# Analysis free oscillations of a non-homogeneous pipe along's thickness and length, taking into account the resistance of the external environment

## Authors

• Gulnar Mirzayeva Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 9 B, Vahabzadeh str., Baku, Republic of Azerbaijan, AZ1143, Ukraine
• Vusala Rzayeva Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 9 B, Vahabzadeh str., Baku, Republic of Azerbaijan, АZ1143, Ukraine

## Keywords:

non-homogeneous pipe, environment with complex properties, Winkler model, Pasternak model, Karnet model.

## Abstract

Structural elements made of heterogeneous natural and artificial materials are widely used in the construction of modern building complexes and in many other areas. Among them is the use of boards and shells of different configurations. Currently, one of the most important requirements for designers and accountants is to properly assess the mechanical properties of the material of the structural element and the impact of the environment in contact during operation. Taking these into account, the mathematical solution of the problem becomes difficult, and if not, serious mistakes can be made. One of the most problematic places is considering the resistance of the external environment.

Objects of research are modern pipes exposed to the external environment.

In the paper is analysis Pasternak is one of the mathematical models that accurately reflects the elastic real properties of the environment, the model Winkler, which is characterized by two constants, the model of Karnet and the model of Rjanitsin. And is analysis their effected.

In the course of the research, the method of separation of variables and then Bubnov-Galerkin method is used, which explain relationship between the dimensionless value of frequency, the parameters that characterize the non-homogeneous of the base, and the pipe. The selection of special frequencies is carried out by selecting the corresponding special equation and boundary conditions. There are nonlinear algebraic equations and their solution using computer technology. It is shown that when the mechanical properties of the pipe vary in length, the above solution method does not work and the determination of the characteristic parameters must be performed using other approximate analytical methods. In engineering practice, it is usually sufficient to find the basic tone of the frequency. As a result of the research it is shown that the external environment effects are important for non-homogeneous pipe and should be considered in the design of the structure-ground interaction. In the future, the proposed approach and should be considered in the design of the structure non-homogenous pipe interaction.

## Author Biographies

### Gulnar Mirzayeva, Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 9 B, Vahabzadeh str., Baku, Republic of Azerbaijan, AZ1143

PhD, Senior Researcher

Department of Theory of Elasticity and Plasticity

### Vusala Rzayeva, Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 9 B, Vahabzadeh str., Baku, Republic of Azerbaijan, АZ1143

Master

Department of Wave Dynamics

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2020-10-30

## How to Cite

Mirzayeva, G., & Rzayeva, V. (2020). Analysis free oscillations of a non-homogeneous pipe along’s thickness and length, taking into account the resistance of the external environment. Technology Audit and Production Reserves, 5(1(55), 24–27. https://doi.org/10.15587/2706-5448.2020.213475

## Section

Reports on research projects