CN112149245B - Flexible shaft-disc system coupling dynamics modeling and analyzing method - Google Patents
Flexible shaft-disc system coupling dynamics modeling and analyzing method Download PDFInfo
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Abstract
The invention provides a flexible shaft-disc system coupling dynamics modeling and analyzing method, and belongs to the technical field of mechanical dynamics. Firstly, establishing a three-dimensional coordinate system, and respectively simulating boundary conditions of a flexible shaft-disk system and a connection coupling relation between a shaft and a disk by adopting an artificial spring; and then establishing an energy equation of the flexible shaft-disk system to further obtain a total kinetic energy equation and a total potential energy equation of the flexible shaft-disk system, calculating discretized total kinetic energy and total potential energy, finally establishing a coupling dynamic model of the flexible shaft-disk system under any boundary condition, and solving the natural frequency and the vibration mode of the flexible shaft-disk system by using the model. The invention can be used for the dynamic characteristic analysis of the flexible shaft-disk-system under any boundary condition. The invention can predict and analyze the dynamic characteristics of the system more accurately, guide the design of the shaft-disk system in the actual engineering and provide further guidance for the vibration control of the system.
Description
Technical Field
The invention belongs to the technical field of mechanical dynamics, and particularly relates to a flexible shaft-disc system coupling dynamics modeling and analyzing method.
Background
The shaft-disc system is a main component of a rotary machine, and the rotary machine is the most widely used machine in industry, and plays a very important role in various fields of production and life (such as aerospace, petrochemical industry, marine manufacturing and the like). Shaft-disk systems are usually operated at high speed, and the vibration and noise problems are not negligible, so that many researchers have studied the dynamic modeling method and dynamic characteristics of the shaft-disk system, and have great significance in reducing the vibration and noise of the shaft-disk system and even the whole rotating machine.
The existing main dynamics modeling methods of the shaft-disc system are a finite element method and a theoretical analytic method. The finite element method is based on a variation principle and a weighted residue method, and the basic solution idea is to divide a calculation domain into finite non-overlapping units, select some proper nodes in each unit as interpolation points of a solution function, rewrite variables in a differential equation into a linear expression composed of node values of each variable or derivatives thereof and the selected interpolation function, and discretely solve the differential equation by means of the variation principle or the weighted residue method. The finite element method consumes very large computing resources such as computing time, memory, disk space and the like for the analysis and calculation of complex problems, and has no good processing method for solving a domain problem and a nonlinear problem infinitely. When the system is subjected to parameter analysis, the method needs to be modeled again from the beginning, and is very inconvenient. Accordingly, theoretical analytical methods solve this problem well.
The theoretical analytical method is to theoretically deduce a kinetic equation of the system through a mechanical theory and establish a kinetic model of the system. However, the existing dynamic model usually considers the shaft and the disc as two independent substructures for research, and one method is to consider the shaft as a spring and the disc as a rigid body; another approach is to consider the flexibility of the shaft, considering the disc as a rigid body or an elastomer. However, in the shaft-disk system having a thick disk, an accurate calculation result can be obtained by regarding the disk as a rigid body, but the design trend of the rotating apparatus is high speed, light weight, large size and automation, and flexibility thereof is unavoidable. Therefore, when the dynamic characteristics of the shaft-disk system are predicted, the flexibility of the shaft and the flexibility of the disk are not negligible, and the errors are large when the dynamic characteristics of the shaft-disk system are predicted by adopting a rigid body model.
At present, researchers have conducted some research on the dynamic modeling method of the flexible shaft-disk system, but most of the work is limited to the classical boundary conditions, namely, simple support or cantilever. However, in engineering applications, the flexible shaft-disc system is usually supported by other components (e.g. bearings, etc.), the boundary conditions of which may not always be classical. In some cases, classical boundary conditions cannot be modeled, and serious errors can result from these modeling methods. In addition, in the prior art, the shaft and the disk are generally regarded as fixed connections, but in engineering practice, the shaft and the disk are connected together through assembling (such as a tensioning sleeve and the like), which is different from the fixed connections, the prior modeling method cannot simulate the coupling conditions of the shaft and the disk, and therefore, the dynamic model of the flexible shaft-disk system established by the coupling conditions of the fixed connections cannot accurately predict the dynamic characteristics of the system.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a flexible shaft-disk system coupling dynamics modeling and analyzing method. The invention not only considers the flexibility of the shaft and the disk, but also considers any boundary condition of a flexible shaft-disk system (hereinafter referred to as the system) and the coupling connection condition between the shaft and the disk, simultaneously considers the centrifugal effect, the Coriolis force and the gyroscopic effect brought by rotation in the model establishing process, establishes a universal and accurate dynamic model of the flexible shaft-disk system, and carries out the analysis of the dynamic characteristics by using the model.
The invention provides a flexible shaft-disc system coupling dynamics modeling and analyzing method which is characterized by comprising the following steps:
1) establishing a three-dimensional coordinate system, which comprises the following specific steps:
1-1) establishing an inertial coordinate system C-XYZ, wherein the C-XYZ is a right-handed system, and an X axis points to the axial direction; translating the C-XYZ to the circular disk of the flexible shaft-disk system to obtain a coordinate system C 1 -X 0 Y 0 Z 0 Wherein, C 1 -X 0 Y 0 Z 0 Origin C of 1 At the center of the disk, X 0 Axis Y 0 Axis Z 0 The directions of the axes are respectively parallel to the X axis, the Y axis and the Z axis of a coordinate system C-XYZ;
1-2) establishing a local reference frame C 1 -X 1 Y 1 Z 1 ，C 1 -X 1 Y 1 Z 1 From C 1 -X 0 Y 0 Z 0 Coordinate system around X 0 The shaft rotates anticlockwise by an angle omega t, C 1 -X 1 Y 1 Z 1 Has an origin of C 1 ，X 1 And X 0 The axes being collinear, wherein Ω denotes the system's rotation about X 0 Constant speed of rotation of the shaft, t representing time;
1-3) establishing a disc coordinate system C 1 -X 2 Y 2 Z 2 ；C 1 -X 2 Y 2 Z 2 Is a local coordinate system fixed on the flexible disk and composed of a coordinate system C 1 -X 1 Y 1 Z 1 Around Y 1 Shaft counterclockwise rotation theta y Rewinding Z 2 Shaft counterclockwise rotation theta z Is obtained wherein theta y And theta z Are all greater than 0 DEG C 1 -X 2 Y 2 Z 2 Has an origin of C 1 ；
2) Establishing a boundary spring and a connecting coupling spring of the flexible shaft-disk system; the method comprises the following specific steps:
2-1) establishing a left end boundary spring and a right end boundary spring of the system;
adopting an artificial spring to simulate any boundary condition of the flexible shaft-disc system; the method comprises the following steps of simulating a left end boundary condition of a shaft by adopting a left end boundary spring, and simulating a right end boundary condition of the shaft by adopting a right end boundary spring; the left end boundary spring and the right end boundary spring are simulated by adopting two groups of springs, one group of springs are arranged along the Y-axis direction, the other group of springs are arranged along the Z-axis direction, and each group of springs comprises a linear spring and a torsion spring;
2-2) establishing a vertical shaft-disc connection coupling spring;
simulating the connection coupling relationship between the shaft and the disc by adopting an artificial spring, wherein the spring is marked as a shaft-disc connection coupling spring, the arrangement mode is that a plurality of groups of springs forming a whole circle are continuously arranged on the combination surface of the shaft and the disc, and each group of springs comprises a linear spring and a torsion spring;
3) establishing an energy equation of the flexible shaft-disc system, which specifically comprises the following steps:
3-1) establishing the strain energy expression of the flexible shaft as follows:
wherein E is S Is the elastic modulus, L, of the flexible shaft S Length of flexible shaft, y S Representing elastic displacement of the flexible axis in a direction parallel to the Y-axis, z S Showing the elastic displacement of the flexible axis in a direction parallel to the Z-axis, I Sy Represents the moment of inertia of the flexible shaft cross section to the Y axis;
3-2) establishing the kinetic energy expression of the flexible shaft as follows:
where ρ is S Being a flexible shaftDegree, A S Cross-sectional area of the flexible shaft, theta Sy Representing angular displacement of the flexible axis parallel to the Y-axis, theta Sz Representing an angular displacement of the flexible axis parallel to the Z-axis direction;
3-3) establishing the total strain energy expression for the flexible disk as follows:
wherein D is D Is the bending rigidity of the flexible disk, and the expression is
E D Is the elastic modulus, h, of the flexible disk D Is the thickness of the flexible disc, v is the Poisson ratio + 2 Is Laplace operator, expression is
u D For elastic deformation of the flexible disk, σ r And σ θ Radial stress and tangential stress, respectively;
3-4) the kinetic energy expression for the flexible disk is established as follows:
where ρ is D Density of flexible disk, M D Mass of flexible disk, J Dx Is the moment of inertia, y, of the cross-section of the flexible disk to the X-axis D Is the elastic displacement of the flexible axis parallel to the direction of the Y axis at the location of the flexible disk, z D Is the elastic displacement of the flexible shaft parallel to the Z-axis direction at the position of the flexible disk;
3-5) establishing a potential energy expression of the shaft-disc connection coupling spring as follows:
wherein k is Du,0 To connect toStiffness of coupled linear spring, k Dtu,0 In order to connect the stiffness of the coupled torsion spring,
is the elastic deformation at the inner diameter of the flexible disk;
3-6) establishing the potential energy expression of the left border spring as follows:
wherein the content of the first and second substances,
the stiffness of the linear spring parallel to the Y-axis direction at the left end boundary,
the stiffness of the linear spring at the left end boundary in parallel to the Z-axis direction,
the stiffness of the torsion spring at the left end boundary in parallel to the Y-axis direction,
the rigidity of the torsion spring in the direction parallel to the Z axis at the left end boundary is calculated x＝0 Indicates the elastic deformation of the flexible shaft at the left end boundary, so y b | x＝0 And z b | x＝0 Respectively, the linear elastic displacement of the flexible axis parallel to the Y-axis and Z-axis directions at the left boundary, Y bt | x＝0 And z bt | x＝0 Respectively representing the torsional elastic displacement of a flexible shaft parallel to the Y-axis and Z-axis directions at the left boundary;
3-7) establishing the potential energy expression of the right end boundary spring as follows:
wherein the content of the first and second substances,
the rigidity of the linear spring parallel to the Y-axis direction at the right-end boundary,
the rigidity of the linear spring parallel to the Z-axis direction at the right-end boundary,
the stiffness of the torsion spring at the right end boundary parallel to the Y-axis direction,
the stiffness of the torsion spring at the right end boundary parallel to the Z-axis direction,
indicating elastic deformation of the flexible shaft at the right end boundary, and
and
respectively showing the linear elastic displacement of the flexible axis parallel to the Y-axis and Z-axis directions at the right-end boundary,
and
respectively representing the torsional elastic displacement of a flexible shaft parallel to the Y-axis and Z-axis directions at the right end boundary;
4) establishing a total kinetic energy equation and a total potential energy equation of the flexible shaft-disc system by using the result of the step 3);
wherein the total kinetic energy equation of the flexible shaft-disk system is:
T＝T S +T D
the total potential energy equation for the flexible shaft-disk system is:
5) respectively dispersing the total kinetic energy equation and the total potential energy equation obtained in the step 4) to obtain discretized total kinetic energy and total potential energy; the method comprises the following specific steps:
5-1) discretizing the displacement as follows:
y S ＝Φ S Q y
z S ＝Φ S Q z
u D ＝Φ D Q D1 cosθ+Φ D Q D2 sinθ
wherein phi S And phi D Allowable functions, Q, of the flexible shaft and the flexible disk, respectively y Is y S Corresponding generalized variable, Q z Is z S Corresponding generalized variable, Q D1 Is u D Corresponding first generalized variable, Q D2 Is u D A corresponding second generalized variable;
5-2) using Gram-Schmidt orthogonal polynomials as a permissive function for the flexible shaft and flexible disk;
5-3) discretizing the intermediate variable in the following way:
θ y ＝Φ SD ′Q y ，θ z ＝Φ SD ′Q z
θ Sy ＝Φ S ′Q y ，θ Sz ＝Φ S ′Q z
wherein the subscript L D Representing values calculated at the location of the flexible disk,' watchShowing a time derivative;
5-4) variable y to be discretized S 、z S 、y D 、z D 、u D 、θ y 、θ z 、θ Sy 、θ Sz Substituting the total kinetic energy equation and the total potential energy equation in the step 4) to respectively obtain discretized total kinetic energy and total potential energy;
6) establishing a coupling dynamic model of the flexible shaft-disk system under any boundary condition by using the discretized total kinetic energy and total potential energy obtained in the step 5) and applying a Lagrange equation;
wherein, the Lagrange equation expression is as follows:
wherein L (T-U) is a lagrange function, and Q (Q) is Q y Q z Q D1 Q D2 ]Is a generalized coordinate of the coordinate system of the general sense,
generalized velocity, F generalized force;
the coupling dynamics model of the flexible shaft-disk system is:
wherein, M, G, K are quality, top and rigidity matrix respectively, and the specific expression is as follows respectively:
7) selecting a geometrical parameter L of the flexible shaft-disk system to be analyzed S 、R i 、R 0 、h D 、L D Material parameter p S 、E S 、ρ D 、E D V, boundary condition parameters
And connection coupling condition parameter k Du,0 、k Dtu,0 And substituting the parameters into the step 6) to obtain a dynamic model, and solving the dynamic model to obtain the natural frequency and the vibration mode of the system.
The invention has the characteristics and beneficial effects that:
the invention is suitable for flexible shaft-disk systems with all structural parameters, and when the structural parameters of the system are changed, the inherent dynamic characteristics of different systems can be obtained without changing the model; the invention considers the influence of the gyro effect, the centrifugal effect and the Coriolis force of the coupling system and can obtain more accurate dynamic inherent characteristics of the rotary flexible shaft-disk system; the invention considers the coupling effect of the flexible shaft and the flexible disk, and can consider the influence of the coupling strength; the invention can be used for the dynamic characteristic analysis of the flexible shaft-disc-system under any boundary condition. The invention can predict and analyze the dynamic characteristics of the system more accurately, guide the design of the shaft-disk system in the actual engineering and provide further guidance for the vibration control of the system.
Drawings
FIG. 1 is an overall flow diagram of the process of the present invention.
FIG. 2 is a schematic diagram of a three-dimensional coordinate system according to the present invention.
FIG. 3 is a schematic diagram of a flexible shaft-disk system constructed in accordance with the present invention using artificial springs to simulate arbitrary boundary conditions.
Fig. 4 is a partially enlarged view of a shaft-disk coupling spring.
FIG. 5 is a diagram illustrating an example of the result of one mode shape under certain parameters and boundary conditions in the embodiment of the present invention.
FIG. 6 is a diagram illustrating an example of the results of a mode shape under certain parameters and boundary conditions in an embodiment of the present invention.
Detailed Description
The invention provides a flexible shaft-disc system coupling dynamics modeling and analyzing method, and the technical scheme of the invention is specifically explained by embodiments and the accompanying drawings. The following description of the embodiments of the present invention with reference to the accompanying drawings is intended to explain the general inventive concept in detail and should not be construed as limiting the invention.
The invention provides a flexible shaft-disc system coupling dynamics modeling and analyzing method, the overall flow is shown as figure 1, and the method comprises the following steps:
s1, establishing a three-dimensional coordinate system, which comprises the following steps:
s11, establishing a global coordinate system C-XYZ; FIG. 2 is a schematic diagram of a three-dimensional coordinate system of the present invention, wherein the global coordinate system of the present invention, i.e., the inertial coordinate system C-XYZ, is a right-handed system, wherein the X-axis points to the axial direction. Translating the coordinate system to a disk of a flexible shaft-disk system to obtain a coordinate system C 1 -X 0 Y 0 Z 0 Wherein, the origin C 1 At the center of the disk, X 0 Axis Y 0 Axis Z 0 The directions of the axes are respectively parallel to the X-axis, Y-axis and Z-axis of the coordinate system C-XYZ.
S12 establishing a local reference frame C 1 -X 1 Y 1 Z 1 ，C 1 -X 1 Y 1 Z 1 From C 1 -X 0 Y 0 Z 0 Coordinate system around X 0 The shaft rotates anticlockwise by an angle omega t, and the origin C 1 At the center of the disk, X 0 And X 1 The axes are collinear with each other,wherein Ω represents the system winding X 0 The constant speed of the shaft rotation (over a range of Ω ≧ 0, where 0 is a non-rotating state and when there is no rotation is a lateral vibration, and when Ω is greater than 0 is a rotating state.) t denotes time.
S13 establishing a disc coordinate system C 1 -X 2 Y 2 Z 2 ；C 1 -X 2 Y 2 Z 2 Is a local coordinate system fixed on the flexible disk and composed of a coordinate system C 1 -X 1 Y 1 Z 1 Around Y 1 Shaft counterclockwise rotation theta y Rewinding Z 2 Shaft counterclockwise rotation theta z Is obtained wherein theta y And theta z All are small angles larger than 0 degree, the higher power of the small angles is regarded as high-order infinitesimal, and the origin C 1 Is positioned at the center of the disc.
S2 establishing boundary springs and connecting coupling springs of the flexible shaft-disk system; the method comprises the following specific steps:
s21 establishes the system left and right border springs. In engineering applications, the flexible shaft-disk system is usually supported by other components (e.g., bearings, etc.), this boundary condition is not a classical boundary but an arbitrary boundary, and the present invention employs artificial springs to simulate the arbitrary boundary condition. The structural schematic diagram of the flexible shaft-disc system after the artificial spring is adopted to simulate any boundary condition is shown in figure 3, and the structural schematic diagram comprises the following components: a left end border spring 1, a flexible shaft 2, a flexible disk 3, a shaft-disk connection coupling spring 4 and a right end border spring 5. In fig. 3, the left end boundary spring 1 is used to simulate the left end boundary condition of the shaft, and the right end boundary spring 5 is used to simulate the right end boundary condition of the shaft. Because the rotation characteristic of the system is considered, the freedom degrees in the Y direction and the Z direction need to be considered in the modeling process, the left end boundary spring 1 and the right end boundary spring 5 are simulated by adopting two groups of springs, one group of springs are arranged along the Y-axis direction, the other group of springs are arranged along the Z-axis direction, and each group of springs comprises a linear spring and a torsion spring.
S22, the shaft-disk connection coupling spring 4 is built, and in order to more clearly show the connection relationship between the shaft and the disk, the enlarged structure of the shaft-disk connection coupling spring 4 in fig. 3 is shown in fig. 4. In engineering practice, the shaft and the disc are connected and coupled together through assembling (such as expanding coupling sleeves and the like), which is different from the fixed connection, the invention adopts the artificial spring to simulate the connection coupling relationship between the shaft and the disc, the shaft-disc connection coupling spring 4 is the spring for connecting the shaft and the disc, the arrangement mode is that a plurality of groups of springs forming a whole circle are continuously arranged on the combination surface of the shaft and the disc, each group of springs comprises a linear spring and a torsion spring, and for convenience of illustration and description, only 3 groups of springs are shown in the shaft-disc connection coupling spring in fig. 3 and 4.
S3 sets up an energy equation for the flexible shaft-disk system taking into account coriolis forces, centrifugal effects, and gyroscopic effects. The method specifically comprises the following steps:
s31 assuming that the flexible shaft is an elongated shaft made of isotropic, uniform and linear elastic material, the strain energy expression of the flexible shaft is obtained by Euler-Bernoulli theory as follows:
wherein E is S Is the elastic modulus, L, of the flexible shaft S Length of flexible shaft, y S And z S Showing the elastic displacement of the flexible axis parallel to the Y-axis and Z-axis, respectively, I Sy Representing the moment of inertia of the flexible shaft cross-section to the Y-axis.
S32 considering the rotational kinetic energy and the translational kinetic energy, the kinetic energy expression of the flexible shaft can be derived as follows:
wherein ρ S Density of the flexible shaft, A S Cross-sectional area of the flexible shaft, theta Sy Representing angular displacement of the flexible axis parallel to the Y-axis, theta Sz Indicating an angular displacement of the flexible axis parallel to the Z-axis direction.
S33 assuming the annular disc is made of an isotropic, uniform linear elastic material, the total strain energy of the flexible disc is:
wherein D is D Is the bending stiffness of the flexible disk, expressed as
E D Is the elastic modulus, h, of the flexible disk D V is the thickness of the flexible disc, the poisson's ratio; v 2 Is Laplace operator, expressed as
u D For elastic deformation of the flexible disk, σ r And σ θ Radial stress and tangential stress, respectively.
S34 after further considering the gyro effect, the kinetic energy expression of the flexible disk is derived as follows:
wherein ρ D Density of flexible disk, M D Mass of flexible disk, J Dx Is the moment of inertia, y, of the cross-section of the flexible disk to the X-axis D Is the elastic displacement of the flexible axis parallel to the Y-axis direction at the location of the flexible disk, z D Is the elastic displacement of the flexible axis parallel to the Z-axis direction at the location of the flexible disk.
S35 the present invention introduces artificial spring technology to simulate the coupling condition of the shaft-disk connection in view of the connection and coupling between the flexible shaft and the flexible disk. According to continuity and equilibrium conditions, the potential energy of the shaft-disk coupling spring is:
wherein k is Du,0 For connecting the stiffness, k, of coupled linear springs Dtu,0 In order to connect the stiffness of the coupled torsion spring,
is an elastic deformation at the inner diameter of the flexible disk.
S36 the invention considers the flexible shaft-disk system with any boundary condition, and introduces artificial spring technology to simulate any boundary condition. The potential energy of the left border spring, according to continuity and equilibrium conditions, is:
wherein the content of the first and second substances,
the stiffness of the linear spring parallel to the Y-axis direction at the left end boundary,
the stiffness of the linear spring parallel to the Z-axis direction at the left end boundary,
the stiffness of the torsion spring at the left end boundary in parallel to the Y-axis direction,
the stiffness of the torsion spring in the direction parallel to the Z axis at the left end boundary x＝0 Indicates the elastic deformation of the flexible shaft at the left end boundary, so y b | x＝0 And z b | x＝0 Respectively, the linear elastic displacement of the flexible axis parallel to the Y-axis and Z-axis directions at the left boundary, Y bt | x＝0 And z bt | x＝0 Showing torsional elastic displacement of the flexible axis parallel to the Y-axis and Z-axis directions at the left boundary, respectively.
The potential energy of the boundary spring at the right end of the S37 is as follows:
wherein，
The rigidity of the linear spring parallel to the Y-axis direction at the right-end boundary,
the stiffness of the linear spring parallel to the Z-axis direction at the right end boundary,
the stiffness of the torsion spring at the right end boundary parallel to the Y-axis direction,
the rigidity of the torsion spring parallel to the Z-axis direction at the right-end boundary,
indicating elastic deformation of the flexible shaft at the right end boundary, and
and
respectively showing the linear elastic displacement of the flexible axis parallel to the Y-axis and Z-axis directions at the right-end boundary,
and
respectively, the torsional elastic displacement of the flexible axis parallel to the Y-axis and Z-axis directions at the right-end boundary.
And S4, summing the energy of each part of the system to obtain a total kinetic energy equation and a total potential energy equation of the flexible shaft-disc system. Wherein, the total kinetic energy equation of the system is T ═ T S +T D (ii) a The total potential energy equation of the system is
S5, dispersing the total kinetic energy equation and the total potential energy equation obtained in S4 respectively to obtain discretized total kinetic energy and total potential energy; the method comprises the following specific steps:
the S51 displacement may be discretized as follows:
y S ＝Φ S Q y
z S ＝Φ S Q z
u D ＝Φ D Q D1 cosθ+Φ D Q D2 sinθ
wherein phi S And phi D Allowable functions, Q, of the flexible shaft and the flexible disk, respectively y Is y S Corresponding generalized variable, Q z Is z S Corresponding generalized variable, Q D1 Is u D Corresponding first generalized variable, Q D2 Is u D A corresponding second generalized variable.
S52 using Gram-Schmidt orthogonal polynomials as admissible function of flexible shaft and flexible disk, Φ S And phi D Are all made of
A collection of forms, wherein each item of the collection is derived by the following process. The expression of Gram-Schmidt orthogonal polynomial is:
When considering arbitrary boundary conditions, the first term, i.e.. psi, satisfying the free-free boundary condition is employed 1 (η) 1. The other terms of the polynomial are constructed according to the following recursion formula:
ψ 2 (η)＝(η-B 1 )ψ 1 (η)，ψ m+1 (η)＝(η-B m )ψ m (η)-C m ψ m-1 (η)，m≥2
wherein the content of the first and second substances,
the discretization of the intermediate variable by S53 can be done as follows:
θ y ＝Φ SD ′Q y ，θ z ＝Φ SD ′Q z
θ Sy ＝Φ S ′Q y ，θ Sz ＝Φ S ′Q z
wherein the subscript L D Representing the value calculated at the flexible disk position and' representing the derivative over time.
S54 discretizing the variables (y) S 、z S 、y D 、z D 、u D 、θ y 、θ z 、θ Sy 、θ Sz ) The discretized total kinetic energy and total potential energy can be obtained by substituting the total kinetic energy equation and the total potential energy equation of S4.
S6 obtains a dynamic equation of the system by applying a Lagrange equation according to the discretized total kinetic energy and total potential energy obtained in S5, and accordingly a coupling dynamic model of the flexible shaft-disk system under any boundary condition is established.
Wherein, Lagrange's equation expression is:
wherein L ═ T-U is Lagrangian function, and Q ═ Q y Q z Q D1 Q D2 ]In the form of a generalized coordinate system,
generalized velocity, F generalized force;
the coupling dynamics model of the flexible shaft-disk system is:
wherein M, G, K are quality, top and rigidity matrix respectively, and the specific expressions are as follows respectively:
s7 selecting the geometrical parameters (L) of the flexible shaft-disk system to be analyzed S 、R i 、R 0 、h D 、L D ) Material parameter (p) S 、E S 、ρ D 、E D V), boundary condition parameters
And connection coupling condition parameter (k) Du,0 、k Dtu,0 ) Substituting the above parameters intoS6, obtaining a dynamic model, and solving the dynamic model to obtain the natural frequency and the mode shape of the system.
If a certain parameter of the model is changed, the dynamic model of S6 is re-introduced, and the solution is carried out again, so that the new natural frequency and mode shape can be obtained. By adopting the process, the influence of the geometric parameters, the material parameters, the boundary conditions and the connection coupling conditions of any flexible shaft-disc system on the dynamic characteristics of the flexible shaft-disc system can be analyzed.
In this embodiment, a set of parameters and partial solution results under the parameters are given, and the influence of the coupling conditions on the natural frequency and the mode shape of a certain order of the flexible shaft-disk system is analyzed. Parameters for a given system: for flexible shafts, length L S 820 mm; for a flexible disk, the inner radius is R i 20mm, outer radius R 0 200mm thick h D 10mm, disc position L D ＝0.3L S (ii) a The flexible disk and the flexible shaft are made of the same material, and the density of the flexible disk and the density of the flexible shaft are 7.82g/cm 3 The elastic modulus is E ═ 206GPa, and the Poisson ratio is ν ═ 0.3; the system winds X by 0 to omega 0 The shaft rotates. The modeling method can simulate any boundary condition, namely can simulate all types of boundary conditions, including classical boundary conditions and non-classical boundary conditions, symmetric boundary conditions and asymmetric boundary conditions, such as: simulating the left-end simple-support boundary condition and the right-end elastic boundary condition to enable
The modeling method can simulate the connection coupling condition between the flexible shaft and the flexible disk, the coupling strength is changed by changing the rigidity of the connection coupling spring, the coupling degree is indicated to be stronger when the rigidity value is larger, and the connection coupling spring and the flexible disk are not coupled when the rigidity values of the connection coupling spring are both 0 limit values. Two sets of values are taken in the example, the first set is that the rigidity of the connecting coupling spring between the flexible disk and the flexible shaft is very high, and the connecting coupling spring is regarded as strong coupling: k is a radical of Du，0 ＝k Dtu，0 ＝1×10 12 And the second group is that the rigidity of a connecting coupling spring between the flexible disk and the flexible shaft is small, and the connecting coupling spring is regarded as weak coupling: k is a radical of Du，0 ＝k Du，0 ＝1×10 6 . The parameters of the example are takenAnd (5) entering a dynamic model, and solving to obtain the natural frequency and the vibration mode. The first four order natural frequency results of the system and the two flexible shaft-disk coupling mode shape results are given in this example as shown in table 1, for example.
TABLE 1 first four orders of Natural Frequency (NF) (unit: Hz) for the inventive examples
FIG. 5 is a schematic diagram of a second step mode under the condition of strong coupling between the flexible shaft and the flexible disk according to the parameters of the present embodiment; FIG. 6 is a schematic diagram of a second step mode under the weak coupling condition between the flexible shaft and the flexible disk in the present embodiment.
Claims (1)
1. A flexible shaft-disk system coupling dynamics modeling and analyzing method is characterized by comprising the following steps:
1) establishing a three-dimensional coordinate system, which comprises the following specific steps:
1-1) establishing an inertial coordinate system C-XYZ, wherein the C-XYZ is a right-handed system, and an X axis points to an axial direction; translating the C-XYZ to a disc of a flexible shaft-disc system to obtain a coordinate system C 1 -X 0 Y 0 Z 0 Wherein, C 1 -X 0 Y 0 Z 0 Of origin C 1 At the center of the disk, X 0 Axis Y 0 Axis Z 0 The directions of the axes are respectively parallel to the X axis, the Y axis and the Z axis of the coordinate system C-XYZ;
1-2) establishing a local reference frame C 1 -X 1 Y 1 Z 1 ，C 1 -X 1 Y 1 Z 1 From C 1 -X 0 Y 0 Z 0 Coordinate system around X 0 The shaft rotates anticlockwise by an angle omega t, C 1 -X 1 Y 1 Z 1 Has an origin of C 1 ，X 1 And X 0 The axes being collinear, wherein Ω denotes the system's rotation about X 0 Constant speed of shaft rotation, t represents time;
1-3) establishing a disc coordinate system C 1 -X 2 Y 2 Z 2 ；C 1 -X 2 Y 2 Z 2 Is a local coordinate system fixed on the flexible disk and composed of a coordinate system C 1 -X 1 Y 1 Z 1 Around Y 1 Shaft counterclockwise rotation theta y Rewinding Z 2 Shaft counterclockwise rotation theta z Is obtained wherein theta y And theta z Are all greater than 0 DEG C 1 -X 2 Y 2 Z 2 Has an origin of C 1 ；
2) Establishing a boundary spring and a connecting coupling spring of the flexible shaft-disk system; the method comprises the following specific steps:
2-1) establishing a left end boundary spring and a right end boundary spring of the system;
adopting an artificial spring to simulate any boundary condition of the flexible shaft-disc system; wherein, the left end boundary spring is adopted to simulate the left end boundary condition of the shaft, and the right end boundary spring is adopted to simulate the right end boundary condition of the shaft; the left end boundary spring and the right end boundary spring are simulated by adopting two groups of springs, one group of springs are arranged along the Y-axis direction, the other group of springs are arranged along the Z-axis direction, and each group of springs comprises a linear spring and a torsion spring;
2-2) establishing a shaft-disc connection coupling spring;
simulating the connection coupling relationship between the shaft and the disc by adopting an artificial spring, wherein the spring is marked as a shaft-disc connection coupling spring, the arrangement mode is that a plurality of groups of springs forming a whole circle are continuously arranged on the combination surface of the shaft and the disc, and each group of springs comprises a linear spring and a torsion spring;
3) establishing an energy equation of the flexible shaft-disc system, which specifically comprises the following steps:
3-1) establishing a strain energy expression for the flexible shaft as follows:
wherein E is S Is the elastic modulus, L, of the flexible shaft S Length of flexible shaft, y S Representing elastic displacement of the flexible axis in a direction parallel to the Y-axis, z S Is shown parallel toElastic displacement of the flexible shaft in the Z-axis direction, I Sy Represents the moment of inertia of the flexible shaft cross section to the Y axis;
3-2) the kinetic energy expression for establishing the flexible shaft is as follows:
where ρ is S Density of the flexible shaft, A S Cross-sectional area of the flexible shaft, theta Sy Representing angular displacement of the flexible axis parallel to the Y-axis, theta Sz Representing an angular displacement of the flexible axis parallel to the Z-axis direction;
3-3) establishing the total strain energy expression of the flexible disk as follows:
wherein D is D Is the bending stiffness of the flexible disk, and the expression is
E D Is the elastic modulus, h, of the flexible disk D Is the thickness of the flexible disc, v is the poisson's ratio,
is Laplace operator, expression is
u D For elastic deformation of flexible discs, σ r And σ θ Radial stress and tangential stress, respectively;
3-4) the kinetic energy expression for the flexible disk is established as follows:
where ρ is D Density of flexible disk, M D Mass of flexible disk, J Dx Is the moment of inertia, y, of the cross-section of the flexible disk to the X-axis D Is the elastic displacement of the flexible axis parallel to the Y-axis direction at the location of the flexible disk, z D Is the elastic displacement of the flexible shaft parallel to the Z-axis direction at the position of the flexible disk;
3-5) establishing the potential energy expression of the shaft-disc connecting coupling spring as follows:
wherein k is Du，0 For connecting the stiffness, k, of coupled linear springs Dtu，0 For connecting the stiffness of the coupled torsion spring, u DRi Is the elastic deformation at the inner diameter of the flexible disk;
3-6) the potential energy expression for the left border spring is established as follows:
wherein the content of the first and second substances,
the stiffness of the linear spring parallel to the Y-axis direction at the left end boundary,
the stiffness of the linear spring parallel to the Z-axis direction at the left end boundary,
the stiffness of the torsion spring at the left end boundary in parallel to the Y-axis direction,
the stiffness of the torsion spring in the direction parallel to the Z axis at the left end boundary x＝0 Indicates the elastic deformation of the flexible shaft at the left end boundary, so y b | x＝0 And z b | x＝0 Respectively, the linear elastic displacement of the flexible axis parallel to the Y-axis and Z-axis directions at the left boundary, Y bt | x＝0 And z bt | x＝0 Respectively representing the torsional elastic displacement of a flexible shaft parallel to the Y-axis and Z-axis directions at the left boundary;
3-7) establishing the potential energy expression of the right end boundary spring as follows:
wherein, the first and the second end of the pipe are connected with each other,
the stiffness of the linear spring parallel to the Y-axis direction at the right end boundary,
the rigidity of the linear spring parallel to the Z-axis direction at the right-end boundary,
the stiffness of the torsion spring at the right end boundary parallel to the Y-axis direction,
the stiffness of the torsion spring at the right end boundary parallel to the Z-axis direction,
indicating elastic deformation of the flexible shaft at the right end boundary, and
and
respectively showing the linear elastic position of the flexible axis parallel to the Y-axis and Z-axis directions at the right end boundaryThe movement of the movable part is carried out,
and
respectively representing the torsional elastic displacement of a flexible shaft parallel to the Y-axis and Z-axis directions at the right end boundary;
4) establishing a total kinetic energy equation and a total potential energy equation of the flexible shaft-disc system by using the result of the step 3);
wherein the total kinetic energy equation of the flexible shaft-disk system is:
T＝T S +T D
the total potential energy equation for the flexible shaft-disk system is:
5) respectively dispersing the total kinetic energy equation and the total potential energy equation obtained in the step 4) to obtain discretized total kinetic energy and total potential energy; the method comprises the following specific steps:
5-1) discretizing the displacement as follows:
y S ＝Φ S Q y
z S ＝Φ S Q z
u D ＝Φ D Q D1 cosθ+Φ D Q D2 sinθ
wherein phi S And phi D Allowable functions, Q, of the flexible shaft and the flexible disk, respectively y Is y S Corresponding generalized variable, Q z Is z S Corresponding generalized variable, Q D1 Is u D Corresponding first generalized variable, Q D2 Is u D A corresponding second generalized variable;
5-2) using Gram-Schmidt orthogonal polynomials as a permissive function for the flexible shaft and flexible disk;
5-3) discretizing the intermediate variable in the following way:
θ y ＝Φ SD ′Q y ，θ z ＝Φ SD ′Q z
θ Sy ＝Φ S ′Q y ，θ Sz ＝Φ S ′Q z
wherein the subscript L D Represents a value calculated at the flexible disk position,' represents a derivative over time;
5-4) variable y to be discretized S 、z S 、y D 、z D 、u D 、θ y 、θ z 、θ Sy 、θ Sz Substituting the total kinetic energy equation and the total potential energy equation in the step 4) to respectively obtain discretized total kinetic energy and total potential energy;
6) establishing a coupling dynamic model of the flexible shaft-disk system under any boundary condition by using the discretized total kinetic energy and total potential energy obtained in the step 5) and applying a Lagrange equation;
wherein, the Lagrange equation expression is as follows:
wherein L (═ T-U) is the lagrange function, wherein L ═ T-U is the lagrange function, and Q ═ Q is [ Q ═ Q [ y Q z Q D1 Q D2 ]In the form of a generalized coordinate system,
generalized velocity, F generalized force;
the coupling dynamics model for the flexible shaft-disk system is:
wherein, M, G, K are quality, top and rigidity matrix respectively, and the specific expression is as follows respectively:
7) selecting a geometrical parameter L of the flexible shaft-disk system to be analyzed S 、R i 、R 0 、h D 、L D Material parameter p S 、E S 、ρ D 、E D V, boundary condition parameters
And connection coupling condition parameter k Du，0 、k Dtu，0 And substituting the parameters into the step 6) to obtain a dynamic model, and solving the dynamic model to obtain the natural frequency and the vibration mode of the system.
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