KR20120132101A - Method for measuring nonharmonic periodic excitation forces in nonlinear damped systems - Google Patents

Method for measuring nonharmonic periodic excitation forces in nonlinear damped systems Download PDF

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KR20120132101A
KR20120132101A KR1020110050716A KR20110050716A KR20120132101A KR 20120132101 A KR20120132101 A KR 20120132101A KR 1020110050716 A KR1020110050716 A KR 1020110050716A KR 20110050716 A KR20110050716 A KR 20110050716A KR 20120132101 A KR20120132101 A KR 20120132101A
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장택수
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부산대학교 산학협력단
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Abstract

The present invention relates to a method of estimating non-harmonic and periodic excitation force in a nonlinear damping system, the method of estimating non-harmonic and periodic external loads through displacements and velocities measured in a nonlinear damping system, comprising: Modeling the system as a second-order nonlinear ordinary differential equation; and (b) converting the ordinary differential equation into an equivalent Volterra integral equation; And (c) deriving an approximate solution of the external load through a regularization method.
According to the present invention, it is possible to estimate the external load acting on the structure from the response (displacement and speed) of the system through the inverse problem approach, and to estimate the unharmonized and periodic external load acting on the system. By using the nonparametric estimation method, it is possible to make an accurate estimation without using the hypothesis. In addition, by using the Tychonov normalization method, Landweaver normalization method and L-shaped curve evaluation equation, there is an advantage that can effectively overcome the instability of the solution due to the indefiniteness generated in the formal inverse problem.

Description

Non-harmonic and Periodic Excitation Estimation Method in Nonlinear Damping System METHODS FOR MEASURING NONHARMONIC PERIODIC EXCITATION FORCES IN NONLINEAR DAMPED SYSTEMS

The present invention relates to a non-harmonic and periodic excitation force estimation method in a nonlinear damping system. More particularly, the present invention relates to a non-harmonic and periodic external load acting on a dynamic system. The present invention relates to a method of estimating nonharmonic and periodic excitation in a nonlinear damping system that can be minimized.

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 fixed 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, estimating the external load acting on the dynamic system of various fields is a very important subject, for example, it is an important factor to estimate the life of the structure. However, it is very difficult to measure the external load directly because it is practically difficult to install the transducer at the time of construction of the structure. Therefore, in order to overcome this difficulty, the recent inverse problem approach is used to estimate the external load acting on the structure from the response of the structure.

In this regard, the prior art mainly estimates the external load of the damping system using a parametric estimation method. Here, the parametric estimation method refers to a method of estimating the external load by estimating the damping and restoring force by finding the coefficient of the hypothesis using the hypothesis equation of the damping or restoring force term of the motion equation to be estimated. It is not an accurate system estimation method because different conclusions are drawn according to the type of hypothesis used. In particular, it is virtually impossible to accurately estimate the external load 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 non-harmonic and periodic external loads acting on a dynamic system, the external loads can be precisely used without using a hypothesis. It is an object of the present invention to provide an incoherent and periodic excitation force estimation method in a nonlinear attenuation system that can be estimated.

It is also another object of the present invention to provide a method for estimating an incoherent and periodic excitation in a nonlinear damping system that can effectively overcome the instability of a solution resulting from a formal inverse problem.

In order to achieve the above object, a non-harmonic and periodic excitation force estimation method in a nonlinear damping system according to the present invention is a method for estimating an unharmonic and periodic external load through a displacement and a velocity measured in a nonlinear damping system. (a) modeling the nonlinear damping system as a second-order nonlinear ordinary differential equation as in Equation 1 below; and (b) an equivalent one Volterra integral equation as shown in Equation 2 below. Converting to; And (c) deriving an approximate solution of the external load through a regularization method.

[Equation 1]

Figure pat00001

(here,

Figure pat00002
Is the mass of the vibrator particle,
Figure pat00003
Is the linear spring constant,
Figure pat00004
Is the nonlinear damping term,
Figure pat00005
Is the nonlinear resilience term,
Figure pat00006
Is the displacement of the system,
Figure pat00007
Is the speed of the system,
Figure pat00008
Is the acceleration of the system.)

&Quot; (2) "

Figure pat00009

Figure pat00010

(here,

Figure pat00011
ego,
Figure pat00012
Is,
Figure pat00013
The
Figure pat00014
Τ is any value in the range of 0 to t.

In this case, the step (c) is preferably by the Tikhonov normalization method of Equation 3 below or by the Landweber-Fridman normalization method of Equation 4 below.

&Quot; (3) "

Figure pat00015

(here,

Figure pat00016
Is the estimated external load,
Figure pat00017
ego,
Figure pat00018
Is operator
Figure pat00019
Transpose of,
Figure pat00020
Is the normalization factor,

Figure pat00021
to be.)

&Quot; (4) "

Figure pat00022

(here,

Figure pat00023
Is the estimated external load,
Figure pat00024
ego,
Figure pat00025
Is operator
Figure pat00026
Transpose of,
Figure pat00027
Is the normalization factor (any positive constant),

Figure pat00028
to be.

Also, (d) determining an optimal normalization coefficient in the Tychonov normalization method or an optimal repetition frequency in the Landweber normalization method through L-curve criterion of Equation 5 below. It is preferable to further include;

[Equation 5]

Figure pat00029

(here,

Figure pat00030
Is the estimated external load,
Figure pat00031
,
Figure pat00032
Is the L2 norm,
Figure pat00033
Is the normalization coefficient (α) in the Tychonov normalization method or the number of iterations in the Landweber normalization method (ｊ),

Figure pat00034
to be.)

According to an embodiment of the present invention as described above, first, by estimating the external load acting on the structure from the response (displacement and speed) of the system through the inverse problem approach, the expected life of the structure in advance There is an effect that can be predicted and reflected in the design.

Second, in estimating non-harmonic and periodic external loads acting on the system, there is an advantage that the external load can be accurately estimated by minimizing errors.

Third, by using the Tychonov normalization method or Landweaver normalization method, there is an advantage that can overcome the instability of the solution due to the indefiniteness generated in the formal inverse problem.

Fourth, by using the L-shaped curve evaluation formula, there is an advantage that the optimal normalization coefficient affecting the accuracy of the approximate solution and the optimal repetition frequency in the Landweaver normalization method can be effectively determined in the Tychonov normalization method.

1 is a flow chart of an incoherent and periodic excitation force estimation method in a nonlinear damping system according to an embodiment of the present invention;
2 is a graph for explaining an L curve evaluation method;
3 is a graph relating to actual loads and measured displacements and velocities used to evaluate the estimation method of the present invention;
4 is a comparison graph of the external load and the actual load estimated through the Tychonov normalization method according to an embodiment of the present invention;
Figure 5 is a comparison graph of the external load and the actual load estimated through the Land Weber normalization method according to an embodiment of the present invention
6 is a graph of an L-curve evaluation equation for obtaining an optimal normalization coefficient in a Tychonov normalization method according to an embodiment of the present invention;
7 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.

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 flowchart illustrating a method of estimating an unharmonized and periodic excitation force in a nonlinear damping system according to an embodiment of the present invention.

Excitation force estimating method (S100) in a nonlinear damping system according to an embodiment of the present invention, (a) modeling the nonlinear damping system in a second-order nonlinear ordinary differential equation (S10); and (b) the ordinary differential equation Converting to the equivalent Volterra integral equation (S20); And (c) deriving an approximate solution of the external load through a regularization method (S30).

In step (a), a non-harmonic and periodic load is applied to the nonlinear damping system.

Figure pat00035
If is given, the governing equation of this system is modeled as a second-order nonlinear ordinary differential equation as

&Quot; (6) "

Figure pat00036

here,

Figure pat00037
Is the mass of the vibrator particle,
Figure pat00038
Is the linear spring constant,
Figure pat00039
Is the nonlinear damping term,
Figure pat00040
Is the nonlinear resilience term,
Figure pat00041
Is the displacement of the system,
Figure pat00042
Is the speed of the system,
Figure pat00043
Is the acceleration of the system.

Step (b) (S20) is a step of converting the nonlinear ordinary differential equation to the equivalent Volterra integral equation of the first kind as shown in Equation 7 below.

[Equation 7]

Figure pat00044

Figure pat00045

here,

Figure pat00046
ego,
Figure pat00047
Is,
Figure pat00048
The
Figure pat00049
Τ is any value in the range of 0 to t.

In this case, the integral equation of Equation 7

Figure pat00050

After converting to an equivalent equation like

Figure pat00051
) And speed (
Figure pat00052
) Is a measurable value, so the left term of the equation is assumed to be known and is pseudo-displacement.
Figure pat00053
If you leave it,

Figure pat00054

Figure pat00055
.

Thus known pseudodisplacement

Figure pat00056
And unknown external loads
Figure pat00057
You can get a linear relationship between them.

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, and the solution is unique in the case of the inverse problem provided with sufficient information. 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 external load solution.

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

Figure pat00058
(1-1)

here

Figure pat00059
Represents a system or model, and represents a well-defined relationship between spaces X and Y. Inverse problem given
Figure pat00060
Unknown value
Figure pat00061
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 pat00062
Is assumed, this is measured data
Figure pat00063
Unique from
Figure pat00064
That means you can get. But actually the right side of equation (1-1)
Figure pat00065
Is not known exactly. This is because the measured data contains errors in any form. The noise level is defined as follows.

Figure pat00066
(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 pat00067
(1-3)

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

Figure pat00068
This operator
Figure pat00069
Range of
Figure pat00070
Is not guaranteed to be included). So the best way to do it is
Figure pat00071
Approximation for
Figure pat00072
To determine.

Approximate in this approximation process

Figure pat00073
Must be given
Figure pat00074
It is necessary to rely on succession. But instability operator
Figure pat00075
Inverse operator of
Figure pat00076
Operators are given because they are generally not bounded
Figure pat00077
The direct inverse of does not yield an approximate solution. Therefore, in order to solve the inverse problem, an appropriate bounded approximation
Figure pat00078
It is necessary to construct. This proper approximation is called the normalization method.

Figure 3 is a graph of the actual load and the measured displacement and velocity used to evaluate the estimation method of the present invention, Figures 4 and 5 is a Tychonov normalization method and Landweber normalization method according to an embodiment of the present invention This is a graph comparing the external load and the actual load estimated through

Step (c) (S30) of the non-harmonic and periodic excitation force estimation method (S100) in the nonlinear damping system according to an embodiment of the present invention is a Tikhonov normalization method or a landweber-Fridman method. By normalization method

First, Tikhonov proposed the following equation with constraints to suppress instability.

Figure pat00079
(2-1)

here

Figure pat00080
Represents the normalization coefficient and affects the convergence performance of Equation (2-1).

Therefore, through the Tychonov normalization method it is possible to derive an approximate equation of the external load of Equation 8 below.

[Equation 8]

Figure pat00081

(here,

Figure pat00082
Is the estimated external load,
Figure pat00083
ego,
Figure pat00084
Is operator
Figure pat00085
Transpose of,
Figure pat00086
Is the normalization factor,

Figure pat00087
to be.)

Landweber and Fridman, on the other hand, proposed an iteration of the following form to solve the equation (1-1).

Figure pat00088

(3-1)

here,

Figure pat00089
Represents an identity operator,
Figure pat00090
Is an operator that
Figure pat00091
Represents an adjoint operator of.

Figure pat00092
(3-2)

Sequence

Figure pat00093
Is a normalization factor that
Figure pat00094
About
Figure pat00095
As it grows, it converges to the solution.

Figure pat00096
(3-3)

(here,

Figure pat00097
Is an operator
Figure pat00098
Represents the norm of.

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

&Quot; (9) "

Figure pat00099

(here,

Figure pat00100
Is the estimated external load,
Figure pat00101
ego,
Figure pat00102
Is operator
Figure pat00103
Transpose of,
Figure pat00104
Is the normalization factor (any positive constant),

Figure pat00105
to be.)

6 and 7 are graphs of an L-curve evaluation equation for obtaining an optimal normalization coefficient and a repetition frequency in the Tychonov normalization method and the Landweaver normalization method according to an embodiment of the present invention.

The non-harmonic and periodic excitation force estimating method (S100) in the nonlinear damping system according to an embodiment of the present invention is (d) the optimum in the Ticonov normalization method through the L-curve criterion The method may further include determining a normalization coefficient or an optimal repetition frequency in the Landweaver normalization method (S40).

When the inverse problem is solved and no error is included in Eq. 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 norm of the normalized solution and the norm of the corresponding residual as the normalization coefficient changes. The L-curve is expressed as follows.

Figure pat00106
(4-1)

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 pat00107
Normalize with
Figure pat00108
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 coefficient in each normalization method

Figure pat00109
And optimal number of iterations
Figure pat00110
In order to find, the L-curve criterion is graphed to find the optimal normalization coefficient and the optimal number of iterations.

Equation 10 below is an equation of the L-shaped curve evaluation method

Figure pat00111
The symbol represents L2norm and takes its logarithmic function.

&Quot; (10) "

Figure pat00112

(here,

Figure pat00113
Is the estimated external load,
Figure pat00114
,
Figure pat00115
Is the L2 norm,
Figure pat00116
Is the normalization coefficient (α) in the Tychonov normalization method or the number of iterations in the Landweber normalization method (ｊ),

Figure pat00117
to be.)

Referring to the drawings, the shape of the graph is indicated by an L-shape, and the coefficients and the number of repetitions in the curved portion (the corner portion) are the optimal coefficients and the number of repetitions. External load through this optimal coefficient and number of iterations

Figure pat00118
Can be estimated accurately.

As described above, the non-harmonic and periodic excitation force estimation method of the present invention in the nonlinear damping system estimates the external load acting on the structure from the response (displacement and speed) of the system through the inverse problem approach. In order to estimate the non-harmonic and periodic external load acting on the system, the error can be minimized by using the nonparametric estimation method. In addition, by using the Tychonov normalization method, Landweaver normalization method and L-shaped curve evaluation equation, there is an advantage that can effectively overcome the instability of the solution generated from the inverse problem.

S100: Incoherent and Periodic Excitation Estimation Method for Nonlinear Damping 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: step of deriving an approximate solution of the external load through a regularization method
S40: determining an optimal normalization coefficient or an optimal repetition number

Claims (3)

A method of estimating unharmonious and periodic external loads from displacements and velocities measured in a nonlinear damping system,
(a) modeling the nonlinear damping system as a second-order nonlinear ordinary differential equation as shown in Equation 11 below; and
(b) converting the ordinary differential equation into an equivalent kind of Volterra integral equation as in Equation 12 below; And
and (c) deriving an approximate solution of the external load through a regularization method.
[Equation 11]
Figure pat00119

(here,
Figure pat00120
Is the mass of the vibrator particle,
Figure pat00121
Is the linear spring constant,
Figure pat00122
Is the nonlinear damping term,
Figure pat00123
Is the nonlinear resilience term,
Figure pat00124
Is the displacement of the system,
Figure pat00125
Is the speed of the system,
Figure pat00126
Is the acceleration of the system.)
[Equation 12]
Figure pat00127

Figure pat00128

(here,
Figure pat00129
ego,
Figure pat00130
Is,
Figure pat00131
The
Figure pat00132
Τ is any value in the range of 0 to t.
The method of claim 1,
The step (c)
An incoherent and periodic excitation force estimation method for a nonlinear damping system, characterized by Tikhonov's normalization method of Equation 13 below or Landweber-Fridman normalization method of Equation 14 below.
&Quot; (13) "
Figure pat00133

(here,
Figure pat00134
Is the estimated external load,
Figure pat00135
ego,
Figure pat00136
Is operator
Figure pat00137
Transpose of,
Figure pat00138
Is the normalization factor,
Figure pat00139
to be.)
&Quot; (14) "
Figure pat00140

(here,
Figure pat00141
Is the estimated external load,
Figure pat00142
ego,
Figure pat00143
Is operator
Figure pat00144
Transpose of,
Figure pat00145
Is the normalization factor (any positive constant),
Figure pat00146
to be.)
The method of claim 2,
(d) determining an optimal normalization coefficient in the Ticonov normalization method or an optimal repetition frequency in the Landweber normalization method through an L-curve criterion of Equation 15 below; The method of estimating non-harmonic and periodic excitation in a non-linear damping system further comprises.
&Quot; (15) "
Figure pat00147

(here,
Figure pat00148
Is the estimated external load,
Figure pat00149
,
Figure pat00150
Is the L2 norm,
Figure pat00151
Is the normalization coefficient (α) in the Tychonov normalization method or the number of iterations in the Landweber normalization method (ｊ),
Figure pat00152
to be.)
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CN105808794A (en) * 2014-12-29 2016-07-27 南京理工大学 Time domain integral equation method for analyzing electromagnetic scattering characteristic of hypersonic flight object
CN106874563A (en) * 2017-01-16 2017-06-20 广西大学 A kind of deformable component Fast design method based on space constraint
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Cited By (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN103279601A (en) * 2013-05-17 2013-09-04 南京理工大学 Method for simulating wide-band electromagnetic scattering property of conductor target
CN103279601B (en) * 2013-05-17 2016-01-20 南京理工大学 The emulation mode of target conductor Wide-band Electromagnetic Scattering
CN104493636A (en) * 2014-11-12 2015-04-08 华中科技大学 Metallic cryogenic tempering method for improving milling stability
CN105808794A (en) * 2014-12-29 2016-07-27 南京理工大学 Time domain integral equation method for analyzing electromagnetic scattering characteristic of hypersonic flight object
CN106874563A (en) * 2017-01-16 2017-06-20 广西大学 A kind of deformable component Fast design method based on space constraint
KR20190031803A (en) 2017-09-18 2019-03-27 부산대학교 산학협력단 Method for Developing Function Algorithm of RLW Equation using Integral Equation Formalism

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