CN113901698B - Method for identifying mechanical parameters of clamp pipeline system - Google Patents
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Abstract
The invention is applicable to the technical field of mechanical structure dynamics, and provides a clamp pipeline system mechanical parameter identification method, which comprises the following steps: step one: constructing a finite element model of a road clamp system, carrying out finite element modeling by using a Timoshenko beam theory, and neglecting axial translation and torsion; step two: constructing a pipeline clamp system dynamics equation; step three: designing the rigidity of a pipe body of the pipe clamp system; step four: designing a system pipeline to simulate damping; according to the optimization method based on the Pareto multi-objective genetic algorithm, frequency deviation and frequency response function deviation are taken as objective functions, and the approximate ranges of the static stiffness and the static damping of the clamp, which are obtained through static hysteresis experiment tests, are taken as constraint conditions, so that the mechanical characteristics of the clamp in the pipeline clamp system are identified; meanwhile, the dependence of the rigidity and damping of the clamp identified in the prior identification technology on frequency is removed, and the mechanical characteristics of key connecting parts of the system are accurately identified by using a reverse push identification method.
Description
Technical Field
The invention belongs to the technical field of mechanical structure dynamics, and particularly relates to a clamp pipeline system mechanical parameter identification method.
Background
The pipeline system is an important component part of the aeroengine, and the main function of the pipeline system is to carry out transportation work of media (such as hydraulic oil), and the pipeline system is generally fixed on the casing through a clamp and a bracket. The mechanical properties of the clip, including the stiffness and damping of the clip, have a significant impact on the dynamics of the aircraft engine piping system. Therefore, the research of the mechanical properties of the clamp is of great importance.
The existing research on the pipeline system of the external clamp of the aeroengine is focused on analysis and optimization of the layout and parameters (positions and numbers) of the clamp on the pipeline. As regards the layout of the yoke, li Xin et al propose to adjust the position of the yoke within a certain range by using a particle swarm optimization algorithm (Particle Swarm Optimization, PSO) to minimize the weighted sum of characteristic impedances at the excitation source frequency points (see document: li Xin, wang Shaoping. Vibration and shock analysis of the hydraulic circuits of an aircraft based on the optimized layout of the yoke [ J ]. Vibration and shock, 2013,32 (01): 14-20); XUDONG LIU et al propose a method for achieving hoop placement based on genetic algorithm to effectively reduce the resonance amplitude of the piping system (see LIU XUDONG et al optimization of pipeline system with multi-hoop supports for avoiding vibration, based on particle swarm algorithm [ J ]. Proceedings of the Institution of Mechanical Engineers, part C: journal of Mechanical Engineering Science,2021,235 (9): 1524-1538). The Xiantao Zhang et al describe an optimization method to obtain the optimal clamp position (see: xiantao Zhang, wei Liu, yamei Zhang, yujie Zhao. Experimental Investigation and Optimization Design of Multi-Support Pipeline System [ J ]. Chinese Journal of Mechanical Engineering,2021,34 (02): 145-159). In addition, there are studies on the mechanical parameter identification of the clamp, such as Hui, etc., the identification study on the metal rubber vibration isolator parameters by using an energy method and a least square method (see documents: hui, jiang Hongyuan, liu Wenjian, a.m. ullannov. The metal rubber vibration isolator parameter identification study with hysteresis nonlinearity [ J ]. Physical theory, 2009,58 (08): 5238-5243), chai Qingdong, etc., the clamp stiffness which cannot be measured is obtained by combining a genetic algorithm with a modal test search (see documents: chai Qing east, strength, ma Hui, han Qingkai, zhang Dazhi, single-duplex clamp pipeline system modeling and dynamic characteristic analysis [ J ]. Vibration and impact, 2020,39 (19): 114-120), and a method for reversely pushing the clamp support stiffness and damping based on a measured Frequency Response Function (FRF) is proposed (see documents: gao, sun Wei, poyuhua, ma Hui. The clamp support stiffness and damping [ J ]. Aviation power, 2019,34 (03): 664-670).
In summary, the stiffness and damping identified by the parameters of the clamp have a certain dependence on frequency, vary with the natural frequency, and do not consider the damping effect of the system structure.
Disclosure of Invention
The invention provides a clamp pipeline system mechanical parameter identification method, and aims to solve the problems of target function dependence and the like in the existing connecting part mechanical characteristic reverse-push identification method.
The invention is realized in such a way that a clamp pipeline system mechanical parameter identification method is characterized by comprising the following steps: step one: constructing finite element model of road clamp system, modeling finite element by using Timoshenko beam theory, neglecting axial translation and torsion to obtain rigidity matrix, such as formula (1), obtaining damping matrix, such as formula (2),
if the rigidity matrix of the pipe body is K guan The damping matrix of the pipe body is C guan The total system stiffness matrix and the total damping matrix can be expressed as shown in equations (3), (4);
K＝K k +K guan (3)
C＝C k +C guan (4)
step two: constructing a kinetic equation of the pipeline clamp system, setting the initial condition as 0, as shown in a formula (5),
Mx+Cx+Kx＝0 (5)
wherein M is the total mass matrix of the system; c, a system total damping matrix; k-a system total stiffness matrix; the damping matrix C comprises two parts: pipeline damping and clamp damping;
step three: designing the rigidity of a pipe body of the pipe clamp system; the stiffness matrix is shown in formula (6):
step four: the system pipeline is designed to simulate damping, and Rayleigh damping is used for simulating, as shown in a formula (7),
C guan ＝αM+βK (7)
wherein, alpha and beta are Rayleigh damping coefficients; alpha and beta can be obtained by the formula (8) and the formula (9),
wherein ζ 1 And zeta 2 Structural damping ratios of 1 st order and 2 nd order; omega 1 And omega 2 Natural frequencies (units: rad/s) of order 1 and order 2; the subscript numbers 1,2 do not necessarily refer to the first two steps, but are instead dependent on the minimum and maximum values of the analytical range of interest; the simulation uses the first two steps.
Preferably, the method further comprises: step five: designing a system displacement frequency response function H s (w) as shown in the formula (10),
wherein ω is the external excitation frequency, ω r Zeta is the natural frequency of the system of the r order r For the system's r-th order modal damping ratio, n is the calculated modal number, and φ is the mode shape matrix composed of n-order mode shape vectors.
Preferably, the method further comprises: step six: converting the displacement frequency response function into an acceleration frequency response function H a (w) as shown in the formula (11),
preferably, the method further comprises: step seven: a Pareto multi-target recognition algorithm is introduced, the recognition process is shown in formulas (12) and (13),
min[f 1 (x),f 2 (x),…f m (x)] (12)
wherein f i (x) Is an objective function to be optimized, and x is a variable to be optimized; constraint conditions: the upper and lower limits of x are lb and ub, respectively, aeq and beq constitute the linear equality constraint of x, and a and b constitute the linear inequality constraint of x.
Preferably, the method further comprises: step eight: constructing a clamp stiffness identification algorithm, obtaining the natural frequency of each order by a modal experiment and simulation analysis, constructing a Pareto stiffness algorithm objective function based on the frequency difference ratio of the two, as shown in a formula (14),
wherein f 1 ，f 2 …f i For the experimentally obtained modal frequencies, f m1 ，f m2 …f mi The frequency obtained for the simulation; the constraint is set to a range of stiffness K, as shown in equation (15),
lb≤K≤ub (15)。
preferably, the method further comprises: step nine: constructing a clamp damping identification algorithm, obtaining frequency response functions corresponding to the inherent frequencies of each order through modal experiments and simulation analysis, constructing a Pareto damping algorithm target function based on the frequency response function difference ratio of the two, as shown in a formula (16),
wherein H is 1 ，H 2 …H i To obtain the frequency response function through experiments, H m1 ，H m2 …H mi A frequency response function obtained for simulation; constraintThe condition is set as the structural damping ratio ζ in damping c and Rayleigh damping 1 And zeta 2 The size, as shown in equation (17),
lb≤c j 、ζ 1 、ζ 2 ≤ub (17)
wherein c j Represents damping in the j (x, y, z) direction.
Preferably, the method further comprises: step ten: and carrying out a statics experiment on the clamp to obtain the clamp rigidity and damping identification range lb and ub.
Preferably, the method further comprises: step eleven: and exciting the clamp pipeline system by adopting a hammering method to obtain a frequency response function of the clamp pipeline system.
Preferably, the method further comprises: step twelve: and taking the target order as an analysis object, and applying a Pareto stiffness identification algorithm to identify and obtain the stiffness parameters under the corresponding conditions of the clamp.
Preferably, the method further comprises: step thirteen: and taking the target order as an analysis object, and applying a Pareto damping identification algorithm to identify and obtain damping parameters corresponding to the clamp.
Compared with the prior art, the invention has the beneficial effects that: the invention provides a clamp pipeline system mechanical parameter identification method, which is an improved optimization method based on a Pareto multi-objective genetic algorithm, wherein frequency deviation and frequency response function deviation are used as objective functions, and the approximate ranges of clamp static stiffness and static damping obtained through static hysteresis experiment tests are used as constraint conditions to identify the mechanical characteristics of a clamp in a pipeline clamp system; meanwhile, the dependence of the rigidity and damping of the clamp identified in the prior identification technology on frequency is removed, and the mechanical characteristics of key connecting parts of the system are accurately identified by using a reverse push identification method.
Drawings
FIG. 1 is a flow chart of a method for identifying stiffness of a clamp pipe system according to the present invention;
FIG. 2 is a flow chart of a method for identifying damping suitable for a clamp pipe system according to the present invention;
fig. 3 is a schematic diagram of a simulation test stand of a 1 DK8 single clamp pipe system in the present invention;
fig. 4 is a schematic diagram of a simulation test stand of a 2 DK8 single clamp pipe system in the present invention;
FIG. 5 is a graph of the y-direction frequency response function after identification by the identification algorithm according to the present invention;
FIG. 6 is a schematic diagram of a simulation test stand of a 1 SK8-8 duplex clamp pipe system in the invention;
FIG. 7 is a flowchart of the operation of Sheffield single-target genetic algorithm;
FIG. 8 is a y-direction search result of a 1 SK8-8 duplex clamp pipe system applying the improved Pareto stiffness identification algorithm;
FIG. 9 is a y-direction search result of a 1 SK8-8 duplex clamp pipe system applying the improved Pareto damping identification algorithm;
FIG. 10 is a second order search result y-wise for a 1 SK8-8 duplex clamp pipe system applying a Sheffield stiffness identification algorithm;
FIG. 11 is a second order search result of 1 SK8-8 duplex clamp pipe system applying Sheffield damping identification algorithm y;
fig. 12 is a finite element model of a typical piping system. In the figure, the end A represents pipeline fixation, the end B represents support by the clamp, and s and t represent positions of the single-connection clamp. The clamp is equivalent to two linear springs connected in parallel on a pipeline, wherein Kz and Ky in the figure represent radial stiffness of the clamp, and Kθz and Kθy represent torsional stiffness of the clamp. cz, cy denote radial bearing damping.
In the figure: 1. a first fixing and supporting clamp; 2. picking up vibration points I; 3. excitation point I; 4. a first clamping hoop; 5. a second clamp; 6. vibration pick-up points II; 7. excitation points II; 8. a second fixing and supporting clamp; 9. vibration picking points III; 10. a third clamp; 11. and a third excitation point.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
The invention will be better explained by the following detailed description of the embodiments with reference to the drawings. The operation flow is shown in figures 1-2, and the specific implementation steps are as follows:
step one: constructing a finite element model of the pipeline clamp system shown in fig. 12, performing finite element modeling by using a Timoshenko beam theory, neglecting axial translation and torsion to obtain a rigidity matrix, such as formula (1), obtaining a damping matrix, such as formula (2),
if the rigidity matrix of the pipe body is K guan The damping matrix of the pipe body is C guan The total system stiffness matrix and the total damping matrix can be expressed as shown in equations (3), (4);
K＝K k +K guan (3)
C＝C k +C guan (4)
step two: constructing a kinetic equation of the pipeline clamp system, setting the initial condition as 0, as shown in a formula (5),
Mx+Cx+Kx＝0 (5)
wherein M is the total mass matrix of the system; c, a system total damping matrix; k-a system total stiffness matrix; the damping matrix C comprises two parts: pipeline damping and clamp damping;
step three: designing the rigidity of a pipe body of the pipe clamp system; the stiffness matrix is shown in formula (6):
step four: the system pipeline is designed to simulate damping, and Rayleigh damping is used for simulating, as shown in a formula (7),
C guan ＝αM+βK (7)
wherein, alpha and beta are Rayleigh damping coefficients; alpha and beta can be obtained by the formula (8) and the formula (9),
wherein ζ 1 And zeta 2 Structural damping ratios of 1 st order and 2 nd order; omega 1 And omega 2 Natural frequencies (units: rad/s) of order 1 and order 2; the subscript numbers 1,2 do not necessarily refer to the first two steps, but are instead dependent on the minimum and maximum values of the analytical range of interest; the simulation uses the first two steps.
Step five: designing a system displacement frequency response function Hs (w), as shown in a formula (10),
wherein ω is the external excitation frequency, ω r Zeta is the natural frequency of the system of the r order r For the system's r-th order modal damping ratio, n is the calculated modal number, and φ is the mode shape matrix composed of n-order mode shape vectors.
Step six: converting the displacement frequency response function into an acceleration frequency response function H a (w) as shown in the formula (11),
step seven: a Pareto multi-target recognition algorithm is introduced, the recognition process is shown in formulas (12) and (13),
min[f 1 (x),f 2 (x),…f m (x)] (12)
wherein f i (x) Is an objective function to be optimized, and x is a variable to be optimized; constraint conditions: the upper and lower limits of x are lb and ub, respectively, aeq and beq constitute the linear equality constraint of x, and a and b constitute the linear inequality constraint of x.
Step eight: constructing a clamp stiffness identification algorithm, obtaining the natural frequency of each order by a modal experiment and simulation analysis, constructing a Pareto stiffness algorithm objective function based on the frequency difference ratio of the two, as shown in a formula (14),
wherein f 1 ，f 2 …f i For the experimentally obtained modal frequencies, f m1 ，f m2 …f mi The frequency obtained for the simulation; the constraint is set to a range of stiffness K, as shown in equation (15),
lb≤K≤ub (15)。
constructing a clamp damping identification algorithm, obtaining frequency response functions corresponding to the inherent frequencies of each order through modal experiments and simulation analysis, constructing a Pareto damping algorithm target function based on the frequency response function difference ratio of the two, as shown in a formula (16),
wherein H is 1 ，H 2 …H i To obtain the frequency response function through experiments, H m1 ，H m2 …H mi A frequency response function obtained for simulation; the constraint is set to the structural damping ratio ζ in damping c and Rayleigh damping 1 And zeta 2 The size, as shown in equation (17),
lb≤c j 、ζ 1 、ζ 2 ≤ub (17)
where cj represents damping in the j (x, y, z) direction.
Step ten: and carrying out a statics experiment on the clamp to obtain the clamp rigidity and damping identification range lb and ub.
Step eleven: and (3) establishing a test platform shown in fig. 3, and exciting the clamp pipeline system in the y-direction and the z-direction by adopting a hammering method to obtain a frequency response function of the clamp pipeline system. Wherein 1 is marked in fig. 3 as a first anchor clamps; 2 is a vibration pickup point I; 3 is an excitation point I; and 4 is a clamp I.
Step twelve: and taking the first two orders as analysis objects, and applying a Pareto stiffness identification algorithm based on matlab platform programming to identify and obtain stiffness parameters under the corresponding conditions of the clamp.
Step thirteen: taking two steps of the damping parameters as analysis objects, and applying a Pareto damping identification algorithm based on matlab platform programming to identify the damping parameters under the corresponding conditions of the clamp.
Step fourteen: and then verifying the rigidity damping parameters of the clamp identified based on the method, constructing a test platform shown in fig. 4, and repeating the steps 1-10. Wherein 5 in FIG. 4 is a second clip; 6 is a vibration pickup point II; and 7 is an excitation point II.
Fifteen steps: taking the first two steps as analysis objects, substituting the clamp identification parameters obtained in the steps 11 and 12 into the system of fig. 4, and analyzing the frequency response function in the y direction as shown in fig. 5, so that the reliable result of the invention in identifying the rigidity and damping parameters of the clamp is verified.
The invention uses the frequency deviation and the frequency response function deviation as target functions, uses rigidity and damping obtained by a statics experiment as constraint conditions, and identifies the rigidity and damping of the clamp in the pipeline-clamp system. The invention mainly eliminates the dependence of clamp rigidity and damping on frequency in the identification, and meanwhile, the invention does not lose the precision of clamp mechanical parameters compared with the clamp mechanical parameters identified by the traditional identification method. The identification method in the prior art usually takes one order to identify the corresponding clamp mechanical parameter, and has the defect that the identified mechanical parameter has dependence on the natural frequency, and the Sheffield single-target genetic algorithm commonly used in the prior art is used as a practical contrast with the invention to build a test platform shown in fig. 6, wherein 8 marked in fig. 6 is a second fixed clamp; 9 is a vibration picking point III; 10 is a clamp III; and 11 is an excitation point three.
The specific implementation steps of the verification are as follows:
repeating the steps 1-9.
And carrying out a statics experiment on the clamp to obtain the rigidity and damping identification range lb and ub.
The system is excited by hammering to obtain frequency response functions in y and z directions.
Taking the first third order as an analysis object, applying a Pareto stiffness identification algorithm based on matlab platform programming, and identifying to obtain the stiffness parameters under the corresponding clamp, wherein the y-direction identification result is shown in figure 8.
Taking the first two steps as analysis objects, applying a Pareto damping identification algorithm based on matlab platform programming, and identifying to obtain damping parameters corresponding to the clamp, wherein the y-direction identification result is shown in figure 9.
Sheffield's genetic algorithm is introduced, the operation flow is shown in FIG. 7, the objective function uses the function obtained in step 1), and the identification range uses the value of step 2), so that the identification range is consistent with the variable of the invention.
Taking the first third order as an analysis object, applying a Sheffield traditional stiffness identification algorithm based on matlab platform programming, identifying to obtain a stiffness parameter corresponding to the clamp, taking the second order as a representative, and converging when the y-direction identification result is 16 th generation as shown in fig. 10.
Taking the first two steps as analysis objects, applying Sheffield traditional damping identification algorithm based on matlab platform programming, identifying to obtain damping parameters corresponding to the clamp, taking the second step as a representative, and converging at the 68 th generation as the y-direction identification result is shown in FIG. 11.
Compared with the Sheffield traditional identification algorithm, the improved algorithm identification method provided by the invention keeps the original identification accuracy, furthermore, the invention eliminates the dependence of clamp rigidity and damping on frequency in the identification on the original identification accuracy, overcomes the defect of clamp parameter identification of a pipeline clamp system in the prior related art, further reflects the mechanical characteristics of the clamp of the pipeline clamp system, and is more convenient for the practical analysis and application of engineering.
The foregoing description of the preferred embodiments of the invention is not intended to be limiting, but rather is intended to cover all modifications, equivalents, and alternatives falling within the spirit and principles of the invention.
Claims (10)
1. The method for identifying the mechanical parameters of the clamp pipeline system is characterized by comprising the following steps of:
step one: constructing finite element model of road clamp system, modeling finite element by using Timoshenko beam theory, neglecting axial translation and torsion to obtain rigidity matrix, such as formula (1), obtaining damping matrix, such as formula (2),
if the rigidity matrix of the pipe body is K guan The damping matrix of the pipe body is C guan The total system stiffness matrix and the total damping matrix can be expressed as shown in equations (3), (4);
K＝K k +K guan (3)
C＝C k +C guan (4)
step two: constructing a kinetic equation of the pipeline clamp system, setting the initial condition as 0, as shown in a formula (5),
Mx+Cx+Kx＝0 (5)
wherein M is the total mass matrix of the system; c, a system total damping matrix; k-a system total stiffness matrix; the damping matrix C comprises two parts: pipeline damping and clamp damping;
step three: designing the rigidity of a pipe body of the pipe clamp system; the stiffness matrix is shown in formula (6):
step four: the system pipeline is designed to simulate damping, and Rayleigh damping is used for simulating, as shown in a formula (7),
C guan ＝αM+βK (7)
wherein, alpha and beta are Rayleigh damping coefficients; alpha and beta can be obtained by the formula (8) and the formula (9),
wherein ζ 1 And zeta 2 Structural damping ratios of 1 st order and 2 nd order; omega 1 And omega 2 Natural frequencies (units: rad/s) of order 1 and order 2; the subscript numbers 1,2 do not necessarily refer to the first two steps, but are instead dependent on the minimum and maximum values of the analytical range of interest; the simulation uses the first two steps.
2. The method for identifying mechanical parameters of a clamp pipe system according to claim 1, wherein,
further comprises:
step five: designing a system displacement frequency response function H s (w) as shown in the formula (10),
wherein ω is the external excitation frequency, ω r Zeta is the natural frequency of the system of the r order r For the system's r-th order modal damping ratio, n is the calculated modal number, and φ is the mode shape matrix composed of n-order mode shape vectors.
3. The method for identifying mechanical parameters of a clamp pipe system according to claim 1, wherein,
further comprises:
step six: converting the displacement frequency response function into an acceleration frequency response function H a (w) as shown in the formula (11),
4. the method for identifying mechanical parameters of a clamp pipe system according to claim 1, wherein,
further comprises:
step seven: a Pareto multi-target recognition algorithm is introduced, the recognition process is shown in formulas (12) and (13),
wherein f i (x) Is an objective function to be optimized, and x is a variable to be optimized; constraint conditions: the upper and lower limits of x are lb and ub, respectively, aeq and beq constitute the linear equality constraint of x, and a and b constitute the linear inequality constraint of x.
5. The method for identifying mechanical parameters of a clamp pipe system according to claim 1, wherein,
further comprises:
step eight: constructing a clamp stiffness identification algorithm, obtaining the natural frequency of each order by a modal experiment and simulation analysis, constructing a Pareto stiffness algorithm objective function based on the frequency difference ratio of the two, as shown in a formula (14),
wherein f 1 ，f 2 …f i For the experimentally obtained modal frequencies, f m1 ，f m2 …f mi The frequency obtained for the simulation; the constraint is set to a range of stiffness K, as shown in equation (15),
lb≤K≤ub (15)。
6. the method for identifying mechanical parameters of a clamp pipe system according to claim 1, wherein,
further comprises:
step nine: constructing a clamp damping identification algorithm, obtaining frequency response functions corresponding to the inherent frequencies of each order through modal experiments and simulation analysis, constructing a Pareto damping algorithm target function based on the frequency response function difference ratio of the two, as shown in a formula (16),
wherein H is 1 ，H 2 …H i To obtain the frequency response function through experiments, H m1 ，H m2 …H mi A frequency response function obtained for simulation; the constraint is set to the structural damping ratio ζ in damping c and Rayleigh damping 1 And zeta 2 The size, as shown in equation (17),
lb≤c j 、ζ 1 、ζ 2 ≤ub (17)
wherein c j Represents damping in the j (x, y, z) direction.
7. The method for identifying mechanical parameters of a clamp pipe system according to claim 1, wherein,
further comprises:
step ten: and carrying out a statics experiment on the clamp to obtain the clamp rigidity and damping identification range lb and ub.
8. The method for identifying mechanical parameters of a clamp pipe system according to claim 1, wherein,
further comprises:
step eleven: and exciting the clamp pipeline system by adopting a hammering method to obtain a frequency response function of the clamp pipeline system.
9. The method for identifying mechanical parameters of a clamp pipe system according to claim 1, wherein,
further comprises:
step twelve: and taking the target order as an analysis object, and applying a Pareto stiffness identification algorithm to identify and obtain the stiffness parameters under the corresponding conditions of the clamp.
10. The method for identifying mechanical parameters of a clamp pipe system according to claim 1, wherein,
further comprises:
step thirteen: and taking the target order as an analysis object, and applying a Pareto damping identification algorithm to identify and obtain damping parameters corresponding to the clamp.
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| CN108572072A (en) * | 2018-04-09 | 2018-09-25 | 上海理工大学 | Ball-screw Drive Systems Identifying Dynamical Parameters method based on frequency response function method |
| CN109766609A (en) * | 2018-12-29 | 2019-05-17 | 东北大学 | A kind of method of clip-pipe-line system Dynamic Modeling |
| CN110188512A (en) * | 2019-07-01 | 2019-08-30 | 东北大学 | A kind of method of list dual card banded penstock road system modelling |
| WO2021158198A2 (en) * | 2020-02-07 | 2021-08-12 | Orta Doğu Tekni̇k Üni̇versi̇tesi̇ | A novel experimental modal analysis method based on response control approach for nonlinear engineering structures |
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| CN108572072A (en) * | 2018-04-09 | 2018-09-25 | 上海理工大学 | Ball-screw Drive Systems Identifying Dynamical Parameters method based on frequency response function method |
| CN109766609A (en) * | 2018-12-29 | 2019-05-17 | 东北大学 | A kind of method of clip-pipe-line system Dynamic Modeling |
| CN110188512A (en) * | 2019-07-01 | 2019-08-30 | 东北大学 | A kind of method of list dual card banded penstock road system modelling |
| WO2021158198A2 (en) * | 2020-02-07 | 2021-08-12 | Orta Doğu Tekni̇k Üni̇versi̇tesi̇ | A novel experimental modal analysis method based on response control approach for nonlinear engineering structures |
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