KR20120132099A - Non-parametric simultaneous identification of both the nonlinear damping and restoring characteristics of nonlinear systems whose dampings depend on velocity alone - Google Patents

Non-parametric simultaneous identification of both the nonlinear damping and restoring characteristics of nonlinear systems whose dampings depend on velocity alone Download PDF

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KR20120132099A
KR20120132099A KR1020110050714A KR20110050714A KR20120132099A KR 20120132099 A KR20120132099 A KR 20120132099A KR 1020110050714 A KR1020110050714 A KR 1020110050714A KR 20110050714 A KR20110050714 A KR 20110050714A KR 20120132099 A KR20120132099 A KR 20120132099A
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nonlinear
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damping
restoring force
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장택수
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부산대학교 산학협력단
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Abstract

PURPOSE: A non-linear system of speed dependence non-linear attenuation and restoration force simultaneous estimation method are provided to accurately estimate non-linear attenuation and a restoration force by using a non-parametric estimation method. CONSTITUTION: A non-linear attenuation system is modeled a non-linear phase differential equation(S10). An ordinary differential equation is converted into an equivalent Volterra integral equation(S20). An approximate solution of nonlinear attenuation and restoration force is deducted through a regularization method(S30). Optimal repeat numbers are determined in an Landweber regularization method through an L-curve criterion(S40). [Reference numerals] (S10) Modeling a non-linear attenuation system as a non-linear phase differential equation; (S20) Converting an ordinary differential equation into an equivalent Volterra integral equation; (S30) Deducting an approximate solution of nonlinear attenuation and restoration force through a regularization method; (S40) Determining optimal repeat numbers in an Landweber regularization method through an L-curve criterion

Description

NON-PARAMETRIC SIMULTANEOUS IDENTIFICATION OF BOTH THE NONLINEAR DAMPING AND RESTORING CHARACTERISTICS OF NONLINEAR SYSTEMS WHOSE DAMPINGS DEPEND ON VELOCITY ALONE}

The present invention relates to a method of simultaneously estimating the velocity dependent nonlinear damping and restoring force of a nonlinear system by a nonparametric method, and more particularly, by using the nonparametric estimation method in estimating the velocity dependent nonlinear damping and restoring force of a nonlinear system. The present invention relates to a method of simultaneously estimating the speed-dependent nonlinear damping and restoring force of a nonlinear system using a nonparametric method that can minimize errors.

In determining the cause of physical phenomena, direct observation of the cause is often impossible. In this case, the inverse problem theory can be applied to estimate the cause of the physics that you want to know indirectly based on externally observed information. This inverse problem began as a field of applied mathematics but is now applied to almost all disciplines.

When some physical phenomena are observed and mathematically formulated, these problems are classified into political problems and indefinite problems according to Hadamard's classification method. In the case of pure problems that result in a given cause of physical phenomena, most are formalized as political problems, but the inverse problem of inferring the cause based on partial or incomplete information on a given physical phenomenon is mostly indefinite. It is formulated as a problem.

When physics is formulated as a political problem, it is easily solved by well-known general numerical solutions such as boundary element method, finite element method, finite difference method, etc. Numerical solutions do not yield meaningful solutions.

In the past, there was no practical and general way to solve this problem, but since the development of the normalization method, a general way to overcome the negative condition of the problem, many scholars have found inverse problems in their fields. It is applied.

On the other hand, in estimating velocity-dependent nonlinear attenuation and restoring force in a nonlinear system, a parametric estimation method is mainly used. Here, the parametric estimation method refers to a method of estimating the damping and restoring force by finding the coefficient of the hypothesis using the attenuation or restoring force term of the motion equation to be estimated, and these existing methods are different depending on the type of hypothesis used. The conclusions cannot be considered as accurate system estimation methods, and in particular, it is virtually impossible to accurately estimate the nonlinear damping and nonlinear resilience of a system with strong nonlinearity.

SUMMARY OF THE INVENTION The present invention has been made to solve the above problems, and by using a nonparametric estimation method in estimating velocity-dependent damping and restoring force of a nonlinear system, it is possible to precisely estimate the nonlinear damping and restoring force without using a hypothesis. It is an object of the present invention to provide a method of simultaneously estimating the speed-dependent nonlinear damping and restoring force of a nonlinear system by a nonparametric method.

Another object of the present invention is to provide a method for simultaneously estimating the speed-dependent nonlinear damping and the restoring force of a nonlinear system by a nonparametric method that can effectively overcome the instability of a solution resulting from a formal inverse problem.

In order to achieve the above object, the simultaneous estimation of the speed dependent nonlinear damping and restoring force of the nonlinear system by the nonparametric method according to the present invention estimates the speed dependent nonlinear damping and restoring force through the displacement and velocity measured in the nonlinear damping system. In the method, (a) modeling the nonlinear damping system as a second-order nonlinear ordinary differential equation as shown in Equation 1 below; and (b) the ordinary differential equation as an equivalent type as shown in Equation 2 below. Converting to Volterra integral equation; And (c) deriving an approximate solution of nonlinear damping and restoring force through a regularization method.

[Equation 1]

Figure pat00001

(here,

Figure pat00002
Silver particle mass of the vibrometer,
Figure pat00003
Is the spring constant,
Figure pat00004
Is the acceleration of the system,
Figure pat00005
Is the speed of the system,
Figure pat00006
Is the displacement of the system,
Figure pat00007
Is a velocity dependent nonlinear damping,
Figure pat00008
Is nonlinear resilience.)

&Quot; (2) "

Figure pat00009

Figure pat00010

Figure pat00011

(here,

Figure pat00012
ego,
Figure pat00013
Is,

Figure pat00014
The
Figure pat00015
Τ is any value in the range of 0 to t.

In this case, the step (c) is preferably by the Landweber-Fridman normalization method of Equation 3 below.

&Quot; (3) "

Figure pat00016

(here,

Figure pat00017
Is the sum of the estimated velocity-dependent nonlinear damping and resilience,

Figure pat00018
ego,
Figure pat00019
Is operator
Figure pat00020
Transpose of,
Figure pat00021
Is the normalization factor (any positive constant),
Figure pat00022
to be.)

In addition, (d) determining the optimal number of repetitions in the Land Weber normalization method through the L-curve criterion of Equation 4 below;

&Quot; (4) "

Figure pat00023

(here,

Figure pat00024
ego,
Figure pat00025
Is the L2 norm,
Figure pat00026
Is the sum of the estimated velocity-dependent nonlinear damping and nonlinear restoring force, and ｊ is the number of iterations in the Landweaver normalization method,
Figure pat00027
to be.)

According to an embodiment of the present invention as described above, first, by using a nonparametric estimation method in estimating the speed-dependent nonlinear damping and restoring force acting on the nonlinear system, the error is minimized to accurately estimate the nonlinear damping and restoring force. It can work.

Second, by using the Landweaver normalization method, there is an advantage to overcome the instability of the solution generated in the formal inverse problem.

Third, by using the L-shaped curve evaluation formula, there is an advantage that can effectively determine the optimal number of repetitions in the Land Weber normalization method.

1 is a flow chart of a method of simultaneously estimating the velocity dependent nonlinear damping and restoring force of a nonlinear system by a nonparametric method according to an embodiment of the present invention;
2 is a graph for explaining an L curve evaluation method;
3 is a graph showing dynamic response data (displacement, velocity) obtained from a nonlinear system;
4 is a graph of an L-curve evaluation equation for obtaining an optimal number of repetitions in the Landweber normalization method according to an embodiment of the present invention;
5 is a graph showing the convergence evaluation of the approximate nonlinear damping and restoring force according to the number of repetitions;
6 is a graph illustrating a result of estimating an optimal nonlinear damping and restoring force according to an estimating method according to an embodiment of the present invention.

Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings, which are intended to describe in detail enough to be able to easily carry out the invention by those skilled in the art to which the present invention belongs, This does not mean that the technical spirit and scope of the present invention is limited.

1 is a flow chart of a method (S100) for simultaneously estimating speed dependent nonlinear damping and restoring force of a nonlinear system by a nonparametric method according to an embodiment of the present invention.

Simultaneously estimating the speed dependent nonlinear damping and restoring force of the nonlinear system according to an embodiment of the present invention (S100), (a) modeling the nonlinear damping system as a second order nonlinear ordinary differential equation (S10); and (b Converting the ordinary differential equation into an equivalent Volterra integral equation (S20); And (c) deriving an approximate solution of nonlinear damping and restoring force through a regularization method (S30).

Step (a) (S10) represents a nonlinear system in which the function form of the attenuation is expressed only by the velocity function as a second-order ordinary differential equation as follows.

[Equation 5]

Figure pat00028

here,

Figure pat00029
Silver particle mass of the vibrometer,
Figure pat00030
Is the spring constant,
Figure pat00031
Is the acceleration of the system,
Figure pat00032
Is the speed of the system,
Figure pat00033
Is the displacement of the system,
Figure pat00034
Is a velocity dependent nonlinear damping,
Figure pat00035
Is nonlinear resilience.

Step (b) (S20) is a step of converting such nonlinear ordinary differential equation into an equivalent Volterra integral equation as shown in Equation 6 below.

&Quot; (6) "

Figure pat00036

Figure pat00037

Figure pat00038

here,

Figure pat00039
ego,
Figure pat00040
Is,

Figure pat00041
The
Figure pat00042
Τ is any value in the range of 0 to t.

At this time, the left side of the integral equation of Equation 6 is as follows.

Figure pat00043
If you leave it,

Figure pat00044

Figure pat00045
.

therefore

Figure pat00046
Wow
Figure pat00047
A linear relational equation can be obtained from which the mathematical function of the desired nonlinear damping and nonlinear restoring force can be obtained.

However, as described above, the inverse problem of estimating the input (system load) from a given output (displacement of the system) has to solve one kind of integral equation, which is one of the most common examples of indefinite problems. It is.

In other words, the inverse problem is mostly classified as an ill-posed problem, and in general, a problem of nonexistence, non-uniqueness or instability of a solution occurs. In this case, it can be regarded that the solution exists if it is formulated with mathematically appropriate conditions to describe the actual physical phenomenon. However, even if a formal inverse problem satisfies both the existence and uniqueness of the solution mentioned above, the instability problem of the solution may occur.

The instability of the solution here means that for a formal inverse problem, a small change in a given input has a significant effect on the result. When the conventional numerical solution is applied to the problem that stability is not satisfied among the indefinite conditions, a large error occurs in the result.

Most of the inverse problems formulated in real physics are more serious in stability than in the existence and uniqueness of solutions. In other words, the inverse problem is generally aimed at measuring the results of physical phenomena and restoring the corresponding cause. The measured data includes noise in some form. Even if the amount of errors involved is very small, inverse problems with instability will cause very serious problems when restoring the solution. The instability problem is that the solution is not continuously dependent on boundary conditions or initial conditions.

Step (c) (S30) is a step of deriving an approximate solution through a regularization method to solve the instability of the nonlinear damping and restoring force solution.

Most inverse problems can be represented by the following equations using operators.

Figure pat00048
(1-1)

here Represents a system or model, and represents a well-defined relationship between spaces X and Y. Inverse problem given

Figure pat00050
Unknown value
Figure pat00051
It can be said to determine the solution of the inverse problem.

First, the only solution of equation (1-1) that does not include the error

Figure pat00052
Is assumed, this is measured data
Figure pat00053
Unique from
Figure pat00054
That means you can get. But actually the right side of equation (1-1)
Figure pat00055
Is not known exactly. This is because the measured data contains errors in any form. The noise level is defined as follows.

Figure pat00056
(1-2)

If the measured data contains errors, in order to solve the inverse problem, the following equation should be solved rather than equation (1-1).

Figure pat00057
(1-3)

In general, however, equation (1-3) cannot be solved. Because measurements with errors

Figure pat00058
This operator
Figure pat00059
Range of
Figure pat00060
Is not guaranteed to be included). So the best way to do it is
Figure pat00061
Approximation for
Figure pat00062
To determine.

Approximate in this approximation process

Figure pat00063
Must be given
Figure pat00064
It is necessary to rely on succession. But instability operator
Figure pat00065
Inverse operator of
Figure pat00066
Operators are given because they are generally not bounded
Figure pat00067
The direct inverse of does not yield an approximate solution. Therefore, in order to solve the inverse problem, an appropriate bounded approximation
Figure pat00068
It is necessary to construct. This proper approximation is called the normalization method.

Step (c) (S30) in the method (S100) of the simultaneous estimation of the speed-dependent nonlinear damping and restoring force of the nonlinear system according to an embodiment of the present invention is based on a landweber-Fridman normalization method.

Landweber and Fridman proposed the following form of iteration to solve the equation (1-1).

Figure pat00069

(2-1)

here,

Figure pat00070
Represents an identity operator,
Figure pat00071
Is an operator that
Figure pat00072
Represents an adjoint operator of.

Figure pat00073
(2-2)

Sequence

Figure pat00074
Is a normalization factor that
Figure pat00075
About
Figure pat00076
As it grows, it converges to the solution.

Figure pat00077
(2-3)

(here,

Figure pat00078
Is an operator
Figure pat00079
Represents the norm of.

Therefore, using the Landweaver normalization method, the following repeated equations can be derived.

[Equation 7]

Figure pat00080

here,

Figure pat00081
Is the sum of the estimated velocity-dependent nonlinear damping and resilience,

Figure pat00082
ego,
Figure pat00083
Is operator
Figure pat00084
Transpose of,
Figure pat00085
Is the normalization factor (any positive constant),
Figure pat00086
to be.

FIG. 3 is a graph showing dynamic response data (displacement and velocity) obtained from a nonlinear system, and FIG. 4 is an L-curve evaluation equation for obtaining an optimal number of repetitions in the Landweber normalization method according to an embodiment of the present invention. 5 is a graph illustrating the convergence evaluation of the approximate nonlinear damping and restoring force according to the number of repetitions, and FIG. 6 is a result of estimating the optimal nonlinear damping and restoring force according to the estimation method according to an embodiment of the present invention. Is a graph.

Simultaneously estimating the speed dependent nonlinear damping and restoring force of the nonlinear system according to an embodiment of the present invention (S100) is (d) an optimum in the Landweber normalization method through the L-curve criterion Determining the number of repetitions (S40); may be further included.

When the inverse problem is solved and no error is included as in Eq. (1-1), the approximation solution is known to be closer to the normal solution when the normalization coefficient approaches zero. In most cases, however, an error is included in the measured value as shown in Equation (1-3). The accuracy of the approximation solution obtained by the normalization method is greatly influenced by the normalization coefficient. Therefore, it is very important to select the proper normalization coefficient to obtain the optimal solution.

Among many methods, Hansen's L-curve method is the most widely used method to find the normalization coefficient. L-curve is a graph representing the noise of the normalized solution and the residual of the corresponding residual according to the change of the normalization coefficient on a log-log scale.

Figure pat00087
(3-1)

This graph typically shows an "L" shape as shown in FIG. Here, the optimum value of the normalization coefficient is an L-shaped corner value at which the curvature changes to the maximum. This corner is the residual

Figure pat00088
Normalize with
Figure pat00089
Express a compromise. While we cannot guarantee that the L-curve method can be applied to all adverse conditions, the L-curve method is now known as the simplest and most powerful way to find the proper normalization coefficient in many applications.

Therefore, the estimation method of the present invention is the optimal number of iterations in the Landweaver normalization method.

Figure pat00090
In order to find the L-curve criterion graph, we find the optimal number of repetitions.

Equation 8 below is an equation of the L-shaped curve evaluation method

Figure pat00091
The symbol represents L2norm and takes its logarithmic function.

[Equation 8]

Figure pat00092

here,

Figure pat00093
ego,
Figure pat00094
Is the L2 norm,
Figure pat00095
Is the sum of the estimated velocity-dependent nonlinear damping and nonlinear restoring force, and ｊ is the number of iterations in the Landweaver normalization method,
Figure pat00096
to be.

Referring to the drawings, the shape of the graph is indicated by an L-shape, and at this time, the number of repetitions in the curved portion (the corner portion) becomes the optimal number of repetitions. This optimal number of iterations allows precise estimation of velocity-dependent nonlinear damping and resilience.

As described above, the method of estimating the speed dependent nonlinear damping and restoring force of the nonlinear system by the nonparametric method according to the present invention is based on the inverse problem approach. The resilience can be estimated and the error can be minimized by using a nonparametric estimation method. In addition, by using the Landweaver normalization method and the L-shape evaluation equation, there is an advantage that can effectively overcome the instability of the solution that occurs in the formal inverse problem.

S100: Simultaneous Estimation of Velocity-Dependent Nonlinear Damping and Restoring Forces for Nonlinear Systems
S10: modeling the nonlinear damping system as a second-order nonlinear ordinary differential equation
S20: converting ordinary differential equation to equivalent Volterra integral equation
S30: deriving an approximate solution of nonlinear damping and restoring force through a regularization method
S40: determining an optimal number of iterations in the Landweaver normalization method

Claims (3)

A method for estimating speed dependent nonlinear attenuation and restoring force through measured displacements and speeds in a nonlinear damping system,
(a) modeling the nonlinear damping system as a second-order nonlinear ordinary differential equation as shown in Equation 9 below; and
(b) converting the ordinary differential equation into an equivalent kind of Volterra integral equation as in Equation 10 below; And
(c) deriving an approximate solution of the nonlinear damping and restoring force through a regularization method; and a method of simultaneously estimating the speed dependent nonlinear damping and restoring force of a nonlinear system by a nonparametric method.
[Equation 9]
Figure pat00097

(here,
Figure pat00098
Silver particle mass of the vibrometer,
Figure pat00099
Is the spring constant,
Figure pat00100
Is the acceleration of the system,
Figure pat00101
Is the speed of the system,
Figure pat00102
Is the displacement of the system,
Figure pat00103
Is a velocity dependent nonlinear damping,
Figure pat00104
Is nonlinear resilience.)
[Equation 10]
Figure pat00105

Figure pat00106

Figure pat00107

(here,
Figure pat00108
ego,
Figure pat00109
Is,
Figure pat00110
The
Figure pat00111
Τ is any value in the range of 0 to t.
The method of claim 1,
The step (c)
A method of simultaneously estimating the speed-dependent nonlinear damping and the restoring force of a nonlinear system by the nonparametric method characterized by the landweber-Fridman normalization method of Equation 11 below.
[Equation 11]
Figure pat00112

(here,
Figure pat00113
Is the sum of the estimated velocity-dependent nonlinear damping and resilience,
Figure pat00114
ego,
Figure pat00115
Is operator
Figure pat00116
Transpose of,
Figure pat00117
Is the normalization factor (any positive constant),
Figure pat00118
to be.)
The method of claim 2,
(d) determining an optimal number of repetitions in the Land Weber normalization method through an L-curve criterion of Equation 12 below. Method for Simultaneous Estimation of Velocity-dependent Nonlinear Damping and Restoring Force of Nonlinear Systems
[Equation 12]
Figure pat00119

(here,
Figure pat00120
ego,
Figure pat00121
Is the L2 norm,
Figure pat00122
Is the sum of the estimated velocity-dependent nonlinear damping and nonlinear restoring force, and ｊ is the number of iterations in the Landweaver normalization method,
Figure pat00123
to be.)
KR1020110050714A 2011-05-27 2011-05-27 Non-parametric simultaneous identification of both the nonlinear damping and restoring characteristics of nonlinear systems whose dampings depend on velocity alone KR20120132099A (en)

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