CN110162733B - Milling stability analysis method based on integral discrete strategy - Google Patents
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Abstract
The invention discloses a milling stability analysis method based on an integral discrete strategy. The method of the invention comprises the following steps: converting a milling time lag dynamics equation of the milling system into a state space equation; solving an analytic solution of the state space equation under the condition that the milling system is in a free vibration process, and solving a periodic coefficient term, a state term and a time lag term of the state space equation by adopting a Lagrange interpolation function under the condition that the milling system is in a forced vibration process; constructing a Floque transition matrix according to the solution of the state space equation; and calculating the spectrum radius of the Floque transition matrix, and judging the stability of the milling system according to the spectrum radius. The method simplifies the calculation process of milling stability analysis, improves the calculation efficiency, and can obtain higher approximation precision compared with the existing semi-discrete method and full-discrete method.
Description
Technical Field
The invention relates to the technical field of milling, in particular to a milling stability analysis method based on an integral discrete strategy.
Background
Milling is a basic and extremely important machining method. High-speed milling technology is one of the core technologies for high-performance machining, and is widely applied to manufacturing of various complex parts due to the advantages of high precision and high efficiency. Regeneration chatter is a self-excited instability phenomenon common to milling processes. The occurrence of chatter has a great negative effect on the surface quality of the workpiece, the machining efficiency and the lifetime of the tool. Therefore, effective technical measures are required to avoid such adverse phenomena. From the perspective of suppressing regeneration chatter, an effective technical means is to select non-chatter process parameters according to a stable lobe diagram of the milling process. Modeling the milling process and corresponding stability analysis are therefore of great practical importance.
Document retrieval of existing methods has found that currently widely accepted main semi-discrete and fully-discrete methods:
(1) Semi-discrete method: as suggested by professor instrger and professor Stepan, representative documents are: semi-discretization method for delayed systems, updated Semi-discretization method for periodic delay-differential equations with discrete delay. The semi-discrete method has universality for milling process, namely, good prediction precision and numerical stability are obtained for large and small radial cutting depths and high and low spindle speeds. The disadvantage of this method is the relatively low computational efficiency, since it requires the calculation of an index matrix when scanning the depth of cut. At the same time, the accuracy of the method remains to be improved.
(2) Full discrete method: ding et al discloses a milling process stability analysis method based on a full discrete method. The method separates the coefficient matrix into a constant matrix and a periodic variation matrix, uses one-time interpolation to independently approximate the periodic term, the state term and the time lag term of the state equation, and adopts a direct integration strategy to construct the Floque transition matrix. The full discrete method has higher computational efficiency than the half discrete method because it does not require calculation of an index matrix when scanning the depth of cut by separating the coefficient matrix into a constant matrix and a periodically varying matrix. However, the accuracy of the full discrete method remains to be improved.
Disclosure of Invention
The invention aims to overcome the defect that the modeling and stability analysis methods of the milling process cannot be compatible with the calculation precision and the calculation efficiency in the prior art, and provides a milling stability analysis method based on an integral discrete strategy.
The invention solves the technical problems by the following technical proposal:
the invention provides a milling stability analysis method based on an integral discrete strategy, which is characterized by comprising the following steps of:
step one, converting a milling time lag dynamics equation of a milling system into a state space equation;
the milling time lag dynamics equation is that,
in the formula (1), M, C, K is a modal parameter matrix of a milling cutter in a milling system, a p For depth of cut, S (T) is a matrix of direction coefficients of the milling system, which has a time lag period T, the specific expression of each element in S (T) is,
in the formula (2), K n And K t The normal cutting force coefficient and the tangential cutting force coefficient, phi j (t) is the angular position of the jth cutter tooth, i.e. φ j (t) = (2Ω/60) t+ (j-1) 2Ω/N, window function g (φ) j (t)) is used to determine the cutting state of the j-th cutter tooth as shown in the following formula (3),
in the formula (3), phi st And phi ex Respectively the start of millingThe entry angle and the end cut angle, which may be determined according to the radial intrusion ratio a/D of the cutter teeth of the milling system and the milling direction as shown in the following equation (4),
introducing matrix transformations
With new state variables->
Converting the milling time-lag dynamics equation into a state space equation:
in the formula (5), the amino acid sequence of the compound,
dividing the milling process into a free vibration process and a forced vibration process according to whether the value of a periodic coefficient term in the state space equation is zero, solving an analytic solution of the state space equation for the situation that the milling system is in the free vibration process, and integrally approximating the periodic coefficient term, the state term and a time lag term of the state space equation by adopting a Lagrange interpolation function for the situation that the milling system is in the forced vibration process, so as to solve the solution of the state space equation for the situation that the milling system is in the forced vibration process;
dividing the milling process T into a free vibration process and a forced vibration process to make T fr The time of the forced vibration process is T fo ＝T-T fr Dispersing the forced vibration period into m subintervals, and setting the discrete step length h as h=T fo Each time point was obtained as shown in the following formula (7),
t n ＝t 0 +T fr +(n-1)h,n＝1,2,…,m+1 (7)
in free-vibrating process [0, T ] for milling systems fr ]The analysis solution of the state space equation (5) is shown in the formula (8),
let t=t 1 Substituting the solution into the above formula (8) to further obtain an analytical solution as shown in formula (9),
for the milling system in forced vibration process [ T ] fr ,T]Let b (t) =
A(t)[y(t)-y(t-T)]For subinterval t n ，t n+1 ]The periodic coefficient term, the state term and the time lag term of the state equation are approximated integrally by a Lagrange interpolation function, the Lagrange interpolation function is shown as the following (10),
substituting the above formula (10) into the state space equation (5) and in subinterval [ t ] n ，t n+1 ]Integrating to obtain a solution to the state space equation (5) for the case where the milling system is in a forced vibration process as shown in the following equation (11):
in the formula (11), the amino acid sequence of the compound,
for the first subinterval t 1 ，t 2 ]The periodic coefficient term, the state term and the time-lag term of the state space equation (5) are approximated integrally by adopting a linear interpolation function instead of the Lagrangian interpolation function, so that the solution of the equation is obtained as shown in the following formula (19),
(-G+E)y 1 +(I+F)y 2 ＝Ey 1-T +Fy 2-T (19)
in the formula (19), the amino acid sequence of the compound,
thirdly, according to the solution of the state space equation obtained in the second step, obtaining a matrix mapping relation between the current state of the milling system and the state of the previous period, and further constructing a Floquet transition matrix;
deriving said matrix mapping between the current state of the milling system and the state of the previous cycle according to the above equations (9), (11) and (19),
in the formula (22), the amino acid sequence of the compound,
the Floque transition matrix ψ is obtained by the above formulas (22), (23) and (24),
Ψ＝U -1 W (25)。
and step four, calculating the spectrum radius of the Floque transition matrix, and judging the stability of the milling system by adopting the Floque theory.
Judging that the milling system is unstable when the spectral radius of the floque transition matrix ψ is larger than 1, judging that the milling system is in critical stability when the spectral radius of the floque transition matrix ψ is equal to 1, and judging that the milling system is stable when the spectral radius of the floque transition matrix ψ is smaller than 1.
Preferably, in the second step, when the system is in the forced vibration process [ T ] fr ,T]At that time, for each subinterval [ t ] n ，t n+1 ]The periodic coefficient term, the state term and the time lag term of the state equation are approximated as a whole by a Lagrange interpolation function of first order or higher.
On the basis of conforming to the common knowledge in the field, the above preferred conditions can be arbitrarily combined to obtain the preferred examples of the invention.
The invention has the positive progress effects that:
according to the milling stability analysis method based on the integral discrete strategy, a milling process is divided into two processes of free vibration and forced vibration, a Lagrange interpolation function is adopted to integrally approximate a periodic coefficient term, a state term and a time lag term of a state equation, a Floquet transition matrix of a milling system is further constructed, and finally a stability boundary of the milling process is obtained. Compared with the prior art, the method of the invention not only greatly simplifies the calculation process, thereby improving the calculation efficiency, but also can obtain higher approximation precision compared with the existing semi-discrete method and full-discrete method, and adopts a one-time construction strategy to avoid the calculation time required by large matrix multiplication when constructing the Floquet transition matrix, thereby further improving the efficiency, and generally, improving the calculation precision and the efficiency compared with the prior art.
Drawings
FIG. 1 is a flow chart of a milling stability analysis method based on an overall discrete strategy according to a preferred embodiment of the present invention.
FIG. 2 is a graph of milling process stability lobes at a radial immersion ratio of 0.6 plotted in an example of an application of a method for milling stability analysis according to a preferred embodiment of the present invention.
Fig. 3 is a diagram showing a stability lobe of a milling process at a radial immersion ratio of 0.1, which is drawn in an application example of the milling stability analysis method according to a preferred embodiment of the present invention.
Fig. 4 is a diagram showing a stability lobe of a milling process when a radial immersion ratio is 0.06, which is drawn in an application example of the milling stability analysis method according to a preferred embodiment of the present invention.
Detailed Description
The following detailed description of the preferred embodiments of the invention, taken in conjunction with the accompanying drawings, is given by way of illustration and not limitation, and any other similar situations are intended to fall within the scope of the invention.
In the following detailed description, directional terms, such as "left", "right", "upper", "lower", "front", "rear", etc., are used with reference to the directions described in the drawings. The components of embodiments of the present invention can be positioned in a number of different orientations and the directional terminology is used for purposes of illustration and is in no way limiting.
Referring to fig. 1, a milling stability analysis method based on an overall discrete strategy according to a preferred embodiment of the present invention includes the following steps:
step one, converting a milling time lag dynamics equation of a milling system into a state space equation;
dividing the milling process into a free vibration process and a forced vibration process according to whether the value of a periodic coefficient term in the state space equation is zero, solving an analytic solution of the state space equation for the situation that the milling system is in the free vibration process, and integrally approximating the periodic coefficient term, the state term and a time lag term of the state space equation by adopting a Lagrange interpolation function for the situation that the milling system is in the forced vibration process, so as to solve the solution of the state space equation for the situation that the milling system is in the forced vibration process;
thirdly, according to the solution of the state space equation obtained in the second step, obtaining a matrix mapping relation between the current state of the milling system and the state of the previous period, and further constructing a Floquet transition matrix;
and step four, calculating the spectrum radius of the Floque transition matrix, and judging the stability of the milling system by adopting the Floque theory.
According to a preferred embodiment of the invention, the analysis method may in particular yield a final milling stability analysis result via the following calculation steps (processes).
First, the milling time lag dynamics equation of the milling system is converted into a state space equation. Wherein, the standard milling time lag dynamics equation is that,
in the formula (1), M, C, K is a modal parameter matrix of a milling cutter in a milling system, a p For depth of cut, S (T) is the direction coefficient matrix of the milling system, which has a time lag period T (i.e., S (T) =s (t+t)), for a standard 2-degree-of-freedom milling system, each element in S (T)The specific expression of (c) is that,
in the formula (2), K n And K t The normal cutting force coefficient and the tangential cutting force coefficient, phi j (t) is the angular position of the jth cutter tooth, i.e. φ j (t) = (2Ω/60) t+ (j-1) 2Ω/N, window function g (φ) j (t)) is used to determine the cutting state of the j-th cutter tooth as shown in the following formula (3),
in the formula (3), phi st And phi ex The start cutting angle and the end cutting angle of the milling, respectively, which can be determined according to the radial intrusion ratio a/D of the cutter teeth of the milling system and the milling direction as shown in the following formula (4),
introducing matrix transformations
With new state variables->
The milling time lag dynamics equation may be converted to a state space equation:
in the formula (5), the amino acid sequence of the compound,
upon conversion to the shape of the milling systemAfter the state space equation, the milling process T is divided into a free vibration process and a forced vibration process, so that T fr The time of the forced vibration process is T fo ＝T-T fr Dispersing the forced vibration period into m subintervals, and setting the discrete step length h as h=T fo Each time point was obtained as shown in the following formula (7),
t n ＝t 0 +T fr +(n-1)h,n＝1,2,…,m+1 (7)
in free-vibrating process [0, T ] for milling systems fr ]The analysis solution of the state space equation (5) is shown in the formula (8),
let t=t 1 Substituting the solution into the above formula (8) to further obtain an analytical solution as shown in formula (9),
for the milling system in forced vibration process [ T ] fr ,T]Let b (T) =a (T) [ y (T) -y (T-T)]For subinterval t n ，t n+1 ]The periodic coefficient term, the state term and the time lag term of the state equation are approximated integrally by a Lagrange interpolation function, the Lagrange interpolation function is shown as the following (10),
substituting the above formula (10) into the state space equation (5) and in subinterval [ t ] n ，t n+1 ]Integrating to obtain a solution to the state space equation (5) for the case where the milling system is in a forced vibration process as shown in the following equation (11):
in the formula (11), the amino acid sequence of the compound,
for the first subinterval t 1 ，t 2 ]The periodic coefficient term, the state term and the time-lag term of the state space equation (5) are approximated integrally by adopting a linear interpolation function instead of the Lagrangian interpolation function, so that the solution of the equation is obtained as shown in the following formula (19),
in the formula (19), the amino acid sequence of the compound,
then, a Floque transition matrix can be constructed, and a matrix mapping relation between the current state and the previous period state of the system can be obtained by combining the equation. In particular, the matrix mapping between the current state of the milling system and the state of the previous cycle is derived from the above equations (9), (11) and (19) as shown in equation (22) below,
in the formula (22), the amino acid sequence of the compound,
the Floque transition matrix ψ is obtained by the above formulas (22), (23) and (24),
Ψ＝U -1 W (25)。
after the floque transition matrix ψ is obtained, the spectrum radius κ (ψ) of the floque transition matrix ψ can be calculated, and the floque theory is adopted to judge the stability of the milling dynamic system, and the judging method is shown in the following formula (26):
the k (ψ) in equation (26) is the spectral radius of the Floquet transition matrix, i.e.,
kappa (ψ) =max (|λ (ψ) |), where λ (ψ) is the eigenvalue of the matrix.
According to some preferred embodiments of the present invention, the milling stability analysis method may further use Matlab software to program according to the steps (i.e. step one to step four) above based on the system parameters of a given milling system, and draw a stability lobe map of the milling process corresponding to each selected radial immersion ratio value.
In a preferred embodiment of the present invention, the given system parameters include: milling direction, milling cutter tooth number, normal cutting force coefficient and tangential cutting force coefficient, modal mass of the system, natural circular frequency of the system and damping ratio of the system.
In an application example of the present invention, the given system parameters are: the milling direction is forward milling, the number of teeth of the milling cutter is N=2, and the normal cutting force coefficient and the tangential cutting force coefficient are respectively K n ＝2×10 8 N/m 2 And K t ＝2×10 8 N/m 2 The modal mass of the system is m tx = 0.03993kg and m ty 0.03993kg, natural circular frequency of system ω nx =922×2 pi rad/s and ω ny =922×2pi rad/s, damping ratio of system ζ x =0.011 and ζ y =0.011. The number of discrete steps in the method is selected to be 20, and the spindle rotation speed and the cutting depth plane are scattered into 200×100 grids.
In the application example, the parameters are used as input, matlab software is used for programming the steps, radial immersion ratios are respectively 0.6, 0.1 and 0.06, and stable leaf patterns in the milling process are drawn and are respectively shown in figures 2-4.
The milling stability analysis method of the invention has the advantages that: the method adopts a Lagrange interpolation function to integrally approximate a periodic coefficient term, a state term and a time-lag term of a state equation, integrates the integrally approximated state equation in each subinterval of a forced vibration stage to construct a system Floquet transition matrix, and finally obtains a stability boundary of a milling process; compared with a strategy of independently approximating a system period coefficient term, a state term and a time-lag term by a semi-discrete method and a full-discrete method, the approximation precision is improved, and a recurrence formula is simpler so as to improve the calculation speed; meanwhile, a once-construction strategy is adopted when the Floquet transition matrix is constructed, so that the calculation time caused by large matrix multiplication is avoided, and the calculation efficiency is further improved.
While specific embodiments of the invention have been described above, it will be appreciated by those skilled in the art that these are by way of example only, and the scope of the invention is defined by the appended claims. Various changes and modifications to these embodiments may be made by those skilled in the art without departing from the principles and spirit of the invention, but such changes and modifications fall within the scope of the invention.
Claims (2)
1. The milling stability analysis method based on the integral discrete strategy is characterized by comprising the following steps of:
step one, converting a milling time lag dynamics equation of a milling system into a state space equation;
wherein the milling time lag dynamics equation is that,
in the formula (1), M, C, K is a modal parameter matrix of a milling cutter in a milling system, a p For depth of cut, S (T) is a matrix of direction coefficients of the milling system, which has a time lag period T, the specific expression of each element in S (T) is,
in the formula (2), K n And K t The normal cutting force coefficient and the tangential cutting force coefficient, phi j (t) is the angular position of the jth cutter tooth, i.e. φ j (t) = (2Ω/60) t+ (j-1) 2Ω/N, window function g (φ) j (t)) is used to determine the cutting state of the j-th cutter tooth as shown in the following formula (3),
in the formula (3), phi st And phi ex The start cutting angle and the end cutting angle of the milling, respectively, which can be determined according to the radial intrusion ratio a/D of the cutter teeth of the milling system and the milling direction as shown in the following formula (4),
introducing matrix transformations
With new state variables->
Converting the milling time-lag dynamics equation into a state space equation:
in the formula (5), the amino acid sequence of the compound,
dividing the milling process into a free vibration process and a forced vibration process according to whether the value of a periodic coefficient term in the state space equation is zero, solving an analytic solution of the state space equation for the situation that the milling system is in the free vibration process, and integrally approximating the periodic coefficient term, the state term and a time lag term of the state space equation by adopting a Lagrange interpolation function for the situation that the milling system is in the forced vibration process, so as to solve the solution of the state space equation for the situation that the milling system is in the forced vibration process;
wherein, the milling process T is divided into a free vibration process and a forced vibration process, so that T fr The time of the forced vibration process is T fo ＝T-T fr Dispersing the forced vibration period into m subintervals, and setting the discrete step length h ash＝T fo Each time point was obtained as shown in the following expression (7),
t n ＝t 0 +T fr +(n-1)h,n＝1,2,…,m+1 (7)
in free-vibrating process [0, T ] for milling systems fr ]The analysis solution of the state space equation (5) is shown in the formula (8),
let t=t 1 Substituting the solution into the above formula (8) to further obtain an analytical solution as shown in formula (9),
for the milling system in forced vibration process [ T ] fr ,T]Let b (T) =a (T) [ y (T) -y (T-T)]For subinterval t n ，t n+1 ]The periodic coefficient term, the state term and the time lag term of the state equation are approximated integrally by a Lagrange interpolation function, the Lagrange interpolation function is shown as the following (10),
substituting the above formula (10) into the state space equation (5) and in subinterval [ t ] n ，t n+1 ]Integrating to obtain a solution to the state space equation (5) for the case where the milling system is in a forced vibration process as shown in the following equation (11):
in the formula (11), the amino acid sequence of the compound,
for the first subinterval t 1 ，t 2 ]The periodic coefficient term, the state term and the time-lag term of the state space equation (5) are approximated integrally by adopting a linear interpolation function instead of the Lagrangian interpolation function, so that the solution of the equation is obtained as shown in the following formula (19),
(-G+E)y 1 +(I+F)y 2 ＝Ey 1-T +Fy 2-T (19)
in the formula (19), the amino acid sequence of the compound,
thirdly, according to the solution of the state space equation obtained in the second step, obtaining a matrix mapping relation between the current state of the milling system and the state of the previous period, and further constructing a Floquet transition matrix;
wherein the matrix mapping relationship between the current state of the milling system and the state of the previous cycle is derived from the above formulas (9), (11) and (19),
in the formula (22), the amino acid sequence of the compound,
the Floque transition matrix ψ is obtained by the above formulas (22), (23) and (24),
Ψ＝U -1 W (25)；
step four, calculating the spectrum radius of the Floque transition matrix, and judging the stability of the milling system by adopting the Floque theory;
and judging that the milling system is unstable when the spectral radius of the floque transition matrix ψ is larger than 1, judging that the milling system is in critical stability when the spectral radius of the floque transition matrix ψ is equal to 1, and judging that the milling system is stable when the spectral radius of the floque transition matrix ψ is smaller than 1.
2. The milling stability analysis method based on the whole discrete strategy according to claim 1, wherein in the second step, when the system is in the forced vibration process [ T ] fr ,T]At that time, for each subinterval [ t ] n ，t n+1 ]The periodic system of the state equation is approximated as a whole by using a Lagrangian interpolation function of first order or higherA number term, a status term, and a time lag term.
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