CN113536595A - High-precision dynamics modeling method for spatial large-scale rigid-flexible coupling system - Google Patents

High-precision dynamics modeling method for spatial large-scale rigid-flexible coupling system Download PDF

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CN113536595A
CN113536595A CN202110915895.1A CN202110915895A CN113536595A CN 113536595 A CN113536595 A CN 113536595A CN 202110915895 A CN202110915895 A CN 202110915895A CN 113536595 A CN113536595 A CN 113536595A
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张帆
侯新宇
黄攀峰
张夷斋
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Northwestern Polytechnical University
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Abstract

The invention relates to a high-precision dynamics modeling method for a large-scale spatial rigid-flexible coupling system, and belongs to the technical field of spaceflight. The method comprises the steps of carrying out continuous displacement modal dispersion by an assumed modal method, determining a generalized coordinate and a generalized force, and finally modeling a system based on a Lagrange principle. The core of the modeling method is to provide a high-precision modeling method for the rigid-flexible coupling system based on the Lagrange principle on the basis of comprehensively considering the truss antenna structure. The model established by the invention comprises a truss equivalent model of the antenna parabolic cylinder structure, so that the accuracy of the truss antenna model is improved, and the defects of the existing truss modeling method are overcome.

Description

High-precision dynamics modeling method for spatial large-scale rigid-flexible coupling system
Technical Field
The invention belongs to the technical field of spaceflight, and particularly relates to a high-precision dynamics modeling method applied to a large-scale spatial rigid-flexible coupling system.
Background
At present, the space vehicles taking large-scale flexible structures as the leading space vehicles are increasing day by day, and along with the increase of the size of the flexible structures, the rigid-flexible coupling characteristics of the system are enhanced, and the traditional rigid body model is not applicable any more. The antenna is used as a spacecraft for communication, the parabolic cylinder connected with the support truss is a flexible structure main body, and the adjustment of the attitude angle of the antenna has important significance for system stability and communication capacity.
However, in the existing documents, such as "a class of long and thin truss type spacecraft dynamics characteristic research" and "a vibration suppression and attitude control method research for intelligent truss satellites", an antenna parabolic cylinder reflector structure model is not considered, and thus, only considering the truss structure is not comprehensive and inaccurate.
Disclosure of Invention
Technical problem to be solved
In order to avoid the defects of the prior art, on the basis of the prior invention, an antenna parabolic cylinder structure and a truss equivalent beam model are established, and a Lagrange principle is used for carrying out high-precision dynamic modeling on a large truss type antenna-satellite rigid-flexible coupling system.
Technical scheme
A high-precision dynamics modeling method for a spatial large-scale rigid-flexible coupling system is characterized by comprising the following steps:
step 1: antenna and truss equivalent model and parameter extraction
The total truss unit strain energy is expressed as:
Figure BDA0003205584570000011
wherein v, w and psi respectively correspond to the positions of all directions in the rod member coordinate system xyz;
the continuous beam strain energy is expressed as:
Figure BDA0003205584570000021
the total kinetic energy of the truss unit is expressed as:
Figure BDA0003205584570000022
the total kinetic energy of the continuous beam is expressed as:
Figure BDA0003205584570000023
based on the energy equivalence principle:
U1＝U2
T1＝T2
where EA represents axial tensile stiffness, EI represents bending stiffness, GJ represents torsional stiffness, l represents the length of the beam, Γ is the strain component on the beam neutral axis, D is the stiffness matrix, ρiShowing the density of the truss member, A showing the cross-sectional area of the truss unit member, Vx,1、Vy,1、Vz,1Three velocity components are shown at the end point of any rod,
Figure BDA0003205584570000024
the velocity component of any point on the neutral axis of the equivalent beam is G, which is a mass matrix;
the following parameters were extracted:
l-length of equivalent beam;
ρ -equivalent beam density;
a is equivalent beam cross-sectional area;
EI-bending stiffness;
step 2: establishing a coordinate system of a rigid-flexible coupling system
Establishing an orbit coordinate system OXYZ as an inertia coordinate system, Oxyz as a body coordinate system, theta as an internal surface angle, phi as an external surface angle, gamma as an attitude angle of a parabolic cylinder of the antenna, and P1Represents the position vector, mu, of any point in the body coordinate system when the beam is deformedpIs the vibrational displacement of the point;
and step 3: determining generalized coordinates and generalized forces
Modal dispersion of flexible body
Figure BDA0003205584570000031
Where q (t) sin (ω t + α), ω is the system coupling vibration frequency,
Figure BDA0003205584570000032
the flexible beam is in a rigid-flexible coupling vibration mode, and alpha is a vibration angle;
obtaining each order mode function according to an assumed mode method:
Figure BDA0003205584570000033
Figure BDA0003205584570000034
cosβil coshβil+1＝βil(sinβil coshβil-cosβil sinhβil)mt/ρfl
Figure BDA0003205584570000035
is composed of
Figure BDA0003205584570000036
In each order of mode, truncating the mode function to NtThe steps of the method are as follows,
Figure BDA0003205584570000037
is 1 x NtOrder vector mode function, Q(t)Is NtA column vector of 1; beta is aiRepresenting the vibration angle, p, of each orderfDenotes the hypothetical truss density, mtRepresenting clipping a Modal function to mtStep (2);
the generalized modal coordinates are expressed as:
Q(t)＝[q1,q2,...qn]T
wherein q isiA column vector representing generalized modal coordinates of the lateral vibration;
the lateral vibration of the equivalent beam can be decomposed into two directions, namely the y direction and the z direction shown in the figure, and then the two directional modal dispersion can be expressed as:
Figure BDA0003205584570000038
Figure BDA0003205584570000039
wherein the content of the first and second substances,
Figure BDA00032055845700000310
the mode of the y-direction is represented,
Figure BDA00032055845700000311
representing a z-direction mode;
the generalized force is expressed as:
Q＝[Q1,Q2,Q3,Q4,Q5]T
Q1,Q2,Q3,Q4,Q5respectively representing unknown generalized force column vectors;
and 4, step 4: system dynamics modeling based on Lagrange principle
Figure BDA0003205584570000041
Figure BDA0003205584570000042
Figure BDA0003205584570000043
K＝f(θ,φ,γ,Q(t))
Where eta is [ theta, phi, gamma, Q ═ Q(t)]TQ is the generalized force and P is the remainderAn expression;
and 5: formula derivation
Coordinates of any point position on the equivalent beam under the system are as follows:
p0＝[x,0,0]T
modal representation of system deformation displacement:
μp＝[0,wy,wz]T
the displacement r of any point on the equivalent beam is expressed as:
r＝ST(p0+μp)
wherein S is a rotation matrix;
modal dispersion of vibration:
Figure BDA0003205584570000044
Figure BDA0003205584570000045
the system kinetic energy is expressed as:
Figure BDA0003205584570000046
where ρ A is the product of the equivalent beam density and the cross-sectional area, m1And m2Respectively the satellite masses on both sides, vlAnd v-lRespectively the tail end speeds of the two sides of the beam;
the system potential energy is expressed as:
Figure BDA0003205584570000047
the kinetic and potential energy of the system are brought into the lagrange equation:
Figure BDA0003205584570000051
L＝T-U
η＝[θ,φ,β,Qy,Qz]T
Q＝[Q1,Q2,Q3,Q4,Q5]T
obtaining a kinetic equation:
Figure BDA0003205584570000052
advantageous effects
The equivalent model comprising the antenna parabolic cylinder reflector structure and the large-scale space truss provided by the invention is characterized in that continuous displacement modal dispersion is carried out by an assumed modal method, a generalized coordinate and a generalized force are determined, and finally, a system is modeled based on the Lagrange principle. The core of the modeling method is to provide a high-precision modeling method for the rigid-flexible coupling system based on the Lagrange principle on the basis of comprehensively considering the truss antenna structure. Compared with the prior art, the invention has the following beneficial effects:
(1) the invention designs a truss equivalent model comprising an antenna parabolic cylinder structure, improves the accuracy of the truss antenna model and makes up the defects of the existing truss modeling method;
(2) the high-precision equivalent dynamics modeling method provided by the invention utilizes the modal dispersion of the hypothesis modal method, reduces the degree of freedom of a large-scale flexible structure in space, improves the working efficiency, intuitively and clearly expresses the motion rule based on the rigid-flexible coupling system dynamic equation established by the Lagrange principle, and is convenient for the design of a subsequent controller.
Drawings
The drawings are only for purposes of illustrating particular embodiments and are not to be construed as limiting the invention, wherein like reference numerals are used to designate like parts throughout.
FIG. 1: an equivalent beam model schematic diagram;
FIG. 2: schematic diagram of coordinate system of "antenna-satellite".
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
The technical scheme adopted by the invention comprises the following steps:
the method comprises the following steps: antenna and truss equivalent model and parameter extraction
Step two: establishing a coordinate system of a rigid-flexible coupling system
Step three: determining generalized coordinates and generalized forces
Step four: system dynamics modeling based on Lagrange principle
Step five: formula derivation
In the first step, the equivalent structure of the antenna and the truss is as shown in the attached figure 1, and the equivalent specific steps are as follows:
(1) separating the periodic units of the truss;
(2) the strain energy of the periodic unit is equal to that of the continuous beam, and an equivalent stiffness matrix is obtained;
(3) the kinetic energy of the periodic unit is equal to that of the continuous beam, and an equivalent mass matrix is obtained;
the total truss unit strain energy is expressed as:
Figure BDA0003205584570000061
wherein v, w and psi respectively correspond to the positions of all directions in the rod member coordinate system xyz;
the continuous beam strain energy is expressed as:
Figure BDA0003205584570000062
the total kinetic energy of the truss unit is expressed as:
Figure BDA0003205584570000063
the total kinetic energy of the continuous beam is expressed as:
Figure BDA0003205584570000071
based on the energy equivalence principle:
U1＝U2
T1＝T2
where EA represents axial tensile stiffness, EI represents bending stiffness, GJ represents torsional stiffness, l represents the length of the beam, Γ is the strain component on the beam neutral axis, D is the stiffness matrix, ρiShowing the density of the truss member, A showing the cross-sectional area of the truss unit member, Vx,1、Vy,1、Vz,1Three velocity components are shown at the end point of any rod,
Figure BDA0003205584570000072
the velocity component of any point on the neutral axis of the equivalent beam is G, which is a mass matrix.
And comparing the calculation result based on the equivalent beam model with the calculation result of the finite element software, verifying the reliability of the equivalent beam, and further proving the reliability of the stiffness matrix and the quality matrix of the equivalent beam model.
The following parameters were extracted:
l-length of equivalent beam;
ρ -equivalent beam density;
a is equivalent beam cross-sectional area;
EI-bending stiffness.
In the second step, a rigid-flexible coupling system coordinate system is established as shown in fig. 2:
establishing an orbit coordinate system OXYZ as an inertia coordinate system, Oxyz as a body coordinate system, theta as an internal surface angle, phi as an external surface angle, gamma as an attitude angle of a parabolic cylinder of the antenna, and P1When the deformation of the beam is shown to occur,position vector, mu, of any point in the body coordinate systempIs the vibrational displacement of the spot.
In the third step, the flexible body mode is dispersed into
Figure BDA0003205584570000073
Where q (t) sin (ω t + α), ω is the system coupling vibration frequency,
Figure BDA0003205584570000074
the flexible beam is in a rigid-flexible coupling vibration mode, and alpha is a vibration angle;
obtaining each order mode function according to an assumed mode method:
Figure BDA0003205584570000081
Figure BDA0003205584570000082
cosβil coshβil+1＝βil(sinβil coshβil-cosβil sinhβil)mt/ρfl
Figure BDA0003205584570000083
is composed of
Figure BDA0003205584570000084
In each order of mode, truncating the mode function to NtThe steps of the method are as follows,
Figure BDA0003205584570000085
is 1 x NtOrder vector mode function, Q(t)Is NtColumn vector of x 1. Beta is aiRepresenting the vibration angle, p, of each orderfDenotes the hypothetical truss density, mtRepresenting clipping a Modal function to mtAnd (4) carrying out step.
The generalized modal coordinates are expressed as:
Q(t)＝[q1,q2,...qn]T
wherein q isiA column vector representing the generalized modal coordinates of the lateral vibration.
The lateral vibration of the equivalent beam can be decomposed into two directions, namely the y direction and the z direction shown in the figure, and then the two directional modal dispersion can be expressed as:
Figure BDA0003205584570000086
Figure BDA0003205584570000087
wherein the calculation method of each variable is the same as the steps.
Figure BDA0003205584570000088
The mode of the y-direction is represented,
Figure BDA0003205584570000089
representing the z-direction mode.
The generalized force is expressed as:
Q＝[Q1,Q2,Q3,Q4,Q5]T
Q1,Q2,Q3,Q4,Q5respectively, representing an unknown generalized force column vector.
In the fourth step, in the 'antenna-satellite' system, the satellite rigidity is far greater than that of the antenna accessory, the satellite is regarded as a rigid body, the satellite can be regarded as an equivalent beam according to the low-order modal characteristic of the antenna, the parameters are obtained in the second step, and the simplified model of the system is shown in the attached figure 2.
The system dynamics model is expressed as:
Figure BDA00032055845700000810
Figure BDA0003205584570000091
Figure BDA0003205584570000092
K＝f(θ,φ,γ,Q(t))
where eta is [ theta, phi, gamma, Q ═ Q(t)]TQ is a generalized force, and P is a residual expression;
in step 5), the formula is derived as follows:
coordinates of any point position on the equivalent beam under the system are as follows:
p0＝[x,0,0]T
modal representation of system deformation displacement:
μp＝[0,wy,wz]T
the displacement r of any point on the equivalent beam is expressed as:
r＝ST(p0+μp)
where S is a rotation matrix.
Modal dispersion of vibration:
Figure BDA0003205584570000093
Figure BDA0003205584570000094
the system kinetic energy is expressed as:
Figure BDA0003205584570000095
where ρ A is the product of the equivalent beam density and the cross-sectional area, m1And m2Respectively the satellite masses on both sides, vlAnd v-lRespectively the terminal velocities at both sides of the beam.
The system potential energy is expressed as:
Figure BDA0003205584570000096
the kinetic and potential energy of the system are brought into the lagrange equation:
Figure BDA0003205584570000097
L＝T-U
η＝[θ,φ,γ,Qy,Qz]T
Q＝[Q1,Q2,Q3,Q4,Q5]T
obtaining a kinetic equation:
Figure BDA0003205584570000101
while the invention has been described with reference to specific embodiments, the invention is not limited thereto, and various equivalent modifications or substitutions can be easily made by those skilled in the art within the technical scope of the present disclosure.

Claims (1)

1. A high-precision dynamics modeling method for a spatial large-scale rigid-flexible coupling system is characterized by comprising the following steps:
step 1: antenna and truss equivalent model and parameter extraction
The total truss unit strain energy is expressed as:
Figure FDA0003205584560000011
wherein v, w and psi respectively correspond to the positions of all directions in the rod piece coordinate system xyz;
the continuous beam strain energy is expressed as:
Figure FDA0003205584560000012
the total kinetic energy of the truss unit is expressed as:
Figure FDA0003205584560000013
the total kinetic energy of the continuous beam is expressed as:
Figure FDA0003205584560000014
based on the energy equivalence principle:
U1＝U2
T1＝T2
where EA represents axial tensile stiffness, EI represents bending stiffness, GJ represents torsional stiffness, l represents the length of the beam, Γ is the strain component on the beam neutral axis, D is the stiffness matrix, ρiShowing the density of the truss member, A showing the cross-sectional area of the truss unit member, Vx,1、Vy,1、Vz,1Three velocity components are shown at the end point of any rod,
Figure FDA0003205584560000015
the velocity component of any point on the neutral axis of the equivalent beam is G, which is a mass matrix;
the following parameters were extracted:
l-length of equivalent beam;
ρ -equivalent beam density;
a is equivalent beam cross-sectional area;
EI-bending stiffness;
step 2: establishing a coordinate system of a rigid-flexible coupling system
Establishing an orbit coordinate system OXYZ as an inertia coordinate system, Oxyz as a body coordinate system, theta as an internal surface angle, phi as an external surface angle, gamma as an attitude angle of a parabolic cylinder of the antenna, and P1Represents the position vector, mu, of any point in the body coordinate system when the beam is deformedpIs the vibrational displacement of the point;
and step 3: determining generalized coordinates and generalized forces
Modal dispersion of flexible body
Figure FDA0003205584560000021
Where q (t) sin (ω t + α), ω is the system coupling vibration frequency,
Figure FDA0003205584560000022
the flexible beam is in a rigid-flexible coupling vibration mode, and alpha is a vibration angle;
obtaining each order mode function according to an assumed mode method:
Figure FDA0003205584560000023
Figure FDA0003205584560000024
cosβilcoshβil+1＝βil(sinβilcoshβil-cosβilsinhβil)mt/ρfl
Figure FDA0003205584560000025
is composed of
Figure FDA0003205584560000026
In each order of mode, truncating the mode function to NtThe steps of the method are as follows,
Figure FDA0003205584560000027
is 1 x NtOrder vector mode function, Q(t)Is NtA column vector of 1; beta is aiRepresenting the vibration angle, p, of each orderfDenotes the hypothetical truss density, mtRepresenting clipping a Modal function to mtStep (2);
the generalized modal coordinates are expressed as:
Q(t)＝[q1,q2,...qn]T
wherein q isiA column vector representing generalized modal coordinates of the lateral vibration;
the lateral vibration of the equivalent beam can be decomposed into two directions, namely the y direction and the z direction shown in the figure, and then the two directional modal dispersion can be expressed as:
Figure FDA0003205584560000031
Figure FDA0003205584560000032
wherein the content of the first and second substances,
Figure FDA0003205584560000033
the mode of the y-direction is represented,
Figure FDA0003205584560000034
representing a z-direction mode;
the generalized force is expressed as:
Q＝[Q1,Q2,Q3,Q4,Q5]T
Q1,Q2,Q3,Q4,Q5respectively representing unknown generalized force column vectors;
and 4, step 4: system dynamics modeling based on Lagrange principle
Figure FDA0003205584560000035
Figure FDA0003205584560000036
Figure FDA0003205584560000037
K＝f(θ,φ,γ,Q(t))
Where eta is [ theta, phi, gamma, Q ═ Q(t)]TQ is a generalized force, and P is a residual expression;
and 5: formula derivation
Coordinates of any point position on the equivalent beam under the system are as follows:
p0＝[x,0,0]T
modal representation of system deformation displacement:
μp＝[0,wy,wz]T
the displacement r of any point on the equivalent beam is expressed as:
r＝ST(p0+μp)
wherein S is a rotation matrix;
modal dispersion of vibration:
Figure FDA0003205584560000038
Figure FDA0003205584560000039
the system kinetic energy is expressed as:
Figure FDA0003205584560000041
where ρ A is the product of the equivalent beam density and the cross-sectional area, m1And m2Respectively the satellite masses on both sides, vlAnd v-lRespectively the tail end speeds of the two sides of the beam;
the system potential energy is expressed as:
Figure FDA0003205584560000042
the kinetic and potential energy of the system are brought into the lagrange equation:
Figure FDA0003205584560000043
L＝T-U
η＝[θ,φ,γ,Qy,Qz]T
Q＝[Q1,Q2,Q3,Q4,Q5]T
obtaining a kinetic equation:
Figure FDA0003205584560000044
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CN117131704A (en) * 2023-09-22 2023-11-28 中山大学 Coupling dynamics modeling method and system for robot on-orbit assembly

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CN114237055A (en) * 2021-12-19 2022-03-25 西北工业大学 Multi-order modal active vibration suppression method for extremely-large space rigid-flexible coupling system
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