CN112906212A - Bare electric power rope system modeling method based on absolute node coordinate method - Google Patents
Bare electric power rope system modeling method based on absolute node coordinate method Download PDFInfo
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Abstract
The invention relates to the technical field of spacecraft flight, in particular to a bare electric power rope system modeling method based on an absolute node coordinate method, which can accurately describe the dynamic behavior of a system. According to the method, firstly, a dynamic equation of a bare electrodynamic force rope unit is deduced by an absolute node coordinate method, then a dynamic equation of a satellite rigid body is deduced by a natural coordinate method, finally, constraint is introduced, all bare electrodynamic force rope units of a bare electrodynamic force rope are assembled to obtain a dynamic equation of the whole system, and therefore model construction of the system is completed. A dynamic model capable of describing the behavior of the space bare electric power rope system is constructed, and the model can accurately describe the inside and outside movement of the space bare electric power rope system, particularly the configuration change of a bare electric power rope.
Description
Technical Field
The invention relates to the technical field of spacecraft flight, in particular to a bare electric power rope system modeling method based on an absolute node coordinate method.
Background
The tethers typically used in space-powered tethered satellite systems are flexible, subject to bending and large deformations, which are typically not negligible in the kinematic modeling. The scholars mainly propose the following more general kinetic models: (1) a rigid rod model that does not consider tether mass; (2) a rigid rod model that takes into account tether mass; (3) a bead model; (4) a continuum model. The former two models are usually used for designing a control algorithm due to low degree of freedom, the latter two models are closer to the real situation and are usually used for analyzing the dynamic behavior of the system and the like, but the degree of freedom is large, and the solution is difficult. In the "dynamics control of tether satellite release and recovery" of Nanjing aerospace university Philippine thesis, a dimensionless dynamics equation of a common electric power tether satellite system is derived by using a Lagrange equation without considering tether quality, and an optimal control solution in the release process is further solved. In a journal 'applied method/statistical optimization for flexible thermal satellite systems' paper of Nonlinear Dynamics, a kinetic equation of a system is deduced by considering the mass of a tether but neglecting the vibration of the tether, the problem of rail mobility is researched, and a method for maintaining the stability of the system by controlling the change of the length of the tether is provided. The bead model is equivalent to numerous particles under the condition of neglecting the mass of the tether, and the flexibility of the tether can be effectively displayed. In the 'dynamics and control of flexible tether satellites in complex space environment' of the doctor paper of Nanjing aerospace university, a dynamic equation of a flexible tether is established by utilizing a Newton equation based on a bead model, the system dynamics behavior of the system in the complex space perturbation environment is calculated, and the influence of electric power on the system is further discussed. In the Journal of guiding, controlling, and Dynamics paper "Model estimation and code conversion for the systematic of electrical Dynamics system", a partial differential equation set is established based on a continuum Model to complete modeling of a common electrical tether, and a hypothetical modal method and a finite element method are respectively utilized to realize dispersion of a system equation and then solve, the result shows that the two methods have better effect under the condition of lower dispersion number of the common electrical tether, and the effect of the finite element method is better along with the increase of the dispersion value.
Modeling research in the prior art shows that the modeling research on the space tether is not complete, and an effective method for realizing accurate model construction on the space bare electric power rope does not exist at present.
Disclosure of Invention
The invention aims to provide a bare electric power rope system modeling method based on an absolute node coordinate method, and a dynamic model capable of describing the behavior of a space bare electric power rope system is constructed, and the model can accurately describe the inside and outside movement of the space bare electric power rope system, particularly the configuration change of a bare electric power rope.
The invention discloses a bare electric power rope system modeling method based on an absolute node coordinate method.
A bare electric power rope system modeling method based on an absolute node coordinate method specifically comprises the following steps:
step 1, constructing a coordinate system.
An inertial coordinate system OXYZ, wherein the OX axis points to the vernal equinox point, the OZ axis is perpendicular to the equatorial plane, the OY axis can be determined by a right-hand system, and the unit vector of each corresponding axis direction is respectively Ex、EyAnd Ez。
The track coordinate system oxyz, the oy axis points to the center of mass position of the main star from the earth center, the ox axis is opposite to the flying direction of the main star, the oz axis is determined by the right-hand system, and unit vectors in the three axis directions are respectively ex、eyAnd ez。
Body axis coordinate system C for consolidating main star and sub-star centroidsjxjyjzjWherein j is 1,2, CjxjShaft, CjyjShaft and CjzjRespectively coincide with the principal axes of the rotational inertia of the star, and the unit vectors in the three axial directions are respectively
And
and 2, deducing an unconstrained bare electric power rope unit kinetic equation by using an absolute node coordinate method by adopting a virtual work principle.
And 3, assembling all bare electric power rope units of the bare electric power rope to obtain a dynamic equation of the bare electric power rope.
And 4, under the condition that the main star and the bare electric power rope and the subsatellite and the bare electric power rope have constraints, deriving the dynamic equations of the rigid subsatellite and the rigid main star by using a natural coordinate method.
And 5, simultaneously establishing a dynamic equation of the bare electric power rope, a main star dynamic equation and a sub star dynamic equation to obtain a dynamic equation of the whole bare electric power rope system, thereby completing the model construction of the system.
Further, in the step 2, an unconstrained bare electric power rope unit kinetic equation is derived by adopting a virtual work principle and an absolute node coordinate method, and the method specifically comprises the following steps.
And 2.1, modeling the bare electric power rope unit based on an absolute node coordinate method described by any Lagrange-Euler.
The bare electric power rope unit is formed by uniformly and equally dividing a bare electric power rope into N sections, wherein each section is a bare electric power rope unit.
Under an inertial coordinate system OXYZ, two end points of a bare electric rope unit are used as two nodes, the two nodes comprise a first node and a second node, and the position vector r of the nodesiSlope vector ri,pAnd material coordinates p of the nodeiA broad coordinate vector q representing a bare electrical power cord unit, where i ═ 1 and 2.
Wherein
Refers to a node position vector riAnd a slope vector ri,pConstructed transposed vectorThe amount of the compound (A) is,
is the coordinate p of matteriConstructed translation vector r1 TIs the transposed vector of the position vector of the front end node of the bare electric power rope unit,
is the transposed vector of the slope vector of the first node of the bare electric power rope unit,
is the transposed vector of the position vector of the second node of the bare electric power cord unit,
is the transposed vector, p, of the slope vector of the second node of the bare electric power cord unit1And p2The arc length from the first node and the second node of the bare electric power rope unit to a reference point is defined as the connection point of the main star and the bare electric power rope.
Step 2.2, deriving an unconstrained bare electric power rope unit kinetic equation;
the bare electrical cord cell constraint equation can be expressed as:
Φp(qe,p1,p2,t)＝0 (8)
wherein q iseRefers to a node position vector riAnd a slope vector ri,pConstructed vector, p1And p2The arc length from the position of a first node and a second node of the bare electric power rope unit to a reference point is shown, and t is time;
the following equation of dynamics of the bare electrodynamic rope unit can be obtained according to the Laplace multiplier method:
wherein phip,qIs a Jacobian matrix of bare electrodynamic rope unit constraint equations, lambdaeIs bareLaplace multiplier vector, M, corresponding to electrodynamic rope uniteFor bare electrodynamic rope unit time-varying mass matrix, QiIs a bare electric power rope unit with an additional inertia force vector, QeIs a generalized elastic force, Q, of a bare electrodynamic rope unitgThe bare electric power rope unit is subjected to the earth generalized gravity,
representing the second derivative of the generalized coordinate vector q of the bare electric power cord unit with respect to time.
And 2.3, deducing expressions of all physical quantities in the bare electrodynamic force rope unit kinetic equation by adopting a virtual work principle.
According to the virtual work principle, the sum of virtual work made by the inertia force, the elastic force and the generalized external force of the bare electric power rope unit is zero, and the following components are available:
δW＝δWI+δWe+δWf＝0 (10)
wherein δ WIRepresents the virtual work, delta W, done by the inertia force of the bare electric rope uniteRepresents the virtual work, delta W, done by the elastic force of the bare electric power rope unitfThe virtual work of the bare electric power rope unit under the external load is shown.
In step 2.3, the virtual work principle is adopted to derive the expressions of all physical quantities in the bare electrodynamic force rope unit kinetic equation, and the specific steps are as follows:
step 2.3.1, virtual work done by the inertia force is deduced, and accordingly the additional inertia force vector Q of the bare electric power rope unit is deducediExpression of, bare electrodynamic rope unit time-varying mass matrix MeAnd a jacobian matrix of additional inertial forces.
Step 2.3.2, deducing virtual work done by the elastic force, thereby deducing the generalized elastic force Q of the bare electric power rope uniteThe expression of (2) is a bare electrodynamic rope unit elastic force jacobian matrix expression.
Step 2.3.3, deducing virtual work done by external force, and further deducing generalized gravity Q of the bare electric power rope unitgThe expression of (2) is a bare electrodynamic rope unit gravity jacobian matrix expression.
Further, in step 4, a dynamic equation of the rigid subsatellite and the rigid main star is derived by using a natural coordinate method, and the specific steps are as follows.
And 4.1, deriving a kinetic equation of the subsatellite under the unconstrained condition.
And 4.2, deducing a subsatellite kinetic equation with constraint conditions.
And 4.3, deriving a principal star kinetic equation with constraint conditions.
Further, in step 5, a kinetic equation of the bare electric power rope, a main star kinetic equation and a sub star kinetic equation are combined, and the kinetic equation of the whole bare electric power rope system can be obtained:
wherein M istIs a generalized mass matrix, Q, of all units of a bare electric power rope systemtIs the generalized force vector of a bare electric power rope system, phitRepresenting the constraint equation of a bare electrical power cord system,
constrained jacobian matrix, λ, representing bare electrical power cord systemtLaval multiplier vector, q, for bare electric power cord systemstIs a generalized coordinate vector of a bare electrodynamic roping system.
Has the advantages that: the bare electric power rope system modeling method based on the absolute node coordinate method can accurately describe the dynamic behavior of the system. The method comprises the steps of firstly deducing a dynamic equation of a bare electrodynamic force rope unit by using an absolute node coordinate method, then deducing a dynamic equation of a satellite rigid body by using a natural coordinate method, finally introducing constraint, assembling each bare electrodynamic force rope unit of the bare electrodynamic force rope to obtain the dynamic equation of the whole system, and thus completing the model construction of the system. A dynamic model capable of describing the behaviors of the space bare electric power rope system is constructed, the model can accurately describe the in-plane and out-plane movement of the space bare electric power rope system, particularly for the bare electric power rope system with current in nonlinear distribution, the model can accurately describe the external force born by each bare electric power rope unit, and therefore the configuration change of a tether is accurately calculated.
Drawings
Fig. 1 is a schematic diagram of a bare electrical power cord system of the present invention.
Fig. 2 is a schematic view of a bare electrical power cord unit of the present invention.
FIG. 3 is a schematic diagram of a rigid body star of the present invention.
FIG. 4 is a schematic diagram of the constraint between the rigid body subsatellite or main star and the bare electrodynamic rope unit.
Fig. 5 is a schematic view of the initial condition of the bare electric power cord system of the present invention.
FIG. 6 shows the track inclination icBare electrodynamic rope dynamics response plot at pi/4.
Fig. 6(a) is a graph of bare electrical power cord configuration change in x-y plane without initial perturbation.
Fig. 6(b) is an x-y in-plane angle diagram of a bare electrical cord system.
Fig. 6(c) is a graph of the change in bare electrical cord configuration without initial perturbation in the y-z plane.
Fig. 6(d) is an out-of-y-z surface angle diagram of a bare electrical cord system.
Fig. 6(e) is a three-dimensional view of a bare electrical cord configuration change.
Detailed Description
The invention discloses a bare electric power rope system modeling method based on an absolute node coordinate method, which specifically comprises the following steps:
step 1, constructing a coordinate system.
As shown in fig. 1, the bare electric power cord system includes a rigid main star M, a rigid subsatex S, and a bare electric power cord connecting the main star M and the subsatex S. The mass of the main star and the sub star respectively adopts mMAnd mSShowing that the track of the main star is an orbital plane, and the included angle between the orbital plane and the equatorial plane is an orbital inclination angle ic. Particularly assuming that the earth is a standard sphere, the following three coordinate reference systems are introduced to better establish the kinetic equations of the system.
(1) An inertial coordinate system OXYZ, wherein an OX axis points to the spring minute point,OZ axis is perpendicular to equatorial plane, OY axis can be determined by right-hand system, and unit vector of each axis direction is Ex、EyAnd EzNumerical simulations were performed under this coordinate system.
(2) The oxiz is an orbit coordinate system, the oy axis points to the centroid position of the main star from the earth center, the ox axis is opposite to the flying direction of the main star, the oz axis is determined by the right-hand system, and unit vectors in the three axis directions are respectively ex、eyAnd ezThe simulation results are all shown in the coordinate system.
(3) Body axis coordinate system C established on consolidated main satellite and sub satellite mass centersjxjyjzjWherein j is 1,2, CjxjShaft, CjyjShaft and CjzjRespectively coincide with the principal axes of the rotational inertia of the star, and the unit vectors in the three axial directions are respectively
And
and 2, deducing an unconstrained bare electric power rope unit kinetic equation by using an absolute node coordinate method by adopting a virtual work principle.
And 2.1, modeling the bare electric power rope unit based on an absolute node coordinate method described by any Lagrange-Euler.
As shown in fig. 2, the following assumptions are made for bare electrical power cord before modeling:
(1) the section of the bare electric power rope is always kept rigid and still kept vertical to the axis after deformation; the axis refers to the symmetrical axis of the length direction of the bare electric power rope unit.
(2) The bare electrical cord cross-section is a circular cross-section.
(3) The torsional stiffness of the bare electric cord and the corresponding moment of inertia are negligible.
Based on the above three assumptions, under an inertial coordinate system OXYZ, two ends of a bare electric power rope unit are adopted as two nodes, namely a first node and a second node, and the position vectors r of the nodesiSlope vector ri,pAnd material coordinates p of the nodeiA generalized coordinate vector q representing a bare electrical power cord unit, where i ═ 1 and 2.
Wherein the content of the first and second substances,
refers to a node position vector riAnd a slope vector ri,pThe constructed transposed vector is then translated into a vector,
is the coordinate p of matteriFormed transposed vector r1 TIs the transposed vector of the position vector of the front end node of the bare electric power rope unit,
is the transposed vector of the slope vector of the first node of the bare electric power rope unit,
is the transposed vector of the position vector of the second node of the bare electric power cord unit,
is the transposed vector, p, of the slope vector of the second node of the bare electrokinetic cable unit1And p2Is the arc length from the position of a first node and a second node of the bare electric power rope unit to a reference point, the reference point is the connection point of a main star and the bare electric power rope, and p1And p2Are all time-varying, and thus the substance coordinate p1And p2Variations of bare electrical cord units can be described. When in use
Indicating an increase in the bare electrical cord unit length, whereas it indicates a decrease in the bare electrical cord unit length, where "·" indicates a derivation over time t. The position vector of any point in the bare electric power cord unit can be expressed as
r＝Se(ζ,t)qe(t) (2)
In the formula qeRefers to a node position vector riAnd a slope vector ri,pConstructed vector, SeIs a shape function of a bare electrodynamic cord unit, writable as
Se＝[S1I3×3 S2I3×3 S3I3×3 S4I3×3] (3)
In the formula I3×3Is a third order identity matrix and the parameters
Le＝p2-p1Indicating the bare electrokinetic cord unit length. Therefore, the speed vector and the acceleration vector of any point on the bare electric rope unit can be expressed as
To facilitate the derivation of subsequent bare electrodynamic rope cell kinetic equations, the following simplifications are made
Here, the
Representing a shape function pair p1The partial derivatives are calculated and the subsequent expressions are processed in this way. Thus, equation (5) can be rewritten into
And 2.2, deriving an unconstrained bare electric power rope unit kinetic equation.
Consideration of bare electrodynamic rope unit constraint equation
Φp(qe,p1,p2,t)＝0 (8)
The following dynamic equation of the bare electrodynamic rope unit can be obtained according to the Laplace multiplier method
Wherein phip,qIs a Jacobian matrix of bare electrodynamic rope unit constraint equations, lambdaeIs the Laval multiplier vector, M, corresponding to the bare electric power rope uniteFor bare electrodynamic rope unit time-varying mass matrix, QiIs a bare electric power rope unit with an additional inertia force vector, QeIs a generalized elastic force, Q, of a bare electrodynamic rope unitgThe bare electric power rope unit is subjected to the earth generalized gravity.
And 2.3, deducing expressions of all physical quantities in the bare electric power rope unit kinetic equation by adopting a virtual work principle.
According to the virtual work principle, the sum of virtual work made by the inertia force, the elastic force and the generalized external force of the bare electric rope unit is zero, namely
δW＝δWI+δWe+δWf＝0 (10)
Wherein δ WIRepresents the virtual work, delta W, done by the inertia force of the bare electric rope uniteRepresents the virtual work, delta W, done by the elastic force of the bare electric power rope unitfRepresenting the virtual work performed by the bare electrical cord unit when subjected to an external load, each term is derived in detail below.
Step 2.3.1, deriving inertial forcesVirtual work, thereby deducing an additional inertia force vector Q of the bare electric rope unitiExpression of, bare electrodynamic rope unit time-varying mass matrix MeAnd a jacobian matrix of additional inertial forces.
For any bare electric power rope unit, the cross section area of the bare electric power rope unit is A, the density of the bare electric power rope is rho, and the virtual work done by the bare electric power rope unit can be obtained
Here note
Wherein QiRepresenting additional inertia force vectors, MeThe time varying mass matrix is for the bare electrical power cord unit.
In order to realize the subsequent kinetic equation solution, the jacobian matrix of the additional inertia force is deduced, and the same deduction is carried out when the elastic force and the external force are processed subsequently. For the second term in equation (10), the jacobian matrix from which the additional inertial force can be derived is
In the formula
Wherein
Step 2.3.2, deducing virtual work done by the elastic force, thereby deducing the generalized elastic force Q of the bare electric power rope uniteThe expression of (2) is a bare electrodynamic rope unit elastic force jacobian matrix expression.
Considering here the axial strain epsilon and the bending curvature kappa of the bare electrodynamic rope unit, the expressions are respectively as follows
The strain energy expression of the bare electrodynamic rope unit is
Wherein U is1Representing axial strain energy, U, of bare electrodynamic rope elements2Represents the bending strain energy of the bare electrodynamic cord unit, E represents the young's modulus of the bare electrodynamic cord unit, and J represents the section moment of inertia of the bare electrodynamic cord unit. The generalized coordinate vector Q is subjected to partial derivation by the formula (15), and the generalized elastic force Q of the bare electric rope unit can be obtainedeIs composed of
Wherein c represents the damping coefficient of a bare electrodynamic rope unit
The virtual work done by the elastic force of the bare electric rope unit
Similarly, according to equation (17), regardless of damping, the expression of the elastic force is written as
The elastic force of the above formula (19) is divided into two parts, a first part Qe1And a second part Qe2Wherein
For the first part Qe1The elastic force jacobian matrix can be written as
Wherein
For the second part Qe2The elastic force jacobian matrix can be written as
Wherein
Each term in the formula is expressed as follows
Step 2.3.3, deducing virtual work done by external force, and further deducing generalized gravity Q of the bare electric power rope unitgThe expression of (2) is a gravity jacobian matrix expression of the bare electric power rope unit.
Under the inertial coordinate system, if the bare electric power rope unit is under the action of the uniformly distributed load f, the virtual work done by the external force on the bare electric power rope unit can be expressed as
Generalized external force vector Q borne by bare electric power rope unitfCan be expressed as
The bare electric rope unit bears the generalized gravity Q of the earthgAn expression can be written as
Wherein
In the formula ofEThe Jacobian matrix representing the parameters of the earth's gravity, which means that the bare electric power rope unit is subjected to the generalized gravity, can be written as
In the formula (33), the reaction mixture,
wherein
In addition, assuming that the electromotive force applied to each bare electromotive force cord unit is uniformly distributed, the current magnitude of the kth bare electromotive force cord unit is set as IkThen the bare electric rope unit generalized electric power Q can be obtainedFThe expression is as follows:
where t iskDenotes the unit direction vector of the kth bare electric cord unit, BkThe magnetic field intensity at the orbit of the system can be expressed as
In the formula ofmV is the true paraxial angle of system operation, icThe angle of the system orbital plane to the equatorial plane. Can be derived as a generalized electrodynamic jacobian matrix of
Wherein
Wherein
Step 3, assembling each bare electric power rope unit of the bare electric power rope to obtain a dynamic equation of the bare electric power rope;
wherein M isEIs an overall mass matrix, Q, of all bare electrodynamic rope unitsEIs the integral additional inertial force vector, Q, of all bare electrodynamic rope unitsGIs the overall generalized elastic force vector, Q, of all bare electrodynamic rope unitsIIs the overall generalized gravity vector of all bare electric power cord units,
is an integral constraining matrix of all bare electrodynamic rope units,
jacobian matrix, λ, being an integral constraint matrix of all bare electrodynamic rope unitsEIs the Las multiplier vector, q, corresponding to all bare electric power rope units as a wholeEIs a generalized coordinate vector of all bare electrodynamic rope units, which can be specifically written as
And 4, under the condition that the main star and the bare electric power rope and the subsatellite and the bare electric power rope have constraints, deriving the dynamic equations of the rigid subsatellite and the rigid main star by using a natural coordinate method.
And 4.1, modeling rigid body main star and rigid body subsatellite dynamics, and deducing a dynamic equation of subsategories under an unconstrained condition.
The kinetic equation of the bare electric power rope unit is derived by using an absolute node coordinate method, but two satellite rigid bodies connected with the bare electric power rope cannot be derived by using the method, so that the kinetic equations of a main star and a sub star are derived by using a natural coordinate method. Under an inertial coordinate system, taking a subsatellite as an example, a centroid point and another fixed point on the subsatellite and two special non-coplanar unit vectors are selected as generalized coordinates of the subsatellite. As shown in FIG. 3, the generalized coordinate vector q of the subsatellitesCan be written as
In the formula, rcAnd rjRespectively as a secondary star centroid C2And the coordinate vector of the fixed point j on the subsatellite. u and v are two non-coplanar unit vectors, chosen here as coordinate axis C2y2And C2z2And the fixed point j is located at C2x2On the shaft. Thus, an expression of any point P on the subsatellite under an inertial coordinate system can be obtained
rP＝[XP YP ZP]T＝＝[(1-c1)I3 c1I3 c2I3 c3I3][rc rj u v]T＝Csqs (47)
In the formula c1、c2And c3Expressed as C in a body axis coordinate system2And j point position and unit vectors u and v
In the formula, the middle and upper horizontal line finger vector is in a body axis coordinate system C2x2y2z2The coordinates of the lower, matrix C for a fixed point P on the rigid bodysThe constant velocity and acceleration expression of any point on the subsatellite can be obtained
The position vector of any point on the subsatellite under the body axis coordinate system can be expressed by the formula (47)
Due to taking the point C2Is the centroid of the child star, j is the fixed point on the child star and is located at C2x2On the shaft, can be obtained
Wherein h issIs the height of the sub-star, from which the calculation can be made
According to the virtual work principle, the inertial force and the generalized external force applied to the subsatellite do virtual work on virtual displacement of 0, so that
δWF+δWI＝0 (53)
Wherein, δ WFAnd δ WIRespectively, the imaginary work of the sub-star under the external force and the inertial force, and the expression is as follows
Here QsAs vectors of external forces acting on the subsatellite, MsIs a rigid body mass matrix of subsategories and has the following expression
The kinetic equation of the subsatellite under the unconstrained condition can be obtained
If the subsatellite is not restrained and only acted by the gravity of the earth, the coordinate of the mass center is rs＝rcExternal force vector Q acting on the subsatellitesIs equal to the generalized gravity of the subsatellite and the external force vector Q of the subsatellitesThe expression of (a) is:
external force vector Q acting on subsatellitesThe corresponding Jacobian matrix is equal to the Jacobian matrix corresponding to the generalized gravity of the subsatellite, and the expression is as follows:
wherein
The dynamical equations of the subsatellite under the unconstrained condition are obtained based on the virtual work principle, but because the space rigid body only has six degrees of freedom, and the generalized coordinates of the system take 12, which means that the generalized coordinates are not independent of each other, the dynamical equations of the subsatellite under the unconstrained condition are storedWriting generalized coordinate vectors of subsategories into q in 6 constraint equations1-q12Then the constraint equation has the following relationship
And 4.2, deducing a subsatellite kinetic equation with constraint conditions.
In step 4.1, the unconstrained kinetic equation of the subsatellite is deduced, but in practice, the constraint between the subsatellite and the bare electrodynamic force rope exists besides the internal constraint, as shown in fig. 4, the subsatellite constraint equation and the bare electrodynamic force rope end are connected in a spherical hinge mode, and then the subsatellite constraint equation can be written as
Wherein r isNAnd rcRespectively is a coordinate vector of the tail end of the bare electrodynamic force rope and the centroid of the subsatellite under an inertial coordinate system, AC2OIs a conversion matrix from a sub-star axis coordinate system to an inertial coordinate system,
and the coordinate vectors of the subsatellite and the bare electric rope contact point under a body axis coordinate system are represented.
The subsatellite kinetic equation with constraint conditions can be obtained by means of the Laplace multiplier method
Wherein M issRigid body quality matrix, Q, representing subsategoriessRepresenting the vectors of external forces acting on the subsatellite,
is the Jacobian matrix, λ, of the subsatellite constraint matrixsIs the corresponding Las multiplier vector of the subsatellite.
And 4.3, deriving a principal star kinetic equation with constraint conditions.
Similar principal star kinetic equations can be written in the same form as the subsategories, and the principal star kinetic equations can be expressed as
Wherein M isMRigid body mass matrix, Q, representing the principal starMRepresenting the external force vector acting on the primary star,
jacobian matrix, λ, representing the principal star constraint matrixMIs the corresponding Las multiplier vector of the primary star.
And 5, simultaneously establishing a dynamic equation of the bare electrodynamic force rope, a main star dynamic equation and a subsatellite dynamic equation to obtain a dynamic equation of the whole rigid-flexible coupling system, namely the dynamic equation of the bare electrodynamic force rope system
Where M istIs a generalized mass matrix of all units of a bare electric power rope system, comprising a rigid mass matrix M of a main satelliteMRigid body mass matrix M of subsatellitesAnd an overall mass matrix M of all bare electrodynamic rope unitsE。
QtIs a generalized force vector of a bare electric power rope system, and comprises an integral additional inertia force vector Q of all bare electric power rope unitsEOverall generalized elastic force vector Q of all bare electrodynamic rope unitsGOverall generalized gravity vector Q for all bare electrodynamic rope unitsIExternal force vector Q acting on subsatellitesAnd the external force vector Q acting on the main starM。
ΦtThe constraint equation for representing the bare electric power rope system comprises a constraint equation of a subsatellite, a constraint equation of a main star and all bare electric power rope unitsIs integrated with the constraint matrix
λtThe Laval multiplier vector of the bare electric rope system comprises a Laval multiplier vector lambda corresponding to all bare electric rope units integrallyERayleigh multiplier vector lambda corresponding to subsatellitesLas multiplier vector lambda corresponding to the main starM。
qtGeneralized coordinate vectors for bare electrodynamic roping systems, comprising qEIs a generalized coordinate vector of all naked electric power rope units and a generalized coordinate vector q of subsatellitesAnd generalized coordinate vector q of the principal starM。
Fig. 5 shows a schematic diagram of initial conditions of a bare electric power rope system in a track coordinate system. While assuming an earth mean radius of RE6378km, the orbit of the bare electric power rope system is a circular orbit with the height being 300km away from the earth H, and other parameters of the bare electric power rope are shown in table 1.
TABLE 1 System parameters
Considering that a main satellite runs on a track plane, the initial moment of a sub-satellite is 1km below the main satellite, and the true approach point angle v of the initial moment of the bare electric power rope system is 0, and the result under the inertial system is transferred to the track system oxyz for observation. It can be found that when the bare electric power rope system is influenced by the electric power, the bare electric power rope system may be influenced by the out-of-plane force, so the system dynamic response should also consider the corresponding out-of-plane angle, and take the track inclination angle i without loss of generalitycPi/4, initial condition θ 0, the remaining parameters are consistent with table 3.1, and the resulting dynamics of the bare electrical rope system are shown in fig. 6.
Fig. 6 shows the result of the configuration change of the bare electric power rope, fig. 6(a) shows the configuration change of the bare electric power rope without initial disturbance in the x-y plane, fig. 6(c) shows the configuration change of the bare electric power rope without initial disturbance in the y-z plane, fig. 6(b) is an x-y in-plane angle diagram of the bare electric power rope system, fig. 6(d) is a y-z out-plane angle diagram of the bare electric power rope system and shows the time course of out-of-plane oscillation of the bare electric power rope system, and fig. 6(f) is a three-dimensional configuration change diagram of the bare electric power rope system, and when the track inclination angle is pi/4, it can be seen from fig. 6(d) and fig. 6(f) that the system generates out-of-plane oscillation except for the in-plane oscillation.
The above examples show that the absolute node coordinate method modeling provided by the patent can accurately describe the dynamic behavior of the bare electric power rope system.
Claims (6)
1. A bare electric power rope system modeling method based on an absolute node coordinate method is characterized by comprising the following steps of:
step 1, constructing a coordinate system;
an inertial coordinate system OXYZ, wherein the OX axis points to the vernal equinox point, the OZ axis is perpendicular to the equatorial plane, the OY axis can be determined by a right-hand system, and the unit vector of each corresponding axis direction is respectively Ex、EyAnd Ez；
Orbital coordinate systemThe axis of ox and oy points to the position of the mass center of the main star from the earth center, the flying direction of the axis of ox and the flying direction of the main star are opposite, the axis of ox and the flying direction of the main star are the same and are determined by a right-hand system, and unit vectors in the directions of the three axes are respectively ex、eyAnd ez；
Body axis coordinate system C for consolidating main star and sub-star centroidsjxjyjzjWherein j is 1,2, CjxjShaft, CjyjShaft and CjzjRespectively coincide with the principal axes of the rotational inertia of the star, and the unit vectors in the three axial directions are respectively
And
step 2, deducing an unconstrained bare electric power rope unit kinetic equation by using an absolute node coordinate method by adopting a virtual work principle;
step 3, assembling each bare electric power rope unit of the bare electric power rope to obtain a dynamic equation of the bare electric power rope;
step 4, deducing the dynamic equations of the rigid subsatellite and the rigid main star by using a natural coordinate method under the condition that the main star and the bare electric power rope and the subsatellite and the bare electric power rope have constraints;
and 5, simultaneously establishing a dynamic equation of the bare electric power rope, a main star dynamic equation and a sub star dynamic equation to obtain a dynamic equation of the whole bare electric power rope system, thereby completing the model construction of the system.
2. The modeling method of the bare electrical power rope system based on the absolute node coordinate method as claimed in claim 1, wherein in step 2, the virtual work principle is adopted, and the absolute node coordinate method is used to derive the unconstrained bare electrical power rope unit kinetic equation, which comprises the following steps;
step 2.1, modeling a bare electric power rope unit based on an absolute node coordinate method described by any Lagrange-Euler; the bare electric power rope unit is formed by uniformly and equally dividing a bare electric power rope into N sections, wherein each section is a bare electric power rope unit;
under an inertial coordinate system OXYZ, two end points of a bare electric rope unit are used as two nodes, the two nodes comprise a first node and a second node, and the position vector r of the nodesiSlope vector ri,pAnd material coordinates p of the nodeiA generalized coordinate vector q representing a bare electrical power cord unit, where i ═ 1 and 2;
wherein
Refers to a node position vector riAnd a slope vector ri,pThe constructed transposed vector is then translated into a vector,
is the coordinate p of matteriFormed transposed vector r1 TIs the transposed vector of the position vector of the front end node of the bare electric power rope unit,
is the transposed vector of the slope vector of the first node of the bare electric power rope unit,
is the transposed vector of the position vector of the second node of the bare electric power cord unit,
is the transposed vector, p, of the slope vector of the second node of the bare electric power cord unit1And p2The arc length from the first node and the second node of the bare electric power rope unit to a reference point is defined as the connection point of the main star and the bare electric power rope;
step 2.2, deriving an unconstrained bare electric power rope unit kinetic equation;
the bare electrical cord cell constraint equation can be expressed as:
Φp(qe,p1,p2,t)＝0 (8)
wherein q iseRefers to a node position vector riAnd a slope vector ri,pConstructed vector, p1And p2The arc length from the position of a first node and a second node of the bare electric power rope unit to a reference point is shown, and t is time;
the following dynamic equation of the bare electrodynamic rope unit can be obtained according to the Laplace multiplier method
Wherein phip,qIs a Jacobian matrix of bare electrodynamic rope unit constraint equations, lambdaeIs the Laval multiplier vector, M, corresponding to the bare electric power rope uniteFor bare electrodynamic rope unit time-varying mass matrix, QiIs a bare electric power rope unit with an additional inertia force vector, QeIs a generalized elastic force, Q, of a bare electrodynamic rope unitgThe bare electric power rope unit is subjected to the earth generalized gravity,
representing the second derivative of the generalized coordinate vector q of the bare electrodynamic rope unit with respect to time;
and 2.3, deducing expressions of all physical quantities in the bare electrodynamic force rope unit kinetic equation by adopting a virtual work principle.
3. The modeling method of the bare electrical power rope system based on the absolute node coordinate method as claimed in claim 2, wherein in step 3, the dynamic equation of the bare electrical power rope is obtained by assembling each bare electrical power rope unit of the bare electrical power rope:
wherein M isEIs an overall mass matrix, Q, of all bare electrodynamic rope unitsEIs the integral additional inertial force vector, Q, of all bare electrodynamic rope unitsGIs the overall generalized elastic force vector, Q, of all bare electrodynamic rope unitsIIs the overall generalized gravity vector of all bare electrodynamic rope units, phi (t, q)E) Is an integral constraining matrix of all bare electrodynamic rope units,
jacobian matrix, λ, being an integral constraint matrix of all bare electrodynamic rope unitsEIs the Laval multiplier vector, q, corresponding to all the bare electric power rope unitsEIs a generalized coordinate vector of all bare electrodynamic rope units, which can be specifically written as:
4. the modeling method of the bare electrical power rope system based on the absolute node coordinate method as claimed in claim 1, wherein in step 4, the equations of dynamics of the rigid subsatellite and the rigid main star are derived by a natural coordinate method, and the specific steps are as follows;
step 4.1, deducing a kinetic equation of the subsatellite under the unconstrained condition;
in an inertial coordinate system, the kinetic equation of the subsatellite under the unconstrained condition is as follows,
step 4.2, deducing a subsatellite kinetic equation with constraint conditions;
there is the restraint between subsatellite and the naked electric power rope, subsatellite and naked electric power rope terminal are connected with the ball pivot mode, then subsatellite restraint equation can write:
wherein r isNAnd rcRespectively are coordinate vectors of the tail end of the bare electrodynamic force rope and the centroid of the subsatellite under an inertial coordinate system,
is a conversion matrix from a subsatellite axis coordinate system to an inertial coordinate system,
representing the coordinate vectors of the subsatellite and the bare electric rope contact point under a body axis coordinate system;
obtaining a subsatellite kinetic equation with constraint conditions by means of a Laplace multiplier method:
wherein M issRigid body quality matrix, Q, representing subsategoriessRepresenting the vectors of external forces acting on the subsatellite,
is the Jacobian of the subsatellite constraint matrix, lambdasIs the Las multiplier vector corresponding to the subsatellite;
4.3, deducing a principal star kinetic equation with constraint conditions;
there is also constraint between main star and naked electric power rope, and similar main star kinetic equation can write into the same form with subsatellite, and main star kinetic equation can be expressed as:
wherein M isMRigid body mass matrix, Q, representing the principal starMRepresenting the external force vector acting on the primary star,
jacobian matrix, λ, representing the principal star constraint matrixMIs the corresponding Las multiplier vector of the primary star.
5. The modeling method of the bare electrical rope system based on the absolute node coordinate method as claimed in claim 1, wherein in step 5, the dynamical equation of the bare electrical rope, the dynamical equation of the main star and the dynamical equation of the subsatellite are combined to obtain the dynamical equation of the whole bare electrical rope system:
wherein M istIs a generalized mass matrix, Q, of all units of a bare electric power rope systemtIs the generalized force vector of a bare electric power rope system, phitRepresenting the constraint equation of a bare electrical power cord system,
constrained jacobian matrix, λ, representing bare electrical power cord systemtLaval multiplier vector, q, for bare electric power cord systemstIs a generalized coordinate vector of a bare electrodynamic roping system.
6. The modeling method of the bare electrical rope system based on the absolute node coordinate method as claimed in claim 3, wherein the sum of virtual work of the inertia force, the elastic force and the generalized external force of the bare electrical rope unit is zero according to the virtual work principle, and the method comprises:
δW＝δWI+δWe+δWf＝0 (10)
wherein δ WIRepresenting the virtual work, delta, done by the inertial force of the bare electrodynamic rope unitWeIndicating the virtual work, δ W, done by the elastic force of the bare electrodynamic rope unitfThe virtual work of the bare electric power rope unit under the external load is represented;
in step 2.3, the virtual work principle is adopted to derive the expressions of all physical quantities in the bare electrodynamic force rope unit kinetic equation, and the specific steps are as follows:
step 2.3.1, virtual work done by the inertia force is deduced, and accordingly the additional inertia force vector Q of the bare electric power rope unit is deducediExpression of (1), bare electrodynamic rope unit time-varying mass matrix MeAnd a jacobian matrix of additional inertial forces;
step 2.3.2, deducing virtual work done by the elastic force, thereby deducing the generalized elastic force Q of the bare electric power rope uniteThe expression of (2), the bare electrodynamic force rope unit elastic force jacobian matrix expression;
step 2.3.3, deducing virtual work done by external force, and further deducing generalized gravity Q of the bare electric power rope unitgThe expression of (2) is a bare electrodynamic rope unit gravity jacobian matrix expression.
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