CN103851125B - Variable mass dynamic vibration absorber transient process emulation mode - Google Patents

Variable mass dynamic vibration absorber transient process emulation mode Download PDF

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CN103851125B
CN103851125B CN201410048496.XA CN201410048496A CN103851125B CN 103851125 B CN103851125 B CN 103851125B CN 201410048496 A CN201410048496 A CN 201410048496A CN 103851125 B CN103851125 B CN 103851125B
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CN103851125A (en
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高强
赵艳青
宋伟志
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Changan University
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Abstract

The invention provides a kind of variable mass dynamic vibration absorber transient process emulation mode, the initial parameters of the t that quality changes is obtained according to variable mass dynamic vibration absorber, calculate the initial acceleration vector of t, provide integration step Δ t and β, γ, and calculate integration constant, calculate effective stiffness matrix, calculate the useful load vector of t+ Δ t, calculate the motion vector of t+ Δ t, calculate vector acceleration and the velocity vector of t+ Δ t, absorber mass m 2+ m vt () sports m 2+ m v(t+ Δ t), motion vector, velocity vector and vector acceleration are revised respectively, finally obtain the motion vector of system any time, velocity vector and vector acceleration.The emulation mode that application the present invention carries considers momentum conservation law in mass change process, makes variable mass process meet the natural law.This emulation mode degree of error is credible, can be obtained by the contrast of following simulation and experiment: both the percentage difference 7% that after absorber mass reduces, main system vibration declines.

Description

Variable mass dynamic vibration absorber transient process emulation mode
Technical field
The invention belongs to Vibration Simulation field, relate to variable mass dynamic vibration absorber, be specifically related to variable mass dynamic vibration absorber transient process emulation mode.
Background technique
In engineer applied, great majority vibration is harmful.Therefore damping technology is one of important content of vibration mechanics research.Dynamic absorber is a kind of conventional vibration control method.Traditional dynamic vibration absorber (dynamicvibrationabsorber, DVA) is made up of quality, rigidity, damping unit, installs on the master system.By appropriate design dynamic vibration absorber parameter, make its natural frequency equal excitation force frequency, then can reduce main system vibration significantly.Dynamic vibration absorber because structure is simple, stable performance and with low cost, be widely applied in the vibration control of mechanical system, engineering structure, building and bridge.But when extraneous excitation force frequency changes, the effectiveness in vibration suppression of bump leveller sharply can be deteriorated due to off resonance.
Self-adapting power bump leveller (adaptivedynamicvibrationabsorber, ADVA) bump leveller natural frequency can be regulated by changing inherent parameters (as rigidity, damping), realize the tracking of excitation force frequency to external world, the vibration that its excitation force being more suitable for blanketing frequency change causes, relative to have active power bump leveller power consumption less, the advantage such as good stability, control is simple, thus obtain and study widely.Up to the present, the self-adapting power bump leveller overwhelming majority of Chinese scholars design is all regulate based on rigidity or damping, Chinese invention patent " shock-absorbing means that a kind of natural frequency is adjustable and have the motor of this shock-absorbing means " (publication No.: CN101639109A) openly knows clearly a kind of dynamic vibration absorber based on quality adjustment newly, this bump leveller is using a fluid box as variable mass unit, by the stereomutation absorber mass of liquid in regulating box, and then change bump leveller natural frequency.Fig. 1 is variable mass dynamic vibration absorber model of vibration, wherein m 1, c 1, k 1represent the quality of main system, damping and rigidity respectively, m 2, m v, c 2, k 2represent the fixed mass of bump leveller, variable-quality, damping and rigidity respectively, F 0sin (ω t) represents dynamic excitation power, bump leveller variable-quality m vcan 0 to m vmaxchange in scope.According to vibration mechanics basic theories, variable mass dynamic vibration absorber natural frequency is:
ω 2 = k 2 m 2 + m v
As variable-quality m v0 to m maxvbetween consecutive variations time, bump leveller natural frequency then exists with between consecutive variations.Vibration attenuation mechanism according to dynamic vibration absorber can obtain, and variable mass dynamic vibration absorber exists arrive frequency band in can realize effective vibration damping.
Up to the present, Chinese scholars has carried out deep theory analysis and simulation study to the dynamic vibration absorber regulated based on quality or damping, there are a large amount of scientific papers and bibliographical information, but for the research of variable mass dynamic vibration absorber, particularly the theory analysis of variable mass transient process and simulation study also extremely rare.The transient motion process of bump leveller and main system in variable mass process is analysed in depth, the main system characteristics of motion now can not only be disclosed, and can as the index passing judgment on control algorithm stability and precision, therefore the research of variable mass transient process has great importance.Bump leveller variable mass process is a typical non-linear process, does not also have the ripe method asking analytic solutions at present, can only emulate by the method for numerical analysis.The most frequently used in vibration system numerical analysis is step by step integration, the basic thought of this method is: the time interval of required response is divided into a series of very short period, within a period, regard linear system as approximate for nonlinear system, and calculate its response.
Summary of the invention
For the shortcomings and deficiencies that prior art exists, the object of the invention is to, a kind of variable mass dynamic vibration absorber transient process emulation mode is provided, analysis emulation is carried out to the time domain response of bump leveller and main system in variable mass process.
In order to realize above-mentioned technical assignment, the present invention adopts following technological scheme to be achieved:
A kind of variable mass dynamic vibration absorber transient process emulation mode, the method specifically comprises the following steps:
Step one, obtains the t main system of quality change and the initial mass matrix of bump leveller system according to variable mass dynamic vibration absorber [ M ] t = m 1 0 0 m 2 + m v , Initial damping matrix [ C ] = c 1 + c 2 - c 2 - c 2 c 2 With initial stiffness matrix [ K ] = k 1 + k 2 - k 2 - k 2 k 2 ;
In formula: m 1represent the quality of main system, c 1represent the damping of main system, k 1represent the rigidity of main system; m 2represent the fixed mass of bump leveller, m vrepresent the variable-quality of bump leveller, c 2represent the damping of bump leveller, k 2represent the rigidity of bump leveller;
Step 2, obtains the t main system of quality change and the initial load vector of bump leveller system according to variable mass dynamic vibration absorber { F } t = F 0 s i n ( ω t ) 0 , Initial displacement vector { X } t = x 1 x 2 With initial velocity vector { X · } t = x · 1 x · 2 ;
In formula: F 0sin (ω t) represents dynamic excitation power, x 1represent the initial displacement of main system, x 2represent the initial displacement of bump leveller, represent the initial velocity of main system, represent the initial velocity of bump leveller;
Step 3, calculates the initial acceleration vector of t
[ M ] t { X ·· } t = { F } t - [ C ] { X · } t - [ K ] { X ·· } t
Step 4, provides integration step Δ t and β, γ, and calculates integration constant:
α 0 = 1 γΔt 2 , α 1 = β γ Δ t , α 2 = 1 γ Δ t , α 3 = 1 2 γ - 1 ,
α 4 = β γ - 1 , α 5 = Δ t 2 ( β γ - 2 ) , α 6 = Δ t ( 1 - β ) , α 7 = β t ;
In formula: Δ t=0.001s,
Step 5, calculates effective stiffness matrix
Step 6, calculates the useful load vector of t+ Δ t:
{ F ‾ } = { F } t + Δ t + [ M ] t ( 1 γΔt 2 { X } t + 1 γ Δ t { X · } t + ( 1 2 γ - 1 ) { X ·· } t ) + [ C ] ( β γ Δ t { X } t + ( β γ - 1 ) { X · } t + ( β 2 γ - 1 ) Δ t { X ·· } t )
Step 7, calculates the motion vector of t+ Δ t:
Step 8, calculates vector acceleration and the velocity vector of t+ Δ t:
{ X ·· } t + Δ t = α 0 ( { X } t + Δ t - { X } t ) - α 2 { X · } t - α 3 { X ·· } t
{ X · } t + Δ t = { X · } t + α 6 { X ·· } t + α 7 { X ·· } t + Δ t
Step 9, absorber mass m 2+ m vt () sports m 2+ m v(t+ Δ t), motion vector, velocity vector and vector acceleration are revised respectively in accordance with the following methods:
(1) shift invariant after absorber mass sudden change,
(2) speed after absorber mass sudden change
In formula:
M 2brepresent the quality before the sudden change of dynamic vibration absorber quality;
M 2arepresent the quality after the sudden change of dynamic vibration absorber quality;
represent the speed before the sudden change of dynamic vibration absorber quality;
represent the speed after the sudden change of dynamic vibration absorber quality;
(3) acceleration after absorber mass sudden change:
x ·· 2 a = - c 2 ( x · 2 a - x · 1 ) - k 2 ( x 2 - x 1 ) m 2 a
In formula: represent the speed of main system;
(3) due to bump leveller damping c 2existence, its velocity jump will cause main system acceleration sudden change, the main system acceleration after absorber mass sudden change can calculate according to following formula:
x ·· 1 a = F 0 s i n ( ω t ) - c 2 ( x · 1 - x · 2 a ) - c 1 x · 1 - k 2 ( x 1 - x 2 ) - k 1 x 1 m 1
Finally obtain the motion vector of t+ Δ t, velocity vector and vector acceleration to be shown below:
[ M ] t + Δ t = m 1 0 0 m 2 + m v ( t + Δ t )
{ X } t + Δ t = [ K ‾ ] t - 1 { F ‾ }
{ X · } t + Δ t = x · 1 ( t + Δ t ) ( m 2 + m v ( t ) ) x · 2 ( t + Δ t ) / m 2 + m v ( t + Δ t )
{ X ·· } t + Δ t = F 0 sin ( ω ( t + Δ t ) ) - c 2 ( x · 1 ( t + Δ t ) - x · 2 a ( t + Δ t ) ) - c 1 x · 1 ( t + Δ t ) - k 2 ( x 1 ( t + Δ t ) - x 2 ( t + Δ t ) ) - k 1 x 1 ( t + Δ t ) / m 1 - c 2 ( x · 2 a ( t + Δ t ) - x · 1 ( t + Δ t ) ) - k 2 ( x 2 ( t + Δ t ) - x 1 ( t + Δ t ) ) / m 2 + m v ( t + Δ t )
Step 10, repeats said process step 5 obtains subsequent time t+2 Δ t motion vector { X} to step 9 t+2 Δ t, velocity vector and vector acceleration finally obtain the motion vector of whole process, velocity vector and vector acceleration.
The present invention compared with prior art has following Advantageous Effects:
The emulation mode that application the present invention carries considers momentum conservation law in mass change process, makes variable mass process meet the natural law.This emulation mode degree of error is credible, can be obtained by the contrast of following simulation and experiment: the rear main system of absorber mass reduction vibrates both the percentage difference 7% declined, and (in emulation, the rear main system of absorber mass reduction vibrates decline 84.3%; And in experiment, main system vibration decline 91.3% after absorber mass reduces).The emulation mode that application the present invention carries can emulate the transient state vibration damping process of variable mass dynamic vibration absorber, thus realizes its quantitative study.The inventive method has certain versatility, for solving other variable-mass dynamics problems, has certain reference.
Accompanying drawing explanation
Fig. 1 is variable mass dynamic vibration absorber model.
Fig. 2 is emulation institute Host Systems frequency response function simulation curve.
Main system acceleration time domain response curve when Fig. 3 is the absorber mass reduction of emulation gained.
Fig. 4 is variable mass dynamic vibration absorber experimental system sketch.
Fig. 5 is experiment institute Host Systems frequency response function curve.
Fig. 6 is experiment institute Host Systems acceleration time domain response curve.
Below in conjunction with drawings and Examples, explanation is further elaborated to technological scheme of the present invention.
Embodiment
Defer to technique scheme, following embodiment provides a kind of variable mass dynamic vibration absorber transient process emulation mode, and the method specifically comprises the following steps:
Step one, obtains the t main system of quality change and the initial mass matrix of bump leveller system according to variable mass dynamic vibration absorber [ M ] t = m 1 0 0 m 2 + m v , Initial damping matrix [ C ] = c 1 + c 2 - c 2 - c 2 c 2 With initial stiffness matrix [ K ] = k 1 + k 2 - k 2 - k 2 k 2 ;
In formula: m 1represent the quality of main system, c 1represent the damping of main system, k 1represent the rigidity of main system; m 2represent the fixed mass of bump leveller, m vrepresent the variable-quality of bump leveller, c 2represent the damping of bump leveller, k 2represent the rigidity of bump leveller;
Step 2, obtains the t main system of quality change and the initial load vector of bump leveller system according to variable mass dynamic vibration absorber { F } t = F 0 s i n ( ω t ) 0 , Initial displacement vector { X } t = x 1 x 2 With initial velocity vector { X · } t = x · 1 x · 2 ;
In formula: F 0sin (ω t) represents dynamic excitation power, x 1represent the initial displacement of main system, x 2represent the initial displacement of bump leveller, represent the initial velocity of main system, represent the initial velocity of bump leveller;
Step 3, calculates the initial acceleration vector of t
[ M ] t { X ·· } t = { F } t - [ C ] { X · } t - [ K ] { X ·· } t
Step 4, provides integration step Δ t and β, γ, and calculates integration constant:
α 0 = 1 γΔt 2 , α 1 = β γ Δ t , α 2 = 1 γ Δ t , α 3 = 1 2 γ - 1 ,
α 4 = β γ - 1 , α 5 = Δ t 2 ( β γ - 2 ) , α 6 = Δ t ( 1 - β ) , α 7 = β Δ t ;
In formula: Δ t=0.001s,
Step 5, calculates effective stiffness matrix
Step 6, calculates the useful load vector of t+ Δ t:
{ F ‾ } = { F } t + Δ t + [ M ] t ( 1 γΔt 2 { X } t + 1 γ Δ t { X · } t + ( 1 2 γ - 1 ) { X ·· } t ) + [ C ] ( β γ Δ t { X } t + ( β γ - 1 ) { X · } t + ( β 2 γ - 1 ) Δ t { X ·· } t )
Step 7, calculates the motion vector of t+ Δ t:
Step 8, calculates vector acceleration and the velocity vector of t+ Δ t:
{ X ·· } t + Δ t = α 0 ( { X } t + Δ t - { X } t ) - α 2 { X · } t - α 3 { X ·· } t
{ X · } t + Δ t = { X · } t + α 6 { X ·· } t + α 7 { X ·· } t + Δ t
Step 9, absorber mass m 2+ m vt () sports m 2+ m v(t+ Δ t), motion vector, velocity vector and vector acceleration are revised respectively in accordance with the following methods:
(1) shift invariant after absorber mass sudden change,
(2) speed after absorber mass sudden change
In formula:
M 2brepresent the quality before the sudden change of dynamic vibration absorber quality;
M 2arepresent the quality after the sudden change of dynamic vibration absorber quality;
represent the speed before the sudden change of dynamic vibration absorber quality;
represent the speed after the sudden change of dynamic vibration absorber quality;
(3) acceleration after absorber mass sudden change:
x ·· 2 a = - c 2 ( x · 2 a - x · 1 ) - k 2 ( x 2 - x 1 ) m 2 a
In formula: represent the speed of main system;
(3) due to bump leveller damping c 2existence, its velocity jump will cause main system acceleration sudden change, the main system acceleration after absorber mass sudden change can calculate according to following formula:
x ·· 1 a = F 0 s i n ( ω t ) - c 2 ( x · 1 - x · 2 a ) - c 1 x · 1 - k 2 ( x 1 - x 2 ) - k 1 x 1 m 1
Finally obtain the motion vector of t+ Δ t, velocity vector and vector acceleration to be shown below:
[ M ] t + Δ t = m 1 0 0 m 2 + m v ( t + Δ t )
{ X } t + Δ t = [ K ‾ ] t - 1 { F ‾ }
{ X · } t + Δ t = x · 1 ( t + Δ t ) ( m 2 + m v ( t ) ) x · 2 ( t + Δ t ) / m 2 + m v ( t + Δ t )
{ X ·· } t + Δ t = F 0 sin ( ω ( t + Δ t ) ) - c 2 ( x · 1 ( t + Δ t ) - x · 2 a ( t + Δ t ) ) - c 1 x · 1 ( t + Δ t ) - k 2 ( x 1 ( t + Δ t ) - x 2 ( t + Δ t ) ) - k 1 x 1 ( t + Δ t ) / m 1 - c 2 ( x · 2 a ( t + Δ t ) - x · 1 ( t + Δ t ) ) - k 2 ( x 2 ( t + Δ t ) - x 1 ( t + Δ t ) ) / m 2 + m v ( t + Δ t )
Step 10, repeats said process step 5 obtains subsequent time t+2 Δ t motion vector { X} to step 9 t+2 Δ t, velocity vector and vector acceleration finally obtain the motion vector of whole process, velocity vector and vector acceleration.
Below provide specific embodiments of the invention, it should be noted that the present invention is not limited to following specific embodiment, all equivalents done on technical scheme basis all fall into protection scope of the present invention.
Emulation embodiment:
Below in conjunction with accompanying drawing, verify a kind of variable mass dynamic vibration absorber transient process emulation mode proposed by the invention by experiment.Apply above-mentioned emulation mode, the system shown in his-and-hers watches 1 emulates.
Table 1 variable mass dynamic vibration absorber parameter
Below the embodiment utilizing emulation mode of the present invention to emulate variable mass dynamic vibration absorber that inventor provides.
First main system frequency response function curve when calculating absorber mass is minimum, as shown in Figure 2.As can be seen from Figure 2, best damping frequency when absorber mass is minimum is at 15Hz place.Parameters integration step Δ t=0.001s, initial displacement and initial velocity are 0, and extraneous excitation force frequency gets f=15Hz (the best damping frequency corresponding in Fig. 2).Variable-quality m vfollowing formula change is pressed in change:
m v = 10 0 &le; t &le; 10 20 - t 10 < t &le; 20 0 20 < t &le; 50
The response of main system acceleration time domain as shown in Figure 3.As can be seen from Figure 3, main system is about 4.01m/s at the acceleration vibration amplitude of front 10s 2, along with quality m from the 10th second vcontinuous reduction and reduce, through several times vibration after be stabilized in 0.63m/s 2left and right.After quality reduces, main system acceleration stable state amplitude probably declines 84.3%.
The experimental verification of simulation result:
Experimental system sketch as shown in Figure 4, main system and bump leveller system all elect overhang vibration system as, main system is formed by overhang and mass block, by elastic steel sheet and be connected with the bottle of injector to form bump leveller system, be connected with the bottle of injector to change the quality of bump leveller system, the initial parameters of experimental system is identical with bump leveller system with the main system in emulation embodiment.Signal generator is connected with power amplifier, and power amplifier is connected with vibration exciter, and main system is arranged on vibration exciter; Vibration exciter and main system are separately installed with acceleration transducer, and acceleration transducer is connected with the input end of charge amplifier, and the output terminal of charge amplifier is connected with data collecting instrument, and data collecting instrument is connected with computer.
In experimentation, the accumulation signal that signal generator produces, exports to vibration exciter after power amplifier amplifies, and vibration exciter drives main system vibration according to excitation pulse; The oscillating signal that main system and vibration exciter export, after the collection of charge type acceleration transducer, charge amplifier amplification, data collecting instrument gather, is accepted by data acquisition software, is stored, and analyzes variable mass dynamic vibration absorber to the effectiveness in vibration suppression of main system.
In order to verify variable mass dynamic vibration absorber transient process emulation mode.This experiment first test absorber mass minimum time main system frequency response function curve as shown in Figure 5.In Fig. 5, abscissa is excitation force frequency, and y coordinate is main system acceleration frequence responses function, and it calculates by following formula:
Wherein be respectively the acceleration stable state amplitude of vibration exciter and main system.As can be seen from Figure 5, the best damping frequency of bump leveller is 15Hz.In experiment, excitation force frequency chooses 15Hz (the best damping frequency corresponding to bump leveller in Fig. 6), and at the uniform velocity draw water 10g and time used are 11.54s, and the response of main system acceleration time domain as shown in Figure 6.As can be seen from Figure 6, at first 10 seconds main system acceleration amplitudes at 4.02m/s 2place, reduced along with absorber mass and reduces, being finally tending towards 0.35m/s from the 10th second 2.Main system vibration decline 91.3% after absorber mass reduces.Can be obtained by contrast and experiment Fig. 6 and simulation result Fig. 3, experiment mesometamorphism amount dynamic vibration absorber transient state effectiveness in vibration suppression and simulation are seemingly.Utilize emulation mode of the present invention can carry out simulation study to the transient process of variable mass dynamic vibration absorber.But still there is certain error, the main cause producing error is: only consider the most important vertical vibration of system in simulation calculation, and in experiment except main vertical vibration, also has and swing and transverse vibration; Ignore ambient conditions impact in simulation calculation, and there is the impacts such as sensor data cable, injector flexible pipe and various apparatus errors in experiment.

Claims (1)

1. a variable mass dynamic vibration absorber transient process emulation mode, is characterized in that: the method specifically comprises the following steps:
Step one, obtains the t main system of quality change and the initial mass matrix of bump leveller system according to variable mass dynamic vibration absorber &lsqb; M &rsqb; t = m 1 0 0 m 2 + m v , Initial damping matrix &lsqb; C &rsqb; = c 1 + c 2 - c 2 - c 2 c 2 With initial stiffness matrix &lsqb; K &rsqb; = k 1 + k 2 - k 2 - k 2 k 2 ;
In formula: m 1represent the quality of main system, c 1represent the damping of main system, k 1represent the rigidity of main system; m 2represent the fixed mass of bump leveller, m vrepresent the variable-quality of bump leveller, c 2represent the damping of bump leveller, k 2represent the rigidity of bump leveller;
Step 2, obtains the t main system of quality change and the initial load vector of bump leveller system according to variable mass dynamic vibration absorber { F } t = F 0 s i n ( &omega; t ) 0 , Initial displacement vector { X } t = x 1 x 2 With initial velocity vector { X &CenterDot; } t = x &CenterDot; 1 x &CenterDot; 2 ;
In formula: F 0sin (ω t) represents dynamic excitation power, x 1represent the initial displacement of main system, x 2represent the initial displacement of bump leveller, represent the initial velocity of main system, represent the initial velocity of bump leveller;
Step 3, calculates the initial acceleration vector of t
&lsqb; M &rsqb; t { X &CenterDot;&CenterDot; } t = { F } t - &lsqb; C &rsqb; { X &CenterDot; } t - &lsqb; K &rsqb; { X &CenterDot;&CenterDot; } t
Step 4, provides integration step Δ t and β, γ, and calculates integration constant:
&alpha; 0 = 1 &gamma;&Delta;t 2 , &alpha; 1 = &beta; &gamma; &Delta; t , &alpha; 2 = 1 &gamma; &Delta; t , &alpha; 3 = 1 2 &gamma; - 1 ,
&alpha; 4 = &beta; &gamma; - 1 , &alpha; 5 = &Delta; t 2 ( &beta; &gamma; - 2 ) , &alpha; 6 = &Delta; t ( 1 - &beta; ) , &alpha; 7 = &beta; &Delta; t ;
In formula: Δ t=0.001s,
Step 5, calculates effective stiffness matrix
Step 6, calculates the useful load vector of t+ Δ t:
{ F &OverBar; } = { F } t + &Delta; t + &lsqb; M &rsqb; t ( 1 &gamma;&Delta;t 2 { X } t + 1 &gamma; &Delta; t { X &CenterDot; } t + ( 1 2 &gamma; - 1 ) { X &CenterDot;&CenterDot; } t ) + &lsqb; C &rsqb; ( &beta; &gamma; &Delta; t { X } t + ( &beta; &gamma; - 1 ) { X &CenterDot; } t + ( &beta; 2 &gamma; - 1 ) &Delta; t { X &CenterDot;&CenterDot; } t )
Step 7, calculates the motion vector of t+ Δ t:
Step 8, calculates vector acceleration and the velocity vector of t+ Δ t:
{ X &CenterDot;&CenterDot; } t + &Delta; t = &alpha; 0 ( { X } t + &Delta; t - { X } t ) - &alpha; 2 { X &CenterDot; } t - &alpha; 3 { X &CenterDot;&CenterDot; } t
{ X &CenterDot; } t + &Delta; t = { X &CenterDot; } t + &alpha; 6 { X &CenterDot;&CenterDot; } t + &alpha; 7 { X &CenterDot;&CenterDot; } t + &Delta; t
Step 9, absorber mass m 2+ m vt () sports m 2+ m v(t+ Δ t), motion vector, velocity vector and vector acceleration are revised respectively in accordance with the following methods:
(1) shift invariant after absorber mass sudden change,
(2) speed after absorber mass sudden change
In formula:
M 2brepresent the quality before the sudden change of dynamic vibration absorber quality;
M 2arepresent the quality after the sudden change of dynamic vibration absorber quality;
represent the speed before the sudden change of dynamic vibration absorber quality;
represent the speed after the sudden change of dynamic vibration absorber quality;
(3) acceleration after absorber mass sudden change:
x &CenterDot;&CenterDot; 2 a = - c 2 ( x &CenterDot; 2 a - x &CenterDot; 1 ) - k 2 ( x 2 - x 1 ) m 2 a
In formula: represent the speed of main system;
(3) due to bump leveller damping c 2existence, its velocity jump will cause main system acceleration sudden change, the main system acceleration after absorber mass sudden change can calculate according to following formula:
x &CenterDot;&CenterDot; 1 a = F 0 s i n ( &omega; t ) - c 2 ( x &CenterDot; 1 - x &CenterDot; 2 a ) - c 1 x &CenterDot; 1 - k 2 ( x 1 - x 2 ) - k 1 x 1 m 1
Finally obtain the motion vector of t+ Δ t, velocity vector and vector acceleration to be shown below:
&lsqb; M &rsqb; t + &Delta; t = m 1 0 0 m 2 + m v ( t + &Delta; t )
{ X } t + &Delta; t = &lsqb; K &OverBar; &rsqb; t - 1 { F &OverBar; }
{ X &CenterDot; } t + &Delta; t = x &CenterDot; 1 ( t + &Delta; t ) ( m 2 + m v ( t ) ) x &CenterDot; 2 ( t + &Delta; t ) / m 2 + m v ( t + &Delta; t )
{ X &CenterDot;&CenterDot; } t + &Delta; t = F 0 sin ( &omega; ( t + &Delta; t ) ) - c 2 ( x &CenterDot; 1 ( t + &Delta; t ) - x &CenterDot; 2 a ( t + &Delta; t ) ) - c 1 x &CenterDot; 1 ( t + &Delta; t ) - k 2 ( x 1 ( t + &Delta; t ) - x 2 ( t + &Delta; t ) ) - k 1 x 1 ( t + &Delta; t ) / m 1 - c 2 ( x &CenterDot; 2 a ( t + &Delta; t ) - x &CenterDot; 1 ( t + &Delta; t ) ) - k 2 ( x 2 ( t + &Delta; t ) - x 1 ( t + &Delta; t ) ) / m 2 + m v ( t + &Delta; t )
Step 10, repeats said process step 5 obtains subsequent time t+2 Δ t motion vector { X} to step 9 t+2 Δ t, velocity vector and vector acceleration finally obtain the motion vector of whole process, velocity vector and vector acceleration.
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