CN112270065B - Dynamic stability prediction method for eccentric rotary annular periodic structure - Google Patents
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Abstract
The invention discloses a dynamic stability prediction method of an eccentric rotary annular periodic structure. And the characteristic value is solved by adopting a classical vibration theory, and the modal characteristics and the instability of the annular periodic structure under different parameter combinations are analyzed. The invention well solves the problems of insufficient consideration of the actual working condition, complex simulation process and inaccurate analysis result prediction of the existing annular periodic structure vibration analysis technology, and provides references and references for parameter design and vibration control of the structure in engineering practice.
Description
Technical Field
The invention relates to the field of vibration of an annular periodic structure, in particular to a method for predicting dynamic stability of an eccentric rotary annular periodic structure.
Background
Various rotating parts are widely applied in engineering practice to realize the functions of transmission, driving, energy conversion and the like, such as gear transmission, rolling bearings, rotating motors and the like. The annular periodic structure used by the component is usually formed by a plurality of symmetrical units with the same configuration and geometric parameters through a circumferential array, such as a gear ring, an inner ring, an outer ring, a stator and a rotor, and the like. The rotationally symmetrical design is beneficial to realizing load balancing and improving structural stability. However, in engineering practice, such structures often exhibit eccentric rotation due to special requirements of the application on the form of movement, or unavoidable manufacturing and installation errors. Under high-speed working conditions, the eccentric rotation generates significant centrifugal force, thereby inducing vibration and noise, and simultaneously affecting the working efficiency. Therefore, the research is particularly important for the modal characteristics and dynamic stability of the rotary annular periodic structure under the influence of eccentric motion.
The literature (HASHEMI M, ASGHARI M. Analytical result of thread-dimensional flexural vibration of micro-rotating shafts with eccentricity utilizing the strain gradient theory [ J ]. Meccanica,2016,51 (6): 1435-1444.) investigated the three-dimensional flexural vibration of eccentric micro-shafts, and the natural frequency of the analytical form was obtained by the galy method, and the influence of mass eccentricity distribution on the vibration behavior was analyzed.
The literature (LIU T, ZHANG W, MAO J.Nonliean breathing vibrations of eccentric rotating composite laminated circular cylindrical shell subjected to temperature, rotating speed and external excitations [ J ]. Mechanical Systems and Signal Processing,2019,127 (15): 463-498.) adopts a multiscale method to study the nonlinear vibration of an eccentric rotary composite material laminated cylindrical shell, and reveals the rule of influence of geometrical parameters such as eccentricity on dynamic behavior.
It should be noted that in the prior art there are relatively few studies on the dynamics of the cyclic structure under eccentric motion. In addition, the prior art adopts a numerical method to predict the dynamic stability, and the method has lower calculation efficiency and cannot reveal the universality rule.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, and provides a power stability prediction method with high applicability, which is specially aimed at an annular periodic structure accounting for eccentric motion. To improve the current situation that the existing research only pays attention to the fixed shaft rotation and ignores the eccentric motion. Under actual working conditions, most annular periodic structures are limited by movement form requirements and manufacturing and installation errors, and ideal fixed-axis rotation is difficult to realize, and eccentric movement occurs. Aiming at the current situation, the invention takes the influence caused by eccentric motion into account, and utilizes the complete model to carry out dynamics analysis, thus obtaining the general guiding theory with wider application range and higher practical value.
The invention aims at realizing the following technical scheme:
a dynamic stability prediction method for an eccentric rotary annular periodic structure comprises the following steps:
(1) On the basis of using the additional mass to simplify the periodic distribution characteristic, the principle of infinitesimal method and superposition is adopted to calculate the internal force distribution of the annular structure during eccentric rotation;
(2) Establishing a dynamic model of the eccentric motion annular periodic structure under an inertial coordinate system by using a Hamilton principle;
(3) According to a classical vibration theory, obtaining an eccentric rotary annular periodic structure characteristic equation; and predicting the modal characteristics and dynamic stability rules of the annular periodic structure under different parameter combinations by solving the eigenvalues.
Further, the internal force profile includes radial and tangential internal force profiles, which are respectively:
further, the dynamic model of the annular periodic structure under eccentric rotation is specifically:
in the middle of
A 0 ＝Ω av +Ω v ，A 2 ＝Ω av Ω v ，
A 6 ＝1+Nm * ，/>
Wherein M is 1 、G 1 、K 1 、D 1 And F is an additional stiffness operator and an excitation operator generated by a quality operator, a gyro operator, a stiffness operator and a support reaction force respectively, and q is as follows 1 Is a displacement vector;A 0 ，A 1 ，A 2 ，A 3 ，A 4 ，A 5 ，A 6 ，f θ respectively refer toThe specific expression has no practical significance; tangential and radial displacements, R, R, of u and v, respectively Δ D, b, h, E and ρ represent the neutral radius, eccentric radius, eccentricity, axial thickness, radial thickness, young's modulus and density, respectively, of the ring; i (i=bh 3 12) is the section moment of inertia of the stator; n is the number of additional masses; omega shape v And omega av The rotation angular velocity and revolution angular velocity are dimensionless, respectively.
Further, a Galerkin method is used to calculate a characteristic equation, and the equation is as follows:
wherein M is 2 、G 2 、K 2 Respectively a mass matrix, a gyro matrix and a rigidity matrix, q 2 Is a feature vector.
Compared with the prior art, the technical scheme of the invention has the following beneficial effects:
1. according to the method, firstly, the infirm force distribution of the annular periodic structure is calculated by adopting a infirm method and a superposition principle, an analytic dynamics model under an inertial coordinate system is obtained by adopting a Hamilton principle and a Galerkin method, characteristic values are solved by adopting a classical vibration theory, and the inherent frequency and dynamic stability rules under different parameter combinations are predicted, so that thought reference and method reference are provided for vibration control of the structure.
2. The invention analyzes the centrifugal force of the annular structure under the eccentric motion based on the kinematics rule. And analyzing single stress situation by a infinitesimal method, and solving the internal force distribution of the annular structure under the action of uniform centrifugal force in an superposition method.
3. Under the inertial coordinate system, the invention derives the steady dynamics equation of the system under different parameter combinations by means of the special property of trigonometric function summation according to the Hamilton principle and the Galerkin method. Solving characteristic values by means of classical vibration theory, and analyzing the modal characteristics and dynamic stability rules of the annular periodic structure under different parameter combinations;
4. the invention has the characteristics of high efficiency, accuracy and universality. According to the technology, the relation between parameters such as eccentricity, periodic distribution characteristics, rotating speed and the like, modal characteristics and dynamic stability can be revealed, vibration conditions under actual working conditions can be estimated in a design stage, and unstable domain boundaries can be determined, so that poor parameter combinations are avoided, dynamic design of an annular periodic structure is guided, and the running stability and reliability of equipment are improved.
Drawings
FIG. 1 is a schematic view of an eccentric rotary annular periodic structure provided by the present invention;
FIGS. 2a and 2b are F obtained according to the method provided by the invention ef1 ，F ef2 The infinitesimal stress under the action;
fig. 3a to 3d show the variation law of natural frequency with rotation speed at low wave number obtained by the method according to the present invention;
fig. 4a and fig. 4b are rules of variation of the imaginary part of the eigenvalue with the rotation speed at high wave numbers obtained by the method provided by the invention;
fig. 5a to 5h are graphs showing the variation law of the imaginary part of the characteristic value of different mass ratios with the rotation speed obtained by the method according to the present invention;
FIGS. 6a to 6d show the variation law of the real part of the characteristic value of different mass ratios with the rotation speed obtained according to the method provided by the invention;
FIGS. 7a to 7d show the variation law of the unstable region with the rotation speed at different eccentricities obtained by the method according to the present invention;
Detailed Description
The invention is described in further detail below with reference to the drawings and the specific examples. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
The invention researches the influence of the periodic distribution characteristic and the eccentricity on the natural frequency and the dynamic stability on the premise of considering the eccentric motion aiming at the annular periodic structure widely applied in the engineering field. The internal force distribution of the structure is calculated by adopting a infinitesimal method and a superposition principle, and a dynamic model is established under an inertial system according to a Hamilton principle. And the characteristic value is solved by adopting a classical vibration theory, and the modal characteristics and the instability under different parameter combinations are analyzed. The result shows that when the periodic distribution characteristic and the wave number meet a certain relation, the natural frequency of the system is split; for varying eccentricity and periodic distribution characteristics, the system exhibits divergent or flutter instabilities at different rotational speeds. The research is helpful for analyzing the dynamic stability of the structure in engineering practice, and provides thought reference and method reference for vibration control. The specific implementation steps are as follows:
(S1) the mathematical model of the eccentric rotary annular periodic structure shown in FIG. 1 consists of a thin ring and N (N is more than or equal to 1) uniformly distributed additional mass blocksComposition is prepared. Wherein the additional mass is considered as particles to simplify the periodic distribution characteristics. The structure rotates about its geometric centroid o at an angular velocity Ω, while following a rigid trajectory at an angular velocity Ω a Revolve around the eccentric o'. o-rθz is a follow-up coordinate system, o '-rθz is an inertial coordinate system, and the initial moment connecting line o' o coincides with the polar axis. p is any point on the neutral circle, u and v are the tangential and radial displacements of the point, R, R, respectively Δ D, b, h, E and ρ represent the neutral radius, eccentric radius, eccentricity, axial thickness, radial thickness, young's modulus and density, respectively, of the ring.
In the inertial frame, the angular position of the first mass is assumed to be θ 1 =0, then the j-th mass is located at θ j =2pi (j-1)/N, and the distribution rule of the additional mass can be expressed as
Where δ (θ) is a dirac function.
According to the simple kinematic relationship, the rotation and revolution of the ring satisfy the following relationship
In a moving state, the rotation centrifugal force and the revolution centrifugal force are generated at the p position and are respectively
Will F r ' radial and tangential resolution of the ring, projection of centrifugal force in radial and tangential directions can be obtained
The centrifugal force at p can be expressed as
In addition, the counter force of the track to the ring is
In order to determine the internal force distribution caused by rotation and revolution centrifugal forces, the single radial force F of the ring is studied ef1 And tangential force F ef2 Internal force distribution upon application. Using a micro-element method, intercepting a micro-segment dθ at p for stress analysis, as shown in FIG. 2a and FIG. 2b, wherein F sf 、F tf And M bm Respectively represent the shearing force, the axial force and the bending moment of the micro-segment.
According to theory of mechanics, F ef2 Can be equivalently applied to F at the geometric centroid e ′ f2 With torque T uniformly distributed in circumferential direction tm2 . In the two stress situations shown in fig. 2a and 2b, uniform virtual forces are respectively distributed on the neutral line of the circular ringAndbalanced with external force, provided with
The virtual forces in fig. 2a and 2b are projected in normal and tangential directions while taking the moment about the center p of the micro-segment, in each case
Considering that the relation between tangential force and bending moment and radial deformation is satisfied
Where μ is poisson's ratio. At the same time, consider the boundary conditions of the stress deformation of the ring in two cases
According to equations (8), (10) and (11), the higher order trace is omitted, and the radial force F can be obtained by an operator method ef1 Distribution of internal forces of a ring under action
Similarly, according to formulas (9), (10) and (12),can obtain tangential force F ef2 Distribution of internal forces of a ring under action
The tangential internal force distribution caused by the radial force and tangential force respectively can be obtained by (13) and (14)
In summary, the tangential internal force of the ring can be expressed as
F θ ＝F θv +F θu +F sv (16)
F in the formula θv 、F θu And F sv Respectively represent tangential internal force distribution caused by radial and tangential components of centrifugal force and track-bearing counterforce, and has
(S2) according to FIG. 1, the position vector at any point p on the circle neutral can be expressed as
Absolute velocity v of point p in inertial coordinate system a Can be expressed as
v a ＝v e +v r (19)
In the formula, v e Velocity (relative velocity), v, of point p with respect to the follow-up coordinate system r Is the velocity (the involvement velocity) of the follower coordinate system relative to the inertial coordinate system, and is known
The kinetic energy of the ring can be expressed as
Considering the characteristics of eccentric motion, the potential energy of the ring includes strain energy caused by elastic vibration and strain energy caused by centrifugal force. Tangential strain at point p is in the plane strain state
ε θ ＝ε θ0 +(r-R)ε θ1 (21)
In the middle of
The strain energy of the ring can be expressed as
Wherein a and I are the area of the circular ring cross section (a=bh) and the main moment of inertia (i=bh, respectively 3 /12)。
(S3) according to the Hamilton principle, the kinetic equation is obtained:
in the middle of
A 0 ＝Ω av +Ω v ，A 2 ＝Ω av Ω v ，
Wherein M is 1 、G 1 、K 1 、D 1 And F is an additional stiffness operator and an excitation operator generated by a quality operator, a gyro operator, a stiffness operator and a support reaction force respectively, and q is as follows 1 Is a displacement vector;A 1 ，A 2 ，A 3 ，A 4 ，A 5 ，A 6 ，f θ respectively refer to specific expressions without practical significance; u and v are the tangential and radial displacements, R, R, respectively, of the point Δ D, b, h, E and ρ represent the neutral radius, eccentric radius, eccentricity, axial thickness, radial thickness, young's modulus and density, respectively, of the ring. I (i=bh 3 And/12) is the section moment of inertia of the stator. And N is the number of additional mass blocks. Omega shape v And omega av The rotation angular velocity and revolution angular velocity are dimensionless, respectively.
(S4) to solve the free vibration response, the tangential and radial vibration displacements are constructed by using Galerkin dispersion
Wherein i is an imaginary unit, "-" represents a complex conjugate operation, U (t) and V (t) are minimum residual unknown functions constructed in discrete operations, and are both complex functions of time, so that it is possible to define
Wherein x is u (t)、y u (t)、x v (t) and y v (t) are all real functions of time. Defining inner product operations
Where x and y are complex functions.
Then with e inθ Performing inner product and separating real and imaginary parts, and finishing to obtain
In the middle of
Wherein M is 2 、G 2 、K 2 Respectively a mass matrix, a gyro matrix and a rigidity matrix, q 2 Is a feature vector. B 0 ，B′ 0 ，B 1 ，B′ 1 ，B 2 ，B′ 2 Specific expressions are respectively referred to, and have no practical meaning.
According to the Euler formula and the trigonometric function arithmetic series summation characteristics, the method comprises the following steps of
To further simplify the kinetic equation, the effect of parameter combinations on the summation term was analyzed herein, with the specific results set forth in Table 1.
Table 1 sum term values under different parameter combinations
Equation (27), which is a kinetic differential equation of the eccentric rotary annular periodic structure, can be used to solve the eigenvalue λ by classical vibration force theory in combination with the parameters described in table 1 results and table 2, thereby predicting the dynamic stability of the structure. If not specified in the following, the eccentricities k=4/3, mass ratio m * ＝0.1。
TABLE 2 basic parameters of cyclic structures
Fig. 3a to 3d illustrate the change rule of the natural frequency of the annular periodic structure along with the revolution speed under the condition of low wave number, which respectively represent the imaginary parts of the first-order and second-order eigenvalues and the corresponding real parts of the eigenvalues, as shown in fig. 3a. The solid line and the dotted line are the forward traveling wave mode and the backward traveling wave mode respectively. It can be seen that at wavenumber 0, the system has instability problems and is only associated with first order vibrations. According to classical vibration theory, if the real part of the eigenvalue is greater than zero, the system will appear unstable, while when the corresponding imaginary part is zero, the divergence will appear unstable. In contrast, when the wave number is 1, the system has better stability, the forward wave and the backward wave of the imaginary part of the first-order eigenvalue are overlapped, and modal transition is generated between the forward wave and the second-order eigenvalue, as shown in the partial enlarged diagrams (1) and (2). It is noted that, in the case where 2N/N is an integer, the cyclic periodic structure is split in natural frequency at rest, and is more remarkable with an increase in the number of added masses.
For the case of high wavenumbers, fig. 4a and 4b depict the effect of rotational speed, wavenumber and additional mass numbers on natural frequency. Obviously, at different additional mass numbers, there is a positive correlation between the stability of the system and the wave number. Similarly, frequency splitting that occurs at rest occurs only in the case where 2N/N is an integer. Moreover, as the number of additional masses increases, the degree of splitting thereof becomes more pronounced.
For a typical case of natural frequency splitting at high wavenumbers, fig. 5a to 6d illustrate the effects of revolution speed, additional mass on the dynamics of the ring structure, taking the parameter combination of { n=2, n=4 } as an example. Fig. 5a shows the result of removing the additional mass, the rest being provided with different additional masses. It can be seen that the system is almost free of instability when the additional mass is removed, and the ring has better stability at medium and high rotational speeds. In contrast, the additional mass causes the system to be unstable in divergence at a low rotating speed, and as the additional mass is continuously increased, the backward wave mode of the first-order characteristic value is sequentially overlapped with the forward wave modes of the first-order characteristic value and the second-order characteristic value, a plurality of wave peaks appear in the real part of the characteristic value in the process of improving the revolution rotating speed of the circular ring, and the vibration instability appears for a plurality of times in succession under the influence of a softening effect.
Considering the influence of eccentricity on eccentric motion, fig. 7a to 7d describe the variation law of unstable regions of the annular periodic structure with eccentricity and revolution speed at different mass ratios. In combination with the natural frequency characteristics described above, it is known that the (1) domain in fig. 7a is the divergent unstable domain at low rotation speed, and the domains (2) and (3) are the flutter unstable domains at medium and high rotation speeds, respectively. It is not difficult to find that as the additional mass increases, the stable domain of the ring at low eccentricity continuously contracts, domain (2) gradually disappears, domain (1) and domain (3) are in contact with each other, and the stability of the system is significantly reduced. It is noted that when the eccentricity is continuously increased (k > 2, the center of revolution moves to the outside of the ring), the stiffening effect caused by revolution significantly affects the softening effect of rotation, and the system reappears a stabilization phenomenon at low rotation speeds and gradually spreads to a wider rotation speed range.
In general, at different rotational speeds, the annular periodic structure is subject to dispersion and flutter instability, the former generally occurring in the low rotational speed parameter domain and the latter often occurring in the medium to high rotational speed parameter domain of high mass ratio. There is a specific dependency between this phenomenon and the eccentricity and periodic distribution characteristic parameters, and there is a significant difference between the low wave number and high wave number cases. By increasing the eccentricity, the stress stiffening effect caused by revolution can significantly suppress instability, thereby improving system stability. The instability suppression method is easy to operate at low mass ratios, but relatively harsh conditions are achieved at high mass ratios.
In summary, the invention provides a method for predicting dynamic stability of an eccentric rotary annular periodic structure. The method fully considers the influence caused by eccentric motion, obtains the characteristic value of the system through an analysis method, improves the accuracy, the calculation efficiency and the universality, and better meets the actual requirements of engineering.
The invention is not limited to the embodiments described above. The above description of specific embodiments is intended to describe and illustrate the technical aspects of the present invention, and is intended to be illustrative only and not limiting. Numerous specific modifications can be made by those skilled in the art without departing from the spirit of the invention and scope of the claims, which are within the scope of the invention.
Claims (1)
1. The method for predicting the dynamic stability of the eccentric rotary annular periodic structure is characterized by comprising the following steps of:
(1) On the basis of using the characteristic of simplifying the periodic distribution of the additional mass, the principle of infinitesimal method and superposition is adopted to calculate the internal force distribution of the annular structure during eccentric rotation, wherein the internal force distribution comprises radial component force and tangential component force; the radial and tangential internal force distribution are respectively:
wherein f θv (θ) is the radial inner force distribution of the annular periodic structure, f θu (θ) is the tangential internal force distribution of the annular periodic structure;
(2) The principle of Hamilton is used for establishing a dynamic model of the eccentric motion annular periodic structure under an inertial coordinate system:wherein M is 1 、G 1 、K 1 、D 1 And F is respectively a mass matrix, a gyro matrix, a rigidity matrix, an additional rigidity matrix generated by a support reaction force and an excitation matrix, q 1 Is a displacement vector;
A 0 ＝Ω av +Ω v ，A 2 ＝Ω av Ω v ，
A 6 ＝1+Nm * ，/>K 1 (11) ，K 1 (12) ，K 1 (21) ，K 1 (22) ，A 0 ，A 1 ，A 2 ，A 3 ，A 4 ，A 5 ，A 6 ，f θ respectively refer to specific expressions without practical significance; tangential and radial displacements, R, R, of u and v, respectively Δ D, b, h, E and ρ represent the neutral radius, eccentric radius, eccentricity, axial thickness, radial thickness, young's modulus and density, respectively, of the ring; i (i=bh 3 12) is the section moment of inertia of the stator; n is the number of additional masses; omega shape v And omega av The rotation angular velocity and revolution angular velocity are dimensionless respectively;
(3) According to classicalAnd (3) obtaining a characteristic equation of the eccentric rotary annular periodic structure according to the vibration theory:wherein M is 2 、G 2 、K 2 Respectively a mass matrix, a gyro matrix and a rigidity matrix, q 2 Is a feature vector; predicting the modal characteristics and dynamic stability rules of the annular periodic structure under different parameter combinations;
B 0 ，B′ 0 ，B 1 ，B′ 1 ，B 2 ，B′ 2 specific expressions are respectively referred to, and have no practical meaning.
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