CN110489904A - A kind of Studying Metal Rubber Isolator System kinetic model and parameter identification method - Google Patents
A kind of Studying Metal Rubber Isolator System kinetic model and parameter identification method Download PDFInfo
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Abstract
The present invention relates to a kind of Studying Metal Rubber Isolator System kinetic model and parameter identification methods, and the model is as the formula (1), by the variable part of metal-rubber elastic restoring force, have the damping force of memory and the damping force of complexity to indicate with fractional order differential item.Model physical meaning of the present invention is clear, can more more accurately describe the constitutive bchavior of metal-rubber, and parameter is greatly reduced.Experiment it is found that the model can fit the hysteresis loop very close with experimental result, illustrate its can accurate description hysteresis loop there is important reference value when making and designing the parameter of the glue vibration isolator containing metal rubber.
Description
Technical field
The present invention relates to Studying Metal Rubber Isolator System kinetic model technical field, specifically a kind of metal-rubber every
Vibrating system kinetic model and parameter identification method.
Background technique
Metal-rubber not only has the good elasticity of rubber like and dissipation a large amount of as a kind of functional structure material
The damping capacity of vibration and impact energy, and have the characteristics that high temperature resistant, corrosion-resistant, non-aging.
The component that metal-rubber is prepared into can satisfy the special requirement of aerospace, spacecraft and civilian goods operating condition,
Damping, vibration damping, filtering, sealing, throttling, heat transfer under the environment such as solution high temperature, low temperature, high pressure, high vacuum and high vibration
The problems such as.Metal-rubber component had not only had the inherent characteristic of selected metal, but also had and be similar to the same elasticity of rubber.In space
It is not evaporated under environment, does not fear high temperature and low temperature, be not afraid of space radiation and particle is hit, select different metals can be with corrosion-resistant
Environment, and the possibility without aging are the best substitutes of traditional rubber.China starts late to the research of metal-rubber component,
But application of the metal-rubber part on spacecraft has caused the attention of authorities at different levels, each scientific research of aerospace field
Department proposes comprehensive demand to metal-rubber component, thus metal-rubber product has a wide range of applications market in China.
In numerous applications, the vibrating isolation system of photoelectric platform is one of metal-rubber damper typical case.With existing
For the needs of military technology development, communications field status of the surveillance information in war is more aobvious to be protruded.It is scouted to make up fixed station
Reconnaissance system is placed in armored ground vehicle, naval vessel, unmanned plane, satellite by the deficiency that the visual field is limited by optical system field angle
Etc. expand the dynamic visual field of optical system and increase receiving information on dynamic loads body.
Dynamic load body inertia photoelectric platform has become scouting weaponry indispensable in modern war, it is integrated with visible
The difference spectral band optical sensor such as photoimaging systems, thermal infrared imager, laser range finder/indicator, passes through remote nothing
Scouting result is returned to the control information station at rear, realizes the accurate commander to battlefield by line electricity link data transfer.Dynamic load body
Photoelectric platform belongs to high-accuracy weaponry, the vibration meeting pair due to caused by the attitudes vibration of carrier, windage and power device etc.
The reliability generation of its photoelectric sensor seriously affects.It, must for the image quality and lasting accuracy for improving dynamic load body photoelectric platform
The vibration of carrier must be inhibited.
Currently, the isolation scheme that photoelectric platform uses is mainly two axis, four frame structure, still, regardless of which kind of shock insulation used
Scheme, to designing and making accurately photoelectricity inertial platform, metal-rubber is as vibration isolation element, the accuracy of mathematical model
The parameter designing of the vibration isolator directly affected with complexity, therefore establish accurate and simple metal-rubber nonlinear constitutive relation
And the kinetic model of shock isolation system has important engineering value.
Although there are many practical application, the correlation engineerings applications such as Dynamic Modeling in relation to metal-rubber for metal-rubber
Technical know-how still needs to further study.
The basic material structure ginseng such as metal-rubber restoring force and wire diameter, spiral coil diameter, material relative density
Number is related, so many scholars study the thin sight constitutive relation of metal-rubber, has obtained including Flexural cantilever model, angle
Different types of micro-mechanical constitutive models such as Based On The Conic Model, curved Beam Model, infinitesimal spring model, to study metal-rubber dynamic system
It lays the foundation.But dynamics macromodel is the main foundation for studying dynamic performance, metal-rubber vibration insulating system moves
Mechanical modeling and parameter identification are always the critical issue of metal-rubber research.Abundant experimental results show metal-rubber element
Stress-strain relation be non-linear hysteresis curve.Numerous scholars describe hysteresis loop using a variety of mathematical models, Bai Hongbai and
Li Dongwei carries out approximation to metal-rubber lag loop using bilinear model, that is, improved Bingham model, can be preferably
The dry friction of metal-rubber is described, but this model only considered linear rigidity and have ignored the non-linear of metal-rubber completely
Characteristic；Zhou Yanguo etc. carries out the research of metal-rubber mechanical property using trace method model, and trace method model is based on average and equivalent original
Reason, restoring force is related with amplitude and frequency, but is had ignored in metal-rubber in model containing the complicated restoring force such as class viscoplasticity
Constitutive relation；Wang Zhidong studies metal-rubber using Bouc-Wen model, and Bouc-Wen model is a kind of relatively warp
The model of the description hysteresis loop of allusion quotation, is made of, precision is higher, but elastic force and damping force table in system two smooth curves
Existing form is indefinite, and parameter is more, is unfavorable for parameter identification.Yang Shaopu et al. is proposed with 3 powers of displacement and speed to stagnant
The advantages of nonlinear system is simulated afterwards, model is that mathematic(al) representation is simple, and undetermined parameter is few, is free of sign function, very
The research of mechanism and characteristic suitable for system hysteresis non-linearity, but it is not particularly suited for the material containing viscoelastic property.Wang Yajie
Reflect that its macroscopic property, Xiao Kun are using SIN function and e index by establishing the hardening broken line constitutive relation of metal-rubber
The functional form that function combines describes damped coefficient with the changing rule of amplitude and frequency, but their constitutive relations for being proposed
The complicated restoring force such as viscoplasticity is not considered.Different metal rubber cast has certain scope of application.But it is high in order to obtain
The model of precision must be provided with enough parameters, increase complexity for Analysis of dynamics performance；And these models are most
Containing sign function, when analysis system dynamic performance, inevitably carries out gradation study, it is difficult to obtain complete continuous dynamic
Mechanical response characteristic.It is therefore desirable to further investigate accurate, simple and readily identified parameter kinetic model.
Summary of the invention
It is an object of the invention to provide a kind of Studying Metal Rubber Isolator System kinetic model and parameter identification method, to
Important reference is provided when making and designing the vibrating isolation system parameter containing metal-rubber.
The present invention is implemented as follows: a kind of Studying Metal Rubber Isolator System kinetic model, the model such as formula (1) institute
Show:
In formula, x (t) is metal-rubber spring deflection, k'1For linear elasticity restoring force constant portion stiffness coefficient, k'3
For cubic non-linearity elastic restoring force stiffness coefficient, c'1For single order viscous damping coefficient, h' is fractional order viscoelastic damping system
Number, p are Fractional Derivative order, and m is mass of system, and F' is sinusoidal excitation amplitude, and ω is excitation angular frequency.
By the variable part of metal-rubber elastic restoring force, there are the damping force of memory and complicated damping force micro- with fractional order
Subitem indicates, to obtain the constitutive relation model of the metal-rubber as shown in formula (2):
The Studying Metal Rubber Isolator System kinetic model can be obtained according to constitutive relation model.
A kind of parameter identification method of above-mentioned Studying Metal Rubber Isolator System kinetic model, comprising the following steps:
A, following substitution of variable is carried out to system:
The Studying Metal Rubber Isolator System as shown in formula (3) being obtained
Kinetics equation:
B, set the solution of kinetics equation asIn formula, xmFor vibration amplitude,For phase angle；It willFractional order differential item is substituted into, and the equation as shown in formula (4) is obtained by the pull-type final abbreviation of transformation
Formula:
C, parameter identification is carried out to equation (4) and linear rigidity coefficient k is determined according to actual condition1, rigidity system three times
Number k3, damped coefficient c1, fractional order damped coefficient h and fractional order order the p functional relation with vibration amplitude and frequency respectively.
Further, in step b, the process of abbreviation are as follows: set fractional order differential item as f (t)=h [DpX (t)], it willFractional order differential item is substituted into, and carries out pull-type transformation L [f (t)]=L { h [DpX (t)] }, it can obtain
Pull-type inverse transformation is carried out to formula (5)It can obtain
The method that pull-type inverse transformation is calculated using residue theorem,Singular point be S=± i ω, can by residue theorem
By formulaBringing formula (7) abbreviation into can obtain
It can similarly obtain
Formula (9) and formula (10), which are substituted into formula (6), to be obtained
Formula (11) are substituted into formula (3) up to formula (4) illustrated equation.
Metal-rubber constitutive relation is extremely complex, to estimating and by its enough vibrational system for its nonlinear constitutive relation
Response computation it is all extremely difficult, be the problem in generally acknowledged vibration engineering field urgently to be resolved.The main of Dynamic Modeling is appointed
Business is exactly experimental study and the theory analysis according to metal-rubber flexible deformation and damping energy dissipation, is gone to portray and be retouched with mathematical model
The kinetic characteristics for stating metallic rubber vibration isolator, the whether reasonable standard of the model established are exactly to see that can it reflect metal rubber
The typical nonlinear characteristic of glue vibration isolator.
Viscoelastic fractional differential term had not only contained the variable part of elastic restoring force in model of the present invention, but also contained multiple
Miscellaneous damping item parts.The model structure is simple, and the model of continuous non-segmentation can be obtained.More importantly fractional order item contains note
Recall characteristic, is ideal mode for describing to have the material of memory characteristic.
Model physical meaning of the present invention is clear, can more more accurately describe the constitutive bchavior of metal-rubber, and parameter is big
To reduce.Experiment illustrates that it can accurately be retouched it is found that the model can fit the hysteresis loop very close with experimental result
Hysteresis loop is stated, when making and designing the parameter of the glue vibration isolator containing metal rubber, there is important reference value.
Detailed description of the invention
Fig. 1 is Studying Metal Rubber Isolator System kinetic model schematic diagram of the present invention.
Fig. 2 is the structure chart of metal-rubber module in embodiment.
Fig. 3 is the structural schematic diagram of metallic rubber vibration isolator in embodiment.
Fig. 4 is empirical curve and models fitting curve comparison figure.
Fig. 5 is stiffness coefficient with amplitude variations curve graph.
Fig. 6 is damped coefficient match value and calculated value comparison diagram.
Fig. 7 is the match value and calculated value comparison diagram of fractional order order and coefficient.
Fig. 8 is empirical curve, matched curve and calculated curve comparison diagram.
Specific embodiment
Below in conjunction with attached drawing, present invention is further described in detail, and those skilled in the art can be as disclosed by this specification
Content realize the present invention.
One, shown in Studying Metal Rubber Isolator System kinetic model of the present invention such as formula (1):
In formula, x (t) is metal-rubber spring deflection, k'1For linear elasticity restoring force constant portion stiffness coefficient, k'3
For cubic non-linearity elastic restoring force stiffness coefficient, c'1For single order viscous damping coefficient, h' is fractional order viscoelastic damping system
Number, p are Fractional Derivative order, and m is mass of system, and F' is sinusoidal excitation amplitude, and ω is excitation angular frequency.
Studying Metal Rubber Isolator System kinetic model is as shown in Figure 1.Above-mentioned model modeling are as follows: it is non-to ignore high order more than three times
Linear elasticity restoring force and high order Nonlinear Damping Term by the variable part of metal-rubber elastic restoring force, have the damping of memory
Power and complicated damping force are indicated with fractional order differential item, to obtain the constitutive relation mould of the metal-rubber as shown in formula (2)
Type:
The Studying Metal Rubber Isolator System kinetic model can be obtained according to constitutive relation model.Viscoplasticity point in model
Number rank differential term had not only contained the variable part of elastic restoring force, but also contained complicated damping item parts.Link will be remembered in formula
It being substituted by fractional order differential item, gained model physical meaning is clear, the constitutive bchavior of metal-rubber can be described relatively accurately,
And parameter is greatly reduced.
Two, following substitution of variable is carried out to system:
The Studying Metal Rubber Isolator System as shown in formula (3) being obtained
Kinetics equation:
Three, set the solution of kinetics equation asIn formula, xmFor vibration amplitude,For phase angle；It willFractional order differential item is substituted into, and is obtained by the pull-type final abbreviation of transformation such as formula (4) illustrated equation:
The process of specific abbreviation are as follows: set fractional order differential item as f (t)=h [DpX (t)], it willGeneration
Enter fractional order differential item, and carries out pull-type transformation L [f (t)]=L { h [DpX (t)] }, it can obtain
Pull-type inverse transformation is carried out to formula (5)It can obtain
The method that pull-type inverse transformation is calculated using residue theorem,Singular point be S=± i ω, can by residue theorem
By formulaBringing formula (7) abbreviation into can obtain
It can similarly obtain
Formula (9) and formula (10), which are substituted into formula (6), to be obtained
Formula (11) are substituted into formula (3) up to formula (4) illustrated equation.
Four, parameter identification is carried out to equation (4)
1, metallic rubber vibration isolator mechanical property tests
Each metal-rubber spring module uses density for 1.6 × 103Kg/mm3, string diameter 0.325mm is highly 32mm
304 (06Cr19Ni10) stainless steel metal wire materials be made, metal-rubber module is as shown in Figure 2.
Test specimen uses the symmetrical metallic rubber vibration isolator structure of typical drawing-pressure, respectively matches 4 pieces of metal-rubbers, knot up and down
Structure is as shown in Figure 3, wherein 1 refers to metal-rubber block.
When initial assembly, metal module connects always when upper and lower module carries out precompressed to guarantee that center pull rod moves up and down
Four pieces of metal-rubbers are considered as entirety and studied by touching, compressive pre-stress 1.96KN.Metal-rubber is tried using fatigue tester
Part carries out sinusoidal load experiment.
Center pull rod does sinusoidal displacement movement under the drive of test machine grip holder near equipoise.Experiment exists respectively
It is carried out under 5 kinds of different amplitudes (0.8mm, 1.5mm, 2mm, 2.5mm, 3mm), wherein 0.8mm and 1.5mm chooses 5 kinds of different frequencies
(1Hz, 2Hz, 3Hz, 4Hz, 5Hz), 2mm choose 4 kinds of different frequencies (1Hz, 2Hz, 3Hz, 4Hz), and 2.5mm and 3mm choose 3 kinds not
Same frequency (1Hz, 2Hz, 3Hz) tests the relationship of test specimen displacement corresponding under different frequency and amplitude and power.
2, the fitting identification of model parameter
Parameter identification is carried out to model using MATLAB GAs Toolbox, is carried out curve fitting to 15 kinds of operating conditions
Afterwards, model and parameters identification can be obtained the results are shown in Table 1.
Table 1: parameter recognition result
Empirical curve and matched curve are compared, as shown in Fig. 4 (a), Fig. 4 (b), wherein (a) be (amplitude 0.8mm,
1.5mm, 3mm, frequency 1Hz), (b) be (amplitude 0.8mm, 1.5mm, 3mm, frequency 2Hz).
As seen from the figure, matched curve and the empirical curve goodness of fit are high, and parameter recognition result is ideal, illustrates that the present invention mentions
Model out can describe the Hysteresis Nonlinear characteristic of metal rubber shock absorber well.
3, the functional relation of parameter and excitation amplitude and frequency identifies
According to the recognition result of test specimen parameter under operating conditions different in table 1, analysis parameter with frequency and amplitude variation tendency,
It determines each parameter and motivates the functional relation expression formula of amplitude and frequency.
3.1, stiffness coefficient k1(xm) and k3(xm) functional relation identification
Linear rigidity coefficient k1, stiffness coefficient k three times3Shown in curve such as Fig. 5 (a), Fig. 5 (b) with amplitude variations.
By Fig. 5 (a) it is found that as amplitude increases, linear rigidity coefficient is gradually reduced, and variation tendency is unrelated with frequency.Cause
With the curve of amplitude variations, i.e., this is fitted it using the polynomial of degree n of amplitude
In formula, n=3, aiFor parameter to be identified.
By Fig. 5 (b) it is found that amplitude is within the scope of 0.8mm~1.5mm, stiffness coefficient declines quickly three times；With amplitude
Continue to increase, stiffness coefficient gradually tends towards stability three times.Therefore using amplitude power function fitting its with amplitude variations curve,
I.e.
In formula, b0、b1For coefficient to be identified.
It can obtain that linear rigidity coefficient, stiffness coefficient function calculation expression is as follows three times by least square method fitting
3.2, damped coefficient c1(xm, f) functional relation identification
Damped coefficient c1Variation with amplitude and frequency is complex, the function shape combined using multinomial with power function
Formula describes damped coefficient, i.e.,
In formula, n=13, di, e be parameter to be identified.
(16) formula is recognized using least square method, it is shown to obtain function calculation expression such as (17).
By damped coefficient calculated value c1(xm, f) and match value c1It compares, as shown in Figure 6.As seen from the figure, the expression formula
Accurately damped coefficient can be calculated.
3.3, fractional order coefficient h (xm, f) functional relation identification
Fractional order coefficient is more complicated with amplitude and frequency variation.By largely calculating analysis, fractional order coefficient and vibration
Width, frequency and fractional order order have relationship, therefore, describe it with vibration in such a way that power function and multinomial combine
The functional relation of width, frequency and order variation, as shown in (18) formula.
In formula, n=13, ji, r be parameter to be identified.
According to the match value p of fractional order order, by multinomial coefficient j in least square method fitting (18) formulaiWith power exponent r
It can obtain
3.4, fractional order order p (xm, f) functional relation identification
By analyzing kinetics equation, the functional relation of fractional order coefficient and order and amplitude and frequency is found.
Enable x (t)=xmCos (ω t+ θ), then equation (3) is
When ω t=pi/2, then equation becomes
(3) formula is solved and can be obtained using the method for average
In formula
C (p)=c1+hωp-1sin(pπ/2) (22)
K (p)=k1+hωpcos(pπ/2) (23)
Therefore, by formula (21), (22) contain only unknown number p, thus can calculate after (23) substitute into (20b) formula in equation
P value.The calculated value of p, which is substituted into (19) formula, to calculate h.Compare fractional order order calculated value p (xm, f) and match value
P, fractional order coefficient calculated value h (xm, f) and match value h respectively as shown in Fig. 7 (a), Fig. 7 (b).
By Fig. 7 (a), Fig. 7 (b) it is found that calculated value p (xm, f) and match value p, calculated value h (xm, f) and match value h is very
It is close, therefore, fractional order order and fractional order coefficient are calculated using (20b) formula, (19) formula.
The function expression of each parameter is substituted into (3) formula, the restoring force that can be calculated under various amplitude and frequency is sluggish bent
Line.Empirical curve, matched curve and calculated curve are compared, as shown in Fig. 8 (a), Fig. 8 (b), wherein (a) is (amplitude
1.5mm, frequency 1Hz), (b) be (amplitude 2mm, frequency 3Hz).
As seen from Figure 8, the vibration isolator restoring force mathematical model proposed by the invention containing fractional order can be good at
Describe restoring force with amplitude, frequency, order changing rule, meet the actual requirement of engineer application.
Claims (4)
1. a kind of Studying Metal Rubber Isolator System kinetic model, characterized in that shown in the model such as formula (1):
In formula, x (t) is metal-rubber spring deflection, k '1For linear elasticity restoring force constant portion stiffness coefficient, k '3It is three
Secondary nonlinear elasticity restoring force stiffness coefficient, c '1For single order viscous damping coefficient, h' is fractional order viscoelastic damping coefficient, and p is
Fractional Derivative order, m are mass of system, and F' is sinusoidal excitation amplitude, and ω is excitation angular frequency.
2. Studying Metal Rubber Isolator System kinetic model according to claim 1, characterized in that metal-rubber elasticity is extensive
The damping force of the variable part of multiple power, the damping force for having memory and complexity is indicated with fractional order differential item, to obtain such as formula (2)
Shown in metal-rubber constitutive relation model:
The Studying Metal Rubber Isolator System kinetic model can be obtained according to constitutive relation model.
3. a kind of parameter identification method of Studying Metal Rubber Isolator System kinetic model described in claim 1, characterized in that packet
Include following steps:
A, following substitution of variable is carried out to system:
The power of the Studying Metal Rubber Isolator System as shown in formula (3) can be obtained
Learn equation:
B, set the solution of kinetics equation asIn formula, xmFor vibration amplitude,For phase angle；It willFractional order differential item is substituted into, and is obtained by the pull-type final abbreviation of transformation such as formula (4) illustrated equation:
C, parameter identification is carried out to equation (4) and linear rigidity coefficient k is determined according to actual condition1, stiffness coefficient k three times3、
Damped coefficient c1, fractional order damped coefficient h and fractional order order the p functional relation with vibration amplitude and frequency respectively.
4. the parameter identification method of Studying Metal Rubber Isolator System kinetic model according to claim 3, characterized in that step
In rapid b, the process of abbreviation are as follows: set fractional order differential item as f (t)=h [DpX (t)], it willSubstitute into score
Rank differential term, and carry out pull-type transformation L [f (t)]=L { h [DpX (t)] }, it can obtain
Pull-type inverse transformation is carried out to formula (5)It can obtain
The method that pull-type inverse transformation is calculated using residue theorem,Singular point be S=± i ω, can be obtained by residue theorem
By formula ip=(eiπ/2)p=eipπ/2 (8)
Bringing formula (7) abbreviation into can obtain
It can similarly obtain
Formula (9) and formula (10), which are substituted into formula (6), to be obtained
Formula (11) are substituted into formula (3) up to formula (4) illustrated equation.
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| CN112114122A (en) * | 2020-09-04 | 2020-12-22 | 昆明学院 | Characterization method of viscoelastic mechanical property of rubber material |
| CN112711836A (en) * | 2020-12-17 | 2021-04-27 | 哈尔滨工程大学 | Method for rapidly obtaining technological parameters of metal rubber component |
| CN112711836B (en) * | 2020-12-17 | 2022-03-11 | 哈尔滨工程大学 | Method for rapidly obtaining technological parameters of metal rubber component |
| CN112528439A (en) * | 2020-12-22 | 2021-03-19 | 中国人民解放军陆军装甲兵学院 | Manganese-copper-based damping alloy constitutive relation analysis method and electronic equipment |
| CN112528439B (en) * | 2020-12-22 | 2024-03-15 | 中国人民解放军陆军装甲兵学院 | Manganese-copper-based damping alloy constitutive relation analysis method and electronic equipment |
| CN114818180A (en) * | 2022-04-19 | 2022-07-29 | 石家庄铁道大学 | Seawater-aged immersed tunnel GINA waterstop time-varying constitutive model construction method |
| CN114818180B (en) * | 2022-04-19 | 2024-05-03 | 石家庄铁道大学 | Construction method of sea water aging immersed tube tunnel GINA water stop time-varying constitutive model |
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