CN110489904B - Dynamic model and parameter identification method of metal rubber vibration isolation system - Google Patents
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Abstract
The invention relates to a dynamic model of a metal rubber vibration isolation system and a parameter identification method, wherein the model is shown in a formula (1), and the variable part of elastic restoring force, the memorized damping force and the complex damping force of the metal rubber are expressed by fractional differential terms. The model has definite physical meaning, can describe the constitutive property of the metal rubber more accurately, and greatly reduces parameters. Experiments show that the model can be fitted with a hysteresis curve very close to an experimental result, so that the hysteresis curve can be accurately described, and the model has important reference value when parameters of the rubber vibration isolator containing the metal rubber are manufactured and designed.
Description
Technical Field
The invention relates to the technical field of dynamic models of metal rubber vibration isolation systems, in particular to a dynamic model of a metal rubber vibration isolation system and a parameter identification method.
Background
As a functional structural material, the metal rubber not only has good elasticity similar to rubber and damping performance of dissipating a large amount of vibration and impact energy, but also has the characteristics of high temperature resistance, corrosion resistance, difficult aging and the like.
The member prepared from the metal rubber can meet the special requirements of working conditions of aerospace, space vehicles and civil products, and solves the problems of damping, vibration reduction, filtering, sealing, throttling, heat conduction and the like in environments such as high temperature, low temperature, high pressure, high vacuum, severe vibration and the like. The metal rubber member has both the inherent characteristics of the selected metal and the elasticity similar to rubber. The rubber is not evaporated in a space environment, is not afraid of high temperature and low temperature, is not afraid of space radiation and particle impact, can resist corrosion environment by selecting different metals, has no possibility of aging, and is an optimal substitute of traditional rubber. The research of the metal rubber component in China starts later, but the application of the metal rubber part on the space vehicle has attracted importance to all levels of authorities, and all scientific research departments in the aerospace field put forward comprehensive demands on the metal rubber component, so that the metal rubber product has a wide application market in China.
Among the numerous applications, vibration isolation systems for photovoltaic platforms are one of the typical applications for metal rubber shock absorbers. With the need of modern military technology development, the status of reconnaissance information in the communication field in war is more prominent. In order to make up for the defect that the reconnaissance visual field of the fixed station is limited by the visual angle of the optical system, the reconnaissance system is arranged on a moving carrier such as a ground armored vehicle, a ship, an unmanned aerial vehicle, a satellite and the like to expand the dynamic visual field of the optical system and increase the accommodation information.
The movable carrier inertial photoelectric platform is indispensable reconnaissance weaponry in modern war, integrates optical sensors with different spectral bands such as a visible light imaging system, a thermal infrared imager, a laser range finder/indicator and the like, and transmits reconnaissance results back to a rear control information station through long-distance radio link data transmission so as to realize accurate command of a battlefield. The dynamic carrier photoelectric platform belongs to high-precision weapon equipment, and the reliability of a photoelectric sensor can be seriously influenced due to the posture change of a carrier, wind resistance, vibration caused by a power device and the like. In order to improve the imaging quality and stability accuracy of the moving carrier optoelectronic stage, vibration of the carrier must be suppressed.
At present, the vibration isolation scheme adopted by the photoelectric platform is mainly of a two-axis four-frame structure, but no matter which vibration isolation scheme is adopted, the accurate photoelectric inertial platform is designed and manufactured, metal rubber is used as a vibration isolation element, and the accuracy and complexity of a mathematical model directly affect the parameter design of the vibration isolator, so that the establishment of an accurate and simple metal rubber nonlinear constitutive relation and a dynamic model of a vibration isolation system have important engineering values.
Although metal rubber has many practical applications, related engineering application technical theories such as dynamics modeling and the like related to metal rubber still need to be further researched.
The restoring force of the metal rubber is related to basic material structure parameters such as the diameter of the metal wire, the diameter of the spiral coil, the relative density of the material and the like, so that a plurality of students research the microscopic constitutive relation of the metal rubber, obtain microscopic constitutive models of different types including a cantilever Liang Moxing, a pyramid model, a curved beam model, a micro-element spring model and the like, and lay a foundation for researching a metal rubber dynamic system. However, the dynamics macroscopic model is a main basis for researching dynamics performance, and the dynamics modeling and parameter identification of the metal rubber vibration reduction system are always key problems of metal rubber research. A large number of experimental results show that the stress-strain relationship of the metal rubber element is a nonlinear hysteresis curve. Many scholars describe hysteresis curves by adopting various mathematical models, bai Hongbai and Li Dongwei approximate a metal rubber hysteresis loop by adopting a double-fold line model, namely an improved Bingham model, and can better describe the dry friction of the metal rubber, but the model only considers the linear rigidity and completely ignores the nonlinear characteristic of the metal rubber; zhou Yanguo and the like perform research on the mechanical properties of the metal rubber by adopting a trace model, wherein the trace model is based on an average and equivalent principle, and the restoring force is related to the amplitude and the frequency, but the constitutive relation of complex restoring force such as similar viscoelasticity and the like contained in the metal rubber is ignored in the model; wang Zhidong the metal rubber is researched by using a Bouc-Wen model, which is a more classical model for describing hysteresis curves, and consists of two smooth curves, so that the precision is higher, but the expression forms of elastic force and damping force in the system are undefined, the parameters are more, and the parameter identification is not facilitated. Yang Shaopu et al propose to simulate a hysteresis nonlinear system with the 3 rd power of displacement and velocity, the model has the advantages of simple mathematical expression, few undetermined parameters, no sign function, and suitability for the study of the mechanism and characteristics of hysteresis nonlinearity of the system, but not for materials containing viscoelastic characteristics. Wang Yajie the macroscopic property of the metal rubber is reflected by establishing a hardening fold constitutive relation of the metal rubber, and the Shokun describes the variation rule of the damping coefficient along with the amplitude and the frequency in a function form of combining a sine function and an e-exponential function, but the constitutive relation proposed by the Shokun does not consider complex restoring forces such as viscoelasticity and the like. Different metal rubber models have certain application ranges. However, in order to obtain a high-precision model, enough parameters must be set, which increases the complexity of dynamic performance analysis; in addition, most of the models contain a sign function, and the sectional research is inevitably carried out when the dynamic performance of the system is analyzed, so that the complete and continuous dynamic response characteristic is difficult to obtain. There is therefore a need to develop in depth a kinetic model that is accurate, simple and easily identifiable in terms of parameters.
Disclosure of Invention
The invention aims to provide a dynamic model and a parameter identification method of a metal rubber vibration isolation system, which are used for providing important references when manufacturing and designing parameters of the vibration isolation system containing metal rubber.
The invention is realized in the following way: a dynamic model of a metal rubber vibration isolation system, which is shown in a formula (1):
wherein x (t) is the deformation of the metal rubber spring, k' 1 Constant part stiffness coefficient, k 'for linear elastic restoring force' 3 Is the rigidity coefficient of the third nonlinear elastic restoring force, c' 1 For the first order viscous damping coefficient, h 'is the fractional order viscoelastic damping coefficient, p is the fractional derivative order, m is the system mass, F' is the sinusoidal excitation amplitude, ω is the excitation angular frequency.
Expressing the variable part of the elastic restoring force of the metal rubber, the memorized damping force and the complex damping force by fractional differential terms, thereby obtaining the constitutive relation model of the metal rubber shown in the formula (2):
and obtaining the dynamic model of the metal rubber vibration isolation system according to the constitutive relation model.
The parameter identification method of the dynamic model of the metal rubber vibration isolation system comprises the following steps:
a. the following variable substitution is performed on the system:
b. let the solution of the kinetic equation be
Wherein x is m For vibration amplitude +.>
Is the phase angle; will be
Substituting fractional order differential terms, and finally simplifying the fractional order differential terms through pull-type transformation to obtain an equation shown in a formula (4):
c. parameter identification is carried out on the equation (4), and the linear stiffness coefficient k is determined according to the actual working condition 1 Coefficient of tertiary stiffness k 3 Damping coefficient c 1 The fractional order damping coefficient h and the fractional order p are respectively in a function relation with vibration amplitude and frequency.
Further, in the step b, the simplification process is as follows: let f (t) =h [ D ] be the fractional order differential term p x(t)]Will be
Substituting fractional order differential term and making pull-type conversion Lf (t)]＝L{h[D p x(t)]}, can obtain
Carrying out pull-type inverse transformation on the (5)
Is available in the form of
A method of calculating the inverse Law transform by adopting the remainder theorem,
the singular point of the matrix is S = + -i omega, which is obtained by the remainder theorem
The formula is given by
Is carried into (7) for simplification
Is available in the same way
Substitution of formula (9) and formula (10) into formula (6) can be obtained
Substituting the formula (11) into the formula (3) to obtain the formula (4).
The metal rubber constitutive relation is extremely complex, the prediction of the nonlinear constitutive relation and the response calculation of a vibration system formed by the metal rubber constitutive relation are quite difficult, and the metal rubber constitutive relation is a recognized difficult problem in the field of vibration engineering to be solved urgently. The main task of dynamic modeling is to describe and describe the dynamic characteristics of the metal rubber vibration isolator by using a mathematical model according to experimental study and theoretical analysis of elastic deformation and damping energy consumption of the metal rubber, and whether the established model is reasonable or not is to see whether the model can reflect the typical nonlinear characteristics of the metal rubber vibration isolator.
The viscoelastic fractional differential term in the model of the invention includes both a variable portion of the elastic restoring force and a complex damping term portion. The model has a simple structure, and can obtain a continuous non-segmented model. More importantly, fractional order terms contain memory properties, which is a very desirable way to describe materials with memory properties.
The model has definite physical meaning, can describe the constitutive property of the metal rubber more accurately, and greatly reduces parameters. Experiments show that the model can be fitted with a hysteresis curve very close to an experimental result, so that the hysteresis curve can be accurately described, and the model has important reference value when parameters of the rubber vibration isolator containing the metal rubber are manufactured and designed.
Drawings
Figure 1 is a schematic diagram of a dynamic model of the metal rubber vibration isolation system of the present invention.
Fig. 2 is a structural view of a metal rubber module in the embodiment.
Fig. 3 is a schematic structural view of a metal rubber vibration isolator in an embodiment.
Fig. 4 is a graph comparing experimental curves and model fitting curves.
Fig. 5 is a graph of stiffness coefficient as a function of amplitude.
FIG. 6 is a graph of damping coefficient fit versus calculated.
FIG. 7 is a graph of the fit values versus calculated values for fractional order and coefficients.
Fig. 8 is a graph comparing experimental curves, fitted curves and calculated curves.
Detailed Description
The present invention is described in further detail below with reference to the drawings, and one skilled in the art can practice the present invention from the disclosure herein.
1. The dynamic model of the metal rubber vibration isolation system is shown as (1):
wherein x (t) is the deformation of the metal rubber spring, k' 1 Constant part stiffness coefficient, k 'for linear elastic restoring force' 3 Is the rigidity coefficient of the third nonlinear elastic restoring force, c' 1 For the first order viscous damping coefficient, h 'is the fractional order viscoelastic damping coefficient, p is the fractional derivative order, m is the system mass, F' is the sinusoidal excitation amplitude, ω is the excitation angular frequency.
The dynamic model of the metal rubber vibration isolation system is shown in figure 1. The above model is modeled as: ignoring the higher nonlinear elastic restoring force and the higher nonlinear damping term more than three times, and expressing the variable part of the elastic restoring force of the metal rubber, the memory damping force and the complex damping force by fractional differential terms, thereby obtaining a constitutive relation model of the metal rubber as shown in the formula (2):
and obtaining the dynamic model of the metal rubber vibration isolation system according to the constitutive relation model. The viscoelastic fractional differential term in the model contains both the variable part of the elastic restoring force and the complex damping term part. In the formula, the memory link is replaced by a fractional differential term, the physical meaning of the obtained model is clear, the constitutive property of the metal rubber can be accurately described, and the parameters are greatly reduced.
2. The following variable substitution is performed on the system:
3. dynamics of settingSolution of equation
Wherein x is m For vibration amplitude +.>
Is the phase angle; will be
Substituting fractional order differential terms, and finally simplifying the fractional order differential terms through pull-type transformation to obtain an equation shown in a formula (4):
the specific simplification process is as follows: let f (t) =h [ D ] be the fractional order differential term p x(t)]Will be
Substituting fractional order differential term and making pull-type conversion Lf (t)]＝L{h[D p x(t)]}, can obtain
Carrying out pull-type inverse transformation on the (5)
Is available in the form of
A method of calculating the inverse Law transform by adopting the remainder theorem,
the singular point of the matrix is S = + -i omega, which is obtained by the remainder theorem
The formula is given by
Is carried into (7) for simplification
Is available in the same way
Substitution of formula (9) and formula (10) into formula (6) can be obtained
Substituting the formula (11) into the formula (3) to obtain the formula (4).
4. Parameter identification for the equation (4)
1. Mechanical property experiment of metal rubber vibration isolator
Each metal rubber spring module adopts the density of 1.6x10 3 Kg/mm 3 The wire diameter is 0.325mm, the height is 32mm, the metal rubber module is shown in figure 2.
The test piece adopts a typical metal rubber vibration isolator structure with symmetrical pull-press, 4 pieces of metal rubber are respectively matched up and down, the structure is shown in figure 3, wherein 1 refers to a metal rubber block.
During initial assembly, the upper module and the lower module are all pre-pressed to ensure that the metal modules are always contacted when the central pull rod moves up and down, four pieces of metal rubber are regarded as a whole for research, and the pre-pressing stress is 1.96KN. And carrying out sinusoidal loading experiments on the metal rubber test piece by adopting a fatigue testing machine.
The central pull rod is driven by the chuck of the tester to perform sinusoidal displacement motion near the static balance position. Experiments were performed at 5 different amplitudes (0.8 mm,1.5mm,2mm,2.5mm,3 mm), wherein 0.8mm and 1.5mm selected 5 different frequencies (1 Hz,2Hz,3Hz,4Hz,5 Hz), 2mm selected 4 different frequencies (1 Hz,2Hz,3Hz,4 Hz), 2.5mm and 3mm selected 3 different frequencies (1 Hz,2Hz,3 Hz), and the test pieces were tested for the corresponding displacement and force relationships at the different frequencies and amplitudes.
2. Fitting identification of model parameters
The model is subjected to parameter identification by utilizing a MATLAB genetic algorithm toolbox, and after curve fitting is performed on 15 working conditions, the model parameter identification result is shown in table 1.
Table 1: parameter identification result
The experimental curves were compared with the fitted curves as shown in FIG. 4 (a) and FIG. 4 (b), where (a) is (amplitude 0.8mm,1.5mm,3mm, frequency 1 Hz) and (b) is (amplitude 0.8mm,1.5mm,3mm, frequency 2 Hz).
The figure shows that the fitting curve and the experimental curve have high fitting degree, and the parameter identification result is ideal, so that the model provided by the invention can well describe the hysteresis nonlinear characteristic of the metal rubber shock absorber.
3. Identification of parameters as a function of excitation amplitude and frequency
According to the identification results of the test piece parameters under different working conditions in table 1, the change trend of the parameters along with the frequency and the amplitude is analyzed, and the functional relation expression of each parameter, the excitation amplitude and the frequency is determined.
3.1 stiffness coefficient k 1 (x m ) And k 3 (x m ) Functional relation identification of (2)
Coefficient of linear stiffness k 1 Coefficient of tertiary stiffness k 3 The curves with amplitude are shown in FIG. 5 (a) and FIG. 5 (b)Shown.
As can be seen from fig. 5 (a), as the amplitude increases, the linear stiffness coefficient gradually decreases, and the trend of change is independent of frequency. Thus fitting its curve with amplitude using an n th order polynomial of amplitude, i.e
Wherein n=3, a i Is the parameter to be identified.
As can be seen from fig. 5 (b), the amplitude is in the range of 0.8mm to 1.5mm, and the tertiary rigidity coefficient is rapidly reduced; as the amplitude continues to increase, the tertiary stiffness coefficient gradually stabilizes. Fitting its curve with amplitude by a power function of amplitude, i.e
Wherein b is 0 、b 1 Is the coefficient to be identified.
The linear rigidity coefficient and the cubic rigidity coefficient function calculation expression can be obtained by least square fitting
3.2 damping coefficient c 1 (x m Functional relation identification of f)
Damping coefficient c 1 The variation with amplitude and frequency is complex, and the damping coefficient is described by a function form of a polynomial combined with a power function, i.e
Wherein n=13, d i And e is a parameter to be identified.
And (3) identifying the formula (16) by adopting a least square method to obtain a function calculation expression as shown in (17).
Calculated value c of damping coefficient 1 (x m F) and fitting value c 1 Comparison is made as shown in fig. 6. From the graph, the expression can accurately calculate the damping coefficient.
3.3, fractional order coefficient h (x m Functional relation identification of f)
Fractional order coefficients are more complex as amplitude and frequency vary. Through a number of computational analyses, the fractional order coefficient is related to the amplitude, frequency and fractional order, so that a combination of a power function and a polynomial is used to describe its functional relationship with the amplitude, frequency and order, as shown in (18).
Wherein n=13, j i And r is a parameter to be identified.
Fitting (18) polynomial coefficients j by least squares according to the fitting value p of fractional order i And power exponent r can be obtained
3.4, fractional order p (x m Functional relation identification of f)
The fractional order coefficient and the functional relation of the order, the amplitude and the frequency are found by analyzing the dynamics equation.
Let x (t) =x m cos (ωt+θ), equation (3) is
When ωt=pi/2, the equation becomes
Solving the (3) by adopting an average method to obtain
In the middle of
c(p)＝c 1 +hω p-1 sin(pπ/2) (22)
k(p)＝k 1 +hω p cos(pπ/2) (23)
Therefore, the equation (20 b) is substituted with the equations (21), (22), and (23), and the equation contains only the unknown number p, whereby the p value can be calculated. H can be calculated by substituting the calculated value of p into the expression (19). Comparison of fractional order calculated values p (x m F) and fitting value p, fractional coefficient calculation value h (x m The fitting values h and f) are shown in fig. 7 (a) and 7 (b), respectively.
As can be seen from fig. 7 (a) and 7 (b), the calculated value p (x) m F) and the fitting value p, the calculated value h (x m F) is very close to the fitting value h, so the fractional order and fractional order coefficients can be calculated using the equation (20 b) and equation (19).
The restoring force hysteresis curves under different amplitudes and frequencies can be calculated by substituting the functional expression of each parameter into the expression (3). The experimental curve, the fitted curve and the calculated curve were compared as shown in fig. 8 (a) and 8 (b), wherein (a) is (amplitude 1.5mm, frequency 1 Hz) and (b) is (amplitude 2mm, frequency 3 Hz).
As can be seen from fig. 8, the mathematical model of the recovery force of the vibration isolator with fractional order provided by the invention can well describe the change rule of the recovery force along with the amplitude, the frequency and the order, and meets the practical requirements of engineering application.
Claims (3)
1. The dynamic model of the metal rubber vibration isolation system is characterized in that a variable part of elastic restoring force of the metal rubber, a memorized damping force and a complex damping force are expressed by fractional differential terms, so that a constitutive relation model of the metal rubber shown in the formula (1) is obtained:
obtaining the dynamic model of the metal rubber vibration isolation system according to the constitutive relation model, wherein the model is shown in the formula (2):
wherein x (t) is the deformation of the metal rubber spring, k 1 ' constant part stiffness coefficient, k, for linear elastic restoring force 3 ' is the rigidity coefficient of the tertiary nonlinear elastic restoring force, c 1 ' is a first order viscous damping coefficient, h ' is a fractional order viscoelastic damping coefficient, p is a fractional derivative order, m is system mass, F ' is sinusoidal excitation amplitude, and ω is excitation angular frequency.
2. A method for identifying parameters of a dynamic model of a metal rubber vibration isolation system according to claim 1, comprising the steps of:
a. the following variable substitution is performed on the system:
b. let the solution of the kinetic equation be
Wherein x is m For vibration amplitude +.>
Is the phase angle; will be
Substituting fractional order differential terms, and finally simplifying the fractional order differential terms through pull-type transformation to obtain an equation shown in a formula (4):
c. parameter identification is carried out on the equation (4), and the linear stiffness coefficient k is determined according to the actual working condition 1 Coefficient of tertiary stiffness k 3 Damping coefficient c 1 The fractional order damping coefficient h and the fractional order p are respectively in a function relation with vibration amplitude and frequency.
3. The method for identifying parameters of a dynamic model of a metal rubber vibration isolation system according to claim 2, wherein in the step b, the simplification process is as follows: let f (t) =h [ D ] be the fractional order differential term p x(t)]Will be
Substituting fractional order differential term and making pull-type conversion Lf (t)]＝L{h[D p x(t)]}, can obtain
Carrying out pull-type inverse transformation on the (5)
Is available in the form of
A method of calculating the inverse Law transform by adopting the remainder theorem,
the singular point of the matrix is S = + -i omega, which is obtained by the remainder theorem
Will be given by equation i p ＝(e iπ/2 ) p ＝e ipπ/2 (8) Is carried into (7) for simplification
Is available in the same way
Substitution of formula (9) and formula (10) into formula (6) can be obtained
Substituting the formula (11) into the formula (3) to obtain the formula (4).
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