CN106695458A - Friction force modeling method for numerical control machine tool feeding system - Google Patents

Abstract

The invention discloses a friction force modeling method for a numerical control machine tool feeding system, and belongs to the technical field of precision manufacturing. The friction force modeling method comprises the following steps: S1, acquiring different rotating speeds of all shafts of a numerical control machine tool in an idling state and sizes of friction forces corresponding to all the different rotating speeds; S2, building a conventional friction force mathematical model according to the sizes of the friction forces which are acquired in the step S1; S3, acquiring critical rotating shafts of all the shafts of the numerical control machine tool under different accelerations; S4, building static friction force mathematical models of all the shafts according to critical rotating speed data acquired in the step S3; S5, integrating the conventional friction force mathematical model obtained in the step S2 and the static friction force mathematical models obtained in the step S4 so as to obtain a general friction force model. According to the friction force modeling method disclosed by the invention, the friction forces on all the shafts of the numerical control machine tool under high-speed operation can be precisely described, and the friction forces on all the shafts of the numerical control machine tool during low-speed reversing can be also accurately described.

Description

A kind of frictional force modeling method of NC machine tool feed system

Technical field

The invention belongs to Precision Manufacturing Technology field, more specifically, being to be related in a kind of Digit Control Machine Tool process each Feed shaft frictional force modeling method.

Background technology

Developed rapidly with manufacturing, processing part more and more tends to precise treatment and complication, to Digit Control Machine Tool in essence Processing request in close manufacture field is also gradually continued to develop to high accuracy direction.So-called high accuracy, profile generally referred to as high Control accuracy.The machining accuracy level of Digit Control Machine Tool directly reflects the technical merit of national equipment manufacturing.Due to Digit Control Machine Tool During part is processed there is frictional force in each feed shaft, and frictional force be influence one of Digit Control Machine Tool machining accuracy it is important Factor, therefore in order to improve the machining accuracy of Digit Control Machine Tool, the influence for reducing or eliminating frictional force is that must have option, and accurate Frictional force Mathematical Modeling be suppress even eliminate the dysgenic precondition of frictional force.

Although existing conventional friction power model can accurately describe the Stribeck phenomenons of frictional force, by It is very complicated in frictional force self character, especially frictional force low speed commutation under the conditions of nonlinearity characteristic, traditional rubs Wiping power model cannot accurately describe friction characteristics when low speed commutates.

Therefore, harmful effect of the frictional force to numerical control machine finish how is suppressed, how to set up accurately to describe Frictional model when low speed commutates, is current a great problem urgently to be resolved hurrily.

The content of the invention

For the disadvantages described above or Improvement requirement of prior art, entered based on Digit Control Machine Tool (CNC) the invention provides one kind To the frictional force modeling method of system, it is a kind of more accurate frictional force modeling method, not only can accurately describe number The suffered frictional force under high-speed cruising of each axle of control lathe, moreover it is possible to which accurate description each axle of Digit Control Machine Tool is when low speed commutates Suffered frictional force.

To achieve the above object, according to one aspect of the present invention, there is provided a kind of friction of NC machine tool feed system Force modeling method, it comprises the following steps：

S1：Each axle of acquisition Digit Control Machine Tool rotating speed different in idling conditions and the corresponding friction of each different rotating speeds Power size；

S2：The frictional force size obtained using step S1, conventional friction power mathematics is constructed in conjunction with numerical method Model；

S3：Obtain the critical speed of Digit Control Machine Tool each axle under different acceleration；

S4：The critical speed data obtained using step S3 construct the stiction Mathematical Modeling of each axle；

S5：The stiction Mathematical Modeling that the conventional friction power Mathematical Modeling that step S2 is obtained is obtained with step S4 is carried out Integrate, to obtain general Frictional model.

Further, in step S1, Digit Control Machine Tool each axle is made to be moved reciprocatingly in different velocity intervals, to obtain Corresponding frictional force size under each axle of Digit Control Machine Tool rotating speed different in idling conditions,

In order to pick out the characteristic of boundary friction and segment fluid flow friction, in interval selection rotational speed omega=± 0.1 of low speed, ± 0.2, ± 0.3, ± 0.5, ± 0.7, ± 1, ± 3 ± 5 ± 10rad/s are used as test speed, while turning in interval selection at a high speed Fast ω=± 25, ± 50, ± 75, ± 100rad/s is used as test speed.

Further, in step S2, the Mathematical Modeling such as following formula of conventional friction power is represented：

In formula, ω represents rotating speed,WithRepresent static friction torque,WithCoulomb moment of friction is represented,WithViscous friction torque is represented,WithRepresent rotating speed constant, the wherein direction of sign ± expression speed, e Represent natural logrithm.

Further, in step S3, the critical speed of Digit Control Machine Tool each axle under different acceleration is obtained, it is first determined " critical displacement x_break" size, critical displacement x_breakIt is as follows：

Wherein, F_cAnd σ_oIt is respectively the frictional force and rigidity suffered by machine spindle, after operating mode determines, F_cAnd σ_oIt is to determine constant , it is also possible to represent critical displacement x with acceleration and time_breakIt is as follows：

Wherein, t_breakIt is rotary shaft shaking off the displacement interval time (maximum shakes off displacement as critical displacement) used, T_V=0It is moment when speed is equal to 0, it is the time variable factor that a (τ) is real time acceleration τ,It is average acceleration,

Acceleration a when by speed equal to 0₀Average acceleration is equivalent to, i.e.,Under then obtaining Formula：

Wherein, v_breakIt is critical speed, by measuring the critical speed under different acceleration, obtains acceleration with critical speed The relation of degree, you can obtain the size of critical displacement,

In order to obtain the relation of acceleration and critical speed, corresponding critical speed is measured under different acceleration, obtained The relation curve of acceleration and critical speed square, and then the size of critical displacement is obtained, under the critical displacement for obtaining is substituted into In formula,

Obtain real-time critical speed under different acceleration.

Further, in step S4, the stiction mathematical modulo of each axle is constructed using the critical speed data for obtaining Type, specific as follows, stiction F (x) size is obtained by following formula：

F (x)=- kx

Wherein, k is the equivalent stiffness on surface of contacting with each other, and x is displacement, and displacement x is represented using following formula：

Wherein, v is reference velocity, is obtained by the once differentiation (difference) to reference locus, a₀When being equal to 0 for speed Acceleration,

Typically use speed variable, and moment of friction to be directly proportional to frictional force in rotary shaft, make scale factor for λ, then formulaIt is transformed to：

Wherein,r_gIt is the gearratio of ball screw, orderThen formulaIt is transformed to down Formula：

T (ω)=κ ω²

By " critical speed " v for trying to achieve_breakSubstitution formulaTry to achieve Critical friction torque T_break, then according to T (ω)=κ ω², obtain the size of parameter κ, the final mathematical modulo for obtaining stiction Type.

Further, in step S5, conventional friction power Mathematical Modeling is integrated with stiction Mathematical Modeling, to obtain General Frictional model specifically, general frictional force Mathematical Modeling is as follows,

In formula, ω represents rotating speed,WithRepresent static friction torque,WithCoulomb moment of friction is represented,WithViscous friction torque is represented,WithRotating speed constant is represented,WithRepresent critical speed, κ⁺And κ^- Stiction constant, the wherein direction of sign ± expression speed are represented, e represents natural logrithm.

In general, by the contemplated above technical scheme of the present invention compared with prior art, can obtain down and show Beneficial effect：

(1) present invention establishes a kind of general frictional force Mathematical Modeling based on each rotating speed of Digit Control Machine Tool, of the invention General Frictional model not only inhibits influence of the frictional force to numerical control machine tooling under high-speed case, has also effectively processed low speed In the case of ill-effect of the nonlinear normal modes to Digit Control Machine Tool.

(2) present invention is simple and easy to apply, can accurately, reliable influence when suppressing frictional force to numerical control machine tooling part, And possessing versatility, it is adaptable to various types of numerical control machine tools (center), the machining accuracy for improving Digit Control Machine Tool has There is good application value.

Brief description of the drawings

Fig. 1 is the frictional force Mathematical Modeling modeling procedure figure of Digit Control Machine Tool (CNC) feed system in the embodiment of the present invention；

Fig. 2 is the Stribeck-Curve curve maps of conventional friction power model in the embodiment of the present invention；

Fig. 3 is critical speed and acceleration relation schematic diagram in the embodiment of the present invention；

Fig. 4 is the Stribeck-Curve curve maps of general Frictional model proposed by the present invention；

Fig. 5 is experimental verification design sketch in the embodiment of the present invention.

Specific embodiment

In order to make the purpose , technical scheme and advantage of the present invention be clearer, it is right below in conjunction with drawings and Examples The present invention is further elaborated.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, and It is not used in the restriction present invention.As long as additionally, technical characteristic involved in invention described below each implementation method Not constituting conflict each other can just be mutually combined.

Fig. 1 is the frictional force Mathematical Modeling modeling procedure figure of Digit Control Machine Tool (CNC) feed system in the embodiment of the present invention, such as Shown in Fig. 1, the method that the present invention builds NC machine tool feed system frictional force Mathematical Modeling is comprised the following steps：

S1：Digit Control Machine Tool each axle rotating speed in the idling condition and corresponding frictional force size are obtained, specifically, such as Shown in Fig. 2, conventional friction power model is divided into four regions, as the increase of speed is followed successively by：Stiction, boundary lubrication, portion Shunting body lubricates and full fluid lubrication.Wherein, interval in partial hydrodynamic lubrication, the size of frictional force subtracts with the increase of speed This small phenomenon is referred to as Stribeck phenomenons.It is emphasized that conventional friction power model does not have substantially describe quiet rubbing Wipe the characteristic of power.

The Mathematical Modeling of conventional friction power is represented by following formula (1)：

In formula (1), ω represents rotating speed,WithRepresent static friction torque,WithExpression coulomb moment of friction ( It is dynamic friction torque),WithViscous friction torque is represented,WithRepresent rotating speed constant, wherein sign The direction of ± expression speed, e represents natural logrithm.

Moment of friction T and frictional force F_fIt is the relation of direct ratio：

T=μ * F_f(2)

In formula (2), μ is the coefficient that frictional force is converted to moment of friction, it is determined that under the conditions of its be a constant value.

Because multiplex moment of friction is represented in ball leading screw driving system, and moment of friction can equally recognize and obtain, Therefore the size of frictional force is represented with moment of friction herein.

In order to obtain the data needed for the Mathematical Modeling for building frictional force, to make Digit Control Machine Tool each axle in different speed In the range of move reciprocatingly.In order to accurately pick out the characteristic of boundary friction and segment fluid flow friction, in the interval selection of low speed Speed omega=± 0.1, ± 0.2, ± 0.3, ± 0.5, ± 0.7, ± 1, ± 3 ± 5 ± 10rad/s is used as test speed；Simultaneously Choose ω=± 25 interval at a high speed, ± 50, ± 75, ± 100rad/s as test speed, due to the lower frictional force of high speed Characteristic is relatively easy, and four groups of test speeds can meet required precision.

13 groups of speed do idle running uniform motion respectively more than respectively to allow Digit Control Machine Tool each axle, and record corresponding rubbing Wipe the size of torque.The record information ginseng of each axle is shown in Table 1.

Frictional force under the friction speed of table 1 corresponding to machine spindle

In above-mentioned table 1, T₁、T₂······、T₁₃The corresponding frictional force of 13 groups of speed is represented respectively.

S2：Conventional friction power Mathematical Modeling is constructed using data application numerical method achieved above, specifically, formula (1) there are ten parameters to be determined in, they are respectively： With

Because ω=0.1rad/s speed is very small, so static friction torque can be used in ω=± 0.1rad/s When measured moment of friction represent, i.e.,：

From formula (1), when speed omega is sufficiently large,WithLevel off to 0, and WithLevel off to constant value respectivelyWithMoment of friction now is directly proportional to speed, and ratio is stickiness Moment of friction.

So, viscous friction torque can be obtained by following formula (4)：

After viscous friction torque is obtained, Coulomb friction torque can be obtained by following formula (5)：

Next four last parameters are determinedWithIn order to determine that this four parameters need to borrow Minimum cost function in numerical method, if minimize cost function V being：

In formula (6), δ T (i) is the difference of Frictional model estimated value and actual measured value, and δ T herein are considered as one Individual entirety, δ represents i-th group of frictional force, i=1,2,313 without practical significance, T (i).

Wherein, Frictional model estimated value is represented with value of delta T (i) of actual measured value using formula (7)：

In formula (7), ω (i) is 13 groups of reciprocating speed, i=1,2,3 ..., 13,It is corresponding i-th Moment of friction obtained by the actual measurement of group,The implication of middle parameters is identical with formula (1).

Allow(Refer toWith) 10rad/s is increased to 0.01rad/s speed since 0.01rad/s,(Refer toWith) 0.01rad/s is increased to 0.01rad/s to speed since -10rad/s, soWithRespectively There are 10000 kinds of packets, every group of Ω is calculated by formula (7)_1,2Corresponding 13 groups of torques estimation and actual error value, and by formula (6) every group of Ω is calculated_1,2Corresponding cost function value, chooses Ω corresponding during cost function value V minimums_1,2Value, as rubs Wipe the parameter value of power Mathematical Modeling.

So, four last parametersWithAlso obtain.In this way, it is finally obtained tradition Frictional force Mathematical Modeling.S3：The critical speed of Digit Control Machine Tool each axle under different acceleration is obtained, specifically, conventional friction The maximum shortcoming of power model, is exactly the characteristic for not describing stiction accurately.However, stiction is Digit Control Machine Tool feeding Generally existing and inevitable phenomenon in system.In order to eliminate the influence of stiction, it is necessary to accurately set up quiet Frictional model.Stiction occurs in " critical displacement " at speed commutation, this " critical displacement x_break" size under Formula (8) determines：

In formula (8), F_cAnd σ_oIt is respectively the frictional force and rigidity suffered by machine spindle.After condition (operating mode) determines, F_cAnd σ_oIt is Determine constant, therefore " critical displacement (alternatively referred to as shaking off displacement) " is also constant constant value.It is also possible to use acceleration " critical displacement x is represented with the time_break”：

In formula (9), t_breakIt is rotary shaft " (maximum shakes off displacement as critical potential to shake off the displacement interval time used Move), T_V=0It is moment when speed is equal to 0, it is the time variable factor that a (τ) is real time acceleration τ,It is average acceleration.

AgainAnd acceleration there is no great changes in " critical dislocation shifting ", thus can Speed is equal to acceleration a when 0₀Average acceleration is equivalent to, i.e.,So formula (9) can be expressed as following formula：

In formula (10), v_breakIt is " critical speed ".From formula (10), " critical speed " square with acceleration into just Than.Because " critical displacement " very little (only about 10 microns, different materials may be more greatly or a little bit smaller), institute Larger error may be produced with direct measurement " critical displacement ".But, from formula (10), can be by measuring different acceleration " critical speed " under degree, obtains the relation of acceleration and " critical speed ", you can obtain the size of " critical displacement ".

In order to obtain the relation of acceleration and " critical speed ", can different acceleration (1,2,3,4,5,6,8,10, 15,20,25,50,75,100rad/s²) under measure and obtain similar to the acceleration shown in Fig. 3 by corresponding " critical speed " With the relation curve of " critical speed " square.Convolution (10), can learn the half i.e. " critical potential of straight slope shown in Fig. 3 The size of shifting ".During " critical displacement " that obtains is substituted into following formula (11)：

Can obtain under different acceleration real-time " critical speed ".

S4：The stiction Mathematical Modeling of each axle is constructed using the critical speed data for obtaining, it is specific as follows, it is quiet to rub Power F (x) size is wiped to be obtained by formula (12)：

F (x)=- kx (12)

In formula (12), k is the equivalent stiffness on surface of contacting with each other, and x is displacement.Displacement x is again：

In formula (13), v is reference velocity, and it can be obtained by the once differentiation (difference) to reference locus, a₀It is speed Degree is equal to acceleration when 0.Because typically using speed variable, and moment of friction to be directly proportional to frictional force in rotary shaft, order ratio The example factor is λ, and then formula (13) of having been changed at formula 2 can be replaced with formula (14)：

In formula (14),r_gIt is the gearratio of ball screw, orderThen formula (14) can be write as following formula

T (ω)=κ ω² (15)

By " critical speed " v for trying to achieve_breakSubstitution formula (1), tries to achieve critical friction torque T_break, then the two Substitution formula (15), you can obtain the size of parameter κ, the final Mathematical Modeling for obtaining stiction.

It is emphasized that：Here critical speed is real-time, because critical acceleration is according to the change of reference locus And change, critical speed can in real time be tried to achieve by formula (11).

S5：Conventional friction power Mathematical Modeling is integrated with stiction Mathematical Modeling, to obtain general frictional force mould Type, specifically,

Convolution (1) and formula (15) can obtain the general frictional force Mathematical Modeling as shown in following formula (16)：

In formula (16), ω represents rotating speed,WithRepresent static friction torque,WithCoulomb moment of friction is represented,WithViscous friction torque is represented,WithRotating speed constant is represented,WithRepresent critical speed, κ⁺And κ^-Stiction constant, the wherein direction of sign ± expression speed are represented, e represents natural logrithm.

The curve map of the frictional force Mathematical Modeling for finally being obtained is such as Fig. 4, as seen from the figure, this Frictional model is not only The Stribeck phenomenons of frictional force can accurately be described, moreover it is possible to accurately describe the characteristic of stiction.

Fig. 5 and Biao 2 are illustrated and entered respectively with general Frictional model proposed by the invention with conventional friction power model The comparing result of row Friction Compensation experiment.Wherein, Fig. 5 (a), Fig. 5 (c) respectively show X and Y-axis in differentiated friction force compensating The contrast of actual tracking error and theoretical tracking error under method, Fig. 5 (b), Fig. 5 (d) compared for the difference of X and Y-axis respectively The deviation of actual tracking error and theoretical tracking error under Friction Compensation method；Table 2 specifically compared for X respectively and Y-axis exists The actual numerical value with the deviation of theoretical tracking error under differentiated friction force compensating method.It is actual straight with the deviation of theoretical tracking error The accuracy for reflecting Frictional model is connect, the smaller then Frictional model of deviation is more accurate.As can be drawn from Figure 5：Compared to biography System Frictional model, the compensation effect of the Frictional model for being proposed is more preferable.By making a concrete analysis of table 2, it can be deduced that：Compared to Conventional friction power model, the Frictional model for being proposed respectively can reduce the deviation of X and the maximum tracking error of Y-axis 75.7% and 57.1%, the deviation of average tracking error reduces 57.6% and 21.9%.Therefore, it can draw and rubbed compared to tradition Power model is wiped, Frictional model proposed by the invention can effectively reduce the harmful effect of stiction.

Under the distinct methods of table 2 the actual deviation ratio with theoretical tracking error compared with

The present invention establishes a kind of general frictional force Mathematical Modeling based on each rotating speed of Digit Control Machine Tool, of the invention general Frictional model not only inhibits influence of the frictional force to numerical control machine tooling under high-speed case, has also effectively processed low-speed situations Under ill-effect of the nonlinear normal modes to Digit Control Machine Tool.The inventive method is simple and easy to apply, can accurately, reliably suppress to rub Influence when power is to numerical control machine tooling part, and possess versatility, it is adaptable to various types of numerical control machine tools (center), Machining accuracy for improving Digit Control Machine Tool has good application value.

As it will be easily appreciated by one skilled in the art that the foregoing is only presently preferred embodiments of the present invention, it is not used to The limitation present invention, all any modification, equivalent and improvement made within the spirit and principles in the present invention etc., all should include Within protection scope of the present invention.

Claims (6)

1. the frictional force modeling method of a kind of NC machine tool feed system, it is characterised in that it comprises the following steps：

S1：Each axle of acquisition Digit Control Machine Tool rotating speed different in idling conditions and the corresponding frictional force of each different rotating speeds are big It is small；

S2：The frictional force size and rotating speed obtained using step S1, construct conventional friction power Mathematical Modeling；

S3：Obtain the critical speed of Digit Control Machine Tool each axle under different acceleration；

S4：The critical speed obtained using step S3 constructs the stiction Mathematical Modeling of each axle；

S5：The conventional friction power Mathematical Modeling that step S2 is obtained is carried out with the stiction Mathematical Modeling that step S4 is obtained whole Close, to obtain general Frictional model.

2. a kind of frictional force modeling method of NC machine tool feed system as claimed in claim 1, it is characterised in that step S1 In, Digit Control Machine Tool each axle is moved reciprocatingly in different velocity intervals, to obtain each axle of Digit Control Machine Tool in idle running shape Corresponding frictional force size under different rotating speeds during state,

In order to pick out the characteristic of boundary friction and segment fluid flow friction, rotational speed omega=± 0.1 is chosen low speed is interval, ± 0.2, ± 0.3, ± 0.5, ± 0.7, ± 1, ± 3 ± 5 ± 10rad/s as test speed, at a high speed it is interval choose rotational speed omega=± 25, ± 50, ± 75, ± 100rad/s is used as test speed.

3. a kind of frictional force modeling method of NC machine tool feed system as claimed in claim 2, it is characterised in that step S2 In, the Mathematical Modeling such as following formula of conventional friction power is represented：

In formula, ω represents rotating speed,WithRepresent static friction torque,WithCoulomb moment of friction is represented,With Viscous friction torque is represented,WithRotating speed constant, the wherein direction of sign ± expression speed are represented, e is represented Natural logrithm.

4. a kind of frictional force modeling method of NC machine tool feed system as claimed in claim 3, it is characterised in that step S3 In, obtain the critical speed of Digit Control Machine Tool each axle under different acceleration, it is first determined " critical displacement x_break" size, face Boundary's displacement x_breakIt is as follows：

$x_{break}=\frac{F_c}{{\mathrm{\σ}}_o}$

Wherein, F_cAnd σ_oIt is respectively the frictional force and rigidity suffered by machine spindle, after operating mode determines, F_cAnd σ_oIt is to determine constant, use Acceleration and time represent critical displacement x_breakIt is as follows：

$x_{break}={\∫}_{T_{v=0}}^{T_{v=0}+t_{break}}{\∫}_{T_{v=0}}^{t}a\left(\mathrm{\τ}\right)d\mathrm{\τ}dt\≈\frac{1}{2}\stackrel{\‾}{a}{t}_{break}^2$

Wherein, t_breakBeing rotary shaft is shaking off the displacement interval time used, and maximum shakes off displacement as critical displacement, T_V=0It is speed Degree is equal to moment when 0, and a (τ) is real time acceleration, and τ is the time variable factor,It is average acceleration, t is the moment,

Due toAcceleration a when by speed equal to 0₀Average acceleration is equivalent to, i.e.,Under then obtaining Formula：

$x_{break}\≈\frac{1}{2}\stackrel{\‾}{a}{t}_{break}^2\≈\frac{1}{2}{a}_0{t}_{break}^2=\frac{v_{break}^2}{2a_0}\→2x_{break}\≈\frac{v_{break}^2}{a_0}$

Wherein, v_breakIt is critical speed, by measuring the critical speed under different acceleration, obtains acceleration with critical speed Relation, you can obtain the size of critical displacement,

In order to obtain the relation of acceleration and critical speed, corresponding critical speed is measured under different acceleration, accelerated Degree and the relation curve of critical speed square, and then obtain the size of critical displacement, during the critical displacement for obtaining is substituted into following formula,

$v_{break}=\sqrt{2a_0{x}_{break}}$

Obtain real-time critical speed under different acceleration.

5. a kind of frictional force modeling method of NC machine tool feed system as claimed in claim 4, it is characterised in that step S4 In, the stiction Mathematical Modeling of each axle, specific as follows, stiction F (x) are constructed using the critical speed data for obtaining Size is obtained by following formula：

F (x)=- kx

Wherein, k is the equivalent stiffness on surface of contacting with each other, and x is displacement, and displacement x is represented using following formula：

$x=\frac{v^2}{2a_0}\left(\right|v|\≤v_{break})$

Wherein, v is reference velocity, is obtained by the once differentiation to reference locus, a₀It is acceleration when speed is equal to 0,

Typically use speed variable, and moment of friction to be directly proportional to frictional force in rotary shaft, make scale factor for λ, then formulaIt is transformed to：

$T\left(\mathrm{\ω}\right)=-k\mathrm{\λ}\frac{{\mathrm{\ω}}^2}{2a_0}$

Wherein,r_gIt is the gearratio of ball screw, orderThen formulaIt is transformed to following formula：

T (ω)=κ ω²

By " critical speed " v for trying to achieve_breakSubstitution formulaWithAsk Obtain critical friction torque T_break, then according to T (ω)=κ ω², obtain the size of parameter κ, the final mathematics for obtaining stiction Model.

6. a kind of frictional force modeling method of NC machine tool feed system as claimed in claim 5, it is characterised in that step S5 In, conventional friction power Mathematical Modeling is integrated with stiction Mathematical Modeling, to obtain general Frictional model specifically, General frictional force Mathematical Modeling is as follows,

$T_{fric}\left(\mathrm{\&omega;}\right)=\left\{\begin{array}{ccc}{T}_f^{+}\left(\mathrm{\&omega;}\right)=T_{stat}^{+}{e}^{-\mathrm{\&omega;}/{\mathrm{\&Omega;}}_1^{+}}+T_{coul}^{+}(1-e^{-\mathrm{\&omega;}/{\mathrm{\&Omega;}}_2^{+}})+T_{visc}^{+}\mathrm{\&omega;}& if& \mathrm{\&omega;}\&GreaterEqual;{\mathrm{\&omega;}}_{break}^{+}\\ T_f^{-}\left(\mathrm{\&omega;}\right)=T_{stat}^{-}{e}^{-\mathrm{\&omega;}/{\mathrm{\&Omega;}}_1^{-}}+T_{coul}^{-}(1-e^{-\mathrm{\&omega;}/{\mathrm{\&Omega;}}_2^{-}})+T_{visc}^{-}\mathrm{\&omega;}& if& \mathrm{\&omega;}\&le;{\mathrm{\&omega;}}_{break}^{-}\\ {\mathrm{\&kappa;}}^{+}{\mathrm{\&omega;}}^2& if& 0\&le;\mathrm{\&omega;}<{\mathrm{\&omega;}}_{break}^{+}\\ {\mathrm{\&kappa;}}^{-}{\mathrm{\&omega;}}^2& if& {\mathrm{\&omega;}}_{break}^{-}\&le;\mathrm{\&omega;}<0\end{array}\right.$

In formula, ω represents rotating speed,WithRepresent static friction torque,WithCoulomb moment of friction is represented,With Viscous friction torque is represented,WithRotating speed constant is represented,WithRepresent critical speed, κ⁺And κ^-Table Show stiction constant, the wherein direction of sign ± expression speed, e represents natural logrithm.