CN107562691A - A kind of micro thrust dynamic testing method based on least square method - Google Patents
A kind of micro thrust dynamic testing method based on least square method Download PDFInfo
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Abstract
The invention discloses a kind of dynamic micro -force measurement method based on least square method, step is：First using functional thought as the whole concept for solving dynamic force；It is then assumed that swing arm generates corresponding displacement x (t) in the presence of power f (t), and f (t) causes governing equation to have analytic solutions；The superposability and continuous function discretization of governing equation are recycled, ifThis is dynamic force model to be asked, and is substituted into governing equation and tried to achieveUsing variance ratio compared with Theory Solution and the similarity degree of experimental data, when x (t) causes variance minimum, f (t) is as the power for producing this experimental data；Finally according to judgment criterion, with the undetermined coefficient f of least square method theory of solving modelτ, finally derive on dynamic force fτSystem of linear equations；And by this system of linear equations sequencing, making dynamic force automatic calculation, this method is rocked the similar system of system governing equation to this and is applicable to all, is not limited to seek power, is less than 2.5% to the system dynamic force measurement error.
Description
Technical field
The present invention relates to micro-ox level dynamic force measurement technical field, specially a kind of micro thrust based on least square method moves
State method of testing.
Background technology
At present, dynamic measurement of the multiple fields to small power proposes demand, such as microsatellite accurately controls neck
Domain, gravity field measurement, biological tissue's Force measurement etc., all the accurate measure of small power or momentum is proposed very high
It is required that.In face of this problem, researcher both domestic and external has carried out substantial amounts of research to the measuring method of small power, such as 2002
The micro-ox level torsional pendulum type dynamometric system of Andre.J.Jamison and E.P.Muntz [1] team research and development of University of Southern California of the U.S.,
The μ N of dynamometry scope 80nN~1；The milli ox level of the H.Koizumi and K.Komurasaki team research and development of Tokyo University in 2004 is rocked
Formula dynamometric system, the μ N~200mN of dynamometry scope 5；A kind of cantilevered dynamometry of French research team's design studies in 2007
The measuring precision is 0.1 μ N；The torsion of the S.Roy and J.Soni team design of university of Florida of U.S. Experiment of Applied Physics room
Pendulum-type dynamometric system, its minimum measured value have reached 10nN.Domestic aspect, the momentum indirect method of BJ University of Aeronautics ＆ Astronautics obtain
The thrust of milli ox；The stand of the three-line pendulum principle of the Ning Zhongxi and Fan Jin Rui research teams research of Harbin Institute of Technology can be surveyed
Obtain power of milli ox magnitude etc..But, above work is more tests for steady state force.
Demand in terms of dynamic force measurement, researcher have carried out basic dynamics research to dynamometric system.Such as
The Keiji Hagiwara team research and development of Japan pass through PID control active dynameter, the S.Ciaralli team in Washington
Displacement transducer linear response section is studied, BrianC.D team is for specific momentum problem to torsional pendulum type dynamometry
System has carried out systematic research, obtains a kind of by way of Numerical Value Derivative, to obtain Impulse details method.Separately there is it
His some specific dynamic analyses etc..These methods add the complexity of system or without versatilities.
Above-mentioned studied torsional pendulum type dynamometric system includes two large divisions：
1) mechanical system part；This part is divided into swing arm system, calibration system, damping system and displacement measurement system again.
Swing arm system is that thrust is converted into the device of displacement, balances the gravity of thruster so as to overcome micro-thruster thrust-weight ratio small
Problem, it is to realize the key that microthrust accurately determines.The principle that power is produced in magnetic field by electrified wire realizes Accurate Calibration,
The μ N of calibration range 1~10000, precision are better than 0.1 μ N.The effect of damping system is to make the swing arm of motion static as early as possible, during saving
Between.Displacement measurement system, the displacement transducer using plane-parallel capacitor as principle is selected, measurement range is 100 μm, resolution ratio
1nm；
2) control section；Including corresponding software and hardware, mainly the equipment such as sensor are controlled to collect data simultaneously.The dynamometry
The major parameter of system is：The μ N of dynamometry scope 5~3000, the μ N of precision 1.In vacuum test, typical case's output of displacement transducer is made an uproar
Sound amplitude is 0.01~0.05V, and typical voltage-power calibration coefficient is 202.15 μ N/V.
With the trend toward miniaturization of space satellite, kept in microsatellite orbit determination, posture regulation and control, position etc. is to satellite
The requirement more and more higher of Push Technology, scientific research personnel are this research and design new thruster of milli ox level and micro-ox level.For solution
The measurement problem of the certainly thrust of micro-thruster, particularly dynamic thrust, the ox dynamometric system that declines that rocks developed is carried out
Dynamics research.By carrying out mathematical analysis to the governing equation of dynamometric system, research one kind is asked by known displacement data
The method that solution produces the dynamic force of the displacement.This method has versatility, to meeting that the Second-order Damped of similar governing equation vibrates
System is applicable, and on the premise of extras are not increased, is had to solve the problems, such as that the measure of current dynamic force provides one
Effect approach.
The content of the invention
In order to overcome the shortcomings of prior art, the technical solution adopted for the present invention to solve the technical problems is：One
Kind carries out mathematical analysis based on governing equation, derives that production is calculated by known displacement data in one kind with reference to least square method
The expression formula of the thrust of the raw displacement data, and by the solution of programming realization dynamic force, comprise the following steps：
S100, the whole concept first using functional thought as solution dynamic force；
S200, after the displacement data of one group of torsional pendulum type swing arm is obtained, it is then assumed that swing arm produces in the presence of power f (t)
Corresponding displacement x (t), and f (t) causes governing equation to have analytic solutions；
S300, because governing equation be linear equation, have analytic solutions using its stackable principle and to Impulse, and
Cause with reference to the method for continuous function discrete approximation：
Wherein δ (t- τ) is the unit impulse effect at τ moment, the momentum size at f is weight i.e. τ moment；
Each δ (t- τ) brings governing equation into and obtains a corresponding x (τ), then obtains：
This x (t) is the approximation theory solution for the power assumed, when two adjacent momentum spacing go to zero, x (t) now
As true Theory Solution；And different f (t) can obtain different x (t)；
The displacement data that S400, the x (t) by Theory Solution obtain with experimental data under identical sample rate with experiment is being counted
It is compared on calculation machine, chooses and cause both immediate f (t) to be as the thrust function for producing experimental data, Rule of judgment
Seek minimum variance between Theory Solution and experimental data i.e.：
Now the Theory Solution model of fitting function has obtained in theory, only poor undetermined coefficient, according to Rule of judgment, profit
Undetermined coefficient is solved with least square method, finally gives an expression formula
The power that f in this expression formula is both undetermined coefficient and we need；And solve equation group and obtain to be asked move
State thrust, dynamic thrust can be obtained by bringing the displacement data that measurement obtains into program.
Preferably, the functional basic thought described in the step S100 is that to be tried to achieve by displacement be the effect of which kind of force function
Displacement function data are produced on platform system, first assume that a force function f (t), the force function f (t) allow the equation of motion to have solution
Analysis solution, different power f (t) answer different displacement x (t), are judged by judging initial data with calculating the variance of solution data
Both similarity, when registration between the two preferably when, f (t) now is used as real thrust.
Preferably, in the step S300, according to the superposability of governing equation, the discrete approximation of continuous function, and
Governing equation has analytic solutions to obtain any Displacement Sequence to Impulse can be expressed as the time shifting of unit impulse wave train
Weighted sum, its expression formula are：
Wherein, τ represents time discrete interval, and its inverse is the sample rate of power；Discrete " rectangle effect " is considered as quasi- punching
Amount is acted on, and each rectangle action effect can be obtained with the difference of two adjacent big step actions of grade.
In the step S400：Theory is judged according to variance, the undetermined coefficient with least square method theory of solving model is
The power solved, while minimum variance is obtained, the expression formula of derived for solving dynamic force simultaneously uses computer programing；
The original definition of least square method and the extreme value solving method of the function of many variables：
The function of many variables ask local derviation to obtain：
Obtained after arrangement:
The formula is Ax=y form；Wherein A is N × N square formation, and x is to wait to seek vector, and y is known column vector；
When the time, discrete interval was intended to zero, the discrete front and rear action effect of continuous function is of equal value, because power is continuous
Effect can be used to functional value corresponding to above discrete point as height by the effect of two neighboring discrete point after discrete, using discrete interval as
Each rectangle action effect is obtained with the difference of the action effect of two adjacent step forces for " the rectangle effect " at bottom, and step masterpiece
Governing equation under has analytic solutions, and solution of the unit step power under the effect of 0 moment is：
Wherein ζ is damped coefficient, ωnIt is intrinsic frequency, ωdIt is the frequency under damping action；
Understand that the effect expression formula of any step force is similar, and the step force acted at different moments is only suitable according to formula (9)
In the difference of initial phase；
The effect of " square wave " represents as follows：
X (τ)=x (t)-x (t- τ) (10)
Bring x (τ) into formulas (5), obtain arbitrary displacement expression formula：
Wherein, x (t) is displacement, XτIt is constant, with fτVariation coefficient be present；
Formula (11) is applied to least square method, obtained：
Formula (12) is a system of linear equations for having unique solution, and solution formula (4) obtains fkDynamic thrust as to be asked.
Preferably, the least square method in the step 400 is the matched curve model for being used to solve acquired f (t)
Undetermined coefficient：The step directly utilizes the base that least square method is associated according to the relational expression for judging Theory Solution and experimental data
This definition, so carry out solution undetermined coefficient using least square method.
Compared with prior art, the beneficial effects of the invention are as follows：The present invention uses functional general principle, with reference to least square
Method, by deriving, solving the system of linear equations relevant with dynamic force to be asked, dynamic force is obtained, this method is simple to operate, without adding
Add miscellaneous equipment, dynamic thrust can be obtained by bringing the displacement data that measurement obtains into program, and this method is to the system dynamometry model
The data enclosed in (10~3000 μ N) are applicable, and dynamic force measurement error is less than 2.5%.
Brief description of the drawings
Fig. 1 is the method flow schematic diagram of the present invention；
Fig. 2 is continuous function discretization schematic diagram in the specific embodiment of the invention；
Fig. 3 is experimental data and result of calculation comparison diagram in the specific embodiment of the invention；
Fig. 4 is amplitude-versus-frequency curve figure in the specific embodiment of the invention；
Fig. 5 is that noise intensity is followed successively by 1%, 5%, 10% from top to bottom in the specific embodiment of the invention, 15% calculating
Result schematic diagram；
Fig. 6 is that the sample rate of displacement data in the specific embodiment of the invention is different, and the sample rate of power is under the same conditions
Schematic diagram of calculation result；
Fig. 7 is the residual plot of result of calculation when noise is 1% in the specific embodiment of the invention.
Embodiment
Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete
Site preparation describes, it is clear that described embodiment is only part of the embodiment of the present invention, rather than whole embodiments.It is based on
Embodiment in the present invention, those of ordinary skill in the art are obtained every other under the premise of creative work is not made
Embodiment, belong to the scope of protection of the invention.
Embodiment：
It is as shown in figure 1, provided by the invention a kind of based on governing equation progress mathematical analysis combination least square method derivation
Go out a kind of expression formula that the thrust for producing the displacement data is calculated by known displacement data, and programmed in fact by matlab
The solution of existing dynamic force comprises the following steps：
Step S100, obtain solving the whole concept of dynamic force by functional basic thought；Described functional basic thought
It is that to be tried to achieve by displacement be which kind of force function acts on platform system generation displacement function data, first assumes a force function f
(t), the force function f (t) allows the equation of motion to have analytic solutions, and different power f (t) answers different displacement x (t), original by judging
The variance that data solve data with calculating judges both similarities, when registration between the two preferably when, f now
(t) it is used as real thrust；
Step S200, after the displacement data of one group of torsional pendulum type swing arm is obtained, it is then assumed that the displacement data is in power f
(t) moving displacement of swing arm in the presence of, and f (t) meets that governing equation has the condition of analytic solutions；
Step S300, because governing equation is linear equation, using its stackable principle and there is parsing to Impulse
Solution, and the method for combining continuous function discrete approximation causes：
Wherein δ (t- τ) is the unit impulse effect at τ moment, the momentum size at f is weight i.e. τ moment；
Each δ (t- τ) brings governing equation into and obtains a corresponding x (τ), then obtains：
This x (t) is the approximation theory solution for the power assumed, when two adjacent momentum spacing go to zero, x (t) now
As true Theory Solution；And different f (t) can obtain different x (t)；
Step S400, the displacement data obtained by the x (t) of Theory Solution with experimental data under identical sample rate with experiment
It is compared on computers, chooses and cause both immediate f (t) as the thrust function for producing experimental data, to judge bar
Part is to seek minimum variance between Theory Solution and experimental data i.e.：
Now the Theory Solution model of fitting function has obtained in theory, only poor undetermined coefficient, according to Rule of judgment, profit
Undetermined coefficient is solved with least square method, the step is directly utilized and associated according to the relational expression for judging Theory Solution and experimental data
The basic definition of least square method, so carrying out solution undetermined coefficient using least square method, finally give an expression formula
The power that f in this expression formula is both undetermined coefficient and we need；And solve equation group and obtain to be asked move
State thrust, dynamic thrust can be obtained by bringing the displacement data that measurement obtains into program.
In above-mentioned steps S300 and S400, according to the superposability of governing equation, the discrete approximation of continuous function, and control
Equation processed have to Impulse analytic solutions obtain any Displacement Sequence can be expressed as unit impulse wave train time shifting plus
Quan He, its expression formula are：
Wherein, τ represents time discrete interval, and its inverse is the sample rate of power；Discrete " rectangle effect " is considered as quasi- punching
Amount is acted on, and each rectangle action effect can be obtained with the difference of two adjacent big step actions of grade.
Theory is judged according to variance, is solved power with the undetermined coefficient of least square method theory of solving model, simultaneously
Minimum variance is obtained, the expression formula of derived for solving dynamic force simultaneously passes through sequencing；
The original definition of least square method and the extreme value solving method of the function of many variables：
The function of many variables ask local derviation to obtain：
Obtained after arrangement:
The formula is Ax=y form；Wherein A is N × N square formation, and x is to wait to seek vector, and y is known column vector；
When the time, discrete interval was intended to zero, the discrete front and rear action effect of continuous function is of equal value, because power is continuous
Effect can be used to functional value corresponding to above discrete point as height by the effect of two neighboring discrete point after discrete, using discrete interval as
Each rectangle action effect is obtained with the difference of the action effect of two adjacent step forces for " the rectangle effect " at bottom, and step masterpiece
Governing equation under has analytic solutions, and solution of the unit step power under the effect of 0 moment is：
Wherein ζ is damped coefficient, ωnIt is intrinsic frequency, ωdIt is the frequency under damping action；
Understand that the effect expression formula of any step force is similar, and the step force acted at different moments is only suitable according to formula (9)
In the difference of initial phase；
The effect of " square wave " represents as follows：
X (τ)=x (t)-x (t- τ) (10)
Bring x (τ) into formulas (5), obtain arbitrary displacement expression formula：
Wherein, x (t) is displacement, XτIt is constant, with fτVariation coefficient be present；
Formula (11) is applied to least square method, obtained：
Formula (12) is a system of linear equations for having unique solution, and solution formula (4) obtains fkDynamic thrust as to be asked.
It should be noted that the present invention is to be based on rocking the ox dynamometric system progress dynamic analysis that declines, the ox that declines is rocked
Dynamometric system is a kind of Second-order Damped vibrational system, and its governing equation is：
Wherein I is swing arm rotary inertia, and C is damped coefficient, and K is the elastic coefficient, and x is displacement (pivot angle), and f is to wait to ask
Dynamic force, L is torque arm length；
This equation is the basis of all dynamic force measurement methods, and our thinking is that the displacement data that will have been obtained is brought into
Governing equation, the dynamic force for producing the displacement is obtained from displacement data backwards calculation.Therefore, it is necessary to demarcate controlling party first
All parameters of journey.
The present invention has carried out checking test to verify the dynamic force measurement method：By means of rocking the ox dynamometric system that declines
Electromagnetism calibration system, using signal generator provide known waveform (sinusoidal, square wave and triangular wave), frequency (0.1~10Hz) and
The electric current of amplitude, produce the active force of known waveform, frequency and size.By the known dynamic force and the above method that contrast loading
Result of calculation, verify applicable frequency range and the dynamometry precision of this method；
Fig. 3 is that the experiment of three kinds of waveform power under 0.1Hz and result of calculation contrast, as seen from Figure 3, the dynamic being calculated
Power almost overlaps with the power applied, illustrates the accuracy of above-mentioned dynamic force measurement method.
Table 1 is under sinusoidal waveform, different voltages, and the amplitude of loading force and the amplitude contrast that power is calculated, both are most
Large deviation is 1.5%.Corresponding to the measurement accuracy of the force measuring method.
The thrust of the experiment loading of table 1, sinusoidal waveform
| Driving voltage V | Loading thrust is shaken μ | Calculate thrust amplitude |
| 2 | 120 | 121.6 |
| 3 | 180 | 179.9 |
| 4 | 240 | 238.6 |
| 5 | 300 | 295.4 |
Second-order Damped vibrational system corresponding to formula (13), there is intrinsic amplitude-frequency response characteristic, and this rocks the ox dynamometry that declines
The amplitude-versus-frequency curve of system is as shown in Figure 4；Abscissa is relative frequency (frequency/intrinsic frequency) in figure, and ordinate amplitude is put
Big rate；It can be seen that when external force frequency is much smaller than stand intrinsic frequency, Amplitude amplification rate is 1, is corresponded toWhen external force frequency
When rate is equal to intrinsic frequency, the stand resonance for the ox dynamometric system that declines is rocked, magnifying power is about 5.0；When external force frequency continues to increase
Add, displacement equations rate strongly reduces
As seen from Figure 4, when relative frequency is more than 5 (corresponding about 7Hz), Amplitude amplification rate is less than 0.05；Now, effective stand
Motion amplitude is suitable with noise amplitude, and dynamic force can be caused to solve error and dramatically increased.
In the solution of dynamic force, the noise in the sample rate and data of displacement data is that force measurements have an impact, right
It carries out simulated experiment, and basic process is：(such as sine wave) is loaded using the dynamic force of hypothesis, the position of stand is calculated
Move response curve；Different degrees of white noise is loaded for this curve, or different sample rates is separated into it；This place
Data conduct after reason " experimental data " brings program solution into, obtains dynamic thrust and is contrasted with initial hypothesis power, assesses sampling
The influence of frequency and noise level for the dynamic force measurement method.
Fig. 5 and Fig. 6 is that the measurement of different noise levels and different sample rates for same sinusoidal load power influences respectively；
Wherein, the displacement data in simulated test corresponds to 200uN amplitudes, sinusoidal frequency 0.1Hz, and sample rate is 100~1000 points/second,
Noise level 1%~15% (the ratio between noise amplitude and former sinusoidal amplitude)；As seen from Figure 5, under same frequency, noise is bigger, meter
The signal to noise ratio for calculating result is poorer；But after being carried out smoothly to force measurements, obtained amplitude is 196.4 μ N, and error only has
1.8%；Illustrate certain noise level, excessive influence will not be produced on low frequency dynamic force, but necessarily upper frequency is influenceed compared with
Greatly.
Fig. 6 is displacement data sample rate when being respectively 100,200,500,1000, the meter under the sample rate of identical power
Calculate result.It can be seen that during same noise level, on the premise of the sample rate of power is constant, displacement data sample rate is higher to be obtained
The noise of thrust is smaller.
According to formula (12) understand, multiple spot sum when can produce average effect because when displacement sample rate it is higher, in unit
The average value of noise more levels off to zero in time, and displacement data to be calculated is closer to " real displacement data ", so dynamometry knot
Fruit can improve.Therefore, displacement data sample rate is properly increased, dynamic force measurement precision can be improved.
Fig. 7 is the dynamometry residual plot under 1% noise intensity, carries out residual analysis to the noise of result of calculation, finds power
Noise is more uniformly distributed tends to 0 compared with the both sides of transverse axis, residual error average value；The power for illustrating to be calculated can be analyzed to sine
Oscillating component and white noise part sum, the governing equation of this and system is that linear equation has substantial connection, is counted in the future for us
Calculation filters out high-frequency noise, retains real part and noise is smoothly removed by multiple spot and provide experimental basis.
In summary, by carrying out mathematical analysis to torsional pendulum type uN dynamometric systems, on the premise of extras are not increased,
Based on least square method, the thrust of the displacement is produced by measuring displacement data backwards calculation；This dynamic force measurement error is 2%
Within, for the dynamometric system in text, highest response frequency 7Hz can be achieved.Being tested by experimental verification proves the essence of this method
Degree, analyzes to obtain the influence of noise and sample rate for dynamometry precision by simulated experiment.As a result show, for low frequency dynamic
Power, noise intensity remain to more accurate dynamometry when within 15%.In addition, it can be carried by improving the method for displacement data sample rate
High calculating resolution and dynamometry precision.
It is obvious to a person skilled in the art that the invention is not restricted to the details of above-mentioned one exemplary embodiment, Er Qie
In the case of without departing substantially from spirit or essential attributes of the invention, the present invention can be realized in other specific forms.Therefore, no matter
From the point of view of which point, embodiment all should be regarded as exemplary, and be nonrestrictive, the scope of the present invention is by appended power
Profit requires rather than described above limits, it is intended that all in the implication and scope of the equivalency of claim by falling
Change is included in the present invention.Any reference in claim should not be considered as to the involved claim of limitation.
Claims (4)
- A kind of 1. micro thrust dynamic testing method based on least square method, it is characterised in that：Comprise the following steps：S100, the whole concept first using functional thought as solution dynamic force；S200, after the displacement data of one group of torsional pendulum type swing arm is obtained, it is then assumed that swing arm generates phase in the presence of power f (t) The displacement x (t) answered, and f (t) causes governing equation to have analytic solutions；S300, because governing equation be linear equation, have analytic solutions using its stackable principle and to Impulse, and combine The method of continuous function discrete approximation causes：<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>&tau;</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>f</mi> <mi>&tau;</mi> </msub> <mo>&CenterDot;</mo> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mi>&tau;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>Wherein δ (t- τ) is the unit impulse effect at τ moment, the momentum size at f is weight i.e. τ moment；Each δ (t- τ) brings governing equation into and obtains a corresponding x (τ), then obtains：<mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>&tau;</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>X</mi> <mi>&tau;</mi> </msub> <mo>&CenterDot;</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>&tau;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>This x (t) is the approximation theory solution for the power assumed, when two adjacent momentum spacing go to zero, x (t) now is True Theory Solution；And different f (t) can obtain different x (t)；The displacement data that S400, the x (t) by Theory Solution obtain with experimental data under identical sample rate with experiment is in computer On be compared, choose the thrust function for causing both immediate f (t) as generation experimental data, Rule of judgment is to seek reason It is by the minimum variance between solution and experimental data：Now the Theory Solution model of fitting function has obtained in theory, only poor undetermined coefficient, according to Rule of judgment, using most Small square law solves undetermined coefficient, finally gives an expression formula：F in this expression formulaτBe both undetermined coefficient but we need power；And solve equation group and obtain dynamic push to be asked Power, dynamic thrust can be obtained by bringing the displacement data that measurement obtains into program.
- A kind of 2. micro thrust dynamic testing method based on least square method according to claim 1, it is characterised in that： Functional basic thought described in the step S100 is, first assumes that a force function f (t), the force function f (t) allow motion side Journey has analytic solutions, and different power f (t) answers different displacement x (t).Pass through the variance for judging initial data with calculating solution data To judge both similarities, when registration between the two preferably when, f (t) now is used as real thrust.
- A kind of 3. micro thrust dynamic testing method based on least square method according to claim 1, it is characterised in that： In the step S300, according to the superposability of governing equation, the discrete approximation of continuous function, and governing equation are made to momentum The weighted sum that any Displacement Sequence can be expressed as the time shifting of unit impulse wave train, its expression formula are obtained with there are analytic solutions For：<mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>&tau;</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>X</mi> <mi>&tau;</mi> </msub> <mo>&CenterDot;</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>&tau;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>Wherein, τ represents time discrete interval, and its inverse is the sample rate of power；Discrete " rectangle effect " is considered as into quasi- momentum to make With each rectangle action effect can be obtained with the difference of two adjacent big step actions of grade.In the step S400：Theory is judged according to variance, it is with the undetermined coefficient of least square method theory of solving model, i.e., required The power of solution, while minimum variance is obtained, the expression formula of derived for solving dynamic force and sequencing；The original definition of least square method and the extreme value solving method of the function of many variables：The function of many variables ask local derviation to obtain：Obtained after arrangement:The formula is Ax=y form；Wherein A is N × N square formation, and x is to wait to seek vector, and y is known column vector；When the time, discrete interval was intended to zero, the discrete front and rear action effect of continuous function is of equal value, because the continuous action of power Functional value corresponding to above discrete point can be used to as height by the effect of discrete rear two neighboring discrete point, using discrete interval the bottom of as " rectangle effect " each rectangle action effect is obtained with the difference of the action effect of two adjacent step forces, and under step force effect Governing equation have an analytic solutions, and unit step power acted at 0 moment under solution be：<mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>t</mi> <mo><</mo> <msup> <mn>0</mn> <mo>-</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn>1</mn> <msubsup> <mi>&omega;</mi> <mi>n</mi> <mn>2</mn> </msubsup> </mfrac> <mo>&lsqb;</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>&xi;&omega;</mi> <mi>n</mi> </msub> </mrow> </msup> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>&omega;</mi> <mi>d</mi> </msub> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mi>&zeta;</mi> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>&zeta;</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>&omega;</mi> <mi>d</mi> </msub> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&rsqb;</mo> </mrow> </mtd> <mtd> <mrow> <mi>t</mi> <mo>&GreaterEqual;</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>Wherein ζ is damped coefficient, ωnIt is intrinsic frequency, ωdIt is the frequency under damping action；Understand that the effect expression formula of any step force is similar according to formula (9), the step force acted at different moments just corresponds to just The difference of beginning phase；The effect of " square wave " represents as follows：X (τ)=x (t)-x (t- τ) (10)Bring x (τ) into formulas (5), obtain arbitrary displacement expression formula：<mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>&Sigma;</mi> <mn>1</mn> <mi>M</mi> </msubsup> <msub> <mi>X</mi> <mi>&tau;</mi> </msub> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>-</mo> <mi>x</mi> <mo>(</mo> <mrow> <mi>t</mi> <mo>-</mo> <mi>&tau;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>Wherein, x (t) is displacement, XτIt is constant, with fτVariation coefficient be present；Formula (11) is applied to least square method, obtained：Formula (12) is a system of linear equations for having unique solution, and solution formula (4) obtains fkDynamic thrust as to be asked, by this Equation sequencing can be achieved to solve.
- A kind of 4. micro thrust dynamic testing method based on least square method according to claim 3, it is characterised in that： Least square method in the step 400 is the undetermined coefficient for being used to solve the matched curve model of acquired f (t)：The step It is rapid directly to utilize the basic definition that least square method is associated according to the relational expression for judging Theory Solution and experimental data, so use Least square method carries out solution undetermined coefficient.
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