## Methodology and Results  
### Vibrations Induced in the Side Branches  
The dangerous self-excited gas pressure oscillations may occur at the T-branch of the gas pipe system due to the following cases:  
* the gas flow rate is sufficiently high,
* there are necessary fluid and geometric conditions for the generation of excited pressure oscillations, and
* there was a resonant match between the excitation frequency and the natural frequency of the closed tap volume (existence of Helmholtz standing waves in the branch--the so-called \"whistle effect\"). Pressure oscillations induced by the standing wave can vibrate the piping system around the T-branch. This effect is even more pronounced with several closed side branches in a row [3]. Oscillation of this type also occurs in bypasses, where the sharp radius of the arc creates conditions close to the blind side branches (see Figure 1). These operating states can be partially predicted in the design phase of the piping system. Figure 1: Principle of self-excited oscillations in closed side branches. D is the diameter of the main pipe, and L (d) is a length (diameter) of a side branch. The critical gas velocity \(v_{n}\) for self-excitation can be calculated according to the relationship  
\[v_{n}=\frac{f_{ne}d}{S_{t}} \tag{1}\]  
where \(S_{t}\) is the Strouhal number for the geometric and fluid ratios in the T-branch, and \(f_{ne}\) is the oscillation frequency given as  
\[f_{ne}=\frac{(0.88\div 1)nc}{4L} \tag{2}\]  
where \(n=1,3,5,...,c\) is the sound velocity in an environment [7]. Due to the unclear geometry of the rounding edge of the T-branch or the change in fluid conditions compared to the design state, it is not always possible to eliminate such oscillations. A suitable device for ensuring operational safety should be installed in a monitoring system that can be based on sensing vibration parameters (accelerometers) or strain sensors. Accelerometers enable detecting quantities such as the velocity of oscillation or acceleration and frequency values at which the pipe oscillates. Allowable values for the velocity of the oscillation are stated by guidelines and industry regulations. As far as the effects of oscillation on the safe operations are concerned, it is essential to know the amplitude of the strain (distortions), which could cause the fatigue of a material. However, this type of information cannot be provided by accelerometers. The need to know the maximum displacement of the oscillating part relative to the base (branch oscillation displacement relative to the main pipe or fixed ground) would require the measuring of accelerations and calculating of displacements in multiple directions and in multiple parts of the pipeline node, as documented in Figure 2.  
In general, the pipelines oscillate in a wide spectrum of frequencies. The integrity of pipelines is put at risk only at frequencies that cause significant magnitude of the strain amplitudes. Having compared the frequency composition of the oscillation shown in Figure 3, measured by accelerometers and bending stress detected by strain gauges on the identical section of the pipeline, it is obvious that significant deformation amplitudes occur only at frequencies up to 3 Hz. The frequency spectrum measured by accelerometer sensors (Figure 3 on the left side) does not reveal the significant magnitudes of strain that can decrease the operational lifetime of a pipe. More reliable results can be obtained by the direct measuring of strains at the critical cross-section. The suitable arrangement of the sensors allows to calculate directly the maximum value of the stress in the most loaded cross-section of the vibrating pipeline part. Figure 4 shows the deployment of a minimum number of strain sensors along the perimeter of the pipeline's cross-section, allowing the calculations of bending, torsion, and normal forces. The methodology of measurements and calculations is elaborated in more detail in the papers [11; 12].