KR101790733B1 - Step length control apparatus with multi-staged clutch damper model and the method thereof - Google Patents

Abstract

The present invention relates to a step length control apparatus and method for analyzing displacement nonlinear behavior characteristics such as a multi-stage clutch damper applied to an actual system,
The step length control device includes a flywheel and a multi-stage clutch damper. The clutch model includes an initial condition and a restriction condition set for step length control including an analysis of an output torque characteristic with respect to an input torque of the multi-stage clutch damper An input unit; A storage unit for storing a program for the initial condition and the constraint condition or the step length control; And a controller for driving the program to generate an initial equation using the initial conditions and the restriction conditions, and to calculate an initial torque, and to calculate an output torque of the clutch using the step length control.

Description

BACKGROUND OF THE INVENTION 1. Field of the Invention [0001] The present invention relates to a step length control device having a multi-stage clutch damper model,

The present invention relates to a step length control apparatus having a multi-stage clutch damper model for analyzing displacement nonlinear behavior characteristics such as a multi-stage clutch damper applied to an actual system, and a method thereof.

In general, the analysis of nonlinear behavior characteristics of real systems involves many difficulties because of the strong nonlinearities that the system itself has. The strong nonlinearity of such a system itself occurs in systems such as multi-stage clutch dampers or the like having asymmetric hysteresis levels at multiple locations.

Conventional examples of the multi-stage clutch damper include a clutch release system of Korean Patent Laid-Open Publication No. 10-1999-0033397, a damper clutch of a torque converter of Korean Patent Laid-open Publication No. 10-2013-0065175, A damper spring having a free length different from that of the main spring in the 'automatic damper clutch control method of an automatic transmission and an automatic transmission' and the 'local dampers of an automotive clutch' of Korean Patent No. 10-1048135 To perform multi-step damping.

In order to analyze the nonlinear dynamic response of a physical system having such a sinusoidal excitation condition, a harmonic balance method (HBM) based on the Galerkin method can be applied. The Galerkin method applied in the present invention assumes that all response characteristics are composed of Fourier series constants. To calculate the exact solution at each step, we use the arc length continuation method with a predictor and a corrector repeatedly. However, applying the step length control method or determining the arc length in the continuous arc method has many difficulties in terms of time efficiency and convergence problem in the calculation of the simulation. In particular, when these issues relate to the interpretation of an actual system with multiple degrees of freedom including extreme nonlinear problems at multiple locations, the optimum step length must be controlled to correspond to the state of the solution at all computational steps.

Korean Patent Publication No. 10-1999-0033397 Korean Patent Publication No. 10-2013-0065175 Korean Patent Publication No. 10-2016-0035297 Korean Patent No. 10-1048135

SUMMARY OF THE INVENTION Accordingly, the present invention has been made keeping in mind the above problems occurring in the prior art, and an object of the present invention is to provide a nonlinear system based on a harmonic balance method (HBM) using a continuous arc method for a nonlinear system such as a multistage clutch damper model, In simulating a continuous method, it is possible to improve the efficiency of simulation and overcome the convergence problem by optimizing the step length by applying the two control methods to calculate the solution between the current step and the next step. An object of the present invention is to provide a step length control apparatus and a method thereof.

According to an aspect of the present invention,

A clutch damper model having a flywheel and a multi-stage clutch damper,

An input unit for receiving an initial condition and a limiting condition set for a step length control including an analysis of an output torque characteristic with respect to an input torque of the multi-stage clutch damper;

A storage unit for storing a program for the initial condition and the constraint condition or the step length control; And

And a controller for driving the program to generate an initial equation using the initial condition and the restriction condition to obtain an initial solution and to calculate an output torque of the clutch using the step length control .

Wherein,

A basic formula generating unit for generating a basic formula consisting of matrices and vectors based on a Galerkin scheme using input initial conditions;

(1, c) and θc (1, c) from the predicted displacement angle θc (1, p) of the initial flywheel by applying the first iterative calculation An initial solution calculator 53 configured to perform the process; And

를 위한 해를 매개변수화된(parameterized) 호장연속법을 사용하여 계산하며, 정규화된(normalized) 주파수

가 정해진

에 도달할 때까지 수행되는 k+1 번째 해 계산부;를 포함하여 구성될 수 있다.Each step (k step)

The solution for the normalized frequency is calculated using a parameterized continuous oscillation method,

Established

And a k + 1 < th >

According to an aspect of the present invention, there is provided a step length control method,

CLAIMS 1. A clutch damper model having a flywheel and a multi-stage clutch damper, comprising: an input unit for receiving initial conditions and limiting conditions set for step length control including analysis of an output torque characteristic with respect to an input torque of a multi-stage clutch damper; A storage unit for storing a program for the initial condition and the constraint condition or the step length control; And a controller for driving the program to generate an initial equation by using the initial condition and the restriction condition, to obtain an initial solution, and to calculate an output torque of the clutch using the step length control, And a k + 1 < th > first harmonic calculation unit, wherein the step length control method comprises:

A basic formula generating step of generating a basic formula consisting of matrices and vectors based on a Galerkin scheme using initial conditions input by the basic formula generating unit;

(1, c) and θc (1, c) from the predicted displacement angle θc (1, p) of the initial flywheel for the application of the continuous arc method by performing the first iteration calculation by the initial- an initial solution calculation process for performing a process of obtaining? And

를 위한 해를 매개변수화된(parameterized) 호장연속법을 사용하여 계산하며, 정규화된(normalized) 주파수

가 정해진

에 도달할 때까지 반복 수행하는 호장연속법 적용 k+1 번째 해 계산과정;을 포함하여 이루어진다.Each step

The solution for the normalized frequency is calculated using a parameterized continuous oscillation method,

Established

And a k + 1-th solution calculation process in which the continuous process is repeated until the k + 1-th solution is reached.

In the basic formula,

는 플라이휠의 관성모멘트,

는 점성 댐핑(viscous damping), ω는 각속도, TEc는 입력토크 푸리에 상수 벡터, H는 이산푸리에변환 행렬, P는 이산푸리에변환 미분 행렬, fnc는 비선형함수 푸리에 상수 벡터를 나타낸다.here,

The moment of inertia of the flywheel,

Is the viscous damping, ω is the angular velocity, TEc is the input torque Fourier constant vector, H is the discrete Fourier transform matrix, P is the discrete Fourier transform differential matrix, and fnc is the nonlinear function Fourier constant vector.

The initial solution calculation process includes:

A predictor calculation process of calculating a predicted displacement angle? C (1, p) of the initial flywheel using an arbitrary arctic length based on an initial condition input torque TE (t);

A Newton-Raphson applied compensator calculation process of calculating a corrected displacement angle? C (1, c) of the initial flywheel by correcting the predicted displacement angle? C (1, p) by applying Newton-Raphson;

A Newton-Raphson time domain differentiation process for performing a derivative in the time domain during the calibrator calculation process and then returning the result;

Determining whether an iteration number ni of the process exceeds a limit number ne given as an initial value;

An initial condition resetting process for resetting the initial condition when ni exceeds ne, and returning to the predictor calculating process;

If ni does not exceed ne, it is determined whether the error (?) Of the calculated correction displacement angle? C (1, c) is smaller than an error? Given as an initial condition, If the error (Ψ) of the corrected displacement angle θc (1, c) is larger than the error (ε) given as the initial condition, an error range determination process for returning to the Newton- ; And

If the error Ψ of the correction displacement angle θc (1, c) is smaller than the error ε given as the initial condition as a result of the determination of the error range determination process, the calculated correction displacement angle θc (1, c) And a correction displacement angle selection process for returning to the process of FIG.

The error (?) Of the corrected displacement angle? C (1, c)

는 플라이휠의 관성모멘트,

는 점성 댐핑(viscous damping), ω는 각속도, TEc는 입력토크 푸리에 상수 벡터, H는 이산푸리에변환 행렬, P는 이산푸리에변환 미분 행렬, fnc는 비선형함수 푸리에 상수 벡터를 나타낸다.here,

The moment of inertia of the flywheel,

Is the viscous damping, ω is the angular velocity, TEc is the input torque Fourier constant vector, H is the discrete Fourier transform matrix, P is the discrete Fourier transform differential matrix, and fnc is the nonlinear function Fourier constant vector.

The k + 1 < th >

(K + 1) -th predictor calculation process for obtaining a (k + 1) th predictor? C (k + 1, p) from the already calculated? C (k, p) using an arbitrary arc length?

Newton-Raphson is applied to calculate the corrected displacement angle θc (k + 1, c) of the (k + 1) th flywheel by applying the Newton-Raphson to correct the predicted displacement angle θc (k + 1, p) k + 1 < th >

K + 1 < th > time domain differentiation process for performing the derivative in the time domain during the k + 1 < th >

K + 1 < th > time domain differentiation process for determining whether the number of iterations ni of the process exceeds a limit number ne given as an initial value;

resetting the ΔSd to return to the (k + 1) -th predictor calculation process after resetting the arbitrary arctangent length ΔSd (ΔS1) when ni exceeds ne as a result of the determination of the step of determining the k + 1th limited repetition number;

If the determination result of the (k + 1) th iteration limit exceeding determination process is that ni does not exceed ne, the error (?) of the calculated k + 1th corrected displacement angle? c (k + 1, c) (k + 1) < th >

If the error (?) of the corrected displacement angle? c (k + 1, c) is smaller than the error? given as the initial condition as a result of the determination of the (k + 1) th error range determination process, k + 1 < th > corrector selection process for selecting (k + 1, c)

(k + 1) -th correction compensator selection process, it is determined whether or not the step is completed after the (k + 1) -th correction displacement angle? c (k + 1, c) is selected; And

As a result of the determination of the step end determination process, if the step is not finished, the arbitrary arctangent length ΔSd (ΔS2) is reset, the k + 2th predictor is calculated, and the process returns to the (k + 1) th compensator calculation process using Newton- And a next predictor calculation process to repeat the predictor calculation process.

The? Sd can be set arbitrarily.

The k + 1 < th >

(? c (k + 1, p) =? c (k, p) +? s1)

Here, p may be a predictor, c may be a subscript indicating a corrector, and a process of calculating a predictor by the initial arctangent length ΔSd of Δs1.

The apparatus and method for controlling the step length of the present invention having the above-described structure can provide an effect of accurately, promptly, and easily analyzing the angular velocity according to the angular displacement of the flywheel for analyzing the input / output torque of the multi-stage clutch damper model to provide.

Further, the present invention provides an effect of reducing the simulation time of the multi-stage clutch damper and improving the efficiency of the step length control by overcoming the convergence problem by reducing the analysis time and the number of steps required in the analysis of the multi-stage clutch damper.

1 is a block diagram of a step length control apparatus 1 of a multi-stage clutch damper model according to an embodiment of the present invention.
BACKGROUND OF THE INVENTION 1. Field of the Invention [0001] The present invention relates to a multi-step clutch damper model,
3 is a flowchart showing a detailed processing procedure of an initialization calculation process (S20) in the process of FIG.
FIG. 4 is a flowchart showing a detailed processing procedure of a (k + 1) th solution calculation step (S30) using a continuous arc method in the processing of FIG.
Fig. 5 is a diagram showing a nonlinear rotating system model with multistage displacement sectional asymmetry (a) a single degree of freedom nonlinear tortional system with a multi-stage clutch damper, (b) a torque for a multi- Tc (? F) profile).
6 is a diagram showing a concept of a predictor and a corrector having a sequence continuation concept of scheme number # 1.
7 is a diagram showing the basic concept of the step length control. ((A) is a step length control by a format number # 1 (scheme 1), and (b) is a step length control by a format number # 2 (scheme)).
FIG. 8 is a graph showing a comparison between simulation with and without step length control ((a) simulation with step length control, and (b) simulation without step length control).
9 is a graph showing the RMS (Root Mean Square) values of the displacements in the frequency domain from the HBM simulation ((a) the result of HBM with step length control (case I), (b) Results from HBM (Case IV).
Figure 10 shows the actual step length values applied during the simulation of the HBM routine ((a) Case I, both Formats # 1 and # 2 applied, (b) Case II, type number # 1 only).
11 shows the actual step length values applied during the simulation of the HBM routine ((a) case III, type number # 2 only, (b) case IV, no step length control).
FIG. 12 shows an example of the actual step length value ((a) case I, both format numbers # 1 and # 2, and (b) case II, format number # 1)
13 shows an actual step length value according to a number of steps ((a) case III, type number # 2 only, (b) case IV, no step length control).

In the following description of the present invention, detailed description of known functions and configurations incorporated herein will be omitted when it may make the subject matter of the present invention rather unclear.

The embodiments according to the concept of the present invention can make various changes and have various forms, so that specific embodiments are illustrated in the drawings and described in detail in this specification or the application. It is to be understood, however, that the intention is not to limit the embodiments according to the concepts of the invention to the specific forms of disclosure, and that the invention includes all modifications, equivalents and alternatives falling within the spirit and scope of the invention. Also, the word "exemplary" is used herein to mean "serving as an example, instance, or illustration." Any aspect described herein as "exemplary " is not necessarily to be construed as preferred or advantageous over other aspects.

It is to be understood that when an element is referred to as being "connected" or "connected" to another element, it may be directly connected or connected to the other element, . On the other hand, when an element is referred to as being "directly connected" or "directly connected" to another element, it should be understood that there are no other elements in between. Other expressions that describe the relationship between components, such as "between" and "between" or "neighboring to" and "directly adjacent to" should be interpreted as well.

The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. The singular expressions include plural expressions unless the context clearly dictates otherwise. In this specification, the terms "comprises ", or" having ", and the like, are intended to specify the presence of stated features, integers, steps, operations, elements, , Steps, operations, components, parts, or combinations thereof, as a matter of principle.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter, the present invention will be described in detail with reference to the accompanying drawings showing embodiments of the present invention.

1 is a block diagram of a step length control apparatus 1 of a multi-stage clutch damper model according to an embodiment of the present invention.

1, the step length control device 1 is a simulation device for performing an analysis on a clutch model (see FIG. 5) having a flywheel 100 and a multi-stage clutch damper 120, A storage unit 20, an output unit 30, a display unit 40, a basic equation generating unit 51, an initial solution calculating unit 53, and a k + 1th solution calculating unit 55 ).

The input unit 10 includes a keyboard input port to which a keyboard receiving an input of an initial condition and a restriction condition set for a step length control including analysis of an output torque characteristic with respect to an input torque of the multi-stage clutch damper, a USB input device, A read-write device such as a disk drive capable of recording and reading data to or from a storage device such as a device or an optical disk storage device, or a file input device capable of receiving a file input. Specifically, the initial condition for the analysis of the multi-stage clutch damper includes the input torque, and the restriction condition includes the calculation repetition limit number (ne) and the given error (epsilon).

The storage unit 20 may comprise a storage device such as a hard disk, a ROM, and an EPROM for storing initial conditions and constraints input through the input unit 10 or a program for controlling the step length.

The output unit 30 is configured to output a simulation result, and may include a printer, a plotter, and the like.

The display unit 40 may be constituted by a display device or the like as an information display device for confirming the driving state of the step length control device 1 such as a simulation process and a calculation result value executed in the step length control device 1. [ have.

The controller 50 controls the step length controller 1 to drive a program for performing a simulation on a system having a displacement section nonlinearity including a multi-stage clutch damper model, And to calculate an output torque of the clutch by using the step length control.

To this end, the basic formula generating unit 51 of the control unit 50 is configured to generate a basic formula consisting of matrices and vectors based on a Galerkin scheme using input initial conditions.

에서의 해를 얻기 위한 처리 절차를 수행하고, 도 2에서 S20으로 표시된 절차를 수행하도록 구성된다. 즉, 초기해계산부(53)는 첫 번째 반복 계산을 수행하여 호장연속법의 적용을 위해 초기플라이휠의 예측 변위각 θc(1,p)로부터 보정 변위각 θc(1,c)와 θc(1,c)에서의 각속도 ω를 구하는 처리과정을 수행하도록 구성된다(도 6 참조). 이때

는 초기 조건에 매우 종속적이며,

는 예측변위가 θc(1,p)와 보정변위각 θc(1,c)을 포함한다.The initial solution calculation unit 53 is configured to perform a process of performing a first iteration calculation,

And to perform the procedure indicated by S20 in Fig. That is, the initial solution calculator 53 performs the first iterative calculation to calculate the corrected displacement angles? C (1, c) and? C (1, p) from the predicted displacement angles? , c) of the angular velocity? (see Fig. 6). At this time

Is highly dependent on initial conditions,

(1, p) and the corrected displacement angle? C (1, c).

를 위한 해를 매개변수화된(parameterized) 호장연속법을 사용하여 계산하며, 정규화된(normalized) 주파수

가 정해진

에 도달할 때까지 수행되도록 구성되고, 도 2에서 S30으로 표시된 절차를 수행하도록 구성된다.The (k + 1) th solution calculation unit 55 calculates

The solution for the normalized frequency is calculated using a parameterized continuous oscillation method,

Established

, And is configured to perform the procedure indicated by S30 in Fig.

FIG. 2 is a flowchart showing a process procedure of a step length control method of a multi-stage clutch damper model according to an embodiment of the present invention. FIG. 3 is a flowchart showing a detailed process of an initialization calculation process (S20) And FIG. 4 is a flowchart showing a detailed processing procedure of the (k + 1) th solution calculation step (S30) using the continuous arc method in the processing of FIG.

2 through 4, the process of the flowchart of FIG. 3 is the scheme number # 1, and the process of the flowchart of FIG. 4 is defined by the scheme number # 2 method do.

As shown in FIG. 2, the step length control method includes a basic formula generation step S10, an initial solution calculation step S20, and a k + 1th solution calculation step S30 using a continuous arc method.

The basic formula generating process S10 is a process of generating a basic formula consisting of matrices and vectors based on a Galerkin scheme using initial conditions input by the basic formula generating unit 51 .

에서의 해를 얻기 위한 처리 절차를 수행하도록 구성된다. 즉, 초기해계산과정(S20)은 초기해계산부(53)가 첫 번째 반복 계산을 수행하여 호장연속법의 적용을 위해 초기플라이휠의 예측 변위각 θc(1,p)로부터 보정 변위각 θc(1,c)와 θc(1,c)에서의 각속도 ω를 구하는 처리과정을 수행하도록 구성된다. 이때

는 초기 조건에 매우 종속적임은 상술한 바와 같다. 여기서 p는 예측자(predictor)임을, c는 보정자(corrector)임을 나타낸다.The initial solution calculation step (S20) is configured to perform a process of performing a first iteration calculation,

In order to obtain a solution at the point of time t. That is, in the initial solution calculation process S20, the initial solution calculation unit 53 performs the first iteration calculation to calculate the corrected displacement angle? C (1, p) from the predicted displacement angle? 1, c) and the angular velocity? In? C (1, c). At this time

Is highly dependent on the initial conditions as described above. Where p is a predictor and c is a corrector.

Referring to FIG. 3, the initial solution calculation step S20 will be described in more detail.

3, the initial solution calculation step S20 includes a predictor calculation step S21, a Newton-Raphson applied compensator calculation step S22, a time domain differentiation step S23, an over-limit repetition determination step S24, An initial condition resetting process S25, an error range determining process S26, and a correction displacement angle selecting process S27.

The predictor calculation process S21 is configured to calculate the predicted displacement angle? C (1, p) of the initial flywheel using an arbitrary arctic length based on the initial condition of the input torque TE (t).

The Newton-Raphson applied compensator calculation process S22 is configured to calculate the corrected displacement angle? C (1, c) of the initial flywheel by correcting the predicted displacement angle? C (1, p) by applying Newton-Raphson .

The time domain differential process S23 is configured to perform a derivative in the time domain of the Newton-Raphson applied compensator calculation process S22, and then return a result value.

The step S24 of judging whether or not the repetition number of times of repetition exceeds the number n2 of repetitions of the above-described processing exceeds the number of repetition ns given as the initial value.

In the initial condition resetting process S25, if it is determined in step S24 that the ni exceeds ne, the initial condition resetting process is performed. Then, the process returns to the predictor calculating process S21 So that the process is repeated.

In the error range determination step S26, if the error psi of the calculated corrected displacement angle? C (1, c) is given as an initial condition when ni does not exceed ne as a result of the determination in the step S24, Is smaller than the error?.

If the error? Of the corrected displacement angle? C (1, c) is greater than the error? Given as the initial condition as a result of the determination in the error range determination step S26, the process returns to the Newton- So that the process is repeated.

 In the correction displacement angle selection process S27, if it is determined in the error range determination process S26 that the error Ψ of the corrected displacement angle θc (1, c) is smaller than the error ε given as the initial condition, After selecting the angle? C (1, c) as the corrector, the process returns to the process of FIG.

를 위한 해를 매개변수화된(parameterized) 호장연속법을 사용하여 계산하며, 정규화된(normalized) 주파수

가 정해진

에 도달할 때까지 수행되도록 구성된다.Referring to FIG. 2 again, the (k + 1) -th calculation process S30 of applying the continuous-sequence method is performed in each step

The solution for the normalized frequency is calculated using a parameterized continuous oscillation method,

Established

As shown in FIG.

Referring to FIG. 4, the k + 1 th calculation process S30 includes a k + 1 th predictor calculation process S31, a Newton-Raphson applied k + 1 th corrector calculation process S32, (K + 1) -th limit repetition number determining step S34, the? Sd resetting step S35, the k + 1th error range determining step S36, (S37), a step end determination process (S38), and a next predictor calculation process (S39).

The k + 1th predictor calculation process S31 calculates the k + 1th predictor θc (k + 1, p) from the already calculated θc (k, p) using an arbitrary length ΔSd (ΔS1) . At this time,? Sd is a value arbitrarily set.

The Newton-Raphson applied k + 1th compensator calculation process (S32) corrects the predicted displacement angle θc (k + 1, p) (predictor) by applying Newton- (K + 1, c).

The (k + 1) -th time domain differentiation process S33 is configured to perform the derivative in the time domain of the (k + 1) th calibrator calculation process S32 of the Newton-Raphson applied and then return the result value.

The k + 1th exceeding limit number exceeding determination process S34 determines whether the number of iterations ni of the above process exceeds the limit number ne given as an initial value.

The? Sd resetting process S35 is a process of resetting the arbitrary arctangent length? Sd (? S1) when ni exceeds ne as a result of the determination in the step S34 of judging the exceeding of the (k + (S31) so as to repeat the process.

The (k + 1) -th error range determining step S36 is a step of determining whether or not the calculated k + 1th corrected displacement angle is configured to determine if the error ([psi] ) of [theta] c (k + 1, c) is less than the given error [epsilon] as an initial condition.

The (k + 1) -th corrector selection process S37 determines whether the error Ψ of the corrected displacement angle θc (k + 1, c) as a result of the k + 1th error range determination process (S36) 1) th correction displacement angle? C (k + 1, c) as the corrector.

After the k + 1th corrected displacement angle? C (k + 1, c) is selected in the (k + 1) th corrector selection process S37, the step end determination process S38 determines whether or not the step is completed. The process returns to the process of FIG. 2.

The next predictor calculation process S39 resets the arbitrary arctangent length DELTA Sd (DELTA S2) when the step is not finished as a result of the determination in the step end determination process (S38), calculates a k + 2 & And then returns to the +1-th compensator calculation process to repeat the process.

<Examples>

The embodiment of the present invention will be described with reference to the processing steps of FIG. 2 to FIG.

Fig. 5 is a diagram showing a nonlinear rotating system model with multistage displacement sectional asymmetry (a) a single degree of freedom nonlinear tortional system with a multi-stage clutch damper, (b) a torque for a multi- Tc (? F) profile).

Figure 5 shows a single degree of freedom system with asymmetric hysteresis with a multi-step clutch damper. In the present invention, the input torque TE (t) is applied to the flywheel 100, and the clutch has four stages of displacement-variable linear springs employed as the multi-stage clutch damper 120. And resistive torque 110 and viscous damping are shown as actual damping 130. [ Hysteresis is caused by friction members located between the friction plates and linear springs configured asymmetrically. The hypothetical dynamic characteristics of the multi-stage clutch damper are based on statistical measurements, and their probabilities are summarized in Table 1.

[Table 1] Characteristics of real-life multi-stage clutch damper

In order to simulate the system response under sinusoidal oscillation in both frequency domain and time domain, a harmonic balance method (HBM) using various techniques can be applied. The various techniques are residual formulations and a continuous method using the Newton-Raphson method.

The actual system of Figure 5 is limited to steady state conditions under sinusoidal oscillation with a single degree of freedom.

Problem formulation

Based on the system given in FIG. 5, the governing equations and the non-linear functions are derived as follows.

[Equation 1]

&Quot; (2) &quot;

&Quot; (3) &quot;

,

,

&Quot; (4) &quot;

&Quot; (5) &quot;

&Quot; (6) &quot;

&Quot; (7) &quot;

&Quot; (8) &quot;

&Quot; (9) &quot;

here,

: 플라이휠의 관성모멘트

: Moment of inertia of flywheel

: 점성 댐핑(viscous damping)

: Viscous damping

: 프리로드(preload)에 의한 전체 토크(torque)

: Total torque by preload

및

: 프리로드에 의해 인가되는 양의 토크와 음의 토크

And

: Positive torque and negative torque applied by the preload

: 프리로드 토크는

의 함수)(

: The preload torque

Function)

: 프리로드 각도(angle located at the preload)

: Angle located at the preload

: 클러치 강성(stiffness)의 i번째 단계

: I-th step of clutch stiffness

및

: i번 째 단계의 강성에 의해 인가된 양과 음의 클러치 토크

And

: positive and negative clutch torque applied by the stiffness of the i-th stage

및

: 양과 음의 i번 째 전이각(transition angle)

And

: I-th transition angle between positive and negative (transition angle)

: 히스테리시스에 의해 인가된 토크

: Torque applied by hysteresis

: i 번째 단계의 히스테리시스

: Hysteresis of the i-th stage

및

: i 번째 단계의 히스테리시스에 의해 인가되는 양과 음의 클러치 토크

And

: Positive and negative clutch torque applied by the hysteresis of the i &lt; th &gt;

: 클러치 토크(테이블 1: 클러치 토크 리스트)

: Clutch torque (Table 1: clutch torque list)

: 플라이휠의 절대 변위(도 5 참조)

: Absolute displacement of the flywheel (see FIG. 5)

to be.

The input torque TE (t) is given by the following equation (10).

&Quot; (10) &quot;

Where Tm and Tpi are the mean and alternating values of the input torque, ωp and φpi are the vibration frequency and phase angle, and Nmax is the number of harmonics.

The present invention can be limited to the sinusoidal frequency condition within the range of the input torque as shown in Equation (11).

&Quot; (11) &quot;

(Reference 1: Yoon JY and Yoon HS. Nonlinear frequency response analysis of a multi-stage clutch damper with multiple nonlinearities. ASME J Computat Nonlinear Dynam 2014; 9: 031007.

Reference 2: Kim, TC, Rook, and Singh, R. Super and sub harmonic response calculations for a torsional system with clearance nonlinearity using the harmonic balanced method. J Sound Vibrat 2005; 281: 965-993.)

FIGS. 2 and 3 are flowcharts showing the process of simulation using the HBM.

Generally, the HBM used in the simulation consists of three parts as shown in Figs. 2 and 3.

The first is a basic formula consisting of matrices and vectors based on a Galerkin scheme described by [Equation 11].

에서의 해를 얻기 위하여 수행된다. 이때

는 초기 조건에 매우 종속적이며, 도 2에서 S20로 표시된 절차이다.The second is that the first iteration of the first step

In order to obtain the solution at. At this time

Lt; / RTI &gt; is highly dependent on the initial conditions and is the procedure labeled S20 in Figure 2.

를 위한 해가 매개변수화된(parameterized) 호장연속법을 사용하여 계산되며, 정규화된(normalized) 주파수

가 정해진

에 도달할 때까지 수행되고, 도 2에서 S30으로 표시된 절차이다.Third, each step

Is calculated using a parameterized continuous oscillation method, and the solution for the normalized frequency &lt; RTI ID = 0.0 &gt;

Established

, And is the procedure indicated by S30 in Fig.

4 is a flowchart showing a simulation procedure for each step in a scheme to which two step length control is applied, which is defined as S20 and S30 in Fig.

의 계산은 유효한 시뮬레이션의 달성을 위해 적합한 스텝 길이를 가지도록 동기화된다.As shown in FIGS. 2, 3 and 4, each Newton-Raphson repeat in each step and

Is synchronized to have an appropriate step length for achieving a valid simulation.

Development of Step Length Control Method with Piecewise Type Nonlinearity

FIGS. 2, 3 and 4 are flow charts showing the overall processing of the HBM for simulating the nonlinear dynamic response characteristics of the system of FIG. FIGS. 3 and 4 show the entire process of steps S20 and S30 of FIG.

을 얻기 위한 절차를 나타낸다. 도 3에서 ni는 실행되는 계산의 반복 차수를 나타내며, ne는 초기조건으로 주어진 제한된 반복차수이다. ε은 [수학식 12]의 나머지 공식(residual formulation)에 대한 주어진 오차(error)이다.Figure 3 shows the

. Fig. In Fig. 3, ni denotes the repetition degree of the calculation being performed, and ne is the limited repetition degree given as the initial condition. epsilon is a given error for the residual formulation of equation (12).

&Quot; (12) &quot;

P denotes a discrete Fourier transform differential matrix, and fnc denotes a nonlinear function Fourier constant vector.

은 정확한

이 발견될 때까지의 다수의 초기 조건을 적용하여 얻어진다. 이 후 k 스텝에서의

항의 해는 도 4와 같이 계산된다.

와

는 예측자와 보정자를 나타낸다.The exact solution of each step during the calculation is obtained based on the concept of a predictor and a corrector in both the frequency domain and the time domain, and the derivative of the nonlinear function is performed in the time domain. here,

Is accurate

Is obtained by applying a number of initial conditions up to the discovery. Then,

The protest solution is calculated as shown in FIG.

Wow

Represents the predictor and the corrector.

While the k-th simulation is being performed, two concepts of a step length control scheme represented by S20 and S30 are employed as shown in Fig. In general, the step length control technique must be applied for a variety of reasons. For example, the step length control technique can reduce computation time and simulation cost. Thus, the step length is increased with small changes in the solution. On the other hand, the step length control technique must reduce the unstable elements resulting from the stiff variation in the interpretation. In particular, the stiff problem arising from a sudden change in parameters due to numerical analysis can result in the analysis result staying at one point, or not leading to the calculation result of the next step, . Therefore, it may not be possible to calculate the solution in the entire frequency domain using only the call method. In order to solve such a problem, in the present invention, two schemes of the step length control technique are considered as shown in FIG.

6 is a diagram showing a concept of a predictor and a corrector having a sequence continuation concept of scheme number # 1.

는 도 6과 같이 제어 인자 α를 곱해서 수정된다. First, the initially given step length value can be controlled corresponding to the number of iterations, as shown in Figure 6, and is defined as scheme # 1. Here, the step length initially given in the k-th step

Is corrected by multiplying the control factor? As shown in FIG.

&Quot; (13) &quot;

는 초기적으로 주어지는 스텝 길이 크기를 가지는 호 길이

Lt; RTI ID = 0.0 &gt; length &lt; / RTI &gt;

는 α를 곱해서 수정된 스텝 길이,

Lt; RTI ID = 0.0 &gt; a &lt; / RTI &gt;

ni or ne is the number of iterations during which the simulation is performed (or a given value of the number of iterations initially given)

는 반복적으로 α가 곱해진다. 본 발명을 위하여 적용된 특성들이 테이블 2에 기재되어 있다.Thus, until the arc length reaches the corrected solution,

Is repeatedly multiplied by a. The characteristics applied for the present invention are described in Table 2.

7 is a diagram showing the basic concept of the step length control. ((A) is a step length control according to a format number # 1 (scheme 1), and (b) is a step length control with a format number # 2 (scheme)).

Second, the step length can be adjusted by the relationship between the arc length and angle, as shown in FIG. 7, and is defined as format number # 2 (scheme # 2).

(또는

)는 도 4와 같이 접선 벡터들

,

(또는

와

)로 구성된다. 따라서 초기에 주어진 호의 길이가 해로 적합하다고 가정되는 한, 다음 단계의 호 길이가

관계식에 의해 자동적으로 적용되어 질 수 있다. 각도

와

의 크기는 반복 횟수 ni에 비례하는 것으로 가정될 수 있다. 그러나 이러한 계산은 (k+1) 번째 스텝의 해가 극단적으로 변하는 때에 불안정하다. 예를 들어, 만일

이고

이면,

는 무한대이다. 또한, 만일

이고,

이면, 스텝 길이는 해로부터 발산되는 반복 횟수를 가지는 매우 높은 레벨에 도달할 수도 있다.Here,

(or

As shown in FIG. 4,

,

(or

Wow

). So, as long as the initial length of the given arc is assumed to be the solution, the arc length of the next step is

Can be automatically applied by a relational expression. Angle

Wow

Can be assumed to be proportional to the number of repetitions ni. However, such a calculation is unstable when the solution of the (k + 1) th step changes extreme. For example,

ego

If so,

Is infinite. Also,

ego,

, The step length may reach a very high level with a repetition frequency that is diverged from the solution.

Constraints can solve these problems.

에는

가 할당된다.

는 첫 번째 스텝에서 정확한 해로 수렴되기에 충분하게 정의된다. 둘 째, 다른 스텝 길이 제어 인자

가

로 정의된다. 여기서

및

는 도 7b에 나타난 바와 같이,

의 하부와 상부 레벨이다. 따라서 각 스텝의 스텝 길이는

에 의해 계산된다.

를 구하기 위해서는, [수학식 14]와 같은 분수함수가 고려될 수 있다. first,

There

.

Is sufficiently defined to converge to the correct solution in the first step. Second, other step length control factors

end

. here

And

As shown in Fig. 7B,

And the upper level. Therefore, the step length of each step is

Lt; / RTI &gt;

, A fractional function such as Equation (14) can be considered.

&Quot; (14) &quot;

=1 x 10¹⁰ 및

= 1.1 이 본 발명의 실시예에 적용되었다. 따라서

=

의 조건이 만족된다면,

= 1 이고, 이는 이전 단계의 스텝 길이를 현재 단계의 스텝 길이와 동일한 값으로 만든다. 그렇지 않으면, 스텝 길이는 [수학식 14]에 의해 항상 적응적으로 변환된다. 따라서 형식 번호 #1과 #2는 각각

와

적응적 크기(adaptive sizes)를 가지는 스텝 길이 제어 방법이다.

= 1 x 10 &lt; 10 &^gt;

= 1.1 was applied to the embodiment of the present invention. therefore

=

Is satisfied,

= 1, which makes the step length of the previous step equal to the step length of the current step. Otherwise, the step length is always adaptively transformed by (14). Therefore, the format numbers # 1 and # 2 are

Wow

This is a step length control method with adaptive sizes.

Result

[Table 2]

와

는 도 4 내지 도 7에 도시되고 [수학식 13] 및 [수학식 14]의 방정식 내에서 기술된 바와 같이 사용된다. 오직

을 적용한 형식 번호 #1은 케이스 II를 위해 사용되고, 오직

를 적용한 형식 번호 #2는 케이스 III을 위해 사용되며, 케이스 IV에서는 스텝 길이 제어 방법이 사용되지 않는다. 또한, 케이스 III과 IV는

의 초기값으로 0.001을 사용하였는데, 이것은 케이스 III과 IV의 경우에는 케이스 I 및 케이스 II 에 적용한 동일한 값들을 가지고서는 수렴 문제를 극복하지 못하기 때문이다.Table 2 shows an example of four step length controls indicated by cases I, II, III and IV. Case I employs format number # 1 and format number # 2 to elaborate the development of a step length control method with a problem formulation and a displacement section nonlinearity part. Adaptive arc length

Wow

Is used as described in Figs. 4 to 7 and described in the equations of [Equation 13] and [Equation 14]. Only

Is used for case II, and only &lt; RTI ID = 0.0 &gt;

Is used for case III, and in case IV, the step length control method is not used. In addition, cases III and IV

In the case of cases III and IV, the same values applied in case I and case II can not overcome the convergence problem.

8 is a graph showing a comparison between simulations with and without step length control ((a) simulation with step length control, (b) simulation without step length control).

=0.01인 케이스 III 및 케이스 IV에서도 발생한다. 따라서 케이스 III 및 IV를 위한 표 2에 묘사된 바와 같이,

=0.001가 이후의 설명에서 사용된다.In Figure 8, two simulation procedures with and without convergence problems are compared. For example, Figure 8 (a) shows the corrector obtained in the third iteration. However, FIG. 8 (b) shows that at any step the simulation never reaches the correct solution after multiple iterations, even if the number of iterations is 20 or more. In addition,

= 0.01 in Case III and Case IV. Thus, as depicted in Table 2 for cases III and IV,

= 0.001 is used in the following description.

9 is a graph showing RMS (root mean square) values of displacement in the frequency domain from the HBM simulation ((a) the result of HBM with step length control (case I), Results of HBM (Case IV).

는 스텝 길이 제어를 가지지 않는 케이스 IV의 값과 비교했을 때, 전체 시스템 응답특성의 분석을 위해 계산하는데 필요한 스텝수가 매우 적게 소요됨을 알 수 있다. 또한, 도 9과 표 3에 비교한 바와 같이, 케이스 1의 결과로부터 비록 분석에 필요한 스텝 수가 줄어들었지만, 계산 시간은 감소되면서도 비선형 동역학 특성을 분석하기 위한 충분한 정보를 포함하고 있는 것을 알 수 있다.As shown in Fig. 9, the RMS value of case I with step length control

Can be seen to require very few steps to compute for the analysis of the overall system response characteristics, as compared to the value of case IV having no step length control. 9 and Table 3, it can be seen from the result of Case 1 that although the number of steps required for the analysis is reduced, the calculation time is reduced and sufficient information for analyzing the nonlinear dynamics characteristic is included.

 [Table 3] Comparison of computation time per step length control case

Table 3 compares the calculation time for each case. Here, the calculation times are processed and measured for each case in order to determine the average value, so that the multiple computing conditions such as the background work of the simulation device itself cause a difference in calculation time for each trial . The approach used in Case I is most effective in reducing computation time compared to other cases. For the evaluation of the performance, the ratios of the calculation times were compared to the normalized values based on the calculation time of the case IV in which the step length control was not used. For example, the ratio of the calculation time of case 1 is very small, less than 1, while case III has a value of 1 or more.

Interesting results were observed in the calculation times of cases III and IV. As shown in FIG. 4, even though format number # 2 was applied to case III for adjustment of the size of the arc length, it did not contribute to reducing the calculation time compared to case IV. This indicates that the involvement of the adaptive step length control technique in a simulation set to a step length sufficient to overcome the convergence problem can increase the computation time and these results can be compared by subsequent simulation results .

Figure 10 is a graph showing the actual step length values applied during the simulation of the HBM routine ((a) Case I, both form numbers # 1 and # 2, (b) Case II, type number # 1 only). And Fig. 11 is a graph showing the actual step length value applied during the simulation of the HBM routine ((a) case III, type number # 2 only, (b) case IV, no step length control).

영역에서의 각 케이스별 실제 적용된 스텝 길이를 비교한다. 케이스 I의 경우, 결과가 검토될 때,

및

급격한 변화는 각각 0.2 이하 및 1로 발생하였다. 특히

는 수렴 및 각 단계의 강성 조건(stiffness condition)에 따라 넓은 범위의 값을 가진다. 그러나 도 10(b)에 보여지는 바와 같이, 케이스 II는 오직

값의 변화만 관측할 수 있고, 이는 [표 2]와 같이 오직

적응적 조건만이 적용되었기 때문이다.Figs. 10 and 11 show the relationship between the adopted step length and the

Compare the actual applied step length for each case in the area. In Case I, when the results are reviewed,

And

Sudden changes occurred less than 0.2 and 1 respectively. Especially

Has a wide range of values depending on the convergence and the stiffness condition of each step. However, as shown in Fig. 10 (b)

Only the change of the value can be observed,

Only adaptive conditions are applied.

=0.001을 가지는

및

의 실제 값들을 비교한다. 도 13(a)에 도시된 바와 같이, 케이스 III에서는 오직

가

=1 이하의 적응적 값만을 나타낸다. 그러나 케이스 III의

의 변화 범위는, 도 10(b)와 같이, 케이스 I에 비해 매우 작으며, 케이스 IV의 경우에는 도 11(b)와 같이,

와

에는 변화가 없었다. 특히, 케이스 III의

는 케이스 I의

보다 한 차수 작으며, 이것이

및

의 변화에 영향을 미쳤다. 따라서

의 작은 값이 적응적(adaptive) 값과

및

의 범위를 감소시키는 것을 알 수 있다. 스텝 길이 값들의 적용은 실제로 적용된 스텝 길이 대 스텝 k의 수의 관계에 기초하여 조사될 수 있다.FIG. 11 is a cross-

= 0.001

And

&Lt; / RTI &gt; As shown in Fig. 13 (a), in case III, only

end

= 1 or less. However, Case III

As shown in Fig. 10 (b), the change range of the case IV is very small as compared with the case I, and as shown in Fig. 11 (b)

Wow

There was no change. In particular, Case III

Case I

Lt; RTI ID = 0.0 &gt;

And

Of the respondents. therefore

The smaller value of &lt; RTI ID = 0.0 &gt;

And

Is reduced. The application of the step length values can be examined based on the relationship of the actually applied step length to the number of step k.

12 is an actual step length value according to a number of steps ((a) Case I, both of format numbers # 1 and # 2, and (b) case II, format number # 1). 13 is an actual step length value according to a number of steps ((a) case III, type number # 2 only, (b) case IV, no step length control).

와

의 변화를 나타낸다. 즉,

(또는

)는 k=1385 이하(또는 이상)에서 큰 변화를 나타냈다. 케이스 II는

가 k=2653 까지 케이스 I보다 극심한 변화의 넓은 범위를 나타냈다. 도 13(a)에 보여지는 바와 같이, 케이스 III의 경우,

만이 k=6912까지의 조정적인(adaptive)인 변화 값을 보이고 있고, 이는 케이스 IV와 비교해서 계산시간의 지연(lag)을 발생시킨다. 결국, 도 12 내지 도 13의 결과로부터,

와

의 조정적인 값들(adaptive values)과 범위들은 초기에 주어지는

과 매우 관련된다. 예를 들어,

가 미세하면, 스텝 길이 제어 방법들은 형식 #3 및 #4에서 묘사된 바와 같이 간단(또는 불필요)해 지나, 시뮬레이션 시간은 증가된다. 반면, 스텝 길이 제어의 적용은 계산 시간을 감소시키는 효율성을 높일 수 있으며, 이는 표 3과 도 12 및 도 13에서 명확히 관찰되었다. 특히, 시뮬레이션이 다양한 매개 변수를 필요로 하는 경우, 형식 #1 및 #2와 같은 스텝 길이 제어 방법은 형식 #3과 #4와 비교하여 효율성을 증가시킬 수 있다. 또한, 형식 #1 및 #2는 해가 강성 영역(stiff range) 내에 존재하는 경우 수렴 문제를 극복한다. 따라서 스텝 길이 제어 방법은 적합한 해의 효율적인 계산을 위하여 조정적으로(adaptive) 정확한 스텝 사이즈를 평가할 수 있도록 한다.FIGS. 12 and 13 show different information in comparison with FIG. 10 and FIG. For example, Case I occurs at each step in the interpretation

Wow

. In other words,

(or

) Showed a large change at k = 1385 or less (or more). Case II

Showed a wider range of extreme changes than case I up to k = 2653. As shown in Fig. 13 (a), in case III,

Only shows an adaptive change value up to k = 6912, which causes a lag of computation time compared to case IV. As a result, from the results shown in Figs. 12 to 13,

Wow

The adaptive values and ranges of &lt; RTI ID = 0.0 &gt;

. E.g,

The step length control methods are simple (or unnecessary) as described in Format # 3 and # 4, but the simulation time is increased. On the other hand, the application of the step length control can increase the efficiency of reducing the calculation time, which is clearly observed in Table 3, Figs. 12 and 13. Particularly, if the simulation requires various parameters, the step length control methods such as formats # 1 and # 2 can increase the efficiency compared to formats # 3 and # 4. Formats # 1 and # 2 also overcome the convergence problem when the solution is within a stiff range. Thus, the step length control method allows for an adaptive and accurate evaluation of the step size for efficient calculation of the solution.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the invention is not limited to the disclosed exemplary embodiments. It will be apparent to those skilled in the art that various modifications may be made without departing from the scope of the present invention. Accordingly, the true scope of the present invention should be determined by the technical idea of the appended claims.

100: flywheel 110: resistance torque
120: multi-stage clutch damper
130: Actual damping (damping value due to the viscous force present between the flywheel and the friction clutch)

Claims (9)

In a clutch model having a flywheel and a multi-stage clutch damper,
An input unit for receiving an initial condition and a limiting condition set for a step length control including an analysis of an output torque characteristic with respect to an input torque of the multi-stage clutch damper;
A storage unit for storing a program for the initial condition and the constraint condition or the step length control; And
And a controller for calculating the output torque of the clutch using the step length control by driving the program to generate an initial equation using the initial condition and the restriction condition,
Wherein,
A basic formula generating unit for generating a basic formula consisting of matrices and vectors based on a Galerkin scheme using input initial conditions;
(1, c) and θc (1, c) from the predicted displacement angle θc (1, p) of the initial flywheel by applying the first iterative calculation An initial solution calculator 53 configured to perform the process; And
Each step (k step)

The solution for the normalized frequency is calculated using a parameterized continuous oscillation method,

Established

And a (k + 1) &lt; th &gt;
delete A clutch model having a flywheel and a multi-stage clutch damper, comprising: an input part for receiving initial conditions and constraints set for step length control including analysis of an output torque characteristic with respect to an input torque of the multi-stage clutch damper; A storage unit for storing a program for the initial condition and the constraint condition or the step length control; And a controller for driving the program to generate an initial equation by using the initial condition and the restriction condition, to obtain an initial solution, and to calculate an output torque of the clutch using the step length control, And a k + 1 &lt; th &gt; first harmonic calculation unit, wherein the step length control method comprises:
A basic formula generating step of generating a basic formula consisting of matrices and vectors based on a Galerkin scheme using initial conditions input by the basic formula generating unit;
(1, c) and θc (1, c) from the predicted displacement angle θc (1, p) of the initial flywheel for the application of the continuous arc method by performing the first iteration calculation by the initial- an initial solution calculation process for performing a process of obtaining? And
Each step

The solution for the normalized frequency is calculated using a parameterized continuous oscillation method,

Established

And a k + 1th-order calculation process of repeating the process until the time k + 1 is reached.
4. The method of claim 3,

here,

The moment of inertia of the flywheel,

Is a viscous damping, ω is angular velocity, TEc is an input torque Fourier constant vector, H is a discrete Fourier transform matrix, P is a discrete Fourier transform differential matrix, and fnc is a nonlinear function Fourier constant vector.
4. The method of claim 3,
A predictor calculation process of calculating a predicted displacement angle? C (1, p) of the initial flywheel using an arbitrary arctic length based on an initial condition input torque TE (t);
A Newton-Raphson applied compensator calculation process of calculating a corrected displacement angle? C (1, c) of the initial flywheel by correcting the predicted displacement angle? C (1, p) by applying Newton-Raphson;
A Newton-Raphson time domain differentiation process for performing a derivative in the time domain during the calibrator calculation process and then returning the result;
Determining whether an iteration number ni of the process exceeds a limit number ne given as an initial value;
An initial condition resetting process for resetting the initial condition when ni exceeds ne, and returning to the predictor calculating process;
If ni does not exceed ne, it is determined whether the error (?) Of the calculated correction displacement angle? C (1, c) is smaller than an error? Given as an initial condition, If the error (Ψ) of the corrected displacement angle θc (1, c) is larger than the error (ε) given as the initial condition, an error range determination process for returning to the Newton- ; And
If the error Ψ of the correction displacement angle θc (1, c) is smaller than the error ε given as the initial condition as a result of the determination of the error range determination process, the calculated correction displacement angle θc (1, c) And a correction displacement angle selecting step of returning to the process of FIG.
The method according to claim 5, wherein Correction displacement angle The error? of? c (1, c)

here,

The moment of inertia of the flywheel,

Is a viscous damping, ω is angular velocity, TEc is an input torque Fourier constant vector, H is a discrete Fourier transform matrix, P is a discrete Fourier transform differential matrix, and fnc is a nonlinear function Fourier constant vector.
4. The method of claim 3, wherein the (k + 1)
(K + 1) -th predictor calculation process for obtaining a (k + 1) th predictor? C (k + 1, p) from the already calculated? C (k, p) using an arbitrary arc length?
Newton-Raphson is applied to calculate the corrected displacement angle θc (k + 1, c) of the (k + 1) th flywheel by applying the Newton-Raphson to correct the predicted displacement angle θc (k + 1, p) k + 1 &lt; th &gt;
K + 1 &lt; th &gt; time domain differentiation process for performing the derivative in the time domain during the k + 1 &lt; th &gt;
K + 1 &lt; th &gt; time domain differentiation process for determining whether the number of iterations ni of the process exceeds a limit number ne given as an initial value;
resetting the ΔSd to return to the (k + 1) -th predictor calculation process after resetting the arbitrary arctangent length ΔSd (ΔS1) when ni exceeds ne as a result of the determination of the step of determining the k + 1th limited repetition number;
If the determination result of the (k + 1) th iteration limit exceeding determination process is that ni does not exceed ne, the error (?) of the calculated k + 1th corrected displacement angle? c (k + 1, c) (k + 1) &lt; th &gt;
If the error (?) of the corrected displacement angle? c (k + 1, c) is smaller than the error? given as the initial condition as a result of the determination of the (k + 1) th error range determination process, k + 1 &lt; th &gt; corrector selection process for selecting (k + 1, c)
(k + 1) -th correction compensator selection process, it is determined whether or not the step is completed after the (k + 1) -th correction displacement angle? c (k + 1, c) is selected; And
As a result of the determination of the step end determination process, if the step is not finished, the arbitrary arctangent length ΔSd (ΔS2) is reset, the k + 2th predictor is calculated, and the process returns to the (k + 1) th compensator calculation process using Newton- And a next predictor calculation step of repeating the step of estimating the step length.
The method of claim 7,
And? Sd is arbitrarily set.
8. The method of claim 7, wherein the (k + 1)
(? c (k + 1, p) =? c (k, p) +? s1)
Here, p is a predictor, c is a subscript indicating a corrector, and? S1 is a process of calculating a predictor by an initial arc length length? Sd.

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