CN106777738A - A kind of numerical computation method for variable mass dynamical system - Google Patents

Abstract

The invention discloses a kind of numerical computation method for variable mass dynamical system, comprise the following steps：Determine integral parameter γ, β (t) and step delta t, calculate integration variable, obtain the system mode of t, obtain t, the systematic parameter of t+ Δ ts, calculate the effective rigidity of t+ Δ ts, calculate the payload of t+ Δ ts, payload is the displacement of t+ Δ ts with the business of effective rigidity, using the system mode of t, the displacement of quality and t+ Δ ts, speed of the Mass Calculation system in t+ Δ ts, by the displacement of t+ Δ ts, the acceleration of speed and Mass Calculation system in t+ Δ ts, and then iterate to calculate, finally try to achieve all vibratory responses in the range of required time-histories.The present invention not only considers influence of the quality to system displacement, speed and acceleration, and in view of integral parameter and the influence factor of step-length, makes algorithm computational accuracy preferable.

Description

A kind of numerical computation method for variable mass dynamical system

Technical field

The present invention relates to a kind of numerical computation method for variable mass dynamical system, belong to numerical analysis simulation technology neck Domain.

Background technology

In recent years, due to the development of aeronautical and space technology and industrial technology, the research of variable mass dynamical system is increasingly received To the attention of researcher, such as rocket, aircraft and vehicle bridge coupling vibration system.Variable mass dynamical system vibration characteristics is different from Constant-quality dynamical system, its research is also long-standing, and early in 1897, Russia's scholar Mi Xie Bielskis were just proposed for grinding Study carefully the dynamic (dynamical) close Bielski equation of having a rest of variable mass particle.

Single-degree-of-freedom variable mass dynamical system, as shown in figure 1, in Fig. 1 m (t), c, k represent respectively mass of system, damping and Rigidity, quality m (t) is the function with time correlation；X and f represent system displacement and extraneous exciting force respectively.Variable mass dynamical system Subordination is in nonlinear dynamic system, if direct solution kinetic equation is relatively difficult, cannot even solve sometimes, therefore researcher Method frequently with numerical computations asks its vibratory response.Chinese invention patent " variable mass dynamic vibration absorber transient process emulation side Method " (publication No.：CN 103851125A) disclose a kind of numerical computations for solving variable mass dynamic vibration absorber transient process Method, the method takes into consideration only influence of the mass change to system speed and acceleration, but does not consider integral parameter With the influence factor of step-length, its computational accuracy has much room for improvement.It is published in《Mechanics journal》Scientific paper on the 5th phase of volume 44 " the adaptive N ewmark methods cooperateed with Structure Dynamic Characteristics " is set up when variable mass dynamical system vibratory response is calculated Vibration equation need to consider influence of the mass change to damping, and the Mathematical Modeling set up is more complicated.

The content of the invention

The technical problems to be solved by the invention are the defects for overcoming prior art, there is provided one kind is used for variable mass dynamical system The vibration processes of variable mass dynamical system can be carried out simulation calculation by the numerical computation method of system.

In order to solve the above technical problems, the present invention provides a kind of numerical computation method for variable mass dynamical system, bag Include following steps：

1) integral parameter γ, β (t) and step delta t are determined：

The integral parameter γ is set to：γ=1/2,

Integral parameter β (t) is defined as：

The step delta t meets：

Wherein,

K represents the rigidity of variable mass dynamical system, m_tThe quality of variable mass dynamical system t is represented, c represents variable mass The damping of dynamical system；

2) integration variable α is calculated₀(t), α₁(t), α₂(t), α₃(t), α₄(t), α₅(t)；

3) the state x of variable mass dynamical system t is obtained_tWithWherein, x_tRepresent variable mass dynamical system t position Move,Represent variable mass dynamical system t speed；It is known for the variable mass dynamical system state that initial time t is 0 Amount, when dynamic response is solved, the displacement of known initial time and speed；For the state at t ≠ 0 moment, by the t-1 moment Calculating obtain；

4) the variable mass power system parameter of t, t+ Δ t is obtained, including：The quality of t variable mass dynamical system m_t, the quality m of t+ Δ t variable mass dynamical systems_t+Δt, the extraneous exciting force f of t variable mass dynamical system_t, during t+ Δ t Carve the extraneous exciting force f of variable mass dynamical system_t+Δt, the damping c and rigidity k of variable mass dynamical system；

5) effective rigidity of t+ Δ ts is calculated

6) the effective extraneous exciting force of t+ Δ ts is calculated

7) displacement x of t+ Δ ts is solved_t+Δt；

8) speed of t+ Δ ts is solvedAnd acceleration

9) t=t+ Δs t is made；

10) repeat step 1) to the system response for 9), calculating subsequent time, finally try to achieve all in the range of required time-histories Vibratory response.

Foregoing step 2) in integration variable computing formula it is as follows：

Foregoing step 5) effective rigidityComputing formula it is as follows：

Foregoing step 6) effectively extraneous exciting forceComputing formula it is as follows：

Wherein,Represent variable mass dynamical system t acceleration.

Foregoing step 7) t+ Δ ts displacement x_t+ΔtComputing formula it is as follows：

Wherein,It is the effective rigidity of t+ Δ ts.

Foregoing step 8) t+ Δ ts speedAnd accelerationComputing formula it is as follows：

The beneficial effect that the present invention is reached：

(1) numerical computation method of the invention when variable mass dynamical system vibratory response is calculated, shake by the system for being used Dynamic equation is not required to influence of the consideration mass change to damping, therefore, build variable mass dynamical system vibration equation and be easier.

(2) it is of the invention in calculating process, the not only influence in view of quality to system displacement, speed and acceleration, and And in view of integral parameter and the influence factor of step-length, make algorithm computational accuracy preferable.

(3) numerical computation method applicability of the invention is wide, can be used for all of variable mass dynamical system.

Brief description of the drawings

Fig. 1 is variable mass dynamical system one degree of freedom modeling；

Fig. 2 is using numerical computation method of the invention and adaptive N ewmark algorithms, tradition Newmark algorithms gained Result of calculation comparison diagram.

Specific embodiment

The invention will be further described below.Following examples are only used for clearly illustrating technical side of the invention Case, and can not be limited the scope of the invention with this.

Numerical computation method for variable mass dynamical system of the invention, specifically includes following steps：

Step one, determines integral parameter γ, β (t) and step delta t：

Integral parameter γ is set to：γ=1/2,

Integral parameter β (t) is defined as：

Step delta t meets：

Wherein,

Wherein, in formula (3),

K represents the rigidity of variable mass dynamical system, m_tThe quality of variable mass dynamical system t is represented, c represents variable mass The damping of dynamical system.

Step 2, calculates integration variable α₀(t), α₁(t), α₂(t), α₃(t), α₄(t), α₅(t)：

Step 3, obtains the state (x of variable mass dynamical system t (initial time t is 0)_t、), x_tRepresent variable mass Dynamical system t displacement,Represent variable mass dynamical system t speed；It is known for the state that initial time t is 0 Amount, when dynamic response is solved, the displacement of known initial time and speed；For the state at t ≠ 0 moment, by the t-1 moment Calculating obtain.

Step 4, obtains the variable mass power system parameter of t, t+ Δ t, including：T variable mass dynamical system Quality m_t, the quality m of t+ Δ t variable mass dynamical systems_t+Δt, the extraneous exciting force f of t variable mass dynamical system_t, t+ The extraneous exciting force f of Δ t variable mass dynamical system_t+Δt, the damping c and rigidity k of variable mass dynamical system.

Step 5, calculates the effective rigidity of t+ Δ tsComputing formula is as follows：

Above-mentioned computing formula is to add rotten secondary element on the basis of Newmark algorithms and change integration variable to obtain 's.

Step 6, calculates the effective extraneous exciting force of t+ Δ tsComputing formula is as follows：

Wherein,Variable mass dynamical system t acceleration is represented,

Above-mentioned computing formula is to add rotten secondary element on the basis of Newmark algorithms and change integration variable to obtain 's.

Step 7, solves the displacement x of t+ Δ ts_t+Δt, computing formula is as follows：

Above-mentioned computing formula is to add rotten secondary element on the basis of Newmark algorithms and change integration variable to obtain 's.

Step 8, solves the speed of t+ Δ tsAnd accelerationComputing formula is as follows：

Above-mentioned computing formula is to add rotten secondary element on the basis of Newmark algorithms and change integration variable to obtain 's.

Step 9, makes t=t+ Δs t；

Step 10, repeats the above steps one to nine, calculates the system response of subsequent time, finally tries to achieve required time-histories scope Interior all vibratory responses.

Embodiment

Below by proposed by the invention to verify with adaptive N ewmark algorithms and the contrast of tradition Newmark algorithms Numerical computation method.Using numerical computation method of the present invention, the single-degree-of-freedom variable mass system model shown in Fig. 1 is imitated It is true to calculate, mass of system m in Fig. 1_t, rigidity k and damping c it is as shown in table 1.

The variable mass system parameter of table 1

From table 1 it follows that mass of system m_tIt is the variable changed with time t, in initial time m₀It is 1.Suffered by system Extraneous exciting force f_tIt is sin10t, initial displacement x₀, speed0 is taken respectively.

Numerical computation method of the invention, adaptive N ewmark algorithms and tradition Newmark algorithm calculated results As shown in Figure 2.Fig. 2 is variable mass system acceleration responsive when material calculation is 0.01s, it can be seen that of the invention Numerical computation method acquired results are consistent with adaptive N ewmark algorithms, tradition Newmark algorithms, and in two algorithms gained Between result, so as to illustrate numerical computation method of the invention for calculating the effective of variable mass dynamic vibration absorber vibratory response Property.In Fig. 2,Inventive algorithm is represented, ----adaptive N ewmark algorithms are represented, --- -- represents tradition Newmark Algorithm.

The above is only the preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art For member, on the premise of the technology of the present invention principle is not departed from, some improvement and deformation can also be made, these improve and deform Also should be regarded as protection scope of the present invention.

Claims (6)

1. a kind of numerical computation method for variable mass dynamical system, it is characterised in that comprise the following steps：

1) integral parameter γ, β (t) and step delta t are determined：

The integral parameter γ is set to：γ=1/2,

Integral parameter β (t) is defined as：

$\mathrm{\β}\left(t\right)=\frac{1}{\mathrm{\χ}\left(t\right)}-\frac{1}{{\mathrm{\ω}}^2\left(t\right){\mathrm{\Δt}}^2}-\frac{\mathrm{\ξ}\left(t\right)}{\mathrm{\ω}\left(t\right)\mathrm{\Δ}\left(t\right)}---\left(1\right)$

The step delta t meets：

$\mathrm{\Δ}t\≤min(\frac{1}{\mathrm{\ω}\left(t\right)\sqrt{0.25-\mathrm{\β}\left(t\right)}},\frac{\mathrm{\π}}{2\mathrm{\ω}\left(t\right)})---\left(2\right)$

Wherein,

$\mathrm{\χ}\left(t\right)=\frac{2}{B^2\left(t\right)}\[\frac{1+B\left(t\right){\tan }^2\mathrm{\ω}\left(t\right)\mathrm{\Δ}t-\sqrt{1+2B\left(t\right){\tan }^2\mathrm{\ω}\left(t\right)\mathrm{\Δ}t-B^2\left(t\right){\tan }^2\mathrm{\ω}\left(t\right)\mathrm{\Δ}t}}{1+{\tan }^2\mathrm{\ω}\left(t\right)\mathrm{\Δ}t}\]---\left(3\right)$

$\mathrm{\ω}\left(t\right)=\sqrt{\frac{k}{m_t}}---\left(4\right)$

$\mathrm{\ξ}\left(t\right)=\frac{c}{2\sqrt{m_{t}k}}---\left(5\right)$

$B\left(t\right)=1+\frac{2\mathrm{\ξ}\left(t\right)}{\mathrm{\ω}\left(t\right)\mathrm{\Δ}t}---\left(6\right)$

K represents the rigidity of variable mass dynamical system, m_tThe quality of variable mass dynamical system t is represented, c represents variable mass power The damping of system；

2) integration variable α is calculated₀(t), α₁(t), α₂(t), α₃(t), α₄(t), α₅(t)；

3) the state x of variable mass dynamical system t is obtained_tWithWherein, x_tVariable mass dynamical system t displacement is represented, Represent variable mass dynamical system t speed；It is known quantity for the variable mass dynamical system state that initial time t is 0, is asking During solution dynamic response, the displacement of known initial time and speed；For the state at t ≠ 0 moment, by the calculating at t-1 moment Obtain；

4) the variable mass power system parameter of t, t+ Δ t is obtained, including：The quality m of t variable mass dynamical system_t, t+ The quality m of Δ t variable mass dynamical system_t+Δt, the extraneous exciting force f of t variable mass dynamical system_t, the change of t+ Δs t The extraneous exciting force f of quality power system_t+Δt, the damping c and rigidity k of variable mass dynamical system；

5) effective rigidity of t+ Δ ts is calculated

6) the effective extraneous exciting force of t+ Δ ts is calculated

7) displacement x of t+ Δ ts is solved_t+Δt；

8) speed of t+ Δ ts is solvedAnd acceleration

9) t=t+ Δs t is made；

10) repeat step 1) to the system response for 9), calculating subsequent time, finally try to achieve all vibrations in the range of required time-histories Response.

2. a kind of numerical computation method for variable mass dynamical system according to claim 1, it is characterised in that described Step 2) in integration variable computing formula it is as follows：

${\mathrm{\α}}_0\left(t\right)=\frac{1}{\mathrm{\β}\left(t\right){\mathrm{\Δt}}^2},{\mathrm{\α}}_1\left(t\right)=\frac{\mathrm{\γ}}{\mathrm{\β}\left(t\right)\mathrm{\Δ}t},$

${\mathrm{\α}}_2\left(t\right)=\frac{1}{\mathrm{\β}\left(t\right)\mathrm{\Δ}t},{\mathrm{\α}}_3\left(t\right)=\frac{1}{2\mathrm{\β}\left(t\right)}-1,$

${\mathrm{\α}}_4\left(t\right)=\frac{\mathrm{\γ}}{\mathrm{\β}\left(t\right)}-1,{\mathrm{\α}}_5\left(t\right)=\frac{\mathrm{\Δ}t}{2}(\frac{\mathrm{\γ}}{\mathrm{\β}\left(t\right)}-2).$

3. a kind of numerical computation method for variable mass dynamical system according to claim 1, it is characterised in that described Step 5) effective rigidityComputing formula it is as follows：

${\stackrel{\‾}{k}}_{t+\mathrm{\Δ}t}=k+{\mathrm{\α}}_0\left(t\right)m_t+{\mathrm{\α}}_1\left(t\right)\frac{m_{t+\mathrm{\Δ}t}}{m_t}c---\left(7\right).$

4. a kind of numerical computation method for variable mass dynamical system according to claim 1, it is characterised in that described Step 6) effectively extraneous exciting forceComputing formula it is as follows：

${\stackrel{\‾}{f}}_{t+\mathrm{\Δ}t}=f_{t+\mathrm{\Δ}t}+m_{t+\mathrm{\Δ}t}\left({\mathrm{\α}}_0\right(t)x_t+{\mathrm{\α}}_2(t){\stackrel{\·}{x}}_t+{\mathrm{\α}}_3(t\left){\stackrel{\·\·}{x}}_t\right)+\frac{m_{t+\mathrm{\Δ}t}}{m_t}c\left({\mathrm{\α}}_1\right(t)x_t+{\mathrm{\α}}_4(t){\stackrel{\·}{x}}_t+{\mathrm{\α}}_5(t\left){\stackrel{\·\·}{x}}_t\right)---\left(8\right).$

Wherein,Represent variable mass dynamical system t acceleration.

5. a kind of numerical computation method for variable mass dynamical system according to claim 4, it is characterised in that described Step 7) t+ Δ ts displacement x_t+ΔtComputing formula it is as follows：

$x_{t+\mathrm{\Δ}t}=\frac{{\stackrel{\‾}{f}}_{t+\mathrm{\Δ}t}}{{\stackrel{\‾}{k}}_{t+\mathrm{\Δ}t}}---\left(9\right)$

Wherein,It is the effective rigidity of t+ Δ ts.

6. a kind of numerical computation method for variable mass dynamical system according to claim 1, it is characterised in that described Step 8) t+ Δ ts speedAnd accelerationComputing formula it is as follows：

${\stackrel{\·}{x}}_{t+\mathrm{\Δ}t}=\frac{m_{t+\mathrm{\Δ}t}}{m_t}\left({\mathrm{\α}}_1\right(t\left)\right(x_{t+\mathrm{\Δ}t}-x_t)-{\mathrm{\α}}_4(t){\stackrel{\·}{x}}_t-{\mathrm{\α}}_5(t\left){\stackrel{\·\·}{x}}_t\right)---\left(10\right)$

${\stackrel{\·\·}{x}}_{t+\mathrm{\Δ}t}=\frac{1}{m_{t+\mathrm{\Δ}t}}(f_{t+\mathrm{\Δ}t}-c{\stackrel{\·}{x}}_{t+\mathrm{\Δ}t}-{\mathrm{kx}}_{t+\mathrm{\Δ}t})---\left(11\right).$