Kunming University of Science and Technology, Kunming, Yunnan 650051)) to more complex piping systems. The calculation is given in the paper, and the effectiveness and correctness of the proposed method are proved by comparison with the existing results and experiments.
Fund Project: Yunnan Basic Applied Research Fund Project (97E003Z); Key Water Conservancy Science and Technology Project of the Ministry of Water Resources (SZ9830). 1. The hydraulic transient induces pipe vibration, and the vibration induces a new hydraulic transient. This coupled dynamic process is called For the fluid structure interaction of the pipeline, FSI for short. The FSI phenomenon is the inherent mechanical characteristics of the pipeline, studying the FSI phenomenon in the pipeline to understand the dynamics of the system, stabilize the system operation, improve the reliability of the operation, and avoid excessive pressure in the pipeline , Stress, vibration and noise have important academic and practical value, and have important significance in many industrial fields such as aerospace, petrochemical industry, water conservancy and power, hydraulic transmission, urban water supply and drainage. Fluid-solid coupled vibration is an inherent mechanical phenomenon of weakly constrained flow pipelines. The mechanism of the coupled vibration can be generally summarized into three forms: friction coupling, Poisson coupling, and joint coupling. 1. The frequency domain analysis method of flow pipelines has been subject to Widely regarded, Laplace transform has become a commonly used mathematical tool in frequency domain analysis. Doebelin 2 established the transfer matrix under various pipeline topologies of classic water hammer, and made a more comprehensive analysis and research. Wilkinson 3 derived the transfer matrix that can consider the axial, transverse and torsional motion of the coupling of the function, but did not consider the Poisson coupling and the friction coupling. The model was verified by a 1m long water-filled L-shaped tube. NanyakkaraandPerreira4 derives the transfer matrix describing the motion of straight and bent pipes. The resulting model can consider unction coupling, but cannot consider Poisson coupling and friction coupling. Deong5 used the transfer matrix method (TMM) to study the one-dimensional fluctuation problem in the frequency domain of the pipeline.
The data obtained prove that even for the simplest system, the pressure spectrum calculated by the transfer matrix considering the FSI effect is much better than the result calculated by the classic water hammer 2., 21 transfer matrix.
The TMM method proposed by Svingen7 is a transfer matrix based on the finite element method (FEM). The model he built considers frequency-dependent damping. The test device is an L-shaped water-filled steel tube that is vertically placed 20m long and 80mm in internal diameter.
ZhangS-10 proposed the L-MOC method based on the Laplace transform, which successfully solved the frequency domain analysis of the single-tube FSI problem.
The characteristic of the transfer matrix method is that the system is divided into several pipe segments, the variables at each cross section are formed into a state problem, and the transfer equation describing the motion of the system is established through the transfer matrix to achieve system analysis. TMM has been successfully used in time domain analysis. Many scholars have tried to generalize the transfer matrix method to FSI frequency domain analysis11, but due to the particularity of the FSI problem, these generalized applications are either just an application assumption, can not give examples, or can not show various coupling mechanisms. 12 Although the transfer matrix of FSI in the frequency domain is given, no further calculations and conclusions have been obtained. In this paper, based on the L-MOC method, the transfer matrix method is used to analyze the frequency response characteristics of the variable diameter pipeline FSI system, and a solution example is given.
2 The transfer matrix of the variable-diameter pipeline In order to facilitate the explanation of the problem of the variable-diameter pipeline studied, the pipeline is simplified into a ladder-shaped system composed of N pipe sections as shown, and the node numbers between the pipe sections are 0, 1, n At the node, the mass can be gathered and the corresponding constraints given if necessary.
2.1 Field transfer matrix For any pipe section, the equation of motion can be uniformly expressed in the form of a moment matrix in the form of 8% 10: the coefficient matrix of the equation of the horizontal and lateral vibrations. For the specific meaning, see 2. Since equation (1) is established for all pipe sections, Therefore, for the i-th pipe segment, the solution of 2 can be used directly, that is, a vector of fixed coefficients. Establish local coordinates according to the pipe segment, that is, separately), then the variables between the left and right ends of the pipe segment are satisfied: where: the field transfer matrix is ​​defined as: then for any pipe segment: 2.2 point transfer matrix The point transfer matrix is ​​essentially two A continuity condition of the variable at the connection of the segment pipeline, which is consistent with the boundary condition. After the Laplace transform, the basic variables at both ends of the connection point are equal or satisfy a simple algebraic relationship. 8. The variables in the i-th pipe segment and the i-th pipe segment can be established by the node connection conditions: the general relationship is: If the pipeline moves laterally, it is the identity matrix. Obtained by formula (8): the point transfer matrix is ​​defined as: the relationship between the variables of the two pipe segments connected by the i-th node is: 2.3 transfer matrix method With the point transfer matrix and the field transfer matrix, the variable at the right end of the i-th pipe segment can be established The relationship with the left end variable of the i1th pipe section is as follows: thus the relationship between the right end variable of the ith pipe section and the i2th pipe section is: then the recursive relationship between the right end variable of the first pipe section and the right end variable of the => 1 pipe section variable is: Reusing the point transfer matrix, the right end point value of the 1st = 1st segment variable and the left end value of the 1st = segment variable satisfy: Then: All the undetermined constants and end values ​​of the middle segment are eliminated. Substitute the four equations satisfied by the 8 unknowns of the right end of the first pipe section and the left end of the = pipe section, and then know the same reason as in equation (4): substituting these two relations into (17), Obtain the 4 algebraic equations satisfied by the 8 undetermined constants of the first and the first pipe section: OR: use a similar derivation to obtain the following equation: the fluid hits the steel rod and finally obtains 8 equations about 8 unknowns (20), (21), (22), which is unified as: where: Then: From: can get the final solution of the first and last pipe segment: using these two solutions, and the reverse process of elimination by the transfer matrix method, You can find a solution for any pipe section.
3 The calculation example uses the single-tube standard of the University of Dandy.
Case 2: Double the pipeline of Dandy University, and the radius of the lengthened pipeline is 16mm. The corresponding curve of the calculated pressure frequency is as shown. It can be seen from the figure that the low-frequency part of the frequency is encrypted, and the first-order frequency is significantly reduced.
Table 1 Physical characteristics of liquid-filled steel pipe of Dandi University Liquid-filled pipe!
table!
The axial natural frequency (Hz) does not include the quality of the pipe end plug. The experimental value of the pipe end plug quality is calculated. 10 4 Conclusion
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