Generally speaking, there are three dynamic modeling methods for flexible multi-body systems: The first is Newton-Euler method, which requires the momentum theorem to clarify the corresponding dynamic equations of all separated bodies, and then use constraints to remove the corresponding constraints. This method meets the requirement that there is a rigid body in the system. Secondly, it refers to the virtual displacement method, which should combine the virtual displacement principle to obtain the first Lagrangian equation, or combine the variational principle to construct the second Lagrangian equation, so as to obtain the dynamic model. Finally, it refers to other typical methods such as IGC, AIGC, NSD, etc.
Is the constraint matrix,
In the above formula, P represents relative or absolute coordinates in the floating coordinate system, and A represents deformation coordinates.
Modeling and numerical simulation are the core content of the study of dynamics of flexible multi-body system. It is very difficult to quickly construct the corresponding motion equations of the system, especially to form the numerical solution equations required by computer operation. Due to the large scale of solving the coefficients and variables of the dynamics equation of flexible multi-body system, the actual derivation process is complicated, so the symbolic calculus method is used to deal with the modeling of the system by using finite element method.
There are many methods to transform algebraic mixed equations into numerical models, such as augmented method, reduced method and mixed method. Based on the analysis of the algebraic equations obtained in the above research, it can be seen that:
Taking the derivative of t using the second expression of the algebraic equations, we can get:
The compatible conditions are as follows:
According to an antenna in the flexible multi-body system dynamics model, the control equation of characteristic analysis, comparison on the relatively mature after the numerical simulation method, the constraint default direct correction method, the precise integration method, the nonlinear programming method acquired two, on the basis of continue to analyze the solving process, actual problems and computational cost, In the end, the direct default correction method of the augmented method is selected for solving analysis. The specific process is as follows:
The above formula conforms to
This condition. For nq+nλ unknown variables q and λ, the above equations belong to closed algebraic equations, and the actual solvability will be affected by the non-singularity of the coefficient matrix. In order to better analyze the application principle of direct default correction method, we can first define the method of constraint stability. In Baumgarte (Baumgarte) constrained default stability method, the following equations can be obtained because of the large error accumulation:
The above formula conforms to
Generally speaking, α, β=5 ~ 50 should be set as the critical error resistance state under the condition of α=β, and the actual response stability is faster. The Lagrange multiplier can be obtained by combining the number of dynamic equations of flexible multi-body system and the actual derivative results:
After sorting, it can be obtained:
The above formula conforms to
On the one hand, coordinate default should be corrected. The numerical solution of the coordinate at time t=t (I) is assumed to be
Taylor expansion of the position constraint equation can obtain:
After sorting, it can be obtained:
In general, because the number of generalized coordinates exceeds the number of constraint equations in the system, in other words
At this point, the minimal norm solution is obtained, as follows:
Minimal norm solution
On the other hand, speed default is corrected. Combined with the above analysis, the hypothesis is consistent
For this condition, the modified item is
The constraint equation of the corresponding velocity is:
By simplifying the conditions, we can get:
Thus, the equation of velocity correction value can be obtained as follows:
The minimal norm solution of the above equation is studied by using the computational method of nonlinear programming method, and the specific results are shown as follows:
In the study of the basic numerical methods of dynamic equations, the following points are specific: First, Treanor method. Assume that the unknown function value at tj is yij (I = 0,1... N-1), then calculate the unknown function value yi of tj+1=t+h, j+1, the formula is:
Where, when PI ≤0 is met, the following equation can be obtained:
In accord with pi> When the condition is 0, we can get:
In the above formula:
Among them, qi should be defined in combination with the following formula:
Second, Gill's law. Kiel's formula for integrating from t to t+h is:
In the above formula:
Where, yi (o) represents the position function value yi (t) at point T, yi (4) represents the unknown function value yi (t+ H) at point T + H after one step of integration, qi (O) represents that the initial value is 0, and qi (4) is regarded as the next qi (O) at each step of integration. In practice, this method can offset the rounding error accumulated in each step, effectively improve the accuracy of integration application, and has strong numerical stability for ill-posed dynamic equations. At the same time, the equation calculation has a lower stage error, compared with other methods can improve the integration speed, the actual calculation is also effectively controlled, the overall program has been improved. Third, the Gear method. The general expression formula is:
Using Taylor expansion, the order is p=k, and the coefficient αj (j= 0,1 …) can be calculated under the condition that k= 1–6. K-1), βk, and the corresponding formula can be obtained:
When K= 1,2, the method is a-stable, and when K= 3 ~ 6, the method is A (α) -stable and rigid stable. The results of A (α) -stable α Max and rigid stable parameters Dmin and θ Max are shown in the table 1 below:
Stability parameter analysis
|A - stable, 90°||86°54′||73°14′||51°50′||18°47′|
There are many countermeasures to improve this method, as shown below:
In the above formula, assuming βk is not equal to 0, and meets βk+β K-1 =1 condition, then we still need to calculate the order P =k in the formula, and βk=β is regarded as a free parameter, using Taylor expansion and k order compatible conditions, accurately calculate the corresponding expression formula.
Assuming β=20, the following formula can be obtained:
Compared with the Gear method of the same order, the actual stability region of these formulas is larger, and the corresponding results are shown in the following table:
Stability region comparison results
|Improve gill method||A - stable||89°55′||85°32′||73°2′||51°23′||18°32′|
|Jill method||A - stable||86°54′||73°13′||51°50′||18°47′||unstable|
It can be seen that the disadvantage of this method is that the order of the equation keeps rising, the initial value is difficult to judge, and the multi-step time discrete has a comprehensive influence on the constraint equation, resulting in the drift of the solution. In other words, the rigidity should not be too high, and the Gear method will lose effect when the integral variable appears high-frequency oscillation, because the Gear method does not have A stability region.
The whole software system design includes the finite element analysis, structural design, the content such as geometry modeling, optimization design module, the deployment dynamics analysis module is the process in control, and to exchange data information of an operation, the corresponding numerical simulation module is master data information during the process control need to know. It can be seen that numerical simulation module is the basic requirement to ensure that the process control module can play an effective role. Flow chart 1 of the system platform constructed for deployable antenna in this paper is shown as follows:
The corresponding design system flow chart 2 is:
Flow chart 3 of the numerical calculation module is:
First, we have to deal with differential equations. In order to calculate the differential equation more quickly, it is necessary to deal with the equation obtained in the above research effectively. Because the equations are nQ-dimensional second order ordinary differential equations, so
Assuming t=0, the initial formula of the system is:
Then the above equation of dynamics can be numerically integrated.
Secondly, we need to deal with numerical simulation methods. In this paper, a numerical integration method library is constructed, which contains a variety of different methods for effective selection during simulation. At present, in addition to several methods proposed in this paper, runge-Kutta method with fixed or variable step size and Gear method with variable step size and variable order are also involved. In addition, input more parameters, such as simulation start time, end time, etc.
Finally, the kinetic equation must be calculated and analyzed. On the one hand, for the multi-body system with closed loop, its dynamics equation is differential mixed equation, which needs to be contracted to obtain the pure differential form of the system dynamics equation, which is consistent with the dynamics equation of the tree multi-body system in form, and can be treated and solved in the same way. On the other hand, when the joints in the system have active drive, the simulation should focus on the justice mixing problem.[9,10]
On the one hand, the study of the dynamics of rigid crank is mainly to verify the accuracy of the numerical simulation model. The Gill method of fixed step length is used to study the movement of rigid crank under the action of constant driving torque. The cross section of homogeneous crank is rectangular and shows a trend of horizontal rotation. At this point, the model parameters of the rigid crank involve the arm length L reached 0.75 meters, the cross-sectional area A was 60×15×10 square meters, and the mass M reached 1.4 kg. The driving torque is 62N•m, the crank is in horizontal position at the initial moment, and the initial conditions are as follows:
Among them, the left figure is based on the results obtained from the theoretical research of traditional fixed-axis rotation mechanics, and the right figure is the results obtained from the theoretical simulation analysis of multi-body system dynamics in this paper. Combined with the model parameters, the actual angular acceleration of the crank is 236.19rad/s2. Therefore, it can be found that, on the one hand, the rigid model obtained according to the dynamic model is valid; on the other hand, the dynamic response obtained by the program is consistent with the traditional calculation, which further indicates the correctness of the design model studied in this paper.
On the other hand, the dynamics of two - link flexible manipulator is simulated. The corresponding model is shown in Figure 6 below:
Assuming that both links are in horizontal position at the initial moment and the actual deformation is 0, the initial conditions are as follows:
Gill method with variable step size was used for analysis. The integral step size was H = LE-4, the simulation accuracy was EPS = le-L2, the simulation time was tend=1.2sec, and the calculation time was 2min56sec. The left and right graphs in FIG. 7 show the relation curves between angular velocity and time of two connecting rods at angles θ1 and θ2 respectively.
The Treanor method with fixed step length is used for analysis. Except that the calculation time is 47sec, other conditions are consistent. The relation between the actual angular velocity and time of the two connecting rods is shown in Figure 8 as follows:
Meanwhile, the above two methods are used to calculate the simulation results of h= LE-5 and H = LE-6 integral step sizes respectively. The actual calculation speed is shown in Table 3 below:
|The integral step||Le-4||Le-5||Le-6|
|Variable step size Gill||2min56sec||48min6sec||2h43min53sec|
|Fixed step length Treanor||47sec||8min21sec||11min20sec|
Combined with the analysis in Table 3 above, it can be seen that Gill method takes a long time to calculate a small integral step. However, Treanor method has no clear requirements on the integral step length, and the actual settlement time is very close. Compared with the corresponding dynamic change graphs, Gill's method has stronger stability for numerical solutions than Treanor's method, with higher actual accuracy. With the increase of calculation time, the angular velocity of the two links will continue to rise, which may cause dynamic hardening and increase the probability of data simulation failure.[11,12]
To sum up, the flexible multi-lift system dynamics studied in this paper integrates multiple disciplines. The actual development not only supplements and summarizes the original discipline, but also expands the thinking space of researchers and scholars, and promotes the construction of practical numerical simulation models to be more diversified. At present, the research work of flexible multibody system dynamics has obtained many research results from modeling scheme, symbolic calculus, tempering effect, numerical simulation and so on. Although related research is still a new subject, with the continuous exploration and data accumulation of domestic and foreign researchers, it is inevitable to start from this in the future development of deployable antenna application optimization. In addition, the future research direction of flexible multi-body dynamics will also change. For example, because the current high-precision integration algorithm cannot meet the requirements of implementation simulation, researchers should continue to discuss the appropriate simulation algorithm. In the calculation strategy, we can choose a new modeling method and computing system, such as recursive model, so as to make use of the parallel processing and vector processing capabilities of multi-processing computing, and conduct in-depth research on the simulation of relevant models.
|The integral step||Le-4||Le-5||Le-6|
|Variable step size Gill||2min56sec||48min6sec||2h43min53sec|
|Fixed step length Treanor||47sec||8min21sec||11min20sec|
Stability parameter analysis
|A - stable, 90°||86°54′||73°14′||51°50′||18°47′|
Stability region comparison results
|Improve gill method||A - stable||89°55′||85°32′||73°2′||51°23′||18°32′|
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ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G) on the set of maximal subgroups of p-groups The internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edge-based finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Research on threat assessment problems of island air defence system based on the leader-follower model Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long short-term memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSR-AHP model A study of local smoothness-informed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H2-N2 atmosphere Algorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model