CN113591233B - Split iteration method for solving dynamic equation of flexible beam system - Google Patents
Split iteration method for solving dynamic equation of flexible beam system Download PDFInfo
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Abstract
The invention relates to a split iteration method for solving a dynamic equation of a flexible beam system, and belongs to the field of dynamics of a multi-flexible beam system. The method has an inner layer and an outer layer of iterative processes. Firstly, in the outer layer iteration process, the generalized coordinates of the system are split into two parts, namely a master coordinate and a slave coordinate; secondly, in the inner layer iteration, expanding the slave coordinate iteration into a Taylor series of a main coordinate and Lagrange multiplier; thirdly, substituting the calculated Taylor series into a system equation, and solving a main coordinate and Lagrange multiplier; fourthly, obtaining slave coordinates; fifthly, judging whether the obtained result meets the precision requirement, if yes, ending the iteration, if not, correcting the obtained generalized coordinates, updating a system equation by using the corrected generalized coordinates, setting the corrected main coordinates and Lagrange multipliers as Taylor series expansion points of the next-round inner-layer iteration, and jumping to the next-round outer-layer iteration until a numerical solution meeting the precision requirement is obtained.
Description
Technical Field
The invention belongs to the field of dynamics of a multi-flexible system, and particularly relates to a method for solving a dynamic equation of a multi-section mutually-coupled flexible beam.
Background
The mutually coupled multi-section flexible beam system has wide application in the engineering field, for example, the working arm frames of cranes and aerial work platforms are flexible arm structures which are mutually sleeved or mutually hinged. In order to improve the safety and stability of hoisting equipment, especially high-no-load man operation equipment, it is necessary to perform dynamic modeling and simulation solving on a multi-section flexible beam system which is mutually coupled.
At present, commercial finite element software on the market cannot effectively simulate and solve the telescopic movement and the compound movement of a flexible beam system which are sleeved with each other. In the corresponding engineering application scene, the telescopic motion of the arm support is mostly analyzed and solved based on a variable length beam model, but the method requires to continuously reconstruct a quality matrix and a rigidity matrix in the dynamic simulation process, so that the calculated amount is increased. For pitching motion and slewing motion of the arm support, a floating coordinate (Floating Frame of Reference, FFR) method is generally adopted for modeling analysis, and elements of a mass matrix of a system obtained by the method are functions of generalized coordinates, so that solving is difficult. The D-H method (Denavit-Hartenberg Method) is combined with commercial finite element software to carry out related analysis by students, and the cost is that the coupling between the rigid motion and the elastic deformation of the arm support is ignored, and the solving precision is reduced. The modeling of the flexible beam systems which are sleeved with each other by utilizing the ANCF (Absolute Nodal Coordinate Formulation) method can obtain a constant mass matrix of the beam, but the method has the defects of more generalized coordinates and larger calculated amount during solving.
Disclosure of Invention
Aiming at the problems existing in the solution process of the dynamic equation of the flexible beam system, the invention provides a solution method, wherein part of generalized coordinates of the flexible beam system are expressed as Taylor series of the other part of generalized coordinates and Lagrange multipliers, so that the number of variables in the system equation is reduced, and the expansion points of the Taylor series are changed in an iterative mode, so that the accurate solution of the dynamic equation of the flexible beam system is continuously approximated.
The static equilibrium equation for a compliant beam system has the form:
Wherein e and K are respectively a generalized coordinate vector and a rigidity matrix of the flexible beam system; lambda is Lagrange multiplier vector; f is a generalized force vector, the elements of which are functions of e and lambda; pi is a constraint equation set related to e.
The kinetic equation for the flexible beam system has the form:
Wherein, Is the generalized acceleration vector of the system,/>The mass matrix of the system is M, and the damping matrix of the system is C. e,/>The elements of M, C, K, λ are all functions of time t. And ii is a constraint equation set related to the generalized coordinates e. The time variable t is discretized, and the equation (2) can be converted into the following static solution form by using a numerical integration method:
In equation set (3), Δt is the step size of the time iteration step, the matrix t+Δt K 'and vector t+Δt F' depend on the selected numerical integration method and the result t e of the last iteration step, The equation set (3) has the same form as the equation set (1), so the solving method is also the same. The following steps are as follows:
1. a split iterative method for solving a dynamic equation of a flexible beam system, comprising the steps of:
s1, modeling a system to obtain a static equilibrium equation (1) or a static solving form (3) of a dynamic equation of the system;
s2, dividing a generalized coordinate e of the system into a master coordinate e M and a slave coordinate e S;
S3, iteratively expanding the slave coordinates e S into Taylor series of a master coordinate e M and Lagrange multiplier lambda;
s4, substituting the Taylor series of the obtained slave coordinates e S about the master coordinates e M and Lagrange multiplier lambda into a static equilibrium equation (1) or a static solving form (3) of a dynamic equation, reducing the variable number, and then solving the approximate solution of the master coordinates e M and Lagrange multiplier lambda;
S5, substituting the main coordinate e M and Lagrange multiplier lambda obtained in the S4 into a Taylor series expression of the auxiliary coordinate e S about the main coordinate and Lagrange multiplier to obtain an approximate solution of the auxiliary coordinate e S;
S6, judging whether the obtained result meets the precision requirement, if so, ending iteration, and if not, jumping to S7;
s7, correcting the master coordinates and the slave coordinates obtained in the S4 and the S5;
S8, updating a rigidity matrix, a constraint equation and a generalized force of the system by using the generalized coordinates obtained in the S7;
S9, setting the main coordinates e M obtained in the S7 and the Lagrange multiplier lambda obtained in the S4 as expansion points of Taylor series required by the next iteration;
S10, jumping to S3;
2. In the step S1, the generalized coordinate e in the formula (1) is split into two parts: a portion is called primary coordinates e M; the other part is called the slave coordinate e S.
(1) Can be rewritten as
3. In the step S2, the equilibrium equation of the system can be written in an iterative form
The iteration coefficient matrices K 11 and K 22 of equations (4) and (5) must be reversible;
4. in the step S3, the slave coordinates in equation (5) in equation 1 are iteratively expanded into Taylor series of the master coordinates and Lagrange multipliers;
The method comprises the steps of expanding a coordinate e S into the Taylor series of a main coordinate e M and Lagrange multiplier lambda, substituting (5) the Taylor series to the right of the 1 st formula, expanding the substitution result into the Taylor series of the main coordinate e M and Lagrange multiplier lambda, and iterating until the coefficients of the calculated Taylor series are converged to meet the precision requirement. The iteration initial value of e S can be taken as the value of e S under the rigid configuration of the system;
5. Substituting the obtained Taylor series of the slave coordinate e S with respect to the master coordinates e M and Lagrange multiplier λ into the expression 2 and the expression 3 in the expression (5) to obtain:
Thereby reducing the number of variables, and then solving the main coordinates e M and Lagrange multiplier lambda by the method (7);
6. Substituting the obtained numerical values of the master coordinates e M and Lagrange multiplier lambda into the Taylor series expression of the slave coordinates e S about the master coordinates and Lagrange multiplier to obtain an approximate solution of the slave coordinates e S;
7. Judging whether the solved result meets the solving precision, if yes, finishing the solving, if not, substituting e M、eS and lambda obtained in the step 6 into the right of the first two formulas of (5) to obtain new e M and e S, and updating the rigidity matrix, the quality matrix, the constraint equation and the generalized force by using the result;
8. Taking the numerical value of the main coordinate e M obtained in the step 7 and the Lagrange multiplier lambda obtained in the step 6 as the expansion point of the Taylor series of the next iteration, namely continuously approaching to the true solution of the function by a method of transforming the expansion point of the Taylor series in the iteration;
9. in the solving process, the master coordinates and the slave coordinates are selected according to the change of the boundary conditions, so that the dynamics equation set of the system is ensured to be not ill when the boundary conditions change.
The invention has the beneficial effects that:
The method of the invention is a double iteration method, which comprises an inner layer iteration process and an outer layer iteration process. In the inner layer iteration process, part of generalized coordinates (from coordinates) in a dynamic equation of the flexible beam system are expanded into the Taylor series of another part of generalized coordinates (main coordinates) and Lagrange multipliers. In the outer layer iteration process, the approximate solution of the main coordinates and Lagrange multipliers is obtained, and the result is used as the expansion point of the Taylor series of the next round of inner layer iteration, namely, the real solution of the equation is continuously approximated by a method of transforming the expansion point of the Taylor series in the iteration process. The method can play a role in reducing the number of variables and simplifying equations. The generalized coordinates are split, and meanwhile, the coefficient matrix of the generalized coordinates is also split, so that the calculated amount of coefficient matrix inversion in the solving process is reduced. In the solving process, the master coordinates and the slave coordinates are selected according to the boundary condition change, so that a kinetic equation set of the system is free from illness, the numerical stability is improved, the repeated generation of grids in the solving process is avoided, the repeated generation of a quality matrix and a rigidity matrix of the system is avoided, and the solving efficiency is improved.
Drawings
Fig. 1 is a summary drawing.
Figure 2 is a simplified diagram of a work arm mechanism for a model aerial work platform.
Fig. 3 is a static solution flow chart.
Fig. 4 is a dynamic solution flow chart.
Fig. 5 shows the boundary conditions of the basic arm during the compound motion.
Fig. 6 is a boundary condition of the intermediate arm during the compound motion.
Fig. 7 is a boundary condition of the forearm during compound motion.
FIG. 8a shows the displacement of the three arm head joints in the direction of Or 1 during the combined lifting, pivoting and extending movement of the telescopic arm mechanism when the forearm head is loaded with 300kg of load.
FIG. 8b shows the displacement of the three arm head joints in the direction of Or 2 during the combined lifting, pivoting and extending movement of the telescopic arm mechanism when the forearm head is loaded with 300kg of load.
FIG. 8c illustrates the displacement of the three arm head joints in the direction of Or 3 during the combined lifting, pivoting, extending and compounding motion of the telescopic arm mechanism when the forearm head is loaded with 300kg of load.
FIG. 9a shows the speed of the three arm head joints in the direction of Or 1 during the combined lifting, pivoting, extending and compounding motion of the telescopic arm mechanism when the forearm head is loaded with 300kg of load.
FIG. 9b shows the speed of the three-joint arm head joint in the direction of Or 2 when the telescopic arm mechanism makes a lifting, swinging, extending and compounding motion when the forearm head is loaded with 300kg of load.
FIG. 9c shows the speed of the three-joint arm head joint in the direction of Or 3 when the telescopic arm mechanism makes a lifting, swinging, extending and compounding motion when the forearm head is loaded with 300kg of load.
FIG. 10a shows the acceleration of the three arm head joints from 5 seconds to 45 seconds in the direction of Or 1 when the telescopic arm mechanism makes a lifting, swinging, extending and compounding motion when the forearm head is loaded with 300kg of load.
Fig. 10b shows the acceleration of the three-arm head node in the direction Or 2 from the 5 th to 45 th seconds when the telescopic arm mechanism makes a lifting, swinging, extending and compounding motion when the forearm head is loaded with 300kg of load.
Fig. 10c shows the acceleration of the three-arm head node in the direction Or 3 from the 5 th to 45 th seconds when the telescopic arm mechanism makes a lifting, swinging, extending and compounding motion when the forearm head is loaded with 300kg of load.
In the figure: 1. basic arm, telescopic cylinder, 3, middle arm tail upper end slider, 4, forearm tail upper end slider, 5, basic arm head upper end slider, 6, middle arm, 7, middle arm head upper end slider, 8, forearm, 9, middle arm head extension pulley, 10, middle arm head lower end slider, 11, wire rope, 12, basic arm head lower end slider, 13, forearm tail lower end slider, 14, middle arm tail lower end slider, 15, luffing cylinder, 16, slewing platform, 17, middle arm tail side slider, 18, basic arm head side slider, 19, forearm tail side slider, 20.
The working arm is a three-section telescopic arm mechanism. The middle arm is sleeved in the basic arm, and the front arm is sleeved in the middle arm. The arms are mutually supported through the sliding blocks, and synchronous stretching movement among the three sections of arms is realized through the telescopic cylinder and the steel wire rope as driving. The back hinge point of the basic arm is hinged with the rotary platform, the front hinge point is hinged with one end of the amplitude changing cylinder, and the other end of the amplitude changing cylinder is hinged with the rotary platform.
Detailed Description
The present invention is further described below with reference to the following examples and the accompanying drawings, in which the working arm mechanism of a certain type of aerial work platform is modeled and solved by using the ANCF method and the generalized alpha method (the modeling method and the numerical integration method are selected only for illustrating the patent content, and the modeling method and the numerical integration method are not limited by the present patent).
The static solving method of the telescopic arm mechanism is as follows:
(1) According to the actual structure of the telescopic arm mechanism, defining the shape parameters and grid division of each section of arm;
(2) Establishing a local coordinate system and an interpolation function of each section of arm, and an interpolation matrix;
(3) Updating Euler angles, stiffness matrixes, generalized forces and static balance equations of all sections of arms in the telescopic arm mechanism;
(4) Dividing generalized coordinates e i (i=1, 2, 3) of the three-section arm into two parts of a master coordinate (e i)M (i=1, 2, 3) and a slave coordinate (e i)S (i=1, 2, 3), correspondingly decomposing a rigidity matrix K i and a generalized force array F i, and rewriting a static balance equation into an iterative form;
(5) In the inner layer iteration, the slave coordinates (e i)S (i=1, 2, 3) are iteratively expanded into the Taylor series of the master coordinates (e i)M (i=1, 2, 3) and Lagrange multiplier λ, and the initial value of the iteration of the expansion point of the master coordinates of each arm segment The value when each working arm is in a rigid configuration can be taken; the iteration initial value lambda [0] of Lagrange multiplier can be taken as zero vector;
(6) In the outer layer iteration, substituting Taylor series of the secondary coordinates with respect to the primary coordinates and Lagrange multipliers into a static balance equation set to obtain the primary coordinates and Lagrange multipliers lambda [n];
(7) Substituting the obtained master coordinates and Lagrange multipliers into Taylor series of the slave coordinates with respect to the master coordinates and Lagrange multipliers to obtain the slave coordinates;
(8) Judging whether the solved result meets the solving precision, if yes, finishing the solving, if not, substituting the main coordinates, the auxiliary coordinates and Lagrange multipliers obtained in the step (6) and the step (7) into the right of the iterative form of the static balance equation to obtain a new main coordinate And from the coordinates/>
(9) Principal coordinates obtained in the nth round of outer layer iterationThe expansion point of the Taylor series of the n+1st iteration is set as the expansion point of the Lagrange multiplier lambda [n], and the step (3) is skipped.
The dynamic solving method of the telescopic boom mechanism is as follows:
(1) Defining the size parameters and grid division of each section of arm according to the actual structure of the telescopic arm mechanism;
(2) Establishing a local coordinate system, an interpolation function, an interpolation matrix and a quality matrix of each section of arm by an ANCF method;
(3) The initial state of the telescopic arm mechanism is obtained by a static solving method;
(4) Defining dynamic solving parameters (parameters such as integral parameters and time step in a numerical integration method);
(5) Establishing a system dynamics equation according to an ANCF method and a numerical integration method;
(6) Adding a time step;
(7) Updating a coefficient matrix K' of Euler angles and generalized coordinates of the telescopic arm mechanism, and a constraint equation set pi and a generalized force array F;
(8) To ensure numerical stability, the principal coordinates must be selected according to a constraint equation set so that the principal coordinates appear in the constraint equation set, the constraint equation set is not ill-conditioned, and the iteration coefficient matrices of the principal and subordinate coordinates are reversible;
(9) Results obtained by iterating the outer layer of the nth round With lambda [n] set as the expansion point of Taylor series of the split iteration method, the iteration initial value/>, of the primary coordinateThe iteration initial value lambda [0] of Lagrange multiplier can be taken as zero vector;
(10) Will be from the coordinates in the inner layer iteration Iterative expansion into principal coordinates/>Taylor series with Lagrange multiplier lambda [n+1];
(11) Substituting the slave coordinates into a dynamics equation set with respect to the Taylor series of the master coordinates and Lagrange multipliers, and solving a master coordinate approximation solution and Lagrange multipliers lambda [n+1] for the next iteration in the outer layer iteration;
(12) Substituting the obtained main coordinate approximation solution and Lagrange multiplier lambda [n+1] into Taylor series of the auxiliary coordinate with respect to the main coordinate and Lagrange multiplier to obtain an approximation solution of the auxiliary coordinate;
(13) Judging whether the solving result meets the precision requirement, if so, jumping to the step (16), otherwise, jumping to the step (14);
(14) Substituting the approximate solution of the primary coordinate and the secondary coordinate obtained in the step (11) and Lagrange multiplier lambda [n+1] into the right side of the iterative form of the system equation to obtain the primary coordinate for the next iteration And slave coordinates
(15) Jumping to the step (7);
(16) Solving the speed and the acceleration according to the selected numerical integration method;
(17) And (5) checking whether the dynamic solution is completed, if yes, ending the dynamic solution, otherwise, jumping to the step (6).
The method will be described in detail with reference to the accompanying drawings, taking the lifting-turning-extending compound motion as an example.
In the lifting-turning-extending composite motion, the relative positions of all the telescopic arms are time-varying, and the relative positions of all the telescopic arms are space motion, and each generalized coordinate is a time-varying function in the directions of three coordinate axes. Meanwhile, in order to ensure the numerical stability in the solving process, the selection of the master (slave) coordinates is also complex. According to the rated setting of the model aerial working platform, the duration of the movement process is set to be 45 seconds, and 45-50 seconds are inertial oscillation time.
According to the steps (1) - (3) in the static solving method, modeling is carried out on a model of aerial work platform by using an ANCF method according to the actual shape and size of the mechanism. As shown in fig. 5-7, the global coordinate system is Or 1r2r3. And (5) meshing the sections of arms. Without losing generality, the three-section arm is provided with 4 grid nodes A 1,i,A2,i,A3,i,A4,i respectively; i=1, 2,3. The generalized coordinates of each arm segment are respectively as follows:
generalized coordinate vector of system is The rigidity matrix of each section of arm is K i, i=1, 2 and 3, the rigidity matrix of the system is K=diag (K 1,K2,K3), and the static balance equation of the telescopic arm mechanism in the horizontal initial state is obtained as follows:
Solving an initial state, and then carrying out dynamic modeling on a model of aerial work platform by combining a generalized alpha method (the numerical integration method is not limited in the patent), wherein the static solving form of a dynamic equation of the telescopic boom mechanism is as follows:
to solve equations (9) and (10), the master-slave coordinates of the system are selected.
As shown in fig. 5, O 2,O1 is located at the midpoint of the line connecting the front and rear hinge points of the base arm. The global coordinate system origin O is located at the intersection of the horizontal pivot axis of the base arm and the O 1O2 in the horizontal state. r Rot is parallel to the base arm tail hinge point line. Phi 2,1 is the horizontal pivot angle of the base arm.And/>The vectors pointing from O 1,O2 to a 1,1,A2,1, respectively. The sizes of the two are H. h 1 is the distance between the midpoint O 1 of the posterior hinge point join line in the horizontal plane and the origin O of the global coordinate system. Or 2 is the horizontal pivot axis of the telescopic arm mechanism. The following coordinate system of the basic arm is A 1,1x1y1z1. When the telescopic arm mechanism performs compound motion, the basic arm rotates around the coordinate axis Or 2, the rotation angle is phi 2,1, the angular speed is omega 2, the pitch angle is phi 3,1, and the angular speed is omega 3.
The cross section of the tail part of the basic arm is welded with a reinforcing rib and a reinforcing plate, and the basic arm has enough structural rigidity, so the following geometric constraint conditions are provided.
The distance from the tail node A 1,1 to O 1 of the basic arm is a fixed value H, which satisfies the following conditions
Vector at base arm tail node A 1,1 Parallel to vector/>Under three-dimensional conditions, there are the following constraints:
as a result of the fact that during the course of the movement, Is not zero, so in the above three formulas, optional e 1,1、e2,1、e3,1 is the primary coordinate.
During the compound movement, the section of A 1,1 has no shear deformation, namelyOrthogonal to vector/>
As a result of the fact that during the course of the movement,Since the value is not zero, e 5,1 is the primary coordinate.
During the compound motion, the cross section at the tail node A 1,1 of the base arm is free from torsional deformation and rigid rotation about the telescopic arm axis, i.e. vectorsParallel to unit vector r Rot, and orthogonal to vector/>I.e.
ObviouslyOptionally the primary coordinates of the base arm. In order to ensure that the principal coordinates appear in the constraint equation and that the equation does not appear in a pathological state, provision is made for:
at the base arm a 2,1 node, there are the following constraints:
Assuming that the cross section at the base arm a 2,1 node does not twist about the axis during the compound motion, so:
as described above, to ensure that the primary coordinates appear in the constraint equation and that the coefficients of the intermediate equation are not ill-conditioned, the primary coordinates may be selected according to the following rules:
The reinforced plate is welded at the joint of the basic arm A 2,1, so that the torsional deformation of the section of the A 2,1 in the movement process is restrained, and the following constraint conditions exist in the compound movement process:
Vector quantity Orthogonal to r Rot＝[-sin(φ2,1) 0 cos(φ2,1)]T:
as above, the primary coordinates are selected according to the following rules:
Distance of point O 2 to node A 2,1 on the axis of the base arm The constant value H:
In the process, take Is the primary coordinate of the base arm.
Vector quantityParallel to vector/>
During the course of the movement of the person,Is not zero, so get/>And/>As the primary coordinates of the base arm.
Vector at node A 2,1 Orthogonal to vector/>
e16,1·e19,1+e17,1·e20,1+e18,1·e21,1＝0 (22)
In the course of the movement of the robot,Component/>Is not zero, so get/>Is the primary coordinate of the base arm.
During compound motion, the vector at the A 2,1 nodeOrthogonal to vector/>Due to/>Parallel to r Rot＝[-sin(φ2,1) 0 cos(φ2,1)]T, there are:
e18,1·cos(φ2,1)-e16,1·sin(φ2,1)＝0 (23)
The primary coordinates are selected according to the following rules:
The reinforcing plate is welded on the head of the basic arm, so that the head structure of the basic arm has enough torsional rigidity, and therefore, at the head node A 4,1, the constraint condition is that:
The component forms are as follows:
e43,1·e46,1+e44,1·e47,1+e45,1·e48,1＝0 (26)
component/> Is not zero during exercise, so the formula above is preferable/>Is the primary coordinate of the base arm.
In summary, the primary coordinates of the base arm are selected as follows:
if cos 2(φ2,1)≥sin2(φ2,1), the primary coordinates of the base arm are:
e1,1,e2,1,e3,1,e5,1e9,1,e10,1,e11,1,e14,1,e15,1,e17,1,e18,1,e19,1,e21,1,e22,1,e23,1,e47,1;
If cos 2(φ2,1)＜sin2(φ2,1), the primary coordinates of the base arm are:
e1,1,e2,1,e3,1,e5,1,e7,1,e11,1,e12,1,e13,1,e14,1,e16,1,e17,1,e19,1,e21,1,e23,1,e24,1,e47,1.
As shown in fig. 6, the trailing coordinate system of the intermediate arm is a 1,2x2y222.L1 which is the base arm length, and L 2 which is the intermediate arm length. The tail node A 1,2 of the middle arm is overlapped with the P 1 point on the axis of the basic arm at the time t and is positioned in the 2 nd interpolation section of the basic arm, and the section number is as follows The head node A 4,1 of the basic arm is overlapped with the P 2 point on the axis of the middle arm and is positioned in the 2 nd interpolation interval of the middle arm, and the interval serial number/>The intermediate arm tail node a 1,2 is a distance l 1 (t) from the base arm tail node a 1,1.
The tail node A 1,2 of the middle arm and the section where the tail node A 1,2 is positioned have the following constraint conditions:
The tail node A 1,2 of the intermediate arm coincides with point P 1 on the base arm axis, i.e
E 1 is the generalized coordinates of the base arm,Is the first/>, corresponding to the basic armInterpolation matrix of each interpolation interval. In the above formula, e k,2 (k=1, 2, 3) can be the primary coordinates of the intermediate arm.
The bearing areas of the upper sliding block and the lower sliding block fixedly connected to the tail part of the middle arm are larger, the relative rotation of the section where the tail end node A 1,2 of the middle arm is located and the section where the point P 1 on the basic arm axis is located is restricted, and the side sliding blocks function, so that the following constraint conditions exist in the space compound motion:
vector of section where point A 1,2 is located Vector parallel to the cross-section taken by point P 1 on the base arm axis
During the course of the movement of the person,Is not zero, so it is advisable to take/>And/>As the primary coordinates of the intermediate arm.
Vector of section where A 1,2 is locatedVector/> orthogonal to the same section
e4,2·e7,2+e5,2·e8,2+e6,2·e9,2＝0 (29)
Due toIs not zero in the movement process, so the/>, can be selectedAs the primary coordinates of the intermediate arm.
Because the tail part of the middle arm is welded with other accessories such as a reinforcing plate, a telescopic cylinder and the like, the section of the A 1,2 is provided with stronger torsional rigidity, and no torsional deformation, namely vector, is generated in the space compound movement processCross-sectional vector/>, orthogonal to the point A 1,2
e10,2·e7,2+e11,2·e8,2+e12,2·e9,2＝0 (30)
Due toIs not zero during the compound movement, so in the above formula, the selection/>As the primary coordinates of the intermediate arm.
Also at time t, the distance between the head node A 4,1 of the basic arm and the tail of the middle arm is L 1-l1 (t), so that the interpolation interval sequence number and the interpolation matrix of the middle arm where A 4,1 is can be determinedLet A 4,1 be located at the first/>, of the intermediate armWithin the interpolation interval (in FIG. 6/>) Coinciding with the point P 2 on the axis of the intermediate arm, there are the following constraints at the overlap of the base arm head and the intermediate arm:
Point P 2 satisfies with forearm head node a 4,1:
e 2 is the generalized coordinates of the intermediate arm, Is the first/>, corresponding to the middle armInterpolation matrix of each interpolation interval. In the above, it is preferable that
Is the primary coordinate of the intermediate arm.
The bearing area of the lower slider fixedly connected to the head part of the basic arm is larger, and the action of the side slider restricts the winding vector of the section where P 2 is located and the section where the forearm node A 4,1 is located/>Is provided for the relative rotation of the two. There are the following constraints.
Vector of section where point P 2 is locatedVector/>, of the section where the A 4,1 point parallel to the base arm is located
As a result of the fact that during the course of the movement,Component/>Is not zero, and e 12k+7,2,e12k+9,2,/>, can be selected respectivelyAs the primary coordinates of the intermediate arm.
The middle arm head is also welded with a reinforcing plate with stronger torsional rigidity, soThe component forms are as follows:
e46,2·e43,2+e47,2·e44,2+e48,2·e45,2＝0 (33)
Because during the course of the movement the person, Since it is not zero, e 47,2 is the primary coordinate of the middle arm in the above equation.
Under the action of the reinforcing plate of the head of the basic arm, the head of the basic arm has enough structural rigidity, the shape of the section is kept unchanged, and the vector of the section where the P 2 point of the middle arm is positionedVector of section where the node A 4,1 of the head of the basic arm is locatedOrthogonalization:
Generalized coordinates of P 2 point Is/>And/>Is a linear combination of (a) and (b). In order to ensure that the primary coordinates appear in the constraint equation while avoiding the repeated selection of the same primary coordinate in (33) and (34), provision is made for:
in summary, the primary coordinates of the intermediate arm are selected as follows:
e1,2,e2,2,e3,2,e7,2,e9,2,e5,2,e11,2,e12k+j,2;j＝1,2,3,e12k+7,2,e12k+9,2,e47,2,
As shown in fig. 7, the forearm's follow-up coordinate system a 1,3x3y3z3.L3 is the forearm length. The forearm tail node A 1,3 is overlapped with the point Q 2 on the middle arm axis at the time t and is positioned in the 2 nd interpolation section of the middle arm, and the section number is as follows The middle arm head node A 4,2 coincides with the point Q 3 on the forearm axis and is located in the 2 nd interpolation interval of the forearm, interval serial number/>The distance from the forearm tail node a 1,3 to the mid-arm tail node a 1,2 is l 2 (t). The boundary conditions and the main degrees of freedom of the forearm are chosen similarly to those of the intermediate arm:
e1,3,e2,3,e3,3,e7,3,e9,3,e5,3,e11,3，j＝1，2，3，e12k+7,3；/>e12k+9,3；e47,3，/>
In the solving process, let t=0, l 1(t)|t＝0 and l 2(t)|t＝0 in the initial state can be obtained, and then the serial numbers of each interpolation interval in the initial state can be obtained, so that constraint conditions are established and primary coordinates are selected.
After the primary coordinates are selected, the initial state can be solved according to a static solving method or the dynamic equation can be solved according to a dynamic solving method, and the process is as described above. In the lifting-turning-extending composite motion, the displacement simulation result of the head node of each arm section is shown in fig. 8a-8 c; the results of the velocity simulation of the head nodes of each arm section are shown in fig. 9a-9 c; the acceleration simulation results at the head node of each arm at 5 th to 45 th seconds after ignoring the impact responses of start (0 th to 5 th seconds) and brake (45 th to 50 th seconds) are shown in fig. 10a-10 c. From simulation results, the method can effectively solve the composite motion of lifting motion, telescopic motion and rotary motion of the flexible beam system. Because the three basic movements of lifting, telescoping and rotating are special cases of compound movements, the method disclosed by the invention can be applied to simulation solution of the three basic movements.
It should be noted that the above embodiments are only preferred embodiments of the present invention for illustrating the technical solution of the present invention, and the present invention is not limited to the above embodiments. Any equivalent or obvious modification made under the guidance of the present specification falls within the substantial scope of the present specification and should be protected by the present invention.
Claims (7)
1. A split iterative method for solving a dynamic equation of a flexible beam system, comprising the steps of:
S1: dividing a generalized coordinate e of a flexible beam system into a main coordinate e M and a slave coordinate e S, wherein the dividing method needs to meet the requirement that an iteration coefficient matrix of e M and e S is a reversible matrix, and accordingly, a system equation is rewritten into an iteration form;
S2: iteratively expanding the slave coordinates e S into Taylor series of the master coordinates e M and Lagrange multiplier λ;
S3: substituting the Taylor series of the obtained slave coordinates e S about the master coordinates e M and Lagrange multiplier lambda into a system equation, and further solving an approximate solution of the master coordinates e M and Lagrange multiplier lambda;
S4: substituting the values of the master coordinates e M and the Lagrange multiplier lambda obtained in the step S3 into Taylor series expressions of the slave coordinates e S about the master coordinates e M and the Lagrange multiplier lambda to obtain an approximate solution of the slave coordinates e S;
S5: judging whether the obtained result meets the precision requirement, if so, ending iteration, and if not, jumping to S6;
S6: substituting the results of S3 and S4 into an iterative form of a system equation to obtain new generalized coordinates;
s7: updating Euler angles, stiffness matrixes, constraint equations and generalized forces by using generalized coordinates obtained in the step S6;
s7: taking the main coordinates e M obtained in the step S6 and Lagrange multiplier lambda obtained in the step S3 as expansion points of Taylor series of the next iteration;
s8: jumping to S2;
In the step S1, the generalized coordinates of the system are divided into two parts, namely a master coordinate and a slave coordinate;
The method comprises an inner layer iteration process and an outer layer iteration process;
In the step S2, that is, in the inner layer iteration process, the slave coordinates are iteratively expanded into Taylor series of the master coordinates and Lagrange multipliers;
In the outer layer iteration process, substituting the Taylor series of the slave coordinates obtained in the step S2 with respect to the main coordinates and Lagrange multipliers into a system equation;
In step S7, the true solution of the function is continuously approximated by a method of transforming the expansion points of the Taylor series during iteration.
2. A split iterative method for solving a dynamic equation of a flexible beam system according to claim 1, wherein: in the step S1, the static form of the static balance equation or the dynamic equation of the flexible beam system is rewritten into an iterative form.
3. A split iterative method for solving a dynamic equation of a flexible beam system according to claim 1 or 2, characterized in that: in the step S2, the slave coordinates are expanded into Taylor series of the master coordinates and Lagrange multiplier, and then the Taylor series is used for performing the next iteration until the coefficient of the Taylor series is converged to meet the precision requirement.
4. A split iterative method for solving a dynamic equation of a flexible beam system according to claim 3, wherein: in the outer layer iteration process, the system equation does not contain the slave coordinates e S any more, only contains the master coordinates e M and Lagrange multiplier lambda, reduces the variable number, and solves the master coordinates e M and Lagrange multiplier lambda.
5. A split iterative method for solving a dynamic equation of a flexible beam system according to claim 4, wherein: in the outer layer iterative process, the obtained values of the master coordinates e M and Lagrange multiplier λ are substituted into the Taylor series expression of the slave coordinate e S about the master coordinates e M and Lagrange multiplier λ, and an approximate solution of the slave coordinate e S is obtained.
6. A split iterative method for solving a dynamic equation of a flexible beam system according to claim 5, wherein: substituting the main coordinates e M, the slave coordinates e S and Lagrange multiplier lambda obtained in the step S3 and the step S4 into the right of the iterative form of the system equation to obtain new e M and e S, updating Euler angles, stiffness matrixes, constraint equations and generalized forces of the system equation based on the results, and re-splitting the generalized coordinates, stiffness matrixes and generalized forces of the system according to the updated constraint equations to obtain a new iterative equation.
7. A split iterative method for solving a dynamic equation of a flexible beam system according to claim 6, wherein: in the outer layer iteration process, a new round of inner layer iteration is started by taking the main coordinate e M obtained in the step S6 and the numerical value of Lagrange multiplier lambda obtained in the step S3 as the expansion point of the Taylor series of the next round of inner layer iteration.
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