CN112149239A - Thin-walled cylinder mirror image cutting modeling method based on shell theory - Google Patents
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Abstract
The invention discloses a shell theory-based thin-walled cylinder mirror image cutting modeling method, which comprises the following steps of: step one, simplifying a mirror image cutting process system of the thin-wall cylinder; the turning tool is equivalent to a rigid body, the boring bar and the thin-wall cylinder are equivalent to a flexible body, and the thin-wall cylinder of a processing object can look like a cylindrical shell; fixing a workpiece, neglecting centripetal force and Coriolis force when the part rotates, and simplifying the machining process into forced vibration of a static cylindrical shell under the excitation of rotating force; step two, establishing a forced vibration equation of the rotary thin-walled cylinder by applying a shell theory; step three, establishing a boring bar vibration dynamics equation; establishing a dynamic cutting force model of the cutting process system; and step five, establishing a thin-wall barrel mirror image cutting dynamic model, and calculating and solving by using a Longge Kutta method.
Description
Technical Field
The invention relates to the field of metal material cutting processing. In particular to a thin-wall cylinder mirror image cutting modeling method based on a shell theory.
Background
At present, thin-wall parts represented by engine blades and thin-wall cylinders have wide application in the fields of aerospace, nuclear industry and the like due to the advantages of light weight, compact structure, superior comprehensive performance and the like. Generally, thin-wall parts are often required to have high machining precision and surface quality, and the rigidity of a thin-wall structure is poor, so that large deformation and vibration can be generated in the machining process. In addition, the machining system can vibrate under certain working conditions, and destructive influences can be generated on the quality of a machined surface, the service life of a cutter and the precision of a machine tool. Therefore, stability studies during machining of thin-walled parts are necessary.
By carrying out dynamic modeling on the cutting process system, the machining state of the cutting process system can be predicted in a theoretical calculation mode, the selection of machining process parameters is guided, and a theoretical basis is provided for the implementation of a vibration suppression scheme.
Currently, the research on mirror image machining of thin-wall parts is mainly focused on milling, and the research on turning and boring is less. Due to the influence of the complex modal shape of workpiece rotation and the thin-wall part, the response of a mirror image processing system of the thin-wall part is more complex, and the existing modeling method cannot well represent the vibration characteristic of the thin-wall cylinder part.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provides a shell theory-based thin-wall barrel mirror image cutting modeling method. The modeling method can well show the vibration characteristics of the thin-wall cylinder, and breaks through the limitation of the traditional mass-damping-rigidity unit and beam theoretical modeling.
The purpose of the invention is realized by the following technical scheme:
a thin-wall cylinder mirror image cutting modeling method based on a shell theory comprises the following steps:
step one, simplifying a mirror image cutting process system of the thin-wall cylinder; the turning tool is equivalent to a rigid body, the boring rod and the thin-wall cylinder are equivalent to a flexible body, and the thin-wall cylinder of a processing object is regarded as a cylindrical shell; fixing the thin-walled cylinder workpiece, neglecting centripetal force and Coriolis force when the thin-walled cylinder workpiece rotates, and simplifying the machining process into forced vibration of a static cylindrical shell under the excitation of rotating force;
step two, establishing a forced vibration equation of the rotary thin-walled cylinder by applying a shell theory;
step three, establishing a boring bar vibration dynamics equation;
establishing a dynamic cutting force model of the cutting process system;
and step five, establishing a thin-wall barrel mirror image cutting dynamic model, and calculating and solving by using a Longge Kutta method.
Further, the second step is as follows:
according to the Love shell theory, a rotating thin-wall barrel forced vibration equation is established:
the thin-wall cylinder is regarded as a cylindrical shell, and the forced vibration equation of the cylindrical shell is as follows:
wherein c is the equivalent damping coefficient, ρ is the material density, P is the external excitation, L is the Love operator, expressed as:
wherein U is the three-dimensional vibration displacement of the particles on the surface of the cylindrical shell, and P is the three-way acting force acting on the surface of the cylindrical shell, namely external excitation; z and theta are positions of particles on the cylindrical shell, and t is time; the specific expression for the L operator is as follows:
wherein E is Young's modulus, mu is Poisson's ratio,
is the stiffness of the membrane of the cylindrical shell,
is the bending stiffness of the cylindrical shell, wherein H is the thickness of the cylindrical shell;
the analytical solution for the vibration U can be expressed as a combination of an axial beam function and a circumferential trigonometric function, as follows:
wherein m is the axial half wave number, n is the circumferential wave number,
is a vibration amplitude coefficient, TmnIs the vibration response under the modal coordinate;
wherein λ ism、CmThe value of (A) is determined by boundary conditions, and can be obtained by fitting the curve by a modal experiment;
converting the machining process into forced vibration of the static cylindrical shell under the rotary load;
the radial cutting force is expressed as:
pz(z,θ,t)＝f(t)(z-z*)(θ-θ*) (7)
for the Dirac function:
z and θ denote the position of the cutting point on the workpiece;
taking equations (3) to (6) into equation (1), a second order differential equation in the main resonance state is obtained:
wherein, ω iswmnIs the natural frequency, xi, of the modewIs modal damping ratio, Fwmn(t) is the external force expressed as:
Mwmnis the modal mass;
the equation translates to the M-C-K form:
wherein,
furthermore, in the third step, the boring bar is simplified into a mass-damping-rigidity model, and each modal parameter is obtained by a frequency response test.
Further, the fourth step is as follows:
radial cutting force F of excircletAnd inner circle radial cutting force FbThe mechanical model of (a) is expressed as:
in the formula, KtcIs the radial cutting force coefficient, K, of the cylindrical turning toolbcThe radial cutting force coefficient of the inner circle boring bar, fcIs the feed amount; the dynamic cut thickness equation is defined as:
in the formula, xw(t) is the instantaneous radial displacement of the workpiece, xw(T-T) is the radial displacement of the previous cut revolution of the workpiece, xb(t) is the instantaneous radial displacement of the boring bar, xb(T-T) is the radial displacement of the previous cutting revolution of the boring bar; a ispt(t) is the actual depth of cut of the cylindrical turning, apbAnd (t) is the actual cutting depth of the inner circle boring. Substituting equation (13) into equation (12), the radial dynamic cutting force formula can be expressed as:
further, the step five is specifically as follows:
during the cutting process, the feeding amount and the spindle rotating speed are kept unchanged, so the position of the cutting point on the surface of the workpiece is represented as follows:
wherein N is the spindle speed.
Note xwFor vibration of the surface of the workpiece in modal coordinates, xbDeducing the vibration amplitude of the workpiece and the boring bar for the vibration of the boring cutter under the modal coordinate:
Ww(t) is the mode shape function of the workpiece at the cutting point expressed as:
coupling the workpiece, the boring bar and the cutting force equation to obtain a dynamic equation of the mirror-oriented processing technology system:
with the nominal depth of cut as input and the vibration of the workpiece and the boring bar as output, the following form is obtained:
wherein,
the second order differential equation is rewritten as a state space representation:
wherein,
the equation solves the output response just according to the Runge Kutta method, and solves the stability by using a semi-discrete method or a full-discrete method.
Compared with the prior art, the technical scheme of the invention has the following beneficial effects:
the invention considers the complex modal shape of the thin-wall cylinder in 2 directions of the circumferential direction and the axial direction and the rotation of a workpiece, and converts the turning/boring processing of the thin-wall cylinder into the vibration response of the cylindrical shell under the rotation load. The results show that the calculated vibration signals have good consistency with the experimental test signals in terms of amplitude and frequency components, including the high frequency of the workpiece self-excited vibration at the natural frequency and the low frequency of the workpiece modal shape coupled with the rotation effect excitation. Compared with the existing cutting process modeling method, the modeling method not only simplifies part of workpieces near the cutting point into simple vibration units, but also introduces a shell theory, considers the vibration characteristic of the thin-walled cylinder and the cutting force change caused by workpiece rotation, and enables the established mathematical model to be closer to a real machining process; the low-frequency vibration caused by the rotation of the workpiece appears on the calculation result, and the experimental test result is more fit. The modeling method can simulate the vibration response of the thin-wall cylindrical part under the mirror image processing technology more truly.
Drawings
FIG. 1 is a schematic diagram of the components of a thin-walled cylinder mirror image cutting process system of the present invention;
FIG. 2 is a simplified model of a thin-walled cylinder mirror image cutting process system of the present invention;
FIG. 3 is a simplified schematic of the thin-walled cylinder of the present invention;
FIG. 4 is a frequency response function of a workpiece in an embodiment of the invention;
FIG. 5 is a comparison of the estimated value and the measured value of the axial mode shape of the workpiece according to the embodiment of the present invention;
FIG. 6 is a frequency response function of the boring bar in an embodiment of the present invention;
FIG. 7 is a time domain comparison of a vibration signal obtained from numerical calculations and experiments in an embodiment of the present invention;
FIG. 8 is a frequency domain comparison of the vibration signal obtained by numerical calculation and experiment in the example of the present invention.
Reference numerals: 1-thin-wall cylinder, 2-boring rod, 3-turning tool, 4-machine tool spindle and 5-machine tool tail top.
Detailed Description
The thin-wall barrel mirror image cutting modeling method based on the shell theory is described in detail below by combining the embodiment and the attached drawings.
As shown in figure 1, the thin-wall mirror image processing process system mainly comprises an external turning tool 3, an internal boring rod 2, a thin-wall cylinder 1, a machine tool spindle 4 and a machine tool tail top 5. The rigidity of the external turning tool 3 is far higher than that of the cantilever boring bar 2 and the thin-wall cylinder 1, and the vibration is small in the cutting system, so that the external turning tool is equivalent to a rigid body, and the internal boring bar and the thin-wall workpiece are equivalent to a flexible body, and the simplified model shown in the figure 2 is obtained.
Modeling the system dynamics of the workpiece:
1. simplified model of a workpiece system
The processing object of the process system is a thin-wall cylinder, the structure is simple, and theoretical modeling can be performed by utilizing a shell vibration theory, as shown in figure 3. The machining process of the part is forced vibration of a rotating cylindrical shell, the workpiece is fixed, the centripetal force and Coriolis force generated when the part rotates are ignored, and the machining process is simplified into forced vibration of a static cylindrical shell under the excitation of rotating force. The clamping modes of the two ends of the part are respectively that the part is fixed on the tail top support through threads, and the boundary conditions of the two ends of the cylindrical shell are simplified, one end of the cylindrical shell is fixed, and the other end of the cylindrical shell is simply supported in consideration of constraint relation.
2. According to Love shell theory, the thin-walled cylinder vibration equation is as follows:
wherein c is the equivalent damping coefficient, ρ is the material density, P is the external excitation, L is the Love operator, expressed as:
where U is the three-dimensional vibrational displacement of the particles on the surface of the cylindrical shell and P is the three-way force acting on the surface of the cylindrical shell, i.e., the external excitation described above. z, θ are the locations of the particles on the cylindrical shell, and t is time. The specific expression for the L operator is as follows:
wherein E is Young's modulus, mu is Poisson's ratio,
is the stiffness of the membrane of the cylindrical shell,
is the bending stiffness of the cylindrical shell, and H is the thickness of the cylindrical shell.
Solving the inherent characteristics of the cylindrical shell;
the analytical solution for the vibration U can be expressed as a combination of an axial beam function and a circumferential trigonometric function, as follows:
wherein m is the axial half wave number, n is the circumferential wave number,
is a vibration amplitude coefficient, TmnIs the vibration response in modal coordinates.
wherein λ ism、CmThe value of (c) is determined by the boundary conditions and can be obtained by fitting a modal experiment to the curve. The thin-walled cylinder axial mode shape is obtained by performing a mode test to verify the boundary condition assumption:
fig. 4 is a result of a modal test, a main peak at 680Hz is taken for curve fitting, and a fitted curve and a theoretical calculation curve are compared as shown in fig. 5, which have good consistency. Solving the inherent characteristic of the thin-wall cylinder in a main vibration state by using a beam function method to obtain a result, wherein the axial half wave number m is 1, and the circumferential wave number n is 2; the natural frequency 681Hz and the modal mass 0.26 Kg.
Converting the machining process into forced vibration of the static cylindrical shell under a rotary load, and establishing a thin-wall cylindrical shell dynamic equation under the rotary load;
the radial vibration has the greatest influence on the machining precision in the machining process, so that the radial cutting force is taken as a main research object. The radial cutting force is expressed as:
pz(z,θ,t)＝f(t)(z-z*)(θ-θ*) (7)
for the Dirac function:
z and theta denote the position of the cutting point on the workpiece.
Taking equations (3) to (6) into equation (1), a second order differential equation in the main resonance state is obtained:
wherein, ω iswmnIs the natural frequency, xi, of the modewIs modal damping ratio, Fwmn(t) is the external force expressed as:
Mwmnis the modal mass.
The equation translates to the M-C-K form:
wherein,
3. dynamic modeling of the boring bar:
the boring bar is complex in structure, and cannot be accurately described by a beam theory, so that the boring bar is simplified into a mass-damping-rigidity model, and each modal parameter is obtained by a frequency response test. The frequency response test result is shown in fig. 6, and the modal parameters of the boring bar obtained through modal feature identification are shown in table 1.
TABLE 1 Modal parameters of boring bar
4. Modeling the dynamic cutting force of the cutting process system;
radial cutting force F of excircletAnd inner circle radial cutting force FbMay be expressed as.
In the formula, KtcIs the radial cutting force coefficient, K, of the cylindrical turning toolbcThe radial cutting force coefficient of the inner circle boring bar, fcIs the feed amount. During the cutting process, the vibration of the boring bar and the workpiece can change the cutting thickness, and the surface vibration lines generated by the vibration of the previous rotation can also cause the change of the cutting thickness, thereby causing the dynamic change of the cutting force. The dynamic cut thickness equation can be defined as:
in the formula, xw(t) is the instantaneous radial displacement of the workpiece, xw(T-T) is the radial displacement of the previous cut revolution of the workpiece, xb(t) is the instantaneous radial displacement of the boring bar, xb(T-T) is the radial displacement of the previous cutting revolution of the boring bar; a ispt(t) is the actual depth of cut of the cylindrical turning, apbAnd (t) is the actual cutting depth of the inner circle boring. Substituting equation (13) into equation (12), the radial dynamic cutting force formula can be expressed as:
5. dynamic modeling of a cutting process system;
during the cutting process, the feeding amount and the spindle rotating speed are kept unchanged, so the position of the cutting point on the surface of the workpiece can be represented as follows:
wherein N is the spindle speed.
Note xwFor vibration of the surface of the workpiece in modal coordinates, xbDeducing the vibration amplitude of the workpiece and the boring bar for the vibration of the boring cutter under the modal coordinate:
Ww(t) is the mode shape function of the workpiece at the cutting point expressed as:
coupling the workpiece, the boring bar and the cutting force equation to obtain a dynamic equation of the mirror-oriented processing technology system:
with the nominal depth of cut as input and the vibration of the workpiece and the boring bar as output, the following form is obtained:
wherein,
the second order differential equation is rewritten as a state space representation:
wherein,
the equation can solve the output response just by the Runge Kutta method, and the stability is solved by a semi-discrete method or a full-discrete method.
Cutting experiments were performed, and the experimental results were compared with the numerical calculation results, and the set experimental parameters are shown in table 2, and the comparison results are shown in fig. 7 and 8.
TABLE 2 processing parameters
As shown in fig. 7 and 8, the result obtained by the modeling method in the present invention has better consistency with the result obtained by the cutting experiment, the vibration of the low frequency band is mainly concentrated at the frequency doubling position of the frequency conversion, the vibration of the high frequency band is mainly concentrated at the natural frequency position of the workpiece, and the interference of impact, noise, etc. received in the actual processing is received, and the simulation result and the experiment result have a certain difference in amplitude. The modeling method can reflect the vibration condition of the thin-wall cylinder in the machining process more truly.
The present invention is not limited to the above-described embodiments. The foregoing description of the specific embodiments is intended to describe and illustrate the technical solutions of the present invention, and the above specific embodiments are merely illustrative and not restrictive. Those skilled in the art can make many changes and modifications to the invention without departing from the spirit and scope of the invention as defined in the appended claims.
Claims (5)
1. A thin-wall cylinder mirror image cutting modeling method based on a shell theory is characterized by comprising the following steps:
step one, simplifying a mirror image cutting process system of the thin-wall cylinder; the turning tool is equivalent to a rigid body, the boring rod and the thin-wall cylinder are equivalent to a flexible body, and the thin-wall cylinder of a processing object is regarded as a cylindrical shell; fixing the thin-walled cylinder workpiece, neglecting centripetal force and Coriolis force when the thin-walled cylinder workpiece rotates, and simplifying the machining process into forced vibration of a static cylindrical shell under the excitation of rotating force;
step two, establishing a forced vibration equation of the rotary thin-walled cylinder by applying a shell theory;
step three, establishing a boring bar vibration dynamics equation;
establishing a dynamic cutting force model of the cutting process system;
and step five, establishing a thin-wall barrel mirror image cutting dynamic model, and calculating and solving by using a Longge Kutta method.
2. The shell theory-based thin-walled cylinder mirror image cutting modeling method according to claim 1, wherein in the second step, a rotating thin-walled cylinder forced vibration equation is established according to Love shell theory:
the thin-wall cylinder is regarded as a cylindrical shell, and the forced vibration equation of the cylindrical shell is as follows:
wherein c is the equivalent damping coefficient, ρ is the material density, P is the external excitation, L is the Love operator, expressed as:
wherein U is the three-dimensional vibration displacement of the particles on the surface of the cylindrical shell, and P is the three-way acting force acting on the surface of the cylindrical shell, namely external excitation; z and theta are positions of particles on the cylindrical shell, and t is time; the specific expression for the L operator is as follows:
wherein E is Young's modulus, mu is Poisson's ratio,
is the stiffness of the membrane of the cylindrical shell,
is the bending stiffness of the cylindrical shell, wherein H is the thickness of the cylindrical shell;
the analytical solution for the vibration U can be expressed as a combination of an axial beam function and a circumferential trigonometric function, as follows:
wherein m is the axial half wave number, n is the circumferential wave number,
is a vibration amplitude coefficient, TmnIs the vibration response under the modal coordinate;
wherein λ ism、CmThe value of (A) is determined by boundary conditions, and can be obtained by fitting the curve by a modal experiment;
converting the machining process into forced vibration of the static cylindrical shell under the rotary load;
the radial cutting force is expressed as:
pz(z,θ,t)＝f(t)(z-z*)(θ-θ*) (7)
for the Dirac function:
z and θ denote the position of the cutting point on the workpiece;
taking equations (3) to (6) into equation (1), a second order differential equation in the main resonance state is obtained:
wherein, ω iswmnIs the natural frequency, xi, of the modewIs modal damping ratio, Fwmn(t) is the external force expressed as:
Mwmnis the modal mass;
the equation translates to the M-C-K form:
wherein,
3. the shell theory-based thin-walled cylinder mirror image cutting modeling method is characterized in that in the third step, the boring bar is simplified into a mass-damping-rigidity model, and each modal parameter is obtained through a frequency response test.
4. The shell theory-based thin-walled cylinder mirror image cutting modeling method is characterized in that the fourth step is as follows:
radial cutting force F of excircletAnd inner circle radial cutting force FbThe mechanical model of (a) is expressed as:
in the formula, KtcIs the radial cutting force coefficient, K, of the cylindrical turning toolbcThe radial cutting force coefficient of the inner circle boring bar, fcIs the feed amount; the dynamic cut thickness equation is defined as:
in the formula, xw(t) is the instantaneous radial displacement of the workpiece, xw(T-T) is the radial displacement of the previous cut revolution of the workpiece, xb(t) is the instantaneous radial displacement of the boring bar, xb(T-T) is the radial displacement of the previous cutting revolution of the boring bar; a ispt(t) is the actual depth of cut of the cylindrical turning, apb(t) is the actual depth of cut of the inner circle boring; substituting equation (13) into equation (12), the radial dynamic cutting force formula can be expressed as:
5. the shell theory-based thin-walled cylinder mirror image cutting modeling method according to claim 1, characterized in that the step five is as follows:
during the cutting process, the feeding amount and the spindle rotating speed are kept unchanged, so the position of the cutting point on the surface of the workpiece is represented as follows:
wherein N is the spindle speed;
note xwFor vibration of the surface of the workpiece in modal coordinates, xbDeducing the vibration amplitude of the workpiece and the boring bar for the vibration of the boring cutter under the modal coordinate:
Ww(t) is the mode shape function of the workpiece at the cutting point expressed as:
coupling the workpiece, the boring bar and the cutting force equation to obtain a dynamic equation of the mirror-oriented processing technology system:
with the nominal depth of cut as input and the vibration of the workpiece and the boring bar as output, the following form is obtained:
wherein,
the second order differential equation is rewritten as a state space representation:
wherein,
the equation solves the output response just according to the Runge Kutta method, and solves the stability by using a semi-discrete method or a full-discrete method.
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