WO2019011025A1 - Finite element based method for identifying distributed random dynamic load - Google Patents

Abstract

A finite element based method for identifying a distributed random dynamic load. The method comprises the following steps: S1. acquiring a set of random vibration response samples at a plurality of points in a structure by means of multiple repeated measurements; S2. establishing a polycondensed finite element model of the structure matched with the degree of freedom of an actually measured dynamic response; S3. resolving the distributed random dynamic load from a random dynamic response at a unit node via KL expansion; and S4. resolving a time-varying statistical characteristic of the random dynamic load distributed over space in the structure. The method solves the problem of identifying, by utilizing an actually measured dynamic response sample of a structure in a time domain, a time-varying statistical characteristic of a random dynamic load distributed over space in a structure, and provides a means of indirectly acquiring a dynamic load for structural design and security evaluation of a project serving under a distributed random dynamic load environment.

Description

A distributed random dynamic load identification method based on finite element Technical field:

The invention relates to a distributed random dynamic load identification method based on finite element, which belongs to the technical field of structural dynamic inverse problems.

Background technique:

Dynamic load information on the engineering structure is the basis for structural design and safety assessment. In many cases, dynamic loads are difficult to obtain by direct measurement. The dynamic response of the structure is often measured directly, and the dynamic load information on the structure is obtained by means of indirect identification.

The traditional dynamic load identification method uses the structural dynamic response data of a single actual measurement to identify the excitation information that causes the secondary dynamic response, and is a deterministic dynamic load identification method. The existing deterministic dynamic load identification method is used to obtain information such as concentrated dynamic load, moving load and distributed dynamic load on the engineering structure. It is worth noting that the distributed dynamic load identification problem is equivalent to identifying an infinite number of concentrated dynamic loads, which is more difficult. Generally, the distributed dynamic load identification problem needs to be reduced in dimension.

The dynamic loads acting on the actual engineering structure, such as the wind load on the building, the wave load on the ocean platform and the aerodynamic load on the surface of the aircraft, are not only distributed on the structure, but also random. When a random dynamic load is applied to the structure, the dynamic response will also appear “randomness”; therefore, the structural dynamic response of a single measured measurement can only be one of the samples of the structural random dynamic response information, and the certainty is utilized based on a certain response sample. The dynamic load information obtained by the dynamic load identification method can only partially reflect the random dynamic load excitation; in addition, the dynamic response error contained in a single measurement is also used as part of the “true response” in the deterministic dynamic load identification, causing the load. Identify the deviation of the results. For the identification of such distributed random dynamic loads, the traditional deterministic distributed dynamic load identification method and the centralized random dynamic load identification method are not applicable. It is necessary to develop a new method for distributed random dynamic load identification.

Summary of the invention

The object of the present invention is to provide a distributed random dynamic load identification method based on finite element, which solves the problem of time-varying statistical characteristics of random dynamic load with spatial distribution in the time-domain using the measured structure dynamic response sample identification structure, for serving in random distribution. The engineering structure design and safety assessment under dynamic load environment provides a means of indirect acquisition of dynamic loads.

The above objectives are achieved by the following technical solutions:

A distributed random dynamic load identification method based on finite element, the method comprising the following steps:

S1. Using multiple repeated measurements to obtain a set of random vibration response samples at multiple points on the structure;

S2. Establish a polycondensed structural finite element model matching the measured dynamic response degrees of freedom;

S3. Using KL expansion to solve the distributed random dynamic load by the random dynamic response at the unit node;

S4: Solving the time-varying statistical characteristics of the random dynamic load with spatial distribution on the structure.

The finite element-based distributed random dynamic load identification method, the specific method for establishing a polycondensed structural finite element model that is matched with the measured dynamic response degree of freedom described in step S2 includes:

S21: Ensure that the dynamic response measurement point is a unit node, and use the known structural parameters and finite element method to establish a structural finite element model to obtain the mass matrix and stiffness matrix of the structure. The damping of the structure adopts Rayleigh damping, and the damping matrix is composed of mass matrix and stiffness. Matrix calculation;

S22: Perform model polycondensation on the structural finite element model, retain only the degree of freedom information of the measured dynamic response, obtain the condensed structural finite element model matching the measured dynamic response degree of freedom, obtain the polycondensed mass matrix, stiffness matrix and damping matrix. .

The finite element-based distributed random dynamic load identification method, the specific method for solving the distributed random dynamic load by using the KL expansion in the step S3 to solve the distributed random dynamic load by the unit node includes:

S31: Solving a covariance matrix of the random dynamic response by using a sample set of the measured random dynamic response;

S32: performing eigenvalue decomposition on the covariance matrix, obtaining a random dynamic response KL vector at each unit node on the finite element model after the polycondensation, and completing KL expansion of the random dynamic response;

S33: Inverting a random dynamic load corresponding vector on each unit node by a random motion response KL vector of each unit node;

S34: Solving the distributed random dynamic load by the stochastic dynamic load corresponding vector on the unit node.

The finite element-based distributed random dynamic load identification method, the specific method for solving the distributed random dynamic load by the random dynamic load corresponding vector on the unit node described in step S34 includes the following steps;

S341: Establish a distributed random dynamic load model based on polynomial expansion;

S342: Solving the deterministic coefficient of the distributed random dynamic load model;

S343: Reconstructing the distributed random dynamic load.

Beneficial effects:

Compared with the prior art, the invention has the following advantages:

1. The existing random dynamic load identification technology can only identify the random concentrated dynamic load on the structure by the measured structure dynamic response sample. The existing distributed random dynamic load identification methods cannot be applied to the identification of non-stationary random dynamic loads. The distributed random dynamic load time domain identification technology provided by the invention can utilize the measured structural dynamic response sample at the limited measurement point to identify the statistical characteristics of the random dynamic load with the spatial distribution, and has certain advancement;

The KL expansion method is used to invert the random dynamic load at the unit node by the random dynamic response at the element node, which has higher computational efficiency than the Monte Carlo method based on the random sample, and reduces the time of load identification.

DRAWINGS

Figure 1 is a logic flow diagram of the method of the present invention.

Figure 2 is a schematic diagram of a simply supported beam under distributed random loads.

Figure 3(a) shows the results of the mean value of the random dynamic load in the beam span.

Figure 3(b) shows the results of the variance of the random dynamic load in the beam span.

Figure 4 shows the results of spatial distribution of random dynamic loads on the beam.

Detailed ways

The technical solutions of the present invention are described in detail below by way of embodiments, but the embodiments are merely preferred embodiments of the present invention, and it should be noted that those skilled in the art can, without departing from the principles of the present invention. It is also possible to make several modifications and equivalent substitutions to the structural and dynamic load forms, and the technical solutions of the present invention are modified and equivalently substituted, and all fall within the scope of the present invention.

Embodiment: The distributed random dynamic load acting on the simply supported beam structure as shown in Fig. 2 is identified by the method of the present invention. Beam length L=40m, cross-sectional area A=4.8m ² , section moment of inertia I=2.5498m ⁴ , the damping of the structure adopts Rayleigh damping, the modal damping ratio of each order is ξ _i =0.02, the elastic modulus of the material E= 5 × 10 ¹⁰ N/m ² , density ρ = 2.5 × 10 ³ kg / m ³ . The trapezoidal distributed random dynamic load distribution function to be identified is:

The stochastic dynamic load component F(t, θ) of distributed random dynamic load is divided into two parts: deterministic dynamic load and random dynamic load.

The deterministic dynamic load portion of F(t, θ):

F _d (t)=20000[1+0.1sin(2πt)]N (2)

The random dynamic load portion of F(t, θ) assumes a zero-mean non-stationary Gaussian random process, and the power spectrum function S(ω, t) is:

S(ω,t)=C _f P _d (t)Φ(ω) (3)

Where: C _f represents the random level, taking C _f =0.2; Φ(ω) represents the power spectral density function of the zero-mean non-stationary Gaussian random process, with Φ(ω)=(1/2π)(2/ω ² +1 ).

The spatial distribution and statistical characteristics of the random dynamic load are identified by the measured dynamic random response sample by using the technique of the present invention, and specifically includes the following steps:

和

其中字符上方的点表示对时间求导。如测量获取的是加速度信号，依然可以利用其导数关系求解速度和位移。 S1: Acquire a set of random vibration response samples at multiple points on the structure by using multiple repeated measurements. The details are as follows: the beam structure is divided into 12 equal parts, and the dynamic displacement of the sensor measurement structure is arranged at other nodes except the two end points, then the number of measurement points is m=11; the number of measurement times is 5000 times; the random dynamic displacement vector R(t) of the node is obtained after measurement. , θ), from the displacement and the derivative relationship between velocity and acceleration, the node velocity and acceleration can be expressed as

with

The point above the character indicates the derivation of time. If the measurement acquires an acceleration signal, the derivative relationship can still be used to solve the velocity and displacement.

S2. Establish a polycondensed structural finite element model that matches the measured dynamic response degrees of freedom, including the following steps:

S21: Ensure that the dynamic response measurement point is a unit node. At this time, the beam structure is divided into 12 beam units, and the structural finite element model is established by using the structural parameters of the beam in the embodiment and the finite element method to obtain the mass matrix M of the structure. The stiffness matrix K, the damping of the structure adopts Rayleigh damping; the damping matrix C is calculated from the mass matrix M and the stiffness matrix K by the following formula;

C=c _M M+c _K K (4)

Where c _M and c _K are constants related to structural parameters. If plane beam element modeling is used, the mass matrix of the beam structure after constraining the boundary is considered. The stiffness matrix and the damping matrix are both 24×24 matrices.

S22: finite element model of the polycondensation to carry out the model, leaving only a degree of freedom of information has measured the dynamic response, acquires the measured dynamic response finite element model of the structure of freedom finisher match, acquires the polycondensation mass matrix M _r, stiffness matrix K _r and the damping matrix C _r . details as follows:

There are 11 dynamic response measuring points, only the degree of freedom of the measuring points is retained. The mass matrix M with the original size of 24×24, the stiffness matrix K and the damping matrix C are condensed into the 11×11 polycondensed mass matrix by the model polycondensation method. M _r , stiffness matrix K _r and damping matrix C _r . Here IRS polycondensation method is adopted. Firstly, the random motion vector R(t, θ) of simply supported beam nodes is divided into two categories:

Where: R _m (t, θ) represents the measured degree of freedom dynamic response information, and R _s (t, θ) represents the unmeasured degree of freedom dynamic response information. Similarly, the mass matrix M, the stiffness matrix K and the damping matrix C of the finite element model of the simply supported beam can be similarly partitioned according to the measured and unmeasured degrees of freedom:

Then, the node random motion vector R(t, θ) has the following relationship with the measured degree of freedom random motion vector R _m (t, θ):

R(t,θ)=WR _m (t,θ) (7)

其中：I表示单位矩阵。 Where W represents the transformation matrix, with W=W _s +W _i ,

Where: I represents the identity matrix.

The mass matrix M _r after the polycondensation, the stiffness matrix K _r and the damping matrix C _r can be expressed as:

M _r = W ^T MW, K _r = W ^T KW, C _r = W ^T CW, where the superscript T represents matrix transposition.

S3. Using KL expansion to solve the distributed random dynamic load by the random dynamic response at the unit node, including the following steps:

S31: Solving the covariance matrix Γ _{R of the} random dynamic response by using the sample set of the measured random dynamic displacement R _m (t, θ);

S32: Perform eigenvalue decomposition on the covariance matrix Γ _R , obtain a random dynamic displacement KL vector y ^(j) (t) at each unit node on the finite element model after the polycondensation, and complete the KL expansion of the random dynamic response, as follows:

求得，其中λ _j为Γ _R的第j个特征值。则各单元节点处的随机速度和随机加速度向量可以分别表示为： Where ξ _j (θ) is a random variable and θ represents a random dimension. When j=0, ξ ₀ (θ)=1, N _KL is the number of KL components retained after the KL expansion is truncated.

It is found that λ _j is the jth eigenvalue of Γ _R . Then the random velocity and random acceleration vector at each unit node can be expressed as:

和

求解各单元节点上的随机动载荷对应向量U ^(j)(t)，如下式： S33: random motion response KL vector y ^(j) (t) by each unit node,

with

Solve the stochastic dynamic load corresponding vector U ^(j) (t) on each unit node, as follows:

S34: Solving the distributed random dynamic load f(x, t, θ) by the random dynamic load corresponding vector U ^(j) (t) on the unit node, comprising the following steps:

S341: Establish a distributed random dynamic load model based on polynomial expansion;

Let the distributed random dynamic load f(x, t, θ) have the following form:

f(x,t,θ)=T(x)·P(t,θ) (9),

Where T(x) represents the spatial distribution function of the random dynamic load, and P(t, θ) represents the stochastic process. Projecting the load distribution function T(x) onto the orthogonal space developed by the Chebyshev orthogonal polynomial, ie T(x)=∑a _k T _k (x), substituting into equation (9), let D _k (t, θ)=a _k P(t,θ), then:

Where N _k represents a polynomial order, and in this embodiment, the polynomial order is equal to the number of points, that is, N _k =11. As can be seen from the above equation, the distributed random dynamic load can be identified as long as the deterministic coefficient a _k and the random component D _k (t, θ) can be identified.

Let the random component D _k (t, θ) of the dynamic load have the following expanded form,

Where z ^(j) _k (t) represents the jth deterministic component of the kth random component. Then the distributed random dynamic load f(x, t, θ) can be expressed as:

S342: Solving the deterministic coefficient of the distributed random dynamic load model;

The coefficient vector Z ^(j) (t) consisting of z ^(j) _k (t) and the stochastic dynamic load corresponding vector U ^(j) (t) on the element node have the following relationship:

Q ^k由Q ^k _e按照有限元原理组合而成，且

N ^e(x)为梁单元的形状函数。 Where Q _r is the transformation matrix and + sign represents the generalized inverse.

Q ^{k is} composed of Q ^k _e according to the finite element principle, and

N ^e (x) is the shape function of the beam element.

S343: reconstructing a distributed random dynamic load;

The random dynamic load corresponding vector U ^(j) (t) and equation (13) on the unit node can be used to calculate z ^(j) _k (t), and then the distributed random dynamic load can be reconstructed according to equation (12).

S4: Solving the time-varying statistical characteristics of the spatial distribution of random dynamic loads on the structure. The time-varying mean μ _f (x, t) and the variance Var _f (x, t) of the random dynamic load with the spatial distribution can be obtained by the following formulas:

Figure 3(a) shows the comparison of the mean value of the random dynamic load in the beam span with time to the true value. Figure 3(b) shows the variance of the random dynamic load in the beam span obtained by the identification. The comparison between the law of change with time and the true value is shown in Fig. 4. The comparison results between the spatial distribution and the true distribution of the random dynamic load on the beam at each time are obtained. It can be seen that the identification method in the present invention can accurately identify the distribution of the random dynamic load with space and the statistical characteristics with time according to the response sample at the limited measurement point, and is suitable for the case of non-stationary random dynamic load; Compared with the Monte Carlo method, when the number of measured response samples is large, there is a significant advantage in calculation efficiency. For example, in the embodiment, when the number of measured samples is equal to 5000, 12 units and 11 points are selected. In the case of the 11th-order polynomial, the calculation time based on the KL expansion recognition method is only 13% based on the Monte Carlo method, which greatly improves the recognition efficiency. In summary, the method proposed by the present invention has certain advancement.

Claims (4)

A finite element based distributed random dynamic load identification method, characterized in that the method comprises the following steps: S1. Using multiple repeated measurements to obtain a set of random vibration response samples at multiple points on the structure; S2. Establish a polycondensed structural finite element model matching the measured dynamic response degrees of freedom; S3. Using KL expansion to solve the distributed random dynamic load by the random dynamic response at the unit node; S4: Solving the time-varying statistical characteristics of the random dynamic load with spatial distribution on the structure. The finite element-based distributed random dynamic load identification method according to claim 1, wherein the specific method for establishing the condensed structural finite element model that matches the measured dynamic response degree of freedom in step S2 comprises: S21: Ensure that the dynamic response measurement point is a unit node, and use the known structural parameters and finite element method to establish a structural finite element model to obtain the mass matrix and stiffness matrix of the structure. The damping of the structure adopts Rayleigh damping, and the damping matrix is composed of mass matrix and stiffness. Matrix calculation; S22: Perform model polycondensation on the structural finite element model, retain only the degree of freedom information of the measured dynamic response, obtain the condensed structural finite element model matching the measured dynamic response degree of freedom, obtain the polycondensed mass matrix, stiffness matrix and damping matrix. . The finite element-based distributed random dynamic load identification method according to claim 1 or 2, wherein the specific method for solving the distributed random dynamic load by using the KL expansion in the step S3 to solve the distributed random dynamic load by the unit node comprises: S31: Solving a covariance matrix of the random dynamic response by using a sample set of the measured random dynamic response; S32: performing eigenvalue decomposition on the covariance matrix, obtaining a random dynamic response KL vector at each unit node on the finite element model after the polycondensation, and completing KL expansion of the random dynamic response; S33: Inverting a random dynamic load corresponding vector on each unit node by a random motion response KL vector of each unit node; S34: Solving the distributed random dynamic load by the stochastic dynamic load corresponding vector on the unit node. The finite element-based distributed random dynamic load identification method according to claim 3, wherein the specific method for solving the distributed random dynamic load by the random dynamic load corresponding vector on the unit node in the step S34 comprises the following steps; S341: Establish a distributed random dynamic load model based on polynomial expansion; S342: Solving the deterministic coefficient of the distributed random dynamic load model; S343: Reconstructing the distributed random dynamic load.

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