CN104809300A - Dynamic simulation method for pulse substructure and finite element rigid-elastic mixed connection - Google Patents

Abstract

The invention discloses a dynamic simulation method and an algorithm for pulse substructure and finite element rigid-elastic mixed connection, relates to a dynamic simulation method and an algorithm, and belongs to the technical field of structural dynamics. The method comprises the following steps: obtaining the pulse response function of a pulse substructure and the dynamic equation of a finite element substructure; establishing the consistency condition of a substructure interface; establishing the dynamic equation of the pulse substructure; establishing the dynamic equation of the finite element substructure; combining the dynamic equation of the pulse substructure and the dynamic equation of the finite element substructure together by utilizing the consistency condition of displacement to solve the response of the whole system and complete the simulation of the structural dynamics. According to the dynamic simulation method and the algorithm for pulse substructure and finite element rigid-elastic mixed connection disclosed by the invention, the pulse substructure and finite element rigid-elastic mixed characteristic is truly reflected, and thus the simulation precision of the structural dynamics in the technical field of spaceflight is improved. Moreover, the application range of a traditional IBS method is expanded.

Description

Pulse minor structure and finite element just-play the dynamics simulation method of Hybrid connections

Technical field

The present invention relates to a kind of dynamics simulation method and algorithm, particularly a kind of pulse minor structure and finite element just-play dynamics simulation method and the algorithm of Hybrid connections, belong to Structural Dynamics technical field.

Background technology

Along with the development of spationautics and improving constantly spacecraft performance requirement, spacecraft structure becomes day by day complicated and huge.When carrying out the Dynamic Response and structure optimization to modern Complex Spacecraft, adopting conventional finite element method to analyze introducing numerous degree of freedom system, consuming a large amount of computational resource; In the face of system goes through the structure optimization of dynamic response analysis, traditional finite element modeling method is difficult to be competent at especially.Meanwhile, the cooperation of spacecraft in development process often between different department, different institutions, for the consideration of technical protection or the difference of modeling pattern, the two sides concerned cannot direct Share Model.Dynamic sub-structure methods can solve the problem to a certain extent.

Dynamic Substructure (Dynamic Substituting is realized first from nineteen sixty Hurty, DS) since, engineering field is widely used in through semicentennial development trend Substructure Techniques, define three class methods: modal synthesis (Component Mode Synthesis, CMS) method, frequency domain minor structure (FrequencyBased Substructuring, FBS) method and minor structure (the Impulse BasedSubstituting based on impulse response function, IBS) i.e. classical time domain substructure method, is called for short pulse substructure method.In first two method, the dynamics of minor structure is described by mode and frequency response function respectively; And pulse substructure method is described by impulse response function, when processing little damping, transient impact problem, there is quick, high-precision advantage.

Afterwards, Paul L.C.van der Valk, Daniel J.Rixen have studied based on the model of impulse response function and the combination of finite element model, but their connected mode is only considered to be rigidly connected, just numerous-bullet Hybrid connections mode is there is, the chain connection such as between lunar orbiter centrosome and solar wing in the analysis of spacecraft dynamic response; Dong Weili have studied a kind of flexible linker structure composition method based on impulse response function, the method considers the elastic property of web member between pulse minor structure, but method fails to consider the connection between pulse minor structure and finite element minor structure, when carrying out system structure optimization, need modify to partial structurtes, adopt each Optimized Iterative of the method all will to recalculate the impulse response function of pulse minor structure, and efficient for the method advantage weakens by impulse response function repeatedly regenerate greatly, optimizing process is made to become loaded down with trivial details consuming time.

Therefore, how under firm-bullet Hybrid connections situation, pulse minor structure and finite element minor structure to be carried out comprehensively, become the problem that must solve in time domain Dynamic Substructure technical development process.

Summary of the invention

Only being suitable for analyzing between pulse minor structure for conventional I BS method is rigidly connected defect.Pulse minor structure disclosed by the invention and finite element just-play dynamics simulation method and the algorithm of Hybrid connections, the technical matters solved be provide pulse minor structure and finite element minor structure just-the dynamics simulation method of elasticity Hybrid connections and algorithm, actual response pulse minor structure and finite element minor structure just-elasticity mixed characteristic, and then improve spationautics field structure dynamics simulation precision.

The object of the invention is to be achieved through the following technical solutions:

Basic ideas of the present invention are: first obtain the impulse response function matrix of pulse minor structure and the kinetics equation of finite element minor structure; Then the consistency condition (comprising interfacial displacement consistency condition and interfacial force consistency condition) of Substructure Interfaces is set up according to the annexation between minor structure; Next utilize interfacial force consistency condition, kinetics equation that impulse response function matrix sets up pulse minor structure; Then utilize interfacial force consistency condition, kinetics equation that finite element equation sets up finite element minor structure; Finally utilize displacement compactibility to be integrated by the kinetics equation of pulse minor structure and finite element minor structure, solve the response of whole system.

A kind of pulse minor structure disclosed by the invention and finite element just-play the dynamics simulation method of Hybrid connections, specific implementation step is as follows:

Step 1: carry out minor structure division.Whole system is divided into pulse minor structure and finite element minor structure two parts.The motion vector u of system represents with the motion vector of all minor structures, and piecemeal is u ^(r), u ^(s), as shown in formula (1):

$u={\left[\begin{array}{cc}{u}^{{\left(r\right)}^T}& u^{{\left(s\right)}^T}\end{array}\right]}^T---\left(1\right)$

In formula, subscript r represents that this variable belongs to r pulse minor structure, and subscript s represents that this variable belongs to s finite element minor structure, T representing matrix transposition.

Step 2: structure Boolean matrix.The effect of Boolean matrix is that each minor structure degree of freedom is projected to junction, interface.

By structure Boolean matrix by the displacement components u of pulse minor structure ^(s)project to linkage interface, Boolean matrix tried to achieve by formula (2)

$\left\{\begin{array}{c}{u}_r^{\left(s\right)}=B_r^{\left(s\right)}{u}^{\left(s\right)}\\ u_k^{\left(s\right)}=B_k^{\left(s\right)}{u}^{\left(s\right)}\end{array}\right.---\left(2\right)$

Wherein subscript r represents that this variable belongs to the place that is rigidly connected, and subscript k represents that this variable belongs to elasticity junction.

Equally, Boolean matrix is constructed by the displacement components u of finite element minor structure ^(r)project to junction, interface, Boolean matrix tried to achieve by formula (3):

$\left\{\begin{array}{c}{u}_r^{\left(r\right)}=-B_r^{\left(r\right)}{u}^{\left(r\right)}\\ u_k^{\left(r\right)}=B_k^{\left(r\right)}{u}^{\left(r\right)}\end{array}\right.---\left(3\right)$

Step 3: introduce Lagrange multiplier λ, λ and represent system interface junction attachment force array, the Boolean matrix obtained by step 2, have the relation in formula (4):

$\left\{\begin{array}{c}{\mathrm{\λ}}_r^{\left(s\right)}=B_r^{{\left(s\right)}^T}\mathrm{\λ}\\ {\mathrm{\λ}}_k^{\left(s\right)}=B_k^{{\left(s\right)}^T}\mathrm{\λ}\\ {\mathrm{\λ}}_r^{\left(r\right)}=B_r^{{\left(r\right)}^T}\mathrm{\λ}\\ {\mathrm{\λ}}_k^{\left(r\right)}=B_k^{{\left(r\right)}^T}\mathrm{\λ}\end{array}\right.---\left(4\right)$

Step 4: set up pulse minor structure kinetics equation, concrete steps are as follows:

Step 4.1, obtains the impulse response function matrix H of each pulse minor structure by numerical integration method Newmark method or test method ^(s)(t), the wherein sequence number of s indicating impulse minor structure, H ^(s)(t) entry of a matrix element the dynamic respond of a jth degree of freedom i-th degree of freedom under pulse excitation of expression system s pulse minor structure.

Step 4.2: set up pulse minor structure kinetics equation by impulse response function matrix:

From Duhamel integration, the kinetics equation of system s pulse minor structure is:

$u^{\left(s\right)}\left(t\right)={\∫}_0^t{H}^{\left(s\right)}(t-\mathrm{\τ})[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}]\mathrm{d\τ}---\left(5\right)$

In formula, u ^(s)t () is the displacement array of s pulse minor structure; f ^(s)(t) external applied load array suffered by pulse minor structure; $B^{\left(s\right)}=\left[\begin{array}{cc}{B}_r^{\left(s\right)}& B_k^{\left(s\right)}\end{array}\right]$ Tried to achieve by step 2 Chinese style (2), from step 3 formula (4), indicating impulse Substructure Interfaces place attachment force array.

Step 5: set up finite element minor structure kinetics equation:

$M^{\left(r\right)}{\stackrel{\·\·}{u}}^{\left(r\right)}\left(t\right)+C^{\left(r\right)}{\stackrel{\·}{u}}^r\left(t\right)+K^{\left(r\right)}{u}^{\left(r\right)}\left(t\right)=f^{\left(r\right)}\left(t\right)+B^{{\left(r\right)}^T}\mathrm{\λ}---\left(6\right)$

In formula: M ^(r), C ^(r), K ^(r)be respectively the quality of r finite element minor structure, damping, stiffness matrix; u ^(r)t () is respectively r finite element minor structure acceleration, speed, displacement array; f ^(r)(t) external drive power array suffered by r finite element minor structure; $B^{\left(r\right)}=\left[\begin{array}{cc}{B}_r^{\left(r\right)}& B_k^{\left(r\right)}\end{array}\right]$ Tried to achieve by step 2 Chinese style (3), from step 3 formula (4), represent the attachment force array at finite element Substructure Interfaces place.

Step 6: determine pulse minor structure and finite element Substructure Interfaces consistency condition.

Described pulse minor structure and finite element Substructure Interfaces consistency condition meet: between the degree of freedom between the minor structure of interface that is rigidly connected identical and elasticity linkage interface virgin structure, degree of freedom meets Hooke's law.

The degree of freedom between the minor structure of interface that is rigidly connected is identical, has following relation, as shown in formula (7):

$u_r^{\left(s\right)}=u_r^{\left(r\right)}---\left(7\right)$

Between elasticity linkage interface virgin structure, degree of freedom meets Hooke's law, has following relation, as shown in formula (8):

$k(u_k^{\left(s\right)}+u_k^{\left(r\right)})={\mathrm{\λ}}_k---\left(8\right)$

Wherein λ _kas shown in formula (9):

${\mathrm{\λ}}_k=\left[\begin{array}{c}{\mathrm{\λ}}_k^{\left(r\right)}\\ {\mathrm{\λ}}_k^{\left(s\right)}\end{array}\right]---\left(9\right)$

Step 7: obtain Substructure Synthesis equation, concrete grammar is: the first formula of (2) in step 2 and first formula of (3) are brought into (7), and the second formula of (2) in step 2 and second formula of (3) are brought into (8), (6) formula in integrating step 2.2 in (5) formula and step 3, obtains Substructure Synthesis equation:

$\left\{\begin{array}{c}{u}^{\left(s\right)}\left(t\right)={\∫}_0^t{H}^{\left(s\right)}(t-\mathrm{\τ})[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}]\mathrm{d\τ}\\ M^{\left(r\right)}{\stackrel{\·\·}{u}}^{\left(r\right)}\left(t\right)+C^{\left(r\right)}{\stackrel{\·}{u}}^{\left(r\right)}\left(t\right)+K^{\left(r\right)}{u}^{\left(r\right)}\left(t\right)=f^{\left(r\right)}\left(t\right)+B^{{\left(r\right)}^T}\mathrm{\λ}\\ B_r^{\left(s\right)}{u}^{\left(s\right)}+B_r^{\left(r\right)}{u}^{\left(r\right)}=0\\ k(B_k^{\left(s\right)}{u}^{\left(s\right)}+B_k^{\left(s\right)}{u}^{\left(s\right)})={\mathrm{\λ}}_k\end{array}\right.---\left(10\right)$

Step 8: carry out time discrete to the Substructure Synthesis equation that step 7 obtains, obtains the dynamic respond u of system _nand interfacial force λ between minor structure _ntime recursion formula, complete pulse minor structure and finite element just-Structural Dynamics that plays Hybrid connections part simulates.Concrete steps are:

Step 8.1: time discrete is carried out to (10) first formulas, has:

$u_n^{\left(s\right)}=\underset{i=0}{\overset{n-1}{\mathrm{\Σ}}}{H}_{n-i}^{\left(s\right)}[f_i^{\left(s\right)}+f_{i+1}^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_i-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{i+1}]\frac{\mathrm{dt}}{2}---\left(11\right)$

In formula, dt is integration step, and subscript represents the moment (as u _n=u (ndt));

Step 8.2: adopt following differential mode discrete to (10) second formulas:

$\left\{\begin{array}{c}{\stackrel{\·}{u}}_n^{\left(r\right)}={\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt}(1-\mathrm{\γ}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt\γ}{\stackrel{\·\·}{u}}_n^{\left(r\right)}\\ u_n^{\left(r\right)}=u_{n-1}^{\left(r\right)}+\mathrm{dt}\·{\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2(1/2-\mathrm{\β}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2\mathrm{\β}{\stackrel{\·\·}{u}}_n^{\left(r\right)}\end{array}\right.---\left(12\right)$

In formula, γ, β are integration constant.

Step 8.3: the speed utilizing Newmark method representation system and acceleration

$\left\{\begin{array}{c}{\stackrel{\·\·}{u}}_n=\frac{1}{{\mathrm{\βdt}}^2}(u_n-u_{n-1})-\frac{1}{\mathrm{\βdt}}{\stackrel{\·}{u}}_{n-1}-(\frac{1}{2\mathrm{\β}}-1){\stackrel{\·\·}{u}}_{n-1}\\ {\stackrel{\·}{u}}_n=\frac{\mathrm{\γ}}{\mathrm{\βdt}}(u_n-u_{n-1})+(1-\frac{\mathrm{\γ}}{\mathrm{\β}}){\stackrel{\·}{u}}_{n-1}+(1-\frac{\mathrm{\γ}}{2\mathrm{\β}}){\stackrel{\·\·}{u}}_{n-1}\mathrm{dt}\end{array}\right.---\left(13\right)$

Step 8.4: comprehensive (13), (12), (11), (10) obtain interfacial force λ between minor structure _nwith the dynamic respond u of system _ntime recursion formula:

$\left\{\begin{array}{c}{\mathrm{\λ}}_n={(E+{\mathrm{KBYB}}^T)}^{-1}\mathrm{KB}{\tilde{u}}_n\\ u_n={\tilde{u}}_n-\mathrm{YB}{\mathrm{\λ}}_n\end{array}\right.---\left(14\right)$

In formula:

$E=\left[\begin{array}{cc}0& 0\\ 0& I_{\mathrm{kk}}\end{array}\right]---\left(15\right)$

$K=\left[\begin{array}{cc}{I}_{\mathrm{rr}}& 0\\ 0& k\end{array}\right]---\left(16\right)$

B＝[B ^(r)B ^(s)] (17)

$Y=\left[\begin{array}{cc}{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}& 0\\ 0& H_1^{\left(s\right)}\mathrm{dt}/2\end{array}\right]---\left(18\right)$

${\tilde{u}}_n={\left[\begin{array}{cc}{R}_n^{{\left(r\right)}^T}& S_n^{{\left(s\right)}^T}\end{array}\right]}^T---\left(19\right)$

concrete recursion is respectively as shown in formula (20), formula (22):

$R_n^{\left(r\right)}={\underset{\‾}{u}}_n^{\left(r\right)}+{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}(f_n^{\left(r\right)}-K^{\left(r\right)}{\underset{\‾}{u}}_n^{\left(r\right)}-C^{\left(r\right)}{\underset{\‾}{\overset{\·}{u}}}_n^{\left(r\right)})---\left(20\right)$

Wherein:

$\left\{\begin{array}{c}{\underset{\‾}{u}}_n^{\left(r\right)}=u_{n-1}^{\left(r\right)}+\mathrm{dt}\·{\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2(1/2-\mathrm{\β}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\\ {\underset{\‾}{M}}^{\left(r\right)}=M^{\left(r\right)}+C^{\left(r\right)}\mathrm{\γdt}+K^{\left(r\right)}{\mathrm{dt}}^2\mathrm{\β}\\ {\underset{\‾}{\overset{\·}{u}}}_n^{\left(r\right)}={\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt}(1-\mathrm{\γ}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\end{array}\right.---\left(21\right)$

$\begin{array}{c}{S}_n^{\left(s\right)}=\underset{i=0}{\overset{n-2}{\mathrm{\Σ}}}{H}_{n-i}^{\left(s\right)}[f_i^{\left(s\right)}+f_{i+1}^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_i-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{i+1}]\frac{\mathrm{dt}}{2}\\ +H_1^{\left(s\right)}(f_{n-1}^{\left(s\right)}+f_n^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{n-1})\frac{\mathrm{dt}}{2}\end{array}---\left(22\right)$

Wherein I _rr, I _kkrepresentation unit matrix, it is identical that exponent number is rigidly connected with junction, interface respectively, elasticity is connected number of degrees of freedom, and k is elasticity junction stiffness matrix.

Try to achieve displacement components u _nafter, obtain acceleration by step 8.3 Chinese style (13) and speed according to formula (14), by time stepping method, obtain the displacement components u of whole simulation process, speed acceleration response results, namely complete true reflection pulse minor structure firm with finite element-Structural Dynamics that plays Hybrid connections part simulates.

A kind of pulse minor structure disclosed by the invention and finite element just-play the dynamics simulation algorithm of Hybrid connections, specific implementation step is as follows:

Step 1: definition fundamental unknown variables is displacement function column vector u (t) of system, velocity function column vector respectively acceleration function column vector and junction, interface minor structure intermolecular forces function row vector λ (t);

Step 2: the impulse response function matrix H being obtained each minor structure by Newmark method or test method ^(s)(t), wherein s represents the sequence number of minor structure; H ^(s)(t) entry of a matrix element the dynamic respond of a jth degree of freedom i-th degree of freedom under pulse excitation of expression system s minor structure;

Step 3: obtain the quality of finite element minor structure, damping, stiffness matrix M by Finite Element Method ^(r), C ^(r), K ^(r), wherein r represents the sequence number of minor structure.

Step 4: according to the annexation between minor structure, tries to achieve Boolean matrix B ^(s), B ^(r), B ^(s)effect be the interface degree of freedom filtering out s pulse minor structure from the degree of freedom of s pulse minor structure, B ^(r)effect be the interface degree of freedom filtering out r pulse minor structure from the degree of freedom of r pulse minor structure.

Step 5: connect degree of freedom according to linkage interface elasticity and set up stiffness matrix k;

Step 6: initialization fundamental unknown variables, namely the initial time system of t=0 is not by External Force Acting, and the fundamental unknown variables in step 1 is all null vector;

Step 7: solving system is at external force f ^(s)(t), f ^(r)response under (t) effect, in step 1, the time recursion formula of four groups of fundamental unknown variables is as follows:

$\left\{\begin{array}{c}{\mathrm{\λ}}_n={(E+{\mathrm{KBYB}}^T)}^{-1}\mathrm{KB}{\tilde{u}}_n\\ u_n={\tilde{u}}_n-\mathrm{YB}{\mathrm{\λ}}_n\\ {\stackrel{..}{u}}_n=\frac{1}{\mathrm{\β}{\mathrm{dt}}^2}(u_n-u_{n-1})-\frac{1}{\mathrm{\βdt}}{\stackrel{.}{u}}_{n-1}-(\frac{1}{2\mathrm{\β}}-1){\stackrel{..}{u}}_{n-1}\\ {\stackrel{.}{u}}_n=\frac{\mathrm{\γ}}{\mathrm{\βdt}}(u_n-u_{n-1})+(1-\frac{\mathrm{\γ}}{\mathrm{\β}}){\stackrel{.}{u}}_{n-1}+(1-\frac{\mathrm{\γ}}{2\mathrm{\β}}){\stackrel{..}{u}}_{n-1}\mathrm{dt}\end{array}\right.---\left(23\right)$

Wherein, dt is the integration step determined, subscript represents the moment, as u _n=u (t=ndt), β and γ is the dimensionless group of Newmark method, in formula:

$E=\left[\begin{array}{cc}0& 0\\ 0& I_{\mathrm{kk}}\end{array}\right]$

$K=\left[\begin{array}{cc}{I}_{\mathrm{rr}}& 0\\ 0& k\end{array}\right]$

B＝[B ^(r)B ^(s)]

$Y=\left[\begin{array}{cc}{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}& 0\\ 0& H_1^{\left(s\right)}\mathrm{dt}/2\end{array}\right]$

${\tilde{u}}_n={\left[\begin{array}{cc}{R}_n^{{\left(r\right)}^T}& S_n^{{\left(s\right)}^T}\end{array}\right]}^T$

I _rr, I _kkrepresentation unit matrix, it is identical that exponent number is rigidly connected with junction, interface respectively, elasticity is connected number of degrees of freedom.

obtained by following:

$R_n^{\left(r\right)}={\underset{\‾}{u}}_n^{\left(r\right)}+{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}(f_n^{\left(r\right)}-K^{\left(r\right)}{\underset{\‾}{u}}_n^{\left(r\right)}-C^{\left(r\right)}{\stackrel{.}{\underset{\‾}{u}}}_n^{\left(r\right)})$

$\begin{array}{c}{S}_n^{\left(s\right)}=\underset{i=0}{\overset{n-2}{\mathrm{\Σ}}}{H}_{n-i}^{\left(s\right)}[f_i^{\left(s\right)}+f_{i+1}^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_i-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{i+1}]\frac{\mathrm{dt}}{2}\\ +H_1^{\left(s\right)}(f_{n-1}^{\left(s\right)}+f_n^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{n-1})\frac{\mathrm{dt}}{2}\end{array}$

Wherein

$\left\{\begin{array}{c}{\underset{\‾}{u}}_n^{\left(r\right)}=u_{n-1}^{\left(r\right)}+\mathrm{dt}\·{\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2(1/2-\mathrm{\β}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\\ {\underset{\‾}{M}}^{\left(r\right)}=M^{\left(r\right)}+C^{\left(r\right)}\mathrm{\γdt}+K^{\left(r\right)}{\mathrm{dt}}^2\mathrm{\β}\\ {\underset{\‾}{\stackrel{\·}{u}}}_n^{\left(r\right)}={\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt}(1-\mathrm{\γ}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\end{array}\right.$

Beneficial effect:

1, based on impulse response function, a kind of pulse minor structure of the present invention and finite element just-play dynamics simulation method and the algorithm of Hybrid connections, overcoming that conventional I BS method is only suitable for analyzing between pulse minor structure is rigidly connected defect, supplement pulse minor structure and finite element minor structure just-play the form of Hybrid connections, more real simulation spationautics field structure dynamical motion process.

2, a kind of pulse minor structure of the present invention and finite element just-play dynamics simulation method and the algorithm of Hybrid connections, expanded the range of application of conventional I BS method.

Accompanying drawing explanation

Fig. 1 be a kind of pulse minor structure and finite element just-play the schematic flow sheet of the dynamics simulation method of Hybrid connections.

Fig. 2 be a kind of pulse minor structure and finite element just-play the schematic flow sheet of the dynamics simulation algorithm of Hybrid connections.

Fig. 3 is 11 degree of freedom spring-dampers-quality systems in embodiment.

Fig. 4 is the external applied load-time curve in embodiment in system the 9th degree of freedom.

Fig. 5 is the acceleration responsive-time curve of system the 10th degree of freedom in embodiment.

Embodiment

Embodiment 1.

In order to better set forth object of the present invention and effect, the dynamic analysis below by the spring-dampers-quality system to 11 degree of freedom is explained in detail the present invention.Analyze step-length and get 1ms, analysis time is 1s.

As shown in Figure 3, the system of the present embodiment is divided into two minor structures.Minor structure 1 comprises 7 degree of freedom, is pulse minor structure; Minor structure 2 comprises 4 degree of freedom, is finite element minor structure.The m of minor structure 1 ₆and m ₇with the m of minor structure 2 ₈elasticity is connected, as shown in Figure 1.The kinetic parameter of minor structure and web member is in table 1.Assuming that external force is only loaded into m ₉on, regulation is to the right positive dirction, and the time curve of external applied load is as shown in Figure 4.

Table 1 spring-mass block system parameter list

As shown in Figure 1, a kind of pulse minor structure of the present embodiment and finite element just-play the dynamics simulation method of Hybrid connections, performing step is as follows:

Step 1: carry out minor structure division.Whole system is divided into pulse minor structure and finite element minor structure two parts.The motion vector u of system represents with the motion vector of all minor structures, and piecemeal is u ^(r), u ^(s).

Step 2: structure Boolean matrix

$B_r^{\left(s\right)}=\left[\begin{array}{ccccccc}0& 0& 0& 0& 1& 0& 0\end{array}\right]$

$B_k^{\left(s\right)}=\left[\begin{array}{ccccccc}0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

$B_r^{\left(r\right)}=\left[\begin{array}{cccc}-1& 0& 0& 0\end{array}\right]$

$B_k^{\left(r\right)}=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right]$

Step 3: introduce Lagrange multiplier λ, λ and represent system interface junction attachment force array.

Step 4: set up pulse minor structure kinetics equation.

Step 4.1, obtains the impulse response function matrix H of each pulse minor structure by numerical integration method Newmark method or test method ^(s)(t), the wherein sequence number of s indicating impulse minor structure, H ^(s)(t) entry of a matrix element the dynamic respond of a jth degree of freedom i-th degree of freedom under pulse excitation of expression system s pulse minor structure.

Step 4.2: set up pulse minor structure kinetics equation by impulse response function matrix:

From Duhamel integration, the kinetics equation of system pulses minor structure is:

$u^{\left(s\right)}\left(t\right)={\∫}_0^t{H}^{\left(s\right)}(t-\mathrm{\τ})[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}]\mathrm{d\τ}---\left(24\right)$

Wherein

$B^{\left(s\right)}=\left[\begin{array}{ccccccc}0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

Step 5: set up finite element minor structure kinetics equation:

$M^{\left(r\right)}{\stackrel{\·\·}{u}}^{\left(r\right)}\left(t\right)+C^{\left(r\right)}{\stackrel{\·}{u}}^{\left(r\right)}\left(t\right)+K^{\left(r\right)}{u}^{\left(r\right)}\left(t\right)=f^{\left(r\right)}\left(t\right)+B^{{\left(r\right)}^T}\mathrm{\λ}---\left(25\right)$

Wherein

$M^{\left(r\right)}=\left[\begin{array}{cccc}{m}_8& 0& 0& 0\\ 0& m_9& 0& 0\\ 0& 0& m_{10}& 0\\ 0& 0& 0& m_{11}\end{array}\right]=\left[\begin{array}{cccc}10& 0& 0& 0\\ 0& 10& 0& 0\\ 0& 0& 5& 0\\ 0& 0& 0& 5\end{array}\right]\mathrm{kg}$

$K^{\left(r\right)}=\left[\begin{array}{cccc}{k}_{11}& -k_{11}& 0& 0\\ -k_{11}& k_{11}+k_{12}+k_{13}& -k_{12}& -k_{13}\\ 0& -k_{12}& k_{12}+k_{14}& 0\\ 0& {-k}_{13}& 0& k_{13}+k_{15}\end{array}\right]=\left[\begin{array}{cccc}2& -1& 0& 0\\ -1& 4& -1& -1\\ 0& -1& 4& 0\\ 0& -1& 0& 4\end{array}\right]\mathrm{kN}/m$

$B^{\left(r\right)}=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right]$

Step 6: determine pulse minor structure and finite element Substructure Interfaces consistency condition.Comprise: between the degree of freedom between the minor structure of interface that is rigidly connected identical and elasticity linkage interface virgin structure, degree of freedom meets Hooke's law.

The degree of freedom between the minor structure of interface that is rigidly connected is identical, meets

$u_r^{\left(s\right)}=u_r^{\left(r\right)}---\left(26\right)$

Between elasticity linkage interface virgin structure, degree of freedom meets Hooke's law, meets

$k(u_k^{\left(s\right)}+u_k^{\left(r\right)})={\mathrm{\λ}}_k---\left(27\right)$

Wherein, linkage interface elasticity connection degree of freedom sets up stiffness matrix k:

$k=\left[\begin{array}{ccc}{k}_9& 0& -k_9\\ 0& k_{10}& -k_{10}\\ -k_9& -k_{10}& k_9+k_{10}\end{array}\right]=\left[\begin{array}{ccc}2& 0& -2\\ 0& 2& -2\\ -2& -2& 4\end{array}\right]\mathrm{kN}/m$

Step 7: obtain Substructure Synthesis equation by formula (24) (25) (26) (27):

$\left\{\begin{array}{c}{u}^{\left(s\right)}\left(t\right)={\∫}_0^t{H}^{\left(s\right)}(t-\mathrm{\τ})[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}]\mathrm{d\τ}\\ M^{\left(r\right)}{\stackrel{\·\·}{u}}^{\left(r\right)}\left(t\right)+C^{\left(r\right)}{\stackrel{\·}{u}}^{\left(r\right)}\left(t\right)+K^{\left(r\right)}{u}^{\left(r\right)}\left(t\right)=f^{\left(r\right)}\left(t\right)+B^{{\left(r\right)}^T}\mathrm{\λ}\\ B_r^{\left(s\right)}{u}^{\left(s\right)}+B_r^{\left(r\right)}{u}^{\left(r\right)}=0\\ k(B_k^{\left(s\right)}{u}^{\left(s\right)}+B_k^{\left(s\right)}{u}^{\left(s\right)})={\mathrm{\λ}}_k\end{array}\right.---\left(28\right)$

Step 8: carry out time discrete to the Substructure Synthesis equation that step 7 obtains, obtains the dynamic respond u of system _nand interfacial force λ between minor structure _ntime recursion formula.

Step 8.1: paired pulses minor structure equation of motion formula (24) carries out time discrete:

$u_n^{\left(s\right)}=\underset{i=0}{\overset{n-1}{\mathrm{\Σ}}}{H}_{n-i}^{\left(s\right)}[f_i^{\left(s\right)}+f_{i+1}^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_i-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{i+1}]\frac{\mathrm{dt}}{2}---\left(29\right)$

In formula, dt is integration step, and subscript represents the moment (as u _n=u (ndt));

Step 8.2: formula (28) second formula adopts following differential mode discrete:

$\left\{\begin{array}{c}{\stackrel{\·}{u}}_n^{\left(r\right)}={\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt}(1-\mathrm{\γ}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt\γ}{\stackrel{\·\·}{u}}_n^{\left(r\right)}\\ u_n^{\left(r\right)}=u_{n-1}^{\left(r\right)}+\mathrm{dt}\·{\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2(1/2-\mathrm{\β}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2\mathrm{\β}{\stackrel{\·\·}{u}}_n^{\left(r\right)}\end{array}\right.---\left(30\right)$

In formula, γ, β are integration constant value 0.5,0.25 respectively.

Step 8.3: the speed utilizing Newmark method representation system and acceleration

$\left\{\begin{array}{c}{\stackrel{\·\·}{u}}_n=\frac{1}{\mathrm{\β}{\mathrm{dt}}^2}(u_n-u_{n-1})-\frac{1}{\mathrm{\βdt}}{\stackrel{\·}{u}}_{n-1}-(\frac{1}{2\mathrm{\β}}-1){\stackrel{\·\·}{u}}_{n-1}\\ {\stackrel{\·}{u}}_n=\frac{\mathrm{\γ}}{\mathrm{\βdt}}(u_n-u_{n-1})+(1-\frac{\mathrm{\γ}}{\mathrm{\β}}){\stackrel{\·}{u}}_{n-1}+(1-\frac{\mathrm{\γ}}{2\mathrm{\β}}){\stackrel{\·\·}{u}}_{n-1}\mathrm{dt}\end{array}\right.---\left(31\right)$

Step 8.4: aggregative formula (31), (30), (29), (28) obtain interfacial force λ between minor structure _nwith the dynamic respond u of system _ntime recursion formula:

$\left\{\begin{array}{c}{\mathrm{\λ}}_n={(E+{\mathrm{KBYB}}^T)}^{-1}\mathrm{KB}{\tilde{u}}_n\\ u_n={\tilde{u}}_n-\mathrm{YB}{\mathrm{\λ}}_n\end{array}\right.---\left(32\right)$

Wherein

$E=\left[\begin{array}{cc}0& 0\\ 0& I_{\mathrm{kk}}\end{array}\right]=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

$K=\left[\begin{array}{cc}{I}_{\mathrm{rr}}& 0\\ 0& k\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 2\mathrm{kN}/m& & -2\mathrm{kN}/m\\ 0& & 2\mathrm{kN}/m& -2\mathrm{kN}/m\\ 0& -2\mathrm{kN}/m& -2\mathrm{kN}/m& 4\mathrm{kN}/m\end{array}\right]$

$B=\left[\begin{array}{cc}{B}^{\left(r\right)}& B^{\left(s\right)}\end{array}\right]=\left[\begin{array}{ccccccccccc}-1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

$Y=\left[\begin{array}{cc}{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}& 0\\ 0& H_1^{\left(s\right)}\mathrm{dt}/2\end{array}\right]$

${\tilde{u}}_n={\left[\begin{array}{cc}{R}_n^{{\left(r\right)}^T}& S_n^{{\left(s\right)}^T}\end{array}\right]}^T$

I _rr, I _kkrepresentation unit matrix, exponent number difference 1 and 3.

obtained by following:

$R_n^{\left(r\right)}={\underset{\‾}{u}}_n^{\left(r\right)}+{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}(f_n^{\left(r\right)}-K^{\left(r\right)}{\underset{\‾}{u}}_n^{\left(r\right)}-C^{\left(r\right)}{\stackrel{\·}{\underset{\‾}{u}}}_n^{\left(r\right)})$

$S_n^{\left(s\right)}=\underset{i=0}{\overset{n-2}{\mathrm{\Σ}}}{H}_{n-i}^{\left(s\right)}[f_i^{\left(s\right)}+f_{i+1}^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_i-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{i+1}]\frac{\mathrm{dt}}{2}+H_1^{\left(s\right)}(f_{n-1}^{\left(s\right)}+f_n^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{n-1})\frac{\mathrm{dt}}{2}$

Wherein

$\left\{\begin{array}{c}{\underset{\‾}{u}}_n^{\left(r\right)}=u_{n-1}^{\left(r\right)}+\mathrm{dt}\·{\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2(1/2-\mathrm{\β}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\\ {\underset{\‾}{M}}^{\left(r\right)}=M^{\left(r\right)}+C^{\left(r\right)}\mathrm{\γdt}+K^{\left(r\right)}{\mathrm{dt}}^2\mathrm{\β}\\ {\underset{\‾}{\overset{\·}{u}}}_n^r={\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt}(1-\mathrm{\γ}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\end{array}\right.$

So far complete pulse minor structure and finite element just-play the dynamics simulation method example of Hybrid connections.

As shown in Figure 2, a kind of pulse minor structure of the present embodiment and finite element just-play the dynamics simulation algorithm of Hybrid connections, performing step is as follows:

Step 1: definition fundamental unknown variables is displacement function column vector u (t) of system, velocity function column vector respectively acceleration function column vector ü (t) and junction, interface minor structure intermolecular forces function row vector λ (t);

Step 2: by the impulse response function matrix H of Newmark method numerically operated part 1 ^(s)(t).

Step 3: obtain the quality of minor structure 2, damping, stiffness matrix by Finite Element Method.Do not consider the damping of structure herein, its damping matrix is set to zero C ^(r)=0.

$M^{\left(r\right)}=\left[\begin{array}{cccc}{m}_8& 0& 0& 0\\ 0& m_9& 0& 0\\ 0& 0& m_{10}& 0\\ 0& 0& 0& m_{11}\end{array}\right]=\left[\begin{array}{cccc}10& 0& 0& 0\\ 0& 10& 0& 0\\ 0& 0& 5& 0\\ 0& 0& 0& 5\end{array}\right]\mathrm{kg}$

$K^{\left(r\right)}=\left[\begin{array}{cccc}{k}_{11}& -k_{11}& 0& 0\\ -k_{11}& k_{11}+k_{12}+k_{13}& -k_{12}& -k_{13}\\ 0& -k_{12}& k_{12}+k_{14}& 0\\ 0& -k_{13}& 0& k_{13}+k_{15}\end{array}\right]=\left[\begin{array}{cccc}2& -1& 0& 0\\ -1& 4& -1& -1\\ 0& -1& 4& 0\\ 0& -1& 0& 4\end{array}\right]\mathrm{kN}/m$

Step 4: according to the annexation between minor structure, tries to achieve Boolean matrix B ^(s), B ^(r).

$B^{\left(s\right)}=\left[\begin{array}{ccccccc}0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

$B^{\left(r\right)}=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right]$

Step 5: connect degree of freedom according to linkage interface elasticity and set up stiffness matrix k:

$k=\left[\begin{array}{ccc}{k}_9& 0& -k_9\\ 0& k_{10}& -k_{10}\\ -k_9& -k_{10}& k_9+k_{10}\end{array}\right]=\left[\begin{array}{ccc}2& 0& -2\\ 0& 2& -2\\ -2& -2& 4\end{array}\right]\mathrm{kN}/m$

Step 6: initialization fundamental unknown variables, namely the initial time system of t=0 is not by External Force Acting, and the fundamental unknown variables in step 1 is all null vector:

λ(0)＝[0 0 0 0] ^T

Step 7: solving system is at external force f ^(s)(t), f ^(r)response under (t) effect, only f in the present embodiment ^(r)t () the 2nd element non-zero, the 2nd element as shown in Figure 2.Get dimensionless group β=0.25 of Newmark method, γ=0.5:

$\left\{\begin{array}{c}{\mathrm{\λ}}_n={(E+{\mathrm{KBYB}}^T)}^{-1}\mathrm{KB}{\tilde{u}}_n\\ u_n={\tilde{u}}_n-\mathrm{YB}{\mathrm{\λ}}_n\\ {\stackrel{\·\·}{u}}_n=\frac{1}{\mathrm{\β}{\mathrm{dt}}^2}(u_n-u_{n-1})-\frac{1}{\mathrm{\βdt}}{\stackrel{\·}{u}}_{n-1}-(\frac{1}{2\mathrm{\β}}-1){\stackrel{\·\·}{u}}_{n-1}\\ {\stackrel{\·}{u}}_n=\frac{\mathrm{\γ}}{\mathrm{\βdt}}(u_n-u_{n-1})+(1-\frac{\mathrm{\γ}}{\mathrm{\β}}){\stackrel{\·}{u}}_{n-1}+(1-\frac{\mathrm{\γ}}{2\mathrm{\β}}){\stackrel{\·\·}{u}}_{n-1}\mathrm{dt}\end{array}\right.---\left(33\right)$

Wherein, dt is 0.001s, and subscript represents the moment, as u _n=u (t=ndt), β and γ is the dimensionless group of Newmark method, in formula:

$E=\left[\begin{array}{cc}0& 0\\ 0& I_{\mathrm{kk}}\end{array}\right]=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

$K=\left[\begin{array}{cc}{I}_{\mathrm{rr}}& 0\\ 0& k\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 2\mathrm{kN}/m& & -2\mathrm{kN}/m\\ 0& & 2\mathrm{kN}/m& -2\mathrm{kN}/m\\ 0& -2\mathrm{kN}/m& -2\mathrm{kN}/m& 4\mathrm{kN}/m\end{array}\right]$

$B=\left[\begin{array}{cc}{B}^{\left(r\right)}& B^{\left(s\right)}\end{array}\right]=\left[\begin{array}{ccccccccccc}-1& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

$Y=\left[\begin{array}{cc}{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}& 0\\ 0& H_1^{\left(s\right)}\mathrm{dt}/2\end{array}\right]$

${\tilde{u}}_n={\left[\begin{array}{cc}{R}_n^{{\left(r\right)}^T}& S_n^{{\left(s\right)}^T}\end{array}\right]}^T$

I _rr, I _kkrepresentation unit matrix, exponent number difference 1 and 3.

obtained by following:

$R_n^{\left(r\right)}={\underset{\‾}{u}}_n^{\left(r\right)}+{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}(f_n^{\left(r\right)}-K^{\left(r\right)}{\underset{\‾}{u}}_n^{\left(r\right)}-C^{\left(r\right)}{\underset{\‾}{\overset{\·}{u}}}_n^{\left(r\right)})$

$S_n^{\left(s\right)}=\underset{i=0}{\overset{n-2}{\mathrm{\Σ}}}{H}_{n-i}^{\left(s\right)}[f_i^{\left(s\right)}+f_{i+1}^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_i-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{i+1}]\frac{\mathrm{dt}}{2}+H_1^{\left(s\right)}(f_{n-1}^{\left(s\right)}+f_n^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{n-1})\frac{\mathrm{dt}}{2}$

Wherein

$\left\{\begin{array}{c}{\underset{\‾}{u}}_n^{\left(r\right)}=u_{n-1}^{\left(r\right)}+\mathrm{dt}\·{\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2(1/2-\mathrm{\β}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\\ {\underset{\‾}{M}}^{\left(r\right)}=M^{\left(r\right)}+C^{\left(r\right)}\mathrm{\γdt}+K^{\left(r\right)}{\mathrm{dt}}^2\mathrm{\β}\\ {\underset{\‾}{\overset{\·}{u}}}_n^{\left(r\right)}={\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt}(1-\mathrm{\γ}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\end{array}\right..$

With m ₁₀response be example, adopt the present embodiment emulate the m obtained ₁₀acceleration responsive in contrast to Fig. 5 with using Newmark method (Newmark method is the algorithm that art-recognized computational accuracy is very high, but this algorithm counting yield the is low) result that obtains as benchmark, relative peak error is 1.05 × 10-11.As can be seen here, the present embodiment can not only process pulse minor structure and finite element minor structure just-play the minor structure problem of Hybrid connections, and have high precision.

Above-described specific descriptions; the object of inventing, technical scheme and beneficial effect are further described; be understood that; the foregoing is only specific embodiments of the invention; for explaining the present invention, the protection domain be not intended to limit the present invention, within the spirit and principles in the present invention all; any amendment of making, equivalent replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (2)

1. pulse minor structure and finite element just-play the dynamics simulation method of Hybrid connections, it is characterized in that: comprise the steps,

Step 1: carry out minor structure division; Whole system is divided into pulse minor structure and finite element minor structure two parts; The motion vector u of system represents with the motion vector of all minor structures, and piecemeal is u ^(r), u ^(s), as shown in formula (1):

$u={\left[\begin{array}{cc}{u}^{{\left(r\right)}^T}& u^{{\left(s\right)}^T}\end{array}\right]}^T---\left(1\right)$

In formula, subscript r represents that this variable belongs to r pulse minor structure, and subscript s represents that this variable belongs to s finite element minor structure, T representing matrix transposition;

Step 2: structure Boolean matrix; The effect of Boolean matrix is that each minor structure degree of freedom is projected to junction, interface;

By structure Boolean matrix by the displacement components u of pulse minor structure ^(s)project to linkage interface, Boolean matrix tried to achieve by formula (2)

$\left\{\begin{array}{c}{u}_r^{\left(s\right)}=B_r^{\left(s\right)}{u}^{\left(s\right)}\\ u_k^{\left(s\right)}=B_k^{\left(s\right)}{u}^{\left(s\right)}\end{array}\right.---\left(2\right)$

Wherein subscript r represents that this variable belongs to the place that is rigidly connected, and subscript k represents that this variable belongs to elasticity junction;

Equally, Boolean matrix is constructed by the displacement components u of finite element minor structure ^(r)project to junction, interface, Boolean matrix tried to achieve by formula (3):

$\left\{\begin{array}{c}{u}_r^{\left(r\right)}=-B_r^{\left(r\right)}{u}^{\left(r\right)}\\ u_k^{\left(r\right)}=B_k^{\left(r\right)}{u}^{\left(r\right)}\end{array}\right.---\left(3\right)$

Step 3: introduce Lagrange multiplier λ, λ and represent system interface junction attachment force array, the Boolean matrix obtained by step 2, have the relation in formula (4):

$\left\{\begin{array}{c}{\mathrm{\λ}}_r^{\left(s\right)}=B_r^{{\left(s\right)}^T}\mathrm{\λ}\\ {\mathrm{\λ}}_k^{\left(s\right)}=B_k^{{\left(s\right)}^T}\mathrm{\λ}\\ {\mathrm{\λ}}_r^{\left(r\right)}=B_r^{{\left(r\right)}^T}\mathrm{\λ}\\ {\mathrm{\λ}}_k^{\left(r\right)}=B_k^{{\left(r\right)}^T}\mathrm{\λ}\end{array}\right.---\left(4\right)$

Step 4: set up pulse minor structure kinetics equation, concrete steps are as follows:

Step 4.1, obtains the impulse response function matrix H of each pulse minor structure by numerical integration method Newmark method or test method ^(s)(t), the wherein sequence number of s indicating impulse minor structure, H ^(s)(t) entry of a matrix element the dynamic respond of a jth degree of freedom i-th degree of freedom under pulse excitation of expression system s pulse minor structure;

Step 4.2: set up pulse minor structure kinetics equation by impulse response function matrix:

From Duhamel integration, the kinetics equation of system s pulse minor structure is:

$u^{\left(s\right)}\left(t\right)={\∫}_0^t{H}^{\left(s\right)}(t-\mathrm{\τ})[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}]\mathrm{d\τ}---\left(5\right)$

In formula, u ^(s)t () is the displacement array of s pulse minor structure; f ^(s)(t) external applied load array suffered by pulse minor structure; $B^{\left(s\right)}=\left[\begin{array}{cc}{B}_r^{\left(s\right)}& B_k^{\left(s\right)}\end{array}\right]$ Tried to achieve by step 2 Chinese style (2), from step 3 formula (4), indicating impulse Substructure Interfaces place attachment force array;

Step 5: set up finite element minor structure kinetics equation:

$M^{\left(r\right)}{\stackrel{\·\·}{u}}^{\left(r\right)}\left(t\right)+C^{\left(r\right)}{\stackrel{\·}{u}}^{\left(r\right)}\left(t\right)+K^{\left(r\right)}{u}^{\left(r\right)}\left(t\right)=f^{\left(r\right)}\left(t\right)+B^{{\left(r\right)}^T}\mathrm{\λ}---\left(6\right)$

In formula: M ^(r), C ^(r), K ^(r)be respectively the quality of r finite element minor structure, damping, stiffness matrix; u ^(r)t () is respectively r finite element minor structure acceleration, speed, displacement array; f ^(r)(t) external drive power array suffered by r finite element minor structure; $B^{\left(r\right)}=\left[\begin{array}{cc}{B}_r^{\left(r\right)}& B_k^{\left(r\right)}\end{array}\right]$ Tried to achieve by step 2 Chinese style (3), from step 3 formula (4), represent the attachment force array at finite element Substructure Interfaces place;

Step 6: determine pulse minor structure and finite element Substructure Interfaces consistency condition;

Described pulse minor structure and finite element Substructure Interfaces consistency condition meet: between the degree of freedom between the minor structure of interface that is rigidly connected identical and elasticity linkage interface virgin structure, degree of freedom meets Hooke's law;

The degree of freedom between the minor structure of interface that is rigidly connected is identical, has following relation, as shown in formula (7):

$u_r^{\left(s\right)}=u_r^{\left(r\right)}---\left(7\right)$

Between elasticity linkage interface virgin structure, degree of freedom meets Hooke's law, has following relation, as shown in formula (8):

$k(u_k^{\left(s\right)}+u_k^{\left(r\right)})={\mathrm{\λ}}_k---\left(8\right)$

Wherein λ _kas shown in formula (9):

${\mathrm{\λ}}_k=\left[\begin{array}{c}{\mathrm{\λ}}_k^{\left(r\right)}\\ {\mathrm{\λ}}_k^{\left(s\right)}\end{array}\right]---\left(9\right)$

Step 7: obtain Substructure Synthesis equation, concrete grammar is: the first formula of (2) in step 2 and first formula of (3) are brought into (7), and the second formula of (2) in step 2 and second formula of (3) are brought into (8), (6) formula in integrating step 2.2 in (5) formula and step 3, obtains Substructure Synthesis equation as shown in formula (10):

$\left\{\begin{array}{c}{u}^{\left(s\right)}\left(t\right)={\∫}_0^t{H}^{\left(s\right)}(t-\mathrm{\τ})[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}]\mathrm{d\τ}\\ M^{\left(r\right)}{\stackrel{\·\·}{u}}^{\left(r\right)}\left(t\right)+C^{\left(r\right)}{\stackrel{\·}{u}}^{\left(r\right)}\left(t\right)+K^{\left(r\right)}{u}^{\left(r\right)}\left(t\right)=f^{\left(r\right)}\left(t\right)+B^{{\left(r\right)}^T}\mathrm{\λ}\\ B_r^{\left(s\right)}{u}^{\left(s\right)}+B_r^{\left(r\right)}{u}^{\left(r\right)}=0\\ k(B_k^{\left(s\right)}{u}^{\left(s\right)}+B_k^{\left(s\right)}{u}^{\left(s\right)})={\mathrm{\λ}}_k\end{array}\right.---\left(10\right)$

Step 8: carry out time discrete to the Substructure Synthesis equation that step 7 obtains, obtains the dynamic respond u of system _nand interfacial force λ between minor structure _ntime recursion formula, complete pulse minor structure and finite element just-Structural Dynamics that plays Hybrid connections part simulates; Concrete steps are:

Step 8.1: time discrete is carried out to (10) first formulas, has:

$u_n^{\left(s\right)}=\underset{i=0}{\overset{n-1}{\mathrm{\Σ}}}{H}_{n-i}^{\left(s\right)}[f_i^{\left(s\right)}+f_{i+1}^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_i-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{i+1}]\frac{\mathrm{dt}}{2}---\left(11\right)$

In formula, dt is integration step, and subscript represents the moment (as u _n=u (ndt));

Step 8.2: adopt following differential mode discrete to (10) second formulas:

$\left\{\begin{array}{c}{\stackrel{\·}{u}}_n^{\left(r\right)}={\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt}(1-\mathrm{\γ}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt\γ}{\stackrel{\·\·}{u}}_n^{\left(r\right)}\\ u_n^{\left(r\right)}=u_{n-1}^{\left(r\right)}+\mathrm{dt}\·{\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2(1/2-\mathrm{\β}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2\mathrm{\β}{\stackrel{\·\·}{u}}_n^{\left(r\right)}\end{array}\right.---\left(12\right)$

In formula, γ, β are integration constant;

Step 8.3: the speed utilizing Newmark method representation system and acceleration

$\left\{\begin{array}{c}{\stackrel{\·\·}{u}}_n=\frac{1}{\mathrm{\β}{\mathrm{dt}}^2}(u_n-u_{n-1})-\frac{1}{\mathrm{\βdt}}{\stackrel{\·}{u}}_{n-1}-(\frac{1}{2\mathrm{\β}}-1){\stackrel{\·\·}{u}}_{n-1}\\ {\stackrel{\·}{u}}_n=\frac{\mathrm{\γ}}{\mathrm{\βdt}}(u_n-u_{n-1})+(1-\frac{\mathrm{\γ}}{\mathrm{\β}}){\stackrel{\·}{u}}_{n-1}+(1-\frac{\mathrm{\γ}}{2\mathrm{\β}}){\stackrel{\·\·}{u}}_{n-1}\mathrm{dt}\end{array}\right.---\left(13\right)$

Step 8.4: comprehensive (13), (12), (11), (10) obtain interfacial force λ between minor structure _nwith the dynamic respond u of system _ntime recursion formula:

$\left\{\begin{array}{c}{\mathrm{\λ}}_n={(E+{\mathrm{KBYB}}^T)}^{-1}\mathrm{KB}{\tilde{u}}_n\\ u_n={\tilde{u}}_n-\mathrm{YB}{\mathrm{\λ}}_n\end{array}\right.---\left(14\right)$

In formula:

$E=\left[\begin{array}{cc}0& 0\\ 0& I_{\mathrm{kk}}\end{array}\right]---\left(15\right)$

$K=\left[\begin{array}{cc}{I}_{\mathrm{rr}}& 0\\ 0& k\end{array}\right]---\left(16\right)$

B＝[B ^(r)B ^(s)] (17)

$Y=\left[\begin{array}{cc}{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}& 0\\ 0& H_1^{\left(s\right)}\mathrm{dt}/2\end{array}\right]---\left(18\right)$

${\tilde{u}}_n={\left[\begin{array}{cc}{R}_n^{{\left(r\right)}^T}& S_n^{{\left(s\right)}^T}\end{array}\right]}^T---\left(19\right)$

concrete recursion is respectively as shown in formula (20), formula (22):

$R_n^{\left(r\right)}={\underset{\‾}{u}}_n^{\left(r\right)}+{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}(f_n^{\left(r\right)}-K^{\left(r\right)}{\underset{\‾}{u}}_n^{\left(r\right)}-C^{\left(r\right)}{\underset{\‾}{\stackrel{\·}{u}}}_n^{\left(r\right)})---\left(20\right)$

Wherein:

$\left\{\begin{array}{c}{\underset{\‾}{u}}_n^{\left(r\right)}=u_{n-1}^{\left(r\right)}+\mathrm{dt}\·{\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2(1/2-\mathrm{\β}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\\ {\underset{\‾}{M}}^{\left(r\right)}=M^{\left(r\right)}+C^{\left(r\right)}\mathrm{\γdt}+K^{\left(r\right)}{\mathrm{dt}}^2\mathrm{\β}\\ {\stackrel{\·}{\underset{\‾}{u}}}_n^{\left(r\right)}={\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt}(1-\mathrm{\γ}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\end{array}\right.---\left(21\right)$

$\begin{array}{c}{S}_n^{\left(s\right)}=\underset{i=0}{\overset{n-2}{\mathrm{\Σ}}}{H}_{n-i}^{\left(s\right)}[f_i^{\left(s\right)}+f_{i+1}^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_i-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{i+1}]\frac{\mathrm{dt}}{2}\\ +H_1^{\left(s\right)}(f_{n-1}^{\left(s\right)}+f_n^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{n-1})\frac{\mathrm{dt}}{2}\end{array}---\left(22\right)$

Wherein I _rr, I _kkrepresentation unit matrix, it is identical that exponent number is rigidly connected with junction, interface respectively, elasticity connects number of degrees of freedom, and k is elasticity junction stiffness matrix;

Try to achieve displacement components u _nafter, obtain acceleration by step 8.3 Chinese style (13) and speed according to formula (14), by time recursion, obtain the displacement components u of whole simulation process, speed acceleration response results, completes Structural Dynamics simulation.

2. pulse minor structure and finite element just-play the dynamics simulation algorithm of Hybrid connections, it is characterized in that: comprise the steps,

Step 1: definition fundamental unknown variables is displacement function column vector u (t) of system, velocity function column vector respectively acceleration function column vector and junction, interface minor structure intermolecular forces function row vector λ (t);

Step 2: the impulse response function matrix H being obtained each minor structure by Newmark method or test method ^(s)(t), wherein s represents the sequence number of minor structure; H ^(s)(t) entry of a matrix element the dynamic respond of a jth degree of freedom i-th degree of freedom under pulse excitation of expression system s minor structure;

Step 3: obtain the quality of finite element minor structure, damping, stiffness matrix M by Finite Element Method ^(r), C ^(r), K ^(r), wherein r represents the sequence number of minor structure;

Step 4: according to the annexation between minor structure, tries to achieve Boolean matrix B ^(s), B ^(r), B ^(s)effect be the interface degree of freedom filtering out s pulse minor structure from the degree of freedom of s pulse minor structure, B ^(r)effect be the interface degree of freedom filtering out r pulse minor structure from the degree of freedom of r pulse minor structure;

Step 5: connect degree of freedom according to linkage interface elasticity and set up stiffness matrix k;

Step 6: initialization fundamental unknown variables, namely the initial time system of t=0 is not by External Force Acting, and the fundamental unknown variables in step 1 is all null vector;

Step 7: solving system is at external force f ^(s)(t), f ^(r)response under (t) effect, in step 1, the time recursion formula of four groups of fundamental unknown variables is as follows:

$\left\{\begin{array}{c}{\mathrm{\λ}}_n={(E+{\mathrm{KBYB}}^T)}^{-1}\mathrm{KB}{\tilde{u}}_n\\ u_n={\tilde{u}}_n-\mathrm{YB}{\mathrm{\λ}}_n\\ {\stackrel{\·\·}{u}}_n=\frac{1}{\mathrm{\β}{\mathrm{dt}}^2}(u_n-u_{n-1})-\frac{1}{\mathrm{\βdt}}{\stackrel{\·}{u}}_{n-1}-(\frac{1}{2\mathrm{\β}}-1){\stackrel{\·\·}{u}}_{n-1}\\ {\stackrel{\·}{u}}_n=\frac{\mathrm{\γ}}{\mathrm{\βdt}}(u_n-u_{n-1})+(1-\frac{\mathrm{\γ}}{\mathrm{\β}}){\stackrel{\·}{u}}_{n-1}+(1-\frac{\mathrm{\γ}}{2\mathrm{\β}}){\stackrel{\·\·}{u}}_{n-1}\mathrm{dt}\end{array}\right.$

Wherein, dt is the integration step determined, subscript represents the moment, as u _n=u (t=ndt), β and γ is the dimensionless group of Newmark method, in formula:

$E=\left[\begin{array}{cc}0& 0\\ 0& I_{\mathrm{kk}}\end{array}\right]$

$K=\left[\begin{array}{cc}{I}_{\mathrm{rr}}& 0\\ 0& k\end{array}\right]$

B＝[B ^(r)B ^(s)]

$Y=\left[\begin{array}{cc}{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}& 0\\ 0& H_1^{\left(s\right)}\mathrm{dt}/2\end{array}\right]$

${\tilde{u}}_n={\left[\begin{array}{cc}{R}_n^{{\left(r\right)}^T}& S_n^{{\left(s\right)}^T}\end{array}\right]}^T$

I _rr, I _kkrepresentation unit matrix, it is identical that exponent number is rigidly connected with junction, interface respectively, elasticity connects number of degrees of freedom;

obtained by following:

$R_n^{\left(r\right)}={\underset{\‾}{u}}_n^{\left(r\right)}+{\mathrm{dt}}^2\mathrm{\β}{\underset{\‾}{M}}^{{\left(r\right)}^{-1}}(f_n^{\left(r\right)}-K^{\left(r\right)}{\underset{\‾}{u}}_n^{\left(r\right)}-C^{\left(r\right)}{\underset{\‾}{\stackrel{\·}{u}}}_n^{\left(r\right)})$

$\begin{array}{c}{S}_n^{\left(s\right)}=\underset{i=0}{\overset{n-2}{\mathrm{\Σ}}}{H}_{n-i}^{\left(s\right)}[f_i^{\left(s\right)}+f_{i+1}^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_i-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{i+1}]\frac{\mathrm{dt}}{2}\\ +H_1^{\left(s\right)}(f_{n-1}^{\left(s\right)}+f_n^{\left(s\right)}-B^{{\left(s\right)}^T}{\mathrm{\λ}}_{n-1})\frac{\mathrm{dt}}{2}\end{array}$

Wherein

$\left\{\begin{array}{c}{\underset{\‾}{u}}_n^{\left(r\right)}=u_{n-1}^{\left(r\right)}+\mathrm{dt}\·{\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+{\mathrm{dt}}^2(1/2-\mathrm{\β}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\\ {\underset{\‾}{M}}^{\left(r\right)}=M^{\left(r\right)}+C^{\left(r\right)}\mathrm{\γdt}+K^{\left(r\right)}{\mathrm{dt}}^2\mathrm{\β}\\ {\stackrel{\·}{\underset{\‾}{u}}}_n^{\left(r\right)}={\stackrel{\·}{u}}_{n-1}^{\left(r\right)}+\mathrm{dt}(1-\mathrm{\γ}){\stackrel{\·\·}{u}}_{n-1}^{\left(r\right)}\end{array}\right..$