CN111832128A - Numerical value-analytic hybrid optimization transient response algorithm for segmented system - Google Patents

Numerical value-analytic hybrid optimization transient response algorithm for segmented system Download PDF

Info

Publication number
CN111832128A
CN111832128A CN202010587348.0A CN202010587348A CN111832128A CN 111832128 A CN111832128 A CN 111832128A CN 202010587348 A CN202010587348 A CN 202010587348A CN 111832128 A CN111832128 A CN 111832128A
Authority
CN
China
Prior art keywords
state
displacement
switching
time
point
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN202010587348.0A
Other languages
Chinese (zh)
Inventor
翁建生
许志远
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Nanjing University of Aeronautics and Astronautics
Original Assignee
Nanjing University of Aeronautics and Astronautics
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Nanjing University of Aeronautics and Astronautics filed Critical Nanjing University of Aeronautics and Astronautics
Priority to CN202010587348.0A priority Critical patent/CN111832128A/en
Publication of CN111832128A publication Critical patent/CN111832128A/en
Pending legal-status Critical Current

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/10—Numerical modelling
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Geometry (AREA)
  • General Physics & Mathematics (AREA)
  • Evolutionary Computation (AREA)
  • General Engineering & Computer Science (AREA)
  • Computer Hardware Design (AREA)
  • Computational Mathematics (AREA)
  • Mathematical Analysis (AREA)
  • Mathematical Optimization (AREA)
  • Pure & Applied Mathematics (AREA)
  • Complex Calculations (AREA)

Abstract

The invention discloses a numerical value-analytic mixed optimization transient response algorithm for a segmented system. The algorithm of the invention is not only suitable for the linear composition of the segmented system, but also suitable for the linear and nonlinear composition of the complex segmented system, and has better applicability to different inputs and different systems. Particularly, the algorithm of the invention analyzes the switching among different states in detail, and identifies and selects corresponding methods for different states; optimizing the method for searching and accurately converting the switching points; and error points generated during state switching are eliminated, so that the calculation efficiency and the calculation precision of the algorithm are improved.

Description

Numerical value-analytic hybrid optimization transient response algorithm for segmented system
Technical Field
The invention belongs to the technical field of mechanical vibration, and particularly relates to a numerical-analytic hybrid optimization transient response algorithm for a segmented system.
Background
For a linear vibration system, the vibration response can be expressed by an accurate analytic solution, for a nonlinear system, analytic means such as a perturbation method, an averaging method, a progressive method, a multi-scale method and a harmonic balance method are successively provided, and numerical algorithms such as a single-step method and a multi-step method, for example, a Runge-Kutta algorithm, are also provided. In particular, in the case of a piecewise nonlinear system, researchers have proposed means such as a seam method and an averaging method to solve the nonlinear system, for example, an analytical solution for solving the nonlinear system composed of linear systems by using a harmonic balance method.
However, for solving the actually existing segmented system formed by mixing a linear system and a nonlinear system, the analytical method is too complex and inconvenient for solving the vibration response, and the numerical method has the condition that the efficiency and the precision cannot be considered at the same time, for example, Runge-Kutta can generate inevitable error points when solving the segmented system, and the calculation time is greatly increased when the precision is improved.
The invention discloses a numerical value-analytic mixed optimization transient response algorithm aiming at a segmented system by combining vibration related knowledge and a computer technology. The algorithm of the invention is not only suitable for the linear composition of the segmented system, but also suitable for the linear and nonlinear composition of the complex segmented system, and has better applicability to different inputs and different systems. Particularly, the algorithm of the invention analyzes the switching among different states in detail, and identifies and selects corresponding methods for different states; optimizing the method for searching and accurately converting the switching points; and error points generated during state switching are eliminated, so that the calculation efficiency and the calculation precision of the algorithm are improved.
For convenience, the algorithm is abbreviated as AMS.
Disclosure of Invention
The invention provides a numerical-analytic hybrid optimization transient response algorithm for a segmented system, which aims to solve the problems in the prior art.
In order to achieve the purpose, the invention adopts the following technical scheme:
a numerical-analytic hybrid optimization transient response algorithm for a segmented system, comprising the steps of:
step 1, establishing a vibration system motion differential equation set:
Figure BDA0002554277250000011
the formula (1) represents a generalized motion differential equation set, the form of the generalized motion differential equation set is determined by a vibration system, and the generalized motion differential equation set is a single-degree-of-freedom system, a multi-degree-of-freedom system (limited to linear parts and capable of being decoupled), a continuous system or a segmented system (segments formed by linearity and segments formed by linearity-nonlinearity);
in equation (1): m is a generalized quality matrix; c is a generalized damping matrix; k is a generalized stiffness matrix; f is a generalized excitation force matrix; t represents a time constant; x is the solution of formula (1) and is a system displacement vector;
Figure BDA0002554277250000012
representing a velocity vector, derived by X over t;
Figure BDA0002554277250000013
as an acceleration vector, composed of
Figure BDA0002554277250000014
Deriving t;
step 2, determining the number of states contained in the system according to the generalized motion differential equation set, and judging whether the system is segmented or not and judging the linear condition of each segment in the segmentation;
step 3, establishing a state corresponding condition and establishing a state changing condition
According to step 2, establishing the linear state and the nonlinear state of the selected system, the displacement vector X and the initial condition X0
Figure BDA0002554277250000015
Establishing a corresponding relationship between the state change and the displacement vector X, and establishing an initial condition X0
Figure BDA0002554277250000016
The corresponding relationship of (a);in this case, equation (1) is rewritten in the case of segmentation as:
Figure BDA0002554277250000021
Figure BDA0002554277250000022
wherein: mlAn equation quality matrix in a linear state; clA damping matrix in a linear state; klA stiffness matrix in a linear state; flAn exciting force matrix in a linear state; mnAn equation quality matrix in a nonlinear state; cnA damping matrix in a nonlinear state; knA rigidity matrix in a nonlinear state; fnAn exciting force matrix in a nonlinear state; o islThe displacement speed set in a linear state; o isnThe displacement speed set in a nonlinear state is obtained; x0
Figure BDA0002554277250000023
Respectively representing an initial displacement vector and an initial velocity vector;
step 4, presetting initial conditions; presetting an initial switching point; presetting a solution formula;
presetting initial conditions, and giving X0
Figure BDA0002554277250000024
Assigning; the time constant t is 0; presetting a calculation time step de;
the initial switching point is preset as t _ o; t _ n; wherein: t _ o; t _ n respectively represents the time of the last state change and the time of the next state change in the current state, and the initial values are all 0;
for the determined linear state portion, the displacement vector and velocity vector in equation (1) are expressed as:
Figure BDA0002554277250000025
Figure BDA0002554277250000026
wherein: u shapel(t)、Vl(t) respectively representing a free vibration matrix of the system caused by the initial displacement of each degree unit and a free vibration matrix caused by the initial speed of each toxic unit; τ is an integral variable; h (t-tau) is a system unit impulse response matrix taking t-tau as a variable;
Figure BDA0002554277250000027
representing the derivation of t.
For the determined part of the nonlinear state, the Runge-Kutta algorithm of order 4 is chosen:
Figure BDA0002554277250000028
wherein: r1、R2、R3、R4Respectively a first coefficient, a second coefficient, a third coefficient and a fourth coefficient of the Runge-Kutta algorithm; t is tiThe ith time of the Runge-Kutta algorithm; the function of f () is a first order differential equation system for reducing order change of formula (3); xiIs tiA value vector consisting of degree displacement and velocity; xi+1Is ti+1A value vector consisting of degree displacement and velocity;
step 5, judging the state
Determining the state of the system at the moment t according to the definition of the step 3;
step 6, selecting a calculation method to calculate the relevant numerical value corresponding to the current time
According to the judgment of the steps 2 and 3, when the system is in a linear part, calculating by using the formulas (4) and (5); when the part is non-linear, the calculation is carried out by using the formula (6);
step 7, judging whether state change occurs or not
Judging whether the state is changed according to the definition of the step 3, and calculating whether the state is changed or not by using a formula (6) each time the initial condition needs to be updated, X0＝Xi+1；
Step 8, searching state switching points
Step 81, when the state switching is from linear state switching, using Fzeros () for reconstructing and optimizing an MATLAB self-contained program fzero () as a switching point searching method, generating a multi-point problem for long-time calculation due to the characteristics of an fzero function to cause non-negligible errors, optimizing and adapting based on the program to accurately search for points and simultaneously solve a plurality of equations to search for the state switching point; the optimization steps are: mixing Xa(ta-t _ o) by tf＝ta-t _ o is rewritten as Xf(tf) Using a loop, solving for X separately by fzero ()f(tf) And returning a vector at a crossing point of each function element near the corresponding element A at the time of t-t _ o to complete the construction of Fzeros (), wherein the solution of multiple equations can be realized, and finally, t _ o is added, and the logic expression is as follows: t _ n ═ Fzeros (X)f(tf) T-t _ o) + t _ o, returning to the next switching time t _ n, and recording the displacement X at the switching pointcX (t _ n), switching point speed
Figure BDA0002554277250000031
Wherein: t is taIs a temporary time variable; xa(ta-t _ o) is taDisplacement vector at the moment-t _ o, in the form of ta-t _ o is an expression of a variable; t is tfIs a temporary difference time variable; xf(tf) Is tfDisplacement vector of time in the form of tfIs an expression of a variable; x (t _ n) displacement vector at time t _ n;
Figure BDA0002554277250000032
a velocity vector at t _ n; a is a displacement threshold value vector in the system switching state determined according to the steps 1, 2 and 3, and Xa(ta-the number of elements in t _ o) is determined according to steps 2, 3, the number of elements may be the same or different from the number of degrees of freedom of the system, the displacement of the degrees of freedom directly related to the state switching during the element selection oscillation;
step 81 is the optimization of the switching point search function under the condition of calculating response for a long time, and the optimization before and after in the algorithm is shown in fig. 3;
step 82, when the state is switched from the nonlinear state, using the step-down search program as the switching point search method, where the calculation logic of the step-down search program is as follows:
for a system with multiple degrees of freedom, the state switching conditions are various, and when the state is switched, only X needs to be aimed ata(taT _ o) the degree of freedom of one element is used for accurately searching the state switching point time, so that the accurate searching of the state switching point of the whole system can be met; the function of the step-down pointing procedure is for Xa(ta-t _ o) the degree of freedom of one of the elements for accurately finding the state switching time thereof;
the specific steps of the step-size-decreasing point finding procedure in step 82 are as follows:
step 8201, input parameters, c is t time point before state change, d is t time point before state change, x1、x2Respectively displacement before the state change of the degree of freedom and displacement after the state change; dx (x)1、dx2The speed before the state of the degree of freedom is changed and the speed after the state is changed are respectively; h isrThe input initial value is de for the calculation step length of the point finding program; error is the accuracy of the point finding procedure, usually given by the user; rA is the threshold of the point finding program, and the value is corresponding to X in Aa(ta-t _ o) element values of degrees of freedom;
step 8202, avoiding endless loop when x is1、x2When the requirement is met, the step 9 can be output and executed, and the situation that the dead loop is trapped is avoided;
step 8203, the initialization is carried out,
te＝d；xe＝x2；dxe＝dx2；tc＝c；xc＝x1；dxc＝dx1；td＝d；xd＝x2；dxd＝dx2wherein: t is tc、xc、dxcAre respectively provided withSwitching left time, switching left displacement and switching left speed; t is td、xd、dxdRespectively switching right time, switching right displacement and switching right speed;
step 8204, judging the cyclic error, when | xe-rA | ≧ error continues to be calculated, otherwise the relevant required value is directly output;
step 8205, reducing the step length, and taking hr＝hr/2；
Step 8206, calculating more accurate x by using Runge-Kutta algorithm method with reference to formula (6)e＝Xe(1)、dxe＝Xe(2) (ii) a Wherein: xeRepresenting a closer point value, Xe(1) Represents XeA first element, Xe(2) Represents XeA second element;
step 8207, judge xexcWhether the rA is on the same side, if not, returning to the step 8204, and if so, executing the step 8208 and the step 8205;
step 8208, redefine tc；xc；dxc；tc＝tc+hr；xc＝xe；dxc＝dxe
Step 8209, when x1、x2Directly executing step 8210 to output t after satisfying the error analysise；xe；dxe；
Step 8210, t satisfying the conditione；xe；dxeOutputting;
wherein: t is te、xe、dxeDefining the time, displacement and acceleration of a closer point near the switching point;
t of final outpute、xe、dxeAre respectively assigned to t _ n, xf，dxf(ii) a Wherein: x is the number offIs Xa(ta-displacement at t _ n of the selected element in t _ o) corresponding degree of freedom, dxfIs Xa(ta-acceleration at t _ n of the selected element in t _ o) corresponding degree of freedom;
through tests, the time for finding one point is less than 0.001s when the highest calculation precision is set, and the method has the characteristics of high efficiency and high precision;
step 9, updating the initial conditions
When the state switching is from a linear state, the initial condition is
Figure BDA0002554277250000051
When the state switching is from a non-linear state, the initial condition is
Figure BDA0002554277250000052
Note that X (t _ n); and,
Figure BDA0002554277250000053
Definitions have been given, but the calculation formula should be system specific.
Step 10, calculating the correct point of the next state to cover the error point of the current calculation, and updating the time of the switching point
When the state t changes, the displacement vector and velocity vector X (t) at time t,
Figure BDA0002554277250000054
Belongs to the point with larger deviation, and is rewritten into the displacement and the speed X obtained by the unswitched calculation method for the convenience of understandingd(t)、
Figure BDA0002554277250000055
The step reselects the algorithm corresponding to the switched state to calculate the real displacement and the speed X of the current time te(t)、
Figure BDA0002554277250000056
And replace Xd(t)、
Figure BDA0002554277250000057
And assigned a value to X (t),
Figure BDA0002554277250000058
Updating t _ o to t _ n;
specifically, the error point in this step is explained, and when a response is calculated by using a numerical method, because the step length exists and the state properties of the two ends of the state switching point of the segmented system are quite different, the response belonging to the latter state is calculated by using the parameter of the former state, and a large error is generated, specifically, as shown in fig. 4, this step is combined with step 801 to specifically eliminate the error point; in the figure PaPdIs the error point caused by the step length, and is the first error point, the second error point, PbPcThe recalculated correct points are respectively a first correct point and a second correct point, and l is a state boundary;
step 11, updating time
t＝t+de。
Compared with the prior art, the invention has the following beneficial effects:
the invention is based on the analytic-numerical transient vibration response hybrid optimization transient response calculation algorithm program of the seam theory, the algorithm in the invention is not only suitable for the segmented system with linear composition, but also suitable for the complex segmented system with linear and nonlinear composition, and has better applicability to different inputs and different systems. The algorithm of the invention analyzes the switching between different states in detail, and identifies and selects corresponding methods for different states; optimizing the method for searching and accurately converting the switching points; and error points generated during state switching are eliminated, so that the calculation efficiency and the calculation precision of the algorithm are improved.
Drawings
FIG. 1 is a logic flow diagram of the present invention;
FIG. 2 is a logic diagram of the step-down point finding procedure of the present invention;
FIG. 3 is a comparison graph of the calculation results before and after optimization of the switching point search function in step 801 according to the long-time calculation response;
FIG. 4 is a diagram illustrating an error point in step 10 according to the present invention;
FIG. 5 is a diagram of a one-degree-of-freedom gapped piecewise linear-linear vibrator model in example 1;
FIG. 6 is a comparison of the calculation results of the AMS and the ode algorithm of example 1 with one (ode accuracy Rel-1 e-9; Abs-1 e-9);
FIG. 7 is a comparison of the calculation results of the AMS and the ode algorithm of example 1 with a second graph (ode accuracy Rel 1 e-15; Abs 1 e-12);
FIG. 8 is a comparison of the calculation results of the AMS and the ode algorithm of example 1 with those of FIG. III (ode accuracy Rel 1 e-15; Abs 1 e-15);
FIG. 9 is a graph comparing the calculation times of the AMS and ode algorithms of example 1;
FIG. 10 is a diagram of a single-degree-of-freedom gap-containing linear-nonlinear oscillator model of example 2;
FIG. 11 is a graph comparing the results of AMS and the calculation of the ode algorithm under periodic excitation in the algorithm of example 2 (with the accuracy of ode Rel being 1e-12 and Abs being 1 e-15).
Fig. 12 is a graph comparing the calculation results of the AMS and the ode algorithm under the step excitation in the algorithm of example 2 (the ode accuracy Rel is 1e-12, and Abs is 1 e-15).
Detailed Description
The algorithm of the present invention is further described below with reference to 3 embodiments.
In the embodiment 1, a single-degree-of-freedom gap-containing piecewise linear-linear model is selected for explaining the calculation precision and the calculation speed;
embodiment 2, a single degree of freedom linear-nonlinear model is selected to perform calculation result display, so that the algorithm is proved to have applicability to a system containing nonlinearity;
example 3, a two-degree-of-freedom model containing rigid body modes is selected to perform calculation results, and the applicability of the algorithm to a special system is demonstrated.
Example 1, single degree of freedom gapped piecewise linear-linear, as shown in figure 5,
1. establishing a differential equation of motion of a vibration system
Figure BDA0002554277250000061
Wherein: m is the mass of the vibrator, k1,k2Is the spring rate, c1,c2Is damping coefficient, D is displacement threshold value in switching state, i.e. gap in the figure, B is excitation amplitude, w is excitation frequency, y is vibrator displacement,
Figure BDA0002554277250000064
is the speed of the vibrator, and the speed of the vibrator,
Figure BDA0002554277250000065
is the acceleration of the vibrator, y0、
Figure BDA0002554277250000066
The initial displacement and the initial speed of the vibrator are respectively.
2. Determining the number of states contained in the system according to an equation
And determining the system state to be 3 sections, 3 sections in the linear state and 0 section in the non-linear state according to the equation.
3. Establishing condition corresponding to state, condition changing
And (3) establishing a state corresponding condition by taking the displacement y as a variable:
linear state 1: others;
linear state 2: y ═ D&dy0＞0||y0＞D；
Linear state 3: y ═ D&dy0<0||y0<-D
Establishing a state change condition:
state 1 → 2: y is greater than D; state 1 → 3: y < -D; state 2 → 1: y < D; state 3 → 1: y > -D
4. Initial condition presetting
Figure BDA0002554277250000062
Presetting t _ n as 0; t _ o is 0, t is 0;
a preset solution formula: the simplified solution formulas of states 1, 2 and 3 are respectively as follows:
Figure BDA0002554277250000063
Figure BDA0002554277250000071
Figure BDA0002554277250000072
acceleration:
Figure BDA0002554277250000073
Figure BDA0002554277250000074
Figure BDA0002554277250000075
wherein: y is1、y2、y3The displacement of the vibrator under the state 1, the displacement of the vibrator under the state 2 and the displacement of the vibrator under the state 3 are respectively,
Figure BDA0002554277250000076
the oscillator speed in the state 1, the oscillator speed in the state 2 and the oscillator speed in the state 3 are respectively, e represents a natural logarithm, p11、p12、p21、p22、p31、p32Respectively representing a state 11 coefficient, a state 12 coefficient, a state 21 coefficient, a state 22 coefficient, a state 31 coefficient, and a state 32 coefficient,
Figure BDA0002554277250000077
the first natural frequency is represented by a first frequency,
Figure BDA0002554277250000078
the second natural frequency is represented by a second frequency,
Figure BDA0002554277250000079
is shown asA damping ratio of the damping medium to the damping medium,
Figure BDA00025542772500000710
representing the second damping ratio, B01＝B/k1Representing a first static displacement, B02＝B/(k1+k2) A second static force displacement is indicated and,
Figure BDA00025542772500000711
a first natural frequency of damping is represented,
Figure BDA00025542772500000712
a second natural frequency of damping is indicated,
Figure BDA00025542772500000713
respectively representing a first solution amplitude and a second solution amplitude.
Figure BDA00025542772500000714
The phase of the first solution is represented,
Figure BDA00025542772500000715
indicating the second solution phase.
5. Judging the state
And (5) combining with the step 3, judging the state at the moment t.
6. Selection calculation method
State 1:
Figure BDA0002554277250000081
and uses equation (7).
State 2:
Figure BDA0002554277250000082
and equation (8) is used.
State 3:
Figure BDA0002554277250000083
and formula (9) is used.
It is to be noted that y0、
Figure BDA0002554277250000084
Is an initial condition that is global and changes as the state changes.
7. Judging whether the state change occurs according to the step 3
8. Finding a switching point
State 1 → 2/3 switching point search is performed using Fzeros function and equations (7) and (10 a).
State 2 → 1 switching point search is performed using Fzeros function and equations (8) and (10 b).
State 3 → 1 switching point search is performed using Fzeros function and equations (9) and (10 c).
9. Updating initial conditions
And a combining step 8:
state 1 → 2:
Figure BDA0002554277250000085
state 1 → 3:
Figure BDA0002554277250000086
state 2 → 1:
Figure BDA0002554277250000087
state 3 → 1:
Figure BDA0002554277250000088
wherein the content of the first and second substances,
Figure BDA0002554277250000089
respectively, at times t _ n-t _ o
Figure BDA00025542772500000810
The numerical value of (c).
10. Calculating correct point/update switch point time
State 1 → 2: using equation (10b) + equation (8),
state 1 → 3: using equation (10c) + equation (9),
state 2/3 → 1: using equation (10a) + equation (7),
t_o＝t_n
specifically, the error point in this step is explained, and when a response is calculated by using a numerical method, because the step length exists and the state properties of the two ends of the state switching point of the segmented system are quite different, the response belonging to the latter state is calculated by using the parameter of the former state, and a large error is generated, specifically, as shown in fig. 4, this step is combined with step 801 to specifically eliminate the error point; in the figure PaPdRespectively a first error point, a second error point, PbPcRespectively a first correct point and a second correct point, wherein l is a state boundary;
11. update time and cycle
t＝t+de
12. And (3) calculating the result: and selecting a calculation result of the ode algorithm under different absolute precisions/relative precisions to be compared with a result of the algorithm so as to illustrate the precision and the calculation efficiency of the algorithm. As shown in fig. 6 to 8, it can be seen that as the relative error and the absolute error limited by the ode algorithm are continuously reduced, the response curve thereof gradually tends to the response curve calculated by the AMS, which proves that the response calculated by the AMS has excellent accuracy, and as the relative error and the absolute error limited by the ode algorithm are continuously reduced, the ode calculation time is greatly increased, the efficiency is gradually reduced, while the calculation time of the AMS is short, the efficiency is extremely high, and the extremely high calculation accuracy is ensured, as can be seen from fig. 9.
Example 2, a single degree of freedom linear-nonlinear model, as shown in figure 10,
the algorithm is characterized by having applicability to a segmented vibration system with nonlinear segments.
1. Establishing a differential equation of motion of a vibration system
Figure BDA0002554277250000091
2. Determining the number of states contained in the system according to an equation
And determining the system state to be 2 sections, linear state 1 section and non-linear state 1 section according to the equation.
3. Establishing condition corresponding to state, condition changing
And (3) establishing a state corresponding condition by taking the displacement y as a variable:
linear state 1:
Figure BDA0002554277250000092
nonlinear state 2:
Figure BDA0002554277250000093
establishing a state change condition:
state 1 → 2: y > D
State 2 → 1: y is less than or equal to D
4. Initial condition presetting
Figure BDA0002554277250000094
Presetting an initial switching point t _ n as 0; t _ o is 0, t is 0;
a preset solution formula: the solution formula for the simplified state 1 is written as:
Figure BDA0002554277250000095
Figure BDA0002554277250000096
5. judging the state
And (4) judging the state of the system at the time t by combining the step 3.
6. Selection calculation method
State 1:
Figure BDA0002554277250000101
and uses equation (12).
State 2: equation (6) is used.
7. And (4) judging whether the state change occurs according to the step (3), and is special. At the end of the calculation using equation (6), the initial condition y is updated without the state change0＝Xi+1(1)，
Figure BDA0002554277250000102
Xi+1(1)、Xi+1(2) Are respectively expressed as vector Xi+1The first element, the second element.
8. Searching a switching point:
state 1 → 2: the switching point search is performed using Fzeros () function and formula (12) and formula (13).
State 2 → 1: the 'step-down finding procedure' is used to perform the switch point finding.
9. Updating initial conditions
Combining step 8.:
state 1 → 2:
Figure BDA0002554277250000103
state 2 → 1: using a step-size-reducing point finding procedure to obtain y0、
Figure BDA0002554277250000104
10. Calculating correct point/update switch point time
State 1 → 2: calculated by using the formula (6) and the calculation step length is t-t _ n
State 2 → 1: calculated by using the formula (12) and the formula (13), and the calculation step length is t-t _ n
t_o＝t_n
11. Update time and cycle
t＝t+de
12. And (3) calculating the result: as shown in fig. 11, it is evident that the calculation results of the two algorithms substantially coincide. Meanwhile, the calculation time of using AMS is 0.4s, and the calculation time of using ode is 1.5s, which proves that the AMS has the advantages of high efficiency and high precision for a segmented system containing nonlinearity.
Example 3, the model of example 2 is selected, the excitation is changed to step excitation F-5, the above calculation process is referred to, the system response under sub-step excitation can be obtained through calculation, and a response comparison graph of the AMS and ode calculation methods in the present algorithm is shown in fig. 12. It is evident from the figure that the results of the calculations of the two algorithms substantially coincide. Meanwhile, the calculation time of using the AMS is 0.2s, and the calculation time of using the ode is 1.3s, which proves that the AMS has the advantages of high efficiency and high precision for a step input segmentation system.
In particular, as described above, the algorithm AMS is applicable to a single degree of freedom system, a multiple degree of freedom system (limited to linear portions that can be decoupled), a continuous system, or a segmented system (segments of linear composition, segments of linear-nonlinear composition), a system with different inputs. In fact, the constraint linear part can be decoupled, and the multi-degree-of-freedom system can be decoupled into a plurality of single-degree-of-freedom systems, so that the algorithm is used for solving the segmented system with the single degree of freedom in embodiment 1 for convenience, and the integral solution of the multi-degree-of-freedom system is not exemplified temporarily.
The greatest advantage of the algorithm is that it is excellent for computing a segmented system with linear-nonlinear components, which the algorithm is solved for using embodiment 2.
For different inputs, all the inputs cannot be considered, the algorithm is solved for the system under the step input in embodiment 3, and examples are not given for other inputs.
The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.

Claims (4)

1. A numerical-analytic hybrid optimization transient response algorithm for a segmented system, comprising the steps of:
step 1, establishing a vibration system motion differential equation set:
Figure FDA0002554277240000011
the formula (1) represents a generalized motion differential equation set, the form of the generalized motion differential equation set is determined by a vibration system, and the generalized motion differential equation set is a single-degree-of-freedom system, a multi-degree-of-freedom system, a continuous system or a segmented system;
in equation (1): m is a generalized quality matrix; c is a generalized damping matrix; k is a generalized stiffness matrix; f is a generalized excitation force matrix; t represents a time constant; x is the solution of formula (1) and is a system displacement vector;
Figure FDA0002554277240000012
representing a velocity vector, derived by X over t;
Figure FDA0002554277240000013
as an acceleration vector, composed of
Figure FDA0002554277240000014
Deriving t;
step 2, determining the number of states contained in the system according to the generalized motion differential equation set, and judging whether the system is segmented or not and judging the linear condition of each segment in the segmentation;
step 3, establishing a state corresponding condition, establishing a state change condition:
according to step 2, establishing the linear state and non-linear state of the selected system and the displacement vector X, initial conditions
Figure FDA0002554277240000015
Establishing the corresponding relationship between the state change and the displacement vector X, the initial condition
Figure FDA0002554277240000016
The corresponding relationship of (a); in this case, equation (1) is rewritten in the case of segmentation as:
Figure FDA0002554277240000017
Figure FDA0002554277240000018
wherein: mlAn equation quality matrix in a linear state; clA damping matrix in a linear state; klA stiffness matrix in a linear state; flAn exciting force matrix in a linear state; mnAn equation quality matrix in a nonlinear state; cnA damping matrix in a nonlinear state; knA rigidity matrix in a nonlinear state; fnAn exciting force matrix in a nonlinear state; o islThe displacement speed set in a linear state; o isnThe displacement speed set in a nonlinear state is obtained;
Figure FDA0002554277240000019
respectively representing an initial displacement vector and an initial velocity vector;
step 4, presetting initial conditions; presetting an initial switching point; presetting a solution formula;
presetting initial conditions to
Figure FDA00025542772400000110
Assigning; the time constant t is 0; calculating a time step de preset;
the initial switching point is preset as t _ o; t _ n; wherein: t _ o; t _ n respectively represents the time of the last state change and the time of the next state change in the current state, and the initial values are all 0;
for the determined linear state portion, the displacement vector and velocity vector in equation (1) are expressed as:
Figure FDA00025542772400000111
Figure FDA00025542772400000112
wherein: u shapel(t)、Vl(t) respectively representing a free vibration matrix of the system caused by the initial displacement of each degree unit and a free vibration matrix caused by the initial speed of each toxic unit; τ is an integral variable; h (t-tau) is a system unit impulse response matrix taking t-tau as a variable;
Figure FDA0002554277240000021
denotes the derivation of t;
for the determined part of the nonlinear state, the Runge-Kutta algorithm of order 4 is chosen:
Figure FDA0002554277240000022
wherein: r1、R2、R3、R4Respectively a first coefficient, a second coefficient, a third coefficient and a fourth coefficient of the Runge-Kutta algorithm; t is tiThe ith time of the Runge-Kutta algorithm; the function of f () is a first order differential equation system for reducing order change of formula (3); xiIs tiA value vector consisting of degree displacement and velocity; xi+1Is ti+1A value vector consisting of degree displacement and velocity;
step 5, judging the state:
determining the state of the system at the moment t according to the definition of the step 3;
and 6, selecting a calculation method to calculate a correlation value corresponding to the current time:
according to the judgment of the steps 2 and 3, when the system is in a linear part, calculating by using the formulas (4) and (5); when the part is non-linear, the calculation is carried out by using the formula (6);
and 7, judging whether state change occurs:
judging whether the state is changed according to the definition of the step 3, and calculating whether the state is changed or not by using a formula (6) each time the initial condition needs to be updated, X0＝Xi+1；
Step 8, searching a state switching point:
step 81, when the state switching is from linear state switching, using Fzeros () for reconstructing and optimizing an MATLAB self-contained program fzero () as a switching point searching method, accurately searching points and simultaneously solving a plurality of equations to search the state switching point;
step 82, when the state is switched from the non-linear state, the step length reduction point finding program is used as the point finding method of the switching point, and the time, the displacement and the acceleration t of the closer point near the switching point are outpute；xe；dxe；
T of final outpute、xe、dxeAre respectively assigned to t _ n, xf，dxf(ii) a Wherein: x is the number offIs Xa(ta-displacement at t _ n of the selected element in t _ o) corresponding degree of freedom, dxfIs Xa(ta-acceleration at t _ n of the selected element in t _ o) corresponding degree of freedom;
step 9, updating initial conditions:
when the state switching is from a linear state, the initial condition is
Figure FDA0002554277240000031
When the state switching is from a non-linear state, the initial condition is
Figure FDA0002554277240000032
Step 10, calculating the correct point of the next state to cover the error point of the current calculation, and updating the time of the switching point:
when the state t changes, the displacement vector and velocity vector X (t) at time t,
Figure FDA0002554277240000033
Belongs to the point with larger deviation, and is rewritten into the displacement and the speed X obtained by the non-switching calculation methodd(t)、
Figure FDA0002554277240000034
Reselecting post-switch state correspondencesCalculates the true displacement and velocity X of the current time te(t)、
Figure FDA0002554277240000035
And replace Xd(t)、
Figure FDA0002554277240000036
And assigned a value to X (t),
Figure FDA0002554277240000037
Updating t _ o to t _ n;
step 11, updating time:
t＝t+de。
2. the numerical-analytic hybrid optimization transient response algorithm for a segmented system of claim 1,
in step 81, the step of optimizing the program Fzeros () construction is: mixing Xa(ta-t _ o) by tf＝ta-t _ o is rewritten as Xf(tf) Separately solving for X by means of MATLAB self-contained program fzero () using a loopf(tf) And returning a vector at a crossing point of each function element near the corresponding element A at the time of t-t _ o to complete the construction of Fzeros (), wherein the solution of multiple equations can be realized, and finally, t _ o is added, and the logic expression is as follows: t _ n ═ Fzeros (X)f(tf) T-t _ o) + t _ o, returning to the next switching time t _ n, and recording the displacement X at the switching pointcX (t _ n), switching point speed
Figure FDA0002554277240000038
Wherein: t is taIs a temporary time variable; xa(ta-t _ o) is taDisplacement vector at the moment-t _ o, in the form of ta-t _ o is an expression of a variable; t is tfIs a temporary difference time variable; xf(tf) Is tfDisplacement vector of time in the form of tfIs a variableAn expression; x (t _ n) displacement vector at time t _ n;
Figure FDA0002554277240000039
a velocity vector at t _ n; a is a displacement threshold value vector in the system switching state determined according to the steps 1, 2 and 3, and Xa(taThe number of elements in t _ o) is determined according to steps 2 and 3, the number of elements is the same as or different from the number of system degrees of freedom, and the elements select the displacement of the degree of freedom directly related to state switching in the vibration process.
3. The numerical-analytic hybrid optimization transient response algorithm for a segmented system of claim 2,
in step 82, the calculation logic of the step-down step size point finding program is as follows:
for a system with multiple degrees of freedom, the state switching conditions are various, and when the state is switched, only X needs to be aimed ata(taT _ o) the degree of freedom of one element is used for accurately searching the state switching point time, so that the accurate searching of the state switching point of the whole system can be met; the function of the step-down pointing procedure is for Xa(taT _ o) the degree of freedom of one of the elements is used for accurately searching the state switching time of the element.
4. The numerical-analytic hybrid optimization transient response algorithm for a segmented system of claim 3,
in step 82, the step-size-decreasing point finding procedure specifically includes the following steps:
step 8201, input parameters, c is t time point before state change, d is t time point before state change, x1、x2Respectively displacement before the state change of the degree of freedom and displacement after the state change; dx (x)1、dx2The speed before the state of the degree of freedom is changed and the speed after the state is changed are respectively; h isrThe input initial value is de for the calculation step length of the point finding program; error is the accuracy of the point finding procedure, usually given by the user; rA is a point finding program valveA value of X in Aa(ta-t _ o) element values of degrees of freedom;
step 8202, avoiding endless loop when x is1、x2When the requirement is met, the step 9 can be output and executed, and the situation that the dead loop is trapped is avoided;
step 8203, the initialization is carried out,
te＝d；xe＝x2；dxe＝dx2；tc＝c；xc＝x1；dxc＝dx1；td＝d；xd＝x2；dxd＝dx2wherein: t is tc、xc、dxcRespectively switching left time, switching left displacement and switching left speed; t is td、xd、dxdRespectively switching right time, switching right displacement and switching right speed;
step 8204, judging the cyclic error, when | xe-rA | ≧ error continues to be calculated, otherwise the relevant required value is directly output;
step 8205, reducing the step length, and taking hr＝hr/2；
Step 8206, calculating more accurate x by using Runge-Kutta algorithm method with reference to formula (6)e＝Xe(1)、dxe＝Xe(2) (ii) a Wherein: xeRepresenting a closer point value, Xe(1) Represents XeA first element, Xe(2) Represents XeA second element;
step 8207, judge xexcWhether the rA is on the same side, if not, returning to the step 8204, and if so, executing the step 8208 and the step 8205;
step 8208, redefine tc＝tc；xc；dxc；tc＝tc+hr；xc＝xe；dxc＝dxe；
Step 8209, when x1、x2Directly executing step 8210 to output t after satisfying the error analysise；xe；dxe；
Step 8210, t satisfying the conditione；xe；dxeOutputting;
wherein: t is te、xe、dxeDefining the time, displacement and acceleration of a closer point near the switching point;
t of final outpute、xe、dxeAre respectively assigned to t _ n, xf，dxf(ii) a Wherein: x is the number offIs Xa(ta-displacement at t _ n of the selected element in t _ o) corresponding degree of freedom, dxfIs Xa(ta-acceleration at t _ n of the selected element in t _ o) corresponding degree of freedom.
CN202010587348.0A 2020-06-24 2020-06-24 Numerical value-analytic hybrid optimization transient response algorithm for segmented system Pending CN111832128A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202010587348.0A CN111832128A (en) 2020-06-24 2020-06-24 Numerical value-analytic hybrid optimization transient response algorithm for segmented system

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202010587348.0A CN111832128A (en) 2020-06-24 2020-06-24 Numerical value-analytic hybrid optimization transient response algorithm for segmented system

Publications (1)

Publication Number Publication Date
CN111832128A true CN111832128A (en) 2020-10-27

Family

ID=72899048

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202010587348.0A Pending CN111832128A (en) 2020-06-24 2020-06-24 Numerical value-analytic hybrid optimization transient response algorithm for segmented system

Country Status (1)

Country Link
CN (1) CN111832128A (en)

Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107729592A (en) * 2017-08-14 2018-02-23 西安理工大学 Traced back the Time variable structure Modal Parameters Identification of track based on broad sense subspace

Patent Citations (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107729592A (en) * 2017-08-14 2018-02-23 西安理工大学 Traced back the Time variable structure Modal Parameters Identification of track based on broad sense subspace

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
何冬东;高强;钟万勰;: "求解周期性分段线性系统动态响应的高效数值方法", 应用数学和力学, no. 07 *
孙小娟;张建润;张宏;: "分段阻尼特性对土方机械驾驶室瞬态响应的影响", 筑路机械与施工机械化, no. 02 *

Similar Documents

Publication Publication Date Title
Liu et al. Immersion and Invariance Adaptive Control of Nonlinearly Parameterized Nonlinear Systems $$
WO2005019949A1 (en) Pid parameter adjustment device
CN110412867B (en) High-precision angular rate control method for magnetically suspended control moment gyroscope frame system
CN109164759B (en) Curve interpolation method, equipment and computer readable storage medium
CN109459934A (en) A method of depression of order automatic disturbance rejection controller parameter is adjusted based on PID controller
Tomljanović et al. Damping optimization of parameter dependent mechanical systems by rational interpolation
Mfoumou et al. Computational algorithms of time series for stick-slip dynamics and time-delayed feedback control of chaos for a class of discontinuous friction systems
Calogeracos et al. On the Quantum Electrodynamics of a Dispersive Mirror.: II. The Boundary Condition and the Applied Force via Dirac′ s Theory of Constraints
CN109491242A (en) A kind of grid reconstruction method of the directly discrete solution of optimal control problem
CN111832128A (en) Numerical value-analytic hybrid optimization transient response algorithm for segmented system
Guo et al. Active disturbance rejection control of valve-controlled cylinder servo systems based on MATLAB-AMESim Cosimulation
CN105224770B (en) High speed light loading mechanism nonlinear dynamic system structural topological optimization method
Axehill et al. A preprocessing algorithm for MIQP solvers with applications to MPC
CN109325260A (en) A kind of kinetic model shows hidden mixed asynchronous long stagger scheme calculation method
JPH10323070A (en) Motor controller
Rothmann et al. FAQ: A flexible accelerator for Q-learning with configurable environment
Mu et al. Profile generation algorithm and implementation for high accuracy motion
CN116882195B (en) Explicit method and system for solving second-order nonlinear dynamics problem
CN113703321B (en) Bezier curve slow-moving processing method for vehicle-mounted photoelectric servo control system
Dhaouadi Nonlinear friction compensation in harmonic drives with hysteresis
CN108712129B (en) Torque calculation optimization method based on direct torque control prediction control
Atinga et al. Application of Heavy and Underestimated Dynamic Models in Adaptive Receding Horizon Control Without Constraints
Liu Ham-based package noph for periodic oscillations of nonlinear dynamic systems
Fu et al. The second order temporal difference error for Sarsa (λ)
Pickl et al. When do microswimmers exit the Stokes regime?

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination