CN106547957B - A parametric elastic vibration analysis method for rotating annular periodic structures - Google Patents
A parametric elastic vibration analysis method for rotating annular periodic structures Download PDFInfo
- Publication number
- CN106547957B CN106547957B CN201610911161.5A CN201610911161A CN106547957B CN 106547957 B CN106547957 B CN 106547957B CN 201610911161 A CN201610911161 A CN 201610911161A CN 106547957 B CN106547957 B CN 106547957B
- Authority
- CN
- China
- Prior art keywords
- equation
- rotating
- ring
- periodic structure
- parametric
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
- 230000000737 periodic effect Effects 0.000 title claims abstract description 46
- 238000004458 analytical method Methods 0.000 title claims abstract description 28
- 239000011159 matrix material Substances 0.000 claims abstract description 32
- 238000000034 method Methods 0.000 claims abstract description 24
- 230000004044 response Effects 0.000 claims abstract description 11
- 230000003068 static effect Effects 0.000 claims description 12
- 230000009466 transformation Effects 0.000 claims description 12
- 230000008878 coupling Effects 0.000 claims description 5
- 238000010168 coupling process Methods 0.000 claims description 5
- 238000005859 coupling reaction Methods 0.000 claims description 5
- 230000005284 excitation Effects 0.000 claims description 2
- 238000012545 processing Methods 0.000 claims description 2
- 238000011426 transformation method Methods 0.000 claims description 2
- 230000008569 process Effects 0.000 abstract description 2
- 241000208340 Araliaceae Species 0.000 abstract 3
- 235000005035 Panax pseudoginseng ssp. pseudoginseng Nutrition 0.000 abstract 3
- 235000003140 Panax quinquefolius Nutrition 0.000 abstract 3
- 235000008434 ginseng Nutrition 0.000 abstract 3
- 238000004364 calculation method Methods 0.000 description 7
- 238000005516 engineering process Methods 0.000 description 5
- 238000010586 diagram Methods 0.000 description 4
- 230000003071 parasitic effect Effects 0.000 description 4
- 230000006399 behavior Effects 0.000 description 2
- 230000005540 biological transmission Effects 0.000 description 2
- 238000013461 design Methods 0.000 description 2
- 238000012986 modification Methods 0.000 description 2
- 230000004048 modification Effects 0.000 description 2
- 238000005312 nonlinear dynamic Methods 0.000 description 2
- 238000003672 processing method Methods 0.000 description 2
- 238000005096 rolling process Methods 0.000 description 2
- 230000008859 change Effects 0.000 description 1
- 125000004122 cyclic group Chemical group 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 238000009987 spinning Methods 0.000 description 1
- 238000006467 substitution reaction Methods 0.000 description 1
- 238000012795 verification Methods 0.000 description 1
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/10—Geometric CAD
- G06F30/17—Mechanical parametric or variational design
Landscapes
- Physics & Mathematics (AREA)
- Geometry (AREA)
- Engineering & Computer Science (AREA)
- Theoretical Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Pure & Applied Mathematics (AREA)
- Mathematical Optimization (AREA)
- Mathematical Analysis (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- General Engineering & Computer Science (AREA)
- Computational Mathematics (AREA)
- Testing Of Devices, Machine Parts, Or Other Structures Thereof (AREA)
Abstract
Description
技术领域technical field
本发明涉及一种振动分析方法。特别是涉及一种旋转环状周期结构参激弹性振动分析方法。The invention relates to a vibration analysis method. In particular, it relates to a parametric elastic vibration analysis method of a rotating annular periodic structure.
背景技术Background technique
旋转环状周期结构系统广泛存在于现代工业中，例如行星齿轮传动系统齿圈、旋转电机的环形定转子、汽轮机叶片和滚动轴承内外圈等。该类结构通常由圆环和旋转载荷组成。在旋转载荷作用下，环状结构将产生振动和噪声问题，尤其对于高速、重载的工作场合。在各类振动行为中，参激振动备受关注。但针对其模态特性和动力稳定性问题的传统分析技术计算效率和精度低下，对高效率和高精度分析技术的需求较为迫切。Rotary annular periodic structure systems widely exist in modern industry, such as planetary gear transmission system ring gear, annular stator and rotor of rotating electrical machines, steam turbine blades, inner and outer rings of rolling bearings, etc. This type of structure usually consists of a ring and a rotating load. Under rotating loads, the ring structure will cause vibration and noise problems, especially for high-speed, heavy-duty work. Among all kinds of vibration behaviors, parasitic vibration has attracted much attention. However, the traditional analysis techniques for its modal characteristics and dynamic stability problems have low computational efficiency and accuracy, and there is an urgent need for high-efficiency and high-precision analysis techniques.
发明人认为，含移动支撑的周期结构本质上为参激振动系统。由于经典振动理论无法对参激系统直接进行解析求解(胡海岩.应用非线性动力学，北京：航空工业出版社，2000)，因此多通过引入小参数，利用近似处理的思想展开分析。现有技术(Canchi S V,Parker R G.Parametric instability of a rotating circular ring with moving,time-varying springs,ASME Journal of Vibration and Acoustics,2006,128(2):231-243；Zhao Z F,Wang S Y.Parametric instability induced by traveling magneticload within permanent magnet motors,Nonlinear Dynamics,2015,80(1):827–843)采用多尺度法分别分析了旋转环状周期结构和永磁电机的参激振动问题，揭示了参激不稳定规律，并给出了数值验证。发明人还采用摄动和模态叠加技术探讨了参激系统的振型耦合与组合不稳定之间的关系(Wang S Y,Sun W J,Wang Y Y.Instantaneous modecontamination and parametric combination instability of spinning cyclicsymmetric ring structures with expanding application to planetary gear ring,Journal of Sound and Vibration,2016,375:366–385)。应当指出，上述分析都是在全局静止坐标系下直接进行的。然而，这些技术通常适用于小参数系统，对于旋转支撑刚度较大的情形不再适用。现有文献还通常采用数值计算方法预测参激振动规律，但计算效率较低，且不利于揭示普适规律。The inventors believe that a periodic structure with moving supports is essentially a parasitic vibration system. Since the classical vibration theory cannot directly solve the parametric system analytically (Hu Haiyan. Applied Nonlinear Dynamics, Beijing: Aviation Industry Press, 2000), the analysis is carried out mostly by introducing small parameters and using the idea of approximation. Prior art (Canchi S V, Parker R G. Parametric instability of a rotating circular ring with moving, time-varying springs, ASME Journal of Vibration and Acoustics, 2006, 128(2): 231-243; Zhao Z F, Wang S Y .Parametric instability induced by traveling magnetic load within permanent magnet motors, Nonlinear Dynamics, 2015, 80(1): 827–843) used the multi-scale method to analyze the parametric vibration of the rotating annular periodic structure and the permanent magnet motor respectively, and revealed that The parametric instability law is given, and the numerical verification is given. The inventors also used perturbation and modal superposition techniques to explore the relationship between the mode coupling and the combined instability of the parametric system (Wang S Y, Sun W J, Wang Y Y. Instantaneous modecontamination and parametric combination instability of spinning cyclic symmetric ring structures. with expanding application to planetary gear ring, Journal of Sound and Vibration, 2016, 375:366–385). It should be pointed out that the above analysis is carried out directly in the global stationary coordinate system. However, these techniques are usually suitable for small parameter systems and are no longer applicable for the case of large rotational support stiffness. In the existing literature, numerical calculation methods are usually used to predict the law of parasitic vibration, but the calculation efficiency is low, and it is not conducive to revealing the universal law.
由于旋转环状周期结构的薄圆环假设，因此圆环会呈现出包括径向和切向弹性参激振动的特点，在全局静止坐标系下得到的系统的动力学方程会含有时变的参数项。由于经典振动理论只能解析分析常微分方程，因此现有技术不能直接解析求解其动力学方程。Due to the thin ring assumption of the rotating annular periodic structure, the ring will exhibit the characteristics of radial and tangential elastic parametric vibrations, and the dynamic equation of the system obtained in the global static coordinate system will contain time-varying parameters item. Since classical vibration theory can only analytically analyze ordinary differential equations, the existing technology cannot directly solve its dynamic equations analytically.
发明内容SUMMARY OF THE INVENTION
本发明所要解决的技术问题是，提供一种可以进行系统模态特性的计算求解，也可以进行系统的动力稳定性和动态响应分析的旋转环状周期结构参激弹性振动分析方法。The technical problem to be solved by the present invention is to provide a parametric elastic vibration analysis method of a rotating annular periodic structure that can calculate and solve the modal characteristics of the system as well as analyze the dynamic stability and dynamic response of the system.
本发明所采用的技术方案是：一种旋转环状周期结构参激弹性振动分析方法，所述旋转环状周期结构由薄圆环和离散旋转支撑弹簧组成，是采用坐标变换方法实现参激系统动力稳定性的解析分析，具体包括以下步骤：The technical scheme adopted in the present invention is: a method for analyzing parametrically excited elastic vibration of a rotating annular periodic structure, wherein the rotating annular periodic structure is composed of a thin ring and a discrete rotating support spring, and a coordinate transformation method is used to realize the parametric excitation system. The analytical analysis of dynamic stability includes the following steps:
1)在全局静止坐标系o-rθz下，根据哈密顿原理建立旋转环状周期结构的刚-弹耦合动力学模型：1) In the global static coordinate system o-rθz, the rigid-elastic coupling dynamic model of the rotating ring periodic structure is established according to the Hamiltonian principle:
式中：where:
β为旋转支撑的方向角；β is the direction angle of the rotating support;
θ为表示旋转支撑位置角的一个空间函数；θ is a space function representing the position angle of the rotating support;
kt为圆环外侧均布切向静止支撑刚度；k t is the uniformly distributed tangential static support stiffness outside the ring;
kr为圆环外侧均布径向静止支撑刚度；k r is the uniformly distributed radial static support stiffness outside the ring;
θj＝2π(j–1)/N，表示第j个旋转支撑的初始位置，N为总的旋转支撑个数；θ j = 2π(j–1)/N, indicating the initial position of the j-th rotating support, and N is the total number of rotating supports;
Ω为旋转支撑的转速；Ω is the rotational speed of the rotating support;
t表示时间；t represents time;
cz＝I/(AR2)为人为引入的一个运算符；c z =I/(AR 2 ) is an operator introduced artificially;
I＝bh3/12为圆环截面惯性矩；I=bh 3 /12 is the moment of inertia of the ring section;
A＝bh为圆环截面面积；A=bh is the cross-sectional area of the ring;
R为圆环中心圆半径；R is the radius of the center circle of the ring;
b为圆环的径向厚度；b is the radial thickness of the ring;
h为圆环的轴向高度；h is the axial height of the ring;
ks为旋转支撑刚度；k s is the rotational support stiffness;
2)引入坐标变换将步骤1)所述动力学模型转换到支撑随动坐标系下，从而消除原方程中的参激项，得到偏微分常系数动力学方程：2) Introduce coordinate transformation Convert the dynamic model described in step 1) to the support-following coordinate system , thereby eliminating the parametric term in the original equation, and obtaining the partial differential constant coefficient dynamic equation:
(M′+K′0+K′1)q＝0；(M'+K' 0 +K' 1 )q=0;
式中：where:
3)采用伽辽金方法对旋转支撑随动坐标系下的偏微分常系数动力学方程进行离散处理，得到常微分矩阵方程：3) Using the Galerkin method for the rotation support follow-up coordinate system The partial differential constant coefficient dynamic equation below is discretized, and the ordinary differential matrix equation is obtained:
式中：where:
4)利用经典振动理论，解析分析步骤3)所述的常微分矩阵方程的特征值；4) utilize classical vibration theory, analytically analyze the eigenvalues of the ordinary differential matrix equation described in step 3);
5)分别利用步骤4)得到的常微分矩阵方程的特征值的虚部和实部对旋转环状周期结构的模态特性和参激弹性振动的动力稳定性规律进行分析；5) using the imaginary part and the real part of the eigenvalue of the ordinary differential matrix equation obtained in step 4) to analyze the modal characteristics of the rotating annular periodic structure and the dynamic stability law of parametric elastic vibration;
步骤3)所述的离散处理方式如下：The discrete processing method described in step 3) is as follows:
首先对偏微分常系数动力学方程的响应u和v设解First, let the responses u and v of the partial differential constant coefficient dynamic equation be solved
式中：where:
U(t)＝x1(t)+iy1(t)和V(t)＝x2(t)+iy2(t)；i为虚数单位，“～”表示复共轭；U(t)=x 1 (t)+iy 1 (t) and V(t)=x 2 (t)+iy 2 (t); i is an imaginary unit, and "~" represents a complex conjugate;
定义内积：Define the inner product:
将所述的设解代入步骤2)所述动力学方程，并与作内积，分离方程的实、虚部，进而得到所述的常微分矩阵方程。Substitute the set solution into the kinetic equation of step 2), and combine with Do the inner product, separate the real and imaginary parts of the equation, and then obtain the ordinary differential matrix equation.
步骤4)包括：Step 4) includes:
(1)根据三角函数的运算性质： (1) According to the operational properties of trigonometric functions:
确定步骤3)所述的常微分矩阵方程的刚度矩阵形式如下：Determine the stiffness matrix form of the ordinary differential matrix equation described in step 3) as follows:
a)当2n/N为整数时a) When 2n/N is an integer
b)当2n/N不为整数时b) When 2n/N is not an integer
式中：where:
(2)将常微分矩阵方程中所有的矩阵输入Matlab软件中，调用Matlab软件中的eig命令，得到常微分矩阵方程的特征值。(2) Input all the matrices in the ordinary differential matrix equation into the Matlab software, and call the eig command in the Matlab software to obtain the eigenvalues of the ordinary differential matrix equation.
步骤5)所述的分析是将特征值的虚部作为旋转对称结构的固有频率；将特征值的实部作为稳定性判据：当特征值的实部大于零，则旋转对称结构出现不稳定现象；当特征值的实部小于或等于零，则旋转对称结构稳定。In the analysis described in step 5), the imaginary part of the eigenvalue is used as the natural frequency of the rotationally symmetric structure; the real part of the eigenvalue is used as the stability criterion: when the real part of the eigenvalue is greater than zero, the rotationally symmetric structure appears unstable. Phenomenon; a rotationally symmetric structure is stable when the real part of the eigenvalue is less than or equal to zero.
本发明的一种旋转环状周期结构参激弹性振动分析方法，是在全局静止坐标系下建立旋转环状周期结构的动力学模型，采用一种坐标变换技术和经典振动理论计算系统的特征值，并预测了模态特性和动力稳定性。本发明可用于旋转机械，如行星齿轮传动系统齿圈、旋转电机的环形定转子、汽轮机叶片和滚动轴承内外圈等旋转机械的动力学解析分析，可以进行系统模态特性的计算求解，也可以进行系统的动力稳定性和动态响应的分析。The method for analyzing parametric elastic vibration of a rotating annular periodic structure of the present invention is to establish a dynamic model of the rotating annular periodic structure in a global static coordinate system, and use a coordinate transformation technology and classical vibration theory to calculate the eigenvalues of the system , and predicted the modal characteristics and dynamic stability. The invention can be used for dynamic analysis of rotating machinery, such as planetary gear transmission system ring gear, annular stator and rotor of rotating electrical machine, steam turbine blades, inner and outer rings of rolling bearing, etc. Analysis of the dynamic stability and dynamic response of the system.
首先针对全局静止坐标系下的动力学方程进行坐标变换，通过引入了陀螺项的方式消除经典振动理论无法解析求解的时变参激刚度项。本发明直观地给出了旋转环状周期结构解析形式的特征值，比现有技术更具有高效性、准确性和普适性，克服了现有技术受小参数的限制或偏于数值计算，效率低下的缺点，使针对类似结构的旋转机械参激振动的研究更加符合工程实际。根据本发明可以揭示参数与系统的模态特性、动力稳定性和动态响应之间的关系，可实现在设计阶段预估旋转机械的振动情况及各阶振动的不稳定区域，据此指导旋转机械的高效结构设计，进而提高稳定性及运行可靠性。Firstly, coordinate transformation is carried out for the dynamic equation in the global static coordinate system, and the time-varying parametric stiffness term that cannot be solved analytically by classical vibration theory is eliminated by introducing the gyro term. The invention intuitively provides the eigenvalues of the analytical form of the rotating ring-shaped periodic structure, is more efficient, accurate and universal than the prior art, overcomes the prior art limited by small parameters or biased towards numerical calculation, The disadvantage of low efficiency makes the research on parasitic vibration of rotating machinery with similar structures more in line with engineering practice. According to the invention, the relationship between the parameters and the modal characteristics, dynamic stability and dynamic response of the system can be revealed, and the vibration situation of the rotating machine and the unstable region of each order of vibration can be estimated in the design stage, and the rotating machine can be guided accordingly. High-efficiency structural design, thereby improving stability and operational reliability.
附图说明Description of drawings
图1是本发明的旋转环状周期结构示意图及两种坐标系；Fig. 1 is the rotating annular periodic structure schematic diagram of the present invention and two kinds of coordinate systems;
图2a是振动波数n＝2，支撑弹簧个数N＝4时，根据本发明的方法求得的旋转环状周期结构固有频率随旋转支撑弹簧转速变化规律图；Fig. 2a is a graph of the variation law of the natural frequency of the rotating annular periodic structure obtained by the method of the present invention with the rotational speed of the rotating support spring when the vibration wave number n=2 and the number of support springs N=4;
图2b是振动波数n＝2，支撑弹簧个数N＝4时，根据本发明的方法求得的旋转环状周期结构稳定性判据随旋转支撑弹簧转速变化规律图；Fig. 2b is a graph showing the variation law of the rotational ring-shaped periodic structure stability criterion obtained by the method of the present invention with the rotational speed of the rotational support spring when the vibration wave number n=2 and the number of support springs N=4;
图3a是本发明的方法和现有数值计算方法对旋转环状周期结构不稳定区域预测对比图；Fig. 3a is the method of the present invention and the existing numerical calculation method to the rotation annular periodic structure unstable region prediction comparison chart;
图3b是旋转环状周期结构发散不稳定型动态响应图；Fig. 3b is the dynamic response diagram of the divergent instability type of the rotating annular periodic structure;
图3c是旋转环状周期结构颤振不稳定型动态响应图；Fig. 3c is the dynamic response diagram of the flutter instability type of the rotating annular periodic structure;
图3d是旋转环状周期结构稳态型动态响应图。Figure 3d is a graph of the steady-state dynamic response of the rotating annular periodic structure.
具体实施方式Detailed ways
下面结合实施例和附图对本发明的一种旋转环状周期结构参激弹性振动分析方法做出详细说明。A method for analyzing parametrically excited elastic vibration of a rotating annular periodic structure of the present invention will be described in detail below with reference to the embodiments and the accompanying drawings.
本发明的一种旋转环状周期结构参激弹性振动分析方法，考虑到经典振动理论的特点，通过引入坐标变换，实现了动力学方程在不同坐标系下的转换，提出了一种旋转环状周期结构参激弹性振动的解析分析技术。A method for analyzing parametric elastic vibration of a rotating annular periodic structure in the present invention, considering the characteristics of classical vibration theory, realizes the transformation of dynamic equations in different coordinate systems by introducing coordinate transformation, and proposes a rotating annular periodic structure. Analytical analysis techniques for parametric elastic vibration of periodic structures.
本发明的一种旋转环状周期结构参激弹性振动分析方法中，所述旋转环状周期结构如图1所示由薄圆环和离散旋转支撑弹簧组成，是采用坐标变换方法实现参激系统动力稳定性的解析分析，方法具体包括以下步骤：In a parametric elastic vibration analysis method of a rotating annular periodic structure of the present invention, the rotating annular periodic structure is composed of a thin ring and a discrete rotating support spring as shown in FIG. The analytical analysis of dynamic stability includes the following steps:
1)在全局静止坐标系o-rθz下，根据哈密顿(Hamilton)原理建立旋转环状周期结构的刚-弹耦合动力学方程：1) In the global stationary coordinate system o-rθz, the rigid-elastic coupling dynamic equation of the rotating ring periodic structure is established according to Hamilton's principle:
式中：where:
β为旋转支撑的方向角；β is the direction angle of the rotating support;
θ为表示旋转支撑位置角的一个空间函数；θ is a space function representing the position angle of the rotating support;
kt为圆环外侧均布切向静止支撑刚度；k t is the uniformly distributed tangential static support stiffness outside the ring;
kr为圆环外侧均布径向静止支撑刚度；k r is the uniformly distributed radial static support stiffness outside the ring;
θj＝2π(j–1)/N，表示第j个旋转支撑的初始位置，N为总的旋转支撑个数；θ j = 2π(j–1)/N, indicating the initial position of the j-th rotating support, and N is the total number of rotating supports;
Ω为旋转支撑的转速；Ω is the rotational speed of the rotating support;
t表示时间；t represents time;
cz＝I/(AR2)为人为引入的一个运算符；c z =I/(AR 2 ) is an operator introduced artificially;
I＝bh3/12为圆环截面惯性矩；I=bh 3 /12 is the moment of inertia of the ring section;
A＝bh为圆环截面面积；A=bh is the cross-sectional area of the ring;
R为圆环中心圆半径；R is the radius of the center circle of the ring;
b为圆环的径向厚度；b is the radial thickness of the ring;
h为圆环的轴向高度；h is the axial height of the ring;
ks为旋转支撑刚度；k s is the rotational support stiffness;
上述刚-弹耦合动力学方程为一个含有时变刚度项的参激振动方程，其动力学响应和系统的特征值无法直接通过经典振动理论解析求得，借助Floquét理论和数值计算技术，可以得到在给定初始条件下的结论，但不具有普适性。The above rigid-elastic coupling dynamic equation is a parametrically excited vibration equation containing a time-varying stiffness term, and its dynamic response and eigenvalues of the system cannot be directly obtained by classical vibration theory. Conclusions given initial conditions, but not universal.
由此，本发明通过引入坐标变换可将在全局静止坐标系o-rθz下所得到的动力学方程转换到支撑随动坐标系下，得到一个不含参激项的定常偏微分方程，进而可以计算系统的特征值以确定固有频率和参激不稳定行为。具体过程如下：Thus, the present invention introduces coordinate transformation by The dynamic equations obtained in the global stationary coordinate system o-rθz can be converted to the support-following coordinate system , a stationary partial differential equation with no parametric term is obtained, and the eigenvalues of the system can then be calculated to determine the natural frequency and parametric instability behavior. The specific process is as follows:
2)引入坐标变换将步骤1)所述动力学模型转换到支撑随动坐标系下，从而消除原方程中的参激项，得到偏微分常系数动力学方程：2) Introduce coordinate transformation Convert the dynamic model described in step 1) to the support-following coordinate system , thereby eliminating the parametric term in the original equation, and obtaining the partial differential constant coefficient dynamic equation:
(M′+K′0+K′1)q＝0；(M'+K' 0 +K' 1 )q=0;
式中：where:
通过本发明所提出的这种坐标变换技术，系统的动力学方程引入了陀螺项，消去了原有方程刚度项的时变性；Through the coordinate transformation technology proposed by the present invention, the gyro term is introduced into the dynamic equation of the system, and the time-varying stiffness term of the original equation is eliminated;
3)采用伽辽金(Galerkin)方法对旋转支撑随动坐标系下的偏微分常系数动力学方程进行离散处理，得到常微分矩阵方程。所述的离散处理方式如下：3) Using the Galerkin method for the rotational support follow-up coordinate system The following partial differential constant coefficient dynamic equations are discretized to obtain ordinary differential matrix equations. The discrete processing method described is as follows:
首先对偏微分常系数动力学方程的响应u和v设解First, let the responses u and v of the partial differential constant coefficient dynamic equation be solved
式中：where:
U(t)＝x1(t)+iy1(t)和V(t)＝x2(t)+iy2(t)；i为虚数单位，“～”表示复共轭；U(t)=x 1 (t)+iy 1 (t) and V(t)=x 2 (t)+iy 2 (t); i is an imaginary unit, and "~" represents a complex conjugate;
定义内积：Define the inner product:
将所述的设解代入步骤2)所述动力学方程，并与作内积，分离方程的实、虚部，进而得到所述的常微分矩阵方程：Substitute the set solution into the kinetic equation of step 2), and combine with Take the inner product to separate the real and imaginary parts of the equation, and then obtain the ordinary differential matrix equation:
式中：where:
4)利用经典振动理论，解析分析步骤3)所述的常微分矩阵方程的特征值；包括：4) utilize classical vibration theory, analytically analyze the eigenvalues of the ordinary differential matrix equation described in step 3); include:
(1)根据三角函数的运算性质： (1) According to the operational properties of trigonometric functions:
确定步骤3)所述的常微分矩阵方程的刚度矩阵形式如下：Determine the stiffness matrix form of the ordinary differential matrix equation described in step 3) as follows:
a)当2n/N为整数时a) When 2n/N is an integer
b)当2n/N不为整数时b) When 2n/N is not an integer
式中：where:
(2)将常微分矩阵方程中所有的矩阵输入Matlab软件中，调用Matlab软件中的eig命令，得到常微分矩阵方程的特征值。(2) Input all the matrices in the ordinary differential matrix equation into the Matlab software, and call the eig command in the Matlab software to obtain the eigenvalues of the ordinary differential matrix equation.
以表1中数据为例，利用经典振动理论计算步骤(S3)中常微分方程的特征值。Taking the data in Table 1 as an example, use the classical vibration theory to calculate the eigenvalues of the ordinary differential equation in step (S3).
表1旋转环状周期结构系统参数Table 1 Parameters of rotating ring periodic structure system
5)分别利用步骤4)得到的常微分矩阵方程的特征值的虚部和实部对旋转环状周期结构的模态特性和参激弹性振动的动力稳定性规律进行分析，所述的分析是将特征值的虚部作为旋转对称结构的固有频率；将特征值的实部作为稳定性判据：当特征值的实部大于零，则旋转对称结构出现不稳定现象；当特征值的实部小于或等于零，则旋转对称结构稳定。解析结论如附图2a、图2b所示。5) Use the imaginary part and the real part of the eigenvalues of the ordinary differential matrix equation obtained in step 4) to analyze the modal characteristics of the rotating annular periodic structure and the dynamic stability law of parametric elastic vibration, and the analysis is: The imaginary part of the eigenvalue is used as the natural frequency of the rotationally symmetric structure; the real part of the eigenvalue is used as the stability criterion: when the real part of the eigenvalue is greater than zero, the rotationally symmetric structure appears unstable; Less than or equal to zero, the rotationally symmetric structure is stable. The analytical conclusions are shown in Figures 2a and 2b.
下面用两种不同的方式对本发明的一种旋转环状周期结构参激弹性振动分析方法进行验证。The following two different ways are used to verify the parametric elastic vibration analysis method of the rotating annular periodic structure of the present invention.
第一种方式：The first way:
利用Floquét理论，对本发明得到的旋转环状周期结构的模态特性和参激弹性振动的动力稳定性规律进行数值验证。以2n/N等于整数为例，针对全局静止坐标系o-rθz下动力学模型，采用本发明步骤3)所述Galerkin离散处理，可得下面的常微分方程：Using the Floquét theory, the modal characteristics of the rotating annular periodic structure obtained by the present invention and the dynamic stability law of parametric elastic vibration are numerically verified. Taking 2n/N equal to an integer as an example, for the dynamic model in the global static coordinate system o-rθz, using the Galerkin discrete processing described in step 3) of the present invention, the following ordinary differential equation can be obtained:
式中：where:
式中：where:
将得到的常微分方程改写为状态空间形式：Rewrite the resulting ordinary differential equation into state-space form:
式中：where:
且A(t+Tm)＝A(t)，Tm是与旋转支撑弹簧转速有关的参激变化周期；0表示4维的零矩阵；I表示4维的单位矩阵。 And A(t+T m )=A(t), T m is the parametric change period related to the rotational speed of the rotating support spring; 0 represents a 4-dimensional zero matrix; I represents a 4-dimensional unit matrix.
根据周期矩阵A(t)可以近似地构造出一个Floquét转换矩阵，并能根据该转换矩阵的特征值来判断系统的稳定性。利用现有的Floquét数值计算方法对本发明方法的旋转环状周期结构的模态特性和参激弹性振动的动力稳定性规律的验证如附图3a所示，图中点状区域为数值计算结果，边界线为本发明所提出的解析技术的计算结果。According to the periodic matrix A(t), a Floquét transformation matrix can be constructed approximately, and the stability of the system can be judged according to the eigenvalues of the transformation matrix. Using the existing Floquét numerical calculation method to verify the modal characteristics of the rotating annular periodic structure and the dynamic stability law of parametric elastic vibration according to the method of the present invention, as shown in Figure 3a, the point-shaped area in the figure is the numerical calculation result, The boundary line is the calculation result of the analytical technique proposed in the present invention.
第二种方式：The second way:
利用变步长Runge-Kutta法，对本发明得到的旋转环状周期结构的模态特性和参激弹性振动的动力稳定性规律进行数值验证。结论如附图3b、图3c、图3d所示。证明本发明的方法不仅可以判断系统的不稳定区域，还可以对系统发生的颤振或者发散不稳定的类型进行判断，可以更好的指导工程实践。Using the variable step-size Runge-Kutta method, the modal characteristics of the rotating annular periodic structure obtained in the present invention and the dynamic stability law of parametric elastic vibration are numerically verified. The conclusions are shown in Figure 3b, Figure 3c, Figure 3d. It is proved that the method of the present invention can not only judge the unstable region of the system, but also judge the type of flutter or divergent instability that occurs in the system, which can better guide engineering practice.
综上所述，本发明实施例提供了一种旋转环状周期结构模态特性和参激振动动力稳定性的解析分析技术，该技术可在数学建模阶段从全局静止坐标系入手，然后通过引入一种坐标变换技术，在动力学方程中引入了陀螺项，消去了无法利用经典振动理论解析求解的时变参激刚度项，进而得到旋转机械系统解析形式的特征值，提高了针对旋转机械模态和稳定性分析的效率和普适性，更好地满足了工程实际的需要。To sum up, the embodiment of the present invention provides an analytical analysis technology for the modal characteristics of a rotating annular periodic structure and the dynamic stability of parametric vibration. A coordinate transformation technology is introduced, and the gyro term is introduced into the dynamic equation, which eliminates the time-varying parametric stiffness term that cannot be solved analytically by classical vibration theory, and then obtains the eigenvalues of the rotating mechanical system in the analytical form, which improves the accuracy of the rotating machinery system. The efficiency and universality of modal and stability analysis better meet the needs of engineering practice.
本领域技术人员可以理解附图只是一个特殊实施例的示意图，并不用以限制本发明。显然，本领域的技术人员可以对本发明进行各种改动和变型而不脱离本发明的精神和范围。凡在本发明的精神和原则之内，所作的任何修改、等同替换和变型等，均应包含在本发明的保护范围之内。Those skilled in the art can understand that the accompanying drawings are only schematic diagrams of a specific embodiment, and are not used to limit the present invention. It will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit and scope of the invention. Any modifications, equivalent substitutions and variations made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.
Claims (4)
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201610911161.5A CN106547957B (en) | 2016-10-19 | 2016-10-19 | A parametric elastic vibration analysis method for rotating annular periodic structures |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201610911161.5A CN106547957B (en) | 2016-10-19 | 2016-10-19 | A parametric elastic vibration analysis method for rotating annular periodic structures |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN106547957A CN106547957A (en) | 2017-03-29 |
| CN106547957B true CN106547957B (en) | 2019-07-26 |
Family
ID=58369117
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201610911161.5A Active CN106547957B (en) | 2016-10-19 | 2016-10-19 | A parametric elastic vibration analysis method for rotating annular periodic structures |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN106547957B (en) |
Families Citing this family (10)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108334699A (en) * | 2018-02-05 | 2018-07-27 | 东北大学 | Bulk material analysis of flow characteristics method under a kind of non-linear uninstalling system of improved unilateral exciting |
| CN108982012B (en) * | 2018-09-25 | 2020-02-07 | 国网新疆电力有限公司电力科学研究院 | Vertical pump dynamic balance test method based on virtual sensor |
| CN109284569B (en) * | 2018-10-15 | 2023-04-07 | 天津大学 | Combined instability analysis and prediction method for periodic stator of permanent magnet motor |
| CN110119532B (en) * | 2019-04-09 | 2023-05-23 | 天津大学 | Elastic vibration calculation method of rotary annular periodic structure |
| CN110555254B (en) * | 2019-08-22 | 2023-06-06 | 天津大学 | A method for reducing out-of-plane vibration instability of permanent magnet motors by grouping magnetic poles |
| CN111382503B (en) * | 2020-02-27 | 2022-05-20 | 中南大学 | Vibration analysis method and system of flexible ring rotating under elastic support |
| CN112270065B (en) * | 2020-08-26 | 2023-09-08 | 天津大学 | Dynamic stability prediction method for eccentric rotary annular periodic structure |
| CN112464391B (en) * | 2020-10-15 | 2022-09-06 | 昆明理工大学 | Method for establishing Hamilton model of diesel generator set |
| CN112347591B (en) * | 2020-11-30 | 2022-07-05 | 天津大学 | Internal force analysis and free vibration modeling method for eccentric rotating ring structure |
| CN113704927B (en) * | 2021-09-10 | 2024-03-08 | 长春光华学院 | Torque and rotating speed dynamics analysis and design method for offset elastic connecting disc |
Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US7630869B2 (en) * | 2003-05-27 | 2009-12-08 | University Of Washington | Method for predicting vibrational characteristics of rotating structures |
| CN102141775A (en) * | 2011-01-25 | 2011-08-03 | 天津大学 | Deep-sea platform mooring system parametric excitation-internal resonance coupled vibration analysis and control method |
| CN105659888B (en) * | 2012-08-24 | 2014-07-09 | 北京空间飞行器总体设计部 | The micro-vibration modeling and analysis of a kind of Control loop based on Grazing condition dummy satellite method |
-
2016
- 2016-10-19 CN CN201610911161.5A patent/CN106547957B/en active Active
Patent Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US7630869B2 (en) * | 2003-05-27 | 2009-12-08 | University Of Washington | Method for predicting vibrational characteristics of rotating structures |
| CN102141775A (en) * | 2011-01-25 | 2011-08-03 | 天津大学 | Deep-sea platform mooring system parametric excitation-internal resonance coupled vibration analysis and control method |
| CN105659888B (en) * | 2012-08-24 | 2014-07-09 | 北京空间飞行器总体设计部 | The micro-vibration modeling and analysis of a kind of Control loop based on Grazing condition dummy satellite method |
Non-Patent Citations (3)
| Title |
|---|
| Instantaneous mode contamination and parametric combination instability of spinning cyclically symmetric ring structures with expanding application to planetary gear ring;Shiyu Wang等;《Journal of Sound and Vibration》;20160423;第366-385页 |
| Vibration of rings on a general elastic foundation;Xionghua Wu等;《Journal of Sound and Vibration》;20060310;第194–213页 |
| 环状旋转周期结构模态摄动分析;徐进友 等;《天津大学学报》;20101130;第43卷(第11期);第1015-1019页 |
Also Published As
| Publication number | Publication date |
|---|---|
| CN106547957A (en) | 2017-03-29 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN106547957B (en) | A parametric elastic vibration analysis method for rotating annular periodic structures | |
| Tiwari et al. | Dynamic response of an unbalanced rotor supported on ball bearings | |
| Harsha et al. | The effect of speed of balanced rotor on nonlinear vibrations associated with ball bearings | |
| Ma et al. | Effects of eccentric phase difference between two discs on oil-film instability in a rotor–bearing system | |
| Li et al. | Multiple frequencies elliptical whirling orbit model and transient RANS solution approach to rotordynamic coefficients of annual gas seals prediction | |
| CN106528959B (en) | A Simplified Analysis Method for the Natural Frequency and Stability of Rotationally Symmetrical Structures | |
| Yang et al. | The varying compliance resonance in a ball bearing rotor system affected by different ball numbers and rotor eccentricities | |
| Cong et al. | Experimental validation of impact energy model for the rub–impact assessment in a rotor system | |
| CN103632006B (en) | Method for determining critical rotating speed of wet rotor of multistage centrifugal pump | |
| von Groll et al. | A mechanism of low subharmonic response in rotor/stator contact—measurements and simulations | |
| Brouwer et al. | Combined explicit finite and discrete element methods for rotor bearing dynamic modeling | |
| CN109284569B (en) | Combined instability analysis and prediction method for periodic stator of permanent magnet motor | |
| Kim et al. | Bifurcation analysis of a rotor supported by five-pad tilting pad journal bearings using numerical continuation | |
| Wang et al. | Characteristics analysis of aero-engine whole vibration response with rolling bearing radial clearance | |
| Cao et al. | Coupled lateral and torsional nonlinear transient rotor–bearing system analysis with applications | |
| Liu et al. | Study on the Dynamic Problems of Double‐Disk Rotor System Supported by Deep Groove Ball Bearing | |
| Wang et al. | Certain type turbofan engine whole vibration model with support looseness fault and casing response characteristics | |
| Lu et al. | Dynamic characteristics analysis of rotor-bearing system considering bearing clearance and hybrid eccentricity | |
| Al-Khazali et al. | The experimental analysis of vibration monitoring in system rotor dynamic with validate results using simulation data | |
| Kumar | Rotor dynamic analysis using ANSYS | |
| Humer et al. | Multi-objective optimization of complex multibody systems by coupling HOTINT with MagOpt | |
| Liu et al. | Dynamics analysis of unbalanced motorized spindles supported on ball bearings | |
| CN106940735A (en) | A kind of induction machine elastic vibration stability prediction method | |
| Ma et al. | Numerical research on rub-impact fault in a blade-rotor-casing coupling system | |
| Harsha | Rolling bearing vibrations—the effects of surface waviness and radial internal clearance |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |