Method for analyzing the stability of turning a frustoconical thin-walled structure

Abstract

This invention relates to a method for analyzing the turning stability of a truncated cone thin-walled structure, belonging to the field of machining technology. First, based on shell theory, a vibration differential equation considering the rotational effects of centrifugal force, Coriolis force, and initial ring tension is established. The cantilever-free boundary condition expression is substituted into the equation, and the natural frequencies and modal parameters of the rotating truncated cone thin-walled structure are obtained by solving the equation using a generalized differential quadrature method. Then, considering the vibration of the tool along the generatrix of the shell and the vibration of the workpiece perpendicular to the generatrix, a flexible workpiece-tool coupled turning dynamics model is established. This model simultaneously considers the influence of shell rotation and multi-mode vibration, achieving a more accurate prediction of the turning process of the truncated cone thin-walled structure.

Description

Technical Field

[0001] This invention relates to a method for analyzing the turning stability of a truncated cone thin-walled structure, and more particularly to a method for analyzing the turning stability of a truncated cone thin-walled structure under cantilever-free boundary conditions that simultaneously considers the effects of rotational Coriolis effect and shell multimodal vibration. Background Technology

[0002] When machining thin-walled truncated cone structures, accurate workpiece modes are needed to more accurately predict the stability of the machining process. However, workpiece rotation introduces Coriolis and centrifugal forces, which alter the workpiece modes. Furthermore, the first three modes of the shell structure can potentially induce vibrations during machining. Therefore, when performing stability analysis on the machining of truncated cone shell structures, the effects of rotational modes and multimodal vibrations of the shell must be considered simultaneously.

[0003] Reference 1, "Artem Gerasimenko, Mikhail Guskov, Alexander Gouskov, Philippe Lorong, Alexander Shokhin, Analytical modeling of a thin-walled cylindrical workpiece during the turning process. Stability analysis of a cutting process, Journal of Vibroengineering 19(2017)5825-5841," discloses an analytical method for predicting the turning stability of thin-walled cylindrical tubes. This method considers the stiffness variations at different axial positions of the thin-walled cylindrical tube and the influence of material removal on modal parameters, obtaining a lobe diagram of the turning stability of the thin-walled cylindrical tube. However, this method does not consider the Coriolis effect of workpiece rotation or the influence of multi-mode vibration of the workpiece on the stability prediction results.

[0004] Reference 2, "Han Qinkai, Chu Fulei, Parametric instability of a rotating truncated conical shell subjected to periodic axial loads, Mechanics Research Communications 53(2013)63-74," discloses an analytical method for parametric instability of a rotating truncated conical shell under axial force. This method uses the generalized differential quadrature principle to obtain the natural frequencies and modal parameter matrices of the rotating truncated conical shell, and performs axial force parametric instability analysis based on the Hill method. However, this method is designed for axial force, which is a periodically continuous force, and its mode and direction of action differ from those of cutting forces. Furthermore, this method does not consider the influence of multi-mode vibrations on parametric instability.

[0005] The typical characteristics of the above literature are: when conducting stability analysis, the influence of the workpiece's rotational Coriolis effect and the workpiece's multi-mode vibration on the stability prediction results were not considered simultaneously, or the influence of periodic force was not considered, resulting in inaccurate prediction results. Summary of the Invention

[0006] Technical problems to be solved

[0007] To improve the accuracy of stability prediction during the turning process of truncated cone thin-walled structures, this invention proposes a method for analyzing the turning stability of truncated cone thin-walled structures under cantilever-free boundary conditions, which simultaneously considers the effects of rotational Coriolis effect and shell multimodal vibration.

[0008] Technical solution

[0009] A method for analyzing the turning stability of a truncated cone thin-walled structure, characterized by the following steps:

[0010] Step 1: Establish the vibration differential equation considering the rotational centrifugal force, Coriolis force, and initial ring tension using the following formula:

[0011]

[0012]

[0013]

[0014]

[0015]

[0016] Where, N x It is the axial force along the generatrix of the workpiece, N θIt is the axial force along the tangent direction of the workpiece, N xθ It is shear force, M x It is the torque along the generatrix of the workpiece, M. θ It is the bending moment along the tangent direction of the workpiece, M xθ Here, r(x) is the torque, r(x) is the mid-surface radius at an axial distance x from the midpoint of the workpiece's small end, u, v, and w are the displacements of any point on the mid-surface of the workpiece along the workpiece's generatrix, tangent, and normal directions, respectively, α is the cone angle of the truncated cone shell, ρ is the workpiece material density, and h is the workpiece thickness. The initial ring tension is obtained using the following formula:

[0017]

[0018] Where Ω is the rotational speed of the workpiece around the axis, and a is the mid-surface radius of the small end of the truncated cone shell;

[0019] Step 2: Express the vibration displacement field vectors in the three directions u, v, and w using the product of circumferential trigonometric functions and axially continuous smooth functions:

[0020]

[0021] Where t is the time variable, n is the circumferential half-wave number, and ω is the natural frequency of the rotating truncated cone thin-walled structure;

[0022] Substituting the displacement field vector of the truncated conical shell into the vibration differential equation in step 1 simplifies the equation to a differential equation containing only the variable x:

[0023] LU=0

[0024] Where L is a 3×3 differential coefficient matrix;

[0025] Step 3: Simplify and solve the vibration differential equation from Step 2. Approximate all differential terms in the equation as a weighted sum of function values ​​at points in the spatial domain using the following formula:

[0026]

[0027] Step 4: The fixed boundary conditions at the small end during the turning of the truncated cone thin-walled structure are calculated using the following formula:

[0028]

[0029] The large-port free boundary condition is calculated using the following formula:

[0030]

[0031] Where R is the radius of the mid-face of the large port;

[0032] Step 5: Substitute the expressions from Step 3 and Step 4 into the vibration differential equation from Step 2, and rearrange them based on the natural frequency ω to express the vibration differential equation as a set of numerical discrete equations:

[0033] [ω 2 [M+ωG+K]d=0

[0034] Where M, G, and K are the mass, damping, and stiffness matrices of the truncated cone thin-walled structure, respectively; d is a 3N-4 order column vector representing the displacement of each discrete point, as shown in the following equation:

[0035]

[0036] Step 6: Transform the discrete equations in Step 5 into standard eigenvalue equations, and solve them to obtain the natural frequencies and mode shapes of the rotating truncated cone thin-walled structure;

[0037]

[0038] Solving the standard eigenvalue equation yields 2N eigenvalues, where positive eigenvalues ​​represent forward wave frequencies and negative eigenvalues ​​represent backward wave frequencies. The two eigenvalues ​​with the smallest absolute values ​​represent the frequencies with axial wave number m = 1, the two eigenvalues ​​with the second smallest absolute values ​​represent the frequencies with axial wave number m = 2, and so on. Substituting the eigenvalues ​​into the numerical discretization equation in step 5 yields a set of linear homogeneous equations. The solutions to the equations are the mode shapes at the corresponding natural frequencies.

[0039] Step 7: After obtaining the natural frequencies and mode shapes of the rotating truncated cone thin-walled structure, establish a turning dynamics model of flexible workpiece-tool coupling. Substitute the first three modal parameters of the shell in the static and rotating states respectively, and use the semi-discrete method to calculate the stability lobe diagram. The lowest envelope of the three lobe diagrams is the final stability prediction result.

[0040] A computer system is characterized by comprising: one or more processors, and a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors cause the one or more processors to implement the method described above.

[0041] A computer-readable storage medium is characterized by storing computer-executable instructions, which, when executed, are used to implement the method described above.

[0042] Beneficial effects

[0043] This invention provides a method for analyzing the turning stability of a truncated conical thin-walled structure. The method first calculates the natural frequencies and modal parameters of the rotating shell based on a shell theoretical model considering the rotating Coriolis effect. Then, a two-degree-of-freedom turning dynamics model with flexible workpiece-tool coupling is established. Substituting the first three modes of the shell, the lowest envelope lines of the three stability lobe diagrams are obtained as the final stability prediction results. Compared with References 1 and 2, this invention considers both the rotating Coriolis effect and the multimodal vibration of the shell, enabling more accurate prediction of the turning process and effectively guiding machining in practical engineering applications. Attached Figure Description

[0044] The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Throughout the drawings, the same reference numerals denote the same parts.

[0045] Figure 1 This is a three-dimensional model of the truncated cone shell part in the method embodiment of the present invention.

[0046] Figure 2 This is a graph showing the relationship between the frequency of the truncated cone shell and the axial and circumferential half-wave numbers in an embodiment of the method of the present invention.

[0047] Figure 3 This is a graph showing the relationship between the frequency and rotational speed of the truncated cone shell in an embodiment of the method of the present invention.

[0048] Figure 4 This is a two-degree-of-freedom dynamic model in the embodiment of the method of the present invention.

[0049] Figure 5 This is a stability lobe diagram that simultaneously considers the effects of rotational Coriolis effect and shell multimodal vibration in the embodiments of the method of the present invention. Detailed Implementation

[0050] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0051] This method first establishes vibration differential equations based on shell theory, considering rotational effects such as centrifugal force, Coriolis force, and initial ring tension. Substituting the cantilever-free boundary condition expression into the equations and solving using generalized differential quadrature methods, the natural frequencies and modal parameters of the rotating truncated cone thin-walled structure are obtained. Subsequently, considering the vibration of the tool along the generatrix of the shell and the vibration of the workpiece perpendicular to the generatrix, a flexible workpiece-tool coupled turning dynamics model is established. This model simultaneously considers the effects of shell rotation and multi-mode vibration, achieving a more accurate prediction of the turning process of the truncated cone thin-walled structure.

[0052] It mainly includes the following steps:

[0053] Step 1: Establish the vibration differential equation considering the rotational centrifugal force, Coriolis force, and initial ring tension using the following formula:

[0054]

[0055]

[0056]

[0057]

[0058]

[0059] Where, N x It is the axial force along the generatrix of the workpiece, N θ It is the axial force along the tangent direction of the workpiece, N xθ It is shear force, M x It is the torque along the generatrix of the workpiece, M. θ It is the bending moment along the tangent direction of the workpiece, M xθ Here, r(x) is the torque, r(x) is the mid-surface radius at an axial distance x from the midpoint of the workpiece's small end, u, v, and w are the displacements of any point on the mid-surface of the workpiece along the workpiece's generatrix, tangent, and normal directions, respectively, α is the cone angle of the truncated cone shell, ρ is the workpiece material density, and h is the workpiece thickness. The initial ring tension is obtained using the following formula:

[0060]

[0061] Where Ω is the rotational speed of the workpiece around the axis, and a is the mid-surface radius of the small end of the truncated cone shell.

[0062] Step 2: Express the vibration displacement field vectors in the three directions u, v, and w using the product of circumferential trigonometric functions and axially continuous smooth functions:

[0063]

[0064] Where t is the time variable, n is the circumferential half-wave number, and ω is the natural frequency of the rotating truncated cone thin-walled structure.

[0065] Substituting the displacement field vector of the truncated conical shell into the vibration differential equation in step one simplifies the equation to a differential equation containing only the variable x:

[0066] LU=0

[0067] Where L is a 3×3 differential coefficient matrix, the specific expression of which is determined with reference to the literature "LiHua, Lam KY, Ng TY, Rotating Shell Dynamics, Elsevier Science Ltd, 2005".

[0068] Step 3: Simplify and solve the vibration differential equation from Step 2. Approximate all differential terms in the equation as a weighted sum of function values ​​at points in the spatial domain using the following formula:

[0069]

[0070] Step 4: The fixed boundary conditions at the small end during the turning of the truncated cone thin-walled structure are calculated using the following formula:

[0071]

[0072] The large-port free boundary condition is calculated using the following formula:

[0073]

[0074] Where R is the radius of the mid-face of the large port.

[0075] Step 5: Substitute the expressions from Steps 3 and 4 into the vibration differential equation from Step 2, and rearrange them based on the natural frequency ω to express the vibration differential equation as a set of numerical discrete equations:

[0076] [ω 2 [M+ωG+K]d=0

[0077] Where M, G, and K are the mass, damping, and stiffness matrices of the truncated cone thin-walled structure, respectively; d is a 3N-4 order column vector representing the displacement of each discrete point, as shown in the following equation:

[0078]

[0079] Step 6: Transform the discrete equations from Step 5 into standard eigenvalue equations:

[0080]

[0081] Solving the standard eigenvalue equation yields 2N eigenvalues, where positive eigenvalues ​​represent forward wave frequencies and negative eigenvalues ​​represent backward wave frequencies. The two eigenvalues ​​with the smallest absolute values ​​represent the frequencies with axial wave number m = 1, the two eigenvalues ​​with the second smallest absolute values ​​represent the frequencies with axial wave number m = 2, and so on. Substituting these eigenvalues ​​into the numerical discretization equations in step five yields a set of linear homogeneous equations. The solutions to these equations are the mode shapes at the corresponding natural frequencies.

[0082] Step 7: After obtaining the natural frequencies and mode shapes of the rotating truncated cone thin-walled structure, refer to the appendix. Figure 4 A turning dynamics model of flexible workpiece-tool coupling was established. The first three modal parameters of the shell in both static and rotating states were substituted into the model. The stability leaflet diagrams were calculated using the semi-discrete method published in the literature "Tamás Insperger, Gábor Stépán, Updated semi-discretization method for periodic delay-differential equations with discrete delay, International Journal for Numerical Methods in Engineering, 61 (2004)". The lowest envelope of the three leaflet diagrams is the final stability prediction result. (Refer to Appendix...) Figure 5 .

[0083] To enable those skilled in the art to better understand the present invention, the present invention will be described in detail below with reference to specific embodiments.

[0084] In the example, the cone angle α of the truncated cone thin-walled structure is 20 degrees, the density ρ is 7850 kg / m³, the thickness h is 3 mm, and the mid-surface radius α of the small port is 38.5 mm.

[0085] Step 1: Substitute the geometric and physical parameters of the above truncated cone thin-walled structure into the vibration differential equation considering the rotational centrifugal force, Coriolis force, and initial ring tension:

[0086]

[0087]

[0088]

[0089]

[0090]

[0091] Where, N x It is the axial force along the generatrix of the workpiece, N θ It is the axial force along the tangent direction of the workpiece, N xθ It is shear force, M x It is the torque along the generatrix of the workpiece, M. θ It is the bending moment along the tangent direction of the workpiece, M xθHere, r(x) is the torque, r(x) is the mid-surface radius at an axial distance x from the midpoint of the workpiece's small end, u, v, and w are the displacements of any point on the mid-surface of the workpiece along the workpiece's generatrix, tangent, and normal directions, respectively, α is the cone angle of the truncated cone shell, ρ is the workpiece material density, and h is the workpiece thickness. The initial ring tension is obtained using the following formula:

[0092]

[0093] Where Ω is the rotational speed of the workpiece around the axis, and a is the mid-surface radius of the small port of the truncated cone thin-walled structure.

[0094] Step 2: Express the vibration displacement field vectors in the three directions u, v, and w using the product of circumferential trigonometric functions and axially continuous smooth functions:

[0095]

[0096] Where t is the time variable, n is the circumferential half-wave number, and ω is the natural frequency of the rotating truncated cone thin-walled structure.

[0097] Substituting the displacement field vector of the truncated cone thin-walled structure into the vibration differential equation in step one simplifies the equation to a differential equation containing only the variable x:

[0098] LU=0

[0099] Where L is a 3×3 differential coefficient matrix, the specific expression of which is determined with reference to the literature "LiHua, Lam KY, Ng TY, Rotating Shell Dynamics, Elsevier Science Ltd, 2005".

[0100] Step 3: Simplify and solve the vibration differential equation from Step 2. Approximate all differential terms in the equation as a weighted sum of function values ​​at points in the spatial domain using the following formula:

[0101]

[0102] Step 4: The fixed boundary conditions at the small end during the turning of the truncated cone thin-walled structure are calculated using the following formula:

[0103]

[0104] The large-port free boundary condition is calculated using the following formula:

[0105]

[0106] Where R is the radius of the mid-face of the large port.

[0107] Step 5: Substitute the expressions from Steps 3 and 4 into the vibration differential equation from Step 2, and rearrange them based on the natural frequency ω to express the vibration differential equation as a set of numerical discrete equations:

[0108] [ω 2 [M+ωG+K]d=0

[0109] Where M, G, and K are the mass, damping, and stiffness matrices of the truncated cone thin-walled structure, respectively; d is a 3N-4 order column vector representing the displacement of each discrete point, as shown in the following equation:

[0110]

[0111] Step Six: Transform the discrete equations from Step Five into standard eigenvalue equations, and solve for the natural frequencies of the rotating truncated cone thin-walled structure. Refer to the appendix for the calculation results. Figure 2 and attached Figure 3 .

[0112]

[0113] Solving the standard eigenvalue equation yields 2N eigenvalues, where positive eigenvalues ​​represent forward wave frequencies and negative eigenvalues ​​represent backward wave frequencies. The two eigenvalues ​​with the smallest absolute values ​​represent the frequencies with axial wave number m = 1, the two eigenvalues ​​with the second smallest absolute values ​​represent the frequencies with axial wave number m = 2, and so on. Substituting these eigenvalues ​​into the numerical discretization equations in step five yields a set of linear homogeneous equations. The solutions to these equations are the mode shapes at the corresponding natural frequencies.

[0114] Step 7: After obtaining the natural frequencies and mode shapes of the rotating truncated cone thin-walled structure, refer to the appendix. Figure 4 A turning dynamics model of flexible workpiece-tool coupling was established. The first three modal parameters of the shell in both static and rotating states were substituted into the model. The stability leaflet diagrams were calculated using the semi-discrete method published in the literature "Tamás Insperger, Gábor Stépán, Updated semi-discretization method for periodic delay-differential equations with discrete delay, International Journal for Numerical Methods in Engineering, 61 (2004)". The lowest envelope of the three leaflet diagrams is the final stability prediction result. (Refer to Appendix...) Figure 5 .

[0115] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the scope of the technology disclosed in the present invention, and such modifications or substitutions should all be covered within the scope of protection of the present invention.

Claims

1. A method for analyzing the stability of turning a truncated cone thin-walled structure, characterized by The steps are as follows: Step 1: the vibration differential equation considering the centrifugal force, Coriolis force and initial ring tension is established by using the following formula: where N x is the axial force along the generatrix of the workpiece, N θ is the axial force along the tangent of the workpiece, N xθ is the shear force, M x is the moment along the generatrix of the workpiece, M θ is the bending moment along the tangent of the workpiece, M xθ is the torque, r(x) is the mean radius at a distance x from the axial midpoint of the small end of the workpiece, u, v, w are the displacements of an arbitrary point on the mean surface of the workpiece along the generatrix, tangent and normal to the workpiece, respectively, a is the cone angle of the truncated cone shell, p is the density of the workpiece material, and h is the thickness of the workpiece, is the initial ring tension, which is obtained by the following equation: Wherein, Ω is the rotating speed of the workpiece around the axis, a is the middle surface radius of the small end port of the truncated cone shell; Step 2: the vibration displacement field vectors in u, v, w three directions are expressed in the form of the product of the circumferential trigonometric function and the axially continuous smooth function: Wherein, t is the time variable, n is the circumferential half wave number, ω is the inherent frequency of the rotating truncated cone thin-walled structure; The truncated cone shell displacement field vector is substituted into the vibration differential equation in step 1, and the equation is simplified into a differential equation containing only variable x: LU = 0 Wherein, L is a 3x3 differential coefficient matrix; Step 3: the vibration differential equation in step 2 is simplified and solved, and all the differential items in the equation are approximately processed into the weighted sum of the function values of the points in the spatial domain by the following formula: Step 4: the fixed boundary condition of the small end port in the turning process of the truncated cone thin-walled structure is calculated by the following formula: u = 0, v = 0, w = 0, The large port free boundary condition is calculated by the following equation: Wherein, R is the middle surface radius of the large end port; Step 5: the expressions of step 3 and step 4 are substituted into the vibration differential equation in step 2, and are rearranged based on the inherent frequency ω, so that the vibration differential equation is expressed into a group of numerical discrete equations: [ω 2 M + ωG + K]d = 0 Wherein, M, G, K are the mass, damping and stiffness matrices of the truncated cone thin-walled structure respectively; d is a 3N-4 order column vector, which represents the displacement of each discrete point, as follows: Step 6: the discrete equation in step 5 is converted into a standard eigenvalue equation, and the inherent frequency and modal shape of the rotating truncated cone thin-walled structure are solved; 2N eigenvalues are obtained by solving the standard eigenvalue equation, the positive eigenvalues represent the forward wave frequency, and the negative eigenvalues represent the backward wave frequency; the two eigenvalues with the smallest absolute values represent the frequency of axial wave number m = 1, the two eigenvalues with the second smallest absolute values represent the frequency of axial wave number m = 2, and so on; the eigenvalues are substituted into the numerical discrete equation in step 5 to obtain a group of linear homogeneous equations, and the solution of the equation group is the modal shape under the corresponding inherent frequency; Step 7: after obtaining the inherent frequency and shape of the rotating truncated cone thin-walled structure, the turning dynamics model of the flexible workpiece-tool coupling is established, the first three order modal parameters are substituted into the shell static and rotating state respectively, and the semi-discrete method is used to calculate the stability lobe diagram, and the lowest envelope line of the three lobe diagrams is the final stability prediction result.

2. A computer system, characterized by Comprise: One or more processors, a computer readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement the method of claim 1.

3. A computer-readable storage medium, characterized in that There are computer executable instructions stored, which are used to implement the method of claim 1 when executed.

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