IDENTIFIABILITY FOR COMPOSITE STRING VIBRATION PROBLEM

Gutman, Semion,Ha, Jun-Hong Korean Mathematical Society 2010 대한수학회지 Vol.47 No.5

The paper considers the identifiability (i.e., the unique identification) of a composite string in the class of piecewise constant parameters. The 1-D string vibration is measured at finitely many observation points. The observations are processed to obtain the first eigenvalue and a constant multiple of the first eigenfunction at the observation points. It is shown that the identification by the Marching Algorithm is continuous with respect to the mean convergence in the admissible set. The result is based on the continuous dependence of eigenvalues, eigenfunctions, and the solutions on the parameters. A numerical algorithm for the identification in the presence of noise is proposed and implemented.

ESTIMATION ALGORITHM FOR PHYSICAL PARAMETERS IN A SHALLOW ARCH

Gutman, Semion,Ha, Junhong,Shon, Sudeok Korean Mathematical Society 2021 대한수학회지 Vol.58 No.3

Design and maintenance of large span roof structures require an analysis of their static and dynamic behavior depending on the physical parameters defining the structures. Therefore, it is highly desirable to estimate the parameters from observations of the system. In this paper we study the parameter estimation problem for damped shallow arches. We discuss both symmetric and non-symmetric shapes and loads, and provide theoretical and numerical studies of the model behavior. Our study of the behavior of such structures shows that it is greatly affected by the existence of critical parameters. A small change in such parameters causes a significant change in the model behavior. The presence of the critical parameters makes it challenging to obtain good estimation. We overcome this difficulty by presenting the Parameter Estimation Algorithm that identifies the unknown parameters sequentially. It is shown numerically that the algorithm achieves a successful parameter estimation for models defined by arbitrary parameters, including the critical ones.

IDENTIFIABILITY FOR COMPOSITE STRING VIBRATION PROBLEM

Semion Gutman,하준홍 대한수학회 2010 대한수학회지 Vol.47 No.5

The paper considers the identifiability (i.e., the unique identification)of a composite string in the class of piecewise constant parameters. The 1-D string vibration is measured at finitely many observation points. The observations are processed to obtain the first eigenvalue and a constant multiple of the first eigenfunction at the observation points. It is shown that the identification by the Marching Algorithm is continuous with respect to the mean convergence in the admissible set. The result is based on the continuous dependence of eigenvalues, eigenfunctions,and the solutions on the parameters. A numerical algorithm for the identification in the presence of noise is proposed and implemented.

Dynamic behavior of cracked beams and shallow arches

Semion Gutman,하준홍,손수덕 대한수학회 2022 대한수학회지 Vol.59 No.5

We develop a rigorous mathematical framework for studying dynamic behavior of cracked beams and shallow arches. The governing equations are derived from the first principles, and stated in terms of the subdifferentials of the bending and the axial potential energies. The existence and the uniqueness of the solutions is established under various conditions. The corresponding mathematical tools dealing with vector-valued functions are comprehensively developed. The motion of beams and arches is studied under the assumptions of the weak and strong damping. The presence of cracks forces weaker regularity results for the arch motion, as compared to the beam case.

SHALLOW ARCHES WITH WEAK AND STRONG DAMPING

Semion Gutman,하준홍 대한수학회 2017 대한수학회지 Vol.54 No.3

The paper develops a rigorous mathematical framework for the behavior of arch and membrane like structures. Our main goal is to incorporate moving point loads. Both the weak and the strong damping cases are considered. First, we prove the existence and the uniqueness of the solutions. Then it is shown that the solution in the weak damping case is the limit of the strong damping solutions, as the strong damping vanishes. The theory is applied to a car moving on a bridge.

SHALLOW ARCHES WITH WEAK AND STRONG DAMPING

Gutman, Semion,Ha, Junhong Korean Mathematical Society 2017 대한수학회지 Vol.54 No.3

The paper develops a rigorous mathematical framework for the behavior of arch and membrane like structures. Our main goal is to incorporate moving point loads. Both the weak and the strong damping cases are considered. First, we prove the existence and the uniqueness of the solutions. Then it is shown that the solution in the weak damping case is the limit of the strong damping solutions, as the strong damping vanishes. The theory is applied to a car moving on a bridge.

EQUATIONS OF MOTION FOR CRACKED BEAMS AND SHALLOW ARCHES

Semion Gutman,Junhong Ha,손수덕 경남대학교 수학교육과 2022 Nonlinear Functional Analysis and Applications Vol.27 No.2

Cracks in beams and shallow arches are modeled by massless rotational springs. First, we introduce a specially designed linear operator that “absorbs” the boundary conditions at the cracks. Then the equations of motion are derived from the first principles using the Extended Hamilton’s Principle, accounting for non-conservative forces. The variational formulation of the equations is stated in terms of the subdifferentials of the bending and axial potential energies. The equations are given in their abstract (weak), as well as in classical forms.

Optimal parameters for a damped sine-Gordon equation

하준홍,Semion Gutman 대한수학회 2009 대한수학회지 Vol.46 No.5

In this paper a parameter identification problem for a damped sine-Gordon equation is studied from the theoretical and numerical perspectives. A spectral method is developed for the solution of the state and the adjoint equations. The Powell's minimization method is used for the numerical parameter identification. The necessary conditions for the optimization problem are shown to yield the bang-bang control law. Numerical results are discussed and the applicability of the necessary conditions is examined. In this paper a parameter identification problem for a damped sine-Gordon equation is studied from the theoretical and numerical perspectives. A spectral method is developed for the solution of the state and the adjoint equations. The Powell's minimization method is used for the numerical parameter identification. The necessary conditions for the optimization problem are shown to yield the bang-bang control law. Numerical results are discussed and the applicability of the necessary conditions is examined.

IDENTIFIABILITY FOR 1-D HEAT CONDUCTION WITH BOUNDARY INPUTS

Jun-Hong HA,Semion GUTMAN 한국산업응용수학회 2008 한국산업응용수학회 학술대회 논문집 Vol.4 No.3

In this lecture, the identifiability (i.e. the unique identification) of conductivity in a heat conduction process is considered in the class of piecewise constant conductivities. The 1-D process may have nonzero boundary inputs as well as distributed inputs. Its measurements are collected at finitely many observation points. They are processed to obtain the first eigenvalue and a constant multiple of the first eigenfunction at the observation points.

OPTIMAL PARAMETERS FOR A DAMPED SINE-GORDON EQUATION

Ha, Jun-Hong,Gutman, Semion Korean Mathematical Society 2009 대한수학회지 Vol.46 No.5

In this paper a parameter identification problem for a damped sine-Gordon equation is studied from the theoretical and numerical perspectives. A spectral method is developed for the solution of the state and the adjoint equations. The Powell's minimization method is used for the numerical parameter identification. The necessary conditions for the optimization problem are shown to yield the bang-bang control law. Numerical results are discussed and the applicability of the necessary conditions is examined.