CN111428409B - Equation solving method for mechanical nonlinear vibration analysis - Google Patents

Abstract

The invention discloses a nonlinear dynamical equation solving method and a nonlinear dynamical equation solving system for mechanical nonlinear vibration analysis, wherein the nonlinear dynamical equation solving method comprises the following steps of: constructing and obtaining a nonlinear finite element kinetic equation on a system time domain according to a mechanical nonlinear vibration system to be analyzed; obtaining a nonlinear finite element kinetic equation on a frequency domain by a multi-harmonic balance method; solving a Jacobian matrix of a nonlinear finite element kinetic equation on a frequency domain by using an automatic differentiation method; solving the nonlinear finite element kinetic equation on the frequency domain by adopting an arc length method to obtain the solution of the nonlinear finite element kinetic equation on the frequency domain; and obtaining the solution of the nonlinear finite element kinetic equation in the time domain through the generalized inverse Fourier transform according to the solution of the nonlinear finite element kinetic equation in the frequency domain, and further obtaining a frequency response curve for mechanical nonlinear vibration analysis.

Description

Equation Solving Methods for Mechanical Nonlinear Vibration Analysis

technical field

The invention belongs to the technical field of mechanical vibration safety analysis and nonlinear dynamic equation solving, and in particular relates to a nonlinear dynamic equation solving method and system for mechanical nonlinear vibration analysis.

Background technique

A large number of strong nonlinear phenomena are involved in the vibration safety analysis of mechanical components, such as cracks in rotor dynamics, dry friction damping of turbine blades, etc. It is difficult to solve dynamic equations containing strong nonlinear forces, and harmonic balance method is usually used. The original equations are transformed into balanced equations of harmonic components, and then the nonlinear equations are solved using the Newton iteration method.

The core step of the Newton iteration method to solve the harmonic balance method is to deduce the Jacobian matrix of the harmonic term of the strong nonlinear force to the harmonic term of the displacement. The comparable matrix may even cause the solution process to fail to converge; because the transformation process of the original equation involves generalized Fourier transform, and the nonlinear force expression is complex, the Jacobian matrix cannot be expressed explicitly, and it is necessary to use the chain differentiation rule to determine the specific non-linear force. The re-derivation of the linear force leads to great limitations in the traditional solution method, which is not universal.

In conclusion, a new method and system for solving nonlinear dynamic equations for nonlinear mechanical vibration analysis are urgently needed.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a nonlinear dynamic equation solving method and system for mechanical nonlinear vibration analysis, so as to solve one or more of the above-mentioned technical problems. The main problem to be solved by the present invention is that the existing solution technology is difficult to carry out the vibration safety analysis of mechanical components containing complex nonlinear forces, and there are shortcomings such as heavy reliance on expert experience, consumption of computing resources, etc. The method of the present invention can quickly deduce strong Jacobian matrix of nonlinear force harmonic terms to displacement harmonic terms, thereby speeding up the solution of mechanical nonlinear vibration analysis equations.

To achieve the above object, the present invention adopts the following technical solutions:

A kind of nonlinear dynamic equation solving method for mechanical nonlinear vibration analysis of the present invention comprises the following steps:

Step 1, according to the mechanical nonlinear vibration system to be analyzed, construct and obtain the nonlinear finite element dynamic equation in the time domain of the system, and the expression is:

In formula (1), k, c, m represent the stiffness, viscous damping and mass matrix of the system, respectively; x represents the displacement vector of all degrees of freedom in the system; f _n (x) represents the nonlinear force vector; f _l (t ) represents the external excitation vector;

Step 2, through the multi-harmonic balance method, perform generalized Fourier transform on x and f _l (t) in formula (1) to obtain the nonlinear finite element dynamic equation in the frequency domain;

Step 3, using the automatic differential method to solve the Jacobian matrix of the nonlinear finite element dynamic equation in the frequency domain obtained in step 2;

Step 4: According to the Jacobian matrix obtained in step 3, the arc length method is used to solve the nonlinear finite element dynamic equation in the frequency domain obtained in step 2, and the solution of the nonlinear finite element dynamic equation in the frequency domain is obtained;

Step 5: According to the solution of the nonlinear finite element dynamic equation in the frequency domain obtained in step 4, the solution of the nonlinear finite element dynamic equation in the time domain is obtained through the generalized inverse Fourier transform; according to the nonlinear finite element dynamic equation in the time domain The solution of the element dynamics equation, the frequency response curve is obtained, and the mechanical nonlinear vibration analysis is carried out.

A further improvement of the present invention is that step 2 specifically includes:

Step 2.1, perform Fourier expansion on x in formula (1), and the obtained expression is:

In formula (2):

D=[I,Icos(ωt),Isin(ωt),...,Icos(N _k ωt),Isin(N _k ωt)];

ω is the excitation frequency;

N _k is the number of selected harmonic components;

Step 2.2, perform Fourier expansion on f _l (t) in formula (1), and obtain the expression as:

In formula (3):

D=[I,Icos(ωt),Isin(ωt),...,Icos(N _k ωt),Isin(N _k ωt)];

ω is the excitation frequency;

N _k is the number of selected harmonic components;

Step 2.3: Substitute the Fourier-expanded x and f _l (t) back into equation (1), and obtain the nonlinear finite element dynamic equation expression in the frequency domain as:

R(X)=Z(ω)X+F _n (X)-F _l =0 (4)

In formula (4), X is the amplitude vector of the displacement harmonic components in the direction of each degree of freedom; Z(ω) is the dynamic stiffness matrix in the frequency domain; F _n (X) is the nonlinear force vector in the frequency domain; F _l is the amplitude vector of the harmonic components of the exciting force in the frequency domain;

where Z(ω) is expressed as:

A further improvement of the present invention is that step 3 specifically includes:

Apply Newton's method to solve equation (4); the iterative process is expressed as:

In the formula, k represents the current number of iteration steps;

Combining Equation (5) and Equation (4) to obtain the expression:

具体步骤包括：In formula (6), the automatic differentiation method is used to solve

Specific steps include:

Perform the inverse Fourier transform on the amplitude vector X of the displacement harmonic component in each degree of freedom direction to obtain x, and the expression is:

According to the displacement-nonlinear force relationship, the nonlinear force vector f _n is obtained; the Fourier transform is performed on f _n to obtain the nonlinear force vector F _n in the frequency domain,

Obtain the computation graph of X→x→ _fn → _Fn ;

Find the value of each node by traversing the computational graph in a forward direction;

By traversing the computational graph in reverse, the partial derivative of each node is calculated to obtain

A further improvement of the present invention is that step 4 specifically includes the following steps:

Step 4.1, add λ to formula (4), and obtain the equation system expression containing λ as:

Ψ(X,λ)=Z(ω)X+F _n (X)-λF _l =0; (7)

Construct an N+1-dimensional vector: ν(X,λ)=[ν ₁ ,ν ₂ ,…,ν _N ,ν _N+1 ] ^T ;

式中，符号∧表示删除该列；in,

In the formula, the symbol ∧ means to delete the column;

ν(X,λ) is the tangent vector of the solution curves X(s) and λ(s) in N+1-dimensional space;

式中，||ν||代表ν的模；The unit tangent vector expression of the solution curve is:

In the formula, ||ν|| represents the mode of ν;

For the mth arc length step Δs, the constraint equation is expressed as:

Combining formula (7) and formula (8), the obtained expression is:

The Jacobian matrix expression of formula (9) is:

Step 4.2, in the first iteration, obtain an initial solution X ^* and λ ^* of formula (7) in the weak nonlinear region by Newton's method;

其中，下标m表示第m个弧长步；Step 4.3, estimation step,

Among them, the subscript m represents the mth arc length step;

Step 4.4, correction step, the expression of the iterative process is formula (11); in each iteration step, the approximate solution increment is projected to the direction of τ _m to obtain the arc length increment;

A further improvement of the present invention is that step 5 specifically includes:

Step 5.1, for each solution point in the time domain, convert the displacement vector x _i of each degree of freedom into the total displacement vector x _s of each node of the system;

_{Step 5.2} , select each ^point of the system frequency response curve, and fit all calculated points , obtain the system frequency response curve;

Step 5.3, analyze the frequency response curve of the system, and conduct quantitative research on the nonlinear vibration characteristics of the system.

A further improvement of the present invention is that, in step 1, the mechanical nonlinear vibration system to be analyzed is a turbomachinery blade-disk vibration system considering damping structure friction;

Among them, according to the geometric structure of the turbine blade-disc, a three-dimensional model is established and meshed to form the overall finite element model of the turbine blade-disc;

Determine the contact interface and friction node pairs of each damping structure in the turbine blade-disc model, and perform static contact analysis to obtain the normal preload force N ₀ between the friction node pairs, and obtain the prestress field;

Through the linear finite element method, according to the boundary constraints of the turbomachinery blade-disk structure, and considering the prestress field and the rotational softening effect, the stiffness matrix and mass matrix m of the turbomachinery blade-disk system are obtained;

Assuming that proportional damping is satisfied, the damping matrix c=αk+βm; among them, the experimental measurement of the turbomachinery blade-disc system is carried out to obtain the first two-order resonance frequency and the first two-order mode shape damping ratio of the system, and determine the coefficients α and β , obtain the damping matrix c of the turbomachinery blade-disk system;

The unsteady three-dimensional flow field analysis of the blade is carried out, and the periodic fluid excitation force vector f _l (t) of the turbine blade-disk system is obtained; according to the Coulomb dry friction model, the displacement is affected by the contact interface of each damping structure and the friction node and the normal preload to obtain the nonlinear force vector in the direction of the degrees of freedom of all friction points, and then obtain the nonlinear force vector f _n (x ).

A further improvement of the present invention lies in the dry friction damping for analyzing cracks in rotor dynamics, turbine blades.

A nonlinear dynamic equation solving system for mechanical nonlinear vibration analysis of the present invention includes:

The equation building module in the time domain is used to construct and obtain the nonlinear finite element dynamic equation of the system in the time domain according to the mechanical nonlinear vibration system to be analyzed. The expression is:

In the formula, k, c, m represent the stiffness, viscous damping and mass matrix of the system, respectively; x represents the displacement vector of all degrees of freedom in the system; f _n (x) represents the nonlinear force vector; f _l (t) represents the external incentive vector;

The equation acquisition module in the frequency domain is used to perform the generalized Fourier transform of x and f _l (t) in the nonlinear finite element dynamic equation in the time domain through the multi-harmonic balance method to obtain the nonlinear finite element in the frequency domain. Metadynamic equation;

The Jacobian matrix acquisition module is used to solve the Jacobian matrix of nonlinear finite element dynamic equations in the frequency domain by using the automatic differentiation method;

The frequency domain solution acquisition module is used to solve the nonlinear finite element dynamic equation in the frequency domain by using the arc length method according to the Jacobian matrix obtained by the Jacobian matrix acquisition module, and obtain the solution of the nonlinear finite element dynamic equation in the frequency domain;

The frequency response curve acquisition module is used to obtain the solution of the nonlinear finite element dynamic equation in the frequency domain obtained by the frequency domain solution acquisition module, and obtain the solution of the nonlinear finite element dynamic equation in the time domain through the generalized inverse Fourier transform ;According to the solution of the nonlinear finite element dynamic equation in the time domain, the frequency response curve is obtained, and the mechanical nonlinear vibration analysis is carried out.

Compared with the prior art, the present invention has the following beneficial effects:

计算量极大，在对微分结果进行评估时难度大，也容易对迭代解的准确性及鲁棒性造成影响；而使用解析的方式进行微分，一方面需要较高的专业素养，另一方面需要所研究的对象能够形成清晰的表达式，均难以满足实际问题的应用。以考虑阻尼结构摩碰的透平机械叶片-轮盘振动系统为例，现有的分析方法对阻尼结构的摩碰做了较大的简化，而真实摩擦力的变化较为复杂，常规求解方法往往会因为计算成本高、微分计算难而造成无法求解的情况。而采用本发明的求解方法可以通过位移进行摩擦力的正向计算求解出其微分项，从而实现系统非线性动力学方程的快速求解，使得求解结果更加符合实际情况。In the current vibration safety analysis of mechanical components, it is difficult to solve the dynamic equations containing strong nonlinear forces. Usually, the harmonic balance method is used to transform the original equation into a balance equation of harmonic components, and then the Newton iteration method is used to solve the nonlinear equation. The solution is not universal. In the current nonlinear dynamic equation solving methods, numerical differential methods such as finite difference are used to solve

The amount of calculation is huge, it is difficult to evaluate the differential results, and it is easy to affect the accuracy and robustness of the iterative solution; and the use of analytical methods for differentiation requires high professional quality on the one hand, and on the other hand. Need to study the object can form a clear expression, are difficult to meet the application of practical problems. Taking the turbine blade-disk vibration system considering the friction of damping structure as an example, the existing analysis methods have greatly simplified the friction of damping structure, but the change of the real friction force is more complicated, and the conventional solution methods often It will cause problems that cannot be solved due to the high computational cost and the difficulty of differential calculation. By using the solution method of the present invention, the differential term of the friction force can be solved by the forward calculation of the displacement, thereby realizing the rapid solution of the nonlinear dynamic equation of the system, and making the solution result more in line with the actual situation.

The nonlinear dynamic equation solving method for mechanical nonlinear vibration analysis of the present invention is a nonlinear dynamic equation solving method based on automatic differential harmonic balance; it combines automatic differential method and harmonic balance method, and uses pseudo-harmonic balance method The arc length extension method solves the nonlinear harmonic balance equation, which has a good convergence effect on nonlinear vibration problems, and can adaptively track nonlinear phenomena such as bifurcation and jumping; it is suitable for the Jacobian matrix calculation of arbitrary nonlinear force harmonics , has a very high promotion.

Automatic differentiation is a method between symbolic differentiation and numerical differentiation: numerical differentiation emphasizes directly substituting numerical approximation solutions at the beginning; symbolic differentiation emphasizes solving algebra directly, and then substituting the numerical value of the problem; automatic differentiation applies the symbolic differentiation method to the most Basic operators, then substitute values, keep intermediate results, and finally apply them to the entire function. So it is quite flexible in application, and since its computation is actually a graph computation, it can be optimized a lot. Therefore, the generalized Fourier transform and the nonlinear calculation process are forwarded and placed in the graph calculation, and the Jacobian matrix is solved by automatic differentiation, and then combined with the arc length method to solve the nonlinear dynamic equation, which can be applied to different types of The problem is highly generalizable.

In the present invention, the frequency response curve is analyzed to obtain the influence of the design parameters on the nonlinear vibration characteristics, thereby providing a reference for the structural design. Adjust one or several design parameters that have the greatest influence on nonlinear vibration characteristics, so that the nonlinear vibration is within the set range.

Description of drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the accompanying drawings used in the description of the embodiments or the prior art; obviously, the accompanying drawings in the following description are For some embodiments of the present invention, for those of ordinary skill in the art, other drawings can also be obtained from these drawings without creative efforts.

1 is a schematic flow chart of a method for solving nonlinear dynamic equations for mechanical nonlinear vibration analysis according to an embodiment of the present invention;

FIG. 2 is a schematic diagram of an automatic differentiation process when solving nonlinear force vector differentiation in an embodiment of the present invention.

Detailed ways

In order to make the purposes, technical effects and technical solutions of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention; are some embodiments of the present invention. Based on the embodiments disclosed in the present invention, other embodiments obtained by persons of ordinary skill in the art without creative work shall fall within the protection scope of the present invention.

Referring to FIG. 1, a method for solving nonlinear dynamic equations for nonlinear mechanical vibration analysis according to an embodiment of the present invention is a method for solving nonlinear dynamic equations based on automatic differential-pseudo-arc length extension, including: The following steps:

Step 1, according to the mechanical vibration safety analysis system, construct and obtain the nonlinear dynamic equation:

In formula (1), k, c, and m represent the stiffness, viscous damping and mass matrix of the system, respectively, x represents the displacement vector of all degrees of freedom in the system, and f _n (x) represents the displacement caused by the node (sometimes also includes the displacement velocity). , acceleration) determined nonlinear force vector, f _l (t) represents the external excitation vector.

Optionally, in step 1, the mechanical vibration safety analysis system is a turbomachinery blade-disk vibration system considering the friction of the damping structure. According to the geometric structure of the actual turbomachinery blade-disk Grid division to form the overall finite element model of the turbomachinery blade-disk; determine the contact interface and friction node pairs of each damping structure in the turbomachinery blade-disk model, and perform static contact analysis; carry out the blade-disk Through static contact analysis, the normal preload force N ₀ between the friction node pairs is obtained, and the prestress field is obtained; through the linear finite element method, according to the boundary constraints of the turbine blade-disk structure, the prestress is considered Field and rotational softening effect, the stiffness matrix and mass matrix m of the turbomachinery blade-disk system are obtained; if proportional damping is satisfied, the damping matrix c=αk+βm, the experimental measurement of the turbomachinery blade-disk system can be Obtain the first two-order resonance frequency and the first two-order mode shape damping ratio of the system, from which the values of the coefficients α and β can be determined, thereby obtaining the damping matrix c of the turbomachinery blade-disc system; the aerodynamic analysis of the blade part is The unsteady three-dimensional flow field analysis is carried out to obtain the periodic fluid excitation force vector f _l (t) of the turbomachinery blade-disk system; according to the Coulomb dry friction model, the displacement and displacement of each damping structure contact interface, friction node can be obtained. The normal preload obtains the nonlinear force vector in the direction of the degrees of freedom of all friction points, and further obtains the nonlinear force vector f _n (x) in the direction of the degrees of freedom of each node in the turbomachinery blade-disk system. .

Step 2, by the multi-harmonic balance method, perform generalized Fourier (Fourier) transformation on x and f _l (t) in equation (1), and obtain a frequency domain equation.

Optionally, step 2 specifically includes:

Step 2.1, Fourier expansion of x in equation (1):

In formula (2):

D=[I,Icos(ωt),Isin(ωt),...,Icos(N _k ωt),Isin(N _k ωt)];

ω is the excitation frequency;

N _k is the number of selected harmonic components;

Then, get

Step 2.2, perform Fourier expansion on f _l (t) in equation (1) to obtain:

In formula (3):

D=[I,Icos(ωt),Isin(ωt),...,Icos(N _k ωt),Isin(N _k ωt)];

ω is the excitation frequency;

N _k is the number of selected harmonic components;

Then, get

Step 2.3: Substitute the Fourier-expanded x and f _l (t) back into equation (1) to obtain:

R(X)=Z(ω)X+F _n (X)-F _l =0 (4)

In formula (4), X is the amplitude vector of the displacement harmonic components in the direction of each degree of freedom, Z(ω) is the dynamic stiffness matrix in the frequency domain, F _n (X) is the nonlinear force vector in the frequency domain, F _l is the amplitude vector of the harmonic components of the excitation force in the frequency domain.

In formula (4), Z(ω) can be expressed as:

Step 3, using automatic differentiation method to solve the Jacobian matrix for equation (4).

Equation (4) represents a nonlinear system of equations based on X; at present, when the Newton-Raphson method is used to solve the system of equations, the iterative process can be expressed by the following formula:

In the formula, k represents the current iteration step number.

Formula (5) and formula (4) can be combined to obtain:

时计算量极大，在对微分结果进行评估时难度大，也容易对迭代解的准确性及鲁棒性造成影响；而使用解析的方式进行微分，一方面需要较高的专业素养，另一方面需要所研究的对象能够形成清晰的表达式，均难以满足实际问题的应用。In formula (6), it is solved by numerical differentiation methods such as finite difference

The calculation amount is huge, and it is difficult to evaluate the differential results, and it is easy to affect the accuracy and robustness of the iterative solution; and using the analytical method for differentiation requires high professional quality on the one hand, and another Aspects require that the object under study can form a clear expression, which is difficult to meet the application of practical problems.

Referring to FIG. 2, in the embodiment of the present invention, an automatic differentiation method is used to solve it, and the specific steps include:

First, perform the inverse Fourier transform on the amplitude vector X of the displacement harmonic component in each degree of freedom direction to obtain x

)，即先通过正向遍历计算图求出每个节点的值。Then according to the displacement-nonlinear force relationship, the nonlinear force vector f _n is obtained, and finally the Fourier transform is performed on f _n to obtain the nonlinear force vector F _n in the frequency domain (refer to step 2.2,

), that is, the value of each node is obtained by forward traversal of the computational graph.

Then, by traversing the entire graph in reverse, the partial derivative of each node is calculated. The principle is the chain rule of calculus. Only one forward propagation (solid line process) and one back propagation (dotted line process) can be obtained. The derivatives of all parameters are obtained, which has high performance and good generalization.

In step 4, the arc-length method is used to solve the frequency-domain nonlinear finite element dynamic equation obtained in step 2.3.

Optionally, step 4 specifically includes the following steps:

Step 4.1, supplementary constraint equation:

First, add λ to formula (4) to obtain a system of equations containing λ:

Ψ(X,λ)=Z(ω)X+F _n (X)-λF _l =0; (7)

Then, construct an N+1-dimensional vector: ν(X,λ)=[ν ₁ ,ν ₂ ,...,ν _N ,ν _N+1 ] ^T ;

式中符号∧表示删除该列。in,

The symbol ∧ in the formula means to delete the column.

(式中||ν||代表ν的模)。ν(X,λ) is the tangent vector of the solution curves X(s) and λ(s) in N+1-dimensional space, and the unit tangent vector of the solution curve is:

(where ||ν|| represents the mode of ν).

For the mth arc length step Δs, the constraint equation is:

Then the new system of solving equations becomes:

Then the Jacobian matrix of equation (9) can be written as:

In step 4.2, in the first iteration, an initial solution X ^* and λ ^* of equation (7) in the weak nonlinear region is obtained by Newton-Raphson.

其中，下标m表示第m个弧长步。Step 4.3, estimation step;

where the subscript m represents the mth arc length step.

Step 4.4, correction step; the iterative process is as follows (11), in each iteration step, the approximate solution increment is projected to the direction of τ _m to obtain the arc length increment.

获得时域下的解点，画出研究问题的频响曲线，并可以进一步对非线性振动特性进行分析。Step 5, according to

The solution points in the time domain are obtained, the frequency response curve of the research problem is drawn, and the nonlinear vibration characteristics can be further analyzed.

Optionally, step 5 specifically includes the following steps:

Step 5.1, for each solution point in the time domain, convert the displacement vector _xi of each degree of freedom into the total displacement vector x _s of each node of the system.

_{Step 5.2} , select each ^point of the system frequency response curve, and fit all calculated points , so as to obtain the system frequency response curve.

In step 5.3, by analyzing the frequency response curve of the system, conduct an accurate quantitative study on the nonlinear vibration characteristics of the system.

The frequency response curve is analyzed to obtain the influence of the design parameters on the nonlinear vibration characteristics, so as to provide a reference for the structural design. Adjust one or several design parameters that have the greatest influence on nonlinear vibration characteristics, so that the nonlinear vibration is within the set range.

A nonlinear dynamic equation solving system for mechanical nonlinear vibration analysis according to an embodiment of the present invention is characterized in that, it includes:

The equation building module in the time domain is used to construct and obtain the nonlinear finite element dynamic equation of the system in the time domain according to the mechanical nonlinear vibration system to be analyzed. The expression is:

In the formula, k, c, m represent the stiffness, viscous damping and mass matrix of the system, respectively; x represents the displacement vector of all degrees of freedom in the system; f _n (x) represents the nonlinear force vector; f _l (t) represents the external incentive vector;

The equation acquisition module in the frequency domain is used to perform the generalized Fourier transform of x and f _l (t) in the nonlinear finite element dynamic equation in the time domain through the multi-harmonic balance method to obtain the nonlinear finite element in the frequency domain. Metadynamic equation;

The Jacobian matrix acquisition module is used to solve the Jacobian matrix of nonlinear finite element dynamic equations in the frequency domain by using the automatic differentiation method;

The frequency domain solution acquisition module is used to solve the nonlinear finite element dynamic equation in the frequency domain by using the arc length method according to the Jacobian matrix obtained by the Jacobian matrix acquisition module, and obtain the solution of the nonlinear finite element dynamic equation in the frequency domain;

The frequency response curve acquisition module is used to obtain the solution of the nonlinear finite element dynamic equation in the frequency domain obtained by the frequency domain solution acquisition module, and obtain the solution of the nonlinear finite element dynamic equation in the time domain through the generalized inverse Fourier transform ;According to the solution of the nonlinear finite element dynamic equation in the time domain, the frequency response curve is obtained, and the mechanical nonlinear vibration analysis is carried out.

As will be appreciated by those skilled in the art, the embodiments of the present application may be provided as a method, a system, or a computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) having computer-usable program code embodied therein.

The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present application. It will be understood that each flow and/or block in the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to the processor of a general purpose computer, special purpose computer, embedded processor or other programmable data processing device to produce a machine such that the instructions executed by the processor of the computer or other programmable data processing device produce Means for implementing the functions specified in a flow or flow of a flowchart and/or a block or blocks of a block diagram.

These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory result in an article of manufacture comprising instruction means, the instructions The apparatus implements the functions specified in the flow or flows of the flowcharts and/or the block or blocks of the block diagrams.

These computer program instructions can also be loaded on a computer or other programmable data processing device to cause a series of operational steps to be performed on the computer or other programmable device to produce a computer-implemented process such that The instructions provide steps for implementing the functions specified in the flow or blocks of the flowcharts and/or the block or blocks of the block diagrams.

The above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the above embodiments, those of ordinary skill in the art can still modify or equivalently replace the specific embodiments of the present invention. , any modifications or equivalent replacements that do not depart from the spirit and scope of the present invention are all within the protection scope of the claims of the present invention for which the application is pending.

Claims (4)

1. a nonlinear dynamic equation solving method for mechanical nonlinear vibration analysis, is characterized in that, comprises the following steps: Step 1, according to the mechanical nonlinear vibration system to be analyzed, construct and obtain the nonlinear finite element dynamic equation in the time domain of the system, and the expression is:

In formula (1), k, c, m represent the stiffness, viscous damping and mass matrix of the system, respectively; x represents the displacement vector of all degrees of freedom in the system; f _n (x) represents the nonlinear force vector; f _l (t ) represents the external excitation vector; Step 2, through the multi-harmonic balance method, perform generalized Fourier transform on x and f _l (t) in formula (1) to obtain a nonlinear finite element dynamic equation in the frequency domain; Step 3, using the automatic differential method to solve the Jacobian matrix of the nonlinear finite element dynamic equation in the frequency domain obtained in step 2; Step 4: According to the Jacobian matrix obtained in step 3, the arc length method is used to solve the nonlinear finite element dynamic equation in the frequency domain obtained in step 2, and the solution of the nonlinear finite element dynamic equation in the frequency domain is obtained; Step 5: According to the solution of the nonlinear finite element dynamic equation in the frequency domain obtained in step 4, the solution of the nonlinear finite element dynamic equation in the time domain is obtained through the generalized inverse Fourier transform; according to the nonlinear finite element dynamic equation in the time domain The solution of the element dynamics equation, the frequency response curve is obtained, and the mechanical nonlinear vibration analysis is carried out; Wherein, step 2 specifically includes: Step 2.1, perform Fourier expansion on x in formula (1), and the obtained expression is:

In formula (2): D=[I,Icos(ωt),Isin(ωt),...,Icos(N _k ωt),Isin(N _k ωt)]; ω is the excitation frequency; N _k is the number of selected harmonic components;

Step 2.2, perform Fourier expansion on f _l (t) in formula (1), and obtain the expression as:

In formula (3): D=[I,Icos(ωt),Isin(ωt),...,Icos(N _k ωt),Isin(N _k ωt)]; ω is the excitation frequency; N _k is the number of selected harmonic components;

Step 2.3: Substitute the Fourier-expanded x and f _l (t) back into equation (1), and obtain the nonlinear finite element dynamic equation expression in the frequency domain as: R(X)=Z(ω)X+F _n (X)-F _l =0 (4) In formula (4), X is the amplitude vector of the displacement harmonic components in the direction of each degree of freedom; Z(ω) is the dynamic stiffness matrix in the frequency domain; F _n (X) is the nonlinear force vector in the frequency domain; F _l is the amplitude vector of the harmonic components of the exciting force in the frequency domain; where Z(ω) is expressed as:

Step 3 specifically includes: Apply Newton's method to solve equation (4); the iterative process is expressed as:

In the formula, k represents the current number of iteration steps; Combining Equation (5) and Equation (4) to obtain the expression:

In formula (6), the automatic differentiation method is used to solve

Specific steps include: Perform the inverse Fourier transform on the amplitude vector X of the displacement harmonic component in each degree of freedom direction to obtain x, and the expression is:

According to the displacement-nonlinear force relationship, the nonlinear force vector f _n (x) is obtained; the Fourier transform is performed on f _n (x) to obtain the nonlinear force vector F _n (X) in the frequency domain,

Obtain the computational graph of X→x→f _n (x)→F _n (X); Find the value of each node by traversing the computational graph in a forward direction; By traversing the computational graph in reverse, the partial derivative of each node is calculated to obtain

Step 4 specifically includes the following steps: Step 4.1, add λ to formula (4), and obtain the equation system expression containing λ as: Ψ(X,λ)=Z(ω)X+F _n (X)-λF _l =0; (7) Construct an N+1-dimensional vector: ν(X,λ)=[ν ₁ ,ν ₂ ,…,ν _N ,ν _N+1 ] ^T ; in,

In the formula, the symbol ∧ means to delete the column; ν(X,λ) is the tangent vector of the solution curves X(s) and λ(s) in N+1-dimensional space; The unit tangent vector expression of the solution curve is:

where ||ν|| represents the mode of ν; For the mth arc length step Δs, the constraint equation is expressed as:

Combining formula (7) and formula (8), the obtained expression is:

The Jacobian matrix expression of formula (9) is:

Step 4.2, in the first iteration, obtain an initial solution X ^* and λ ^* of formula (7) in the weak nonlinear region by Newton's method; Step 4.3, estimation step,

Among them, the subscript m represents the mth arc length step; Step 4.4, correction step, the expression of the iterative process is formula (11); in each iteration step, the approximate solution increment is projected to the direction of τ _m to obtain the arc length increment;

2. a kind of nonlinear dynamic equation solution method for nonlinear mechanical vibration analysis according to claim 1, is characterized in that, step 5 specifically comprises: Step 5.1, for each solution point in the time domain, convert the displacement vector x _i of each degree of freedom into the total displacement vector x _s of each node of the system; _{Step 5.2} , select each ^point of the system frequency response curve, and fit all calculated points , obtain the system frequency response curve; Step 5.3, analyze the frequency response curve of the system, and conduct quantitative research on the nonlinear vibration characteristics of the system. 3. a kind of nonlinear dynamic equation solution method for nonlinear mechanical vibration analysis according to claim 1, is characterized in that, in step 1, the mechanical nonlinear vibration system to be analyzed is considering damping structure friction Turbomachinery blade-disk vibration system; Among them, according to the geometric structure of the turbine blade-disc, a three-dimensional model is established and meshed to form the overall finite element model of the turbine blade-disc; Determine the contact interface and friction node pairs of each damping structure in the turbine blade-disc model, and perform static contact analysis to obtain the normal preload force N ₀ between the friction node pairs, and obtain the prestress field; Through the linear finite element method, according to the boundary constraints of the turbomachinery blade-disk structure, and considering the prestress field and rotational softening effect, the stiffness matrix and mass matrix m of the turbomachinery blade-disk system are obtained; Assuming that proportional damping is satisfied, the damping matrix c=αk+βm; among them, the experimental measurement of the turbomachinery blade-disc system is carried out to obtain the first two-order resonance frequency and the first two-order mode shape damping ratio of the system, and determine the coefficients α and β The value of , obtains the damping matrix c of the turbomachinery blade-disk system; The unsteady three-dimensional flow field analysis of the blade is carried out, and the periodic fluid excitation force vector f _l (t) of the turbine blade-disk system is obtained; according to the Coulomb dry friction model, the displacement is affected by the contact interface of each damping structure and the friction node and the normal preload to obtain the nonlinear force vector in the direction of the degrees of freedom of all friction points, and then obtain the nonlinear force vector f _n (x ). 4 . The method for solving nonlinear dynamic equations for nonlinear mechanical vibration analysis according to claim 1 , wherein the method is used for analyzing cracks in rotor dynamics and dry friction damping of turbine blades. 5 .

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