematically shown in FIG. 2;
s2, dividing a finite element grid for the gas film of the gas static pressure main shaft by using a finite element method based on a gas lubrication principle to obtain pressure distribution of a flow field applied to the gas static pressure main shaft;
the finite element meshing of the gas film of the aerostatic spindle is shown in fig. 3, and the radial pressure distribution of the aerostatic spindle is shown in fig. 4, wherein the abscissa is the radial coordinate of the aerostatic bearing, and the ordinate is the distribution pressure.
In this step, the whole solution domain is simulated by dividing the finite mesh into the gas film of the aerostatic spindle system by using the gas lubrication principle and the finite element method, and the method for obtaining the pressure distribution data W of the aerostatic spindle by numerical calculation is as follows:
and (2) applying a fluid control equation to obtain the pressure distribution condition of the gas film of the gas static pressure main shaft, wherein the fluid control equation is a continuity equation, a momentum equation and an energy equation:
Figure BDA0002668805700000041
where ρ is the fluid density, t is the time,
Figure BDA0002668805700000042
is the speed;
p is the pressure on the fluid micro-elements,
Figure BDA0002668805700000043
is unit mass force, mu is gas viscosity coefficient;
e is the total energy of the fluid micelle, including internal energy, kinetic energy and potential energy; k is a radical ofeffFor effective heat transfer coefficient, T is temperature, JjIs the diffusion flux of component j, τeffFor effective viscous stress, ShIs a volumetric heat source term, hjIs the enthalpy of the fluid portion;
div () is the divergence solution, grad is the gradient solution,
Figure BDA0002668805700000051
Is a gradient operator,
Figure BDA0002668805700000052
Is the temperature difference;
the control equation is rewritten into the form of a generic variable equation:
Figure BDA0002668805700000053
wherein the content of the first and second substances,
Figure BDA0002668805700000054
are general variables, represent
Figure BDA0002668805700000055
Or T;
in the case of a generalized diffusion coefficient,
Figure BDA0002668805700000056
is a generalized source term;
carrying out grid division on a gas film flow field calculation domain by utilizing a finite element method discrete fluid control equation, wherein the discrete equation is as follows:
Figure BDA0002668805700000057
v represents a infinitesimal integration volume;
integrating the differential equations in the control volume, wherein the corresponding steady-state solving equations and the corresponding unsteady-state solving equations are respectively as follows:
Figure BDA0002668805700000058
Figure BDA0002668805700000059
is a normal vector of a infinitesimal body, A is the integral area of the infinitesimal, and delta t is the time variation;
and (3) solving the gas film pressure distribution p by using a solving equation according to the analysis condition, and further solving the pressure distribution W:
Figure BDA00026688057000000510
s is the gas film calculation area, r1Is the inner diameter of the gas film, r2For the gas film outer diameter, r and θ are the conversions of the integration of the area integral over the column coordinate by ds, as shown in FIG. 3.
In order to accurately predict the vibration characteristics of the bearing, it is necessary to theoretically calculate the internal pressure distribution of the air film. Because the greatest difference between the studied ultra-precise bearing and the conventional bearing is that air is used as a lubricating medium, the theoretical calculation of the bearing depends on a gas lubrication equation. The basic equation suitable for fluid lubrication can be obtained under reasonable assumption conditions by adding the gas state equation to the conservation equation of mass, momentum and energy, and the basic equation is used as a theoretical equation for numerical solution, and the direct numerical solution of the high-dimensional partial differential equation is extremely difficult. According to the finite element concept, a discrete fluid control equation, a flow equation and boundary conditions are reflected on a model, namely, the model is divided into a finite number of nodes, and the physical quantities of the divided nodes are used for approximately fitting the physical quantities of the whole solution domain. According to the method, the partial differential terms are converted into a differential format by adopting a finite volume method in the process of a discrete fluid control equation, so that a relational expression among nodes in the air film is expressed, and a discrete form of pressure distribution is obtained.
S3, based on a solid control equation, firstly, obtaining the system natural frequency of the gas static pressure spindle system under the fluid-solid coupling effect by using a modal method, and then establishing a motion control equation of the gas static pressure spindle;
the method for describing the physical motion of the main shaft by using the modal shape calculation method and the solid control equation is as follows:
the differential equation of motion for a mass system is:
Figure BDA0002668805700000061
wherein [ M]Is the quality matrix of the spindle system, [ C ]]Is the damping matrix of the spindle system, [ K ]]Is the stiffness matrix of the spindle system, [ F ]]Is the external excitation matrix to which the principal axis is subjected,
Figure BDA0002668805700000062
is the acceleration of the motion of the spindle system,
Figure BDA0002668805700000063
the movement speed of the main shaft system is shown, and x is the movement displacement of the main shaft system;
when there is no external load and damping is neglected, equation (6) is reduced to the free vibration equation
Figure BDA0002668805700000064
When in use
Figure BDA0002668805700000065
When the system is in a normal state, the equation has a non-zero solution, and the vibration equation at the moment is the mode shape of the system, wherein omeganIs the system natural frequency;
the solids control equation is:
Figure BDA0002668805700000066
where ρ issIs the density of the solid, and is,
Figure BDA0002668805700000067
is the solid-domain local acceleration vector, σsIs the Cauchy stress tensor, fsIs the volume force vector, whose governing equation is derived from newton's second law;
for the energy equation of the solid part, the thermal deformation term caused by the temperature difference is added:
Figure BDA0002668805700000068
wherein alpha isTIs the temperature dependent coefficient of thermal expansion.
S4, based on the fluid-solid coupling control equation, coupling the pressure distribution data obtained in the step S2 and the motion control equation in the step S3 together, and establishing a finite element model of the gas static pressure spindle, as shown in FIG. 5;
the obtained pressure distribution data is coupled with the motion control equation in the third step, and the method for establishing the finite element model of the gas static pressure main shaft comprises the following steps:
Figure BDA0002668805700000069
wherein tau is stress, d is displacement, q is heat flow, T is temperature, subscript f represents fluid, subscript s represents solid, and the vibration characteristic of the main shaft system can be calculated by a fluid-solid coupling control equation.
S5, calculating to obtain the self-excited vibration characteristic of the aerostatic spindle according to the finite element model of the aerostatic spindle, wherein the vibration amplitude-frequency characteristic is shown in FIG. 6, the abscissa is frequency, and the ordinate is amplitude-frequency characteristic; by comparing the system natural frequency obtained in step S3, it can be found that the two frequencies are approximately coincident, indicating that resonance of the spindle system occurs. And obtaining a self-excited vibration curve of the aerostatic spindle, as shown in fig. 7, wherein the abscissa is time, and the ordinate respectively represents: the upper graph is the rotor vibration speed, and the lower graph is the rotor vibration displacement. The speed and displacement curve measuring range of the main shaft system accords with the actual measuring result of engineering, the vibration amplitude is in the order of mum, and the two self-excited vibration characteristic curves are used for predicting whether the self-excited vibration of the main shaft system occurs or not.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.

Claims (4)

1. The prediction method for the self-excited vibration of the gas static pressure spindle is characterized by comprising the following steps of:
s1, establishing a physical model of the gas static pressure spindle system, wherein the physical model comprises a gas film of the gas static pressure spindle and a rotor of the gas static pressure spindle;
s2, dividing a finite element grid for the gas film of the gas static pressure main shaft by using a finite element method based on a gas lubrication principle to obtain pressure distribution of a flow field applied to the gas static pressure main shaft;
s3, based on a solid control equation, firstly, obtaining the system natural frequency of the gas static pressure spindle system under the fluid-solid coupling effect by using a modal method, and then establishing a motion control equation of the gas static pressure spindle;
s4, based on the fluid-solid coupling control equation, coupling the pressure distribution data obtained in the step S2 and the motion control equation in the step S3 together, and establishing a finite element model of the gas static pressure spindle;
and S5, calculating to obtain the self-excited vibration characteristic of the aerostatic spindle according to the finite element model of the aerostatic spindle, and obtaining the self-excited vibration curve of the aerostatic spindle.
2. A method for predicting self-excited vibration of an aerostatic spindle according to claim 1, wherein in step S2, the entire solution domain is simulated by dividing a finite mesh into gas films of the aerostatic spindle system by using a gas lubrication principle and a finite element method, and the method of obtaining the pressure distribution data W of the aerostatic spindle through numerical calculation is as follows:
and (2) applying a fluid control equation to obtain the pressure distribution condition of the gas film of the gas static pressure main shaft, wherein the fluid control equation is a continuity equation, a momentum equation and an energy equation:
Figure FDA0002668805690000011
where ρ is the fluid density, t is the time,
Figure FDA0002668805690000012
is the speed;
p is the pressure on the fluid micro-elements,
Figure FDA0002668805690000013
is unit mass force, mu is gas viscosity coefficient;
e is the total energy of the fluid micelle, including internal energy, kinetic energy and potential energy; k is a radical ofeffFor effective heat transfer coefficient, T is temperature, JjIs the diffusion flux of component j, τeffFor effective viscous stress, ShIs a volumetric heat source term, hjIs the enthalpy of the fluid portion;
div () is the divergence solution, grad is the gradient solution,
Figure FDA0002668805690000014
Is a gradient operator,
Figure FDA0002668805690000015
Is the temperature difference;
the control equation is rewritten into the form of a generic variable equation:
Figure FDA0002668805690000016
wherein the content of the first and second substances,
Figure FDA0002668805690000017
are general variables, represent
Figure FDA0002668805690000018
Or T;
in the case of a generalized diffusion coefficient,
Figure FDA0002668805690000021
is a generalized source term;
carrying out grid division on a gas film flow field calculation domain by utilizing a finite element method discrete fluid control equation, wherein the discrete equation is as follows:
Figure FDA0002668805690000022
v represents a infinitesimal integration volume;
integrating the differential equations in the control volume, wherein the corresponding steady-state solving equations and the corresponding unsteady-state solving equations are respectively as follows:
Figure FDA0002668805690000023
Figure FDA0002668805690000024
is a normal vector of a infinitesimal body, A is the integral area of the infinitesimal, and delta t is the time variation;
and (3) solving the gas film pressure distribution p by using a solving equation according to the analysis condition, and further solving the pressure distribution W:
Figure FDA0002668805690000025
s is the gas film calculation area, r1Is the inner diameter of the gas film, r2Is the outer diameter of the air film,r and θ are the transformation of the integration of the area integral over ds in the cylindrical coordinates.
3. A prediction method for aerostatic spindle self-excited vibration according to claim 1, wherein the method for describing the physical motion of the spindle by using the modal shape calculation method and the solid state control equation in step S3 is as follows:
the differential equation of motion for a mass system is:
Figure FDA0002668805690000026
wherein [ M]Is the quality matrix of the spindle system, [ C ]]Is the damping matrix of the spindle system, [ K ]]Is the stiffness matrix of the spindle system, [ F ]]Is the external excitation matrix to which the principal axis is subjected,
Figure FDA0002668805690000027
is the acceleration of the motion of the spindle system,
Figure FDA0002668805690000028
the movement speed of the main shaft system is shown, and x is the movement displacement of the main shaft system;
when there is no external load and damping is neglected, equation (6) is reduced to the free vibration equation
Figure FDA0002668805690000029
When in use
Figure FDA00026688056900000210
When the system is in a normal state, the equation has a non-zero solution, and the vibration equation at the moment is the mode shape of the system, wherein omeganIs the system natural frequency;
the solids control equation is:
Figure FDA00026688056900000211
where ρ issIs the density of the solid, and is,
Figure FDA00026688056900000212
is the solid-domain local acceleration vector, σsIs the Cauchy stress tensor, fsIs the volume force vector, whose governing equation is derived from newton's second law;
for the energy equation of the solid part, the thermal deformation term caused by the temperature difference is added:
Figure FDA0002668805690000031
wherein alpha isTIs the temperature dependent coefficient of thermal expansion.
4. A prediction method for self-excited vibration of a aerostatic spindle according to claim 1, characterized in that the pressure distribution data obtained in step S4 and the motion control equations in step three are coupled together and a method of establishing a finite element model of the aerostatic spindle is as follows:
Figure FDA0002668805690000032
wherein tau is stress, d is displacement, q is heat flow, T is temperature, subscript f represents fluid, subscript s represents solid, and the vibration characteristic of the main shaft system can be calculated by a fluid-solid coupling control equation.
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