CN109557875A - Double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order - Google Patents
Double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order Download PDFInfo
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Abstract
Double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order of invention, the stress model that fractional order item is added is made first, grinding model is obtained by the calculation expression of grinding force, it solves without fractional model stability and draws boundary of stability, Numerical Simulation Program is worked out with Runge-Kutta method iteration in matlab program, numbered analog simulation is carried out, for one group of Reasonable Parameters, according to simulation result, influence of the fractional order item to grinding system is added in concrete analysis；Different fractional order orders and corresponding fractional order coefficient are had studied in the grinding system being added after fractional order item to system response and the influence of stability, Rational choice fractional order parameter can effectively improve the stability that system disturbs Regenerative Chatter, the application and optimization that result of study can be fractional order damper in roll dressing system provide basis, it can be used for the optimization of roll dressing system parameter, moreover it is possible to provide theoretical direction to actual processing process.
Description
Technical field
The invention belongs to accurate grinding technical fields, are related to double circle grindings Regenerative Chatter numerical analysis side of the item containing fractional order
Method.
Background technique
Fractional order differential is widely used in the fields such as signal processing, image procossing, hydrodynamics, viscoelastic damper,
Mechanical kinetics system Zhong Shenyong army introduces the concept of fractional order differential, and it is right mainly to have studied introducing fractional order differential item
The influence of duffing system dynamics, since fractional order differential can characterize the global dependence to historical data, and
Fractional order differential item can pass through equivalent linear damping and equivalent line under the influence of different fractional order coefficients and fractional order order
Property rigidity influences the amplitude and frequency characteristic of system respectively, the characteristics of relative to fractional order item, compares with roll dressing, roll
The generally acknowledged one of model of grinding process flutter is Regenerative Chatter caused by time lag, due to the influence of time lag, roll dressing
Journey historical data overall situation dependence is big, more meets reality, can be improved accuracy for flutter prediction, and roll dressing needs
Amplitude and frequency to system are adjusted inhibition flutter, and the angle based on practical application has had investigated score at this stage
Rank damper, installation application of the fractional order damper in mechanical system be not also extensive, controllable fractional order order and score
Level number is control system parameter spread thinking, to obtain better ground effect, grinding surface quality is improved, to roll
Grinding trembling is of great significance in terms of inhibiting aspect to provide fundamental basis with Engineering Guidance.
The meaning of analytic solutions is big not as good as numerical solution, but it provides theoretical basis for the research of numerical solution, in practice numerical value
Solution application is relatively broad, also lacks fractional order item in the prior art to the grinding system of mechanical system especially high-precision requirement
Analysis, does not couple the method analyzed with Regenerative Chatter caused by time lag to fractional order item, and roll dressing is typical double
Circle grinding process is relatively lagged in grinding area research progress, can not analytically or the angle of numerical value is to fractional order item and double circle mill
Regenerative Chatter impact analysis is cut, no correlation theory is instructed, and fractional order damper answering in roll dressing system is lacked
With with optimization basis, it is even more impossible to the roll dressing system parameter optimization designs further to the item containing fractional order.
Summary of the invention
In order to solve the above technical problems, the object of the present invention is to provide double circle grinding Regenerative Chatter numerical value of the item containing fractional order
Analysis method provides theoretical direction for the optimization of roll dressing system parameter, and to actual processing process.
Double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order, include the following steps:
Step 1: constructing the grinding stress model of the addition fractional order item between the grinding wheel of roll grinder and roll；
Step 2: the calculation expression of grinding force is derived according to grinding stress model, resettle the item containing fractional order it is double when
Stagnant roll dressing model；
Step 3: according to double time lag roll dressing models of the item containing fractional order, establishing the model for being free of fractional order item, solve
The model stability, and draw boundary of stability；
Step 4: according to double time lag roll dressing models of the item containing fractional order, relying on classical four step Runge-Kutta, structure
Nested iterative relation is made, numerical simulation program, the boundary of stability acquired according to analytic solutions in step 3 are worked out based on matlab
It is verified；
Step 5: changing input parameter, compare research, and grind according to the roll that result of study analyzes the item containing fractional order
The dynamic response of system is cut, and studies influence of each parameter to the roll dressing system of the item containing fractional order, for optimizing
The design parameter of roll grinder and the control parameter of fractional order differential.
Step 1 specifically:
The physical model for simplifying roll grinding process, since roll quality is larger and the both ends of roll pass through roll neck and support
Watt cooperation positioning simplifies unnecessary details, makes the assumption that condition according to roll dressing feature, obtains grinding stress model, should
Stress model is ground on the basis of roller neck, the displacement of grinding carriage is as vibration free end, and grinding force containing time lag is as sharp
Vibration power；
Where it is assumed that condition 1 are as follows: the axis shift amount of roll is zero；Assumed condition 2 are as follows: flexible deformation does not occur for roll.
Step 2 includes:
Step 2.1: the expression formula of thickness of cutting, thickness of cutting calculation method are calculated according to stress model are as follows:
δr=μ1x3(t-τr)-x3(t) (1)
δg=x2(t-τg)-x2(t) (2)
Wherein, δrFor the grinding thichness of roll, μ1To be ground overlapping coefficient, x3(t) change for roller surface grinding thichness
Amount, τrThe time lag amount of regeneration efficity is caused for roller surface；δgFor grinding wheel thickness of cutting, x2(t) become for the thickness of wheel face
Change, τgFor the time lag amount of grinding wheel；
Step 2.2: grinding force being associated with thickness of cutting, and then seeks the calculation expression of grinding force, is become according to elasticity
Shape condition, under constant rotational speed grinding condition, the calculation method of grinding force are as follows:
Pn=k0Pt (4)
Wherein, PtFor the tangential component of grinding force, uchFor grinding ratio energy, VrFor roller surface linear velocity, b is that grinding wheel is wide
Degree, VgFor wheel face linear velocity；PnFor the horizontal component of grinding force, k0For the conversion coefficient of grinding force and normal pressure；
Step 2.3: the expression formula between grinding force and regeneration disturbance is obtained according to formula (3) (4):
Introduce grinding coefficient G:
G=roll wear amount volume/abrasion of grinding wheel amount volume=b δrVr/bδgVg (6)
x1(t)=x2(t)+x3(t) (7)
Obtain grinding wheel depth of cut δg=Vrδr/VgG, since grinding coefficient generally takes G=1000, the linear velocity of grinding wheel is high
In the linear velocity of roll, the grinding abrasion amount x of grinding wheel is obtained2(t) it is less than roll dressing abrasion loss x3(t) one thousandth, x1
(t) displacement of grinding carriage is represented, obtains the simplified expression between grinding force and regeneration disturbance accordingly:
Step 2.4: double circles containing fractional order being obtained according to the simplified expression between grinding force and regeneration disturbance and were ground
The Regenerative Chatter model of journey are as follows:
Wherein, m represents grinding carriage quality, and c represents grinding carriage damping, and k represents grinding carriage rigidity, KDα[x1(t)] it represents and divides
Number rank, K are fractional order coefficient, and α is fractional order order.
Step 3 includes:
Step 3.1: the model for being free of fractional order item is established, expression formula is as follows:
According to this model definition status vectorThen by without fractional order item model be rewritten as state to
The matrix form of amount:
Step 3.2: establishing characteristic equation, Analytical Solution is carried out by mathmatica, according to the solution of characteristic equation, draw
Boundary of stability.
Step 4 includes:
Step 4.1: relying on classical four step Runge-Kutta, construct nested iterative relation, numerical simulation iterative algorithm
Are as follows:
Wherein, z is time lag item, represent rotation lap time intrinsic displacement amount be null matrix, later time lag item according to x rule
Rule movement, l, n, i, j respectively represent the number of iterations, DαFor the approximate expression of fractional order item,For fractional order binomial coefficient,
Wherein ti=ih is time sampling point, and h is time step, y andRespectively auxiliary construction function defines iterative initial value y respectively
(t0) andk1Represent starting point slope, k2Represent k1The midpoint slope of decision, k3Represent k2The midpoint slope of decision, k4It represents
Ending slope, distribution weight iterative calculation, obtains Numerical Simulation Results；
Step 4.2: the boundary of stability acquired according to step 3 chooses three points near boundary of stability, is respectively as follows:
Point positioned at boundary of stability's point of proximity, in boundary of stability and the point outside boundary of stability, K value are 0, step
4.1 degenerations are the numerical solution containing only time lag item, and other parameters are constant, determine revolving speed VgWith the value of grinding wheel width b, step is substituted into
4.1 carry out numerical solution；
Step 4.3: showing neutrality characteristic positioned at boundary of stability's point of proximity, the point in boundary of stability
Convergence stability characteristic (quality) is shown, the point outside boundary of stability shows divergence instability characteristic, analytic solutionand numerical solution
The correctness of solution procedure is demonstrated mutually.
In step 5, when selection order is 0, since fractional order item embodies very strong spring performance, the number of fractional order coefficient
Magnitude and grinding carriage rigidity are consistent, and in effectively research section, stability first reduces with the increase of fractional order term coefficient
Increase again, it was demonstrated that linear relationship is not present between stability and fractional order item, the choosing for considering fractional order coefficient should be segmented
It takes.
In step 5, when selection order is 0.25, fractional order item has both spring performance and damping characteristic, but is more biased towards in bullet
Spring characteristic, the order of magnitude and the grinding carriage rigidity of fractional order coefficient are consistent, and in effectively research section, stability is with score
It is exactly the opposite when the increase first increases and then decreases of rank with order is zero, it was demonstrated that have in this section very strong non-linear
Correlation.
In step 5, when selection order is 0.75, fractional order item has both spring performance and damping characteristic, but is more biased towards in resistance
Damping characteristics, the order of magnitude and the grinding carriage damping of fractional order coefficient are consistent, and in effectively research section, stability is with score
The increase of rank increases always, and the gradient only changed is smaller, it was demonstrated that has linear dependence in this section.
In step 5, when selection order is 1, fractional order item shows very strong damping characteristic, the order of magnitude of fractional order coefficient
It is consistent with grinding carriage damping, in effectively research section, stability increases always with the increase of fractional order item, variation
Gradient relative to order be 0.75 when obviously increase, it was demonstrated that in this section have linear dependence.
Double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order of the invention at least have the advantages that
1, Regenerative Chatter numerical analysis methods are ground according to double circles, according to parsing result and numerical simulation study change grinding wheel
Influence, change grinding wheel speed influence to system stability of the width to system stability, obtains the boundary of stability of system respectively,
For instructing basic numerical parameter to choose, it is ensured that parameters value is fallen in boundary of stability；
2, effect of the expression parameters that can be explained the profound in simple terms by model played in grinding system, convenient for deeply
The double circle grinding Regenerative Chatter mechanism for understanding the item containing fractional order, have studied not respectively in the grinding system being added after fractional order item
With fractional order order and corresponding fractional order coefficient to system response and the influence of stability, Rational choice fractional order parameter can
Effectively improve the stability that system disturbs Regenerative Chatter；
3, the application and optimization that result of study can be fractional order damper in roll dressing system provide basis, are not
Stability mentions to merely the higher the better, it will be seen that changing the parameter of fractional order item in certain model in numerical analysis
Enclose it is interior there are benign effectiveness in vibration suppression, there is also pernicious execution within the scope of other, the parameter of fractional order item is different,
On piecewise interval, both there is linear dependence, there is also non-linear dependencies, vibratory output is finally controlled in reasonable range,
Secondary destruction caused by impact is avoided the occurrence of again；
4, result of study can be used for the optimization of roll dressing system parameter in turn, improve roll dressing system medium plain emery wheel
Width, grinding wheel quality, grinding carriage rigidity, grinding carriage damping parameter provide reference for the design theory of roll grinder, keep it more suitable
The grinding system of the item containing fractional order is closed, influence of the Regenerative Chatter disturbance to system is reduced；
5, under the premise of not changing prior art parameter, according to analytical expression adjust grinding wheel speed, roll rotational speed with
The stability of grinding system is improved, can the variation of some control parameters theoretical direction is provided to actual processing process, be avoided
Degree depends on operating experience, and with the variation of grinding process and grinding machine model, grinding response, the accumulation of failed operation experience is often
It pays a high price for, according to reasonable theory, more accurately adjusts variable technique parameter when unstability occurs.
Detailed description of the invention
Fig. 1 is the flow chart of double circle grinding Regenerative Chatter numerical analysis methods of the item of the invention containing fractional order；
Fig. 2 is physical model rough schematic view of the invention；
Fig. 3 is stress model schematic diagram of the invention；
Fig. 4 is grinding wheel speed boundary of stability figure；
Fig. 5 is grinding wheel width boundary of stability figure；
Fig. 6 is that grinding wheel width boundary of stability closes on point value calculating figure；
Fig. 7 is that the outer numerical value of grinding wheel width boundary of stability calculates figure；
Fig. 8 is that numerical value calculates figure in grinding wheel width boundary of stability；
Fig. 9 is that fractional order order takes 0 numerical analysis figure；
Figure 10 is that fractional order order takes 0.25 numerical analysis figure；
Figure 11 is that fractional order order takes 0.75 numerical analysis figure；
Figure 12 is that fractional order order takes 1 numerical analysis figure.
Specific embodiment
As shown in Figure 1, double circle grinding Regenerative Chatter numerical analysis methods of the item of the invention containing fractional order, including walk as follows
It is rapid:
Step 1: in order to which emphasis describes the stress and wheel grinding amounts of thickness variation, roll dressing thickness change of grinding area
Amount, physical model rough schematic view according to fig. 2 construct stress model schematic diagram as shown in Figure 3, R in figuregRepresent grinding wheel
Radius, RrThe radius for representing roll constructs the grinding stress mould of the addition fractional order item between the grinding wheel of roll grinder and roll
Type, step 1 specifically:
The physical model for simplifying roll grinding process simplifies unnecessary details, makes the assumption that according to roll dressing feature
Condition obtains grinding stress model, and for the grinding stress model on the basis of roller neck, the displacement of grinding carriage is used as vibration certainly
By holding, grinding force containing time lag is as exciting force, since roll quality is larger and the both ends of roll pass through roll neck and Tuo Wa cooperation
Positioning；Where it is assumed that condition 1 are as follows: the axis shift amount of roll is zero；Assumed condition 2 are as follows: flexible deformation does not occur for roll.
Step 2: the calculation expression of grinding force is derived according to grinding stress model, resettle the item containing fractional order it is double when
Stagnant roll dressing model, step 2 include:
Step 2.1: the expression formula of thickness of cutting, thickness of cutting calculation method are calculated according to stress model are as follows:
δr=μ1x3(t-τr)-x3(t) (1)
δg=x2(t-τg)-x2(t) (2)
Wherein, δrFor the grinding thichness of roll, μ1To be ground overlapping coefficient, x3(t) change for roller surface grinding thichness
Amount, τrThe time lag amount of regeneration efficity is caused for roller surface；δgFor grinding wheel thickness of cutting, x2(t) become for the thickness of wheel face
Change, τgFor the time lag amount of grinding wheel；
Step 2.2: grinding force being associated with thickness of cutting, and then seeks the calculation expression of grinding force, is become according to elasticity
Shape condition, under constant rotational speed grinding condition, the calculation method of grinding force are as follows:
Pn=k0Pt (4)
Wherein, PtFor the tangential component of grinding force, uchFor grinding ratio energy, VrFor roller surface linear velocity, b is that grinding wheel is wide
Degree, VgFor wheel face linear velocity；PnFor the horizontal component of grinding force, k0For the conversion coefficient of grinding force and normal pressure；
Step 2.3: the simplified expression between grinding force and regeneration disturbance is obtained according to formula (3) (4):
Introduce grinding coefficient G:
G=roll wear amount volume/abrasion of grinding wheel amount volume=b δrVr/bδgVg (6)
x1(t)=x2(t)+x3(t) (7)
Obtain grinding wheel depth of cut δg=Vrδr/VgG, since grinding coefficient generally takes G=1000, the linear velocity of grinding wheel is high
In the linear velocity of roll, available, the grinding abrasion amount x of grinding wheel is analyzed2(t) it is less than roll dressing abrasion loss x3(t) thousand points
One of, x1(t) displacement of grinding carriage is represented, obtains the simplified expression between grinding force and regeneration disturbance accordingly:
Step 2.4: double circles containing fractional order being obtained according to the simplified expression between grinding force and regeneration disturbance and were ground
The Regenerative Chatter model of journey are as follows:
Wherein, m represents grinding carriage quality, and c represents grinding carriage damping, and k represents grinding carriage rigidity, KDα[x1(t)] it represents and divides
Number rank, K are fractional order coefficient, and α is fractional order order.
Step 3: according to double time lag roll dressing models of the item containing fractional order, establishing the model for being free of fractional order item, solve
The model stability, and boundary of stability is drawn, step 3 includes:
Step 3.1: the model for being free of fractional order item is established, expression formula is as follows:
According to this model definition status vectorThen by without fractional order item model be rewritten as state to
The matrix form of amount:
Step 3.2: establishing characteristic equation, Analytical Solution is carried out by mathmatica, according to the solution of characteristic equation, draw
Boundary of stability.Fig. 4 is that the grinding wheel speed boundary of stability drawn according to step 3 method schemes, and horizontal axis represents roll rotational speed, unit
For m/s, the longitudinal axis represents grinding wheel speed, and unit m/s, Fig. 5 are the grinding wheel width boundary of stability drawn according to step 3 method
Figure, horizontal axis represent roll rotational speed, unit m/s, and the longitudinal axis represents grinding wheel width, unit mm.
Step 4: according to double time lag roll dressing models of the item containing fractional order, relying on classical four step Runge-Kutta, structure
Nested iterative relation is made, numerical simulation program, the boundary of stability acquired according to analytic solutions in step 3 are worked out based on matlab
It is verified, step 4 includes:
Step 4.1: relying on classical four step Runge-Kutta, construct nested iterative relation, numerical simulation iterative algorithm
Are as follows:
Wherein, z is time lag item, represent rotation lap time intrinsic displacement amount be null matrix, later time lag item according to x rule
Rule movement, l, n, i, j respectively represent the number of iterations, DαFor the approximate expression of fractional order item,For fractional order binomial coefficient,
Wherein ti=ih is time sampling point, and h is time step, y andRespectively auxiliary construction function defines iterative initial value y respectively
(t0) andk1Represent starting point slope, k2Represent k1The midpoint slope of decision, k3Represent k2The midpoint slope of decision, k4It represents
Ending slope, distribution weight iterative calculation, obtains Numerical Simulation Results；
Step 4.2: the boundary of stability acquired according to step 3 chooses three points near boundary of stability, is respectively as follows:
Point positioned at boundary of stability's point of proximity, in boundary of stability and the point outside boundary of stability, K value are 0, step
4.1 degenerations are the numerical solution containing only time lag item, and other parameters are constant, determine revolving speed VgWith the value of grinding wheel width b, step is substituted into
4.1 carry out numerical solution；
Step 4.3: the wheel vibration numerical solution in step 4.2 exports in the form of images, closes on positioned at boundary of stability
Point vibration characteristics shows neutrality characteristic as shown in fig. 6, being located at boundary of stability's point of proximity, is located in boundary of stability
Point vibration characteristics as shown in figure 8, be located at boundary of stability in point show convergence stability characteristic (quality), be located at boundary of stability
Outer point vibration characteristics is as shown in fig. 7, the point being located at outside boundary of stability shows divergence instability characteristic, analytic solutions and numerical value
Solution demonstrates mutually the correctness of solution procedure.
Step 5: changing input parameter, compare research, and grind according to the roll that result of study analyzes the item containing fractional order
The dynamic response of system is cut, and studies influence of each parameter to the roll dressing system of the item containing fractional order, for optimizing
The design parameter of roll grinder and the control parameter of fractional order differential.
When it is implemented, when selection order is 0, since fractional order item embodies very strong spring performance, fractional order coefficient
The order of magnitude and grinding carriage rigidity be consistent, in effectively research section, stability is first with the increase of fractional order term coefficient
Reduction increases again, it was demonstrated that linear relationship is not present between stability and fractional order item, should be segmented and consider fractional order coefficient
Selection.Fig. 9 is that fractional order order takes 0 numerical analysis figure.
When it is implemented, choose order be 0.25 when, fractional order item has both spring performance and damping characteristic, but be more biased towards in
Spring performance, the order of magnitude and the grinding carriage rigidity of fractional order coefficient are consistent, effectively research section in, stability with point
It is exactly the opposite when the increase first increases and then decreases of number ranks with order is zero, it was demonstrated that have in this section very strong non-thread
Property correlation.Figure 10 is that fractional order order takes 0.25 numerical analysis figure.
When it is implemented, choose order be 0.75 when, fractional order item has both spring performance and damping characteristic, but be more biased towards in
Damping characteristic, the order of magnitude of fractional order coefficient and grinding carriage damping are consistent, in effectively research section, stability with point
The increase of number rank increases always, and the gradient only changed is smaller, it was demonstrated that has linear dependence in this section.Figure 11 is score
Rank order takes 0.75 numerical analysis figure；
When it is implemented, fractional order item shows very strong damping characteristic, the number of fractional order coefficient when selection order is 1
Magnitude is consistent with grinding carriage damping, and in effectively research section, stability increases always with the increase of fractional order item, is become
The gradient of change relative to order be 0.75 when obviously increase, it was demonstrated that in this section have linear dependence.Figure 12 is fractional order rank
Number takes 1 numerical analysis figure.
According to double circles Regenerative Chatter numerical analysis methods are ground, it is wide according to parsing result and numerical simulation study change grinding wheel
It spends the influence to system stability, become influence of the grinding wheel speed to system stability, obtain the boundary of stability of system respectively, use
It is chosen in the basic numerical parameter of guidance, it is ensured that parameters value is fallen in boundary of stability；
Have studied different fractional order orders and corresponding score level respectively in the grinding system being added after fractional order item
Several pairs of system responses and the influence of stability, the expression parameters that can be explained the profound in simple terms by model are played in grinding system
The effect arrived, convenient for the deep double circle grinding Regenerative Chatter mechanism for understanding the item containing fractional order；
The application and optimization that result of study can be fractional order damper in roll dressing system provide basis, are not single
It is pure that stability mentions to the higher the better, it will be seen that changing the parameter of fractional order item in a certain range in numerical analysis
Inside there is benign effectiveness in vibration suppression, there is also pernicious executions within the scope of other, and the parameter of fractional order item is different, are dividing
On section section, both there is linear dependence, there is also non-linear dependencies, the final vibratory output that controls is in reasonable range, again
Avoid the occurrence of secondary destruction caused by impact；
Result of study can be used for the optimization of roll dressing system parameter in turn, and it is wide to improve roll dressing system medium plain emery wheel
Degree, grinding wheel quality, grinding carriage rigidity, grinding carriage damping parameter, provide reference for the design theory of roll grinder, are more suitable for it
The grinding system of the item containing fractional order reduces influence of the Regenerative Chatter disturbance to system；
Under the premise of not changing prior art parameter, grinding wheel speed, roll rotational speed are adjusted to mention according to analytical expression
The variation of some control parameters can be provided theoretical direction to actual processing process, avoided excessively by the stability of high grinding system
Dependent on operating experience, with the variation of grinding process and grinding machine model, grinding response, the accumulation of failed operation experience is often wanted
It pays a high price for, according to reasonable theory, more accurately adjusts variable technique parameter when unstability occurs.
The foregoing is merely presently preferred embodiments of the present invention, the thought being not intended to limit the invention, all of the invention
Within spirit and principle, any modification, equivalent replacement, improvement and so on be should all be included in the protection scope of the present invention.
Claims (9)
1. double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order, which comprises the steps of:
Step 1: constructing the grinding stress model of the addition fractional order item between the grinding wheel of roll grinder and roll；
Step 2: the calculation expression of grinding force is derived according to grinding stress model, the double time lags for resettling the item containing fractional order are rolled
Roller grinding model；
Step 3: according to double time lag roll dressing models of the item containing fractional order, establishing the model for being free of fractional order item, solve the mould
Type stability, and draw boundary of stability；
Step 4: according to double time lag roll dressing models of the item containing fractional order, relying on classical four step Runge-Kutta, construct embedding
The iterative relation of set works out numerical simulation program based on matlab, is carried out according to the boundary of stability that analytic solutions in step 3 acquire
Verifying；
Step 5: changing input parameter, compare research, and analyze the roll dressing system of the item containing fractional order according to result of study
The dynamic response of system, and influence of each parameter to the roll dressing system of the item containing fractional order is studied, for optimizing roll
The design parameter of grinding machine and the control parameter of fractional order differential.
2. double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order as described in claim 1, which is characterized in that step
Rapid 1 specifically:
The physical model for simplifying roll grinding process, since roll quality is larger and the both ends of roll are matched by roll neck and Tuo Wa
Positioning is closed, according to roll dressing feature, simplifies unnecessary details, makes the assumption that condition, obtains grinding stress model, the grinding
Stress model is on the basis of roller neck, and the displacement of grinding carriage is as vibration free end, and grinding force containing time lag is as exciting force；
Where it is assumed that condition 1 are as follows: the axis shift amount of roll is zero；Assumed condition 2 are as follows: flexible deformation does not occur for roll.
3. double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order as described in claim 1, which is characterized in that step
Rapid 2 include:
Step 2.1: the expression formula of thickness of cutting, thickness of cutting calculation method are calculated according to stress model are as follows:
δr=μ1x3(t-τr)-x3(t) (1)
δg=x2(t-τg)-x2(t) (2)
Wherein, δrFor the grinding thichness of roll, μ1To be ground overlapping coefficient, x3It (t) is roller surface grinding thichness variable quantity, τrFor
Roller surface causes the time lag amount of regeneration efficity；δgFor grinding wheel thickness of cutting, x2It (t) is the thickness change of wheel face, τgFor sand
The time lag amount of wheel；
Step 2.2: grinding force being associated with thickness of cutting, and then seeks the calculation expression of grinding force, according to flexible deformation item
Part, under constant rotational speed grinding condition, the calculation method of grinding force are as follows:
Pn=k0Pt (4)
Wherein, PtFor the tangential component of grinding force, uchFor grinding ratio energy, VrFor roller surface linear velocity, b is grinding wheel width, VgFor
Wheel face linear velocity；PnFor the horizontal component of grinding force, k0For the conversion coefficient of grinding force and normal pressure；
Step 2.3: the expression formula between grinding force and regeneration disturbance is obtained according to formula (3) (4):
Introduce grinding coefficient G:
G=roll wear amount volume/abrasion of grinding wheel amount volume=b δrVr/bδgVg (6)
x1(t)=x2(t)+x3(t) (7)
Obtain grinding wheel depth of cut δg=Vrδr/VgG, since grinding coefficient generally takes G=1000, the linear velocity of grinding wheel is higher than roll
Linear velocity, obtain the grinding abrasion amount x of grinding wheel2(t) it is less than roll dressing abrasion loss x3(t) one thousandth, x1(t) it represents
The displacement of grinding carriage obtains the simplified expression between grinding force and regeneration disturbance accordingly:
Step 2.4: double circle grinding process containing fractional order are obtained according to the simplified expression between grinding force and regeneration disturbance
Regenerative Chatter model are as follows:
Wherein, m represents grinding carriage quality, and c represents grinding carriage damping, and k represents grinding carriage rigidity, KDα[x1(t)] representative fraction rank
, K is fractional order coefficient, and α is fractional order order.
4. double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order as described in claim 1, which is characterized in that step
Rapid 3 include:
Step 3.1: the model for being free of fractional order item is established, expression formula is as follows:
According to this model definition status vectorThen the model without fractional order item is rewritten as state vector
Matrix form:
Step 3.2: establishing characteristic equation, Analytical Solution is carried out by mathmatica, according to the solution of characteristic equation, draw and stablize
Property boundary.
5. double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order as described in claim 1, which is characterized in that step
Rapid 4 include:
Step 4.1: relying on classical four step Runge-Kutta, construct nested iterative relation, numerical simulation iterative algorithm are as follows:
Wherein, z is time lag item, and representing rotation lap time intrinsic displacement amount is null matrix, and time lag item is transported according to the rule of x later
Dynamic, l, n, i, j respectively represent the number of iterations, DαFor the approximate expression of fractional order item,For fractional order binomial coefficient, wherein
ti=ih is time sampling point, and h is time step, y andRespectively auxiliary construction function defines iterative initial value y (t respectively0) andk1Represent starting point slope, k2Represent k1The midpoint slope of decision, k3Represent k2The midpoint slope of decision, k4It is oblique to represent terminal
Rate, distribution weight iterative calculation, obtains Numerical Simulation Results；
Step 4.2: the boundary of stability acquired according to step 3 chooses three points near boundary of stability, is respectively as follows: and is located at
Boundary of stability's point of proximity, the point in boundary of stability and the point outside boundary of stability, K value are 0, step 4.1
Degeneration is the numerical solution containing only time lag item, and other parameters are constant, determine revolving speed VgWith the value of grinding wheel width b, step 4.1 is substituted into
Carry out numerical solution；
Step 4.3: showing neutrality characteristic positioned at boundary of stability's point of proximity, the point performance in boundary of stability
Convergence stability characteristic (quality) is gone out, the point outside boundary of stability shows divergence instability characteristic, and analytic solutionand numerical solution is mutual
Demonstrate the correctness of solution procedure.
6. double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order as described in claim 1, which is characterized in that institute
State in step 5, when to choose order be 0, since fractional order item embodies very strong spring performance, the order of magnitude of fractional order coefficient and
Grinding carriage rigidity is consistent, and in effectively research section, stability increases again as the increase of fractional order term coefficient first reduces,
Demonstrating there is no linear relationship, should be segmented the selection for considering fractional order coefficient between stability and fractional order item.
7. double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order as described in claim 1, which is characterized in that institute
It states in step 5, when selection order is 0.25, fractional order item has both spring performance and damping characteristic, but is more biased towards in spring performance,
The order of magnitude of fractional order coefficient is consistent with grinding carriage rigidity, and in effectively research section, stability is with fractional order item
Increase first increases and then decreases, it is exactly the opposite when with order being zero, it was demonstrated that there are very strong non-linear dependencies in this section.
8. double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order as described in claim 1, which is characterized in that institute
It states in step 5, when selection order is 0.75, fractional order item has both spring performance and damping characteristic, but is more biased towards in damping characteristic,
The order of magnitude of fractional order coefficient is consistent with grinding carriage damping, and in effectively research section, stability is with fractional order item
Increase increases always, and the gradient only changed is smaller, it was demonstrated that has linear dependence in this section.
9. double circle grinding Regenerative Chatter numerical analysis methods of the item containing fractional order as described in claim 1, which is characterized in that institute
It states in step 5, when selection order is 1, fractional order item shows very strong damping characteristic, the order of magnitude and grinding wheel of fractional order coefficient
Frame damping is consistent, and in effectively research section, stability increases always with the increase of fractional order item, the gradient phase of variation
It is obviously increased when being 0.75 for order, it was demonstrated that there is linear dependence in this section.
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112733074A (en) * | 2020-12-24 | 2021-04-30 | 中国航空工业集团公司金城南京机电液压工程研究中心 | Method for drawing nonlinear stability and flutter process of machine tool cutting |
| CN113761678A (en) * | 2021-08-17 | 2021-12-07 | 上海机床厂有限公司 | Cylindrical grinding flutter general model and stability analysis method |
| CN115617398A (en) * | 2022-10-28 | 2023-01-17 | 电子科技大学 | Real-time reconfigurable fractional order computing system based on FPGA |
| CN116309547A (en) * | 2023-05-10 | 2023-06-23 | 成都飞机工业（集团）有限责任公司 | Part surface quality detection method, device, equipment and medium |
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Cited By (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112733074A (en) * | 2020-12-24 | 2021-04-30 | 中国航空工业集团公司金城南京机电液压工程研究中心 | Method for drawing nonlinear stability and flutter process of machine tool cutting |
| CN113761678A (en) * | 2021-08-17 | 2021-12-07 | 上海机床厂有限公司 | Cylindrical grinding flutter general model and stability analysis method |
| CN113761678B (en) * | 2021-08-17 | 2023-06-20 | 上海机床厂有限公司 | Cylindrical grinding flutter general model and stability analysis method |
| CN115617398A (en) * | 2022-10-28 | 2023-01-17 | 电子科技大学 | Real-time reconfigurable fractional order computing system based on FPGA |
| CN116309547A (en) * | 2023-05-10 | 2023-06-23 | 成都飞机工业（集团）有限责任公司 | Part surface quality detection method, device, equipment and medium |
| CN116309547B (en) * | 2023-05-10 | 2023-09-19 | 成都飞机工业（集团）有限责任公司 | Part surface quality detection method, device, equipment and medium |
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