CN104809300B - Pulse minor structure and finite element have just played the dynamics simulation method of Hybrid connections - Google Patents

Abstract

Pulse minor structure disclosed by the invention and finite element have just played dynamics simulation method and the algorithm of Hybrid connections, relate to dynamics simulation method and algorithm, belong to Structural Dynamics technical field.Basic ideas of the present invention are: obtain impulse response function matrix and the kinetics equation of finite element minor structure of pulse minor structure；Set up the consistency condition of Substructure Interfaces；Set up the kinetics equation of pulse minor structure；Set up the kinetics equation of finite element minor structure；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, complete Structural Dynamics simulation.The present invention provides pulse minor structure and the dynamics simulation method of finite element minor structure just elastic Hybrid connections and algorithm, actual response pulse minor structure and finite element minor structure just elastic mixed characteristic, and then improves space technology field structure dynamics simulation precision.Additionally, the present invention has expanded the range of application of traditional IBS method.

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 to 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 space technology and improving constantly spacecraft performance requirement, spacecraft structure becomes the most multiple Miscellaneous and huge.When modern Complex Spacecraft is carried out the Dynamic Response and structure optimization, use conventional finite element method System is analyzed being introduced into numerous degree of freedom, consumes and calculate resource in a large number；In the face of system goes through the knot that dynamic response is analyzed Structure optimizes, and traditional finite element modeling method is difficult to be competent at especially.Meanwhile, spacecraft is often difference portion in development process Cooperation between door, different institutions, for consideration or the difference of modeling pattern of technical protection, the two sides concerned cannot directly be total to Enjoy model.Dynamic sub-structure methods can solve the problem that the problems referred to above to a certain extent.

Since nineteen sixty Hurty realizes Dynamic Substructure (Dynamic Substituting, DS) first, through half The development trend Substructure Techniques in century has been widely used for engineering field, defines three class methods: modal synthesis (Component Mode Synthesis, CMS) method, frequency domain minor structure (Frequency Based Substructuring, FBS) the i.e. classical time domain of method and minor structure based on impulse response function (Impulse Based Substituting, IBS) Substructure method, is called for short pulse substructure method.In first two method, the dynamics of minor structure is respectively by mode and frequency Receptance function describes；And pulse substructure method is described by impulse response function, the tool when processing little damping, transient impact problem There is advantage quick, high-precision.

Afterwards, Paul L.C.van der Valk, Daniel J.Rixen have studied mould based on impulse response function Type and the combination of FEM (finite element) model, but their connected mode only considers to be rigidly connected, and deposits in spacecraft dynamic response analysis Numerous just-play Hybrid connections mode, the such as chain connection between lunar orbiter centrosome and solar wing；Dong Weili studies A kind of flexible linker structure composition method based on impulse response function, the method considers connector between pulse minor structure Elastic characteristic, but method fails to consider the connection between pulse minor structure and finite element minor structure, is carrying out system structure optimization Time, partial structurtes need to be modified, use each Optimized Iterative of the method all the pulse recalculating pulse minor structure to be rung Answer function, and efficient for the method advantage is weakened by impulse response function repeatedly regenerate significantly, make optimization process become Loaded down with trivial details time-consumingly.

Therefore, how to carry out comprehensively, becoming by pulse minor structure and finite element minor structure under firm-bullet Hybrid connections situation The problem that must solve in time domain Dynamic Substructure technology development process.

Summary of the invention

Only be suitable to analyze between pulse minor structure for rigidly connected defect for tradition IBS method.Arteries and veins disclosed by the invention Punch structure and finite element just-play dynamics simulation method and the algorithm of Hybrid connections, to solve the technical problem that it is to provide arteries and veins Punch structure and finite element minor structure just-the dynamics simulation method of elastic Hybrid connections and algorithm, actual response pulse is tied Firm with finite element minor structure-elastic mixed characteristic of structure, and then improve space technology field structure dynamics simulation precision.

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

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

A kind of pulse minor structure disclosed by the invention and finite element just-play the dynamics simulation method of Hybrid connections, specifically Realize step 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 of all minor structures of the motion vector u of system represents, 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 the r pulse minor structure, and it is individual limited that subscript s represents that this variable belongs to s Unit's 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 matrixDisplacement components u by pulse minor structure^(s)Project to linkage interface, Bu Erju Battle arrayTried 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 elastic junction.

Equally, structure Boolean matrixDisplacement components u by finite element minor structure^(r)Project to junction, interface, cloth That matrixTried 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: introducing Lagrange multiplier λ, λ represents system interface junction attachment force array, step 2 cloth obtained Your matrix, has a 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, specifically comprise the following steps that

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

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

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

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

In formula, u^(s)T () is the displacement array of the s pulse minor structure；f^(s)T () is the external applied load suffered by pulse minor structure Array；Parameter t, τ represent the time, and d τ is the differential of time.Tried to achieve, by step by step 2 Chinese style (2) Rapid 3 formulas (4) understand,Represent attachment force array at pulse Substructure Interfaces.

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)It is respectively the quality of the r finite element minor structure, damping, stiffness matrix；u^(r)T () is respectively the r finite element minor structure acceleration, speed, displacement array；f^(r)T () is r External drive power array suffered by individual finite element minor structure；Tried to achieve, by step by step 2 Chinese style (3) Rapid 3 formulas (4) understand,Represent the attachment force array at finite element Substructure Interfaces.

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: be rigidly connected between the minor structure of interface At the identical and elastic linkage interface of degree of freedom, between minor 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):

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

At elastic linkage interface, between minor structure, degree of freedom meets Hooke's law, has following relation, as shown in formula (8):

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

K represents stiffness matrix at elastic linkage interface, λ_kRepresent attachment force array at elastic linkage interface.

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, method particularly includes: by first formula of (2) in step 2 and the first of (3) Formula is brought into (7), and second formula of (2) in step 2 and second formula of (3) is brought into (8), (5) formula and step in integrating step 2.2 (6) formula in rapid 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})\[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}\]d\mathrm{\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(r\right)}{u}^{\left(r\right)})={\mathrm{\λ}}_k\end{array}\right.---\left(10\right)$

Step 8: the Substructure Synthesis equation obtaining step 7 carries out time discrete, obtains the dynamic respond u of system_nWith Interfacial force λ between minor structure_nTime recurrence formula, complete pulse minor structure and finite element just-structure that plays Hybrid connections part moves Mechanical simulation.Concretely comprise the following steps:

Step 8.1: (10) first formulas are carried out time discrete, 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{dt}{2}---\left(11\right)$

In formula, dt is integration step, and subscript represents the moment (such as u_n=u (ndt))；Parameter n represents that this variable is in n-th Calculate on time step.

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

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

In formula, γ, β are integral constant.

Step 8.3: utilize the speed of Newmark method representation systemAnd acceleration

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

Step 8.4: comprehensive (13), (12), (11), (10) obtain interfacial force λ between minor structure_nDynamic respond u with system_n Time recurrence formula:

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

In formula:

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

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

B=[B^(r)B^(s)] (17)

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

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

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:

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

$\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{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{dt}{2}\end{array}---\left(22\right)$

Wherein I_rr、I_kkRepresentation unit matrix, exponent number is rigidly connected with junction, interface respectively, elastic connection number of degrees of freedom, Identical, k is elastic junction stiffness matrix.

Try to achieve displacement components u_nAfter, obtain acceleration by step 8.3 Chinese style (13)And speedAccording to formula (14), logical Cross time stepping method, obtain the displacement components u of whole simulation process, speedAccelerationResponse results, i.e. completes truly to reflect pulse Minor structure and finite element just-play the Structural Dynamics simulation of Hybrid connections part.

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

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

Step 2: obtained the impulse response function matrix H of 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 elementThe jth of expression s minor structure of system is certainly By degree dynamic respond of i-th degree of freedom under pulse excitation；

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 from s The degree of freedom of pulse minor structure filters out the interface degree of freedom of the s pulse minor structure, B^(r)Effect be from the r pulse The degree of freedom of minor structure filters out the interface degree of freedom of the r pulse minor structure.

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

Step 6: initialize the initial time system of fundamental unknown variables, i.e. t=0 not basic by External Force Acting, step 1 The all null vectors of unknown quantity；

Step 7: solving system is in external force f^(s)(t)、f^(r)Response under (t) effect, four groups of fundamental unknown variables in step 1 Time recurrence formula is as follows:

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

Wherein, dt is the integration step determined, subscript represents the moment, such as u_n=u (t=ndt), β and γ are Newmark methods Dimensionless group, in formula:

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

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

B=[B^(r)B^(s)]

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

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

I_rr、I_kkRepresentation unit matrix, exponent number is rigidly connected with junction, interface respectively, elastic connection number of degrees of freedom, is identical.

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{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{dt}{2}\end{array}$

Wherein

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

Beneficial effect:

1, based on impulse response function, a kind of pulse minor structure of the present invention and finite element just-play Hybrid connections Dynamics simulation method and algorithm, overcome tradition IBS method being only suitable to analyze 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 truly simulation space technology 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, Expand the range of application of traditional IBS method.

Accompanying drawing explanation

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

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

Fig. 3 is the 11 degree of freedom spring-dampers-quality system in detailed description of the invention.

Fig. 4 is the external applied load-time graph in detailed description of the invention on the 9th degree of freedom of system.

Fig. 5 is the acceleration responsive-time graph of the 10th degree of freedom of system in detailed description of the invention.

Detailed description of the invention

Embodiment 1:

In order to preferably illustrate the purpose of the present invention and effect, below by the spring-dampers of 11 degree of freedom- The present invention is explained in detail by the dynamic analysis of quality system.Analyzing step-length and take 1ms, analysis time is 1s.

As it is shown on figure 3, the system of the present embodiment is divided into two minor structures.Minor structure 1 comprises 7 degree of freedom, for pulse Structure；Minor structure 2 comprises 4 degree of freedom, for finite element minor structure.The m of minor structure 1₆And m₇It is connected with the m8 elasticity of minor structure 2, As shown in Figure 1.The kinetic parameter of minor structure and connector is shown in Table 1.Assuming that external force is only loaded into m₉On, it is stipulated that it is to the right square To, the time graph of external applied load is as shown in Figure 4.

Table 1 spring-mass block system parameter list

As it is shown in figure 1, a kind of pulse minor structure of the present embodiment and finite element just-play the dynamics simulation side of Hybrid connections Method, it is achieved 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 of all minor structures of the motion vector u of system represents, piecemeal is u^(r)、u^(s)。

Step 2: structure Boolean matrix

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

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

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

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

Step 3: introducing Lagrange multiplier λ, λ represents system interface junction attachment force array.

Step 4: set up pulse minor structure kinetics equation.

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

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})\[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}\]d\mathrm{\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

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

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

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

Step 6: determine pulse minor structure and finite element Substructure Interfaces consistency condition.Including: be rigidly connected interface virgin Between structure, at the identical and elastic linkage interface of degree of freedom, between minor 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

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

At elastic linkage interface, between minor 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 connects degree of freedom and sets up stiffness matrix k:

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

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

$\left\{\begin{array}{c}{u}^{\left(s\right)}\left(t\right)={\∫}_0^t{H}^{\left(s\right)}(t-\mathrm{\τ})\[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}\]d\mathrm{\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(r\right)}{u}^{\left(r\right)})={\mathrm{\λ}}_k\end{array}\right.---\left(28\right)$

Step 8: the Substructure Synthesis equation obtaining step 7 carries out time discrete, obtains the dynamic respond u of system_nWith Interfacial force λ between minor structure_nTime recurrence formula.

Step 8.1: pulse minor structure equation of motion formula (24) is carried 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{dt}{2}---\left(29\right)$

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

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

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

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

Step 8.3: utilize the speed of Newmark method representation systemAnd acceleration

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

Step 8.4: aggregative formula (31), (30), (29), (28) obtain interfacial force λ between minor structure_nDisplacement with system rings Answer u_nTime recurrence formula:

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

Wherein

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

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

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

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

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

I_rr、I_kkRepresentation unit matrix, exponent number difference 1 and 3.

Obtained by following:

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

$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{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{dt}{2}$

Wherein

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

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

As in figure 2 it is shown, a kind of pulse minor structure of the present embodiment and finite element just-dynamics simulation that plays Hybrid connections calculates Method, it is achieved step is as follows:

Step 1: definition fundamental unknown variables, is displacement function column vector u (t) of system, velocity function column vector respectivelyAcceleration function column vectorAnd junction minor structure intermolecular forces function row vector λ, interface (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.The most do not consider structure Damping, is set to zero C by its damping matrix^(r)=0.

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

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

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:

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

Step 6: initialize the initial time system of fundamental unknown variables, i.e. t=0 not basic by External Force Acting, step 1 The all null vectors of unknown quantity:

Step 7: solving system is in 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 is as shown in Figure 2.Take dimensionless group β=0.25 of Newmark method, γ=0.5:

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

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

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

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

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

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

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

I_rr、I_kkRepresentation unit matrix, exponent number difference 1 and 3.

Obtained by following:

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

$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{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{dt}{2}$

Wherein

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

With m₁₀Response as a example by, use the present embodiment to emulate the m that obtains₁₀Acceleration responsive with use Newmark method The result conduct that (Newmark method is the algorithm that art-recognized computational accuracy is the highest, but this algorithm computational efficiency is low) obtains Benchmark in contrast to Fig. 5, and 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, have been carried out the most specifically purpose, technical scheme and the beneficial effect of invention Bright, be it should be understood that the specific embodiment that the foregoing is only the present invention, be used for explaining the present invention, be not used to limit this Invention protection domain, all within the spirit and principles in the present invention, any modification, equivalent substitution and improvement etc. done, all should Within being included in protection scope of the present invention.

Claims (1)

1. a pulse minor structure and finite element just-play the dynamics simulation method of Hybrid connections, it is characterised in that: include as follows Step,

Step 1: carry out minor structure division；Whole system is divided into pulse minor structure and finite element minor structure two parts；System The motion vector of all minor structures of motion vector u represent, 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 the r finite element minor structure, and subscript s represents that this variable belongs to the s pulse 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 matrixDisplacement components u by pulse minor structure^(s)Project to linkage interface, Boolean matrixTried 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 elastic junction；

Equally, structure Boolean matrixDisplacement components u by finite element minor structure^(r)Project to junction, interface, Bu Erju Battle arrayTried 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: introducing Lagrange multiplier λ, λ represents system interface junction attachment force array, step 2 the boolean's square obtained Battle array, has a 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, specifically comprise the following steps that

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

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

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

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

In formula, u^(s)T () is the displacement array of the s pulse minor structure；f^(s)T () is the external applied load row suffered by pulse minor structure Battle array；Parameter t, τ represent the time, and d τ is the differential of time；Tried to achieve, by step by step 2 Chinese style (2) 3 formulas (4) understand,Represent attachment force array at pulse Substructure Interfaces；

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)It is respectively the quality of the r finite element minor structure, damping, stiffness matrix；u^(r)T () is respectively the r finite element minor structure acceleration, speed, displacement array；f^(r)T () is r External drive power array suffered by individual finite element minor structure；Tried to achieve, by step by step 2 Chinese style (3) Rapid 3 formulas (4) understand,Represent the attachment force array at finite element Substructure Interfaces；

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: be rigidly connected between the minor structure of interface freely Spend at identical and elastic linkage interface degree of freedom between minor structure and meet 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):

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

At elastic linkage interface, between minor 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, k represents stiffness matrix at elastic linkage interface, λ_kRepresent attachment force array at elastic linkage interface；

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, method particularly includes: by the first formula band of first formula of (2) in step 2 He (3) Enter (7), and second formula of (2) in step 2 and second formula of (3) are brought into (8), (5) formula and step 5 in integrating step 4.2 In (6) formula, obtain shown in Substructure Synthesis equation such as formula (10):

$\left\{\begin{array}{c}{u}^{\left(s\right)}\left(t\right)={\∫}_0^t{H}^{\left(s\right)}(t-\mathrm{\τ})\[f^{\left(s\right)}\left(\mathrm{\τ}\right)+B^{{\left(s\right)}^T}\mathrm{\λ}\]d\mathrm{\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(r\right)}{u}^{\left(r\right)})={\mathrm{\λ}}_k\end{array}\right.---\left(10\right)$

Step 8: the Substructure Synthesis equation obtaining step 7 carries out time discrete, obtains the dynamic respond u of system_nAnd minor structure Between interfacial force λ_nTime recurrence formula, complete pulse minor structure and finite element just-play the Structural Dynamics mould of Hybrid connections part Intend；Concretely comprise the following steps:

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

$u_n^{\left(s\right)}=\underset{i=0}{\overset{n-1}{\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{dt}{2}---\left(11\right)$

In formula, dt is integration step, and subscript represents the moment；Parameter n represents that this variable is on the n-th calculating time step；

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

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

In formula, γ, β are integral constant；

Step 8.3: utilize the speed of Newmark method representation systemAnd acceleration

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

Step 8.4: comprehensive (13), (12), (11), (10) obtain interfacial force λ between minor structure_nDynamic respond u with system_nTime Between recurrence formula:

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

In formula:

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

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

B=[B^(r) B^(s)] (17)

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

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

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:

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

$\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{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{dt}{2}\end{array}---\left(22\right)$

Wherein I_rr、I_kkRepresentation unit matrix, exponent number is rigidly connected with junction, interface respectively, elastic connection number of degrees of freedom, is identical, k For elastic junction stiffness matrix；

Try to achieve displacement components u_nAfter, obtain acceleration by step 8.3 Chinese style (13)And speedAccording to formula (14), when passing through Between recursion, obtain the displacement components u of whole simulation process, speedAccelerationResponse results, completes Structural Dynamics simulation.