A Simple Method to Design and Analyze Dynamic Vibration Absorber of Pipeline Structure Using Dimensional Analysis

Vibrations due to mechanical excitation and internal and external fluid flow can cause fatigue in pipelines and leaks in fittings. A beam-based dynamic vibration absorber (beam DVA) is a device comprising an L-shaped beam with a concentrated mass at its free end that can be used to absorb and dissipate vibrations in the pipeline. In this paper, a mathematical equation is extracted to design the beam DVA using the dimensional analysis (DA) method and data recorded from 120 experimental tests. In the experimental studies, the pipes are fabricated in 1-inch, 2-inch, and 3-inch sizes. Each pipe is subjected to harmonic excitation at different frequencies, and the amplitude of vibration of the pipe is evaluated by changes in the geometric characteristics of beam DVA and concentrated mass. (e proposed methodology is validated using the finite element method and simulation in the SIMULINK/MATLAB. (e results showed that, out of the nine effective dimensionless parameters identified in pipe vibration control, mass ratio and stiffness ratio have the highest and lowest impacts on pipe vibration absorption, respectively.


Introduction
Pipeline structures are used in oil and gas industries, petroleum industries, power plants, and chemical factories. Due to the wide application of pipes in various industries, vulnerability assessment and improvement of piping systems and pipelines in the field of vibration are of great importance. Usually, piping systems are designed against static load cases only. However, often, dynamic loads occur. Vibration excitations in a pipe are induced by turbulent flow in the pipe, water and liquid hammers, and pressure pulsations, caused by driven machines. If the frequency of excitation is close to the natural frequency of the piping system, resonation occurs, causing material fatigue, making them prone to leaking joints, causing safety hazards in the industries, and finally leading to damage and failure. In this situation, the vibration can be reduced using a dynamic vibration absorber. Many studies have been conducted in this regard, each one with advantages and disadvantages. Accordingly, the mathematical modeling of the vibrational behavior of the pipes and finding the optimum parameters of the vibration absorbers in the pipelines have been of interest to researchers in recent years. Norris et al. [1] worked on the application of adjustable vibration absorbers in controlling pipeline vibrations caused by wind force excitation. Although the amplitude of vibration caused by wind power is relatively small, repeated cyclic vibrations can cause fatigue in pipeline connections. e vibration absorber consists of a mass and a flexible element. is paper considers the impedance at the absorber connection point against the impedance of the primary system and the absorber system. In addition, the amplitude reduction factor of the primary structure in excitation frequency is used to design the required absorber mass. e absorber consists of two mass-held elastomeric discs. Elastomeric discs act as a rotating spring. e masses can be easily added and subtracted to adjust the absorber. e absorber is mounted on 60 different pipeline structures with 2″ to 24″ in diameter over more than 10 years in Northern Alaska. Nabeel et al. [2] investigated the vibrational and stability properties of a welding pipe. Welding in pipes causes residual stresses which affect the vibrational properties of the pipe. e Euler-Bernoulli beam theory and plug-flow model were used to model the pipe and the effects of residual stress on the vibrational properties of the pipe, respectively. e fluid velocity and pressure distribution inside the pipe were extracted using the finite element method. e results show that the natural frequencies of the welding pipe decrease with increasing fluid velocity. Welding in the fluid carrier pipe results in a decrease in the normal frequencies; changes in frequencies depend on the location of the welding. Senalp et al. [3] investigated the dynamic response of the Euler-Bernoulli beam to the linear and nonlinear viscoelastic support with respect to centralized impetus. Equations of motion were extracted using mass and stiffness matrices in Euler-Bernoulli theory, and the model was analyzed using the finite element method. Demir and Civalek [4] investigated the vibration of the nanoscaled beam embedded in an elastic matrix under the effect of thermal environments.
is document describes the elemental mass and stiffness matrices via Hermitian cubic shape functions for the nanoscaled beam. e influences of the Pasternak foundation parameter, small scale parameter, and mechanical properties of material and thermal effect on vibrational frequency are investigated. e results show that the difference between the frequencies of classical beam model and those of the nonlocal beam model increases as the mode number and nonlocal parameter increase. Lats [5] studied the vibrations of a beam connected to a rotational transient vibrational absorber system. e beam was subjected to harmonic and central excitation forces; the Euler-Bernoulli beam theory was used for modeling. e Fourier method was applied to analyze the model, and the displacement amplitude of the beam arbitrary points at the desired frequencies was obtained using the Laplace transform. is article aims to investigate the performance of a translational-rotational absorber compared with a translational absorber. Numerical simulations show a significant improvement in vibration reduction when using translational-rotational absorbers. Civalek [6] derived the frequencies and buckling loads of composite plates using the discrete singular convolution (DSC). is method is a good level of accuracy dynamic analysis method for single structures. However, there are also limitations. In this study, the straight-sided quadrilateral domain is mapped into natural coordinates, and the frequencies and buckling loads are accurately predicted. e results are compared with different numerical methods. Syaiful et al. [7] presented a simple support beam theory model with dynamic mass absorber, spring, and damper. e Euler-Bernoulli theory was applied to obtain the displacement amplitude of the beam from the governing equations, and the beam displacement equations were obtained with the absorber using the receptance theory at the beam connection point. In addition, the governing equations of the absorber were extracted using equations of motion and calculated according to the boundary conditions of the displacement to force ratio. MATLAB was used to draw the frequency response function (FRF) diagrams, and ANSYS was used to validate the results. A comparison of theoretical solutions and drawing results with MATLAB and simulation with ANSYS revealed a small error of about 0.1%. e results show that the displacement amplitude decreases by about 30% with increasing vibration absorber to the beam. Zainulabidin et al. [8] investigated the effect of vibration absorbers mounted on beams with clamped support. e spring-mass vibration absorber was mounted on four beams with clamped support and analyzed using ANSYS. First, the natural frequencies and mode shapes in the beam were measured using modal analysis; then, the effect of the location of the absorbers on the beam was studied in terms of vibrational properties. e results show that the location of the absorber strongly influences the amplitude of the vibration of the beam. Also, a high percentage of amplitude reduction is obtained when absorbers are not located on the node. When the vibration absorber is near the fixed supports, the amplitude of vibration decreases to 9.99% and 99.8% in the first and second modes, respectively. Sutar et al. [9] developed a mathematical model for dynamics and stability of the pipe conveying the fluid with guided supports and compared it with the finite element method (FEM). e Euler-Bernoulli theory and Hamilton's energy expressions were applied to obtain the equation of motion and fundamental transverse natural frequency of vibration. e influence of fluid velocity on natural frequencies was extracted using the ABAQUS software for three different pipe support materials: thermocol, foam, and springs with a stiffness of 54 N/mm. e results show that, for foam type of support, the frequencies tend to vary with increase in flow velocities, and for the other type of supports, it is seen that the frequencies reduce for the fluid flow condition only when compared with the other two conditions. Song et al. [10] investigated the vibration control of a pipeline using a pounding tuned mass damper (PTMD) absorber. In this study, an M-shaped pipeline was constructed with the PTMD absorber mounted on it. Due to the lack of empirical studies on the fluid carrier pipe because of safety and operational issues in the working industries, the pipe was considered as free and fluid-independent; also, the fluid effects on the vibrational properties of the pipe were ignored. e PTMD absorber consists of two parts: the absorber part and the separator part. e vibration absorber part consists of a cantilever beam, with a concentrated mass in the free head, mounted horizontally and parallel to the pipe axis. e separator part is a sheet from which a circular hole is cut, the inner surface of which is covered with a viscoelastic material. When the mass fluctuations exceed a certain level, the beam bends and hits the viscoelastic material. As a result, it depletes the kinetic energy and reduces the pipeline vibration. e pipeline modeling is performed using FEM as well as mass and stiffness matrices. e Rayleigh model is used to obtain the attenuation matrix, and the Hertz model is used to obtain the nonlinear shock force model in viscoelastic materials. Since the response of the system under other stimuli (e.g., vortex fluid flow and internal and external fluid flows) may be very different from the response under harmonic excitation, harmonic excitation was considered in this study. Experimental results show the high effectiveness of PTMD in reducing vibration. Steady-state displacement and pipe acceleration responses decreased by 86.6% and 83.6%, respectively. An acceptable concordance was obtained by comparing the numerical results with the experimental data. Fong and Mohd Yassin [11] investigated the vibrational properties of oil-bearing pipelines considering the effects of fluid inside the pipe. In this paper, FEM was applied to calculate the natural frequency of a fluid carrier pipeline located on a viscoelastic foundation. e simple support is modeled using the Euler-Bernoulli beam theory. In addition, mass, stiffness, and damping matrices were extracted using the Galerkin weighted residual method.
e results show that an increase in the foundation stiffness causes an increase in the natural frequency of the pipeline, while an increase in the foundation damping and fluid velocity inside the pipe causes a decrease in the natural frequency of the pipeline. e results were validated using the data from the previous studies. Chang et al. [12] investigated the vibration control of a nuclear power plant's piping system using an earthquake Stockbridge damper. e piping system analysis was performed using SAP2000, both with and without a damper. e damper design was performed in two stages: (1) calculation of mass ratio, frequency ratio, and attenuation ratio and (2) absorber design. e best damper design parameters were extracted according to the fixed-point method. e results show that using vibration absorbers can be effective in controlling the vibration response of the piping system in the nuclear industry. Hua et al. [13] examined the model of a cantilever beam as the primary structure by connecting another cantilever beam at its end as a dynamic vibration absorber. e combination of receptance and fixed-point methods was used to optimally extract the vibration absorber characteristics. e analytical results were validated using ABACUS. Finally, a comparison was made between beams as vibration absorbers and traditional spring damper absorbers. e optimum mass ratio and frequency ratio in the traditional absorber was obtained using the fixed-point method. Also, the maximum amplitude was calculated at two points. e results show that the maximum amplitude is lower when the beam is used as vibration absorbers. Tyng et al. [14] performed a study on a laboratory-scale piping system to investigate the flow-induced vibration problems under different operating conditions. e study of fluid flow-induced vibrations in pipes in real piping systems at operating sites raises safety and operational concerns; therefore, a laboratory-scale piping system is preferred to investigate the flow-induced vibration problem under various operating conditions. A smaller-scale model employs dimensional analysis as well as geometric, kinematic, and dynamic similarities to obtain reliable results. In the geometric similarity, both real and scaled models must have the same length to diameter ratio, as required by Buckingham's theorem. In addition, the mode shape of the scaled model must be like that of the real model in natural frequency. e flow pattern of the models can be used to justify the kinematic similarity. is pattern is similar when the Reynolds number in the real and scalable model is the same. Real and scaled Reynolds numbers are derived from computational fluid dynamics (CFD) simulation results. Fluid-structure interaction (FSI) or CFD analysis should be performed to investigate the dynamic similarity. When the force properties of the fluid and the wall of the pipe are similar, it can be concluded that the dynamic similarity is obtained. e geometric, kinematic, and dynamic similarities were obtained in a case study between the scaled and real models. e dimensional analysis is one of the powerful tools in engineering that enables the rapid analysis of physical systems. Although dimensional analysis is traditionally used in fluid mechanics, this method is useful in all physical and engineering sciences. Moreover, its prominent applications are found in the social sciences and biology. e idea of experimenting with unrealistic dimensions has been suggested by several individuals. Newton was one of those scientists who defined the term "principles of similarity". Later, Maxwell, a Scottish physicist, played an important role in establishing the basic units of mass, length, and time that are used in the creation of other units. Rayleigh was probably the first who applied dimensional analysis to find relationships between physical quantities. In addition, he was the first who used dimensionless parameters to analyze experimental data. e turning point of the dimensional analysis was made by Buckingham, who expressed dimensional analysis and rules of similarity more systematically. Reddy and Reddy [15] investigated different properties of ball-bearing parameters using dimensional analysis. In this study, a dimensional analysis was performed using Buckingham's eorem. For this purpose, 20 dimensionless quantities in ball bearings were identified, and 13 dimensionless terms affecting ball bearings performance were extracted. Ball-bearing length-to-diameter (L/D) ratio diagrams were plotted in each dimensionless quantity at different values of convective heat transfer coefficients. is method can be used to find the effect of slight changes in the convective heat transfer coefficients on ball bearings. Allamraju and Srikanth [16] extracted the equations governing the PZT-5H transducer using Buckingham's eorem. Hassani et al. [17] developed new equations to predict the thermal conductivity of nanofluid based on experimental data using Buckingham's eorem. is author developed a new equation that can predict thermal conductivity for a range of nanofluids. e thermal conductivity of nanofluids depends on the base fluid, temperature, thermal conductivity, viscosity, density, or physical properties of the nanoparticles. akare et al. [18] extracted a mathematical model to predict the displacement of the upper roller of the 3-roll bending machine using dimensional analysis. ey used valid experimental data in the developed model and obtained the error rate of 1.237% for the upper roller displacement, suggesting that the values obtained from the mathematical model are close to the actual values in the experimental tests.
One of the most important issues in using vibration absorbers is the optimal design of their parameters. If the vibration absorber parameters are inappropriate, the existence of the absorber could even cause an increase in the amplitude of vibrations. In this paper, the authors propose Shock and Vibration the use of experimental tests and dimensional analysis to optimally design beam DVA in the piping systems. e use of Buckingham's theorem allows the identification of effective and less effective dimensionless numbers involved in the amplitude of vibration of the pipe and the derivation of functional, simple, and reliable equations in system analysis.

Experimental Model
According to Figures 1 and 2, the experimental model consists of four parts: pipeline structure, motor with unbalanced weight, vibrometer, and beam DVA. e pipes are mounted on two T-post supports that behave like simple support. A noteworthy point in this regard is that it is not possible to carry out practical tests on operating units on fluid carrier pipelines, the necessity of changing process conditions in the system, and the need for a series of measurements and safety concerns. Hence, theoretical studies and experimental tests are performed considering a fluid-free pipe with harmonic force excitation, which may be due to mechanical excitation. In the experiments, three different sizes of pipe 1-inch, 2-inch, and 3-inch are used. Table 1 shows the geometric characteristics and physical properties of the pipeline structure used in the experimental model. e second part is the experimental model of motor with unbalanced weight, which is used for excitation. Harmonic force is applied to the pipe by a rigid rod as a result of motor. e motor is connected to a variable-frequency drive (i.e., an inverter) to allow the desired frequency to be applied. On the velocity control unit, the motor frequency can be increased or decreased by adjusting the screws. e third part is the experimental vibrometer model. A portable vibrometer with a frequency range of 2 to 10,000 Hz is used to measure the frequency, acceleration, velocity, and displacement as well as drawing an FFT diagram. e fourth part is the experimental beam DVA model. is device consists of an L-shaped beam, with a concentrated mass added to its end, which can move in both horizontal and vertical directions and absorb kinetic energy. e use of threaded rod types in beam construction provides high flexibility in design. is flexibility is because of the ability to adjust its length under varying conditions and to modify cross-sectional dimensions. Table 2 illustrates the geometric characteristics and physical properties of the threaded rods used. e beam DVA is installed in the middle of the pipe with the most displacement, and the tests are performed with or without the beam DVA.
As illustrated in Figure 2 Each pipe is subjected to harmonic excitation at different frequencies. Moreover, the amplitude of vibration of the pipe is recorded and evaluated by changes in geometric characteristics of beam DVA and concentrated mass. Finally, 120 experiments were performed in three pipe sizes: Forty items in a 1″ pipe in an excitation frequency range of 8-12.5 Hz, concentrated mass range of 0.8-5 kg, and threaded rods M6, M8, M10, and M12. Forty items in a 2″ pipe in an excitation frequency range of 10-18.5 Hz, concentrated mass range of 2.5-10 kg, and threaded rods M6, M8, M10, and M12. Forty items in a 3″ pipe in an excitation frequency range of 17-25 Hz, concentrated mass range of 2.5-14.5 kg, and threaded rods M8, M10, and M12.

Determination of Natural Frequencies
To derive the natural frequencies of the pipe using the Euler-Bernoulli beam theory, first, the equations of motion of the transverse pipe vibrations are extracted as partial differential equations (PDEs) and the boundary conditions governing simple support are applied. e natural frequencies are obtained by solving the differential equations by the separation of variables. e first two natural frequencies of 1″, 2″, and 3″ pipes with the specifications listed in Table 1 are 9.43 (37.72) Hz, 17.02 (68.08) Hz, and 25.40 (101.60) Hz, respectively. e first mode has the lowest frequency and the largest displacement. erefore, this paper studies the first vibration mode, which is also the most important one.
To extract the natural frequency of the pipe in experimental tests, first, the frequency of the vibrator motor is slowly increased by using the inverter to cause resonance in the system to obtain the natural frequency of the system. Using the data recorded by the accelerometer, the first natural frequency for the 1″, 2″, and 3″ pipes are 10.02, 16.9, and 23.7 Hz, respectively. A comparison of the results shows that the estimation of the first natural frequency of the pipe is very well approximated in the experimental tests, which proves the accuracy and efficiency of the experimental tests. Figure 3 shows the comparison of the 1″ pipe FRF diagram in theory and the data recorded by the vibrometer.

Formulation of Mathematical Model
Using DA Several parameters affect the performance of the beam DVA, which can lead to the complexity of problem analysis and design. As a very powerful tool, dimensional analysis eliminates the complexity and reduces the number of effective variables by compressing some parameters. For this purpose, it is necessary to determine the parameters affecting the performance of the system and reducing the vibrations, which are of high importance. All the variables affecting this phenomenon are f x p , x ps , L a , L p , M a , w, I a , I p , A a , A p , E a , E p , ρ a , ρ p � 0. (1) Determining the function of equation (1) requires a great deal of time and many experimental tests. Using dimensional analysis, this phenomenon can be formulated using an equation between only a few dimensionless groups. Each of these dimensionless groups is called π. Now, the question arises as to how many π can be formed for this number of variables known to be effective in the beam DVA. e Buckingham π theorem is used to respond to this question. According to this theory, the number of the independent 4 Shock and Vibration    Shock and Vibration 5 dimensionless groups that can be used to describe a phenomenon is n-r, where n is the number of the variables influencing the phenomenon and r is the number of the main dimensions required for dimensional. Since the values of modulus of elasticity and density in the pipe and beam DVA are very close to each other in experimental tests, 10 parameters are selected from the 14 effective parameters, where there are 7 independent groups π and one dependent group π. Tables 3 and 4 represent the parameters along with the main M, L, and T quantities. Each dependent π i term is a function of independent π i where the amplitude ratio of the pipeline π S can be shown by dimensionless parameters as (2) at could be expressed in the following form: π S � f π 1 , π 2 , π 3 , π 4 , π 5 , π 6 , π 7 .
A probable exact mathematical form for the dimensional equations of the phenomenon could be assumed to be of the exponential form as follows: (4) where a 1 -a 7 and curve constant k 1 are derived from data collected from experimental tests. First, equation (4) is transformed into the logarithmic form as follows: log π S � log k 1 + a 1 log π 1 + a 2 log π 2 + a 3 log π 3 + a 4 log π 4 + a 5 log π 5 + a 6 log π 6 + a 7 log π 7 , where Z � log π S , K � log k 1 , A � log π 1 , B � log π 2 , C � log π 3 , D � log π 4 , E � log π 5 , F � log k 6 , and G � log π 7 .
Putting the values in equation (5), the same can be written as In this step, regression analysis is utilized. equation (6) is a regression equation of Z in A, B, C, D, E, F, and G. By multiplying equation (6) by A, B, C, D, E, F, and G, separately, a set of simultaneous equations is obtained:  From the data recorded from 120 experimental tests, the unknown coefficients of equation (6) can be derived by Cramer's rule in MATLAB by replacing the derived unknown coefficients in equation (6), which expresses the correlation between the amplitude ratio of the pipeline without and with the beam DVA (π S ) and other dimensionless parameters. is correlation represents a new correlation of the amplitude ratio π S as a function of other dimensionless parameters π of the system. e coefficient of determination R 2 is used to estimate the accuracy between the values of equation (8) and experimental results. e following diagram in Figure 4 shows a comparison between the values of the experimental test data (x pa /x p ) and the calculated π S .
To identify the compact and simpler correlations that can help design and tuning beam DVA for pipeline structures, the above steps are considered with other dimensionless parameters π and the unknown coefficients are calculated. e collection of sample data used to establish the amplitude ratio π S by π i terms and compare it with experimental results (x pa /x p ) is shown in Table 5.

Indices of π i
Terms for π S . In this section, according to Equation (8), the effects of dimensionless parameters on amplitude ratio π S of pipeline structures are discussed. As it is indicated in Figure 5, π 1 has the greatest effect on amplitude ratio.
(i) π 1 is 0.7684, which is the greatest. us, π 1 has the greatest effect on amplitude ratio. e positive value shows that π 1 has a direct relationship with π S . (ii) π 5 is −1.3772, which is the lowest. us, π 5 has the lowest effect on ratio amplitude. e negative value demonstrates that π 5 has an inverse relationship with π S . (iii) e curve constant in the model is 0.44668, which shows the effects of other factors on ratio amplitude.

Numerical Model
e finite element (FE) model is used to validate the methodology presented. First, the equations governing the motion of the pipeline are extracted using the Euler-Bernoulli beam element, as shown in Figure 6. e nodes are under the displacement and rotation of w 1 (t), w 2 (t), w 3 (t), and w 4 (t). e forces f 1 (t) and f 3 (t) are proportional to the displacements of w 1 (t) and w 3 (t), and the moments f 2 (t) and f 4 (t) are proportional to the rotations of w 2 (t) and w 4 (t), respectively. e transverse beam displacement is assumed to be proportional to a cubic equation. After obtaining the shape functions using kinetic energy, the bending strain energy and the virtual work of the mass and stiffness matrices are obtained as follows: Area of the pipe Modulus of elasticity of the pipe Modulus of elasticity of the threaded rod Density of the pipe Density of the threaded rod Frequency of exciting force w rad/s Displacement at the middle of the pipe without beam DVA Displacement at the middle of the pipe with beam DVA Table 4: List of π i groups of the beam DVA attached to the pipeline structure.
No. Dimensionless groups (π i terms) Shock and Vibration Figure 7 shows a schematic of the finite element model of a beam DVA installing pipeline structure containing 10 nodes and 8 elements. In the absorber-free model, the number of nodes and elements is considered to be 7 and 6, respectively. e mass M s and stiffness K s matrices of the system are extracted by combining the mass and stiffness matrices of each element. It should be noted that damping is neglected because of its insignificance. e equations of motion of the pipe-beam DVA system can be written as

Equation of Motion of Pipe-Beam DVA System.
e mass and stiffness matrices of the pipeline are of the order of 12 with no vibration absorber and of the order of 16 with the installation of beam DVA in the piping system. erefore, 12 equations and 16 coupled equations are extracted, respectively. e force vectors are obtained by shape coefficients.
Simulink provides the ability to analyze the system response well by changing the input parameters. e simulation consists of two parts. e first part is the simulation of  Shock and Vibration a pipeline, by which 12 coupled differential equations are analyzed. All geometric characteristics and physical properties of the pipeline and the amplitude and frequency of the sinusoidal force are defined as program inputs by means of block diagrams. Finally, the "displacement of the midpoint of the pipe to the time" is plotted. In the second part of the simulation, the 16 extracted equations are analyzed and the displacement of the midpoint of the pipe is plotted and compared with the first case. e ODE45 variable-step solver is used in model analysis. e accuracy of the FEM model can be improved by further nodes and elements; however, this may reduce the computational efficiency. Several models with a different number of elements were investigated. is section compares the FEM pipeline model without the beam DVA model with analytical beam motion equations in the Euler-Bernoulli theory on the simple support applied by a centralized force in the middle of the beam. e transverse vibration motion of the beams is in the form of a fourthorder partial differential equation as follows: e response of the pipeline structure using the mode superposition principle is Figure 8 shows the displacement of the midpoint of the 1-inch pipeline structure with respect to time and an excitation frequency of 22 Hz, using equation (12) and simulating the FE model. As can be seen, there is an acceptable match.

Comparison of Experimental Study: Finite Element Model with DA
To establish the amplitude ratio of the pipeline without and with the beam DVA, three cases were analyzed using   Shock and Vibration experimental results, numerical results, and methodology described in this paper. In each case, the beam DVA with the same specification was designed for comparison.

Case 1.
In the 1-inch pipe with the specifications listed in Table 1.
Beam DVA: threaded rod M8 with length of 0.18 m and mass of 2.35 kg.
Frequency of exciting force of 12.5 Hz. Numerical and experimental results are presented in Figures 9 and 10. As shown in the figures, the amplitude ratios of the pipeline (x pa /x p ) are 0.475 and 0.397, respectively. Also, according to equation (8), π S � 0.3814.

Case 2.
In the 2-inch pipe with the specifications listed in Table 1.
Beam DVA: threaded rod M16 with length of 0.245 m and mass of 10 kg. Frequency of exciting force of 16 Hz. Numerical and experimental results are presented in Figures 11 and 12. As shown in the figures, the amplitude ratios of the pipeline (x pa /x p ) are 0.457 and 0.427, respectively. Also, according to Equation (8), π S � 0.401.

Case 3.
In the 3-inch pipe with the specifications listed in Table 1.
Beam DVA: threaded rod M16 with length of 0.19 m and massof 10 kg.
Frequency of exciting force of 22 Hz.   Numerical and experimental results are presented in Figures 13 and 14. As shown in the figures, the amplitude ratios of the pipeline (x pa /x p ) are 0.499 and 0.56, respectively. Also, according to equation (8), π S � 0.499.
As can be seen, the proposed method agree quite well with experimental, numerical, and DA results. Also, there are some errors between the numerical and experimental results. ese differences can referred to the restrictions of the motor.

Conclusions
In this paper, the beam DVA was proposed to reduce the vibration of the pipeline structures. For this purpose, dimensional analysis technique was developed for the prediction of the amplitude ratio of the pipeline without and with the beam DVA. Using the dimensional analysis technique, a new, simple, and reliable method was introduced to design dynamic vibration absorbers for pipeline structures. Buckingham's theory in dimensional analysis makes it possible to identify nine dimensionless groups. In the experimental study, a beam DVA was installed on 1-inch, 2inch, and 3-inch pipes, and harmonic force was applied to the pipeline. Using the results obtained from the experimental study and applying the multiple linear regression method, new correlation was derived for designing optimal beam DVA. Also, the results obtained are validated with FEM and indicated that the method based on DA proposed is a good alternative solution for the quick determination of the amplitude ratio of the pipeline structure without excessive computational costs or user expertise.
In addition, nondimensionalization of an equation allows the identification of significant and insignificant terms, and this is very effective in the design of beam DVA. e mass ratio has the greatest effect on amplitude ratio of the pipeline and stiffness ratio has the lowest effect.
Data Availability e data used support the findings of this study are available from the corresponding author upon reasonable request.