# Dynamics of Structures

## Dynamics of Structures

(also called the theory of vibration of structures), the study of the vibration of structures, methods of designing structures subjected to dynamic loads, and means of reducing vibration; a branch of structural mechanics. Dynamic loads on a structure are characterized by such rapid change with time in their size, direction, or point of application that they cause vibrations of the structure, which must be considered during its design. Loads occurring during the operation of machines with unbalanced moving masses, upon impacts of massive bodies, and during earthquakes and explosions are examples of such loads. Not only the movements of structural points but also internal stresses and strains in structural elements may be of a vibrational nature.

The basic content of the dynamic design of a structure consists in the determination of the expected amplitudes of movements and internal strains and stresses in a structure during its vibrations under the influence of a dynamic load (that is, during” forced vibration) and their comparison with permissible levels. Permissible levels of the amplitudes of inner stresses depend on the strength and durability requirements of structural members. The amplitude values for the speed and acceleration of the vibration of buildings and structures housing people or an industry with precision technology depend on requirements of noninjurious influence of vibrations on human health and on the quality of the products being manufactured.

The dynamics of structures is closely associated with structural analysis, which is a fundamental branch of structural mechanics. The problem of the strength and durability of structures is solved on the basis of static calculations (static loads) and dynamic calculations. The dynamics of structures uses highly developed methods of structural analysis but significantly extends them by means of d’Alembert’s principle, introducing time into the equation as a new variable. Experimental and theoretical dynamics of structures are distinguished according to their research methods.

By means of actual experiments and simulations, experimental dynamics of structures studies dynamic loads on structures (from stationary and moving machines and mechanisms, seismic and wind loads, and pulsations of pressure of liquids and gases in water pipes and boilers) and the dynamic properties of materials and structures (dynamic elastic moduli, internal friction and external resistances, endurance limits of materials and joints of riveted and welded elements, and ultimate strength and yield limit at high speeds of deformation caused by powerful impacts), and also tests the reliability of design schemes of structures and the effectiveness of methods of reducing vibration.

Theoretical dynamics of structures, based on the research results of experimental dynamics of structures, develops analytical and numerical methods of determining the amplitude of forced vibrations (the fundamental problem of dynamics of structures), as well as the frequency and shape of free (or intrinsic) vibrations of structures. Methods of solving the basic problems depend on the type of dynamic load and the design scheme of the structure. Dynamic loads are divided into determined loads, which change with time according to a definite law, and random loads, which change irregularly with time and are characterized by statistical values. Depending on the type of design scheme of the structure (beam, truss, frame, arch, plate, dome, or casing), the corresponding method is used to determine the amplitude of vibrations as a function of the coordinates of points of the structure. Methods for determining the frequency and shape of vibrations depend only on the design scheme of the structure. Knowledge of the frequency and shape of the intrinsic vibrations of a structure makes possible, before designing for dynamic load, the prediction of the qualitative pattern of the forced vibrations, the maximum reduction of the estimation, and the revelation of disadvantageous frequency levels for periodic loads and durations for short-term loads.

The dynamics of structures was conceived as a science in the 1920’s; its development was a result of the practical needs of construction and the significant increase of dynamic loads on structures (an increase in the power and speed of movement of machines and a rise in the speeds of traveling loads). However, the evolution of the dynamics of structures in the 1920’s lagged significantly behind its theoretical base (the theory of vibrations and structural mechanics) and the factual information provided by dynamic testing of structures and building materials and by the study of operational and dynamic loads.

The traditional method of computing the influence of dynamic load that was used at that time (the introduction of the dynamic load coefficient into static calculations of the structure) was incomplete; it disregarded the dynamic properties of structures and loads. In the 1930’s the dynamics of structures began to develop rapidly, using experimental data and sufficiently rigorous theory (D. D. Barkan, N. I. Bezukhov, S. A. Bernshtein, V. V. Bolotin, K. S. Zavriev, Iu. A. Nilender, A. F. Smirnov, and I. M. Rabinovich). Progress in postwar computer technology gave new impetus to the development of the dynamics of structures, making possible by means of electronic computers the practical solution of more complex problems (the Central Research Institute of Structural Elements and the Moscow Engineering Institute for Railroad Transportation).

In the 1950’s and 1960’s the USSR was the first country in which specifications were published on dynamic structural designs (developed by the Central Research Institute of Structural Members and the Scientific Research Institute of Foundations and Underground Structures). The specifications reflected the high level of development of the dynamics of structures in the USSR. During this period new, important directions in the dynamics of structures were developed: the dynamic design of structures with nonlinear elastic or dissipative properties (Ia. G. Panovko, G. S. Pisarenko, and E. S. Sorokin), with the consideration of plastic deformations (A. R. Rzhanitsin and others), the dynamic design of structures lying or standing on an elastic inertial foundation (N. M. Borodachev and B. G. Korenev), and the dynamic design of structures subjected to random loads by means of methods of statistical dynamics or the theory of random processes (M. F. Barshtein, V. V. Bolotin, I. I. Gol’denblat, and N. A. Nikolaenko).

Research on problems of the dynamics of structures is published in the journal Stroitel’naia mekhanika i raschet sooruzhenii (Construction Mechanics and the Design of Structures; Moscow, since 1959), in the collection Issledovaniia po teorii sooruzhenii (Research on the Theory of Structures), and in the works of the dynamics laboratory of the Central Research Institute of Structural Elements and the department of theoretical mechanics of the Moscow Engineering Institute for Railroad Transportation.

### REFERENCES

Sorokin, E. S. Dinamicheskii raschet nesushchikh konstruktsii zdanii. Moscow, 1956.
Smirnov, A. F. Ustoichivost’ i kolebaniia sooruzhenii. Moscow, 1958.
Bolotin, V. V. Statisticheskie melody v stroitel’noi mekhanike, 2nd ed. Moscow, 1965.
Nowacky, W. Dinamika sooruzhenii. Moscow, 1963. (Translated from Polish.)

E. S. SOROKIN

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