CN113358308B - Combined structure transverse displacement determination method based on limited measuring points and global mode - Google Patents

Combined structure transverse displacement determination method based on limited measuring points and global mode Download PDF

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CN113358308B
CN113358308B CN202110619854.8A CN202110619854A CN113358308B CN 113358308 B CN113358308 B CN 113358308B CN 202110619854 A CN202110619854 A CN 202110619854A CN 113358308 B CN113358308 B CN 113358308B
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global
combined structure
displacement
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CN113358308A (en
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曹登庆
陈帅
邵崇晖
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Harbin Qinglin Jiecheng Technology Co ltd
Harbin Institute of Technology
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Harbin Institute of Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M7/00—Vibration-testing of structures; Shock-testing of structures
    • G01M7/02—Vibration-testing by means of a shake table
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M7/00—Vibration-testing of structures; Shock-testing of structures
    • G01M7/02—Vibration-testing by means of a shake table
    • G01M7/025—Measuring arrangements
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization

Abstract

A method for determining the transverse displacement of a combined structure based on limited measuring points and a global mode belongs to the field of methods for determining the transverse vibration displacement of combined structures. The invention solves the problems of complex effective measurement and experimental equipment, high cost, low data result precision and the like of physical fields such as transverse displacement, speed and the like of the combined structure in experiments or actual engineering. The device used by the invention is simple, low in cost, high in calculation efficiency and high in relative precision, the construction of the combined structure full displacement field can be realized through the transverse displacement data of a limited number of measuring points and the global mode of the combined structure, and the device has rich application prospect and practical significance.

Description

Combined structure transverse displacement determination method based on limited measuring points and global mode
Technical Field
The invention belongs to the field of a transverse vibration displacement determination method of a combined structure, and particularly relates to a combined structure transverse displacement determination method based on a limited measuring point and a global mode.
Background
A composite structure is a multi-body system formed by connecting a plurality of objects (including flexible bodies, rigid bodies, etc.) in some way, and is widely used in the engineering fields of aerospace, machinery, civil engineering, etc., such as space assemblies with solar sailboards, space flexible mechanical arms, large rotating machinery with long and thin blades, etc. In a combined structure, due to vibration coupling effects among the components, the dynamic behavior of a single flexible component in the system under the static boundary of cantilever, simple support and free is greatly different from that of the whole structure. The research on the vibration characteristics of the combined structure has great significance for meeting the increasingly growing structural requirements of engineering and improving the safety of a system.
In a pre-research experiment or actual engineering health monitoring of a composite structure, physical quantities such as displacement and speed of vibration of the composite structure are necessary data. For some special combined structures, the deformation and vibration of the workpiece are influenced by factors such as harsh working environment, complex load and the like, the deformation and vibration of the workpiece have temporal-spatial non-uniformity and strong temporal variability, only single-point or limited-point measurement information is insufficient to describe actual working conditions, and field information of physical quantities such as transverse displacement and speed is required to be provided. For example, in a combined structure type aeroelastic analysis ground experiment of an aircraft, aerodynamic force borne by each point of the structure is a function of transverse displacement and speed of each point of the corresponding structure, so that an accurate aerodynamic field can be provided only by determining a transverse displacement field and a speed field of the structure.
There are three current ways to determine the structure lateral displacement field: firstly, measuring the lateral displacement of a part of points on the structure, and then determining the displacement field of the whole structure by an interpolation method. Although this method is easy to operate and the amount of computation is not large, it is difficult to estimate the error, and the interpolation effect depends on the number of sampling points, i.e. a large number of sampling points are required if an accurate result is to be obtained. In addition, due to the characteristics of the algorithm, different interpolation methods have the specific defects, such as the displacement value of a displacement interpolation method (CBDI) based on curvature has oscillation effect in an edge point area, so that the obtained result has poor precision; and secondly, the laser vibration meter is utilized to realize the measurement of the transverse displacement of the structure with full coverage. The laser vibration measurement has the advantages of high precision, fast response, large dynamic range and the like. However, the device is extremely harsh to the use environment, the measurement process is greatly influenced by other stray light, the device is complex, and the manufacturing cost is high. In addition, when the structure amplitude is large, the problems of mining leakage and distortion can occur in the measurement of the laser vibration meter on the structure corners; and thirdly, a plate/beam structure transverse displacement determining method based on limited measuring points and vibration modes is a simple method with good precision and effect for measuring the displacement and speed field of the plate/beam structure, but serious errors can occur when the method is applied to a combined structure. In the combined structure, due to the vibration coupling effect among the components, the mode of a single flexible component in the system under the static boundary of cantilever, simple support and free is greatly different from the mode of the whole structure. The use of component-level modes may result in a reduction in the accuracy of the resulting displacement and velocity fields.
In conclusion, in the combined structure transverse vibration experiment or the actual engineering application, the problems of complex effective measurement and experiment equipment, high cost, low data result precision and the like of physical fields such as the transverse displacement, the speed and the like of the combined structure exist.
Disclosure of Invention
The invention solves the problems of complex equipment, high cost, low data result precision and the like of effective measurement and experiment on physical fields such as transverse displacement, speed and the like of a combined structure in experiment or actual engineering, and provides the method for determining the transverse displacement of the combined structure based on limited measuring points and global modes.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
the method for determining the transverse displacement of the combined structure based on the limited measuring points and the global mode comprises the following specific processes:
in the combined structure vibration experiment, m sensors are arranged on the surface or the lower surface of the structure for measuring the transverse displacement w (x, y, t) of the combined structure, and the laying positions of the m sensors are s n (x n ,y n ) Measuring the displacement w n And velocity
Figure BDA0003099368690000027
Wherein n =1 \ 8230m, x n Is the abscissa, y, of the sensor n Is the ordinate of the sensor;
for a given composite structure, the lateral displacement of any point on the kth structure can be expressed as
Figure BDA0003099368690000021
Wherein phi ik (x, y) is a global mode shape function satisfying a composite structure matching condition and a boundary condition, q i (t) is a number corresponding to phi i Generalized coordinates of (x, y);
displacement w measured by the sensor n And velocity
Figure BDA0003099368690000022
Can be expressed as:
Figure BDA0003099368690000023
m is a matrix formed by M-order global modal vectors in the front of the composite structure, and each-order global modal vector comprises the modes of M measuring points; it should be noted here that global modes which are equal to the number of the measurement points and are not related to each other need to be intercepted to ensure that M is a reversible square matrix; the expression of M is as follows:
Figure BDA0003099368690000024
at this time, q can be solved according to the above formula i And
Figure BDA0003099368690000025
about w n And
Figure BDA0003099368690000026
expression (2)
Figure BDA0003099368690000031
Further calculate any point (x) on the combined structure r ,y r ) Transverse displacement and velocity of
Figure BDA0003099368690000032
Compared with the prior art, the invention has the beneficial effects that: the device used by the invention is simple, low in cost, high in calculation efficiency and high in relative precision, the construction of the combined structure full displacement field can be realized through the transverse displacement data of a limited number of measuring points and the global mode of the combined structure, and the device has rich application prospect and practical significance.
Drawings
FIG. 1 is a schematic flow chart of the present invention;
FIG. 2 is a schematic view showing the arrangement of the hinge-connected multi-beam structure and the eddy current sensor in embodiment 1;
fig. 3 is a schematic view of the hinge-connected multi-plate structure and the arrangement of the eddy current sensor in embodiment 2.
Detailed Description
Specific embodiments of the present invention will be described in more detail below with reference to fig. 1 to 3. It should be noted, however, that the present invention may be embodied in various forms and should not be limited by the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.
The global mode refers to the mode of the combined structure containing the rigid body motion of the system and the elastic vibration of all flexible parts.
The first specific implementation way is as follows: the embodiment describes a method for determining the transverse displacement of a combined structure based on a limited measuring point and a global mode, which comprises the following specific processes:
in the composite structure vibration experiment, measurement of the transverse displacement w (x, y, t) of the composite structure is performed, wherein w (x, y, t) represents the transverse displacement on the composite structure at (x, y, z) at time t, w represents the transverse displacement, x and y are coordinates on the composite structure, and t represents time. The surface or the lower surface of the structure is provided with m sensors, and the laying positions of the m sensors are s n (x n ,y n ) Measuring the displacement w n And velocity
Figure BDA0003099368690000033
Wherein n =1 \8230m, x n Is the abscissa, y, of the position of the sensor n Is the ordinate of the position of the sensor;
for a given composite structure, the lateral displacement of any point on the kth structure can be expressed as
Figure BDA0003099368690000034
Wherein phi ik (x, y) is a global mode shape function satisfying a composite structure matching condition and a boundary condition, q i (t) is a radical corresponding to phi i Generalized coordinates of (x, y); for example, the following steps are carried out:
Figure BDA0003099368690000041
is represented by phi 1 (x,y)q 1 (t)+Φ 2 (x,y)q 2 (t)+Φ 3 (x,y)q 3 (t))。
Displacement w measured by a sensor n And velocity
Figure BDA0003099368690000042
Can be expressed as:
Figure BDA0003099368690000043
wherein, the derivative of the variable to time is represented by the upper band point of the variable, M is a matrix composed of M-order global modal vectors before the composite structure, and each-order global modal vector comprises the modes of M measuring points; the global mode of the composite structure is obtained by theoretical calculation, finite element simulation or mode test. It should be noted here that global modes which are equal to the number of the measured points and are not related to each other need to be intercepted to ensure that M is a reversible square matrix; the expression of M is as follows:
Figure BDA0003099368690000044
Φ i (x, y) represents the modal values of order i at the (x, y) position on the composite structure, so it can be seen that the above matrix is made up of the modal values of order n first at n measurement points on the panel. At this time, q can be solved according to the above formula i And
Figure BDA0003099368690000045
with respect to w n And
Figure BDA0003099368690000046
expression (2)
Figure BDA0003099368690000047
Further calculate any point (x) on the combined structure r ,y r ) Transverse displacement and velocity of
Figure BDA0003099368690000048
The second embodiment is as follows: in the method for determining the lateral displacement of the combined structure based on the finite measuring points and the global mode in the first embodiment, the sensors are uniformly arranged on the structure at equal intervals.
The third concrete implementation mode: in the method for determining the lateral displacement of the combined structure based on the limited measuring points and the global mode in the second embodiment, the arrangement mode of the sensors is improved according to the structural characteristics, and the node position of the structural mode is avoided.
Example 1:
the method is applied to a transverse vibration experiment of a multi-beam structure connected by a hinge, and is described by combining figure 1, and the method is specifically realized according to the following operations:
as shown in figure 2,3 eddy current sensors are arranged on the upper surfaces of the three beams connected by the hinge for measuring the transverse movement displacement and speed, and the 3 eddy current sensors are arranged at the positions
Figure BDA0003099368690000051
Measuring displacement
Figure BDA0003099368690000052
And velocity
Figure BDA0003099368690000053
Wherein n =1,2,3.
The lateral displacement of any point on the kth beam can be expressed as
Figure BDA0003099368690000054
Wherein phi ik (x)＝A k cos(β k x)+B k sin(β k x)+C k cosh(β k x)+D k sinh(β k x), k =1,2,3 is a global mode shape function of the multi-beam structure satisfying matching conditions and boundary conditions. A, B, C, D are constants determined by the boundary conditions of the beam, and β is a constant determined by the parameters of the beam;
the transverse displacement w of each measuring point on the combined beam measured by the eddy current sensor n And velocity
Figure BDA0003099368690000055
Can be expressed as:
Figure BDA0003099368690000056
wherein M is a matrix composed of front 3-order modal vectors of the beam structure, and each order modal vector comprises the modal of 3 measuring points. The mode of the beam can be obtained through theoretical calculation, finite element simulation or mode test. It should be noted here that the modes equal to the number of stations and independent of each other need to be intercepted to ensure that M is a reversible square matrix. The expression of M is as follows:
Figure BDA0003099368690000057
at this time, q can be solved according to the above formula i And
Figure BDA0003099368690000058
with respect to w k And
Figure BDA0003099368690000059
expression (2)
Figure BDA00030993686900000510
Further, any point s on the kth beam can be calculated and solved r (x r ) Transverse displacement and velocity of
Figure BDA0003099368690000061
Example 2:
the method is applied to a vibration experiment of a hinge connection multi-plate structure, and the embodiment is described by combining the figure 1, and the method is specifically realized according to the following operations:
for three plates connected by a hinge, 6 eddy current sensors are arranged on the (lower) surface of the plate to measure the lateral displacement and speed, and as shown in fig. 3, the 6 eddy current sensors are laid at the following positions: on the first plate
Figure BDA0003099368690000062
On the second plate
Figure BDA0003099368690000063
On the third plate
Figure BDA0003099368690000064
Measuring the displacement w n And velocity
Figure BDA0003099368690000065
Wherein n =1 \ 8230and 6.
The lateral displacement of any point on the kth plate can be expressed as
Figure BDA0003099368690000066
Wherein
Figure BDA0003099368690000067
The global mode shape function is a global mode shape function which satisfies the multi-plate matching condition and the boundary condition.
Transverse displacement w measured by eddy current sensor n And velocity
Figure BDA0003099368690000068
Can be expressed as:
Figure BDA0003099368690000069
wherein M is a matrix composed of modal vectors of first 6 orders of the structure, each order of the modal vectors including the modes of 6 measurement points. The global mode of the multi-plate can be obtained by theoretical calculation, finite element simulation or mode test. It should be noted here that global modes equal to the number of stations and independent of each other need to be intercepted to ensure that M is a reversible square matrix. The M expression is as follows:
Figure BDA0003099368690000071
at this time, q can be solved according to the above formula i And
Figure BDA0003099368690000072
with respect to w n And
Figure BDA0003099368690000073
expression (2)
Figure BDA0003099368690000074
Further, any point s on the k-th plate can be calculated r (x r ,y r ) Transverse displacement and velocity of
Figure BDA0003099368690000075
Although the invention has been described above with reference to specific embodiments thereof, which are intended to be illustrative and instructive only and not to be limiting, the invention is not limited to the two specific embodiments and the fields of application set forth above. Those skilled in the art, having the benefit of this disclosure, may effect numerous modifications thereto and changes may be made without departing from the scope of the invention as defined by the appended claims.

Claims (4)

1. The method for determining the transverse displacement of the combined structure based on the limited measuring points and the global mode is characterized by comprising the following steps of: the method comprises the following specific processes:
in the combined structure vibration experiment, m sensors are arranged on the surface or the lower surface of the structure for measuring the transverse displacement w (x, y, t) of the combined structure, and the laying positions of the m sensors are s n (x n ,y n ) Measuring the displacement w n And velocity
Figure FDA0003833242390000011
Wherein n =1 \ 8230m, x n Is the abscissa, y, of the sensor n Is the ordinate of the sensor;
for a given composite structure, the lateral displacement of any point on the kth structure is expressed as
Figure FDA0003833242390000012
For a multi-beam structure with n beams, the global mode shape function satisfying the multi-beam matching condition and the boundary condition is as follows
Φ ik (x)＝A k cos(β k x)+B k sin(β k x)+C k cosh(β k x)+D k sinh(β k x),k＝1,2,…,n (2)
A, B, C, D are constants determined by the boundary conditions of the beams, β is a constant determined by the parameters of the beams;
for a multi-plate structure with n plates, the global mode shape function satisfying the multi-plate matching condition and the boundary condition is
Figure FDA0003833242390000013
Wherein phi ik (x, y) is a global mode shape function satisfying a composite structure matching condition and a boundary condition, q i (t) is a radical corresponding to phi ik Generalized coordinates of (x, y);
displacement w measured by a sensor n And velocity
Figure FDA0003833242390000014
Expressed as:
Figure FDA0003833242390000015
m is a matrix formed by M-order global modal vectors in the front of the composite structure, and each-order global modal vector comprises the modes of M measuring points; it should be noted here that global modes which are equal to the number of the measured points and are not related to each other need to be intercepted to ensure that M is a reversible square matrix; the M expression is as follows:
Figure FDA0003833242390000016
at this time, q is solved according to the above formula (4) i And
Figure FDA0003833242390000017
with respect to w n And
Figure FDA0003833242390000018
expression (2)
Figure FDA0003833242390000021
Further calculate any point (x) on the combined structure r ,y r ) Transverse displacement and velocity of
Figure FDA0003833242390000022
2. The method for determining the lateral displacement of a combined structure based on finite measuring points and global modes according to claim 1, wherein: the sensor is not limited to a conventional sensor, and includes a non-contact sensor such as an eddy current sensor or a laser sensor.
3. The method for determining the lateral displacement of a combined structure based on finite measuring points and global modes according to claim 1, wherein: the sensors are arranged equidistantly and uniformly on the structure.
4. The method for determining the lateral displacement of a combined structure based on finite measuring points and global modes according to claim 1, wherein: and improving the arrangement mode of the sensors according to the structural characteristics, and avoiding the node position of the structural vibration mode.
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