Dynamic response of tube containing water subjected to impact loading
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Dynamic response of a round aluminum tube supported at both ends was investigated when impacted at its center by an external mechanical loading. The tube was subjected to different conditions. First, the empty tube was tested. Then the tube was filled with different amounts of stationary water (i.e., 25, 50, 75, and 100% full), based on the inner volume of the tube. Finally, flowing water through the tube was considered. Different magnitudes of impact loading were also applied. The study was primarily conducted experimentally with some additional numerical studies completed to further understand the results. The impact force as well as strain along the tube were measured for all described test conditions. Their results were compared. Additionally, the vibrational frequency and damping of the system were examined using strain–time histories. The results showed that the dynamic behavior of the tube was significantly dependent on the amount of internal water as well as its flow condition.
KeywordsFluid structure interaction Impact loading Vibration Pipe with fluid Dynamic analysis
Tubes and pipes are common in many industrial applications. Most have a round shape and contain fluid either inside and/or surrounding them, depending on the application. As a result, there has been extensive research on dynamic motion of tubes and pipes carrying fluids. For example, flow-induced vibration (FIV) has been investigated extensively (Blevin 1994; Naudascher and Rockwell 2005; Kaneko et al. 2014; Weaver et al. 2000).
Vibration of a fluid-transporting pipe was studied in the early 1950s (Ashley and Haviland 1950; Housner 1952). Since then much more research has been completed. Later researchers considered instability of a pipe depending on different end supports, steady and unsteady flows, straight or curved pipes, etc. A good summary of this subsequent research was published in Refs. (Paidoussis 1987; Li et al. 2015). While those particular studies focus on the internal flow of a pipe, there is also research in external flow over a pipe such as vortex-induced vibrations (VIV). Some of the latter works are given in Chen (1986), Au-Yang et al. (1991), Griffin (1980) and Weaver et al. (2000).
However, not all of those papers considered mechanical impact loading of the pipe. Impact studies on circular cylinders were conducted in Christoforou and Swanson (1990) and Alaei et al. (2019). One paper developed an analytical solution of a simply supported orthotropic cylindrical shell subjected to lateral impact loading (Christoforou and Swanson 1990) while the other investigated structural coupling of two concentric composite circular cylinders containing fluid inside their annulus and subjected to impact loading (Alaei et al. 2019).
The objective of this study is to investigate the dynamic response of a round tube containing either stationary or moving fluid while supported at both ends and impacted at the center between the supports. The subsequent section describes the experimental setup and test procedures. Finally, the results and discussion are provided, followed by conclusions.
2 Description of experiments
The tube was tested in various conditions. The base condition consisted of the dry tube which did not contain any fluid inside. The next set of cases included varying amounts of stationary water inside the tube. In terms of the inside volume of the tube, the water levels were set to 25, 50, 75, and 100%, respectively. Both ends of the tube were sealed by a rubber laboratory stopper to retain the water inside of the tube during the tests. The final case considered steady-state water flow through the pipe. The Reynolds number of the flow rate using a water pump was calculated as 5.21 × 105 based on the inner diameter of the tube.
Each test case was repeated ten times to ensure repeatability of the test results. During each test, the impact force and strain data were recorded at a data sampling frequency of 1000 Hz. This rate was selected to make sure that sufficient data were collected for accurate measurement of the primary vibrational frequency of the tube.
3 Results and discussion
Strain plots for five selected positions along the length of the tube are also included in Fig. 4. Stain gage locations are denoted as left (L), right (R) plus a number from 1 to 5. Figure 2 depicts these positions along the length of the tube with L1 being nearest to the impact point and L5 closest to the end support on the left side of tube. The enumeration pattern is mirrored on the right side of the tube, while C indicates the center location opposite the impact point.
The maximum tensile strain of 1.12 × 10−3 m/m occurred at the center strain gage location. In general, tensile strain was found to decrease with an increase in distance from the tube center span location while compressive strain increased as the gage location approached the supported end. A maximum compressive strain of 4.62 × 10−4 m/m was observed at the ‘L5’ strain gage which was located very near to one of the supported ends. This strain characteristic suggests that the 3-D printed support provided the bending moment to the tube as expected. However, the support did not provide a fully clamped condition because the support structure was not completely rigid. The tube was inserted snuggly into the cylindrical opening of the support structure, but not otherwise fixed in place, with the exception of a 6 mm set screw at each end support. The strain responses were symmetric in terms of the tube center.
To represent the actual boundary condition as in the experimental setup, a torsional spring was attached to a simply supported tube beam structure. When the torsional spring is zero, the tube is simply supported, while it is fully clamped when the torsional spring is infinite or very stiff. The experimental investigation thus falls between these two boundary conditions. When a torsional spring of 6300 N-m was used in the model, the frequency from the finite element analysis (FEA) using 40 beam elements resulted in 143 Hz which matched the experimental frequency. All subsequent FEA used the same torsional springs at the boundary condition.
As expected, the peak impact force increased along with the initial height. Internal water, whether stationary or flowing, increased the impact force as well. The difference between the stationary and flowing water cases was minor for the peak impact force. When the peak force was plotted for the square root of the initial height, the former was almost linear to the latter for the flowing water case. In other words, the peak impact velocity was linearly proportional to the initial impact velocity. The stationary water case was similar to the flowing water case. However, the dry tube showed a nonlinear behavior between the impact force and impact height.
The vibrational frequency of the tube with flowing water was 81 Hz, which is lower than the frequency of the tube without and with stationary water. The experimental frequency with flowing water was very close to the numerical frequency of the dry tube while the water mass was added to the tube mass without any modification of its stiffness. The stationary water contributed the effective tube stiffness and mass while the flowing water only contributed to the effective mass in terms of the vibrational frequency.
Dynamic motions of a tube were studied when the tube was filled to varying levels of fullness based on internal volume of stationary water as well as with flowing water. The dynamic response was measured using strain gages attached along the length of the tube while it was impacted using a pendulum motion. The impact force was also recorded. The stationary water level inside the tube influenced the dynamic motion of the tube significantly. Both impact loading and the resultant strains increased along with the water level as the initial impact condition remained the same. The peak impact force as well as the maximum strain at the center increased more than 50% when the tube was full of stationary water as compared to the no-water case.
The peak impact force was more or less similar between the full stationary and flowing water cases. However, the tube response was not necessarily similar between the two cases. For example, the vibrational frequency was quite different depending on whether or not the water was stationary or flowing. The former case had 20% greater frequency than the latter case. As a result, the strain responses were also different.
A numerical study was conducted to model the vibration of the tube. To simulate the actual boundary condition of the physical experiment, a torsional spring was attached to each end while the tube was simply supported at each end. If the torsional spring constant is zero, the boundary is simply supported. If the spring constant is very large, i.e., infinite, the boundary condition is clamped. The actual boundary condition was between these two extreme cases, i.e., partially clamped. With the proper selection of a spring constant, the numerical model gave a frequency which agreed very well with the experimentally measured value.
The numerical model was run for the tube with water using the properly adjusted boundary condition. The internal water was added to the tube mass in the numerical model to modify the density of the tube. When the stiffness of the tube was modified with the stationary water, the results agreed better with the experimental frequency. On the other hand, the numerical frequency with flowing water was better without modification of the tube stiffness.
Damping resulting from the internal water increased with the addition of stationary water. However, the flowing water case resulted in lower damping than the stationary water. In conclusion, the amount of stationary internal water as well as flowing water greatly influenced the dynamic behavior of the tube subjected to external impact loading.
This work was supported by Office of Naval Research (ONR), and the Program Manager is Dr. Yapa Rajapakse.
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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