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 PDFInfo
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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
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]
(here,
Is the mass of the vibrator particle, Is the linear spring constant, Is the nonlinear damping term, Is the nonlinear resilience term, Is the displacement of the system, Is the speed of the system, Is the acceleration of the system.)&Quot; (2) "
(here,
ego, Is, The 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) "
(here,
Is the estimated external load, ego, Is operator Transpose of, Is the normalization factor,to be.)
&Quot; (4) "
(here,
Is the estimated external load, ego, Is operator Transpose of, Is the normalization factor (any positive constant),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]
(here,
Is the estimated external load, , Is the L2 norm, Is the normalization coefficient (α) in the Tychonov normalization method or the number of iterations in the Landweber normalization method (j),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.
If is given, the governing equation of this system is modeled as a second-order nonlinear ordinary differential equation as&Quot; (6) "
here,
Is the mass of the vibrator particle, Is the linear spring constant, Is the nonlinear damping term, Is the nonlinear resilience term, Is the displacement of the system, Is the speed of the system, 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]
here,
ego, Is, The Τ is any value in the range of 0 to t.In this case, the integral equation of Equation 7
After converting to an equivalent equation like
) And speed ( ) Is a measurable value, so the left term of the equation is assumed to be known and is pseudo-displacement. If you leave it,
.
Thus known pseudodisplacement
And unknown external loads 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.
(1-1)
here
Represents a system or model, and represents a well-defined relationship between spaces X and Y. Inverse problem given Unknown value 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
Is assumed, this is measured data Unique from That means you can get. But actually the right side of equation (1-1) Is not known exactly. This is because the measured data contains errors in any form. The noise level is defined as follows.(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).
(1-3)
In general, however, equation (1-3) cannot be solved. Because measurements with errors
This operator Range of Is not guaranteed to be included). So the best way to do it is Approximation for To determine.Approximate in this approximation process
Must be given It is necessary to rely on succession. But instability operator Inverse operator of Operators are given because they are generally not bounded The direct inverse of does not yield an approximate solution. Therefore, in order to solve the inverse problem, an appropriate bounded approximation 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.
(2-1)
here
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]
(here,
Is the estimated external load, ego, Is operator Transpose of, Is the normalization factor,to be.)
Landweber and Fridman, on the other hand, proposed an iteration of the following form to solve the equation (1-1).
(3-1)
here,
Represents an identity operator, Is an operator that Represents an adjoint operator of.(3-2)
Sequence
Is a normalization factor that About As it grows, it converges to the solution.(3-3)
(here,
Is an operator Represents the norm of.Therefore, using the Landweaver normalization method, the following repeated equations can be derived.
&Quot; (9) "
(here,
Is the estimated external load, ego, Is operator Transpose of, Is the normalization factor (any positive constant),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.
(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
Normalize with 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
And optimal number of iterations In order to find, the L-curve criterion is graphed to find the optimal normalization coefficient and the optimal number of iterations.&Quot; (10) "
(here,
Is the estimated external load, , Is the L2 norm, Is the normalization coefficient (α) in the Tychonov normalization method or the number of iterations in the Landweber normalization method (j),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
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) 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]
(here, Is the mass of the vibrator particle, Is the linear spring constant, Is the nonlinear damping term, Is the nonlinear resilience term, Is the displacement of the system, Is the speed of the system, Is the acceleration of the system.)
[Equation 12]
(here, ego, Is, The Τ is any value in the range of 0 to t.
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) "
(here, Is the estimated external load, ego, Is operator Transpose of, Is the normalization factor,
to be.)
&Quot; (14) "
(here, Is the estimated external load, ego, Is operator Transpose of, Is the normalization factor (any positive constant),
to be.)
(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) "
(here, Is the estimated external load, , Is the L2 norm, Is the normalization coefficient (α) in the Tychonov normalization method or the number of iterations in the Landweber normalization method (j),
to be.)
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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 |
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