Method for Solving Eigenvalues of Blade Mode Shape Function

Abstract

A method for solving the eigenvalue of the blade mode shape function is disclosed. In the method, the blade vibration is measured and the mode shape function is established. The interval of the solutions of the characteristic equation corresponding to the mode shape function is separated by the bisection method, so that each separated interval contains and only contains one possible solution. The solution of the corresponding characteristic equation is obtained according to the Newton iteration method on each separated interval. The solutions of the corresponding characteristic equation are screened to obtain the final solution as the eigenvalue of the mode shape function.

Description

Technical Field

[0001] The present invention belongs to the technical field of vibration detection of turbine blades, and particularly relates to a method for solving the eigenvalue of the vibration mode function of blades. Background Art

[0002] Turbine blades are key components of the entire system, and their vibration conditions are related to the operating state of the entire system. Through blade vibration, it can be understood whether there are faults in the blades, such as cracks and chipping, and it can also be known whether faults such as surging occur in the turbine machinery. In order to study the vibration of blades, an analytical model can be established for the blades, and the modal superposition method can be used for solution. However, the modal superposition method requires obtaining the eigenvalues of each modal vibration mode function to perform accurate calculations. The characteristic equation corresponding to the eigenvalue of the vibration mode function is usually a non-linear equation, and it is difficult to obtain an analytical solution. The method of converging with random initial values in the traversed interval has low calculation efficiency and cannot guarantee obtaining all solutions. In view of the above problems, a method that can quickly and without redundancy or omission obtain the eigenvalues of the vibration mode function is needed.

[0003] The above information disclosed in the background art section is only used to enhance the understanding of the background of the present invention, and thus may include information that does not constitute the prior art known to those of ordinary skill in the art in this country. Summary of the Invention

[0004] In view of the problems existing in the prior art, the present invention proposes a method for solving the eigenvalue of the vibration mode function of blades, which solves the problems that the existing method cannot obtain an analytical solution and the random convergence method is slow in calculation and cannot guarantee obtaining all solutions.

[0005] The object of the present invention is achieved through the following technical solutions. A method for solving the eigenvalue of the vibration mode function of blades includes the following steps:

[0006] Step 1: Measure the blade vibration and establish a vibration mode function. Divide the interval of the solution of the characteristic equation corresponding to the vibration mode function by the bisection method, so that each divided interval contains and only contains one possible solution. The vibration mode function is a function of the amplitude ratio relationship in each mode.

[0007] Step 2: Obtain the solution of the corresponding characteristic equation on each divided interval according to the Newton iteration method.

[0008] Step 3: Screen the solution of the corresponding characteristic equation to obtain the final solution as the eigenvalue of the vibration mode function.

[0009] In the above method, in Step 1,

[0010] Step 1.1: Obtain the iteration step size h of the bisection method, where the iteration step size h is less than the absolute value of the difference between any two solutions in the interval D.

[0011] Step 1.2: Given a small interval \(a = 0\), \(b = h\) that is less than the interval \(D\), calculate the function values \(f(a)\) and \(f(b)\), where \(a\) is the left endpoint of the interval and \(b\) is the right endpoint of the interval;

[0012] Step 1.3: Determine the sign of \(f(a) * f(b)\). If it is negative, record the current small interval \([a, b]\);

[0013] Step 1.4: Move the small interval by \(+h\) units to obtain a new small interval \([a, b]\). Repeat steps 1.2 - 1.4 until the entire solution interval \(D\) is traversed.

[0014] In the method described above, in step 2,

[0015] Step 2.1: In the Newton - Raphson method, set the iteration convergence thresholds \(\theta_1\) and \(\theta_2\) and the maximum number of iterations \(K\);

[0016] Step 2.2: Calculate the Jacobian matrix \(J_f\), which is the reciprocal of the first - order derivative of the eigen - shape function;

[0017] Step 2.3: Calculate the function value \(f(x (k) )\) and \(J_f(x (k) ), where the initial position is the left endpoint of the interval \(D\), and \(x(k)\) refers to the value of \(x\) at the \(k\) - th iteration. Calculate the function value and the value of the Jacobian matrix at this step by substituting \(x(k)\) into the previously obtained function and Jacobian matrix;

[0018] Step 2.4: Solve the linear equation system \(J_f(x (k) )\Delta x (k) =-f(x (k) ), where \(x(k)\) refers to the value of \(x\) at the \(k\) - th iteration, and \(\Delta x(k)=x(k + 1)-x(k)\);

[0019] Step 2.5: Let \(x (k+1) =x (k) +\Delta x (k) ;

[0020] Step 2.6: If \(\|\Delta x (k) \| / \|x (k) \|<\theta_1\), or \(\|f(x (k+1) )\|<\theta_2\), then select \(x(k + 1)\) as the solution; if \(k = K\), output the information that the accuracy requirement is not met after \(K\) iterations and stop the calculation; otherwise, take the next small interval and repeat steps 2.3 to 2.6;

[0021] Step 2.7: Replace with the next interval and repeat steps 2.3 - 2.7 until all the recorded intervals have been calculated.

[0022] In the method described above, in step 3,

[0023] Step 3.1: Screen out the solutions that are not within the solution interval D;

[0024] Step 3.2: Screen out the duplicate solutions to obtain all the solutions that are non-redundant and non-omissive within the solution interval D as the eigenvalue of the blade vibration mode function.

[0025] Beneficial effects

[0026] Compared with the existing technical methods, this method has simple steps, good stability, simple calculation, high precision, and good popularization. Description of the drawings

[0027] By reading the detailed description in the preferred specific embodiments below, various advantages and benefits of the present invention will become clear to those of ordinary skill in the art. The drawings in the specification are only for the purpose of showing the preferred embodiments and are not considered to be a limitation of the present invention. Obviously, the drawings described below are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts. Moreover, throughout the drawings, the same reference numerals are used to represent the same components.

[0028] In the drawings:

[0029] Figure 1 is a process block diagram of a method for quickly solving the eigenvalue of the blade vibration mode function provided by an embodiment of the present invention;

[0030] Figure 2 is Figure 1 a schematic diagram of the calculation speed of a method for quickly solving the eigenvalue of the blade vibration mode function shown and the convergence of random initial values in the traversal interval.

[0031] The present invention will be further explained below with reference to the drawings and embodiments. Specific embodiments

[0032] The following refers to the attached Figures 1 to 2 The specific embodiments of the present invention will be described in more detail. Although the specific embodiments of the present invention are shown in the drawings, it should be understood that the present invention can be implemented in various forms and should not be limited by the embodiments described here. On the contrary, these embodiments are provided to enable a more thorough understanding of the present invention and to fully convey the scope of the present invention to those skilled in the art.

[0033] It should be noted that certain terms are used in the specification and claims to refer to specific components. Those skilled in the art should understand that technicians may use different nouns to refer to the same component. The specification and claims do not distinguish components by the difference in nouns, but by the difference in the functions of the components. As mentioned throughout the specification and claims, the term "comprising" or "including" is an open-ended term and should be interpreted as "including but not limited to". The subsequent description in the specification is the preferred implementation manner for implementing the present invention, but the description is for the purpose of the general principles of the specification and is not used to limit the scope of the present invention. The protection scope of the present invention shall be subject to what is defined by the appended claims.

[0034] For the convenience of understanding the embodiments of the present invention, the following will further explain with specific embodiments in conjunction with the accompanying drawings, and the accompanying drawings do not constitute a limitation to the embodiments of the present invention.

[0035] A method for quickly solving the eigenvalue of a blade vibration mode function, the specific steps include:

[0036] Step 1: By the bisection method, divide the interval of the solution of the characteristic equation corresponding to the vibration mode function, so that each divided interval contains and only contains one possible solution. The vibration mode can be understood as a proportional relationship between amplitudes, that is, when the blade vibrates, the proportional relationship of the vibration amplitudes of each point. And the vibration mode function is the function of the amplitude proportional relationship in each mode. The mode refers to a certain vibration form, such as the first-order bending mode, the second-order torsional mode, etc. The characteristic equation is an equation for obtaining the variables corresponding to different modes in the vibration mode function. For example, if the vibration mode function is sin(ax), a is the variable corresponding to the mode. When it is the first-order bending, a is 1, then the first-order bending vibration mode is sin(x), and the amplitudes of each point are arranged according to sin(x). The principle of the bisection method can ensure that the divided interval must contain a solution, and by iterating the interval length, it can be ensured that each divided interval contains only one solution;

[0037] Step 2: Apply the Newton iteration method to each divided interval to obtain the corresponding solution;

[0038] Step 3: Screen the solutions obtained in Step 3 to obtain the final solution, that is, the eigenvalue of the vibration mode function.

[0039] Preferably, the specific steps of the said Step 1 include:

[0040] Step 1.1: According to prior knowledge, obtain the iteration step size h of the bisection method, where h is generally less than the absolute value of the difference between any two solutions within the solution interval D. The intersection of the function corresponding to the equation and the x-axis in the drawn image is the solution to be obtained, and the approximate gap between the solutions within the interval can be observed to obtain the iteration step size;

[0041] Step 1.2: Given a small interval [a = 0, b = h], calculate the function values f(a) and f(b). The expression of f(x) is determined according to the characteristic equation corresponding to the vibration mode function;

[0042] Step 1.3: Judge the sign of f(a)*f(b). If it is negative, record the interval [a, b] at this time;

[0043] Step 1.4: Move the small interval by +h units to obtain a new interval [a, b], and repeat Steps 1.2 - 1.4 until the entire solution interval D is traversed.

[0044] Preferably, in Step 1.1, the iteration step size can be obtained according to prior knowledge, or by observing the distribution of solutions through graphing, or a sufficiently small value can be directly set;

[0045] Preferably, the interval obtained in Step 1 contains and only contains one possible solution. Since the solution interval D is not necessarily continuous while the movement of the small interval is continuous, this possible solution can be an eigenvalue or not. Preferably, the specific steps of Step 2 include:

[0046] Step 2.1: Set the iteration convergence thresholds θ1 and θ2, and the maximum number of iterations K;

[0047] Step 2.2: Calculate the Jacobian matrix Jf, which is the reciprocal of the first derivative of the vibration mode function at this time. The Jacobian matrix is calculated using the reciprocal of the first derivative of the vibration mode function. Since the vibration mode function is a function of a single variable, the Jacobian matrix here is a 1x1 matrix, that is, a number;

[0048] Step 2.3: Calculate the function value f(x (k) ) and Jf(x (k) ), where the initial position is the left endpoint of the small interval obtained in Step 1;

[0049] Step 2.4: Solve the linear equation system Jf(x (k) )Δx (k) =-f(x (k) );

[0050] Step 2.5: Let x (k+1) =x (k) +Δx (k) ;

[0051] Step 2.6: If ||Δx (k) || / ||x (k) ||<θ1, or ||f(x (k+1) )||<θ2, then take x (k+1)is the root of the equation; if k = K, output the information that the K - th iteration does not meet the accuracy requirements. The accuracy requirements refer to ||Δx(k)|| / ||x(k)|| < θ1 and ||f(x(k + 1))|| < θ2, where θ1 and θ2 are self - given thresholds, and stop the calculation; otherwise, k = k + 1, and repeat steps 2.3 to 2.6.

[0052] Step 2.7: Replace with the next interval and repeat steps 2.3 - 2.7 until all intervals recorded in step 1 are calculated.

[0053] Preferably, the specific steps described in step 3 include:

[0054] Step 3.1: Screen out the solutions that are not within the solution interval D.

[0055] Step 3.2: Screen out the duplicate solutions to obtain all non - redundant and non - omitted solutions on the solution interval D, that is, the eigenvalues.

[0056] In one embodiment, the characteristic equation corresponding to the eigenvalue of the blade vibration mode function:

[0057] f = 1 + cos(x)*cosh(x)+ms / rho / A / L*x*(cos(x)*sinh(x)-sin(x)*cosh(x)) = 0

[0058] where x is a symbolic variable and the rest are known parameters.

[0059] First, according to prior knowledge, it is known that the eigenvalue distribution distance is usually greater than 1, and determine the bisection method interval step size h = 0.1; within the given interval D = [0 100], starting from the interval [0 1], gradually calculate the values f(a) and f(b) at the two endpoints [a b] of the interval; judge the sign of f(a)*f(b). If it is positive, it means that there is no root of the equation on this small interval, and discard this small interval; if it is negative, it means that there is a root of the equation on this small interval, and record this small interval [a b].

[0060] Move the interval by step size h until the given interval D is traversed.

[0061] Set the thresholds for iterative convergence to 1e - 6 and 1e - 6, and the maximum number of iterations to 10000 times;

[0062] Calculate the Jacobian matrix Jf. Since the given characteristic equation is a univariate equation, the Jacobian matrix is

[0063] Jf = 1 / (df / dx)

[0064] Given a small interval, let x be the value of the left - hand endpoint of the small interval, calculate the function value and the value of the Jacobian matrix at x, and solve the linear equation system

[0065] Jf(x (k) )Δx (k) =-f(x (k) )

[0066] Obtain Δx(k)

[0067] If ||Δx(k)|| / ||x(k)|| < θ1, or ||f(x(k + 1))|| < θ2, then take x (k+1) as the root of the equation; if k = 10000, output that the accuracy requirement is not met after 10000 iterations and stop the calculation; otherwise, k = k + 1, and use the updated x (k) to repeat solving the linear equations and the subsequent steps.

[0068] Perform iterative calculations on all the small intervals recorded by the bisection method until all calculations are completed, and output the obtained roots; compare the range of the interval D and delete the roots that converge outside the interval;

[0069] According to the prior conditions, the distance between any two roots within the interval D is greater than 1. Take the integer part of all roots and delete the repeated roots to ensure the non - redundancy of the solutions. At this time, the obtained solutions are the non - redundant and non - omitted solutions within the interval D, that is, the characteristic roots of the characteristic equation. When dividing the interval by the bisection method in step 1, it can be ensured that every small interval where there may be a solution is not missed on the solution interval D. The bisection method only requires that the function values at the two endpoints have different signs, which proves that there may be a solution between these two endpoints. The reason why it is only possible rather than certain to have a solution is that the function is not necessarily continuous. For example, the function values of tanx on both sides of pi / 2 have different signs, but pi / 2 is not its zero solution.

[0070] Although the embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the above - mentioned specific embodiments and application fields. The above - mentioned specific embodiments are only illustrative and guiding, rather than restrictive. Those of ordinary skill in the art can also make many forms under the inspiration of this specification and without departing from the scope protected by the claims of the present invention, and these all belong to the scope of protection of the present invention.

Claims

1. A method for solving the eigenvalue of the blade vibration mode function, the method comprising the following steps: Step 1: Measure the blade vibration and establish the vibration mode function. Divide the solution interval of the characteristic equation corresponding to the vibration mode function by the bisection method, so that each divided interval contains and only contains one possible solution. The vibration mode function is a function of the amplitude ratio relationship in each mode; Step 2: Obtain the solution of the corresponding characteristic equation on each divided interval according to the Newton iteration method; Step 3: Screen the solutions of the corresponding characteristic equation to obtain the final solution as the eigenvalue of the vibration mode function; Among them, in Step 2, Step 2.1: In the Newton iteration method, set the iteration convergence thresholds θ1 and θ2 and the maximum number of iterations K; Step 2.2: Calculate the Jacobian matrix Jf, which is the reciprocal of the first derivative of the vibration mode function; Step 2.3: Calculate the function value f(x (k) ) and Jf(x (k) ), where the initial position is the left endpoint of the interval D, and x (k) refers to the value of x at the k-th iteration step. Calculate the function value and the Jacobian matrix value at this step by substituting x (k) into the previously obtained function and Jacobian matrix; Step 2.4: Solve the linear equation system Jf(x (k) )Δx (k) = -f(x (k) ), where x (k) refers to the value of x at the k-th iteration step, and Δx (k) = x (k+1) - x (k) ; Step 2.5: Let x (k+1) = x (k) + Δx (k) ; Step 2.6: If ||Δx (k) || / ||x (k) || < θ1, or ||f(x (k+1) ) || < θ2, then select x (k+1) as the solution; if k = K, output the information that the accuracy requirement is not met after K iterations, and stop the calculation; otherwise, take the next small interval and repeat Steps 2.3 to 2.6; Step 2.7: Replace with the next interval and repeat Steps 2.3 - 2.7 until all the recorded intervals are calculated; In Step 3, Step 3.1: Screen out the solutions not within the solution interval D; Step 3.2: Screen out the repeated solutions to obtain all the non-redundant and non-omissive solutions on the solution interval D as the eigenvalues of the blade vibration mode function.

2. The method according to claim 1, wherein In Step 1, Step 1.1: Obtain the iteration step size h of the bisection method, where the iteration step size h is less than the absolute value of the difference between any two solutions within the interval D; Step 1.2: Given a small interval a = 0 and b = h smaller than the interval D, calculate the function values f(a) and f(b), where a is the left endpoint of the interval and b is the right endpoint of the interval; Step 1.3: Judge the sign of f(a)*f(b). If it is negative, record the small interval [a, b] at this time; Step 1.4: Move the small interval by +h units to obtain a new small interval [a, b], and repeat Steps 1.2 - 1.4 until the entire solution interval D is traversed.

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