CN106295228A

CN106295228A - A kind of Methods of Chaotic Forecasting of asymmetric zonal cooling nonlinear restriction system

Info

Publication number: CN106295228A
Application number: CN201610761669.1A
Authority: CN
Inventors: 刘飞; 刘彬; 刘浩然; 姜甲浩; 李鹏
Original assignee: Xian University of Science and Technology
Current assignee: Xian University of Science and Technology
Priority date: 2016-08-29
Filing date: 2016-08-29
Publication date: 2017-01-04

Abstract

The invention discloses a chaos prediction method for an asymmetric piecewise continuous nonlinear constraint system, which relates to the technical field of vibration behavior prediction of dynamic systems under piecewise nonlinear constraints, and is used to solve a kind of vibration caused by piecewise nonlinear constraints The problem of judging the chaotic motion state of the system. This method: according to the asymmetric piecewise continuous nonlinear constraint model and the general formula of the dynamic model, the dynamic model of the asymmetric piecewise continuous nonlinear constraint system is determined; according to the dynamic model, by analyzing the asymmetric piecewise continuous nonlinear constraint The saddle point of the system, determine the homoclinic orbital parameter equation of the saddle point; according to the dynamic model and the general formula of the Melnikov function, determine the Melnikov function of the asymmetric piecewise continuous nonlinear constraint system; according to the asymmetric piecewise continuous nonlinear constraint model and the homoclinic orbital parameters Equation, to determine the upper and lower limits of the integral of the Melnikov function; according to the Melnikov function and the upper and lower limits of the integral, determine the critical condition of the asymmetric piecewise continuous nonlinear constraint system in chaotic motion.

Description

A Chaos Prediction Method for Asymmetric Piecewise Continuous Nonlinear Constrained Systems

技术领域technical field

本发明涉及分段非线性约束下的动力学系统振动行为预测技术领域，更具体的涉及一种非对称分段连续非线性约束系统的混沌预测方法。The invention relates to the technical field of vibration behavior prediction of a dynamic system under a piecewise nonlinear constraint, and more specifically relates to a chaos prediction method for an asymmetric piecewise continuous nonlinear constraint system.

背景技术Background technique

工程实践中的承载结构常常会因为弹性预紧变形受到一类分段非线性约束作用，主要表现为在弹性元件组成的承载结构中，当受到静态载荷作用时，承载结构中的弹性元件会产生相应的弹性初始变形，当受到的载荷作用发生动态变化时，弹性预紧变形的交替恢复使得系统所受约束不断发生变化，系统受到这类分段非线性约束的作用会出现分岔、混沌等运动状态，处在混沌运动状态的动力学系统的行为对初值敏感而且混乱、无序，这将对系统的稳定性构成威胁。比如在金属加工领域，作为主要工艺的板带轧制过程中会由于液压压下系统的固液弹性模量差异较大，在较大压力作用下液压油产生的预紧变形在振动过程中就不可忽略。这些预紧弹性变形的交替恢复就会导致辊系所受到的约束不断发生变化，进而出现复杂的非线性现象，不仅威胁系统运行稳定性，还会造成轧制产品质量缺陷。然而，目前对于这一类分段非线性约束作用的振动系统的混沌运动状态的判断尚未有所研究。The load-bearing structures in engineering practice are often subjected to a kind of piecewise nonlinear constraints due to elastic preload deformation, which is mainly manifested in the load-bearing structure composed of elastic elements. When subjected to static loads, the elastic elements in the load-bearing structure will produce For the corresponding elastic initial deformation, when the applied load changes dynamically, the alternate recovery of elastic preload deformation makes the constraints of the system constantly change, and the system will appear bifurcation, chaos, etc. under the action of such segmental nonlinear constraints In the state of motion, the behavior of the dynamical system in the state of chaotic motion is sensitive to the initial value and chaotic and disorderly, which will pose a threat to the stability of the system. For example, in the field of metal processing, in the process of strip rolling as the main process, due to the large difference in the elastic modulus of the solid and liquid in the hydraulic pressing system, the pre-tightening deformation generated by the hydraulic oil under the action of a large pressure will be reduced during the vibration process. Can not be ignored. The alternate recovery of these preloaded elastic deformations will lead to continuous changes in the constraints on the roll system, and then complex nonlinear phenomena will appear, which not only threatens the stability of the system, but also causes quality defects in rolled products. However, the judging of the chaotic motion state of this kind of vibrating system with piecewise nonlinear constraints has not been studied at present.

综上所述，现有技术中，存在一类分段非线性约束作用的振动系统的混沌运动状态的判断尚未有所研究的问题。To sum up, in the prior art, there is a problem of judging the chaotic motion state of a vibrating system with piecewise nonlinear constraints that has not yet been studied.

发明内容Contents of the invention

本发明实施例提供一种非对称分段连续非线性约束系统的混沌预测方法，用以解决一类分段非线性约束作用的振动系统的混沌运动状态的判断的问题。An embodiment of the present invention provides a chaos prediction method for an asymmetric piecewise continuous nonlinear constraint system, which is used to solve the problem of judging the chaotic motion state of a vibrating system with piecewise nonlinear constraints.

本发明实施例提供一种非对称分段连续非线性约束系统的混沌预测方法，包括：An embodiment of the present invention provides a chaos prediction method for an asymmetric piecewise continuous nonlinear constraint system, including:

根据非对称分段连续非线性约束模型和动力学模型通式，确定非对称分段连续非线性约束系统的动力学模型；According to the asymmetric piecewise continuous nonlinear constraint model and the general formula of the dynamic model, the dynamic model of the asymmetric piecewise continuous nonlinear constraint system is determined;

根据所述动力学模型，通过分析所述非对称分段连续非线性约束系统的鞍点，确定所述鞍点的同宿轨道参数方程；According to the dynamic model, by analyzing the saddle point of the asymmetric piecewise continuous nonlinear constraint system, determine the homoclinic orbit parameter equation of the saddle point;

根据所述动力学模型和Melnikov函数通式，通过公式(1)，确定所述非对称分段连续非线性约束系统的Melnikov函数；According to described dynamic model and Melnikov function general formula, by formula (1), determine the Melnikov function of described asymmetric piecewise continuous nonlinear constraint system;

根据所述非对称分段连续非线性约束模型和所述同宿轨道参数方程，确定所述Melnikov函数的积分上下限；Determine the integral upper and lower limits of the Melnikov function according to the asymmetric piecewise continuous nonlinear constraint model and the homoclinic orbital parameter equation;

根据所述Melnikov函数和所述积分上下限，通过公式(2)，确定所述非对称分段连续非线性约束系统在混沌运动时的临界条件；According to the Melnikov function and the upper and lower limits of the integral, by formula (2), determine the critical condition of the asymmetric piecewise continuous nonlinear constraint system during chaotic motion;

根据等效刚度系数、非线性刚度系数、刚度、质量、阻尼比和弹性变形量的实测数据，以及所述混沌运动时的临界条件，确定所述非对称分段连续非线性约束系统在混沌运动时的周期外激励幅值条件；According to the measured data of equivalent stiffness coefficient, nonlinear stiffness coefficient, stiffness, mass, damping ratio and elastic deformation, and the critical conditions during the chaotic motion, determine the asymmetric piecewise continuous nonlinear constraint system in the chaotic motion The condition of the excitation amplitude outside the period;

根据周期外激励幅值的实测数据和混沌运动时的周期外激励幅值条件，确定所述非对称分段连续非线性约束系统的混沌运动状态；According to the measured data of the excitation amplitude outside the period and the excitation amplitude condition outside the period during the chaotic motion, the chaotic motion state of the asymmetric piecewise continuous nonlinear constraint system is determined;

公式(1)如下所示：Formula (1) is as follows:

公式(2)如下所示：Formula (2) is as follows:

其中，t₀为初始时间，且为任意实数； e₁和e₂为弹性变形量，为阻尼比，qsin(ωt)为周期外激励，q为周期外激励振幅，ω为周期外激励角频率，k为刚度，m为质量，μ＝1/m，κ₁、κ₂和κ₃为等效刚度系数，γ₁、γ₂和γ₃为非线性刚度系数，t₁和t₂分别为Melnikov函数的积分上下限，χ＝-μ(k+κ₂)。Among them, t ₀ is the initial time, and is any real number; e ₁ and e ₂ are the amount of elastic deformation, is the damping ratio, qsin(ωt) is the external excitation of the period, q is the amplitude of the external excitation of the period, ω is the angular frequency of the external excitation of the period, k is stiffness, m is mass, μ=1/m, κ ₁ , κ ₂ and κ ₃ are equivalent stiffness coefficients, γ ₁ , γ ₂ and γ ₃ are nonlinear stiffness coefficients, t ₁ and t ₂ are Melnikov The integral upper and lower limits of the function, χ=-μ(k+κ ₂ ).

较佳地，所述根据非对称分段连续非线性约束模型和动力学模型通式，确定非对称分段连续非线性约束系统的动力学模型，包括：Preferably, according to the asymmetric piecewise continuous nonlinear constraint model and the general formula of the dynamic model, determining the dynamic model of the asymmetric piecewise continuous nonlinear constraint system includes:

所述非对称分段连续非线性约束模型，由下式确定：The asymmetric piecewise continuous nonlinear constraint model is determined by the following formula:

$G G ((x x)) = = \{\begin{matrix} {κ κ}_{11} x x + + {γ γ}_{11} {x x}^{33} & x x > > - - {e e}_{22} \\ {κ κ}_{22} x x + + {γ γ}_{22} {x x}^{33} & - - {e e}_{11} \leq \leq x x \leq \leq - - {e e}_{22} \\ {κ κ}_{33} x x + + {γ γ}_{33} {x x}^{33} & x x < < - - {e e}_{11} \end{matrix}$

所述非对称分段连续非线性约束模型在分段点具有连续性的条件为：The condition that the asymmetric piecewise continuous nonlinear constraint model has continuity at the piecewise point is:

$\{\begin{matrix} \underset{x x &RightArrow; &Right Arrow; - - {e e}_{22}}{lim lim} {κ κ}_{11} x x + + {γ γ}_{11} {x x}^{33} = = - - {κ κ}_{22} {e e}_{22} - - {γ γ}_{22} {e e}_{22}^{33} \\ \underset{x x &RightArrow; &Right Arrow; - - {e e}_{11}}{lim lim} {κ κ}_{33} x x + + {γ γ}_{33} {x x}^{33} = = - - {κ κ}_{22} {e e}_{11} - - {γ γ}_{22} {e e}_{11}^{33} \end{matrix}$

所述动力学模型，由下式确定：The kinetic model is determined by the following formula:

其中，x为约束条件下的位移，为x的一阶导数，为x的二阶导数，G(x)为约束力。Among them, x is the displacement under the constraints, is the first derivative of x, is the second derivative of x, and G(x) is the binding force.

较佳地，所述根据所述动力学模型，通过分析所述非对称分段连续非线性约束系统的鞍点，确定所述鞍点的同宿轨道参数方程，包括：Preferably, according to the dynamic model, by analyzing the saddle point of the asymmetric piecewise continuous nonlinear constraint system, the homoclinic orbit parameter equation of the saddle point is determined, including:

根据所述动力学模型确定所述动力学模型的状态方程，所述动力学模型的状态方程为：Determine the equation of state of the kinetic model according to the kinetic model, the equation of state of the kinetic model is:

根据所述动力学模型的状态方程，确定所述非对称分段连续非线性约束系统的Hamilton函数，所述Hamilton函数为：According to the equation of state of the dynamic model, determine the Hamilton function of the asymmetric piecewise continuous nonlinear constraint system, the Hamilton function is:

$H h ((x x,, y the y)) = = \frac{11}{22} {y the y}^{22} + + \frac{11}{22} {ω ω}_{00}^{22} {x x}^{22} + + \{\begin{matrix} \frac{11}{22} {μκ μκ}_{11} {x x}^{22} + + \frac{11}{44} {μγ μγ}_{11} {x x}^{44} & x x > > - - {e e}_{22} \\ \frac{11}{22} {μκ μκ}_{22} {x x}^{22} + + \frac{11}{44} {μγ μγ}_{22} {x x}^{44} & - - {e e}_{11} \leq \leq x x \leq \leq - - {e e}_{22} \\ \frac{11}{22} {μκ μκ}_{33} {x x}^{22} + + \frac{11}{44} {μγ μγ}_{33} {x x}^{44} & x x < < - - {e e}_{11} \end{matrix}$

当所述Hamilton函数满足H(x,y)＝0，根据所述非对称分段连续非线性约束系统不同约束情形下中心点的位置，通过下式确定所述鞍点的同宿轨道参数方程：When the Hamilton function satisfies H(x, y)=0, according to the position of the central point under the different constraint situations of the asymmetric piecewise continuous nonlinear constraint system, the homoclinic orbit parameter equation of the saddle point is determined by the following formula:

$(\begin{matrix} {x x}_{&PlusMinus; &PlusMinus;} ((t t)) \\ {y the y}_{&PlusMinus; &PlusMinus;} ((t t)) \end{matrix}) = = (\begin{matrix} &PlusMinus; &PlusMinus; \sqrt{- - \frac{22 ((k k + + {κ κ}_{i i}))}{{γ γ}_{i i}}} sec sec h h ((\sqrt{- - μ μ ((k k + + {κ κ}_{i i}))} t t)) \end{matrix})$

其中，为y的一阶导数，ε为常数，且0＜ε＜＜1，i＝1,2,3。in, is the first derivative of y, ε is a constant, and 0<ε<<1, i=1,2,3.

本发明实施例提供一种非对称分段连续非线性约束系统的混沌预测装置，包括：An embodiment of the present invention provides a chaos prediction device for an asymmetric piecewise continuous nonlinear constraint system, including:

第一确定单元，用于根据非对称分段连续非线性约束模型和动力学模型通式，确定非对称分段连续非线性约束系统的动力学模型；The first determination unit is used to determine the dynamic model of the asymmetric piecewise continuous nonlinear constraint system according to the asymmetric piecewise continuous nonlinear constraint model and the general formula of the dynamic model;

第二确定单元，用于根据所述动力学模型，通过分析所述非对称分段连续非线性约束系统的鞍点，确定所述鞍点的同宿轨道参数方程；The second determination unit is used to determine the homoclinic orbit parameter equation of the saddle point by analyzing the saddle point of the asymmetric piecewise continuous nonlinear constraint system according to the dynamic model;

第三确定单元，用于根据所述动力学模型和Melnikov函数通式，通过公式(1)，确定所述非对称分段连续非线性约束系统的Melnikov函数；The third determining unit is used to determine the Melnikov function of the asymmetric piecewise continuous nonlinear constraint system according to the dynamic model and the general formula of the Melnikov function through formula (1);

第四确定单元，用于根据所述非对称分段连续非线性约束模型和所述同宿轨道参数方程，确定所述Melnikov函数的积分上下限；The fourth determination unit is used to determine the integral upper and lower limits of the Melnikov function according to the asymmetric piecewise continuous nonlinear constraint model and the homoclinic orbit parameter equation;

第五确定单元，用于根据所述Melnikov函数和所述积分上下限，通过公式(2)，确定所述非对称分段连续非线性约束系统在混沌运动时的临界条件；The fifth determination unit is used to determine the critical condition of the asymmetric piecewise continuous nonlinear constraint system during chaotic motion according to the Melnikov function and the upper and lower limits of the integral, through formula (2);

第六确定单元，用于根据等效刚度系数、非线性刚度系数、刚度、质量、阻尼比和弹性变形量的实测数据，以及所述混沌运动时的临界条件，确定所述非对称分段连续非线性约束系统在混沌运动时的周期外激励幅值条件；The sixth determining unit is used to determine the asymmetric piecewise continuous Conditions of excitation amplitude outside the period of nonlinear constrained system in chaotic motion;

第七确定单元，用于根据周期外激励幅值的实测数据和混沌运动时的周期外激励幅值条件，确定所述非对称分段连续非线性约束系统的混沌运动状态；The seventh determination unit is used to determine the chaotic motion state of the asymmetric piecewise continuous nonlinear constraint system according to the measured data of the excitation amplitude outside the period and the excitation amplitude condition outside the period during chaotic motion;

公式(1)如下所示：Formula (1) is as follows:

公式(2)如下所示：Formula (2) is as follows:

较佳地，所述第一确定单元具体用于：Preferably, the first determining unit is specifically configured to:

较佳地，所述第二确定单元具体用于：Preferably, the second determining unit is specifically configured to:

根据所述动力学模型的状态方程，确定所述非对称分段连续非线性约束系统的Hamilton函数，所述Hamilton函数为：According to the equation of state of the dynamic model, the Hamilton function of the asymmetric piecewise continuous nonlinear constraint system is determined, and the Hamilton function is:

本发明实施例中，提供一种非对称分段连续非线性约束系统的混沌预测方法，该方法给出了系统在鞍点处由三段不同轨道组成的同宿轨道参数方程，根据约束的分段条件确定积分上下限并通过分段积分得到Melnikov函数，解决了非对称分段非线性约束系统无法直接得到Melnikov函数来预测混沌的难点；进一步，非对称分段非线性约束系统发生混沌时的临界条件解析表达式，对于具有非对称分段非线性约束特征的系统混沌运动状态的预测问题提供了一种快速、有效的解析方法，根据预测结果能够通过调节控制参数对系统混沌运动行为进行控制，避免由于出现混沌运动状态而导致系统失稳。In the embodiment of the present invention, a chaos prediction method for an asymmetric segmented continuous nonlinear constrained system is provided. The method provides a homoclinic orbit parameter equation composed of three different orbits at the saddle point of the system. According to the segmental condition of the constraint Determine the upper and lower limits of the integral and obtain the Melnikov function through piecewise integration, which solves the difficulty that the asymmetric piecewise nonlinear constraint system cannot directly obtain the Melnikov function to predict chaos; further, the critical condition for the asymmetric piecewise nonlinear constraint system when chaos occurs The analytical expression provides a fast and effective analytical method for the prediction of the chaotic motion state of the system with the characteristics of asymmetric piecewise nonlinear constraints. According to the prediction results, the chaotic motion behavior of the system can be controlled by adjusting the control parameters to avoid The system is unstable due to the chaotic motion state.

附图说明Description of drawings

图1为本发明实施例提供的一种非对称分段连续非线性约束系统的混沌预测方法流程图；Fig. 1 is a flow chart of a chaos prediction method for an asymmetric piecewise continuous nonlinear constrained system provided by an embodiment of the present invention;

图2(a)为本发明实施例提供的当k+κ_i＜0时非对称分段非线性约束系统的势函数曲线；Fig. 2 (a) is the potential function curve of the asymmetric piecewise nonlinear constraint system provided by the embodiment of the present invention when _k +κi<0;

图2(b)为本发明实施例提供的当k+κ_i＜0时非对称分段非线性约束系统的相轨迹；Figure 2(b) is the phase trajectory of the asymmetric piecewise nonlinear constraint system when _k +κi <0 provided by the embodiment of the present invention;

图3(a)为本发明实施例提供的当k+κ_i＞0时非对称分段非线性约束系统的势函数曲线；Fig. 3 (a) is the potential function curve of the asymmetric piecewise nonlinear constraint system provided by the embodiment of the present invention when _k +κi>0;

图3(b)为本发明实施例提供的当k+κ_i＞0时非对称分段非线性约束系统的相轨迹；Fig. 3(b) is the phase trajectory of the asymmetric piecewise nonlinear confinement system when _k +κi>0 provided by the embodiment of the present invention;

图4为本发明实施例提供的当δ＝5,τ＝0.5时非对称分段非线性约束系统的分岔图及最大Lyapunov指数曲线。FIG. 4 is a bifurcation diagram and a maximum Lyapunov exponent curve of an asymmetric piecewise nonlinear constraint system provided by an embodiment of the present invention when δ=5 and τ=0.5.

图5为本发明实施例提供的一种非对称分段连续非线性约束系统的混沌预测装置结构示意图。Fig. 5 is a schematic structural diagram of a chaos prediction device for an asymmetric piecewise continuous nonlinear constraint system provided by an embodiment of the present invention.

具体实施方式detailed description

下面将结合本发明实施例中的附图，对本发明实施例中的技术方案进行清楚、完整地描述，显然，所描述的实施例仅仅是本发明一部分实施例，而不是全部的实施例。基于本发明中的实施例，本领域普通技术人员在没有做出创造性劳动前提下所获得的所有其他实施例，都属于本发明保护的范围。The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some, not all, embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts belong to the protection scope of the present invention.

图1示例性的示出了本发明实施例提供的一种非对称分段连续非线性约束系统的混沌预测方法流程图。如图1所示，本发明实施例提供的一种非对称分段连续非线性约束系统的混沌预测方法包括以下步骤：Fig. 1 exemplarily shows a flow chart of a chaos prediction method for an asymmetric piecewise continuous nonlinear constrained system provided by an embodiment of the present invention. As shown in Figure 1, a method for predicting chaos of an asymmetric piecewise continuous nonlinear constrained system provided by an embodiment of the present invention includes the following steps:

步骤S101：根据非对称分段连续非线性约束模型和动力学模型通式，确定非对称分段连续非线性约束系统的动力学模型。Step S101: According to the asymmetric piecewise continuous nonlinear constraint model and the general formula of the dynamic model, determine the dynamic model of the asymmetric piecewise continuous nonlinear constraint system.

具体地，将分段非线性模型各部分进行级数展开得到相应分段非线性的约束形式，从而得到如公式(3)所示的一种非对称分段连续非线性约束模型：Specifically, the series expansion of each part of the piecewise nonlinear model is carried out to obtain the corresponding piecewise nonlinear constraint form, so as to obtain an asymmetric piecewise continuous nonlinear constraint model as shown in formula (3):

$G G ((x x)) = = \{\begin{matrix} {κ κ}_{11} x x + + {γ γ}_{11} {x x}^{33} & x x > > - - {e e}_{22} \\ {κ κ}_{22} x x + + {γ γ}_{22} {x x}^{33} & - - {e e}_{11} \leq \leq x x \leq \leq - - {e e}_{22} \\ {κ κ}_{33} x x + + {γ γ}_{33} {x x}^{33} & x x < < - - {e e}_{11} \end{matrix} - - - - - - ((22))$

在(3)式中：x为约束条件下的位移；G(x)为约束力；κ_i(i＝1,2,3)为等效刚度系数；γ_i(i＝1,2,3)为非线性刚度系数；e₁和e₂为两种弹性体由于预紧力所产生的弹性变形量。In formula (3): x is the displacement under constraint conditions; G(x) is the constraint force; κ _i (i=1,2,3) is the equivalent stiffness coefficient; γ _i (i=1,2,3 ) is the nonlinear stiffness coefficient; e ₁ and e ₂ are the elastic deformation of the two elastic bodies due to the pre-tightening force.

需要说明的是，由于受到约束作用表现为多分段非线性，在振动过程中系统所受到的弹性力随位移的变化也表现出分段非线性特征，并且在x＝-e₁和x＝-e₂处分别有明显的弹性力变化，各段具有不同非线性弹性力，分段处(x＝-e₂,x＝-e₁)具有连续性，并且约束参数满足条件It should be noted that, due to the restraint effect, which is multi-segment nonlinear, the change of the elastic force on the system with displacement also shows a segment nonlinear feature during the vibration process, and at x=-e ₁ and x= There are obvious elastic force changes at -e ₂ respectively, and each segment has different nonlinear elastic forces, and the segmental position (x=-e ₂ , x=-e ₁ ) has continuity, and the constraint parameters meet the conditions

$\{\begin{matrix} \underset{x x &RightArrow; &Right Arrow; - - {e e}_{22}}{lim lim} {κ κ}_{11} x x + + {γ γ}_{11} {x x}^{33} = = - - {κ κ}_{22} {e e}_{22} - - {γ γ}_{22} {e e}_{22}^{33} \\ \underset{x x &RightArrow; &Right Arrow; - - {e e}_{11}}{lim lim} {κ κ}_{33} x x + + {γ γ}_{33} {x x}^{33} = = - - {κ κ}_{22} {e e}_{11} - - {γ γ}_{22} {e e}_{11}^{33} \end{matrix} - - - - - - ((44))$

需要说明的是，将(3)式引入振动系统的动力学模型中，确定非对称分段连续非线性约束系统的动力学模型：It should be noted that the dynamic model of the asymmetric piecewise continuous nonlinear constraint system is determined by introducing formula (3) into the dynamic model of the vibration system:

在(5)式中：为x的一阶导数，为x的二阶导数，为系统阻尼比；k为刚度，m为质量，μ＝1/m；qsin(ωt)为周期外激励，q为周期外激励振幅，ω为周期外激励角频率，t为时间。In formula (5): is the first derivative of x, is the second derivative of x, is the system damping ratio; k is the stiffness, m is the mass, μ=1/m; qsin(ωt) is the external excitation, q is the amplitude of the external excitation, ω is the angular frequency of the external excitation, and t is time.

需要说明的是，动力学模型通式指的是振动系统的动力学模型，即(5)式中的G(x)为任意函数时为动力学模型通式。It should be noted that the general formula of the dynamic model refers to the dynamic model of the vibration system, that is, when G(x) in formula (5) is an arbitrary function, it is the general formula of the dynamic model.

步骤S102：根据动力学模型，通过分析非对称分段连续非线性约束系统的鞍点，确定鞍点的同宿轨道参数方程。Step S102: According to the dynamic model, by analyzing the saddle point of the asymmetric piecewise continuous nonlinear constraint system, determine the homoclinic orbit parameter equation of the saddle point.

需要说明的是，将非对称分段连续非线性约束系统的动力学模型的微分方程转换为非对称分段连续非线性约束系统的动力学模型的状态方程。It should be noted that the differential equation of the dynamic model of the asymmetric piecewise continuous nonlinear constraint system is transformed into the state equation of the dynamic model of the asymmetric piecewise continuous nonlinear constraint system.

具体地，将(5)式写作状态方程形式：Specifically, formula (5) is written in the state equation form:

在(6)式中：为y的一阶导数，ε为常数，且0＜ε＜＜1。In formula (6): is the first derivative of y, ε is a constant, and 0<ε<<1.

需要说明的是，根据非对称分段连续非线性约束系统的动力学模型的状态方程，确定非对称分段连续非线性约束系统的Hamilton函数。It should be noted that, according to the state equation of the dynamic model of the asymmetric piecewise continuous nonlinear constraint system, the Hamilton function of the asymmetric piecewise continuous nonlinear constraint system is determined.

具体地，根据(6)式确定系统的Hamilton量：Specifically, the Hamilton quantity of the system is determined according to formula (6):

$H h ((x x,, y the y)) = = \frac{11}{22} {y the y}^{22} + + \frac{11}{22} {ω ω}_{00}^{22} {x x}^{22} + + \{\begin{matrix} \frac{11}{22} {μκ μκ}_{11} {x x}^{22} + + \frac{11}{44} {μγ μγ}_{11} {x x}^{44} & x x > > - - {e e}_{22} \\ \frac{11}{22} {μκ μκ}_{22} {x x}^{22} + + \frac{11}{44} {μγ μγ}_{22} {x x}^{44} & - - {e e}_{11} \leq \leq x x \leq \leq - - {e e}_{22} \\ \frac{11}{22} {μκ μκ}_{33} {x x}^{22} + + \frac{11}{44} {μγ μγ}_{33} {x x}^{44} & x x < < - - {e e}_{11} \end{matrix} - - - - - - ((77))$

需要说明的是，根据非对称分段连续非线性约束系统的Hamilton函数，通过分析非对称分段连续非线性约束系统的鞍点，确定非对称分段连续非线性约束系统在鞍点的同宿轨道参数方程。It should be noted that, according to the Hamilton function of the asymmetric piecewise continuous nonlinear constraint system, by analyzing the saddle point of the asymmetric piecewise continuous nonlinear constraint system, the homoclinic orbit parameter equation of the asymmetric piecewise continuous nonlinear constraint system at the saddle point is determined .

具体地，图2(a)为本发明实施例提供的当k+κ_i＜0时非对称分段非线性约束系统的势函数曲线。图2(b)为本发明实施例提供的当k+κ_i＜0时非对称分段非线性约束系统的相轨迹。当k+κ_i＜0时，奇点(0,0)为鞍点，为中心点，根据图2(a)和图2(b)可获知，中心点在三段约束区间之间相应切换。Specifically, Fig. 2(a) is a potential function curve of an asymmetric piecewise nonlinear constraint system provided by an embodiment of the present invention when k+κ _i <0. FIG. 2( b ) is the phase trajectory of the asymmetric piecewise nonlinear constrained system when k+κ _i <0 provided by the embodiment of the present invention. When k+κ _i <0, the singular point (0,0) is a saddle point, is the center point, according to Figure 2(a) and Figure 2(b), it can be known that the center point switches between the three constraint intervals accordingly.

具体地，图3(a)为本发明实施例提供的当k+κ_i＞0时非对称分段非线性约束系统的势函数曲线。图3(b)为本发明实施例提供的当k+κ_i＞0时非对称分段非线性约束系统的相轨迹。当k+κ_i＞0时，系统只有一个奇点(0,0)为中心点，根据图3(a)和图3(b)可获知，系统为稳定的周期运动，不存在混沌运动。Specifically, Fig. 3(a) is a potential function curve of an asymmetric piecewise nonlinear constraint system provided by an embodiment of the present invention when k+κ _i >0. Fig. 3(b) is the phase trajectory of the asymmetric piecewise nonlinear constrained system provided by the embodiment of the present invention when k+κ _i >0. When k+κ _i >0, the system has only one singular point (0,0) as the center point. According to Figure 3(a) and Figure 3(b), it can be known that the system is a stable periodic motion without chaotic motion.

因此，只需分析当k+κ_i＜0时的情况，双曲鞍点(0,0)的稳定流形和不稳定流形重合构成的同宿轨道满足微分方程：Therefore, it is only necessary to analyze the situation when k+κ _i <0, the homoclinic orbit formed by the coincidence of the stable manifold and the unstable manifold of the hyperbolic saddle point (0,0) satisfies the differential equation:

H(x,y)＝0 (8)H(x,y)=0 (8)

设起始t＝0时由(8)式可解得(i＝1,2,3)，并且存在Let start t=0 From (8) can be solved (i=1,2,3), and there exists

${&Integral; &Integral;}_{{x x}_{00 i i}}^{x x} \frac{11}{&PlusMinus; &PlusMinus; x x \sqrt{- - μ μ ((k k + + {κ κ}_{i i})) - - \frac{11}{22} {μγ μγ}_{i i} {x x}^{22}}} d d x x = = t t - - - - - - ((99))$

根据系统不同约束情形下中心点，对(9)式积分和简化之后，得到系统在鞍点(0,0)的同宿轨道参数方程为：According to the central point of the system under different constraints, after integrating and simplifying equation (9), the homoclinic orbit parameter equation of the system at the saddle point (0,0) is obtained as:

在(10)式中，i＝1,2,3。In formula (10), i=1,2,3.

步骤S103：根据动力学模型和Melnikov函数通式，确定非对称分段连续非线性约束系统的Melnikov函数。Step S103: According to the dynamic model and the general formula of the Melnikov function, determine the Melnikov function of the asymmetric piecewise continuous nonlinear constraint system.

具体地，结合(6)式，定义系统同宿轨道Melnikov函数Specifically, combined with formula (6), the Melnikov function of the homoclinic orbit of the system is defined

在(11)式中， In formula (11),

需要说明的是，初始时刻t₀为任意实数，根据(11)式得系统Melnikov函数为：It should be noted that the initial time _t0 is any real number, and the Melnikov function of the system according to formula (11) is:

需要说明的是，f(x,y)和g(x,y)两函数为任意函数时候(11)式为Me l n i kov函数通式。It should be noted that when the two functions f(x, y) and g(x, y) are arbitrary functions, formula (11) is the general formula of Mel n i kov function.

需要说明的是，本发明实施例中步骤S103和步骤S104在执行时没有先后顺序的限制。It should be noted that there is no restriction on the order of execution of step S103 and step S104 in the embodiment of the present invention.

步骤S104：根据非对称分段连续非线性约束模型和同宿轨道参数方程，确定Melnikov函数的积分上下限。Step S104: According to the asymmetric piecewise continuous nonlinear constraint model and the homoclinic orbit parameter equation, determine the integral upper and lower limits of the Melnikov function.

需要说明的是，结合(4)式和(10)式，根据系统约束分段条件(x＞-e₂)确定积分上限根据系统约束分段条件(x＜-e₁)确定积分下限 It should be noted that, in combination with formula (4) and formula (10), the upper limit of integration is determined according to the subsection condition of system constraints (x>-e ₂ ) Determine the lower limit of the integral according to the subsection condition of the system constraint (x<-e ₁ )

对(12)式的Melnikov函数进一步求解可得：To further solve the Melnikov function of formula (12), we can get:

在(1)式中， In formula (1),

${Q Q}_{22} = = {&Integral; &Integral;}_{- - ∞ ∞}^{{t t}_{11}} {y the y}_{- -} ((t t)) s the s i i n no ((ω ω t t)) d d t t$

${Q Q}_{33} = = {&Integral; &Integral;}_{{t t}_{22}}^{∞ ∞} {y the y}_{+ +} ((t t)) c c o o s the s ((ω ω t t)) d d t t$

${Q Q}_{44} = = {&Integral; &Integral;}_{{t t}_{22}}^{∞ ∞} {y the y}_{+ +} ((t t)) s the s i i n no ((ω ω t t)) d d t t . .$

将积分上、下限代入(1)式，令χ＝-μ(k+κ₂)，简化可得：Substituting the upper and lower limits of the integral into formula (1), let χ=-μ(k+κ ₂ ), simplify and get:

$\begin{matrix} {&Integral; &Integral;}_{- - ∞ ∞}^{{t t}_{11}} {x x}_{- -} ((t t - - τ τ)) {y the y}_{- -} ((t t)) d d t t = = - - \frac{22}{{μγ μγ}_{22}} \sqrt{{χ χ}^{33}} {&Integral; &Integral;}_{- - ∞ ∞}^{{t t}_{11}} sec sec h h ((\sqrt{χ χ} ((t t - - τ τ)))) sec sec h h ((\sqrt{χ χ} t t)) tanh tanh ((\sqrt{χ χ} t t)) d d t t \\ = = - - \frac{22 {e e}^{τ τ}}{{μγ μγ}_{22}} \sqrt{{χ χ}^{33}} {&Integral; &Integral;}_{- - ∞ ∞}^{{t t}_{11}} {sech sech}^{22} ((\sqrt{χ χ} t t)) tanh tanh ((\sqrt{χ χ} t t)) d d t t \\ = = - - \frac{22 {e e}^{τ τ} χ χ}{{μγ μγ}_{22}} {&Integral; &Integral;}_{- - ∞ ∞}^{\sqrt{χ χ} {t t}_{11}} {sech sech}^{22} ((\sqrt{χ χ} t t)) tanh tanh ((\sqrt{χ χ} t t)) d d ((\sqrt{χ χ} t t)) \\ = = \frac{22 {e e}^{τ τ} χ χ}{{μγ μγ}_{22}} {&Integral; &Integral;}_{00}^{sec sec h h ((\sqrt{χ χ} {t t}_{11}))} sec sec h h ((\sqrt{χ χ} t t)) d d ((sec sec h h ((\sqrt{χ χ} t t)))) \\ = = \frac{{e e}^{τ τ} χ χ}{{μγ μγ}_{22}} {sech sech}^{22} ((\sqrt{χ χ} {t t}_{11})) \end{matrix}$

$\begin{matrix} {&Integral; &Integral;}_{{t t}_{22}}^{∞ ∞} {x x}_{+ +} ((t t - - τ τ)) {y the y}_{+ +} ((t t)) d d t t = = - - \frac{22}{{μγ μγ}_{22}} \sqrt{{χ χ}^{33}} {&Integral; &Integral;}_{{t t}_{22}}^{∞ ∞} sec sec h h ((\sqrt{χ χ} ((t t - - τ τ)))) sec sec h h ((\sqrt{χ χ} t t)) tanh tanh ((\sqrt{χ χ} t t)) d d t t \\ = = - - \frac{22 {e e}^{τ τ}}{{μγ μγ}_{22}} \sqrt{{χ χ}^{33}} {&Integral; &Integral;}_{{t t}_{22}}^{∞ ∞} {sech sech}^{22} ((\sqrt{χ χ} t t)) tanh tanh ((\sqrt{χ χ} t t)) d d t t \\ = = - - \frac{22 {e e}^{τ τ} χ χ}{{μγ μγ}_{22}} {&Integral; &Integral;}_{\sqrt{χ χ} {t t}_{22}}^{∞ ∞} {sech sech}^{22} ((\sqrt{χ χ} t t)) tanh tanh ((\sqrt{χ χ} t t)) d d ((\sqrt{χ χ} t t)) \\ = = \frac{22 {e e}^{τ τ} χ χ}{{μγ μγ}_{22}} {&Integral; &Integral;}_{sec sec h h ((\sqrt{χ χ} {t t}_{22}))}^{∞ ∞} sec sec h h ((\sqrt{χ χ} t t)) d d ((sec sec h h ((\sqrt{χ χ} t t)))) \\ = = - - \frac{{e e}^{τ τ} χ χ}{{μγ μγ}_{22}} {sech sech}^{22} ((\sqrt{χ χ} {t t}_{22})) \end{matrix}$

根据上面两个中间推导式，可解得：According to the above two intermediate derivations, we can get:

${J J}_{11} = = {&Integral; &Integral;}_{- - ∞ ∞}^{{t t}_{11}} {(({y the y}_{- -} ((t t))))}^{22} d d t t = = \frac{22 \sqrt{{χ χ}^{33}} (({cosh cosh}^{33} ((\sqrt{χ χ} {t t}_{11})) - - sinh sinh ((\sqrt{χ χ} {t t}_{11})) + + sinh sinh ((\sqrt{χ χ} {t t}_{11})) cosh cosh ((\sqrt{χ χ} {t t}_{11}))))}{- - 33 {μγ μγ}_{22} {cosh cosh}^{33} ((\sqrt{χ χ} {t t}_{11}))}$

${J J}_{22} = = {&Integral; &Integral;}_{{t t}_{22}}^{∞ ∞} {(({y the y}_{+ +} ((t t))))}^{22} d d t t = = \frac{22 \sqrt{{χ χ}^{33}} (({cosh cosh}^{33} ((\sqrt{χ χ} {t t}_{22})) + + sinh sinh ((\sqrt{χ χ} {t t}_{22})) - - sinh sinh ((\sqrt{χ χ} {t t}_{22})) cosh cosh ((\sqrt{χ χ} {t t}_{22}))))}{- - 33 {μγ μγ}_{22} {cosh cosh}^{33} ((\sqrt{χ χ} {t t}_{22}))}$

步骤S105：根据Melnikov函数和积分上下限，通过公式(2)，确定非对称分段连续非线性约束系统在混沌运动时的临界条件。Step S105: According to the Melnikov function and the upper and lower limits of the integral, through the formula (2), determine the critical condition of the asymmetric piecewise continuous nonlinear constraint system during the chaotic motion.

具体地，当M(t₀)＝0时，将(1)式变形可得：Specifically, when M(t ₀ )=0, the formula (1) can be transformed to get:

根据正弦函数的基本特征，由(13)式可得到系统发生混沌运动时的临界条件解析表达式：According to the basic characteristics of the sine function, the analytical expression of the critical condition when the system occurs chaotic motion can be obtained from formula (13):

需要说明的是，根据非对称分段连续非线性约束系统发生混沌运动时的临界条件表达式，确定非对称分段连续非线性约束系统的混沌运动状态。It should be noted that the chaotic motion state of the asymmetric piecewise continuous nonlinear constraint system is determined according to the critical condition expression when the chaotic motion occurs in the asymmetric piecewise continuous nonlinear constraint system.

步骤S106：根据等效刚度系数、非线性刚度系数、刚度、质量、阻尼比和弹性变形量的实测数据，以及所述混沌运动时的临界条件，确定所述非对称分段连续非线性约束系统在混沌运动时的周期外激励幅值条件。Step S106: According to the measured data of equivalent stiffness coefficient, nonlinear stiffness coefficient, stiffness, mass, damping ratio and elastic deformation, and the critical conditions of the chaotic motion, determine the asymmetric piecewise continuous nonlinear constraint system Out-of-period excitation amplitude conditions during chaotic motion.

步骤S107：根据周期外激励幅值的实测数据和混沌运动时的周期外激励幅值条件，确定所述非对称分段连续非线性约束系统的混沌运动状态。Step S107: According to the measured data of the extra-period excitation amplitude and the condition of the extra-period excitation amplitude during chaotic motion, determine the chaotic motion state of the asymmetric piecewise continuous nonlinear constrained system.

本发明实施例提供的一种非对称分段连续非线性约束系统的混沌预测方法，具体实施例如下：An embodiment of the present invention provides a chaos prediction method for an asymmetric piecewise continuous nonlinear constraint system, the specific examples are as follows:

在板带轧机的液压系统中，液压缸和弯辊缸分布于轧辊上、下两侧，液压缸在静态条件下受到预紧力的作用出现的预紧变形量，动态条件下会使辊系受到非对称分段非线性约束。受这种约束作用的影响，轧机辊系出现的振动行为不仅使轧制产品质量难以保证，还会对轧制系统稳定性构成威胁。以受到这种非对称分段非线性约束的板带轧机上辊系为对象，采用集中质量法建立相应的动力学模型，预测该动力学系统出现混沌的临界条件。预测结果可指导工程技术人员进行参数设定和调节，避免因混沌状态的出现所导致的系统失稳。In the hydraulic system of the strip mill, the hydraulic cylinder and the roll bending cylinder are distributed on the upper and lower sides of the roll. The pre-tightening deformation of the hydraulic cylinder under the action of the pre-tightening force under static conditions will make the roll system Subject to asymmetric piecewise nonlinear constraints. Affected by this constraint, the vibration behavior of the rolling mill roll system not only makes it difficult to guarantee the quality of rolled products, but also poses a threat to the stability of the rolling system. Taking the upper roll system of the strip mill subject to the asymmetric piecewise nonlinear constraint as the object, the corresponding dynamic model is established by using the lumped mass method, and the critical condition of chaos in the dynamic system is predicted. The prediction results can guide engineers and technicians to set and adjust parameters to avoid system instability caused by the chaotic state.

选取不同时滞量τ和时滞反馈增益δ，可以得到不同参数条件下振动系统发生混沌运动的临界条件。具体地，系统参数取值κ₁＝-500，γ₁＝1800，κ₂＝-200，γ₂＝600，κ₃＝-584，γ₃＝1200，m＝1，k＝100，μ＝1，e₁＝0.8，e₂＝-0.5，当时滞反馈增益δ＝5，时滞量τ＝0.5时，根据临界条件表达式(13)计算得到系统发生混沌的外激励幅值条件为q≥82.8。By selecting different time-delay values τ and time-delay feedback gains δ, the critical conditions for chaotic motion of vibration systems under different parameter conditions can be obtained. Specifically, the system parameters take values κ ₁ =-500, γ ₁ =1800, κ ₂ =-200, γ ₂ =600, κ ₃ =-584, γ ₃ =1200, m=1, k=100, μ= 1, e ₁ =0.8, e ₂ =-0.5, when the time-delay feedback gain δ=5 and the time-delay τ=0.5, according to the critical condition expression (13), the external excitation amplitude condition for system chaos to occur is q≥82.8 .

图4为本发明实施例提供的当δ＝5，τ＝0.5时非对称分段非线性约束系统的分岔图及最大Lyapunov指数曲线。如图4所示的分岔图和最大Lyapunov指数曲线(MLE)，通过对照分析可知，系统在q≥82.8时最大Lyapunov指数变为正数，表明系统进入了混沌运动状态，验证了在该参数条件下系统发生混沌的条件的预测结果q≥82.8。FIG. 4 is a bifurcation diagram and a maximum Lyapunov exponent curve of an asymmetric piecewise nonlinear constraint system provided by an embodiment of the present invention when δ=5 and τ=0.5. As shown in Figure 4, the bifurcation diagram and the maximum Lyapunov exponent curve (MLE), through comparative analysis, the maximum Lyapunov exponent of the system becomes positive when q ≥ 82.8, indicating that the system has entered a state of chaotic motion. Under the condition that the system is chaotic, the prediction result q≥82.8.

基于同一发明构思，本发明实施例提供了一种非对称分段连续非线性约束系统的混沌预测装置，由于该装置解决技术问题的原理与一种非对称分段连续非线性约束系统的混沌预测装置方法相似，因此该装置的实施可以参见方法的实施，重复之处不再赘述。Based on the same inventive concept, the embodiment of the present invention provides a chaos prediction device for an asymmetric piecewise continuous nonlinear constraint system, because the principle of the device for solving technical problems is the same as that of an asymmetric piecewise continuous nonlinear constraint system. The device and method are similar, so the implementation of the device can refer to the implementation of the method, and the repetition will not be repeated.

图5为本发明实施例提供的一种非对称分段连续非线性约束系统的混沌预测装置结构示意图。如图5所示，本发明实施例提供的一种非对称分段连续非线性约束系统的混沌预测装置包括：Fig. 5 is a schematic structural diagram of a chaos prediction device for an asymmetric piecewise continuous nonlinear constraint system provided by an embodiment of the present invention. As shown in Figure 5, an asymmetric piecewise continuous nonlinear constraint system chaos prediction device provided by an embodiment of the present invention includes:

第一确定单元51，用于根据非对称分段连续非线性约束模型和动力学模型通式，确定非对称分段连续非线性约束系统的动力学模型；The first determination unit 51 is used to determine the dynamic model of the asymmetric piecewise continuous nonlinear constraint system according to the asymmetric piecewise continuous nonlinear constraint model and the general formula of the dynamic model;

第二确定单元52，用于根据所述动力学模型，通过分析所述非对称分段连续非线性约束系统的鞍点，确定所述鞍点的同宿轨道参数方程；The second determination unit 52 is used to determine the homoclinic orbit parameter equation of the saddle point by analyzing the saddle point of the asymmetric piecewise continuous nonlinear constraint system according to the dynamic model;

第三确定单元53，用于根据所述动力学模型和Melnikov函数通式，通过公式(1)，确定所述非对称分段连续非线性约束系统的Melnikov函数；The third determining unit 53 is used to determine the Melnikov function of the asymmetric piecewise continuous nonlinear constraint system according to the dynamic model and the general formula of the Melnikov function through formula (1);

第四确定单元54，用于根据所述非对称分段连续非线性约束模型和所述同宿轨道参数方程，确定所述Melnikov函数的积分上下限；The fourth determination unit 54 is used to determine the upper and lower integral limits of the Melnikov function according to the asymmetric piecewise continuous nonlinear constraint model and the homoclinic orbital parameter equation;

第五确定单元55，用于根据所述Melnikov函数和所述积分上下限，通过公式(2)，确定所述非对称分段连续非线性约束系统在混沌运动时的临界条件；The fifth determining unit 55 is used to determine the critical condition of the asymmetric piecewise continuous nonlinear constraint system during chaotic motion according to the Melnikov function and the upper and lower limits of the integral, by formula (2);

第六确定单元56，用于根据等效刚度系数、非线性刚度系数、刚度、质量、阻尼比和弹性变形量的实测数据，以及所述混沌运动时的临界条件，确定所述非对称分段连续非线性约束系统在混沌运动时的周期外激励幅值条件；The sixth determination unit 56 is used to determine the asymmetric segment according to the measured data of equivalent stiffness coefficient, nonlinear stiffness coefficient, stiffness, mass, damping ratio and elastic deformation, and critical conditions during the chaotic motion Conditions of excitation amplitude outside the period of continuous nonlinear constrained system in chaotic motion;

第七确定单元57，用于根据周期外激励幅值的实测数据和混沌运动时的周期外激励幅值条件，确定所述非对称分段连续非线性约束系统的混沌运动状态；The seventh determination unit 57 is used to determine the chaotic motion state of the asymmetric piecewise continuous nonlinear constraint system according to the measured data of the excitation amplitude outside the period and the excitation amplitude condition outside the period during chaotic motion;

公式(1)如下所示：Formula (1) is as follows:

公式(2)如下所示：Formula (2) is as follows:

较佳地，所述第一确定单元51具体用于：Preferably, the first determining unit 51 is specifically configured to:

较佳地，所述第二确定单元52具体用于：Preferably, the second determining unit 52 is specifically configured to:

应当理解，以上非对称分段连续非线性约束系统的混沌预测装置包括的单元仅为根据该设备装置实现的功能进行的逻辑划分，实际应用中，可以进行上述单元的叠加或拆分。并且该实施例提供的非对称分段连续非线性约束系统的混沌预测装置所实现的功能与上述实施例提供的非对称分段连续非线性约束系统的混沌预测装置方法一一对应，对于该装置所实现的更为详细的处理流程，在上述方法实施例一中已做详细描述，此处不再详细描述。It should be understood that the units included in the chaos prediction device for the above asymmetric piecewise continuous nonlinear constraint system are only logically divided according to the functions realized by the device, and in practical applications, the above units can be superimposed or split. And the functions realized by the chaos prediction device of the asymmetric piecewise continuous nonlinear constraint system provided by this embodiment correspond one-to-one to the method of the chaos prediction device of the asymmetric piecewise continuous nonlinear constraint system provided by the above-mentioned embodiment, for the device The more detailed processing flow implemented has been described in detail in the first method embodiment above, and will not be described in detail here.

综上所述，本发明实施例提供的一种非对称分段连续非线性约束系统的混沌预测方法，该方法给出了系统在鞍点处由三段不同轨道组成的同宿轨道参数方程，根据约束的分段条件确定积分上下限并通过分段积分得到Melnikov函数，很好的解决了非对称分段非线性约束系统无法直接得到Melnikov函数来预测混沌的难点；进一步，非对称分段非线性约束系统发生混沌时的临界条件解析表达式，对于具有非对称分段非线性约束特征的系统混沌运动状态的预测问题提供了一种快速、有效的解析方法，根据预测结果能够通过调节控制参数对系统混沌运动行为进行控制，避免由于出现混沌运动状态而导致系统失稳。对于非对称分段连续非线性约束系统的混沌预测结果可以指导设计人员优化轧制设备结构和工艺参数，指导工程技术人员进行参数设定和调节，避免因混沌状态的出现所导致的系统失稳。To sum up, the embodiment of the present invention provides a chaos prediction method for an asymmetric piecewise continuous nonlinear constrained system. This method gives the homoclinic orbit parameter equation composed of three different orbits at the saddle point of the system. According to the constraint The segmental conditions determine the upper and lower limits of the integral and obtain the Melnikov function by segmental integration, which solves the difficulty that the asymmetric segmental nonlinear constraint system cannot directly obtain the Melnikov function to predict chaos; further, the asymmetric segmental nonlinear constraint The analytical expression of the critical condition when the system is chaotic provides a fast and effective analytical method for the prediction of the chaotic motion state of the system with the characteristics of asymmetric piecewise nonlinear constraints. According to the prediction results, the system can be adjusted by adjusting the control parameters The chaotic motion behavior is controlled to avoid system instability due to the chaotic motion state. The prediction results of chaos for asymmetric piecewise continuous nonlinear constraint systems can guide designers to optimize the structure and process parameters of rolling equipment, guide engineering and technical personnel to set and adjust parameters, and avoid system instability caused by the emergence of chaotic states .

以上公开的仅为本发明的几个具体实施例，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，倘若本发明的这些修改和变型属于本发明权利要求及其等同技术的范围之内，则本发明也意图包含这些改动和变型在内。The above disclosures are only a few specific embodiments of the present invention, and those skilled in the art can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention, provided that these modifications and modifications of the present invention belong to the rights of the present invention The present invention also intends to include these modifications and variations within the scope of the requirements and their technical equivalents.

Claims

1. A chaotic prediction method of an asymmetric piecewise continuous nonlinear constraint system is characterized by comprising the following steps:

determining a dynamic model of the asymmetric piecewise continuous nonlinear constraint system according to the asymmetric piecewise continuous nonlinear constraint model and the general formula of the dynamic model;

according to the dynamic model, determining a co-homing orbit parameter equation of the saddle point by analyzing the saddle point of the asymmetric piecewise continuous nonlinear constraint system;

determining the Melnikov function of the asymmetric piecewise continuous nonlinear constraint system through a formula (1) according to the dynamic model and the Melnikov function general formula;

determining an upper and lower integral limit of the Melnikov function according to the asymmetric piecewise continuous nonlinear constraint model and the co-homing orbit parameter equation;

determining the critical condition of the asymmetric piecewise continuous nonlinear constraint system in chaotic motion through a formula (2) according to the Melnikov function and the upper and lower integral limits;

according to the measured data of the equivalent stiffness coefficient, the nonlinear stiffness coefficient, the stiffness, the mass, the damping ratio and the elastic deformation and the critical condition during the chaotic motion, determining the out-of-period excitation amplitude condition of the asymmetric piecewise continuous nonlinear constraint system during the chaotic motion;

determining the chaotic motion state of the asymmetric piecewise continuous nonlinear constraint system according to the measured data of the periodic external excitation amplitude and the periodic external excitation amplitude condition during chaotic motion;

equation (1) is as follows:

equation (2) is as follows:

wherein, t₀Is an initial time and is any real number; e₁and e₂In order to have an elastic deformation amount,for damping ratio, qsin (ω t) is the out-of-period excitation, q is the out-of-period excitation amplitude, ω is the out-of-period excitation angular frequency,k is stiffness, m is mass,. mu.1/m,. kappa₁、κ₂And kappa₃Is an equivalent stiffness coefficient, γ₁、γ₂And gamma₃Is a nonlinear stiffness coefficient, t₁And t₂Are the upper and lower limits of the integral of the Melnikov function, χ ═ μ (k + κ), respectively₂)。

2. The chaotic prediction method for the asymmetric piecewise continuous non-linear constraint system according to claim 1, wherein the determining the dynamical model of the asymmetric piecewise continuous non-linear constraint system according to the asymmetric piecewise continuous non-linear constraint model and the dynamical model general formula comprises:

the asymmetric piecewise continuous nonlinear constraint model is determined by the following formula:

G (x) = \{\begin{matrix} κ_{1} x + γ_{1} x^{3} & x > - e_{2} \\ κ_{2} x + γ_{2} x^{3} & - e_{1} \leq x \leq - e_{2} \\ κ_{3} x + γ_{3} x^{3} & x < - e_{1} \end{matrix}

the condition that the asymmetric piecewise continuous nonlinear constraint model has continuity at the piecewise point is as follows:

\{\begin{matrix} \lim_{x &RightArrow; - e_{2}} κ_{1} x + γ_{1} x^{3} = - κ_{2} e_{2} - γ_{2} e_{2}^{3} \\ \lim_{x &RightArrow; - e_{1}} κ_{3} x + γ_{3} x^{3} = - κ_{2} e_{1} - γ_{2} e_{1}^{3} \end{matrix}

the kinetic model is determined by the following formula:

wherein x is the displacement under the constraint condition,is the first derivative of x and is,is the second derivative of x, and G (x) is the restraining force.

3. The chaotic prediction method for the asymmetric piecewise continuous non-linear constraint system according to claim 1, wherein the determining the co-homing orbit parameter equation of the saddle point by analyzing the saddle point of the asymmetric piecewise continuous non-linear constraint system according to the dynamical model comprises:

determining a state equation of the dynamic model according to the dynamic model, wherein the state equation of the dynamic model is as follows:

determining a Hamilton function of the asymmetric piecewise continuous nonlinear constraint system according to a state equation of the dynamic model, wherein the Hamilton function is as follows:

H (x, y) = \frac{1}{2} y^{2} + \frac{1}{2} ω_{0}^{2} x^{2} + \{\begin{matrix} \frac{1}{2} {μκ}_{1} x^{2} + \frac{1}{4} {μγ}_{1} x^{4} & x > - e_{2} \\ \frac{1}{2} {μκ}_{2} x^{2} + \frac{1}{4} {μγ}_{2} x^{4} & - e_{1} \leq x \leq - e_{2} \\ \frac{1}{2} {μκ}_{3} x^{2} + \frac{1}{4} {μγ}_{3} x^{4} & x < - e_{1} \end{matrix}

when the Hamilton function meets H (x, y) being 0, determining the co-homing orbit parameter equation of the saddle point according to the position of the central point under different constraint conditions of the asymmetric piecewise continuous nonlinear constraint system by the following formula:

wherein,is the first derivative of y, is a constant, and 0 < 1, i 1,2, 3.

4. A chaos prediction device of an asymmetric piecewise continuous nonlinear constraint system is characterized by comprising:

the first determining unit is used for determining a dynamic model of the asymmetric piecewise continuous non-linear constraint system according to the asymmetric piecewise continuous non-linear constraint model and the general formula of the dynamic model;

the second determining unit is used for determining a co-homing orbit parameter equation of the saddle point by analyzing the saddle point of the asymmetric piecewise continuous nonlinear constraint system according to the dynamic model;

the third determining unit is used for determining the Melnikov function of the asymmetric piecewise continuous nonlinear constraint system through a formula (1) according to the dynamic model and the Melnikov function general formula;

a fourth determining unit, configured to determine an upper and lower integral limit of the Melnikov function according to the asymmetric piecewise continuous nonlinear constraint model and the co-homing orbit parameter equation;

a fifth determining unit, configured to determine, according to the Melnikov function and the upper and lower integration limits, a critical condition of the asymmetric piecewise continuous nonlinear constraint system during chaotic motion by using a formula (2);

a sixth determining unit, configured to determine, according to the measured data of the equivalent stiffness coefficient, the nonlinear stiffness coefficient, the stiffness, the mass, the damping ratio, and the elastic deformation, and the critical condition during the chaotic motion, an out-of-cycle excitation amplitude condition of the asymmetric piecewise continuous nonlinear constraint system during the chaotic motion;

a seventh determining unit, configured to determine a chaotic motion state of the asymmetric piecewise continuous nonlinear constraint system according to measured data of the periodic external excitation amplitude and a periodic external excitation amplitude condition during chaotic motion;

equation (1) is as follows:

equation (2) is as follows:

wherein, t₀Is an initial time and is any real number; e₁and e₂In order to have an elastic deformation amount,for damping ratio, qsin (ω t) is the out-of-period excitation, q is the out-of-period excitation amplitude, ω is the out-of-period excitation angular frequency,k is stiffness, m is mass,. mu.1/m,. kappa₁、κ₂And kappa₃Is an equivalent stiffness systemNumber, gamma₁、γ₂And gamma₃Is a nonlinear stiffness coefficient, t₁And t₂Are the upper and lower limits of the integral of the Melnikov function, χ ═ μ (k + κ), respectively₂)。

5. The chaotic prediction device of the asymmetric piecewise-continuous nonlinear constraint system according to claim 4, wherein the first determining unit is specifically configured to:

G (x) = \{\begin{matrix} κ_{1} x + γ_{1} x^{3} & x > - e_{2} \\ κ_{2} x + γ_{2} x^{3} & - e_{1} \leq x \leq - e_{2} \\ κ_{3} x + γ_{3} x^{3} & x < - e_{1} \end{matrix}

\{\begin{matrix} \lim_{x &RightArrow; - e_{2}} κ_{1} x + γ_{1} x^{3} = - κ_{2} e_{2} - γ_{2} e_{2}^{3} \\ \lim_{x &RightArrow; - e_{1}} κ_{3} x + γ_{3} x^{3} = - κ_{2} e_{1} - γ_{2} e_{1}^{3} \end{matrix}

the kinetic model is determined by the following formula:

wherein x is a constraint stripThe displacement under the element is carried out,is the first derivative of x and is,is the second derivative of x, and G (x) is the restraining force.

6. The chaotic prediction device of the asymmetric piecewise-continuous nonlinear constraint system according to claim 4, wherein the second determining unit is specifically configured to:

H (x, y) = \frac{1}{2} y^{2} + \frac{1}{2} ω_{0}^{2} x^{2} + \{\begin{matrix} \frac{1}{2} {μκ}_{1} x^{2} + \frac{1}{4} {μγ}_{1} x^{4} & x > - e_{2} \\ \frac{1}{2} {μκ}_{2} x^{2} + \frac{1}{4} {μγ}_{2} x^{4} & - e_{1} \leq x \leq - e_{2} \\ \frac{1}{2} {μκ}_{3} x^{2} + \frac{1}{4} {μγ}_{3} x^{4} & x < - e_{1} \end{matrix}

wherein,is the first derivative of y, is a constant, and 0 < 1, i 1,2, 3.