CN114952831A - Robot milling machining stability prediction method considering body structure vibration - Google Patents
Robot milling machining stability prediction method considering body structure vibration Download PDFInfo
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Abstract
The invention belongs to the field of milling, and particularly discloses a robot milling stability prediction method considering body structure vibration, which comprises the following steps: constructing a dynamic cutting force model considering damping force brought by cutter separation and time-varying process damping, and obtaining a discrete graph of a multiple time-lag cutting state according to the dynamic cutting force model and by combining a robot dynamic equation; calculating transition matrixes of all tooth through periods in a low-frequency vibration period of the robot structure according to a discrete graph of a multiple time-lag cutting state, and calculating a mode of a characteristic value of the transition matrix according to milling parameters of the robot; if the moduli of the characteristic values are all less than 1, the system is stable, otherwise the system is unstable. According to the method, a cutter workpiece separation model and a time-varying process damping model are established, a relevant time lag coefficient of a contact cutting state is provided, a milling regeneration flutter stability model considering the structural vibration of the robot body is established based on the relevant time lag coefficient, and the accurate prediction of the milling stability of the robot is realized.
Description
Technical Field
The invention belongs to the field of milling, and particularly relates to a robot milling stability prediction method considering body structure vibration.
Background
In the current research and report on the milling stability of the robot, the main shaft-cutter or robot mode is focused on the performance of different stability models and the difference of flutter at different rotating speeds. The influence of the low frequency vibrations of the structure of a weakly rigid robotic milling system on the stability has not been considered. However, the robot milling process has structural low-frequency vibration with large amplitude, and the effect of the component on the processing stability cannot be considered in the conventional dynamic model. Therefore, the action mechanism of the low-frequency vibration on the interaction force and the geometric relation of the tool and the workpiece needs to be researched, a milling stability prediction model considering the low-frequency vibration of the robot structure is established, and the prediction of the flutter-free cutting parameters of the robot is facilitated.
At present, the research on milling stability of a robot considering low-frequency vibration is quite deficient, and only Mohammadi and the like carry out related research on the influence of axial low-frequency vibration on milling regeneration vibration. The axial low-frequency vibration of the tool caused by the vibration of the robot is considered, the vibration is directly introduced into the stability solving process, and corresponding stability modeling and solving are achieved. However, since only axial vibrations are considered, there is still a large difference between the predicted results of the new model and the stability in actual machining.
Disclosure of Invention
Aiming at the defects or the improvement requirements of the prior art, the invention provides a robot milling stability prediction method considering body structure vibration, and aims to accurately predict the milling stability of a robot by considering the low-frequency vibration of the robot structure.
In order to achieve the purpose, the invention provides a robot milling stability prediction method considering body structure vibration, which comprises the following steps:
s1, obtaining a discrete graph of a multiple time-lag cutting state according to the dynamic cutting force model and by combining a robot dynamics equation;
the dynamic cutting force model is specifically as follows:
wherein, F x 、F y Dynamic cutting force in x and y directions, x (t), y (t) dynamic displacement in x and y directions at time t, x direction is a feeding direction, and y direction is a normal direction; c pd,ij (t) is the process damping coefficient, A ij Is a dynamic cutting force coefficient, B ij Damping coefficient of dynamic process, tau is time lag coefficient, i represents ith cutting element, j represents jth cutter tooth, T tp The tooth on cycle time is M, the total cutting micro-element number is N, the total cutter tooth number is N, and dz is the axial micro-element height;
for the present radial contact angle, the radial contact angle,
a unit step function of the current cutting state;
s2, calculating transition matrixes of all tooth pass cycles in the low-frequency vibration cycle of the robot structure according to the discrete graph of the multiple time-lag cutting state, and calculating a mode of a characteristic value of the transition matrix according to the milling parameters of the robot; if the moduli of the characteristic values are all less than 1, the system is stable, otherwise the system is unstable.
As a further preference, the unit step function of the current cutting state
Wherein the content of the first and second substances,
the unit step function is used for indicating whether the current cutting edge infinitesimal participates in cutting or not;
is a second unit step function for indicating whether the radial tool workpiece is separated or not.
More preferably, the first unit step function is specifically as follows:
wherein the content of the first and second substances,
for the present radial contact angle, the radial contact angle,
in order to make the angle of incidence,
to cut out the corners.
More preferably, the second unit step function is specifically as follows:
wherein h is rv,ij For the thickness change of the robot structure caused by low-frequency vibration, h ij And (t) is the current cutting thickness.
More preferably, the current cutting thickness is calculated as follows:
wherein f is t For the feed per tooth, T is the current time, T tp Is the tooth on cycle time, T rv For a single low-frequency vibration cycle time, sign"\ indicates a remainder operation.
More preferably, the skew coefficient τ is determined as follows:
wherein h is dy,ij For the variation of cutting thickness, T is the current time, T tp Is the tooth through period time, n is the number of the tooth through periods to be analyzed,
T rv for a single low-frequency vibration cycle time, sign
Indicating a ceiling operation.
As a further preference, the process damping coefficient C pd,ij (t) damping determination from a time-varying process;
the time-varying process damping calculation equation is as follows:
wherein the content of the first and second substances,
damping of time-varying processes, S, radial and tangential respectively rd,ij (t) is the time-varying total indentation area, K d Is the indentation coefficient associated with the material, α is a scaling factor representing the actual plunge, and μ represents the ratio of the tangential to radial cutting force coefficients.
More preferably, the total indentation area S varies with time rd,ij (t) is calculated as follows:
S rd,ij (t)＝S rst,ij (t)+S rdy,ij (t)
wherein S is rst,ij (t) is the time-varying dynamic indentation area, S rdy,ij (t) is the time-varying static indentation area; r is h Is the cutting edge radius of the tool, beta rs,ij Is the dynamic separation angle, gamma is the cutter relief angle; l is a radical of an alcohol rpd,ij (t) is the dynamic indentation length, r v,ij (t) represents the radial vibration velocity of the cutter teeth, V ctp Representing the cutter tooth tangential velocity.
Preferably, in step S3, a transition matrix in a single time period is constructed from a series of discrete graphs of multiple time-lag cutting states, and then the transition matrices of all tooth pass periods in the low-frequency vibration period of the robot structure are calculated.
Generally, compared with the prior art, the technical scheme conceived by the invention mainly has the following technical advantages:
1. the invention takes the low-frequency vibration of the two degrees of freedom in the tangential direction of the cutter caused by the low-frequency vibration of the robot structure into consideration, and the change of the contact state of the cutter workpiece into consideration, constructs a dynamic cutting force model taking the damping force brought by the separation of the cutter and the damping in the time-varying process into consideration, and establishes a milling regeneration flutter stability model taking the vibration of the robot body structure into consideration on the basis of the dynamic cutting force model, thereby realizing the accurate prediction of the milling stability of the robot and laying a theoretical foundation for the optimization of the robot stability.
2. The invention establishes a specific radial and tangential tool workpiece separation model and a time-varying process damping model, and describes the change mechanism of the contact state of the tool workpiece; and a time lag coefficient depending on a contact state is provided, and a stability prediction model considering the vibration of the robot body structure is established on the basis of the time lag coefficient.
3. The method researches the influence of the robot structure vibration on the narrow stable region under different postures, finds the robot posture with higher flexibility, has more obvious cutter-workpiece separation, larger time-varying process damping and more prominent narrow stable region, can accurately select the main shaft rotating speed with better stability in the narrow stable region during robot milling, and realizes the stability optimization of the robot milling.
Drawings
FIG. 1 is a schematic diagram of cutting thickness considering low-frequency vibration of a robot structure according to an embodiment of the present invention, in which (a) is the cutting thickness of the first three teeth through cycles, and (b) to (d) are the cutting times of the first to third teeth;
fig. 2 is a diagram illustrating an analysis of the influence of low-frequency vibration of a robot structure on a cutting process according to an embodiment of the present invention, where (a) is a cutting region, and (b) to (e) respectively represent T ═ T tp 、2T tp 、3T tp 、4T tp The change of the blade track and the cutting thickness at any moment;
FIG. 3 is a process damping model for low frequency vibration of a robot structure not considered in an embodiment of the present invention;
fig. 4 is a damping model of a time-varying process considering low-frequency vibration of a robot structure according to an embodiment of the present invention, where (a) is a vibration trajectory change in a whole low-frequency vibration period, (b) is an indentation model considering vibration of the robot structure, and (c) to (e) are indentation models of A, B, C three typical cutting areas considering vibration of the robot structure;
FIG. 5 is a comparison of the area of the process damping indentation area per unit radial vibration velocity for embodiments of the present invention, wherein (a) is x rv ＝y rv 15 μm, (b) x rv ＝y rv ＝5μm；
FIG. 6 is a schematic view of a cutting force model according to an embodiment of the present invention;
fig. 7 is a schematic diagram of the calculation of the stability of the robot structure in consideration of the low-frequency vibration according to the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
According to the robot milling stability prediction method considering the body structure vibration, provided by the embodiment of the invention, a tool workpiece separation model and a time-varying process damping model are established by considering the low-frequency vibration of two degrees of freedom in the tangential direction of a tool caused by the low-frequency vibration of the robot structure, the change of the contact state of the tool workpiece is considered, the relevant time lag coefficient of the contact state is provided, a milling regeneration flutter stability model considering the body structure vibration of the robot is finally established on the basis, and the robot milling stability prediction is realized.
The specific process is as follows:
1.1 tool-workpiece contact mechanism modeling considering robot structural vibration
Due to the fact that the robot structure is weak in rigidity, low-frequency vibration is prone to occurring in milling, the phenomenon that cutter workpiece separation and process damping change are possibly caused by large amplitude can occur, and the phenomenon can affect machining stability. To achieve the above object, modeling of the tool-workpiece contact mechanism considering tool-workpiece separation and process damping variation is first required.
1.1.1 tool workpiece separation
According to the difference of the low-frequency vibration direction of the robot, the robot cutter workpiece separation can be divided into radial cutter workpiece separation (RTWS) and tangential cutter workpiece separation (TTWS). Firstly, RTWS unfolding mechanism analysis and modeling are carried out.
The low-frequency vibration of the robot structure has the characteristic of long vibration period, namely a single low-frequency vibration period T rv There will be a plurality of tooth through periods T tp And the low frequency vibration has a large difference in the effect on the magnitude of the cut thickness in each tooth pass period. Therefore, after considering the low-frequency vibration of the robot structure, the stability should be analyzed with the low-frequency vibration period as a unit time.
Fig. 1 (a) is a schematic diagram of the cutting thickness under the influence of low-frequency vibration of the robot, the abscissa is the rotation angle (or time history) of the tool, the ordinate is the cutting thickness, and the number of teeth of the tool is 3 (see table 1, the parameters in table 1 are only used as examples). Is cut byThe range influence, namely the tooth-through cycle time T, is calculated by considering the cutting lag time when the cutting thickness of the robot structure under the influence of low-frequency vibration is considered tp . Since there is a possibility that the ratio of the low frequency oscillation period to the tooth pass period is not an integer, the last remainder portion in the figure will appear in general. When the robot structure is considered to vibrate at low frequency, the stability change in the whole vibration period needs to be completely analyzed, so that the last section of remainder part needs to be backwards extended to a complete tooth-through period for calculation. The number of tooth-on cycles to be analyzed and calculated finally is as follows:
TABLE 1 parameter settings
Further, for ease of understanding, the first three tooth pass periods, i.e., the first spindle rotation period, in fig. 1 (a) will be described in detail. As shown in fig. 1 (b) to (d), the first cutting tooth cutting time is when the tool rotation angle θ is 0 ° to 120 °, and so on, and the second and third cutting tooth cutting times are when θ is 120 ° to 240 ° and 240 ° to 360 °. The cutting time of each cutter tooth is one tooth-on period time T tp I.e. the lag time.
It is obvious that there is a difference in the effect of the low frequency vibration of the robot structure on each tooth pass period. Instantaneous radial contact angle
The radial vibration of the cutter teeth caused by the low-frequency vibration can be expressed as:
wherein x is rv (t) and y rv (t) is lower in the X and Y directions respectivelyAnd vibrating at a high frequency. The variation in the cut thickness caused by the radial vibration can be expressed as:
h rv,ij (t)＝r rv,ij (t)-r rv,ij (t-T tp )(0≤t＜T rv ) (3)
when the low-frequency vibration of the robot structure is not considered, the cutting thickness is the same in 9 tooth pass periods, and the cutting thickness at any time can be expressed as the feed amount f per tooth t Instantaneous radial contact angle
Function of (c):
when the robot structure is considered to vibrate at low frequency, the cutting thickness will change, and the specific expression is as follows:
h dy,ij (t)＝h ij (t)+h rv,ij (t)(0≤t＜T rv ) (5)
when the thickness of the cut caused by the low-frequency vibration of the robot structure changes h rv,ij Is negative and its absolute value is greater than the current cutting thickness h ij And (t), the cutting and contacting state of the cutter workpiece is a radial cutter workpiece separation state (RTWS), and the RTWS is mainly influenced by the low-frequency vibration size and the feeding amount of each tooth of the robot structure, as shown in formulas (3) and (4). When in RTWS, the current cutting force is zero and no cutting is occurring with the current cutter tooth. The judgment index can be expressed by the following unit step function.
For convenience of expression, the cutting contact state of the tool workpiece is classified into two types, i.e., an RTWS state in which the tool workpiece is separated and a cutting state in which the tool workpiece is not separated, according to whether or not radial tool workpiece separation occurs. In the cutting state, the cutting state is subdivided by judging whether or not radial tool workpiece separation has occurred in the previous tooth and several times of radial tool workpiece separation has occurred in succession. A cutting state in which the radial tool piece separation has not occurred in the previous tooth is referred to as a single time lag cutting State (STDC), and a cutting state in which the radial tool piece separation has occurred in the previous tooth or the radial tool piece separation has occurred in succession in the previous teeth is referred to as a multiple time lag cutting state (MTDC).
The representation in fig. 1 is used to show the contact state of the tool workpiece in different tooth pass cycles, as shown in fig. 2. Wherein the vertical axis is the cutting time process of the cutter teeth, and the horizontal axis is the rotation angle theta of the cutter. And analyzing and modeling the contact state of the cutter workpiece considering the low-frequency vibration of the robot structure by recording the cutting thickness change of different tooth through periods in the same low-frequency vibration period. In fig. 2, (b) to (e) each represent T ═ T tp ～4T tp The change of the blade path and the cutting thickness at that time correspond to the cutting areas 1 to 4 in fig. 2 (a), respectively. In the figure, a black solid line and a black dotted line respectively represent a current edge trajectory and a nominal edge trajectory, a black dotted line represents an edge trajectory at a historical time, an orange dotted line represents a workpiece surface profile, a shaded part represents a cut region, and a grid part is a cut region at the current time. For convenience of understanding, in fig. 2, (b) to (e) show the entry point and the contact point of the same cutting path of the blade by the same color, and different colors show different cutting paths of the blade. When not in the RTWS state, in which the tool workpiece contact state is in the cutting state, the two cutter tooth cutting trajectories constituting the cutting region are marked below the cutting state notation in the drawing. As shown in fig. 2 (c), the green and purple cutter cutting trajectories constitute the cutting thickness at this time.
The cutting region 1 was STDC, and the cutting thickness pattern is shown in fig. 2 (b). The kth cutter tooth and the (k +1) th cutter tooth are not separated from each other in the radial direction, the current cutting force is not zero, the cutting thickness is influenced by the vibration of the two adjacent cutter teeth (the kth cutter tooth and the (k +1) th cutter tooth), and the time lag time is a single tooth pass cycle. This state only needs to take into account the effect of low frequency vibrations on the cut thickness.
The cutting region 2 has two cutting states, STDC and RTWS, and the cutting thickness model is shown in fig. 2 (c). The radial tool piece separation occurs in the second half of the cutting process for the first tooth k +2, at which point the cutting force is zero.
The cutting region 3 is RTWS, and the cutting thickness pattern is shown in fig. 2 (d). Radial tool workpiece separation occurs at the (k + 3) th tooth, at which point the cutting force is zero.
The cutting region 4 was MTDC, and the cutting thickness model is shown in fig. 2 (e). The cutting thickness is influenced by the vibration of the (k +4) th cutter tooth and the (k +2) th cutter tooth, and the time lag time is ((k +4) - (k + 2)). T tp ＝2T tp . Radial cutter workpiece separation occurs on the k +3 th cutter tooth and the k +2 th cutter tooth in the second half, the cutting surface of the k +4 th cutter tooth is the cutting surface left by the k +1 th cutter tooth which is closest to the cutter tooth without radial cutter workpiece separation, the cutting thickness is influenced by vibration of the k +4 th cutter tooth and the k +1 th cutter tooth, and the time lag time is 3T tp . By analogy, when the jth cutter tooth is not separated from the radial cutter workpiece, and the jth cutter tooth from the jth cutter tooth to the jth cutter tooth from the jth-1 cutter tooth to the jth cutter tooth are separated from the radial cutter workpiece, the current cutting force is not zero, the cutting thickness is influenced by the vibration of the jth cutter tooth and the jth cutter tooth from the jth cutter tooth to the jth cutter tooth from the jth tooth tp ＝(n+1)T tp 。
In summary, if the radial tool workpiece separation phenomenon continuously occurs in the previous n (n ≧ 1) periods of the current cutting period, the workpiece surface to be cut at this time is caused by the cutting process with the interval of n +1 periods, and the lag time of the cutting at this time is n +1 times of the tooth pass period, i.e., (n +1) T tp . The ratio of the lag time of the current cutting to the tooth through time is expressed as a lag coefficient tau, and the corresponding judgment function is as follows:
according to formula (6) and formula (7), when g 2 When the cutting force is equal to 0, the radial cutter workpiece separation phenomenon occurs at the moment, and the cutting force is zero; when g is 2 When the tau is more than or equal to 1, the cutting thickness is determined by the current cutting edgeThe cutting path of the cutting edge which is closest to the cutting edge and is not separated from the radial tool workpiece is determined, the time lag time is determined by an equation (7), and the corresponding stability modeling and solving process is detailed in the subsequent section 1.2.
1.1.2 time-varying Process damping
The process damping is a damping force generated when a tool contacts with a vibration pattern on the surface of a workpiece in low-speed milling, and generally represents an inhibiting effect on machining vibration. Since the magnitude of process damping is inversely proportional to the tooth tangential velocity, it is generally only considered in low speed machining. However, low frequency vibrations are present in the robot process, which will cause the process damping setback region to change, thereby affecting the magnitude of the process damping. Therefore, low frequency vibrations of the robot structure may cause process damping phenomena in machining at higher rotational speeds.
In the process damping model, the process damping force is modeled as a function of the indented region, and the radial and tangential process damping are expressed as follows:
wherein S is d,ij (t) is the volume of the indentation area, which is the product of the area of the indentation area and the cut width; k d Is the indentation coefficient associated with the material; alpha is a scale factor representing the actual amount of retraction; mu represents the ratio of tangential to radial cutting force coefficients, where K d α, μ are constant terms related to experimental conditions.
Indentation area S d,ij Divided into static indentation zones S st (t) and dynamic indentation area S dy,ij (t) as shown in FIG. 3. Both regions have an influence on the processing stability. When the low frequency vibration is not considered, the separation point SP is determined only by the structure of the tool, and the calculation formula of the static indentation area and the dynamic indentation area is as follows:
S d,ij ＝S st +S dy,ij (11)
wherein r is v,ij Indicating the radial vibration speed, L, of the cutter teeth pd For the indentation length, as shown in fig. 3, the specific calculation expression is as follows:
L pd ＝r h [sinβ s +sinγ+(cosγ-cosβ s )/tanγ] (13)
wherein r is h Is the radius of the cutting edge of the tool, beta s Is a separation angle for defining a Separation Point (SP), gamma is a tool relief angle, V ctp Is the tangential velocity of the cutter teeth.
When the robot structure is considered to vibrate at a low frequency, the separation point SP changes along with the change of the separation angle, and the indentation area changes dynamically and has different changes in different cutter tooth periods. As shown in fig. 4, the influence of the low-frequency vibration of the robot structure on the separation point SP and the dynamic indentation area in different tooth pass periods is described by describing the vibration trajectory change in the whole low-frequency vibration period. Wherein the gray dashed line represents the blade trajectory without considering the low frequency vibration of the robot structure, the black solid line represents the blade trajectory with considering the low frequency vibration of the robot structure, and the orange dashed line represents the low frequency vibration trajectory of the robot structure. A, B, C three typical cutting areas were selected for analysis, and the detail diagrams are shown in fig. 4 (b) - (e).
In the cutting area A, the low-frequency vibration of the robot structure causes the cutter to be pressed into the surface of a workpiece, and the separation angle beta s The separation point SP moves upward as the size increases, and the indentation area increases accordingly. At this time, the process damping increases, and the pair of indentation patterns is shown in fig. 4 (b) and (c). In the cutting area B, the vibration amplitude of the robot structure is small, and the influence on the area of the indentation area is small. At the moment, the process damping can be approximately regarded as unchanged, and the impression model is comparedAs shown in fig. 4 (b) and (d). In the cutting area C, the low frequency vibration of the robot structure causes the tool to be far away from the surface of the workpiece, and the separation angle beta s When the separation point SP becomes smaller, the indentation area becomes smaller. At this time, the process damping is reduced, and the impression model pair is shown in fig. 4 (b) and (e).
Using the parameters in table 1, the area of the indentation region with and without considering the low-frequency vibration of the robot structure is calculated at a unit radial vibration velocity, and the calculation result is shown in fig. 5 (a). The figure shows the area of the indentation region at a low frequency vibration period (T) rv ) The process of change in. Corresponding to (a) in FIG. 4, at T rv In the first half of the process, the low-frequency vibration of the robot structure causes the cutter to be pressed into the surface of the workpiece, the area of an indentation area is increased, and the process damping is increased. At T rv In the second half of the process, the low-frequency vibration of the robot structure causes the cutter to be far away from the surface of the workpiece, the area of the indentation area is reduced, and the process damping is reduced. When the low frequency vibration amplitude is reduced to 5 μm, the influence of the low frequency vibration on the area of the indentation region is also reduced, as shown in (b) of fig. 5. Therefore, the damping in the time-varying process is related to the low-frequency vibration amplitude of the robot, and when the low-frequency vibration amplitude is larger, the damping in the time-varying process is larger, so that the stability boundary is also influenced more obviously.
In summary, the separation angle β depends on the low-frequency vibration of the robot structure s And length L of indentation pd Will be changed accordingly. Dynamic separation angle beta rs,ij (t) and dynamic indentation Length L rpd,ij (t) the computational expression is as follows:
L rpd,ij (t)＝r h [sin(β rs,ij (t))+sinγ+(cosγ-cos(β rs,ij (t)))/tanγ] (15)
wherein r is rv,ij And (t) represents the radial vibration of the cutter teeth caused by low-frequency vibration, as shown in the formula (2). Influence of low-frequency vibration of robot structure on damping in milling process and magnitude and square of radial vibration of robotIn the correlation, the area expression of the damping indentation area in the time-varying process considering the low-frequency vibration of the robot structure is as follows:
S rd,ij (t)＝S rst,ij (t)+S rdy,ij (t) (18)
wherein S is rst,ij (t) is a time-varying dynamic indentation area, S rdy,ij (t) is a time-varying static indentation area, S rd,ij (t) is the time-varying total indentation area. r is v,ij (t) represents the radial vibration velocity of the cutter teeth, V ctp Indicating the tangential velocity (V) of the cutter teeth ctp 2 pi nR/60, n is the rotational speed, R is the tool radius). Therefore, the calculation expression for the damping of the time-varying process considering the low-frequency vibration of the robot structure is as follows:
1.2 stability modeling of structural vibration effects of a robot
1.2.1 dynamic cutting force modeling
As shown in fig. 6, when the low frequency vibration of the robot structure is not considered, the dynamic cut-thickness model is as follows:
wherein the content of the first and second substances,
is the instantaneous radial contact angle, x (T), y (T) represents the dynamic displacement in the x and y directions, respectively, T tp Representing the tooth on cycle time.
As analyzed in section 1.1.1, the occurrence of a cutter separation will change the way the dynamic cutting force is calculated. When the cutter separation occurs, the cutting force of the current infinitesimal is 0, the combined type cutting force is combined with the radial and axial dynamic cutting force infinitesimal expression as follows:
wherein the content of the first and second substances,
is a unit step function and is used for indicating whether the current cutting edge infinitesimal participates in cutting or not. The expression is as follows:
wherein the content of the first and second substances,
and
respectively, the cutting-in and cutting-out angles. On the other hand, in calculating the dynamic cut thickness, it is necessary to consider the influence of a time lag coefficient depending on the cutting state due to the separation of the cutting tools. The combined type (7) and the formula (20) consider the dynamic shear thickness expression of the low-frequency vibration of the robot structure as follows:
wherein, tau is a time lag coefficient, and the judgment function is shown as the formula (7). In summary, the dynamic cutting force infinitesimal calculation expression considering the low-frequency vibration of the robot structure is as follows:
as analyzed in section 1.1.2, the magnitude of the damping force in the time-varying process will change with the change of the low-frequency vibration of the robot structure in the cutting process, and considering the damping force brought by the damping in the time-varying process as shown in equation (19), the radial and axial dynamic cutting force infinitesimal expression in equation (24) is modified as follows:
through coordinate transformation, integration along the axial direction and summation of each cutter tooth, the following dynamic cutting force components in a rectangular coordinate system can be obtained:
the combined type (23), the formula (25) and the formula (26), the final dynamic cutting force can be expressed as follows:
wherein:
for ease of expression, equation (27) is rewritten as follows:
wherein w represents the cut width:
1.2.2 stability modeling
The dynamic equation considering the low-frequency vibration effect of the robot structure is as follows:
wherein m is ij (i, j ═ x, y) represents the modal mass of the system excited in the j direction, with the response in the i direction; c. C ij (i, j ═ x, y) represents the modal damping of the system excitation in the j direction, i direction response; k is a radical of ij And (i, j ═ x, y) represents the modal stiffness of the system excited in the j direction and responded in the i direction.
The combination of (32) and (34) can be obtained:
wherein q (t) ═ x (t) y (t)] T M, C and K are modal mass, damping and rigidity matrixes, and tau is a time lag coefficient.
Based on a fully discrete solution idea, a dynamic equation is rewritten into a state space form, a tau time lag phenomenon caused by cutter separation is considered, and the formula (35) can be rewritten into the following form:
x(t)＝A 0 x(t)+A(t)x(t)-A(t)x(t-τT tp ) (36)
wherein:
discretizing a kinetic equation, and performing a tooth penetration period T by adopting a direct integration method and a linear approximation method tp Equally dividing into m discrete times at intervals of tau, and within each time interval k tau is not more than t not more than (k +1) tau (k is 0, …, m), x can be obtained k+1 The expression of (a) is as follows:
wherein:
since a (T) is a periodic function, i.e., a (T) ═ a (T-T) tp )＝A(t - τT tp ) Thus, there is equation F in the formula m-1 ＝F τm-1 ,F m ＝F τm . As can be seen from equation (40), the definition of the discrete graph is:
wherein the content of the first and second substances,
a discrete graph showing a single-lag cut state,
a dispersion map representing a multiple time-lag cutting state, the corresponding expression being as follows:
y k ＝col(x k x k-1 … x k+1-m x k-m … x k+1-2m x k-2m … x k+1-τm x k-τm ) (42)
and constructing a transition matrix phi in a single time period through a series of discrete graphs, and judging the stability of the system in the current gear-through period by judging whether the modulus of the characteristic value of the transition matrix is less than 1. The transition matrix expression is as follows:
as shown in fig. 1, since the low-frequency vibration cycle time of the robot structure is longer than the tooth-through cycle time, the transition matrix of a single tooth-through cycle cannot reflect the influence of the vibration of the robot structure on the dynamic cutting force. Therefore, as shown in fig. 7, when the vibration of the robot structure is considered, the transition matrix Φ of all the tooth-through periods in the low-frequency vibration period of the robot structure needs to be calculated according to equation (45) η And judging the system stability according to the eigenvalues of all the transition matrixes.
The invention introduces the establishment process of the milling stability prediction model of the vibration robot considering the body structure and a milling stability judgment formula convenient for the model in detail. When stability prediction is carried out through the robot milling device, firstly, milling parameters of the robot are collected or preset, and then the following steps are carried out:
s1, calculating and considering dynamic cutting force of the damping force brought by cutter separation and time-varying process damping according to the formula (27);
s2, considering dynamic cutting force, establishing a dynamic equation considering the low-frequency vibration influence of the robot structure, and further deducing a specific formula of a discrete graph of a multiple time-lag cutting state
S3, according to the discrete graph
Is calculated by the specific formula (46) of the transition matrix phi of all the tooth-through periods in the low-frequency vibration period of the robot structure η And calculating the moduli of the eigenvalues of all the transition matrixes, and according to the Floquet theory, if the moduli of the eigenvalues are all smaller than 1, the system is stable, otherwise, the system is unstable.
In conclusion, the invention considers the low-frequency vibration of the robot structure, establishes a cutter workpiece separation model and a time-varying process damping model, provides a relevant time lag coefficient of a contact cutting state, and analyzes the influence of the low-frequency vibration of the robot structure on the regenerative chatter vibration based on the time lag coefficient. Considering the change of the contact state of a cutter workpiece, and establishing a milling regeneration flutter stability model considering the structural vibration of the robot body based on the relevant time lag coefficient of the contact state. And (3) carrying out experimental verification and analysis on the established stability model by combining a milling experiment, comparing stability lobe graphs of different postures by combining the pose dependence of robot structure vibration, and providing guidance for selecting the rotation speed posture for optimizing the stability of the robot.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (9)
1. A robot milling processing stability prediction method considering body structure vibration is characterized by comprising the following steps:
s1, obtaining a discrete graph of a multiple time-lag cutting state according to the dynamic cutting force model and by combining a robot dynamics equation;
the dynamic cutting force model is specifically as follows:
wherein, F x 、F y Dynamic cutting force in x and y directions, x (t), y (t) dynamic displacement in x and y directions at time t, wherein x direction is normal direction, and y direction is feeding direction; c pd,ij (t) is the process damping coefficient, A ij Is a dynamic coefficient of cutting force, B ij Is a dynamic process damping coefficient, tau is a time lag coefficient, i represents the ith cutting element, j represents the jth cutter tooth, T tp The tooth on cycle time is, M is the total cutting infinitesimal number, N is the total cutter tooth number, and dz is the axial infinitesimal height;
for the present radial contact angle, the radial contact angle,
a unit step function of the current cutting state;
s2, calculating transition matrixes of all tooth pass cycles in the low-frequency vibration cycle of the robot structure according to the discrete graph of the multiple time-lag cutting state, and calculating a mode of a characteristic value of the transition matrix according to the milling parameters of the robot; if the moduli of the characteristic values are all smaller than 1, the system is stable, otherwise the system is unstable.
2. The method for predicting milling process stability of a robot considering vibration of a body structure as set forth in claim 1, wherein a unit step function of a current cutting state
Wherein the content of the first and second substances,
the unit step function is used for indicating whether the current cutting edge infinitesimal participates in cutting or not;
is a second unit step function for indicating whether the radial tool workpiece is separated or not.
3. The method for predicting the milling stability of a robot considering the vibration of the body structure as claimed in claim 2, wherein the first unit step function is specifically as follows:
wherein the content of the first and second substances,
for the present radial contact angle, the radial contact angle,
in order to make the angle of incidence,
to cut out the corners.
4. The method for predicting the milling stability of a robot considering the vibration of the body structure as claimed in claim 2, wherein the second unit step function is specifically as follows:
wherein h is rv,ij For the thickness change of the robot structure caused by low-frequency vibration, h ij And (t) is the current cutting thickness.
5. A method for predicting milling stability of a robot considering vibration of a body structure as set forth in claim 4, wherein the current cutting thickness is calculated as follows:
wherein f is t For the feed per tooth, T is the current time, T tp Is the tooth on cycle time, T rv The symbol "\\" represents a remainder operation for a single low frequency vibration cycle time.
6. The method for predicting the milling stability of a robot considering the vibration of the body structure as set forth in claim 1, wherein the time lag coefficient τ is determined as follows:
wherein h is dy,ij For the variation of cutting thickness, T is the current time, T tp Is the tooth on period time, n is the number of tooth on periods to be analyzed,
T rv for a single low-frequency vibration cycle time, sign
Indicating a ceiling operation.
7. The method for predicting milling process stability of robot considering vibration of body structure as set forth in claim 1, wherein a process damping coefficient C pd,ij (t) damping determination from a time-varying process;
the time-varying process damping calculation is as follows:
wherein the content of the first and second substances,
are respectively radial,Tangential time-varying process damping, S rd,ij (t) is the time-varying total indentation area, K d Is the indentation coefficient associated with the material, alpha is a scaling factor representing the actual plunge, and mu represents the tangential to radial cutting force coefficient ratio.
8. The method for predicting milling process stability of robot considering vibration of body structure as set forth in claim 7, wherein the total indentation area S varies in time rd,ij (t) is calculated as follows:
S rd,ij (t)＝S rst,ij (t)+S rdy,ij (t)
wherein S is rst,ij (t) is the time-varying dynamic indentation area, S rdy,ij (t) is the time-varying static indentation area; r is h Is the cutting edge radius of the tool, beta rs,ij Is the dynamic separation angle, gamma is the cutter relief angle; l is rpd,ij (t) is the dynamic indentation length, r v,ij (t) represents the radial vibration velocity of the cutter teeth, V ctp Representing the cutter tooth tangential velocity.
9. A robot milling machining stability prediction method considering body structure vibration as claimed in any one of claims 1-8, characterized in that in step S3, a transition matrix in a single time period is constructed through a series of discrete graphs of multiple time-lag cutting states, and then the transition matrix of all tooth pass periods in the low-frequency vibration period of the robot structure is calculated.
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