CN114065429B - Method for solving inherent characteristics of single-symmetrical-section wind turbine blade - Google Patents
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Abstract
A method for solving the inherent characteristics of a wind turbine blade with a single symmetrical section belongs to a method for solving the inherent characteristics of a wind turbine blade. At present, most wind turbine blades adopt a single-symmetrical-section structural form, so the blades can be regarded as single-symmetrical-section Eubernoulli cantilever beams, the existing inherent characteristic solving method cannot solve the problem of the analytic mode and the inherent frequency of the single-symmetrical Eubernoulli cantilever beams, the method provided by the invention can effectively solve the problem, the inherent frequency and the mode of the accurate analytic form can be obtained, and the problem of inherent characteristic analysis of the wind turbine blades is well solved.
Description
Technical Field
The invention relates to a method for solving the intrinsic characteristic of a wind turbine blade, in particular to a method for solving the intrinsic characteristic of a wind turbine blade with a single symmetrical section.
Background
The wind power generation is a new green, low-carbon and environment-friendly energy source. However, the blade vibrates due to the adverse effects of sand, dust, gravel, high wind speed and other adverse conditions during the service period of the wind turbine, and when the frequency of the exciting force is the same as the natural frequency of the blade, resonance occurs. This not only presents a safety hazard but also reduces the blade life. For this reason, it is necessary to perform a kinetic analysis, including an analysis of the intrinsic characteristics of the blade structure, when designing a wind turbine.
The blade has three main vibration modes, flap, lag and torsional vibration. Flapping refers to bending vibration of the blade in a direction perpendicular to the plane of rotation; shimmy refers to the bending vibration of the blade in the plane of rotation; torsional vibration refers to torsional vibration of the blade about its pitch axis. In the dynamic analysis, the blade can be regarded as a cantilever beam, as shown in fig. 1, there are many methods for calculating the natural frequency and the array type of the cantilever beam, mainly including the rayleigh Lei Fa, deng Kelai method, the litz method, etc., the former two methods are only used for approximating the natural frequency of the first order of the system, and the latter method is used for an approximation solution for reducing the degree of freedom of the system. Therefore, the existing cantilever beam inherent characteristic solving method has great limitation, and the analytic mode of the cantilever beam with the complex section cannot be solved.
Disclosure of Invention
The invention aims to solve the problem that the existing inherent characteristic solving method cannot solve the analytic mode and the inherent frequency of a single-symmetrical-section Eubernoulli cantilever beam, and provides the inherent characteristic solving method of the single-symmetrical-section wind turbine blade.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a method for solving intrinsic characteristics of a single-symmetrical-section wind turbine blade specifically comprises the following steps:
the blade is regarded as a single symmetrical section Eubernoulli cantilever beam, two coordinate systems are used in the derivation process, one is an inertia orthogonal coordinate system x, y, z, and the origin is located at the Liang Genbu cross section shearing center; the other is a cross section coordinate system xi, eta, zeta, the origin of which is positioned at the shearing center of the cross section; the section of the Euler Bernoulli cantilever beam is symmetrical about a y-axis, and a dynamic equation set under a free vibration condition is written as the following by considering the bending vibration of the Euler Bernoulli cantilever beam to the y-axis and the torsional vibration of the Euler Bernoulli cantilever beam to the x-axis
Wherein, the first and the second end of the pipe are connected with each other,
in the formula, D η Represents bending stiffness, w represents bending displacement of the beam in the z-axis direction, and is a function of position and time, namely w (x, t), theta represents torsion displacement of the beam in the x-axis direction, and is a function of position and time, namely theta (x, t), w' represents a fourth-order partial derivative of bending displacement of the beam in the z-axis direction to position x, m represents line density, and>
represents the second-order partial derivative, eta, of the bending displacement of the beam in the z-axis direction to the time t c Representing eccentricity, <' > or>
Represents the second order partial derivative of the beam's torsional displacement in the x-axis direction with respect to time t, J η Representing moment of inertia about the y-axis>
Representing the second order partial derivative of the beam z-axis direction bending displacement with respect to time t and the second order partial derivative with respect to position x, D ξ Representing torsional stiffness, [ theta ] representing the second order partial derivative of the beam's x-axis torsional displacement versus position x, J ξ Representing the moment of inertia about the x-axis, a representing the cross-sectional area, ρ representing the bulk density, E representing the young's modulus, G representing the shear modulus, ψ representing the warp displacement field function, ξ, η, ζ representing the three directional coordinates in the cross-sectional coordinate system;
since periodic vibration of the blade is of interest, the non-periodic vibration portion of the solution to the system of equations is not considered, and accordingly, the solution to the system of equations (1) is assumed to be
In the formula, W (x) represents a mode shape function of bending vibration of the beam in the z-axis direction, Θ (x) represents a mode shape function of torsional vibration of the beam in the x-axis direction, e is a natural number, i is an imaginary number, ω is a circular frequency, and t is time, and for convenience of description, W (x) and Θ (x) are respectively denoted as W and Θ;
substituting equation (2) into equation (1) can realize the separation of time variable and space variable to obtain ordinary differential equation system about space variable x
Equation (3) can be combined into one sixth order ordinary differential equation applicable to both W (x) and Θ (x), as follows:
(D 6 +aD 4 -bD 2 -c)Φ(x)＝0 (4)
wherein D is a partial derivative symbol, and is recorded as
Phi (x) = W (x) or theta (x), a, b and c are intermediate parameters, the expression is,
the solution of formula (4) is
A 1 ～A 6 、B 1 ～B 6 Alpha, beta and gamma are intermediate parameters for undetermined coefficients, cosh () is a hyperbolic cosine function, sinh () is a hyperbolic sine function, expressed specifically as,
wherein the expressions of the parameters q and lambda are as follows,
as can be seen from the formula (5), the bending and torsional vibration type functions have 12 undetermined coefficients in total, but the bending and torsional coupling are not independent, so that the undetermined coefficients have a certain relation,the formula (5) is substituted into the formula (3), and the coefficient A to be determined in the bending and torsional vibration type function can be obtained by utilizing the identity relation i 、B i The relationship between them is as follows:
in the formula, the parameter k α 、k β 、k γ Is expressed as
The frequency equation of the system is obtained from the boundary conditions, and for the cantilever model, there are three geometrical boundary conditions at the fixed end as follows:
w＝0,θ＝0,w'＝0 (7)
the free end has three force (moment) boundary conditions as follows:
the equations (5), (6), (7) and (8) are combined, and the system of the equations can be obtained as follows:
wherein L is the length of the beam, parameter h α 、h β 、h γ Is an intermediate parameter, has no specific meaning, and has the expression:
h α ＝D η α 2 +J η ω 2
h β ＝D η β 2 -J η ω 2
h γ ＝D η γ 2 -J η ω 2
the equation set has no zero solution, and the determinant of the coefficient matrix is 0 as known from the linear algebraic theory, namely:
equation (10) is the frequency equation of the system, and the solution of the equation is the natural frequency of each order of the system;
performing row transformation on the coefficient matrix in the formula (9) by using a Gaussian elimination method to obtain an upper triangular matrix, thereby obtaining A i The relationship between them, set
C 3 ＝-γ 2 sin(γL)-αγsinh(αL)
C 5 ＝-β 2 sin(βL)-αβsinh(αL)
C 6 ＝k γ -k α
C 7 ＝k β -k α
With A 6 Is a free parameter, get
The natural frequency omega of each order i Substituting the value of (A) into the above formula and setting the free parameter A 6 =1, the bending and torsional vibration type function of each order can be obtained, until the analysis of the inherent characteristics of the linear equation is finished;
orthogonality of bending vibration and torsional vibration modes of the Euler Bernoulli cantilever beam with the single symmetrical section is proved as follows, for the ith order vibration type function, a dynamic equation set after separation of variables is written into a matrix form
Wherein, W i Represents the bending vibration mode function of the ith order z-axis direction of the beam theta i Representing the ith order of the beam in the direction of the x axis torsional vibration mode function;
left-multiplying (W) by equation (12) j Θ j ) To obtain
Is unfolded to obtain
Integrated along the whole beam
The first term, the second term and the third term on the left side of the equation are zero according to the boundary condition; therefore, it is not only easy to use
Similarly, for the jth order mode equation
Subtracting the two formulae to obtain
When i ≠ j, ω i 2 ≠ω j 2 (ii) a Is inherently
Thereby having
When i = j, the ith order modal quality is
Modal stiffness of order i
The ith order natural frequency of
Compared with the prior art, the invention has the beneficial effects that: at present, most wind turbine blades adopt a single-symmetrical-section structural form, so the blades can be regarded as single-symmetrical-section Eubernoulli cantilever beams, the existing inherent characteristic solving method cannot solve the problem of the analytic mode and the inherent frequency of the single-symmetrical Eubernoulli cantilever beams, the method provided by the invention can effectively solve the problem, the inherent frequency and the mode of the accurate analytic form can be obtained, and the problem of inherent characteristic analysis of the wind turbine blades is well solved.
Drawings
FIG. 1 is a schematic diagram of a single-symmetric section Euler-Bernoulli beam deformation;
FIG. 2 is a diagram of the natural frequency and mode shape of the 1 st order of a single symmetric Euler-Bernoulli beam;
FIG. 3 is a graph of 2 nd order natural frequency and mode shape of a single-symmetric section Euler-Bernoulli beam;
FIG. 4 is a graph of the 3 rd order natural frequency and mode shape of a single-symmetric section Euler-Bernoulli beam;
FIG. 5 is a graph of the 4 th order natural frequency and mode shape of a single-symmetric section Euler-Bernoulli beam;
FIG. 6 is a graph of the 5 th order natural frequency and mode shape of a single-symmetric section Euler-Bernoulli beam;
FIG. 7 is a graph of the 6 th order natural frequency and mode shape of a single-symmetric section Euler-Bernoulli beam.
Detailed Description
The technical solutions of the present invention are further described below with reference to the drawings and the embodiments, but the present invention is not limited thereto, and modifications or equivalent substitutions may be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention.
Example 1:
a method for solving intrinsic characteristics of a single-symmetrical-section wind turbine blade specifically comprises the following steps:
the fundamental parameters of the euler-bernoulli beam with the single symmetrical section shown in table 1 are adopted, and the natural frequency and the mode of the euler-bernoulli beam are obtained by the solving method provided by the invention.
TABLE 1 parameters of Euler-Bernoulli Liang Jiben single symmetric section
Regarding the blade as a single symmetrical section Euler Bernoulli cantilever beam, as shown in FIG. 1, two coordinate systems are used in the derivation process, one is an inertial orthogonal coordinate system x, y, z, and the origin is located at the cross-section shearing center Liang Genbu; the other is a cross section orthogonal coordinate system xi, eta, zeta, the origin of which is positioned at the shearing center of the cross section; the section of the Euler Bernoulli cantilever beam is symmetrical about a y-axis, and a dynamic equation set under a free vibration condition is written as the following by considering the bending vibration of the Euler Bernoulli cantilever beam to the y-axis and the torsional vibration of the Euler Bernoulli cantilever beam to the x-axis
Wherein the content of the first and second substances,
in the formula D η Represents bending stiffness, w represents bending displacement of the beam in the z-axis direction, and is a function of position and time, namely w (x, t), theta represents torsion displacement of the beam in the x-axis direction, and is a function of position and time, namely theta (x, t), w' represents a fourth-order partial derivative of bending displacement of the beam in the z-axis direction to position x, m represents line density, and/or>
Representing the second-order partial derivative, eta, of the bending displacement of the beam in the z-axis direction with respect to time t c Representing eccentricity, <' > or>
Representing the second order partial derivative, J, of the beam's torsional displacement in the x-axis direction versus time t η Representing moment of inertia about the y-axis>
Representing the second order partial derivative of the beam z-axis direction bending displacement with respect to time t and the second order partial derivative with respect to position x, D ξ Representing torsional stiffness, [ theta ] "representing the second order partial derivative of beam torsional displacement in the x-axis direction with respect to position x, J ξ Representing the moment of inertia about the x-axis, a representing the cross-sectional area, ρ representing the bulk density, E representing the young's modulus, G representing the shear modulus, ψ representing the warp displacement field function, ξ, η, ζ representing the three directional coordinates in the cross-sectional coordinate system; the upper right (') of the variable symbol indicates the partial derivative of the variable with respect to position, and the point above the variable symbol indicates the partial derivative of the variable with respect to time;
since periodic vibration of the blade is of interest, the non-periodic vibration portion of the solution to the system of equations is not considered, and accordingly, the solution to the system of equations (1) is assumed to be
In the formula, W (x) represents a mode shape function of bending vibration of the beam in the z-axis direction, Θ (x) represents a mode shape function of torsional vibration of the beam in the x-axis direction, e is a natural number, i is an imaginary number, ω is a circular frequency, and t is time;
the separation of the time variable and the space variable can be realized by substituting the formula (2) into the formula (1), and the ordinary differential equation system related to the space variable x is obtained as
Equation (3) can be combined into one sixth order ordinary differential equation applicable to both W (x) and Θ (x), as follows:
(D 6 +aD 4 -bD 2 -c)Φ(x)＝0 (4)
wherein D is a partial derivative symbol, and is recorded as
Phi (x) = W (x) or theta (x), a, b and c are intermediate parameters, the expression is,
the solution of formula (4) is
A 1 ～A 6 、B 1 ～B 6 Alpha, beta and gamma are intermediate parameters for undetermined coefficients, cosh () is a hyperbolic cosine function, sinh () is a hyperbolic sine function, expressed specifically as,
wherein the expressions of the parameters q and lambda are as follows,
as can be seen from the equation (5), the bending and torsional vibration type functions have 12 undetermined coefficients in total, but the bending and torsional coupling are not independent, so that the undetermined coefficients have a certain relation, the equation (5) is substituted into the equation (3), and the undetermined coefficients A in the bending and torsional vibration type functions can be obtained by utilizing the identity relation i 、B i The relationship between them is as follows:
in the formula, the parameter k α 、k β 、k γ Is expressed as
The frequency equation of the system is obtained from the boundary conditions, and for the cantilever model, there are three geometrical boundary conditions at the fixed end as follows:
w＝0,θ＝0,w'＝0 (7)
the free end has three force (moment) boundary conditions as follows:
combining (5), (6), (7) and (8), the following equations can be obtained:
wherein L is the length of the beam, parameter h α 、h β 、h γ Is an intermediate parameter, has no specific meaning, and has the expression:
h α ＝D η α 2 +J η ω 2
h β ＝D η β 2 -J η ω 2
h γ ＝D η γ 2 -J η ω 2
the equation set has no zero solution, and the determinant of the coefficient matrix is 0 as known from the linear algebraic theory, namely:
equation (10) is the frequency equation of the system, and the solution of the equation is the natural frequency of each order of the system;
performing line transformation on the coefficient matrix in the formula (9) by using a Gaussian elimination method to obtain an upper triangular matrix, thereby obtaining A i The relationship between them, set
C 3 ＝-γ 2 sin(γL)-αγsinh(αL)
C 5 ＝-β 2 sin(βL)-αβsinh(αL)
C 6 ＝k γ -k α
C 7 ＝k β -k α
With A 6 Is a free parameter, get
The natural frequency omega of each order i Substituting the value of (A) into the above formula and setting the free parameter A 6 =1, the bending and torsional vibration type function of each order can be obtained, until the analysis of the inherent characteristics of the linear equation is finished; the basic data of the beams in table 1 are adopted to obtain the first six-order natural frequency and the mode shape as shown in fig. 2 to 7;
orthogonality of bending vibration and torsional vibration modes of the Euler Bernoulli cantilever beam with the single symmetrical section is proved as follows, for the ith order vibration type function, a dynamic equation set after separation of variables is written into a matrix form
Wherein, W i Represents the i-th order z-axis direction bending vibration mode shape function of the beam i Representing the ith order of the beam in the direction of the x axis torsional vibration mode function;
left-multiplying (W) by equation (12) j Θ j ) To obtain
Is unfolded to obtain
Integrated along the whole beam
The first term, the second term and the third term on the left side of the equation are zero according to the boundary condition; therefore, it is possible to
Similarly, for the jth order mode equation
Subtracting the two formulas to obtain
When i ≠ j, ω i 2 ≠ω j 2 (ii) a Is inherently provided with
Thereby is provided with
When i = j, the ith order modal quality is
Modal stiffness of order i
The ith natural frequency of
Claims (1)
1. A method for solving intrinsic characteristics of a single-symmetrical-section wind turbine blade is characterized by comprising the following steps: the method specifically comprises the following steps:
the blade is regarded as a single symmetrical section Eubernoulli cantilever beam, two coordinate systems are used in the derivation process, one is an inertia orthogonal coordinate system x, y, z, and the origin is located at the Liang Genbu cross section shearing center; the other is a cross section coordinate system xi, eta, zeta, the origin of which is positioned at the shearing center of the cross section; the section of the Euler Bernoulli cantilever beam is symmetrical about a y axis, the bending vibration of the Euler Bernoulli cantilever beam to the y axis and the torsional vibration of the Euler Bernoulli cantilever beam to the x axis are considered, and a dynamic equation system under the free vibration condition is written as
Wherein the content of the first and second substances,
in the formula, D η Representing flexural rigidity, w representing beam z-axis direction bending displacement as a function of position and time, i.e., w (x, t), θ representing beam x-axis direction torsional displacement, and also as a function of position and time, i.e., θ (x, t), w "" representing beam z-axis direction bending displacement versus position x fourth order partial derivative, m representing linear density,
represents the second-order partial derivative, eta, of the bending displacement of the beam in the z-axis direction to the time t c Representing eccentricity, <' > or>
Represents the second order partial derivative of the beam's torsional displacement in the x-axis direction with respect to time t, J η Represents a moment of inertia about the y-axis>
Representing the second order partial derivative of the beam z-axis direction bending displacement with respect to time t and the second order partial derivative with respect to position x, D ξ Representing torsional stiffness, [ theta ] "representing the second order partial derivative of beam torsional displacement in the x-axis direction with respect to position x, J ξ Representing the moment of inertia about the x-axis, a representing the cross-sectional area, ρ representing the bulk density, E representing the young's modulus, G representing the shear modulus, ψ representing the warp displacement field function, ξ, η, ζ representing the three directional coordinates in the cross-sectional coordinate system;
since periodic vibration of the blade is of interest, the non-periodic vibration portion of the solution to the system of equations is not considered, and accordingly, the solution to the system of equations (1) is assumed to be
In the formula, W (x) represents a mode shape function of bending vibration of the beam in the z-axis direction, Θ (x) represents a mode shape function of torsional vibration of the beam in the x-axis direction, e is a natural number, i is an imaginary number, ω is a circular frequency, and t is time, and for convenience of description, W (x) and Θ (x) are respectively denoted as W and Θ;
substituting equation (2) into equation (1) can realize the separation of time variable and space variable to obtain ordinary differential equation system about space variable x
Equation (3) can be combined into one sixth order ordinary differential equation applicable to both W (x) and Θ (x), as follows:
(D 6 +aD 4 -bD 2 -c)Φ(x)＝0 (4)
wherein D is a partial derivative symbol, and is recorded as
Phi (x) = W (x) or theta (x), a, b and c are intermediate parameters, and the expression is represented as +>
The solution of formula (4) is
A 1 ～A 6 、B 1 ～B 6 Alpha, beta and gamma are intermediate parameters for undetermined coefficients, cosh () is a hyperbolic cosine function, sinh () is a hyperbolic sine function, expressed specifically as,
wherein the expressions of the parameters q and lambda are as follows,
as can be seen from the formula (5), the bending and torsional vibration type functions have 12 undetermined coefficients in total, but the bending and torsional coupling performance does not cause the bending and torsional vibration to be independent, so that the undetermined coefficients have a certain relation, the formula (5) is replaced by the formula (3), and the undetermined coefficients in the bending and torsional vibration type functions can be obtained by utilizing the identity relationA i 、B i The relationship between them is as follows:
in the formula, the parameter k α 、k β 、k γ Is expressed as
The frequency equation of the system is obtained from the boundary conditions, and for the cantilever model, there are three geometrical boundary conditions at the fixed end as follows:
w＝0,θ＝0,w'＝0 (7)
the free end has three force (moment) boundary conditions as follows:
combining (5), (6), (7) and (8), the following equations can be obtained:
wherein L is the length of the beam, parameter h α 、h β 、h γ Is an intermediate parameter, has no specific meaning, and has the expression:
h α ＝D η α 2 +J η ω 2
h β ＝D η β 2 -J η ω 2
h γ ＝D η γ 2 -J η ω 2
the equation set has no zero solution, and the determinant of the coefficient matrix is 0 as known from the linear algebraic theory, namely:
equation (10) is a frequency equation of the system, and the solution of the equation is the natural frequency of each order of the system;
performing row transformation on the coefficient matrix in the formula (9) by using a Gaussian elimination method to obtain an upper triangular matrix, thereby obtaining A i The relationship between (A) and (B)
C 3 ＝-γ 2 sin(γL)-αγsinh(αL)
C 5 ＝-β 2 sin(βL)-αβsinh(αL)
C 6 ＝k γ -k α
C 7 ＝k β -k α
With A 6 Is a free parameter, get
The natural frequency omega of each order i Substituting the value of (b) into the above formula, and setting the free parameter A 6 =1, the bending and torsional vibration type function of each order can be obtained, until the analysis of the inherent characteristics of the linear equation is finished;
orthogonality of bending vibration and torsional vibration modes of the Euler Bernoulli cantilever beam with the single symmetrical section is proved as follows, for the ith order vibration type function, a dynamic equation set after separation of variables is written into a matrix form
Wherein, W i Represents the i-th order z-axis direction bending vibration mode shape function of the beam i Representing the ith order of the beam in the direction of the x axis torsional vibration mode function;
left-multiplying (W) by equation (12) j Θ j ) To obtain
Is unfolded to obtain
Integrated along the whole beam
According to boundary conditions, the first term, the second term and the third term on the left side of the equation are zero; therefore, it is possible to
Similarly, for the jth order mode equation
Subtracting the two formulae to obtain
When i ≠ j, ω i 2 ≠ω j 2 (ii) a Is inherently provided with
Thereby is provided with
When i = j, the ith order modal quality is
Modal stiffness of order i
The ith order natural frequency of
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