2. Vibration structure

It is used as a reliability factor of the structure under vibrations [1, 3]. Under harmonic vibrations, the vibration velocity can serve as a reliability criterion for the pipeline.

The harmonic vibrations are characterized by two parameters: frequency of vibrations and displacement amplitude:

$$y = Y\_0 \sin \omega t \tag{1}$$

M zð Þ¼ EIy00ð Þ¼� <sup>z</sup> EI <sup>i</sup>

<sup>σ</sup>makc <sup>¼</sup> EI <sup>i</sup>

<sup>ω</sup><sup>i</sup> <sup>¼</sup> <sup>i</sup> 2 π2 L2

2 π2

At initial approximation, certain typical ideal forms of vibrations are used for vibration analysis. Then, the natural vibration frequency of the pipeline is expressed by the following formula:

According to [3, 5], the maximum vibration velocity is determined from the following equa-

where vbase is a vibration velocity of the base plate; η is a dynamic magnification factor; k is a

Expansion bends are regarded as vibration damping elements for the pipelines to ensure their vibration resistance. The expansion bends prevent the transfer of vibrations along the pipeline.

Stresses across the cross sections of the pipeline under natural vibrations can be determined

<sup>0</sup> ydz= Ð 1 <sup>0</sup> <sup>y</sup><sup>2</sup> ffiffiffiffiffiffiffi EIg rF

s

where I is a moment of inertia.

Maximum stresses in the pipeline:

then W is a moment of resistance in the pipeline.

where F is a square section of the pipeline.

4. Analysis of pipeline vibrations

form engagement factor: <sup>k</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

tions and current coordinate.

from the following equation:

tion:

2 π2

Figure 1. Force excitation of vibrations. (а) Excitation through an elastic support; (b) kinematic excitation of vibrations.

<sup>L</sup><sup>2</sup> <sup>Y</sup><sup>0</sup> sin <sup>i</sup>π<sup>z</sup>

<sup>L</sup> (5)

Vibration Strength of Pipelines

25

http://dx.doi.org/10.5772/intechopen.72794

(7)

WL<sup>2</sup> <sup>Y</sup>0, (6)

Vmax ¼ vbaseKη (8)

dz, y, z are dimensionless forms of pipeline vibra-

Vibration velocity and vibration acceleration are expressed as follows:

$$
\omega v\_0 = \omega Y\_0; \quad \mathcal{g}\_0 = \omega^2 Y\_0 \tag{2}
$$
