Multi-time-delay processing system stability analysis method based on updating precise integral method

Abstract

The invention discloses a multi-time-delay processing system stability analysis method based on an updating precise integral method. The method comprises the following steps: S1, obtaining a state space according to a multi-time-delay n-dimensional linear dynamic system equation; s2, obtaining continuous output of the multi-time-delay processing system according to the ordinary differential equation solution; s3, obtaining a discrete output expression of the multi-time-delay processing system according to the second-order Newton polynomial; s4, further modeling the discrete output expression obtained in the step S3; s5, constructing a state transition matrix irrelevant to the initial value according to the Floquet theory; s6, judging the stability of the equation according to the characteristic value of the transfer matrix phi of the system; and S7, analyzing the stability of the multi-time-delay processing system based on the updated accurate integral. According to the method, high-order discretization is adopted, so that the calculation precision is improved; and only one matrix instead of m matrixes is constructed in one period, so that a large amount of calculation time can be saved, and the processing efficiency is improved.

Description

Technical Field

[0001] The present invention relates to the field of mechanical vibration, and particularly to a method for analyzing the stability of a multi-time-delay machining system based on the updated precise integration method. Background Art

[0002] The time-delay effect widely exists in mechanical dynamic systems, such as active / passive control and cutting processes. The multi-time-delay dynamic system can be described by delay differential equations. By solving the DDE, the stability or instability of the dynamic system can be estimated. During the machining process, the cutting thickness change caused by the workpiece-tool vibration induces the time-delay effect. The machining process is described as a DDE that combines the cutting force coefficient, tool size, and machining parameters, etc.

[0003] The methods for analyzing the stability of a machining system with time delay are mainly divided into two categories: the frequency domain and the time domain. The frequency domain method represents the output of the machining system using a frequency domain model by applying the time-delay effect, and obtains the chatter-free machining parameters by judging the characteristic roots of the cutting closed-loop system. The time-domain-based method focuses on obtaining the vibration displacement of the machining system on the discrete time scale, then constructing the state transition matrix to relate the displacements of two adjacent cycles, and finally using the Floquet theory to determine the stability of the DDE.

[0004] Currently, as a high-precision calculation method, the precise integration method (PIM) is widely used in dynamic analysis and has also been extended to the prediction of the stability of machining system chatter. In the first-order SDM and GPIM, the displacement relationship matrix between adjacent moments is established through the iterative expression of the system output, that is, the relationship matrix is continuously multiplied from the beginning to the end to obtain the transition matrix. That is to say, the relationship matrix needs to be established in each calculation step, and m relationship matrices are established within one cycle (m is the number of discretizations of the time period), which will consume a lot of time. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide a method for analyzing the stability of a multi-time-delay machining system based on the updated precise integration method, so as to improve the calculation accuracy and machining efficiency.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is: a method for analyzing the stability of a multi-time-delay machining system based on the updated precise integration method, comprising the following steps:

[0007] S1. Obtain its state space according to the n-dimensional linear dynamic system equation with multi-time delay;

[0008] S2. Obtain the continuous output of the multi-time-delay machining system according to the solution of the ordinary differential equation;

[0009] S3. Obtain the discrete output expression of the multi-time-delay machining system according to the second-order Newton polynomial;

[0010] S4. Further model the discrete output expression obtained in step S3;

[0011] S5. Construct a state transition matrix independent of the initial value according to the Floquet theory;

[0012] S6. Judge the stability of the equation according to the eigenvalues of the transfer matrix Φ of the system;

[0013] S7. Stability analysis of the multi-time-delay processing system based on updated exact integration.

[0014] Preferably, the specific operation method of step S1 is:

[0015] Describe the n-dimensional linear dynamic system with multi-time delays as:

[0016]

[0017] where q(t) is the vibration displacement in the n-dimensional direction, M is the mass, C is the damping, K is the stiffness, U j (t), U 1,j (t), U 2,j (t) is the coefficient matrix of the jth unit, T 1,j , T 2,j is the time delay of the jth unit, and j takes 1, 2,..., N k (k = 1, 2, 3);

[0018] Substitute Establish the state space form of equation (1) as follows:

[0019]

[0020] where,

[0021] Preferably, the specific operation method of step S2 is:

[0022] Express the non-homogeneous term as g(t), and using the knowledge of the solution of ordinary differential equations, obtain the continuous output of v(t) in the following equation (3), where v(t p ) represents the value of v(t) at time t p and

[0023]

[0024] Express v(t p+1 ) as the output of v(t) at time (t p+1 ), that is, v(t p+1 ) is expressed as:

[0025]

[0026] Convert Equation (4) to:

[0027]

[0028] Preferably, the specific operation method of step S3 is: In order to obtain the discrete solution of v(t) suitable for performing numerical calculations in software, first, g(t) is discretized. Therefore, the period T is first evenly divided into m elements, and the size of each discrete time element is τ, that is, T = mτ.

[0029] Expand g(υ + t p ) in Equation (5) using a second-order Newton polynomial as:

[0030] g(υ + t p ) = r 0 + r 1 υ + r 2 (υ - τ)υ (6)

[0031] where r 0 = g(t p+1 ), r 1 = [g(t p+1 ) - g(t p )] / τ, r 2 = [g(t p+1 ) - 2g(t p ) + g(t p-1 )] / (2τ 2 ). Replace g(t p-1 ), g(t p ) and g(t p+1 ) with g p-1 , g p and g p+1 ), that is

[0032] r 0 = g p+1 , r 1 = (g p+1 - g p ) / τ, r 2 = (g p+1 - 2g p + g p-1 ) / (2τ 2 );

[0033] Substitute Equation (6) into Equation (5), and the discrete output of v(t p+1 ) is calculated by the following formula:

[0034] v(t p+1 ) = Mv(tp ) + M 0 r 0 + M 1 r 1 + M 2 r 2 (7)

[0035] where M = e Aτ ,

[0036] Replace r in equation (7) i (i = 0, 1, 2) as follows:

[0037] v(t p+1 ) = Mv(t p ) + G 0 g p+1 + G 1 g p + G 2 g p-1 (8)

[0038] where G 0 = M 0 + M 1 / τ + M 2 / (2τ 2 ), G 1 = -M 1 / τ - M 2 / τ 2 , G 2 = M 2 / (2τ 2 ).

[0039] Preferably, the specific operation method of step S4 is: Because

[0040] Equation (8) is further modeled as:

[0041]

[0042] where v p and v p+1 are abbreviations of v(t p ) and v(t p+1 ); B j.p+i , B 1,j,p+i and B 2,j,p+i are respectively

[0043] B j (t p+i ), B 1,j (t p+i ) and B 2,j (t p+i) is an abbreviation, where i = -1, 0, 1. Also, since v(t p-i -T k,j ) = v(t p-i -m k,j τ), v(t p-i -T k,j ) can be expressed as and m k,j = fix(T k,j / τ + 0.5), where fix(x) is a function that rounds x to zero, j = 1, 0, -1; K = 1, 2. Substituting the above equation into equation (9) can be transformed into:

[0044]

[0045]

[0046] Further transform equation (10) into:

[0047]

[0048] Preferably, the specific operation method of step S5 is:

[0049] In order to use Floquet theory to determine the stability of DDE, a state transition matrix independent of the initial value must be established. First, construct equation (12) from the cyclic relationship described in equation (11).

[0050]

[0051] In the formula:

[0052]

[0053] column

[0054] (m - m i,j + 1)

[0055]

[0056] Preferably, the specific operation method of step S6 is:

[0057] It can be clearly seen from equation (12) that the historical data caused by different time delays, that is, converting and in equation (11) into linear combinations of the vibration displacement data of the previous and the entire cycle respectively, that is, [v 1 , v 2 ,..., v m+1 , v m+2 T and [v​1-T , v 2-T ,..., v m+1-T , v m+2-T T ,

[0058] The conversion of Equation (12) is as follows:

[0059]

[0060] Finally, the transition matrix Φ of the system is defined as:

[0061]

[0062] When is irreversible, the extended Moore - Penrose matrix can be used as an alternative,

[0063] The stability of the multiple - delay equation described by Equation (1) can be determined by Floquet theory. More specifically, when the maximum modulus of the eigenvalues of the state - transition matrix Φ is less than 1, the DDE is stable; otherwise, it is unstable.

[0064] Preferably, the specific operation method of step S7 is as follows: The stability analysis method of the multi - time - delay dynamic system based on the updated precise integration method PIM first describes the multi - time - delay system as a delay differential equation DDE in the state - space form including homogeneous and non - homogeneous terms. Then, using the knowledge of the solution of ordinary differential equations, the non - homogeneous equation is expanded by a second - order Newton polynomial, and the discrete displacement output of the dynamic system is evenly divided into several parts at uniform intervals. Then, the historical terms of the recurrence displacement expressions corresponding to different time - delay periods are transformed into a combination of the vibration displacement data of the entire previous period and the current period; finally, a transition matrix representing the vibration displacement mapping between two adjacent periods is constructed to determine the stability of the DDE.

[0065] The beneficial effects of adopting the above - mentioned technical solutions are as follows: The stability analysis method of the multi - time - delay processing system based on the improved precise integration method proposed by the present invention, on the one hand, adopts high - order discretization, improving the calculation accuracy; on the other hand, this method transforms the historical terms corresponding to different time - delay periods into a combination of the vibration displacement data of the entire previous period and the current period, and constructs a transition matrix representing the vibration displacement mapping between two adjacent cycles. Only one matrix is constructed within one cycle instead of m matrices, which can save a large amount of calculation time and improve the processing efficiency. Detailed implementation mode

[0066] The stability analysis method of the multi - time - delay processing system based on the updated precise integration method of the present invention includes the following steps:

[0067] S1. Obtain its state - space according to the n - dimensional linear dynamic system equation with multiple time - delays:​

[0068] Describe the n - dimensional linear dynamic system with multiple time - delays as follows:

[0069]

[0070] where \(q(t)\) is the vibration displacement in the \(n\) - dimensional direction, \(M\) is the mass, \(C\) is the damping, \(K\) is the stiffness, \(U\) j (t), \(U\) 1,j (t), \(U\) 2,j (t) is the coefficient matrix of the \(j\) - th unit, \(T\) 1,j , \(T\) 2,j is the time - delay of the \(j\) - th unit, and \(j\) takes values of \(1,2,\cdots,N\) k (k = 1,2,3);

[0071] Substitute Establish the state - space form of Equation (1) as follows:

[0072]

[0073] where,

[0074] S2. Obtain the continuous output of the multi - time - delay processing system according to the solution of the ordinary differential equation:

[0075] Express the non - homogeneous term as \(g(t)\). Using the knowledge related to the solution of ordinary differential equations, obtain the continuous output of \(v(t)\) in Equation (3) below, where \(v(t\) p ) represents the value of \(v(t)\) at time \(t\) p .

[0076]

[0077] Use \(v(t\) p+1 ) to represent the output of \(v(t)\) at time \((t\) p+1 ), that is, \(v(t\) p+1 ) is expressed as:

[0078]

[0079] Transform Equation (4) into:

[0080]

[0081] S3. Obtain the discrete - output expression of the multi - time - delay processing system according to the second - order Newton polynomial: To obtain the discrete solution of \(v(t)\) suitable for performing numerical calculations in software, first discretize \(g(t)\). Therefore, first divide the period \(T\) evenly into \(m\) elements, and the size of each discrete - time element is \(\tau\), that is, \(T = m\tau\).

[0082] Expand \(g(\upsilon + t\) p ) into a second-order Newton polynomial as follows:

[0083] g(\upsilon + t\) p ) = r 0 + r 1 \(\upsilon + r\) 2 \((\upsilon - \tau)\upsilon\ (6)\)

[0084] where \(r\) 0 = g(t p+1 ), \(r\) 1 = [g(t p+1 ) - g(t p )] / \(\tau\), \(r\) 2 = [g(t p+1 ) - 2g(t p ) + g(t p-1 )] / (2\(\tau\) 2 ), Replace g(t p-1 ), g(t p ), and g(t p+1 ) with \(g\) p-1 , \(g\) p , and \(g\) p+1 ), that is

[0085] \(r\) 0 = \(g\) p+1 , \(r\) 1 = (\(g\) p+1 - \(g\) p ) / \(\tau\), \(r\) 2 = (\(g\) p+1 - 2\(g\) p + \(g\) p-1 ) / (2\(\tau\) 2 );

[0086] Substitute equation (6) into equation (5), the discrete output of \(v(t\) p+1 ) is calculated by the following formula:

[0087] \(v(t\) p+1 ) = Mv(t\) p ) + M 0 \(r\) 0 + M 1 \(r\) 1 + M 2 \(r\) 2 (7)

[0088] where \(M = e\) Aτ ,

[0089] Replace \(r\) i (\(i = 0, 1, 2\)) in equation (7) as follows:

[0090] v(t p+1 )=Mv(t p )+G 0 g p+1 +G 1 g p +G 2 g p-1 (8)

[0091] where, G 0 =M 0 +M 1 / τ+M 2 / (2τ 2 ), G 1 =-M 1 / τ-M 2 / τ 2 , G 2 =M 2 / (2τ 2 )。

[0092] S4. Further model the discrete output expression obtained in step S3: Since (Equation (8)) is further modeled as:

[0093]

[0094] where, v p and v p+1 are abbreviations of v(t p ) and v(t p+1 ) respectively; B j.p+i , B 1,j,p+i and B 2,j,p+i are abbreviations of

[0095] B j (t p+i ), B 1,j (t p+i ) and B 2,j (t p+i ) respectively, where i = -1, 0, 1. Also, since v(t p-i -T k,j )=v(t p-i -m k,j τ), v(t p-i -T k,j ) can be expressed as and m k,j =fix(T k,j / τ+0.5), where fix(x) is a function that rounds x to zero, j = 1, 0, -1; K = 1, 2. Substituting the above formula into formula (9), it can be transformed into:

[0096]

[0097]

[0098] Further transform Equation (10) into:

[0099]

[0100] S5. Construct a state transition matrix independent of the initial value according to Floquet theory:

[0101] In order to determine the stability of the DDE using Floquet theory, a state transition matrix independent of the initial value must be established. First, construct Equation (12) from the cyclic relationship described in Equation (11).

[0102]

[0103] Where:

[0104]

[0105] column

[0106] (m - m i,j + 1)

[0107]

[0108] S6. Judge the stability of the equation according to the eigenvalues of the system's transition matrix Φ:

[0109] It can be clearly seen from Equation (12) that the historical data caused by different time delays, that is, transform the

[0110] and in Equation (11) into linear combinations of the vibration displacement data of the previous and the whole cycle respectively, that is, [v 1 , v 2 ,..., v m+1 , v m+2 T and [v 1-T , v 2-T ,..., v m+1-T , v m+2-T T ,

[0111] The transformation of Equation (12) is:

[0112]

[0113] Finally, define the system's transition matrix Φ as:

[0114]

[0115] When irreversible, the extended Moore-Penrose matrix can be used as an alternative,

[0116] The stability of the multiple-delay equation described by Equation (1) can be determined by Floquet theory. More specifically, the DDE is stable when the maximum modulus of the eigenvalues of the state-transition matrix Φ is less than 1; otherwise, it is unstable.

[0117] S7. Stability analysis of a multi-time-delay machining system based on updated precise integration: The stability analysis method of a multi-time-delay dynamic system based on the updated precise integration method PIM first describes the multi-time-delay system as a delay differential equation DDE in the state-space form containing homogeneous and inhomogeneous terms. Then, using the knowledge of the solution of ordinary differential equations, the inhomogeneous equation is expanded by a second-order Newton polynomial, and the discrete displacement output of the dynamic system is divided into several parts at uniform intervals. Then, the historical terms of the recurrence displacement expressions corresponding to different time-delay periods are transformed into a combination of the vibration displacement data of the entire previous period and the current period. Finally, a transition matrix representing the vibration displacement mapping between two adjacent periods is constructed to determine the stability of the DDE.

[0118] The above description is only proposed as a technically feasible solution of the present invention and does not serve as a single limiting condition for its technical solution itself.

Claims

1. A stability analysis method for multi-time-delay machining systems based on the updated exact integration method, characterized in that: The following steps are involved: S1. Obtain the state space of the n-dimensional linear dynamic system equation with multiple time delays; S2, obtain the continuous output of the multi-time-delay processing system according to the solution of ordinary differential equations; S3, obtain the discrete output expression of the multi-time-delay processing system based on the second-order Newton polynomial; S4, further modeling the discrete output expression obtained in step S3; S5. Construct a state transfer matrix independent of the initial value based on Floquet theory; S6. Determine the stability of the equation based on the eigenvalue of the system's transfer matrix Φ; S7. Stability analysis of multi-delay machining systems based on updated exact integration.

2. The analysis method according to claim 1, characterized in that: The specific operation method of step S1 is: The n-dimensional linear dynamic system with multiple time delays is described as: Among them, q(t) is the vibration displacement in the n-dimensional direction, M is the mass, C is the damping, K is the stiffness, and U is the j (t),U 1,j (t),U 2,j (t) is the coefficient matrix of the jth unit, T 1,j ,T 2,j is the delay of the jth unit, j is 1, 2, ..., N k (k=1,2,3); Substitution The state space form of formula (1) is established as follows: in, 3. The analysis method according to claim 1, characterized in that: The specific operation method of step S2 is: The inhomogeneous terms Expressed as g(t), using the knowledge of ordinary differential equation solutions, we can obtain the continuous output of v(t) in equation (3), where v(t p ) represents v(t) at time t p The value of Use v(t p+1 ) indicates that v(t) at time (t p+1 ) output, namely v(t p+1 ) is expressed as: Transform formula (4) into:

4. The analysis method according to claim 1, characterized in that: The specific operation method of step S3 is: in order to obtain a discrete solution of v(t) suitable for performing numerical calculations in software, firstly, g(t) is discretized, so, First, divide the period T into m elements on average. The size of each discrete time element is τ, that is, T = mτ. In formula (5), g(υ+t p ) is expanded into a second-order Newton polynomial: g(u+t p )=r0+r1υ+r2(υ-τ)υ (6) Where r0 = g(t p+1 ),r1=[g(t p+1 )-g(t p )] / τ,r2=[g(t p+1 )-2g(t p )+g(t p-1 )] / (2τ 2 ), use g p-1 ,g p and g p+1 Replace g(t p-1 ),g(t p ) and g(t p+1 ),Right now r0=g p+1 ,r1=(g p+1 -g p ) / τ,r2=(g p+1 -2g p +g p-1 ) / (2τ 2 ); Substituting equation (6) into equation (5), v(t p+1 ) is calculated by the following formula: v(t p+1 )=Mv(t p )+M0r0+M1r1+M2r2 (7) in, Replace r in formula (7) i (i=0,1,2) is replaced as follows: v(t p+1 )=Mv(t p )+G0g p+1 +G1g p +G2g p-1 (8) where, G0 = M0 + M1 / τ + M2 / (2τ 2 ), G1 = -M1 / τ - M2 / τ 2 , G2 = M2 / (2τ 2 )。 5. The analysis method according to claim 1, characterized in that: The specific operation method of step S4 is: Formula (8) is further modeled as: Among them, v p and v p+1 They are v(t p ) and v(t p+1 ) abbreviation; B j.p+i ,B 1,j,p+i and B 2,j,p+i B j (t p+i ),B 1,j (t p+i ) and B 2,j (t p+i ), where i = -1, 0, 1, and because v(t p-i -T k,j )=v(t p-i -m k,j τ), we can convert v(t p-i -T k,j ) is expressed as And m k,j =fix(T k,j / τ+0.5), fix(x) is a function that rounds x to zero, j = 1, 0, -1; K = 1, 2, Substituting into formula (9), it can be transformed into: Formula (10) can be further transformed into:

6. The analysis method according to claim 1, characterized in that: The specific operation method of step S5 is: In order to determine the stability of DDE using Floquet theory, it is necessary to establish a state transition matrix that is independent of the initial value. First, the cyclic relationship described in equation (11) is used to construct equation (12): Where:

7. The analysis method according to claim 1, characterized in that: The specific operation method of step S6 is: It can be clearly seen from formula (12) that the historical data caused by different time delays, that is, and are respectively transformed into linear combinations of the vibration displacement data of the previous and entire cycles, namely [v1,v2,...,v m+1 ,v m+2 ] T and [v 1-T ,v 2-T ,...,v m+1-T ,v m+2-T ] T , the conversion of formula (12) is: Finally, the transfer matrix Φ of the system is defined as: when Irreversible, the extended Moore-Penrose matrix can be used as an alternative, The stability of the multi-delay equation described by equation (1) can be determined using Floquet theory. More specifically, when the maximum modulus of the eigenvalues ​​of the state transfer matrix Φ is less than 1, the DDE is stable; otherwise, it is unstable.

8. The analysis method according to claim 1, characterized in that: The specific operation method of step S7 is: the stability analysis method of multi-delay dynamic system based on the updated exact integration method PIM first describes the multi-delay system as a time-delay differential equation DDE in the state space form containing homogeneous terms and inhomogeneous terms, then uses the knowledge of ordinary differential equation solutions, uses second-order Newton polynomials to expand the inhomogeneous equation, divides the discrete displacement output of the dynamic system into several parts at uniform intervals, and then converts the historical terms of the recursive displacement expression corresponding to different time-delay periods into a combination of the vibration displacement data of the entire previous period and the current period; finally, constructs a transition matrix representing the vibration displacement mapping of two adjacent periods to determine the stability of the DDE.

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