CN112270065B - Dynamic stability prediction method for eccentric rotary annular periodic structure - Google Patents

Abstract

The invention discloses a method for predicting the dynamic stability of an eccentric rotating annular periodic structure. On the premise of considering the eccentric motion, the invention uses the micro-element method and the superposition principle to calculate the internal force distribution of the structure, and obtains the distribution of the internal force through the Hamilton principle and the Galerkin method. Analytical dynamics model in inertial coordinate system. Classical vibration theory is used to solve the eigenvalues, and the modal characteristics and instability of the ring periodic structure under different parameter combinations are analyzed. The invention solves the problems of insufficient consideration of actual working conditions, redundant and complex simulation process, and inaccurate prediction of analytical results in the existing annular periodic structure vibration analysis technology, and provides a solution for the parameter design and vibration control of this type of structure in actual engineering. Reference and reference.

Description

A method for predicting dynamic stability of eccentrically rotating annular periodic structures

Technical field

The present invention relates to the field of vibration of annular periodic structures, and is particularly directed to a method for predicting the dynamic stability of an eccentrically rotating annular periodic structure.

Background technique

Various types of rotating components are widely used in engineering practice to achieve functions such as transmission, drive and energy conversion, such as gear transmission, rolling bearings and rotating motors. The annular periodic structure used in this type of component is usually formed by a circular array of several symmetrical units with the same configuration and geometric parameters, such as ring gears, inner and outer rings, stator and rotor, etc. This rotationally symmetrical design is conducive to achieving load balance and improving structural stability. However, in engineering practice, due to the special requirements for the form of motion in the application, or unavoidable manufacturing and installation errors, this type of structure usually exhibits an eccentric rotation state. Under high-speed working conditions, eccentric rotation will generate significant centrifugal force, causing vibration and noise, and affecting work efficiency. Therefore, it is particularly important to study the modal characteristics and dynamic stability of this type of rotating annular periodic structure under the influence of eccentric motion.

The literature (HASHEMI M, ASGHARI M. Analytical study of three-dimensional flexural vibration of micro-rotating shafts with eccentricity utilizing thestrain gradient theory [J]. Meccanica, 2016, 51 (6): 1435-1444.) studied the eccentric micro-rotating shafts For three-dimensional bending vibration, the natural frequency in analytical form was obtained using Galerkin method, and the influence of mass eccentric distribution on vibration behavior was analyzed.

Literature (LIU T, ZHANG W, MAO J. Nonlinear breathing vibrations of eccentricrotating composite laminated circular cylindrical shell subjected to temperature, rotating speed and external excitations[J]. Mechanical Systems and Signal Processing, 2019, 127(15): 463-498.) The multi-scale method was used to study the nonlinear vibration of an eccentrically rotating composite laminated cylindrical shell, revealing the influence of geometric parameters such as eccentricity on the dynamic behavior.

It should be noted that there are relatively few studies on the dynamics of ring-shaped periodic structures under eccentric motion in the existing literature. In addition, the existing technology uses numerical methods to predict dynamic stability, which has low computational efficiency and cannot reveal universal laws.

Contents of the invention

The purpose of the present invention is to overcome the deficiencies in the prior art and provide a dynamic stability prediction method with strong applicability, specifically for annular periodic structures that take eccentric motion into account. In order to improve the current situation that existing research only focuses on fixed-axis rotation and ignores eccentric motion. Under actual working conditions, most annular periodic structures are limited by motion form requirements and manufacturing and installation errors, making it difficult to achieve ideal fixed-axis rotation and causing eccentric motion. In response to this current situation, the present invention takes into account the impact of eccentric motion, uses a complete model to conduct dynamic analysis, and obtains a general guiding theory with wide application and high practical value.

The purpose of the present invention is achieved through the following technical solutions:

A method for predicting dynamic stability of eccentrically rotating annular periodic structures, including the following steps:

(1) On the basis of using additional mass to simplify the periodic distribution characteristics, the microelement method and the superposition principle are used to calculate the internal force distribution of the ring structure during eccentric rotation;

(2) Use Hamilton's principle to establish a dynamic model of the eccentric motion ring periodic structure in the inertial coordinate system;

(3) According to the classical vibration theory, the characteristic equation of the eccentric rotating annular periodic structure is obtained; by solving the characteristic values, the modal characteristics and dynamic stability rules of the annular periodic structure under different parameter combinations are predicted.

Further, the internal force distribution includes radial and tangential internal force distribution. The radial and tangential internal force distributions are respectively:

Furthermore, the dynamic model of the annular periodic structure under eccentric rotation is specifically:

In the formula

A ₀ =Ω _av +Ω _v , A ₂ =Ω _av Ω _v ,

A ₆ =1+Nm ^* ,/>

In the formula, M ₁ , G ₁ , K ₁ , D ₁ and F are the mass operator, gyro operator, stiffness operator, additional stiffness operator generated by the support reaction force and excitation operator respectively, q ₁ is the displacement vector; A ₀ , A ₁ , A ₂ , A ₃ , A ₄ , A ₅ , A ₆ , f _θ respectively refer to specific expressions and have no practical meaning; u and v are respectively the tangential and radial displacement, R, R _Δ , d, b, h, E and ρ represent the neutral circle radius, eccentric radius, eccentricity, axial thickness, radial thickness, Young's modulus and density of the ring respectively; I (I＝bh ³ /12 ) is the cross-sectional moment of inertia of the stator; N is the number of additional mass blocks; Ω _v and Ω _av are the dimensionless rotation angular velocity and revolution angular velocity respectively.

Further, the Galerkin method is used to obtain the characteristic equation, which is as follows:

In the formula, M ₂ , G ₂ , and K ₂ are the mass matrix, gyro matrix, and stiffness matrix respectively, and q ₂ is the eigenvector.

Compared with the existing technology, the beneficial effects brought by the technical solution of the present invention are:

1. This method first uses the microelement method and the superposition principle to calculate the internal force distribution of the annular periodic structure, obtains the analytical dynamic model in the inertial coordinate system through the Hamilton principle and the Galerkin method, and uses the classical vibration theory to solve the eigenvalues and predict The natural frequency and dynamic stability rules under different parameter combinations are clarified, which provides an idea reference and method reference for the vibration control of this type of structure.

2. Based on the laws of kinematics, the present invention analyzes the centrifugal force of the annular structure under eccentric motion. A single force situation is analyzed by the microelement method, and the internal force distribution of the ring structure under the action of uniform centrifugal force is calculated by the superposition method.

3. In the inertial coordinate system, based on Hamilton's principle and Galerkin's method, and with the help of the special properties of trigonometric function summation, the present invention derives the steady dynamic equations of the system under different parameter combinations. Using classical vibration theory to solve the eigenvalues, the modal characteristics and dynamic stability rules of the annular periodic structure under different parameter combinations are analyzed;

4. The present invention is efficient, accurate and universal. This technology can reveal the relationship between parameters such as eccentricity, periodic distribution characteristics and rotational speed, modal characteristics and dynamic stability, and enable the prediction of vibration conditions under actual working conditions during the design stage and determine the boundary of the unstable domain, thereby avoiding Poor parameter combination to guide the dynamic design of the ring periodic structure and improve the stability and reliability of equipment operation.

Description of the drawings

Figure 1 is a schematic diagram of the eccentric rotating annular periodic structure provided by the present invention;

Figure 2a and Figure 2b are F _ef1 obtained according to the method provided by the present invention, and the force on the micro-element under the action of F _ef2 ;

Figures 3a to 3d show the variation of the natural frequency with the rotational speed at low wave numbers obtained according to the method provided by the present invention;

Figures 4a and 4b show the variation pattern of the imaginary part of the characteristic value with the rotation speed at high wave numbers obtained according to the method provided by the present invention;

Figures 5a to 5h show how the imaginary parts of different mass ratio characteristic values obtained according to the method provided by the present invention change with the rotation speed;

Figures 6a to 6d show how the real parts of different mass ratio characteristic values obtained according to the method provided by the present invention change with the rotational speed;

Figures 7a to 7d show the variation of the unstable domain with the rotational speed under different eccentricities obtained according to the method provided by the present invention;

Detailed ways

The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described here are only used to explain the present invention and are not intended to limit the present invention.

Aiming at the annular periodic structure widely used in the engineering field, the present invention studies the influence of periodic distribution characteristics and eccentricity on natural frequency and dynamic stability, taking into account eccentric motion. The microelement method and the superposition principle were used to calculate the internal force distribution of the structure, and a dynamic model was established in the inertial frame based on Hamilton's principle. Classical vibration theory is used to solve the eigenvalues, and the modal characteristics and instability under different parameter combinations are analyzed. The results show that when the periodic distribution characteristics and wave number satisfy a certain relationship, the natural frequency of the system splits; for changing eccentricity and periodic distribution characteristics, the system exhibits divergence or flutter instability at different rotational speeds. This research helps analyze the dynamic stability of this type of structure in actual engineering, and provides a reference for ideas and methods for its vibration control. The specific implementation steps are as follows:

(S1) The mathematical model of the eccentric rotating annular periodic structure shown in Figure 1 is composed of a thin ring and N (N≥1) uniformly distributed additional mass blocks. composition. Among them, the additional mass block is regarded as a particle to simplify the periodic distribution characteristics. The structure rotates around its geometric center o at an angular velocity Ω, and simultaneously revolves along a rigid orbit around the eccentricity o′ at an angular velocity Ω _a . o-rθz is the following coordinate system, o′-rθz is the inertial coordinate system, and the center line o′o coincides with the polar axis at the initial moment. p is any point on the neutral circle, u and v are the tangential and radial displacements of the point respectively, R, R _Δ , d, b, h, E and ρ represent the neutral circle radius and eccentricity of the ring respectively. Radius, eccentricity, axial thickness, radial thickness, Young's modulus and density.

In the inertial coordinate system, assuming that the angular coordinate of the first mass block is θ ₁ =0, then the jth mass block is located at θ _j =2π(j-1)/N, and the distribution law of the additional mass can be expressed as

where δ(θ) is the Dirac function.

According to the simple kinematics relationship, the rotation and revolution of the ring satisfy the following relationship

In the state of motion, the rotational centrifugal force and the revolutional centrifugal force will be generated at p, respectively:

By decomposing F _r ′ along the radial and tangential directions of the ring, the projection of the centrifugal force in the radial and tangential directions can be obtained

Therefore, the centrifugal force at position p can be expressed as

In addition, the reaction force exerted by the orbit on the ring is

In order to determine the internal force distribution caused by the centrifugal force of rotation and revolution, the internal force distribution when the ring is acted upon by a single radial force F _ef1 and a tangential force F _ef2 is first studied. Using the micro-element method, the micro-segment dθ at p is intercepted for stress analysis, as shown in Figure 2a and Figure 2b, where F _sf , F _tf and M _bm respectively represent the shear force, axial force and bending moment of the micro-segment.

According to theoretical mechanics, F _ef2 can be equivalent to F _e ′ _f2 applied at the geometric centroid and the torque T _tm2 uniformly distributed in the circumferential direction. In the two stress situations shown in Figure 2a and Figure 2b, uniform virtual forces are distributed on the neutral line of the ring. and Balanced with external forces, there is

Project the virtual force in Figure 2a and Figure 2b along the normal and tangential directions, and at the same time take the moment about the micro-segment center p. In the two cases, there are

Considering that the relationship between tangential force, bending moment and radial deformation satisfies

where μ is Poisson's ratio. At the same time, considering the boundary conditions of the force deformation of the ring in the two situations

According to formulas (8), (10) and (11), the internal force distribution of the ring under the action of the radial force F _ef1 can be solved by using the operator method by omitting the high-order traces.

Similarly, according to equations (9), (10) and (12), the internal force distribution of the ring under the action of tangential force F _ef2 can be obtained

From equations (13) and (14), we can get the distribution of tangential internal forces caused by radial force and tangential force respectively.

To sum up, the tangential internal force of the ring can be expressed as

F _θ ＝F _θv +F _θu +F _sv (16)

In the formula, F _θv , F _θu and F _sv respectively represent the tangential internal force distribution caused by the radial and tangential components of the centrifugal force and the orbital reaction force, and have

(S2) According to Figure 1, the position vector at any point p on the neutral circle of the ring can be expressed as

The absolute velocity v _a of point p in the inertial coordinate system can be expressed as

v _a = v _e + v _r (19)

In the formula, v _e is the speed of point p relative to the following coordinate system (relative speed), v _r is the speed of the following coordinate system relative to the inertial coordinate system (involved speed), and we know

Then the kinetic energy of the ring can be expressed as

Considering the characteristics of eccentric motion, the potential energy of the ring includes the strain energy caused by elastic vibration and the strain energy caused by centrifugal force. In the plane strain state, the tangential strain at point p is

ε _θ =ε _θ0 +(rR)ε _θ1 (21)

In the formula

Then the strain energy of the ring can be expressed as

In the formula, A and I are the area of the circular ring section (A=bh) and the principal moment of inertia (I=bh ³ /12) respectively.

(S3) According to Hamilton’s principle, the kinetic equation can be obtained:

In the formula

A ₀ =Ω _av +Ω _v , A ₂ =Ω _av Ω _v ,

In the formula, M ₁ , G ₁ , K ₁ , D ₁ and F are the mass operator, gyro operator, stiffness operator, additional stiffness operator generated by the support reaction force and excitation operator respectively, q ₁ is the displacement vector; A ₁ , A ₂ , A ₃ , A ₄ , A ₅ , A ₆ and f _θ respectively refer to specific expressions and have no practical meaning; u and v are the tangential and radial displacements of the point respectively, R and R _Δ , d, b, h, E and ρ represent the neutral circle radius, eccentric radius, eccentricity, axial thickness, radial thickness, Young's modulus and density of the ring respectively. I (I=bh ³ /12) is the cross-sectional moment of inertia of the stator. N is the number of additional mass blocks. Ω _v and Ω _av are the dimensionless rotation angular velocity and revolution angular velocity respectively.

(S4) In order to solve the free vibration response, Galerkin discretization is used to construct the tangential and radial vibration displacements.

In the formula, i is the imaginary unit, "~" represents the complex conjugate operation, U(t) and V(t) are the unknown minimum residual force functions constructed in the discrete operation, and they are both complex functions of time, so they can be defined

In the formula, x _u (t), y _u (t), x _v (t) and y _v (t) are all real functions of time. Define inner product operation

In the formula, x and y are both complex functions.

Then do the inner product with e ^inθ and separate the real and imaginary parts. After sorting, we can get

In the formula

In the formula, M ₂ , G ₂ , and K ₂ are the mass matrix, gyro matrix, and stiffness matrix respectively, and q ₂ is the eigenvector. B ₀ , B′ ₀ , B ₁ , B′ ₁ , B ₂ and B′ ₂ respectively refer to specific expressions and have no practical meaning.

According to Euler's formula and the summation characteristics of arithmetic sequence of trigonometric functions, we have

In order to further simplify the dynamic equation, this paper analyzes the influence of parameter combination on the summation term. The specific results are shown in Table 1.

Table 1 Values of the summation term under different parameter combinations

Equation (27) is the dynamic differential equation of the eccentric rotating annular periodic structure. Combining the results in Table 1 and the parameters described in Table 2, the classical vibration force theory can be used to solve the characteristic value λ, thereby predicting the dynamic stability of the structure. If there is no special explanation below, the eccentricity k=4/3 and the mass ratio m ^* =0.1 are assumed.

Table 2 Basic parameters of ring periodic structure

Figures 3a to 3d describe the change of the natural frequency of the annular periodic structure with the revolution speed under low wave number conditions, respectively representing the imaginary part of the first-order and second-order eigenvalues and the corresponding real part of the eigenvalue, as shown in Figure 3a. The solid lines and dashed lines represent the forward and backward traveling wave modes respectively. It can be seen that when the wave number is 0, the system has instability problems and is only related to first-order vibration. According to the classical vibration theory, if the real part of the eigenvalue is greater than zero, the system will be unstable, and when the corresponding imaginary part is zero, divergent instability will occur. In contrast, when the wave number is 1, the system has better stability. The modes of the forward and backward traveling waves in the imaginary part of the first-order eigenvalue coincide with each other, and a mode transition occurs between them and the second-order eigenvalue. , as shown in the partial enlarged pictures ① and ②. It is worth noting that for the case where 2n/N is an integer, the natural frequency splitting of the ring periodic structure occurs at rest, and it becomes more significant as the additional mass number increases.

For the case of high wave numbers, Figure 4a and Figure 4b describe the effects of rotational speed, wave number and additional mass on the natural frequency. Obviously, there is a positive correlation between the stability of the system and the wave number under different additional mass numbers. Similarly, frequency splitting that occurs at rest only occurs when 2n/N is an integer. Moreover, as the additional mass number increases, the degree of fragmentation becomes more significant.

In view of the typical situation of natural frequency splitting at high wave numbers, Figures 5a to 6d take the parameter combination {n=2, N=4} as an example to describe the effects of revolution speed and additional mass on the dynamic characteristics of the ring structure. Figure 5a shows the result of removing the additional mass, and the rest have different additional masses. It can be seen that when the additional mass is removed, there is almost no instability in the system, and the ring has good stability at medium and high speeds. In contrast, the additional mass causes the system to exhibit divergent instability at low rotational speeds, and as the additional mass continues to increase, the backward traveling wave mode of the first-order eigenvalue and the forward-moving wave mode of the first-order and second-order eigenvalues successively The wave modes overlap, and the real part of the eigenvalue appears with multiple wave peaks as the ring revolution speed increases. Under the influence of the softening effect, multiple flutter instabilities appear one after another.

Considering the influence of eccentricity on eccentric motion, Figures 7a to 7d describe the variation of the unstable domain of the annular periodic structure with the eccentricity and revolution speed under different mass ratios. Combined with the natural frequency characteristics described previously, it can be seen that domain ① in Figure 7a is the divergence unstable domain at low speeds, while domains ② and ③ are flutter unstable domains at medium and high speeds respectively. It is not difficult to find that as the added mass increases, the stable domain of the ring at low eccentricity continues to shrink, domain ② gradually disappears, domain ① and domain ③ contact each other, and the stability of the system decreases significantly. It is worth noting that when the eccentricity continues to increase (k>2, the center of the revolution circle moves to the outside of the ring), the stiffening effect caused by the revolution significantly affects the softening effect of the rotation, and the system reappears stable at low speeds. And gradually spread to a wider speed range.

In general, at different rotational speeds, the annular periodic structure has divergence and flutter instability. The former generally appears in the low-speed parameter domain, while the latter often appears in the medium-to-high rotational speed parameter domain with a high mass ratio. There is a specific dependence between this phenomenon and the characteristic parameters of eccentricity and periodic distribution, and there are clear differences between low and high wavenumber cases. By increasing the eccentricity, the stress stiffening effect caused by revolution can significantly suppress instability, thereby improving system stability. This instability suppression method is easy to operate at low mass ratios, but the implementation conditions are relatively harsh at high mass ratios.

To sum up, the present invention provides a method for predicting the dynamic stability of an eccentrically rotating annular periodic structure. This method fully considers the impact of eccentric motion, obtains the eigenvalues of the system through analytical methods, improves accuracy, calculation efficiency and universality, and better meets the actual needs of engineering.

The invention is not limited to the embodiments described above. The above description of the specific embodiments is intended to describe and illustrate the technical solution of the present invention. The above specific embodiments are only illustrative and not restrictive. Without departing from the spirit of the present invention and the scope protected by the claims, those of ordinary skill in the art can make many specific changes based on the inspiration of the present invention, and these all fall within the protection scope of the present invention.

Claims (1)

1. A method for predicting the dynamic stability of an eccentrically rotating annular periodic structure, which is characterized by including the following steps: (1) On the basis of using additional mass to simplify the periodic distribution characteristics, the microelement method and the superposition principle are used to calculate the internal force distribution of the ring structure during eccentric rotation. The internal force distribution includes radial components and tangential components; the radial sum The tangential internal force distributions are: In the formula, f _θv (θ) is the radial internal force distribution of the annular periodic structure, and f _θu (θ) is the tangential internal force distribution of the annular periodic structure; (2) Use Hamilton's principle to establish a dynamic model of the eccentric motion ring periodic structure in the inertial coordinate system: In the formula, M ₁ , G ₁ , K ₁ , D ₁ and F are the mass matrix, gyro matrix, stiffness matrix, additional stiffness matrix generated by the support reaction force and the excitation matrix respectively, q ₁ is the 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 refer to respectively The specific expression has no practical meaning; u and v are the tangential and radial displacements of respectively, R, R _Δ , d, b, h, E and ρ respectively represent the neutral circle radius, eccentric radius and eccentric distance of the ring. , axial thickness, radial thickness, Young's modulus and density; I (I=bh ³ /12) is the cross-sectional moment of inertia of the stator; N is the number of additional mass blocks; Ω _v and Ω _av are dimensionless respectively. Rotation angular velocity and revolution angular velocity; (3) According to the classical vibration theory, the characteristic equation of the eccentric rotating ring periodic structure is obtained: In the formula, M ₂ , G ₂ , K ₂ are the mass matrix, gyro matrix, and stiffness matrix respectively, and q ₂ is the eigenvector; predict the modal characteristics and dynamic stability rules of the ring periodic structure under different parameter combinations; B ₀ , B′ ₀ , B ₁ , B′ ₁ , B ₂ and B′ ₂ respectively refer to specific expressions and have no practical meaning.