, MATHEMATICAL MODELS FOR INVERSE PROBLEMS OF DEFICIENT CONSTRUCTIONS DYNAMICS // BSU bulletin. Mathematics, Informatics. - 2019. №3. . - С. 77-86.

MATHEMATICAL MODELS FOR INVERSE PROBLEMS OF DEFICIENT CONSTRUCTIONS DYNAMICS

The article presents the solution for residual stiffnesses determination in various structural elements, which is based on the results of instrumental measurements of the parameters of natural vibrations carried out using high-precision instruments. Our research is extremely relevant to determine the degree of deficiencies in build- ings, supporting structures of aircraft, ship and other systems with a certain opera- tional cycle or structures that have been subjected to intense impacts. The use of dynamic methods for analyzing the state of load-bearing structures has undeniable advantages, since they exclude the need for a detailed surveying, often associated with the opening of enclosure structures. The advantages of using this approach are especially evident when surveying a large area of residential structures located in heterogeneous operating conditions. The article considers the problems of determin- ing the stiffness properties of structural elements with the predominance of shearing strain in continuity and discreteness of inertial parameters distribution. There is a need to determine the level of defects accumulation, so we have calculated the ratio of real stiffness parameters, defined in the process of dynamic tests, to some initial ones, inherent to the constructions without defects. The article shows the invariance of such relations with account of discreteness and continuity properties of the mathematical models for structural elements. These methods have been tested in survey of 1-335 series high-rise residential buildings in Irkutsk.

residual stiffness; dynamic inverse problem; frequency of natural vibrations; shearing strain; mass distribution; discrete parameters.

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Cawley Р., and Adams R. D. The Location of Defects in Structures from Meas- urements of Natural Frequencies. The Journal of Strain Analysis for Engineering De- sign. 1979. V. 14. Iss. 2. Pp. 49–57.

Berman А. System Identification of Structural Dynamic Models – Theoretical and Practical Bounds. Proceedings of the AIAA/ASME/ASCE/AHS 25th Structures, Structural Dynamics and Materials Conference. California, Palm Springs, 1984. Pp. 123–129.

Sobolev V. I., Pinus B. I. Opredelenie parametrov ostatochnoi zhestkosti defektnykh zdanii na osnove lazernykh otobrazhenii kolebanii i resheniya obratnoi zadachi dinamiki [Determination of the Parameters of Deficient Buildings Residual Stiff- ness Based on Laser Imaging of Vibrations and Solving the Dynamic Inverse Problem]. Vestnik VSGUTU. 2019. No. 1 (72). Pp. 55–67.

Amelkin V. V. Differentsialnye uravneniya v prilozheniyakh [Differential Equa- tions in Applications]. Moscow: Nauka Publ., 1987. 160 p.

Clough R. W., Penzien J. Dynamics of Structures. McGraw-Hill College, 1975. 634 p.

Pinus B. I., Morgayev D. E. Otsenka ostatochnogo resursa sei-smostoikosti zdanii serii 1-335 ks v gorode Irkutske [Assessment of the Residual Seismic Stability of 1-335 ks Series Buildings in Irkutsk]. Tezisy dokladov V Rossiiskoi Natsional'noi konferentsii po seismostoikomu stroitel'stvu i seismicheskomu raionirovaniyu s mezhdunarodnym uchastiem. Moscow, 2003. 81 p.

Sobolev V. I. Raschet mnogoetazhnykh zdanii razlichnykh konstruktivnykh sis- tem na gorizontalnoye seismicheskoye vozdeistvie s uchetom prostranstvennogo de- formirovaniya [Calculation of Multistory Buildings of Various Structural Systems for Horizontal Earthquake Action Taking into Account Spatial Deformation]. Mate- maticheskoye modelirovaniye v mekhanike sploshnykh sred na osnove metodov granichnykh i konechnykh elementov. Proc. 18th Int. conf. St Petersburg, 2000. V. 1. 217 p.

Argyris J. H., Boni В., Hinderlang V. Finite Element Analysis of Two- and Three Dimensional Elastoplastic Frames — the Natural Approach. Comp. Meth. Appl. Mech. 1982. V. 35. No. 2. Pp. 221–248.

Aizenberg Ya. M. Razvitie kontseptsii i norm antiseismicheskogo proektiro- vaniya [Development of the Concepts and Regulations for Earthquake-Proof Engineer- ing]. Moscow: Stroitelstvo Publ., 1997. 73 p.

Davies E. B., Gladwell G. M. L., Leydold J. S., Peter F. Discrete Nodal Domain Theorems. Linear Algebra Appl. 2001. No. 336. Pp. 51–60.

Ikramov Kh. D. Nesimmetrichnaya problema sobstvennykh znachenii. Chislen- nye metody [Asymmetric Eigenvalue Problem. Numerical Methods]. Moscow: Nauka Publ., 1991. 240 p.

Koloushek V. Dinamika stroitelnykh konstruktsii [Dynamics of Building Con- structions]. Moscow: Building Literature Publishing Department, 1965. 632 p.

Farlow S. J. Partial Differential Equations for Scientists and Engineers. New York: Wiley, 1982. 402 p.

Krylov A. N. O nekotorykh differentsialnykh uravneniyakh matematicheskoi fiziki, imeyuschikh prilozhenie v tekhnicheskikh voprosakh [About Some Differential Equations of Mathematical Physics, Having Application in Engineering]. Moscow, Leningrad: Gostekhizdat Publ., 1950. 368 p.

Cawley Р., and Adams R. D. The Location of Defects in Structures from Meas- urements of Natural Frequencies. The Journal of Strain Analysis for Engineering De- sign. 1979. V. 14. Iss. 2. Pp. 49–57.

Berman А. System Identification of Structural Dynamic Models – Theoretical and Practical Bounds. Proceedings of the AIAA/ASME/ASCE/AHS 25th Structures, Structural Dynamics and Materials Conference. California, Palm Springs, 1984. Pp. 123–129.

Sobolev V. I., Pinus B. I. Opredelenie parametrov ostatochnoi zhestkosti defektnykh zdanii na osnove lazernykh otobrazhenii kolebanii i resheniya obratnoi zadachi dinamiki [Determination of the Parameters of Deficient Buildings Residual Stiff- ness Based on Laser Imaging of Vibrations and Solving the Dynamic Inverse Problem]. Vestnik VSGUTU. 2019. No. 1 (72). Pp. 55–67.

Amelkin V. V. Differentsialnye uravneniya v prilozheniyakh [Differential Equa- tions in Applications]. Moscow: Nauka Publ., 1987. 160 p.

Clough R. W., Penzien J. Dynamics of Structures. McGraw-Hill College, 1975. 634 p.

Pinus B. I., Morgayev D. E. Otsenka ostatochnogo resursa sei-smostoikosti zdanii serii 1-335 ks v gorode Irkutske [Assessment of the Residual Seismic Stability of 1-335 ks Series Buildings in Irkutsk]. Tezisy dokladov V Rossiiskoi Natsional'noi konferentsii po seismostoikomu stroitel'stvu i seismicheskomu raionirovaniyu s mezhdunarodnym uchastiem. Moscow, 2003. 81 p.

Sobolev V. I. Raschet mnogoetazhnykh zdanii razlichnykh konstruktivnykh sis- tem na gorizontalnoye seismicheskoye vozdeistvie s uchetom prostranstvennogo de- formirovaniya [Calculation of Multistory Buildings of Various Structural Systems for Horizontal Earthquake Action Taking into Account Spatial Deformation]. Mate- maticheskoye modelirovaniye v mekhanike sploshnykh sred na osnove metodov granichnykh i konechnykh elementov. Proc. 18th Int. conf. St Petersburg, 2000. V. 1. 217 p.

Argyris J. H., Boni В., Hinderlang V. Finite Element Analysis of Two- and Three Dimensional Elastoplastic Frames — the Natural Approach. Comp. Meth. Appl. Mech. 1982. V. 35. No. 2. Pp. 221–248.

Aizenberg Ya. M. Razvitie kontseptsii i norm antiseismicheskogo proektiro- vaniya [Development of the Concepts and Regulations for Earthquake-Proof Engineer- ing]. Moscow: Stroitelstvo Publ., 1997. 73 p.

Davies E. B., Gladwell G. M. L., Leydold J. S., Peter F. Discrete Nodal Domain Theorems. Linear Algebra Appl. 2001. No. 336. Pp. 51–60.

Ikramov Kh. D. Nesimmetrichnaya problema sobstvennykh znachenii. Chislen- nye metody [Asymmetric Eigenvalue Problem. Numerical Methods]. Moscow: Nauka Publ., 1991. 240 p.

Koloushek V. Dinamika stroitelnykh konstruktsii [Dynamics of Building Con- structions]. Moscow: Building Literature Publishing Department, 1965. 632 p.

Farlow S. J. Partial Differential Equations for Scientists and Engineers. New York: Wiley, 1982. 402 p.

Krylov A. N. O nekotorykh differentsialnykh uravneniyakh matematicheskoi fiziki, imeyuschikh prilozhenie v tekhnicheskikh voprosakh [About Some Differential Equations of Mathematical Physics, Having Application in Engineering]. Moscow, Leningrad: Gostekhizdat Publ., 1950. 368 p.