ANSYS APDL-based triple clamp pipeline system dynamics modeling method

Abstract

The invention relates to an ANSYS APDL-based triple clamp pipeline system dynamic modeling method, which comprises the following steps of: discretizing a pipeline system by adopting a beam unit to obtain a finite element model of the pipeline system; a space spring unit is adopted to carry out rigidity equivalent discretization processing on the triple clamp, and a finite element model of the triple clamp is established; the equivalent stiffness initial value of the triple clamp is obtained through a static stiffness test, the equivalent stiffness initial value is corrected through a modal test and a stiffness parameter correction method, and the equivalent stiffness of the triple clamp is obtained; and endowing a space spring unit corresponding to the finite element model of the triple clamp with the equivalent stiffness of the triple clamp, and introducing the finite element model of the triple clamp into the finite element model of the pipe system, so that the finite element model of the pipe system of the triple clamp is established. By adopting the method for modeling, the dynamic characteristics of the triple clamp can be accurately simulated.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of pipeline system, and relates to a triple clamp pipeline system dynamics modeling method based on ANSYS APDL, which is suitable for complex spatial pipeline finite element modeling and modal analysis containing triple clamps. BACKGROUND

[0002] In a small-diameter pipeline system of a liquid rocket engine, a triple clamp is often used to constrain multiple pipelines to increase the stiffness of the system. The triple clamp structure and coordinate system definition are as follows Figure 2The results show that the clamp modeling accuracy is an important factor affecting the accuracy of the pipeline system modeling and the accuracy of the dynamic characteristics analysis. Therefore, in the existing research on the dynamic modeling technology of the clamp pipeline system, the main research is on the dynamic modeling method of single clamp and double clamp. For single clamp, Optimization of hoop layouts for reducing vibration amplitude of pipeline system using the semi-analytical model and genetic algorithm (IEEE Access, 2020: 224394-224408), Optimization of pipeline system with multi-hoop supports for avoiding vibration, based on particle swarm algorithm (Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2021(9): 1524-1538) simulate the constraint of single clamp on the pipeline with one or more sets of linear stiffness springs and angular stiffness springs; Impedance analysis and clamp locations optimization of hydraulic pipeline system in aircraft: 2015 International Conference on Fluid Power and Mechatronics (Harbin, China, 2015) uses a fixed beam element at one end to simulate the mechanical properties of single clamp.For the double clamp, "Finite Element Calculation and Experimental Determination of Clamp Stiffness" (Journal of Aerospace Power, 1999 (2): 68-71) studied the stiffness of the double clamp. The finite element model was established using solid elements to calculate the stiffness in two directions. At the same time, the stiffness of the double clamp in two directions was measured through experiments. The results showed that the deviation between the calculated and measured stiffness in both directions exceeded 10%; "A Method for Modeling Single and Double Clamp Pipeline Systems" (CN110188512A) discretized the double clamp into two linear stiffness springs and two angular stiffness springs, and built a test device for the lateral stiffness and angular stiffness of the clamp and conducted experiments. The clamp stiffness in that direction was obtained by fitting the hysteresis loop measured in the experiment. For stiffness parameters in certain directions that cannot be directly measured through experiments, a genetic algorithm combined with a modal test was used to search for the optimal solution to determine its stiffness value.

[0003] However, the relevant method is not applicable to triple clamps and more complex multi-clamps, mainly due to the following problems:

[0004] (1) Single clamps are used to connect pipelines to foundation structures and do not involve mutual constraints and coupling between multiple pipes;

[0005] (2) For double clamps, solid units or spring units are currently mainly used to simulate the connection relationship between pipes. Solid units are difficult to accurately simulate the contact stiffness between the clamp and the pipe, resulting in low modeling accuracy. When using spring units to simulate the clamp effect, multiple single-degree-of-freedom spring units are usually used to simulate the translation or rotation between the pipes. If this method is used to simulate triple clamps (or multi-clamps), on the one hand, the use of single-degree-of-freedom springs makes it impossible to accurately simulate the degree-of-freedom coupling effect between the pipes. On the other hand, as the number of pipes increases, the number of single-degree-of-freedom springs increases rapidly, making it difficult to obtain the spring stiffness parameters between the pipes.

[0006] Therefore, for the piping system containing triple clamps that is widely present in liquid rocket engines, there is an urgent need for a piping system dynamics modeling method that can accurately simulate the dynamic characteristics of the triple clamps. Summary of the Invention

[0007] The technical problem solved by the present invention is to overcome the deficiencies of the prior art and propose a dynamic modeling method for a triple clamp pipeline system based on ANSYS APDL.

[0008] The solution of the present invention is:

[0009] A dynamic modeling method for a triple clamp piping system based on ANSYS APDL, including:

[0010] The pipe system is discretized using beam elements to obtain a finite element model of the pipe system, on which a triple clamp can be placed;

[0011] The spatial spring unit is used to perform equivalent discretization of the stiffness of the triple clamp and a finite element model of the triple clamp is established.

[0012] The initial value of the equivalent stiffness of the triple clamp is obtained through static stiffness test, and the initial value of the equivalent stiffness is corrected by modal test and stiffness parameter correction method to obtain the equivalent stiffness of the triple clamp.

[0013] The equivalent stiffness of the triple clamp is assigned to the spatial spring unit corresponding to the triple clamp finite element model, and the triple clamp finite element model is introduced into the piping system finite element model, thereby establishing a triple clamp piping system finite element model.

[0014] Preferably, performing stiffness equivalent discretization processing on the triple clamp includes:

[0015] The stiffness of the triple clamp is equivalently discretized into three spatial spring units. Each spatial spring unit has a node at both ends, and each node has 6 degrees of freedom. The three spatial spring units share a node at one end, and the three nodes at the other end are connected one-to-one with the three pipes at the center of the triple clamp hole. The spatial spring unit is used to simulate the stiffness relationship between the triple clamp and the pipeline. The stiffness parameter k of the spatial spring unit is defined by the following formula:

[0016] k=[k u k v k w k θx k θy k θz ]

[0017] Among them, k u 、k v 、k w Represents the stiffness of the translational freedom in the x, y, and z coordinate axes respectively; k θx 、k θy 、k θz They represent the stiffness of the rotational freedom around the x, y, and z coordinate axes respectively.

[0018] Preferably, the unit stiffness matrix K of the spatial spring unit is e as follows:

[0019]

[0020] l is the distance between the nodes at both ends of the spatial spring.

[0021] Preferably, the initial equivalent stiffness value of the triple clamp is obtained through a static stiffness test, and the method is as follows:

[0022] The translation stiffness test system is used to measure the initial values ​​of the translation stiffness of the first pipe outside the triple clamp in the x, y, and z coordinate directions. The rotational stiffness test system is used to measure the initial value of the rotational stiffness of the first pipe outside the triple clamp around the three coordinate axes of x, y, and z. Thus, the initial value of the spatial spring unit stiffness corresponding to the first pipe outside the triple clamp is obtained

[0023] The initial value of the spatial spring unit stiffness corresponding to the second pipe in the middle of the triple clamp is obtained by measuring the translation stiffness test system and the rotation stiffness test system respectively.

[0024] The triple clamp has a symmetrical structure, so the initial value of the space spring unit stiffness corresponding to the third pipe of the triple clamp is equal to the initial value of the space spring unit stiffness corresponding to the first pipe.

[0025] Preferably, the translational stiffness testing system includes a mounting plate, a stepper motor, a lead screw, a slide rail, a slider, a force sensor, a digital micrometer, a loading tool and a fixed support tool; the slide rail and the stepper motor are fixedly mounted on the mounting plate, the output end of the stepper motor is connected to the lead screw, the slider is mounted on the slide rail, the slider is connected to the lead screw through a threaded pair, the upper part of the slider is connected to the loading tool through a force sensor, the loading tool is connected to the measured pipeline for applying load, the triple clamp is mounted on the fixed support tool by bolts, the fixed support tool is fixedly mounted on the mounting plate, and two digital micrometers respectively measure the translational displacement at both ends of the loading tool.

[0026] Preferably, when the translation stiffness test system is used to measure the initial value of the translation stiffness of the clamp, the stepper motor drives the lead screw to rotate, causing the slider to translate along the slide rail and push the loading fixture through the force sensor, thereby applying translation tensile loads in the x, y, and z directions to both ends of the pipeline, respectively, measuring the translation displacement of the outer tube through a digital dial indicator, and obtaining the magnitude of the applied tensile load through the force sensor;

[0027] Considering the clamp constraint position as the fixed support boundary, the bending deformation of the pipeline itself is calculated by the following formula:

[0028]

[0029] Where: p is the calculated value of the pipeline's own bending deformation, F is the force sensor measurement value, L is the distance between the clamp constraint position and the load application position, E is the elastic modulus of the material, and I is the moment of inertia of the pipeline section.

[0030] The deformation of the clamp is calculated by the following formula:

[0031] δc = δ t - δ p

[0032] wherein, δ c is the deformation of the clamp, δ t is the average value of the deformation results measured by two digital micrometers;

[0033] By changing the force load amplitude, multiple loadings are carried out, multiple groups of data points of F and the deformation δ c of the clamp are obtained, the data points are linearly fitted, and the initial value of the translational stiffness of the clamp in the corresponding direction is calculated by the slope of the curve.

[0034] Preferably, the rotational stiffness test system comprises a mounting plate, a stepping motor, a supporting bearing and a support, a rotating shaft, a torque sensor, an inclination sensor, a loading tool and a fixed support tool; the stepping motor, the supporting bearing and the support, and the fixed support tool are fixedly installed on the mounting plate, the output end of the stepping motor is connected with the rotating shaft, the rotating shaft is supported by the supporting bearing and the support, the end of the rotating shaft is connected with the torque sensor, the torque sensor is connected with the loading tool, the inclination sensor is installed on the loading tool, the loading tool clamps the pipeline, the three-joint clamp is installed on the fixed support tool by bolts, and the inclination sensor measures the rotation angle of the pipeline.

[0035] Preferably, when the rotational stiffness of the clamp is measured by the rotational stiffness measurement system, the stepping motor drives the rotating shaft to rotate, the rotating shaft rotates the loading tool through the torque sensor, so as to realize the application of the rotational torque load around the x, y and z directions to the pipeline respectively, and the rotation angle deformation of the clamp is measured by the inclination sensor.

[0036] The constraint position of the clamp is regarded as the fixed boundary, and the rotation angle deformation of the pipeline itself is calculated by the following formula:

[0037]

[0038] wherein: θ p is the calculation value of the rotation angle deformation of the pipeline, M is the measurement value of the torque sensor, L is the distance between the constraint position of the clamp and the position of the torque load, E is the elastic modulus of the material, and I is the moment of inertia of the cross section of the pipeline.

[0039] The rotation angle deformation of the clamp is obtained by the following formula:

[0040] θ c = θ t - θ p

[0041] wherein, θ c is the rotation angle deformation of the clamp, and θ t is the measurement value of the inclination sensor.

[0042] Through multiple loadings by changing the torque load amplitude, multiple groups of torque and clamp rotation deformation θ c Data points are linearly fitted, and the curve slope is calculated to obtain the initial value of the clamp rotation stiffness in the corresponding direction.

[0043] Preferably, the equivalent stiffness initial value is corrected by using a modal test and a stiffness parameter correction method, and the method is as follows:

[0044] A three-clamp three-pipe modal test pipe system is constructed, and modal tests are carried out, and for the outer pipe and the intermediate pipe, the characteristic modal frequency test values of the corresponding characteristic vibration modes reflecting six stiffness parameters of the spatial spring unit are obtained;

[0045] A finite element model of the three-clamp three-pipe modal test pipe system is established, wherein the pipe part is modeled by using a beam element, and the clamp is equivalently modeled by using three spatial spring elements, the equivalent stiffness initial value of the three-clamp is respectively assigned to the three spatial spring elements, the modal calculation of the three-clamp three-pipe modal test pipe system is carried out, and the characteristic modal frequency simulation values corresponding to the characteristic vibration modes are obtained;

[0046] The characteristic modal frequency test values obtained in the modal test are analyzed in relation to the six stiffness parameters, and the main influence stiffness parameters of each characteristic modal frequency are identified;

[0047] Based on the finite element model of the three-clamp three-pipe modal test pipe system, the main influence stiffness parameters in the spatial spring element corresponding to the characteristic modal frequency are corrected in a single parameter, so that the relative deviation between the corrected characteristic modal frequency simulation value and the characteristic modal frequency test value is less than 1%.

[0048] The six stiffness parameters of the spatial spring element corresponding to the outer pipe and the intermediate pipe are corrected one by one, and the corrected spatial spring element stiffness parameters, i.e., the equivalent stiffness of the three-clamp, are obtained.

[0049] Preferably, the three-clamp three-pipe modal test pipe system is modal calculated by using the following undamped characteristic equation:

[0050] (K-ΛM)U=0

[0051] Wherein, K is the stiffness matrix of the three-clamp pipe system, M is the mass matrix of the three-clamp pipe system, Λ is the eigenvalue matrix of the three-clamp pipe system, and U is the vibration mode matrix of the three-clamp pipe system.

[0052] The beneficial effects of the present application compared with the prior art are:

[0053] The application provides a triple clamp pipeline system dynamics modeling method, which can simulate the coupling effect between the clamp and the pipeline, and on the basis of obtaining the triple clamp stiffness, the triple clamp pipeline system dynamics model established can more accurately predict the modal frequency characteristics of the pipeline system, and the degree of freedom of the finite element model is far lower than that of the physical model, which greatly improves the calculation efficiency. BRIEF DESCRIPTION OF DRAWINGS

[0054] Figure 1 Triple clamp pipeline system dynamics modeling flow;

[0055] Figure 2 Triple clamp structure and coordinate system definition;

[0056] Figure 3 Triple clamp equivalent discrete model based on generalized spatial spring element;

[0057] Figure 4 Spatial spring element and its local coordinate system schematic diagram;

[0058] Figure 5 Translation stiffness test system;

[0059] Figure 6 Rotation stiffness test system;

[0060] Figure 7 Three-pipe test pipeline system and its characteristic mode schematic diagram;

[0061] Figure 8 Characteristic modal frequency and stiffness parameter component correlation curve;

[0062] Figure 9 Equivalent discrete finite element model of complex spatial pipeline system containing triple clamp schematic diagram;

[0063] Figure 10 Complex spatial pipeline system containing triple clamp entity finite element model schematic diagram;

[0064] Figure 11 Complex spatial pipeline system containing triple clamp modal frequency simulation and test deviation. DETAILED DESCRIPTION

[0065] The application will be further described below in combination with the embodiments and the drawings.

[0066] Embodiment Figure 1 A triple clamp pipeline system dynamics modeling method provided by an embodiment of the application has a flowchart, and the method comprises the following steps:

[0067] As Figure 1As shown, in step 1, a complex spatial piping system consisting of four pipes is discretized using beam elements to obtain a finite element model of the complex spatial piping system consisting of four pipes. There are two pipe sections on the four pipes that can be used to place triple clamps.

[0068] Step 2: Use the space spring unit to perform equivalent discretization of the stiffness of the triple clamp and establish a finite element model of the triple clamp, such as Figure 3 As shown;

[0069] Step 3: Using a combined stiffness test combining a static stiffness test with a modal test and stiffness parameter correction to obtain the equivalent stiffness of the triple clamp;

[0070] Step 4: By assigning the corrected equivalent stiffness parameters to the spatial spring unit corresponding to the triple clamp finite element model and introducing the triple clamp finite element model into the four-pipe system finite element model, a triple clamp piping system finite element model is established.

[0071] The following Figure 1 The specific implementation of each step of the embodiment shown is described in detail:

[0072] In step 1, the geometric model of the four pipes is established in ANSYS APDL by connecting key points. Based on the established geometric model, the four pipes are meshed with a grid size of 1 mm, and the finite element model of the four pipes is further established using the Beam188 beam element.

[0073] The geometric model of the four pipelines is that the first pipe is composed of 8 straight pipe sections connected by 7 elbows, the second pipe is composed of 7 straight pipe sections connected by 6 elbows, the third pipe is composed of 8 straight pipe sections connected by 7 elbows, and the fourth pipe is composed of 7 straight pipe sections connected by 6 elbows.

[0074] There are two sections on the four pipes that can be used to place triple clamps. Each section consists of straight sections of three adjacent pipes, with the three adjacent straight sections located in the same plane. The axial distance between the two outer straight sections and the middle straight section is equal to 20 mm. The length of the pipe section for placing the clamps is not less than 50 mm.

[0075] The pipeline geometric model is established by connecting key points. In ANSYS APDL, the K command is used to establish key points through point coordinates. For the straight section of the pipeline, the L command is used to connect the key points in a straight line. For the curved section of the pipeline, the LFILLT command is used and the bending radius of the elbow is input to make an arc-shaped elbow connection between the two straight sections.

[0076] The four pipes are made of stainless steel with an elastic modulus of 200 GPa and a density of 7850 kg / m 3Poisson's ratio is 0.3. In ANSYS APDL, material properties can be defined by MP command.

[0077] In ANSYS APDL, the grid size is defined by LESIZE and LMESH commands and the grid is divided for the pipe geometry model.

[0078] The cross-section parameters of the four pipes are the same, the pipe outer diameter is 6 mm, the pipe wall thickness is 1 mm, and the bend radius of the bend pipe part is 23 mm. In ANSYS APDL, the beam element interface parameters are defined by SECTYPE and SECDATA commands.

[0079] In ANSYS APDL, the element type is defined as BEAM188 by ET command. The BEAM188 beam element used is based on Timoshenko beam theory. The Beam188 beam element is a two-node three-dimensional element, and each node contains six degrees of freedom, i.e. translational degrees of freedom along x, y, z three orthogonal directions and rotational degrees of freedom around x, y, z three directions.

[0080] In step 2, the three-way clamp is discretely processed by spatial spring element for stiffness equivalent.

[0081] Figure 2 The structure diagram of the three-way clamp (including pipes) is shown in Figure 3 The equivalent discretization model diagram of the three-way clamp is shown in Figure 4 The spatial spring element and its local coordinate system are shown in

[0082] In ANSYS APDL, the three-way clamp is discretely processed by three spatial spring elements COMBI250 for equivalent, so as to simulate the stiffness relationship between the clamp and the connected three pipes.

[0083] As shown in Figure 2 , the center of the clamp is taken as the origin of the local coordinate system, the length direction of the clamp is taken as the X axis, the hole axis direction of the clamp is taken as the Y axis, and the Z axis is determined according to the right-hand rule.

[0084] The COMBI250 spatial spring element is a two-node three-dimensional element, and each node contains six degrees of freedom, i.e. translational degrees of freedom along x, y, z three orthogonal directions and rotational degrees of freedom around x, y, z three directions. The stiffness characteristics of COMBI250 element are defined by the following formula:

[0085] k = [k u k v k w k θx k θy k θz ] (1)

[0086] Among them, k u 、k v 、k w Represents the translational degree of freedom stiffness in the x, y, and z coordinate axes respectively; k θx 、k θy 、k θz Represents the stiffness of the rotational freedom around the x, y, and z axes. The stiffness of the COMBI250 element is described based on the local coordinate system. The local coordinate system is defined in the same way as Figure 2 The local coordinate system for the three-way clamp is the same: the clamp centroid is used as the origin, the length of the clamp is defined as the X-axis, the axial direction of the clamp hole is defined as the Y-axis, and the Z-axis is determined by the right-hand rule. The mass properties of the COMBI250 element are simulated using a mass point located at the element centroid.

[0087] The element stiffness matrix K of the spatial spring element e It can be written as formula (2).

[0088]

[0089] According to the unit stiffness matrix of the space spring unit, there is coupling between the translation along the y-axis and the rotation around the z-axis; there is coupling between the translation along the z-axis and the rotation around the y-axis. Figure 2 For example, when the pipe outside the center clamp is subjected to a force along the Y axis, its translational motion along the Z axis and rotational motion around the Y axis are simultaneously constrained by the multi-joint clamp, creating a coupling between these two degrees of freedom. Compared to a spring group consisting of multi-directional, single-degree-of-freedom, one-dimensional springs, the generalized spatial spring simulates the multi-joint clamp more closely. l is the distance between the nodes at the two ends of the spatial spring.

[0090] In ANSYS APDL, the ET command is used to select the COMBI250 element. When simulating the clamp stiffness characteristics, the RMORE command is used to define the stiffness parameters of the COMBI250 element shown in Equation (1). When simulating the clamp mass characteristics, the R command is used to define the mass parameters of the COMBI250 element. To simulate the spatial distribution of the clamp mass, the mass of each COMBI250 element is defined as 1 / 3 of the total clamp mass.

[0091] In one embodiment of the present invention, the triple clamp is made of 5A06 aluminum alloy and has a mass of 28 g.

[0092] In step 3, a combined stiffness test combining a static stiffness test with a modal test and stiffness parameter correction are used to obtain the equivalent stiffness of the triple clamp.

[0093] Figure 5 The figure shows the translational stiffness test system.Figure 6 The rotation stiffness test system is shown.

[0094] The translation stiffness test system comprises a mounting plate, a stepper motor, a screw rod, a slide rail, a slide block, a force sensor, a digital dial gauge, a loading tool and a fixed support tool; the slide rail and the stepper motor are fixedly installed on the mounting plate, the output end of the stepper motor is connected with the screw rod, the slide block is installed on the slide rail, the slide block is connected with the screw rod through a threaded pair, the upper part of the slide block is connected with the loading tool through the force sensor, the loading tool is connected with the measured pipeline for applying load, the three-connector clamp is installed on the fixed support tool through bolts, the fixed support tool is fixedly installed on the mounting plate, and two digital dial gauges measure the translation displacement of the two ends of the loading tool, respectively.

[0095] When the translation stiffness of the clamp is measured by the translation stiffness test system, the stepper motor drives the screw rod to rotate, so that the slide block translates along the slide rail and pushes the loading tool through the force sensor, so that the translation tensile load in the x, y and z directions is applied to the two ends of the pipeline, respectively, the translation displacement of the outer pipeline is measured by the digital dial gauge, and the size of the applied tensile load is obtained by the force sensor; the translation stiffness test of the clamp aims to obtain the deformation of the clamp under the load, but the actual measurement result simultaneously contains the deformation of the clamp and the deformation of the pipeline. In order to eliminate the deformation of the pipeline, the deformation measured in the test is subtracted from the bending deformation of the pipeline itself calculated by the analytical method, and the difference is the deformation of the clamp. The slope of the fitting curve is calculated by linear fitting the load-deformation result, which is the initial value of the stiffness in the direction.

[0096] When the pipeline is calculated as a cantilever beam by the analytical method, the clamp constraint position is regarded as the fixed boundary, and the bending deformation of the pipeline itself is calculated by the following formula:

[0097]

[0098] Wherein: δ p is the calculation value of the bending deformation of the pipeline itself, F is the measurement value of the force sensor, L is the distance between the clamp constraint position and the load application position, E is the elastic modulus of the material, and I is the moment of inertia of the cross section of the pipeline.

[0099] The deformation of the clamp is obtained by the following formula:

[0100] δ c = δ t - δ p

[0101] Wherein: δ c is the deformation of the clamp, and δ t is the average value of the deformation measured by the two digital dial gauges.

[0102] A plurality of loads F and clamp deformations δc After performing linear fitting on the data points, the slope of the curve is calculated to obtain the initial value of the clamp translational stiffness in the corresponding direction.

[0103] Figure 6 The rotational stiffness test system shown includes a mounting plate, a stepper motor, support bearings and supports, a rotating shaft, a torque sensor, an inclination sensor, a loading fixture, and a fixing fixture. The stepper motor, support bearings and supports, and the fixing fixture are all fixedly mounted on the mounting plate. The output end of the stepper motor is connected to the rotating shaft, which is supported by the support bearings and supports. The end of the rotating shaft is connected to the torque sensor, which is connected to the loading fixture. The inclination sensor is mounted on the loading fixture, which clamps the pipeline. The triple clamp is mounted on the fixing fixture via bolts. The inclination sensor measures the rotation angle of the pipeline.

[0104] When using a rotational stiffness measurement system to measure the initial value of a clamp's rotational stiffness, a stepper motor drives the shaft, which, via a torque sensor, rotates the loading fixture, thereby applying rotational torque loads to the pipeline in the x, y, and z directions. The clamp's angular deformation is then measured using an inclination sensor. Similar to the clamp's static and dynamic stiffness test, the clamp's rotational stiffness test aims to determine the clamp's own angular deformation under torsional load. However, the actual measurement results include both the clamp's angular deformation and the pipeline's angular deformation. To eliminate the pipeline's angular deformation, the measured angular deformation is subtracted from the analytically calculated angular deformation of the pipeline as a cantilever beam. The difference is the clamp's angular deformation. A linear fit is performed on the load-deformation results, and the slope of the fitted curve is calculated as the initial value of the stiffness in that direction.

[0105] When the pipeline is treated as a cantilever beam and the analytical method is used to calculate the pipeline's own angular deformation, the clamp constraint position is regarded as the fixed support boundary. The pipeline's own angular deformation is calculated using the following formula:

[0106]

[0107] Where: θ p is the calculated value of the pipeline's own angular deformation, M is the torque sensor measurement value, L is the distance between the clamp constraint position and the torque load application position; E is the elastic modulus of the material; I is the moment of inertia of the pipeline section.

[0108] The angular deformation of the clamp is calculated by the following formula:

[0109] θ c =θ t -θ p

[0110] Among them, θ c is the angular deformation of the clamp, θ t The measured value of the tilt sensor.

[0111] Through multiple loading by changing the torque load amplitude, multiple sets of torque M and clamp corner deformation θ data points are obtained, and the initial value of the clamp rotation stiffness in the corresponding direction is calculated by linear fitting of the data points and calculating the slope of the curve. c

[0112] In step 3, first, the stiffness test system shown in Figure 5 and Figure 6 is used to obtain the initial value of the stiffness parameter of the space spring unit corresponding to the triple clamp, and the specific process is as follows:

[0113] (1) The translational stiffness test system is used to measure the initial values of the translational stiffness k u , k v , k w of the first pipe outside the triple clamp in the x, y, z three coordinate axis directions, and the rotational stiffness test system is used to measure the initial values of the rotational stiffness k θx , k θy , k θz of the first pipe outside the triple clamp around the x, y, z three coordinate axis directions. Thus, the initial value of the space spring unit stiffness corresponding to the first pipe outside the triple clamp is obtained.

[0114] (2) The translational stiffness test system and the rotational stiffness test system are used to measure the initial value of the spring unit stiffness

[0115] (3) The triple clamp is a symmetrical structure, so the initial value of the space spring unit stiffness corresponding to the third pipe of the triple clamp is equal to the initial value of the space spring unit stiffness corresponding to the first pipe, which is

[0116] The space spring unit stiffness parameters for the outer hole site (corresponding to the first pipe) and the middle hole site (corresponding to the second pipe) measured by the translational stiffness test system and the rotational stiffness test system are shown in Table 1.

[0117] Table 1 Space spring unit stiffness parameters obtained based on the stiffness test system

[0118]

[0119] The equivalent stiffness initial value is corrected by using the modal test and stiffness parameter correction method, and the method is as follows:

[0120] A triple clamp three-pipe modal test pipe system is constructed and a modal test is carried out, and for the outer pipe and the middle pipe, the characteristic modal frequency test values of the corresponding characteristic modes reflecting the six stiffness parameters of the space spring unit are obtained; the triple clamp three-pipe modal test pipe system usually consists of a triple clamp, a straight pipe and an L-shaped elbow pipe.​

[0121] A finite element model of the piping system for the triple clamp and three-tube modal test was established. The pipeline part was modeled using beam elements, and the clamp was equivalently modeled using three spatial spring elements. The initial values ​​of the equivalent stiffness of the triple clamp were assigned to the three spatial spring elements respectively. Modal calculation was performed on the piping system for the triple clamp and three-tube modal test to obtain the simulation values ​​of the characteristic modal frequencies corresponding to the characteristic vibration shapes.

[0122] The correlation analysis of the six stiffness parameters was performed on the experimental values ​​of the characteristic modal frequencies obtained in the modal test, and the main influencing stiffness parameters of each characteristic modal frequency were identified.

[0123] Based on the finite element model of the piping system with triple clamps and three pipes in the modal test, a single parameter correction is performed on the main influencing stiffness parameters in the spatial spring unit corresponding to the identified eigenmodal frequency, so that the relative deviation between the corrected eigenmodal frequency simulation value and the eigenmodal frequency test value is less than 1%.

[0124] The six stiffness parameters of the space spring unit corresponding to the outer pipeline and the middle pipeline are corrected one by one to obtain the corrected stiffness parameters of the space spring unit.

[0125] In this embodiment, after obtaining the initial values ​​of the stiffness parameters of the three spatial spring units On this basis, by constructing five groups of three-tube test pipe systems and conducting modal tests, the characteristic modal frequency test values ​​of the corresponding characteristic vibration modes that can reflect the stiffness characteristics of the six degrees of freedom directions are obtained for the outer holes and the middle holes of the triple clamp respectively. The specific process is as follows:

[0126] Structure as Figure 7 The three-tube test systems (a) through (e) shown here comprise five three-tube test systems, each undergoing six modal tests. Two modal tests were conducted on the three-tube test system (a), obtaining the eigenmodal frequency test values ​​corresponding to eigenmodes (I) and (II), respectively. One modal test was conducted on each of the three-tube test systems (b) through (e), obtaining the eigenmodal frequency test values ​​corresponding to eigenmodes (III) through (VI), respectively.

[0127] The three-tube test system (a) consists of a long straight tube, two short straight tubes, and a triple clamp. The three straight tubes are located in the XY plane. The characteristic vibration mode (I) is the vibration mode of the outer long straight tube rotating about the X axis, which mainly reflects the stiffness parameter k of the interaction between the triple clamp and the outer pipe. 1,θx The characteristic vibration mode (II) is the vibration mode of the outer pipeline rotating around the Z axis, which mainly reflects the stiffness parameter k of the interaction between the triple clamp and the first outer pipeline. 1,θz Because the triple clamp is symmetrical, k1,θx =k 3θx, , k 1,θz =k 3,θz , where the subscripts 1, 2, and 3 represent the pipeline numbers connected by the triple clamp, respectively. By placing the long straight pipe in the middle hole of the triple clamp, the stiffness parameter k reflecting the interaction between the triple clamp and the middle pipeline can be obtained similarly. 2,θx 、k 2,θz The corresponding characteristic vibration mode and the corresponding characteristic modal frequency test value can be obtained accordingly.

[0128] The three-tube test system (b) consists of a straight tube and an L-shaped tube on the outside, a straight tube in the middle, and a triple clamp. All three tubes are located in the XY plane. The characteristic vibration mode (III) is the vibration mode of the outer L-shaped tube rotating about the Y axis, which mainly reflects the stiffness parameter k of the interaction between the triple clamp and the outer pipe. 1,θy (=k 3,θy ). By placing the L-shaped tube in the middle hole of the triple clamp, similarly, the stiffness parameter k reflecting the interaction between the triple clamp and the middle pipe can be obtained. 2,θy The corresponding characteristic vibration mode and the corresponding characteristic modal frequency test value can be obtained accordingly.

[0129] The three-tube test system (c) consists of two outer L-shaped tubes, a straight tube in the middle, and a triple clamp. All three tubes are located in the XY plane. When both L-shaped tubes are located in the outer holes, the characteristic vibration mode (IV) is manifested as the vibration mode of the triple clamp moving along the Y axis, which mainly reflects the stiffness parameter k of the interaction between the triple clamp and the outer pipe. 1,u (=k 3,u ) If the two L-shaped tubes are located at the outer hole and the middle hole respectively, similarly, the stiffness parameter k that reflects the interaction between the triple clamp and the middle pipe can be obtained 2,u The corresponding characteristic vibration mode and the corresponding characteristic modal frequency test value can be obtained accordingly.

[0130] The three-tube test system (d) consists of two long straight tubes, one short straight tube, and a triple clamp. All three tubes are located in the XY plane. When the two long straight tubes are located in the outer holes, the characteristic vibration mode (V) is manifested as the vibration mode of the triple clamp rotating around the Y axis, which mainly reflects the stiffness parameter k of the interaction between the triple clamp and the outer pipe. 1,w (=k 3,w If two long straight pipes are located at the outer hole and the middle hole respectively, similarly, the stiffness parameter k reflecting the interaction between the triple clamp and the middle pipe can be obtained. 2,w The corresponding characteristic vibration mode and the corresponding characteristic modal frequency test value can be obtained accordingly.

[0131] The three-pipe test pipe system (e) is composed of two L-shaped pipes, one short straight pipe and a three-pipe clamp, and the three pipes are located in the YZ plane. When the two L-shaped pipes are located at the outer hole positions, the characteristic vibration mode (VI) is a vibration mode in which the three-pipe clamp moves along the Y axis, mainly reflecting the effect of the stiffness parameter k 1,v (=k 3,v ) of the interaction between the three-pipe clamp and the outer pipe. If the two L-shaped pipes are respectively arranged at the outer hole positions and the middle hole positions, similarly, the stiffness parameter k 2,v corresponding to the interaction between the three-pipe clamp and the middle pipe can be obtained. Correspondingly, the corresponding characteristic modal frequency test value can be obtained.

[0132] Then, by establishing a finite element model of the three-pipe test pipe system, modal calculation is performed on the three-pipe test pipe system to obtain the characteristic modal frequency simulation value corresponding to the characteristic vibration mode. The specific process is as follows:

[0133] According to the three-pipe test pipe system structure shown in Figure 7 , a corresponding finite element model of the three-pipe test pipe system is established. Among them, the BEAM188 beam element is used to establish the finite element model of the pipe part, and the three spatial spring elements COMBI250 are used to establish the equivalent finite element model of the three-pipe clamp. The obtained initial value of the stiffness parameter k is respectively given to the corresponding spatial spring element, and on this basis, the finite element model of the three-pipe clamp is introduced into the finite element model of the pipe, thereby establishing the finite element model of the three-pipe test pipe system. Based on the finite element model of the three-pipe test pipe system, the corresponding boundary conditions are applied by using the D command, the solution type is set to modal analysis by using the ANTYPE command, the Block Lanczos method is selected for modal solution by using the MODOPT command, the ANSYS APDL post-processing module is entered by using the / POST1 command, and the vibration mode is selected and drawn by using the SET command and the PLNSOL command, thereby obtaining the characteristic modal frequency simulation value corresponding to the characteristic vibration mode.

[0134] When the finite element model of the three-pipe clamp is introduced into the finite element model of the pipe, in order to simulate the influence of the width of the clamp structure, the CERIG command is used, the nodes of the pipe near the center of the contact area between the pipe and the clamp are taken as the main nodes, and the nodes of the pipe in the contact area are rigidified to the main nodes. In the finite element model of the three-pipe test pipe system, three main nodes are formed respectively by the contact areas corresponding to the three pipes.

[0135] When the triple clamp finite element model is introduced into the pipeline finite element model, one end of the three COMBI250 spatial spring units in the triple clamp equivalent finite element model shares a node. The node is established using the N command, and the coordinates are located at the centroid of the triple clamp. The other end nodes of the three COMBI250 units are actually the three main nodes.

[0136] The width of the triple clamp structure is 12mm.

[0137] Finally, based on the obtained eigenmodes and eigenmodal frequency simulation values ​​of the three-tube test pipe system finite element model corresponding to the outer or middle hole positions, a correlation analysis of the eigenfrequency simulation values ​​with the spatial spring unit stiffness parameters is carried out to identify the main influencing stiffness parameters corresponding to the eigenmodal frequencies and eigenmodes. On this basis, a single parameter correction is carried out for the main influencing stiffness parameters, so that the relative deviation between the corrected eigenmodal frequency simulation value and the eigenmodal frequency test value is less than 1%, thereby obtaining the corrected spatial spring unit stiffness. The specific process is as follows:

[0138] (1) Let’s take the eigenmode and eigenmodal frequency simulation values ​​of the three-tube test pipe system finite element model corresponding to the outer hole position as an example. At this time, the stiffness of the three spatial spring units is still the initial value. For the characteristic vibration mode (I), analyze the characteristic modal frequency simulation value with respect to the stiffness parameter component The stiffness parameter component that has the greatest impact on the eigenmode frequency simulation value is selected as the main influencing stiffness parameter component.

[0139] (2) Let’s assume is the main influencing stiffness parameter component, and the other five stiffness parameter components remain unchanged. Single parameter correction was carried out. After N1 rounds of correction, the relative deviation between the simulated value of the characteristic modal frequency of the characteristic vibration mode (I) and the experimental value of the characteristic modal frequency was less than 1%. The stiffness parameter k 1,θx The corrected value is at this time,

[0140] (3) Based on the characteristic vibration mode (II), the characteristic modal frequency simulation value is analyzed with respect to the remaining 5 stiffness parameter components. The stiffness parameter component that has the greatest impact on the eigenmode frequency simulation value is selected as the main influencing stiffness parameter component.

[0141] (4) Let’s assume The main influencing stiffness parameter component is the stiffness parameter component, and the other four stiffness parameter components remain unchanged. After N2 rounds of correction, the relative deviation of the characteristic modal frequency simulation value of the characteristic mode (II) from the characteristic modal frequency test value is less than 1%, and the stiffness parameter k 1,θz The corrected value is At this time,

[0142] (5) Similarly, the stiffness parameter component correlation analysis and stiffness parameter correction are sequentially performed for the remaining characteristic modes (III) to (VI), so as to obtain the corrected stiffness of the outer space spring unit

[0143] (6) Repeating (1) to (5), the corrected stiffness of the middle space spring unit is obtained

[0144] In an embodiment of the present application, taking the outer hole site as an example, the correlation analysis result of the characteristic frequency simulation value with respect to the stiffness parameter of the space spring unit is as follows Figure 8 For the characteristic mode (I) and the corresponding characteristic modal frequency, k 1,θx (=k 3,θx ) is identified as the main influence stiffness parameter; for the characteristic mode (II) and the corresponding characteristic modal frequency, k 1,θz (=k 3,θz ) is identified as the main influence stiffness parameter; for the characteristic mode (III) and the corresponding characteristic modal frequency, k 1,θy (=k 3,θy ) is identified as the main influence stiffness parameter; for the characteristic mode (IV) and the corresponding characteristic modal frequency, k 1,u (=k 3,u ) is identified as the main influence stiffness parameter; for the characteristic mode (V) and the corresponding characteristic modal frequency, k 1,w (=k 3,w ) is identified as the main influence stiffness parameter; for the characteristic mode (VI) and the corresponding characteristic modal frequency, only k 1,v (=k 3,v ) is the main influence stiffness parameter

[0145] After correction, the spring unit stiffness parameters for the outer hole site (corresponding to the first pipeline) and for the middle hole site (corresponding to the second pipeline) are shown in Table 2.

[0146] Table 2 Corrected spring unit stiffness parameters

[0147]

[0148] In step 4, the corrected stiffness parameters k 1,mod , k 2,mod , k3,mod The three spatial spring units corresponding to the equivalent finite element model of the triple clamp are assigned, and the triple clamp finite element model is introduced into the complex spatial piping finite element model, thereby establishing a complex spatial piping finite element model containing the triple clamp.

[0149] Example

[0150] Establish an equivalent discrete finite element model of a complex spatial piping system containing triple clamps, such as Figure 9 As shown in the figure, the modal frequency simulation value of the equivalent discrete model was obtained through modal simulation; as a comparison, a complex spatial piping solid finite element model containing a triple clamp was established, as shown in the figure. Figure 10 As shown in the figure, the modal frequency simulation value of the entity model was obtained through modal simulation; in addition, for the same entity model, the modal frequency test value was obtained through modal test. In the frequency range of 1000Hz, the first 41 modal frequency simulation values ​​of the equivalent discrete model and the entity model were compared with the modal frequency test values ​​of the entity model to verify the accuracy of the simulation model. The results show that the average deviation between the modal frequency simulation value of the equivalent discrete model established by the method of the present invention and the modal frequency test value of the entity model is only 1.46%, and the maximum deviation does not exceed 6%. When the entity model is used for modeling, the average deviation between the modal frequency simulation value of the entity model and the modal frequency test value is 10.27%, and the maximum deviation is 22.79%. The deviation comparison results are shown in the figure. Figure 11 The modeling method of the present invention can be used in engineering practice for analyzing vibration characteristics of a piping system containing a triple clamp.

[0151] In the above embodiment, the solid finite element model of the piping system has a total of 2,512,692 degrees of freedom, while the equivalent discrete finite element model of the piping system has a total of 18,918 degrees of freedom. Compared to the solid finite element model, the equivalent discrete finite element model of the piping system established using the method provided by the present invention can significantly reduce the model size and improve solution efficiency.

[0152] Parts of the present invention that are not described in detail belong to common knowledge among those skilled in the art.

Claims

1. A dynamic modeling method for a triple clamp piping system based on ANSYS APDL, characterized in that: include: The pipe system is discretized using beam elements to obtain a finite element model of the pipe system, on which a triple clamp can be placed; The spatial spring unit is used to discretize the stiffness of the triple clamp equivalently, and the finite element model of the triple clamp is established. The initial value of the equivalent stiffness of the triple clamp is obtained through static stiffness test, and the initial value of the equivalent stiffness is corrected by modal test and stiffness parameter correction method to obtain the equivalent stiffness of the triple clamp. The equivalent stiffness of the triple clamp is assigned to the spatial spring unit corresponding to the triple clamp finite element model, and the triple clamp finite element model is introduced into the piping system finite element model, thereby establishing a triple clamp piping system finite element model.

2. The method for dynamic modeling of a triple clamp piping system based on ANSYS APDL according to claim 1, characterized in that: The stiffness equivalent discretization of the triple clamp is performed, including: The stiffness of the triple clamp is equivalently discretized into three spatial spring units. Each spatial spring unit has a node at both ends, and each node has 6 degrees of freedom. The three spatial spring units share a node at one end, and the three nodes at the other end are connected one-to-one with the three pipes at the center of the triple clamp hole. The spatial spring unit is used to simulate the stiffness relationship between the triple clamp and the pipeline. The stiffness parameter k of the spatial spring unit is defined by the following formula: k=[k u k v k w k θx k θy k θz ] Among them, k u 、k v 、k w Represents the stiffness of the translational freedom in the x, y, and z coordinate axes respectively; k θx 、k θy 、k θz They represent the stiffness of the rotational freedom around the x, y, and z coordinate axes respectively.

3. The dynamic modeling method of a triple clamp piping system based on ANSYS APDL according to claim 2 is characterized by: The element stiffness matrix K of the spatial spring element e as follows: l is the distance between the nodes at both ends of the spatial spring.

4. The method for dynamic modeling of a triple clamp piping system based on ANSYS APDL according to claim 1, characterized in that: The initial value of the equivalent stiffness of the triple clamp is obtained through a static stiffness test. The method is as follows: The translation stiffness test system is used to measure the initial values ​​of the translation stiffness of the first pipe outside the triple clamp in the x, y, and z coordinate directions. The rotational stiffness test system is used to measure the initial value of the rotational stiffness of the first pipe outside the triple clamp around the three coordinate axes of x, y, and z. Thus, the initial value of the spatial spring unit stiffness corresponding to the first pipe outside the triple clamp is obtained The initial value of the spatial spring unit stiffness corresponding to the second pipe in the middle of the triple clamp is obtained by measuring the translation stiffness test system and the rotation stiffness test system respectively. The triple clamp has a symmetrical structure, so the initial value of the space spring unit stiffness corresponding to the third pipe of the triple clamp is equal to the initial value of the space spring unit stiffness corresponding to the first pipe.

5. The method for dynamic modeling of a triple clamp piping system based on ANSYS APDL according to claim 4, characterized in that: The translational stiffness testing system includes a mounting plate, a stepper motor, a lead screw, a slide rail, a slider, a force sensor, a digital dial indicator, a loading fixture, and a fixed support fixture; the slide rail and the stepper motor are fixedly mounted on the mounting plate, the output end of the stepper motor is connected to the lead screw, the slider is mounted on the slide rail, the slider is connected to the lead screw through a threaded pair, the upper part of the slider is connected to the loading fixture through a force sensor, the loading fixture is connected to the measured pipeline for applying a load, the triple clamp is mounted on the fixed support fixture through bolts, the fixed support fixture is fixedly mounted on the mounting plate, and two digital dial indicators respectively measure the translational displacement at both ends of the loading fixture.

6. The method for dynamic modeling of a triple clamp piping system based on ANSYS APDL according to claim 5, characterized in that: When the translational stiffness test system is used to measure the initial value of the clamp's translational stiffness, the stepper motor drives the lead screw to rotate, causing the slider to translate along the slide rail and push the loading fixture through the force sensor, thereby applying translational tensile loads in the x, y, and z directions to both ends of the pipeline. The translational displacement of the outer tube is measured using a digital dial indicator, and the magnitude of the applied tensile load is obtained through the force sensor. Considering the clamp constraint position as the fixed support boundary, the bending deformation of the pipeline itself is calculated by the following formula: Where: p is the calculated value of the pipe's own bending deformation, F is the value measured by the force sensor, L is the distance between the clamp's constraint position and the load application position, E is the elastic modulus of the material, and I is the moment of inertia of the pipe section; The deformation of the clamp is calculated by the following formula: d c =d t -d p Among them, δ c is the deformation of the clamp, δ t It is the average value of the deformation results measured by two digital dial indicators; By changing the force load amplitude and performing multiple loadings, multiple sets of F and clamp deformation δ are obtained. c After performing linear fitting on the data points, the slope of the curve is calculated to obtain the initial value of the clamp translational stiffness in the corresponding direction.

7. The method for dynamic modeling of a triple clamp piping system based on ANSYS APDL according to claim 4, characterized in that: The rotational stiffness test system includes a mounting plate, a stepper motor, supporting bearings and supports, a rotating shaft, a torque sensor, an inclination sensor, a loading tooling and a fixing tooling; the stepper motor, supporting bearings and supports, and the fixing tooling are all fixedly installed on the mounting plate, the output end of the stepper motor is connected to the rotating shaft, the rotating shaft is supported by the supporting bearings and supports, the end of the rotating shaft is connected to the torque sensor, the torque sensor is connected to the loading tooling, the inclination sensor is installed on the loading tooling, the loading tooling clamps the pipeline, the triple clamp is installed on the fixing tooling by bolts, and the inclination sensor measures the pipeline angle.

8. The method for dynamic modeling of a triple clamp piping system based on ANSYS APDL according to claim 7, characterized in that: When the rotational stiffness measurement system is used to measure the initial value of the clamp's rotational stiffness, the stepper motor drives the shaft to rotate, and the shaft rotates the loading fixture through the torque sensor, thereby applying rotational torque loads around the x, y, and z directions to the pipeline, and measuring the angular deformation of the clamp through the inclination sensor; Considering the clamp constraint position as the fixed support boundary, the pipe's own angular deformation is calculated by the following formula: Where: θ p is the calculated value of the pipeline's own angular deformation, M is the torque sensor measurement value, L is the distance between the clamp constraint position and the torque load application position; E is the elastic modulus of the material; I is the moment of inertia of the pipeline section; The angular deformation of the clamp is calculated by the following formula: i c =θ t -θ p Among them, θ c is the angular deformation of the clamp, θ t is the measured value of the tilt sensor; By changing the torque load amplitude and performing multiple loadings, multiple sets of torque and clamp angular deformation θ are obtained. c After performing linear fitting on the data points, the slope of the curve is calculated to obtain the initial value of the clamp rotational stiffness in the corresponding direction.

9. The method for dynamic modeling of a triple clamp piping system based on ANSYS APDL according to claim 4, characterized in that: The modal test and stiffness parameter correction method are used to correct the initial value of the equivalent stiffness. The method is as follows: A three-tube modal test system with three clamps was constructed and modal tests were carried out. The test values ​​of the characteristic modal frequencies of the corresponding characteristic vibration modes, which can reflect the six stiffness parameters of the spatial spring unit, were obtained for the outer and middle pipes respectively. A finite element model of the piping system for the triple clamp and three-tube modal test was established. The pipeline was modeled using beam elements, while the clamps were equivalently modeled using three spatial spring elements. The initial values ​​of the triple clamp's equivalent stiffness were assigned to the three spatial spring elements. Modal calculations were performed on the piping system for the triple clamp and three-tube modal test to obtain the simulated eigenmodal frequencies corresponding to the eigenmode shapes. The correlation analysis of the six stiffness parameters was performed on the test values ​​of the characteristic modal frequencies obtained in the modal test to identify the main influencing stiffness parameters of each characteristic modal frequency; Based on the finite element model of the piping system with triple clamps and three pipes in the modal test, a single parameter correction was performed on the main influencing stiffness parameters in the spatial spring unit corresponding to the identified eigenmodal frequency, so that the relative deviation between the simulated and tested eigenmodal frequency values ​​after correction was less than 1%. The six stiffness parameters of the space spring unit corresponding to the outer pipeline and the middle pipeline are corrected one by one to obtain the corrected stiffness parameters of the space spring unit, that is, the equivalent stiffness of the triple clamp.

10. The method for dynamic modeling of a triple clamp piping system based on ANSYS APDL according to claim 9, characterized in that: The following undamped characteristic equation is used to perform modal calculations on the piping system for the triple clamp and three-tube modal test: (K-ΛM)U=0 Among them, K is the stiffness matrix of the triple clamp piping system, M is the mass matrix of the triple clamp piping system, Λ is the eigenvalue matrix of the triple clamp piping system, and U is the vibration mode matrix of the triple clamp piping system.

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