CN110020460B - Method for analyzing uncertainty of cylindrical shell structure frequency response function of bolt connecting flange - Google Patents
Method for analyzing uncertainty of cylindrical shell structure frequency response function of bolt connecting flange Download PDFInfo
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Abstract
The invention belongs to the technical field of mechanical dynamics, and particularly relates to a method for analyzing uncertainty of a cylindrical shell structure frequency response function of a bolt connecting flange. The method adopts 8-node degraded shell units, and establishes a dynamic model of the bolted flange-cylindrical shell through a finite element method; the spring unit is used for modeling and simulating bolt connection in a discretization mode, and the effectiveness of the built model is verified through a modal test; considering the uncertainty of the bolt connection rigidity, respectively solving frequency response function interval ranges of 5 directions under a cylindrical coordinate system when the connection rigidity is uncertainty parameters based on a Chebyshev polynomial agent model and an interval analysis method, adopting a Monte-Carlo sampling method to solve the frequency response function interval ranges under the certainty inherent frequency, and comparing the solving precision and the efficiency of the two methods; and finally, solving the system frequency response function interval range of multi-parameter uncertainty.
Description
Technical Field
The invention belongs to the technical field of mechanical dynamics, and particularly relates to a method for analyzing uncertainty of a frequency response function of a cylindrical shell structure with a flange in a bolt connection mode.
Background
The cylindrical shell has the advantages of high structural strength, large rigidity, light weight and the like, so the cylindrical shell is widely applied to the fields of aerospace, ocean engineering, pipelines, large dams, cooling towers and the like. In practical engineering, cylindrical shells are generally connected through bolts, but the dynamic characteristics of the bolt connection structure become more complicated due to various uncertainty sources of the bolt connection structure, and therefore the bolt connection uncertainty research is of great significance. The pretightening force of the bolt, the manufacturing error of the contact surface, the bolt looseness and the like are uncertainty sources, and finally the connection rigidity of the bolt has uncertainty.
At present, the dynamics certainty of the cylindrical shell is researched more, and a finite element method, an analytic method and a transfer matrix method are adopted in a modeling method. In addition, the study of the dynamic characteristics of the cylindrical shell structure connected by the bolt is also carried out by the scholars. Yao and the like perform parametric modeling by adopting a thin-layer unit simulation bolt connection structure based on a finite element method, thereby reducing the complexity of a finite element model of a multi-bolt connected aircraft engine case. Marc et al propose a new block model for nonlinear dynamics analysis of bolt flanges in aero-engine casings, which is more computationally efficient than models based on contact cell building. Tang et al established a nonlinear dynamics model of a bolted cylindrical shell structure by adopting a semi-analytical method, and analyzed the influence of boundary connection parameters on the dynamic characteristics of the model.
However, in actual operation, many parameters in the cylindrical shell dynamic model have uncertainties along with changes of the operating environment, so that the dynamic characteristics of the cylindrical shell dynamic model have corresponding uncertainties. Silva and the like analyze the influence of uncertainty of physical and geometric parameters on nonlinear vibration and stability of the simple cylindrical shell based on a probability method. Hakula et al solved the frequency response analysis of the complex shell under the uncertainty parameters using a stochastic method and verified it using the Monte Carlo method. It is worth noting that when a probabilistic method is used for uncertainty analysis, an accurate probability distribution of uncertainty parameters is difficult to obtain. Therefore, a non-probabilistic method, interval analysis, has been proposed by many scholars and used for dynamic uncertainty analysis of rotors, vehicles, gears, etc. The interval analysis method does not make any assumption on the specific probability distribution condition, and only needs to know the upper and lower bounds of the uncertain parameters, namely the parameters have the characteristic of 'uncertain but bounded'.
Based on the documents, the deterministic dynamics of the cylindrical shell is researched more, only a small amount of documents carry out the uncertain research on the cylindrical shell based on a probability method, and less documents adopt an interval analysis method to carry out the uncertain research on the vibration characteristic of the cylindrical shell.
Disclosure of Invention
Technical problem to be solved
The invention provides a cylindrical shell structure frequency response function uncertainty analysis method of a bolt connection flange, aiming at the technical problem that uncertainty analysis on the vibration characteristic of a cylindrical shell is few in the prior art.
(II) technical scheme
In order to achieve the purpose, the invention adopts the main technical scheme that:
a cylindrical shell structure frequency response function uncertainty analysis method of a bolt connecting flange adopts 8-node degraded shell units, and establishes a dynamic model of the bolt connecting flange-cylindrical shell through a finite element method;
the spring unit is used for modeling and simulating bolt connection in a discretization mode, and the effectiveness of the built model is verified through a modal test;
considering the uncertainty of the bolt connection rigidity, respectively solving frequency response function interval ranges of 5 directions under a cylindrical coordinate system when the connection rigidity is uncertainty parameters based on a Chebyshev polynomial agent model and an interval analysis method, adopting a Monte-Carlo sampling method to solve the frequency response function interval ranges under the certainty inherent frequency, and comparing the solving precision and the efficiency of the two methods;
and finally, solving the system frequency response function interval range of multi-parameter uncertainty.
Preferably, the establishing of the dynamical model comprises establishing a finite element model, and the overall coordinate of any point in the shell passes through the overall coordinate (x) of each node i ,y i ,z i ) And the interpolated form of the natural coordinates (xi, eta, zeta) is expressed as
Wherein, V 3i Is the position vector of the vertex relative to the bottom point in the thickness direction of the node i, N i Is a shape function of the cell;
at the corner nodes i equal to 1,2,3 and 4, the shape function is
The edge midpoints i are 5, 6, 7 and 8, and the shape function is
The displacement of any point in the cell in the global coordinate system can be expressed as
Wherein, t i Is the thickness of the shell at node i, l 3i 、m 3i And n 3i Respectively being cosine of an included angle between a local coordinate system at the node i and a coordinate axis of the whole coordinate system;
formula (4) can be converted into
Wherein the content of the first and second substances,
the strain vector of the unit is
ε＝GH (7)
Wherein
Relationship of unit strain vector epsilon capable of being converted into displacement
ε＝Ba (8)
The strain matrix B is
In the formula
Wherein, J -1 mn (m, n is 1,2,3) is the number of the mth row and the nth column after the Jacobian matrix J is inverted, I is a 3 x 3-order identity matrix,
the material elastic matrix of the cell under the local coordinate system is
Wherein, the first and the second end of the pipe are connected with each other,
e and mu are respectively the elastic modulus and Poisson's ratio of the material, and the coefficient k is 1.2;
the elastic matrix of the material is converted into a whole coordinate system from a local coordinate system
D＝T T D′T (11)
Wherein T is a transformation matrix between the local coordinate system and the global coordinate system;
the matrix of cell stiffness and the matrix of cell mass are
The expressions (12) and (13) are obtained by gaussian integration of 2 × 2 × 2.
Preferably, the establishing of the dynamic model comprises dynamic modeling and frequency response function analysis, and specifically comprises the following steps: when the shell unit is used for simulating a flange-cylindrical shell, the structure is dispersed into a limited number of unit bodies;
simulating a bolted connection by applying a space spring unit at each node on the flange;
the space spring applied on the flange shell has three translational stiffness and two rotational stiffness under a cylindrical coordinate system, wherein the three translational stiffness and the two rotational stiffness are respectively the translational stiffness of a radial u ' shaft, a tangential v ' shaft and an axial w ' shaft, and the rotational stiffness around the u ' shaft and the v ' shaft;
since the shell theory is 5 degrees of freedom in a cylindrical coordinate system, and each node corresponding to the space spring unit is 6 degrees of freedom, the corresponding row and column in the stiffness matrix of the spring unit in the other rotation direction are filled with 0;
wherein the spring unit matrix is
In the formula
k 1 ＝diag[k u′ k v′ k w′ k θu′ k θv′ 0]
Wherein k is u′ 、k v′ And k w′ Translational stiffness in u ', v ' and w ' directions, respectively, k θu′ And k θv′ Rotational stiffness around the u 'and v' directions, respectively;
spring unit stiffness matrix k 'in column coordinate system' spr And a unit stiffness matrix k under the global coordinate system spr The conversion relationship between the two is as follows
k spr ＝(T spr ) T k′ spr T spr (15)
Transformation matrix T spr Is composed of
T spr ＝diag[t spr t spr t spr t spr ] (16)
In the formula
C is a cosine value of an included angle between the coordinate axes of the integral coordinate system and the cylindrical coordinate system of the node; the rigidity matrix and the mass matrix of the structure are obtained through unit set, and the dynamic equation of the bolt connecting flange-cylindrical shell is
In the formula, M, C, K, K spr 、
q and f are respectively a mass matrix, a damping matrix, a structural rigidity matrix, a boundary connection rigidity matrix, an acceleration vector, a velocity vector, a displacement vector and an external excitation vector of the system;
proportional damping C ═ α M + β (K + K) spr ) Alpha and beta are constants of
In the formula, omega 1 And ω 7 Natural angular frequencies, ξ, of the 1 st and 7 th orders of the system, respectively 1 And xi 7 Modal damping ratio of two natural frequencies, respectively, where xi is taken 1 0.005 and xi 7 ＝0.001；
Fourier transform is performed on the formula (17) to obtain a displacement response
X(ω)＝(-Mω 2 +jωC+(K+K spr )) -1 F(ω) (19)
Wherein the content of the first and second substances,
the displacement frequency response function can be obtained
H d (ω)＝(-Mω 2 +jωC+(K+K spr )) -1 (20)
The mode shape matrix of the system can be expressed as
u＝[u 1 u 2 … u n ] (21)
Each column of the mode shape matrix is a mode shape vector and is a regularized mode shape vector, and the modal mass matrix is an identity matrix, i.e., an identity matrix
Thereby decoupling equation (17) into n independent equations of motion with an acceleration frequency response function of
Therefore, the acceleration frequency response function of the point b at the excitation point a is
Wherein, ω is i Is the ith order natural angular frequency, u i Is the ith order mode vector, and m is the mode order.
Preferably, the recursion formula of the Chebyshev polynomial in the uncertainty parameter interval analysis based on the Chebyshev polynomial is
T n+1 (x)＝2xT n (x)-T n-1 (x),n＝1,2,3,… (25)
Wherein, T 0 (x)＝1，T 1 (x)＝x；
For multidimensional parameters, the Chebyshev polynomial approximates to
Wherein
In the formula
Wherein x is 1 ,…,x g For uncertainty parameters in the standard interval [ -1,1 [)]G is the number of uncertainty parameters, u is i in the Chebyshev polynomial 1 ,…,i g The number of zero;
the multiple integral of the formula (26) is converted into numerical integral, the Gauss-Chebyshev numerical integral is adopted to convert the complex multiple integral into the numerical integral, the numerical integral is suitable for MATLAB numerical calculation software to operate, the solving speed is obviously improved under the condition of ensuring the solving precision, and the method can obtain
Wherein (x) 1 ,…,x m ) The number of Gaussian points in each dimension is h, and the Gaussian points are Gaussian points
X is a one-dimensional Gaussian point X 1 ,…,x m Is the product of the tensors of (X) is
Preferably, the analysis of the frequency response function interval in the analysis of the uncertain parameter interval based on the Chebyshev polynomial considers that the parameter vector a has uncertainty, the available interval is used for representing the fluctuation range of the uncertain parameter, no assumption is made on the specific probability distribution condition, and only the upper and lower bounds of the uncertain parameter are needed to be known, namely the parameter has the characteristic of uncertainty but limitation;
the uncertainty vector of the system is represented by intervals, and the uncertainty parameter of the g dimension can be represented as
Wherein
a m And
respectively, is an uncertainty parameter a m According to the interval mathematical theory, the uncertainty parameter vector a of the system is a g-dimensional interval combination;
intermediate values of the uncertain parameter of
The upper and lower bounds of the uncertainty parameter can be expressed as
Wherein the content of the first and second substances,
as a parameter of uncertainty m The fluctuation coefficient of (a);
therefore, the corresponding configuration points can be obtained through the formula (28), and then the frequency response function proxy model can be obtained through the Chebyshev polynomial;
consider that the Chebyshev polynomial approximation function is in the interval [ -1,1]Performing analysis while rotating the cylinder housingDeterministic parameters are arbitrarily spaced, so uncertainty parameters can be spaced by inter-region transformation
Transition to the interval [ -1,1]Is a
The range of the acceleration frequency response function interval of the excitation point a and the measuring point b of the bolt connecting flange-cylindrical shell structure is
The lower and upper bounds of the known frequency response function can be expressed as
The formulas (45) and (46) satisfy
The interval range of the frequency response function is difficult to solve directly, and only a small number of samples are needed by using a Chebyshev polynomial proxy model, so that the solving formulas (33) and (34) are converted into extreme values of the solving formula (25).
Preferably, the model analysis is based on the analysis of uncertainty parameter intervals of the Chebyshev polynomial
When the connection rigidity of the bolt connection flange-cylindrical shell has uncertainty, the uncertainty of a frequency response function is caused, and the connection rigidity k in a certain direction under a cylindrical coordinate system j Is expressed by interval mathematics as
Wherein the content of the first and second substances,
for uncertainty of connection stiffness k j The coefficient of fluctuation of (a).
The Chebyshev polynomial adopts 14-order expansion, the number of Gaussian integral points is 15, the fluctuation coefficient of the connection rigidity is 10%, and the interval range of the frequency response function is analyzed.
And respectively solving the frequency response function interval range under the cylindrical coordinate system when the single-parameter uncertainty of the rigidity in 5 directions exists, and comparing and analyzing the influence of the connection rigidity in each direction on the frequency response function interval range.
(III) advantageous effects
The beneficial effects of the invention are: the method provided by the invention has the following effects:
(1) the Chebyshev polynomial proxy model has higher solving precision and calculation efficiency, and the axial connection rigidity has the largest influence on the frequency response function of the system;
(2) the bolted stiffness mainly results in a large resonance band at the system's 1 st and 3 rd order natural frequencies.
Drawings
FIG. 1 is a schematic view of a degenerate case cell provided in accordance with the present invention;
FIG. 2 is a schematic diagram of a bolted flange-cylindrical shell boundary condition simulation provided by the present invention;
FIG. 3 is a test piece and test system provided by the present invention;
FIG. 4 is a schematic view of a flange-cylindrical shell structure provided by the present invention;
FIG. 5 is a comparison of frequency response functions of the present application and experiments provided by the present invention;
FIG. 6 is a frequency response function when translational stiffness in the u' direction is uncertain;
FIG. 7 is a frequency response function for uncertain translational stiffness in the v' direction;
FIG. 8 shows the frequency response function for uncertain axial stiffness in the w' direction;
FIG. 9 is a frequency response function when the rotational stiffness in the u' direction is uncertain;
FIG. 10 is a frequency response function when the rotational stiffness in the v' direction is uncertain;
FIG. 11 is a frequency response function for a multi-parameter uncertainty.
Detailed Description
For the purpose of better explaining the present invention and to facilitate understanding, the present invention will be described in detail by way of specific embodiments with reference to the accompanying drawings.
The invention discloses a method for analyzing uncertainty of a cylindrical shell structure frequency response function of a bolt connecting flange, which comprises the following steps of:
an 8-node degraded shell unit is adopted, and a dynamic model of the bolt connection flange-cylindrical shell is established through a finite element method;
the spring unit is used for modeling and simulating bolt connection in a discretization mode, and the effectiveness of the built model is verified through a modal test;
considering the uncertainty of the bolt connection rigidity, respectively solving frequency response function interval ranges of 5 directions under a cylindrical coordinate system when the connection rigidity is uncertainty parameters based on a Chebyshev polynomial agent model and an interval analysis method, adopting a Monte-Carlo sampling method to solve the frequency response function interval ranges under the certainty inherent frequency, and comparing the solving precision and the efficiency of the two methods;
and finally, solving the system frequency response function interval range of multi-parameter uncertainty.
The method is specifically as follows:
1 model establishment and experimental verification
1.1 finite element model
FIG. 1 shows a 8-node degenerate case unit, where the global coordinate of any point in the case passes through the global coordinate (x) of each node i ,y i ,z i ) And the interpolated form of the natural coordinates (xi, eta, zeta) as
Wherein, V 3i Is the position vector of the vertex relative to the bottom point in the thickness direction of the node i, N i Is a shape function of the cell.
At the angular nodes i 1,2,3 and 4, the shape function is
Edge midpoints i are 5, 6, 7 and 8, and the shape function is
The displacement of any point in the unit in the global coordinate system can be expressed as
Wherein, t i Is the thickness of the shell at node i, l 3i 、m 3i And n 3i Respectively, the cosine of the included angle between the local coordinate system at the node i and the coordinate axis of the global coordinate system.
Formula (4) can be converted into
Wherein
a i ＝[u i v i w i θ xi θ yi θ zi ] T ，(i＝1,2,…,8)
The strain vector of the unit is
ε＝GH (7)
Wherein
Relationship by which element strain vector epsilon can be converted into displacement
ε＝Ba (8)
The strain matrix B is
In the formula
Wherein, J -1 mn (m, n is 1,2,3) is the number of the mth row and the nth column after the Jacobian matrix J is inverted, I is a 3 x 3-order identity matrix,
the material elastic matrix of the cell under the local coordinate system is
Wherein the content of the first and second substances,
e and mu are respectively the elastic modulus and Poisson's ratio of the material, and the coefficient k is 1.2.
The elastic matrix of the material is converted into a whole coordinate system from a local coordinate system
D＝T T D′T (11)
Wherein, T is a transformation matrix between the local coordinate system and the global coordinate system.
The matrix of cell stiffness and the matrix of cell mass are
The expressions (12) and (13) are obtained by gaussian integration of 2 × 2 × 2.
1.2 kinetic modeling and frequency response function analysis
When the shell unit is used for simulating a flange-cylindrical shell, the structure is dispersed into a limited number of unit bodies. Further, the bolted connection is simulated herein by applying a space spring unit at each node on the flange. The space spring applied on the flange shell has three translational stiffness and two rotational stiffness under a cylindrical coordinate system, namely the translational stiffness of a radial u ' axis, a tangential v ' axis and an axial w ' axis, and the rotational stiffness around the u ' axis and the v ' axis, as shown in fig. 2. Since the housing theory herein is 5 degrees of freedom in a cylindrical coordinate system and each node corresponding to a spatial spring element is 6 degrees of freedom, the other rotational direction is filled with 0 in the corresponding row and column of the spring element stiffness matrix.
The spring unit matrix is
In the formula
k 1 ＝diag[k u′ k v′ k w′ k θu′ k θv′ 0]
Wherein k is u′ 、k v′ And k w′ Translational stiffness in u ', v ' and w ' directions, respectively, k θu′ And k θv′ Rotational stiffness around the u 'and v' directions, respectively. Spring unit stiffness matrix k 'in column coordinate system' spr And a unit stiffness matrix k in a global coordinate system spr The conversion relationship between the two is as follows
k spr ＝(T spr ) T k′ spr T spr (15)
Transformation matrix T spr Is composed of
T spr ＝diag[t spr t spr t spr t spr ] (16)
In the formula
And C is an included angle cosine value of the node between the coordinate axes of the whole coordinate system and the cylindrical coordinate system.
The flange-cylindrical shell finite element meshing obtains a rigidity matrix and a quality matrix of the structure through unit set, and the dynamic equation of the bolted flange-cylindrical shell is
In the formula, M, C, K, K spr 、
And q and f are respectively a mass matrix, a damping matrix, a structural rigidity matrix, a boundary connection rigidity matrix, an acceleration vector, a velocity vector, a displacement vector and an external excitation vector of the system.
Proportional damping C ═ α M + β (K + K) spr ) Alpha and beta are constants of
In the formula, ω 1 And ω 7 The natural angular frequencies of the 1 st and 7 th orders of the system, respectively,ξ 1 And xi 7 Modal damping ratios of two natural frequencies, respectively, where ξ is taken 1 0.005 and xi 7 ＝0.001。
Fourier transform is performed on the formula (17) to obtain a displacement response
X(ω)＝(-Mω 2 +jωC+(K+K spr )) -1 F(ω) (19)
Wherein the content of the first and second substances,
the displacement frequency response function can be obtained
H d (ω)＝(-Mω 2 +jωC+(K+K spr )) -1 (20)
The mode matrix of the system can be expressed as
u＝[u 1 u 2 … u n ] (21)
Each column of the mode shape matrix is a mode shape vector and is a regularized mode shape vector, and the modal mass matrix is an identity matrix, i.e., an identity matrix
Thus, equation (17) is decoupled into n independent equations of motion with a frequency response function of acceleration of
Therefore, the acceleration frequency response function of the point b at the excitation point a is
Wherein, ω is i Is the ith order natural angular frequency, u i Is the ith order mode vector, and m is the mode order.
1.3 Experimental validation
In order to verify the correctness of the established model, the frequency response function is verified experimentally, a hammering method is adopted for modal testing, and the experiment mainly uses a Donghua vibration testing system for data acquisition and analysis. FIG. 3 shows a test piece and a test system for modal testing, in which a flange-cylindrical shell is connected to a base by 12 bolts, and each bolt is applied with an equal pre-tightening force.
The schematic diagram of the flange-cylindrical shell structure is shown in fig. 4, wherein the influence of bolt holes on the flange is ignored, the geometric parameters are shown in table 1, and the radius of the middle surface of the cylindrical shell is R ═ R (R) 1 +r 2 ) And/2, the material parameters of the structure are as follows: elastic modulus E is 200GPa, Poisson's ratio mu is 0.26, density rho is 7850kg/m 3 。
TABLE 1 Flange-cylindrical Shell Structure geometry parameters
In the experiment, a unidirectional acceleration sensor is used, and modal test is carried out on the unidirectional acceleration sensor by adopting single-point input and single-point output, so that a frequency response function of the flange-cylindrical shell structure is obtained. In addition, the validity of the created model was also verified herein using the shell281 unit in ANSYS (see Table 2), which also applies space spring unit simulation boundary conditions in the cylindrical coordinate system of the unit in ANSYS software. It is worth explaining that the stiffness of each spring in each direction is k under a cylindrical coordinate system u′ ＝9.5×10 7 N/m、k v′ ＝7.36×10 6 N/m、k w′ ＝9.8×10 6 N/m、 k θu′ ＝1.8×10 4 N·m/rad、k θv′ 1.1 × 104N · m/rad. Table 2 shows the first 7 th order natural frequency comparison, fig. 5 shows the frequency response function comparison of the present document and the experimental test, where the coordinate position of the acceleration sensor is (R, 0, L) and the excitation point is also (R, 0, L).
TABLE 2 first 7 natural frequency contrast
Note: error 1 and error 2 are the error from the text and experiment and the error from ANSYS
As can be seen from Table 2, the maximum error of the natural frequency of the spring unit discretization modeling simulation bolt connection is 3 rd order, the error is only 1.4%, and the maximum error of the spring unit discretization modeling simulation bolt connection is 0.48% in ANSYS.
As can be seen from fig. 5, the trend of the frequency response function obtained by simulation and experimental test in this document is well matched, and only a certain error exists on the peak value, which is mainly because proportional damping is used in this document, and it is difficult to truly reflect each order modal damping of the system, so that an error exists on the peak value of the frequency response function.
2 uncertainty parameter interval analysis based on Chebyshev polynomial
2.1 multidimensional Chebyshev polynomial
The recursion formula of the Chebyshev polynomial is
T n+1 (x)＝2xT n (x)-T n-1 (x),n＝1,2,3,… (25)
Wherein, T 0 (x)＝1，T 1 (x)＝x。
For multidimensional parameters, the Chebyshev polynomial approximates to
Wherein
In the formula
Wherein x is 1 ,…,x g For uncertainty parameters in the standard interval [ -1,1 [)]G is the number of uncertainty parameters, u is i in the Chebyshev polynomial 1 ,…,i g The number of zeros.
Converting the multiple integral of the formula (26) into a numerical integral, converting the complex multiple integral into the numerical integral by adopting Gauss-Chebyshev numerical integral, wherein the numerical integral is suitable for MATLAB numerical calculation software to carry out operation, the solving speed is also obviously improved under the condition of ensuring the solving precision, and the solving speed can be obtained
Wherein (x) 1 ,…,x m ) The Gaussian point is a Gaussian point of Gaussian integral, the number of Gaussian points in each dimension is h, and the Gaussian point
X is a one-dimensional Gaussian point X 1 ,…,x m Is the product of the tensors, X is
2.2 frequency response function Interval analysis
Considering that the parameter vector a has uncertainty, the available interval is used for representing the fluctuation range of the uncertainty parameter, no assumption is made on the specific probability distribution condition, and only the upper and lower bounds of the uncertainty parameter are needed to be known, namely the parameter has the characteristic of 'uncertain but bounded'. The uncertainty vector of the system is represented by intervals, and the uncertainty parameter of the g dimension can be represented as
Wherein
a m And
respectively, is an uncertainty parameter a m The lower and upper bounds of (1), the uncertainty parameter vector a of the system is a g-dimensional interval combination according to the interval mathematical theory.
Intermediate values of the uncertain parameter of
The upper and lower bounds of the uncertainty parameter can be expressed as
Wherein, the first and the second end of the pipe are connected with each other,
as a parameter of uncertainty a m The fluctuation coefficient of (2).
Therefore, the corresponding configuration point number can be obtained by the equation (28), and then the frequency response function proxy model can be obtained by the Chebyshev polynomial. Consider that the Chebyshev polynomial approximation function is in the interval [ -1,1]The uncertainty parameter in the rotating cylindrical shell is arbitrarily divided, so that the uncertainty parameter can be divided by interval transformation
Transition to the interval [ -1,1 [)]Is a
The range of the acceleration frequency response function interval of the excitation point a and the measuring point b of the bolt connecting flange-cylindrical shell structure is
The lower and upper bounds of the known frequency response function can be expressed as
The formulas (45) and (46) satisfy
The interval range of the frequency response function is difficult to directly solve, and only a small number of samples are needed by using a Chebyshev polynomial proxy model, so that the solving formulas (33) and (34) are converted into the extreme value of the solving formula (25).
2.3 model analysis
When the connection rigidity of the bolt connection flange-cylindrical shell has uncertainty, the uncertainty of a frequency response function is caused, and the connection rigidity k in a certain direction under a cylindrical coordinate system j Is expressed by interval mathematics as
Wherein the content of the first and second substances,
for uncertainty of connection stiffness k j The coefficient of fluctuation of (a).
The Chebyshev polynomial adopts 14-order expansion, the number of Gaussian integral points is 15, the fluctuation coefficient of the connection rigidity is 10%, and the interval range of the frequency response function is analyzed.
And respectively solving frequency response function interval ranges under the cylindrical coordinate system when the single-parameter uncertainty of the stiffness in 5 directions exists, and comparing and analyzing the influence of the connection stiffness in each direction on the frequency response function interval ranges.
As can be seen from FIGS. 6-10, when the fluctuation coefficient of the uncertainty parameter is 10%, the influence of the w' direction axial connection stiffness on the frequency response function of the system is the largest (see FIG. 8), and the frequency response function fluctuation around the 3 rd order natural frequency is larger, and the original single resonance peak becomes a resonance band, so that the frequency shift phenomenon occurs, and the natural frequency is approximately between 1291-1349 Hz. Secondly, the rotational stiffness in the v' direction also has certain influence on the frequency response function (see fig. 10), the 3 rd order natural frequency is about 1316-. It is worth to be noted that the interval of the frequency response function is not symmetrical about the deterministic frequency response function, the maximum peak value at the resonance band is basically the same as the deterministic resonance peak value, and the minimum peak value becomes very small; the minimum peak at the anti-resonance band does not vary much from the deterministic anti-resonance peak, but the maximum peak becomes much larger.
And simultaneously, solving a frequency response function interval range at the deterministic inherent frequency when the axial rigidity in the w' direction is an uncertain parameter by adopting a Monte-Carlo sampling method, wherein the number of samples of the Monte-Carlo sampling method is 500, finally obtaining the maximum value and the minimum value of the frequency response function, and then comparing the frequency response function interval range obtained by the method with that obtained by the Monte-Carlo sampling method and comparing the time consumption of the two methods. Table 3 shows the upper and lower bounds of the frequency response function corresponding to the system deterministic natural frequency, and it can be seen that the Chebyshev polynomial proxy model has a better comparison with the upper and lower bounds results obtained by the sampling method, and the error is very small, wherein the time consumed by the Monte-Carlo sampling method is 2286.2s, the time consumed by the Chebyshev polynomial is 85.2 s, and is about 3.7% of the calculation time of the sampling method, thereby verifying that the method has higher precision and solving efficiency.
TABLE 3 frequency response function interval ranges of two methods
Because the axial connection rigidity in the w 'direction and the rotational rigidity in the v' direction have larger influence on the frequency response function of the system, and the connection rigidity in the other three directions have smaller influence, the axial connection rigidity in the w 'direction and the rotational rigidity in the v' direction are considered as uncertain parameters at the same time, the fluctuation coefficients are all 10 percent, the frequency response function interval range of the system is solved, and the influence of multi-parameter uncertainty on the frequency response function is analyzed. As can be seen from fig. 11, the uncertainty of the bolt connection stiffness mainly causes the frequency response function of the system to fluctuate greatly around the 1 st order natural frequency and the 3 rd order natural frequency, and the 3 rd order natural frequency is approximately between 1286 and 1354Hz, which indicates that the range of the frequency response function is expanded by the multi-parameter uncertainty.
3. Conclusion
A dynamic model of a bolted flange-cylindrical shell is established based on a finite element method, the effectiveness of the established model is verified through tests and ANSYS simulation, secondly, the influence of uncertainty of connection rigidity on a system frequency response function is analyzed based on a Chebyshev polynomial proxy model, and the verification is carried out through a Monte-Carlo sampling method, so that the following conclusion is obtained:
(1) under the condition of ensuring the solving precision, the Chebyshev polynomial proxy model has higher efficiency of solving the range of the system frequency response function interval, the axial connection rigidity of the flange-cylindrical shell has larger influence on the frequency response function of the system, and particularly, a larger resonance band is formed at the 3 rd order natural frequency, so that the frequency shift phenomenon occurs.
(2) The connection rigidity mainly causes larger resonance bands to be formed at the 1 st order and 3 rd order natural frequencies of the system, so that the influence of the bolt connection rigidity on the dynamic characteristics of the flange-cylindrical shell is larger, and the uncertainty of the connection rigidity is fully considered in the design and manufacture.
The foregoing description of the principles of the invention has been presented in connection with specific embodiments and is made only for the purpose of illustrating the principles of the invention and is not to be construed as limiting the scope of the invention in any way. Based on the teachings herein, those skilled in the art will be able to conceive of other specific embodiments of the present invention without any creative effort, and these embodiments will fall within the protection scope of the present invention.
Claims (4)
1. A method for analyzing uncertainty of a cylindrical shell structure frequency response function of a bolt connecting flange is characterized by comprising the following steps:
an 8-node degraded shell unit is adopted, and a dynamic model of the bolted flange-cylindrical shell is established by a finite element method;
the spring unit is used for modeling and simulating bolt connection in a discretization mode, and the effectiveness of the built model is verified through a modal test;
considering the uncertainty of the bolt connection rigidity, respectively solving frequency response function interval ranges of 5 directions under a cylindrical coordinate system when the connection rigidity is uncertainty parameters based on a Chebyshev polynomial agent model and an interval analysis method, adopting a Monte-Carlo sampling method to solve the frequency response function interval ranges under the certainty inherent frequency, and comparing the solving precision and the efficiency of the two methods;
finally, solving a system frequency response function interval range of multi-parameter uncertainty;
the recursion formula of the Chebyshev polynomial in the uncertainty parameter interval analysis based on the Chebyshev polynomial is
T n+1 (x)＝2xT n (x)-T n-1 (x),n＝1,2,3,…(1)
Wherein, T 0 (x)＝1，T 1 (x)＝x；
For multidimensional parameters, the Chebyshev polynomial approximates to
Wherein
In the formula
Wherein，x 1 ,…,x g For uncertainty parameters in the standard interval [ -1,1 [)]G is the number of uncertainty parameters, u is i in the Chebyshev polynomial 1 ,…,i g The number of zero;
converting the multiple integral of the formula (2) into a numerical integral, converting the complex multiple integral into the numerical integral by adopting Gauss-Chebyshev numerical integral, wherein the numerical integral is suitable for MATLAB numerical calculation software to carry out operation, the solving speed is also obviously improved under the condition of ensuring the solving precision, and the method can obtain
Wherein (x) 1 ,…,x m ) The Gaussian point is a Gaussian point of Gaussian integral, the number of Gaussian points in each dimension is h, and the Gaussian point
X is a one-dimensional Gaussian point X 1 ,…,x m Is the product of the tensors of (X) is
In the analysis of the frequency response function interval in the uncertainty parameter interval based on the Chebyshev polynomial, the parameter vector a is considered to have uncertainty, the interval can be used for representing the fluctuation range of uncertainty parameters, no assumption is made on the specific probability distribution condition, and only the upper and lower bounds of the uncertainty parameters are needed to be known, namely the parameters have the characteristic of uncertainty but being bounded;
the system uncertainty vector is represented by intervals, and the g-dimensional uncertainty parameter can be represented as
Wherein
a m And
respectively an uncertain parameter a m According to the interval mathematical theory, the uncertainty parameter vector a of the system is a g-dimensional interval combination;
intermediate values of the uncertain parameter of
The upper and lower bounds of the uncertainty parameter can be expressed as
Wherein the content of the first and second substances,
as a parameter of uncertainty a m The fluctuation coefficient of (a);
therefore, the corresponding configuration point number can be obtained through the formula (4), and then the frequency response function proxy model can be obtained through the Chebyshev polynomial;
consider that the Chebyshev polynomial approximation function is in the interval [ -1,1]The uncertainty parameter in the rotating cylindrical shell is arbitrarily divided, so that the uncertainty parameter can be divided by region change
Transition to the interval [ -1,1 [)]Is a
The range of the acceleration frequency response function interval of the excitation point a and the measuring point b of the bolt connecting flange-cylindrical shell structure is
The lower and upper bounds of the known frequency response function can be expressed as
The formulas (10) and (11) satisfy
The interval range of the frequency response function is difficult to solve directly, and only a small number of samples are needed by using a Chebyshev polynomial agent model, so that the solving equations (9) and (10) are converted into the extreme value of the solving equation (1).
2. The method of claim 1, wherein the step of modeling the dynamics includes modeling finite elements, and the global coordinates of any point in the casing are determined by the global coordinates (x) of each node i ,y i ,z i ) And the interpolated form of the natural coordinates (xi, eta, zeta) as
Wherein, V 3i Is a position vector of a vertex relative to a bottom point in the thickness direction of the node i, N i Is a shape function of the cell;
at the angular nodes i 1,2,3 and 4, the shape function is
Edge midpoints i are 5, 6, 7 and 8, and the shape function is
The displacement of any point in the unit in the global coordinate system can be expressed as
Wherein, t i Is the thickness of the shell at node i, /) 3i 、m 3i And n 3i Respectively being cosine of an included angle between a local coordinate system at the node i and a coordinate axis of the whole coordinate system;
formula (16) can be converted to
Wherein, the first and the second end of the pipe are connected with each other,
the strain vector of the unit is
ε＝GH (19)
Wherein
Relationship by which element strain vector epsilon can be converted into displacement
ε＝Ba (20)
The strain matrix B is
In the formula
Wherein, J -1 mn (m, n is 1,2,3) is the number of the mth row and the nth column after the Jacobian matrix J is inverted, I is a 3 x 3-order identity matrix,
the elastic matrix of the material of the unit under the local coordinate system is
Wherein the content of the first and second substances,
e and mu are respectively the elastic modulus and Poisson's ratio of the material, and the coefficient k is 1.2;
the elastic matrix of the material is converted into a whole coordinate system from a local coordinate system
D＝T T D′T (23)
Wherein T is a transformation matrix between the local coordinate system and the global coordinate system;
the matrix of cell stiffness and the matrix of cell mass are
The equations (24) and (25) are obtained by gaussian integration of 2 × 2 × 2;
where ρ is the density and N is the haplotype function.
3. The method for analyzing uncertainty of the frequency response function of the cylindrical shell structure of the bolted flange according to claim 2, wherein the step of establishing a dynamic model comprises dynamic modeling and frequency response function analysis, and specifically comprises the following steps: when the shell unit is used for simulating a flange-cylindrical shell, the structure is dispersed into a limited unit body;
simulating a bolted connection by applying a spatial spring unit at each node on the flange;
the space spring applied on the flange shell has three translational stiffness and two rotational stiffness under a cylindrical coordinate system, wherein the three translational stiffness and the two rotational stiffness are respectively the translational stiffness of a radial u ' shaft, a tangential v ' shaft and an axial w ' shaft, and the rotational stiffness around the u ' shaft and the v ' shaft;
since the shell theory is 5 degrees of freedom in a cylindrical coordinate system, and each node corresponding to the space spring unit is 6 degrees of freedom, the corresponding rows and columns in the stiffness matrix of the spring unit in the other rotation direction are filled with 0;
wherein the spring unit matrix is
In the formula
k 1 ＝diag[k u′ k v′ k w′ k θu′ k θv′ 0]
Wherein k is u′ 、k v′ And k w′ Translational stiffness in u ', v ' and w ' directions, respectively, k θu′ And k θv′ Rotational stiffness around the u 'and v' directions, respectively;
spring unit stiffness matrix k 'in column coordinate system' spr And a unit stiffness matrix k under an overall coordinate system spr The conversion relationship between the two is as follows
k spr ＝(T spr ) T k′ spr T spr (27)
Transformation matrix T spr Is composed of
T spr ＝diag[t spr t spr t spr t spr ] (28)
In the formula
C is a cosine value of an included angle between the coordinate axes of the integral coordinate system and the cylindrical coordinate system of the node;
the rigidity matrix and the mass matrix of the structure are obtained through unit set, and the dynamic equation of the bolt connecting flange-cylindrical shell is
In the formula, M, C, K, K spr 、
q and f are respectively a mass matrix, a damping matrix, a structural rigidity matrix, a boundary connection rigidity matrix, an acceleration vector, a velocity vector, a displacement vector and an external excitation vector of the system;
proportional damping C ═ α M + β (K + K) spr ) Alpha and beta are constants of
In the formula, ω 1 And omega 7 Natural angular frequencies, ξ, of the 1 st and 7 th orders of the system, respectively 1 And xi 7 Modal damping ratios, here ξ, for two natural frequencies respectively 1 0.005 and xi 7 ＝0.001；
Fourier transform is performed on the formula (29) to obtain a displacement response
X(ω)＝(-Mω 2 +jωC+(K+K spr )) -1 F(ω) (31)
Wherein, the first and the second end of the pipe are connected with each other,
the displacement frequency response function can be obtained
H d (ω)＝(-Mω 2 +jωC+(K+K spr )) -1 (32)
The mode matrix of the system can be expressed as
u＝[u 1 u 2 …u n ] (33)
Each column of the mode shape matrix is a mode shape vector and is a regularized mode shape vector, and the mode quality matrix is an identity matrix, i.e.
Thus, equation (29) is decoupled into n independent equations of motion, with the acceleration frequency response function being
Therefore, the acceleration frequency response function of the point b at the excitation point a is
Wherein, ω is i Is the ith order natural angular frequency, u i Is the ith order mode vector, and m is the mode order.
4. The method of claim 1, wherein the method comprises a model analysis in an analysis of uncertainty parameter intervals based on a Chebyshev polynomial
When the connection rigidity of the bolt connection flange-cylindrical shell has uncertainty, the uncertainty of a frequency response function is caused, and the connection rigidity k in a certain direction under a cylindrical coordinate system j Is expressed by interval mathematics as
Wherein the content of the first and second substances,
for uncertainty of connection stiffness k j The fluctuation coefficient of (a);
the Chebyshev polynomial adopts 14-order expansion, the number of Gaussian integration points is 15, the fluctuation coefficient of the connection rigidity is 10%, and the interval range of the frequency response function is analyzed;
and respectively solving the frequency response function interval range under the cylindrical coordinate system when the single-parameter uncertainty of the rigidity in 5 directions exists, and comparing and analyzing the influence of the connection rigidity in each direction on the frequency response function interval range.
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