End Of String Research Articles

Non-relativistic expansion of open strings and D-branes

We expand the relativistic open bosonic string in powers of 1/c2 where c is the speed of light. We perform this expansion to next-to-leading order in 1/c2 and relate our results to known descriptions of non-relativistic open strings obtained by taking limits. Just as for closed strings the non-relativistic expansion is well-defined if the open string winds a circle in the target space. This direction must satisfy Dirichlet boundary conditions. It is shown that the endpoints of the open string behave as Bargmann particles in the non-relativistic regime. These open strings end on nrDp-branes with p ≤ 24. When these nrDp-branes do not fluctuate they correspond to (p + 1)-dimensional Newton-Cartan submanifolds of the target space. When we include fluctuations and worldvolume gauge fields their dynamics is described by a non-relativistic version of the DBI action whose form we derive from symmetry considerations. The worldvolume gauge field and scalar field of a nrD24-brane make up the field content of Galilean electrodynamics (GED), and the effective theory on the nrD24-brane is precisely a non-linear version of GED. We generalise these results to actions for any nrDp-brane by demanding that they have the same target space gauge symmetries that the non-relativistic open and closed string actions have. Finally, we show that the nrDp-brane action is transverse T-duality covariant. Our results agree with the findings of Gomis, Yan and Yu in [1].

Polymer encapsulation impact on potential-induced degradation in PV modules revealed by a multi-modal field study

Potential-induced degradation is a significant issue affecting the performance and reliability of photovoltaic systems, particularly in large-scale installations. Understanding the mechanisms underlying the formation of potential-induced degradation is crucial for developing effective mitigation strategies and ensuring the long-term sustainability of solar energy technologies. During the inspection of a 1.2 MWp photovoltaic power plant, various imaging methods, electrical measurements, spectroscopic characterization methods, and monitoring data analyses were utilized and combined. We observed that two percent of the photovoltaic modules at the string ends exhibited the characteristic checkerboard pattern in infrared or electroluminescence imaging, implicating issues on twelve percent of all strings. This uneven distribution of potential-induced degradation-affected modules suggests additional key factors are at play. Investigations of the backsheet and ethylene-vinyl acetate using near-infrared reflectance analysis revealed that one backsheet type and two different ethylene-vinyl acetate types were present in the installed solar panels. Yet, only one type of ethylene-vinyl acetate correlated with potential-induced degradation. The same type also exhibited a higher rate of oxidative degradation. Scanning electron microscopy and energy-dispersive X-ray spectroscopy revealed a significantly increased sodium ion content in the ethylene-vinyl acetate and glass of the potential-induced degradation-affected modules. Our findings highlight the crucial role of polymer encapsulation in the development of potential-induced degradation and underscore the importance of careful polymer selection in the design of photovoltaic module technologies.

Physical Simulation of Steam Absorption Characteristics in Horizontal Wells of Super Heavy Oil Reservoir

In the process of dual Horizontal well SAGD development of super heavy oil, the steam absorption balance of upper horizontal steam injection well is an important factor to determine the development effect of SAGD. The factors affecting the steam absorption of upper horizontal wells involve reservoir permeability, steam injection technology, steam injection parameters, etc. The physical simulation method is used to study the influence degree of each specific factor and improvement measures. Based on the similarity criterion, the reservoir scale model is designed and made according to the oilfield prototype, and the simulation experiment is carried out on the steam absorption characteristics and influencing factors along the horizontal wellbore. The experimental results show that the steam injection rate, non condensable gas, the location of the steam injection string and the reservoir permeability have great effects on the steam absorption of the reservoir. The heating uniformity along the horizontal well can be improved by properly increasing the steam injection speed, adding a certain amount of non condensable gas to the steam, and the position of the steam injection string end in the relatively low permeability area of the reservoir. This achievement has a strong guiding significance for improving the steam absorption balance of horizontal wells and improving the development level of dual level SAGD.

Loop-string-hadron formulation of an SU(3) gauge theory with dynamical quarks

Towards the goal of quantum computing for lattice quantum chromodynamics, we present a loop-string-hadron (LSH) framework in 1+1 dimensions for describing the dynamics of SU(3) gauge fields coupled to staggered fermions. This novel framework was previously developed for an SU(2) lattice gauge theory in $d\leq3$ spatial dimensions and its advantages for classical and quantum algorithms have thus far been demonstrated in $d=1$. The LSH approach uses gauge invariant degrees of freedoms such as loop segments, string ends, and on-site hadrons, it is free of all nonabelian gauge redundancy, and it is described by a Hamiltonian containing only local interactions. In this work, the SU(3) LSH framework is systematically derived from the reformulation of Hamiltonian lattice gauge theory in terms of irreducible Schwinger bosons, including the addition of staggered quarks. Furthermore, the superselection rules governing the LSH dynamics are identified directly from the form of the Hamiltonian. The SU(3) LSH Hamiltonian with open boundary conditions has been numerically confirmed to agree with the completely gauge-fixed Hamiltonian, which contains long-range interactions and does not generalize to either periodic boundary conditions or to $d>1$.

Research on the arcing horn with active arc extinguishing function for HVDC grounding electrode line

AbstractThe grounding electrode line is an important part of the Ultra high voltage direct current (UHVDC) transmission system. It acts as the channel to conduct the unbalanced current and the grounding current.The arcing horn consists of a pair of electrodes installed in parallel at both ends of the insulator string, which provides a discharge channel for the fault current by an air gap to protect the insulators. In this paper, the voltage and current distributions of the arcing horn were studied when the grounding electrode line was struck by lightning. It is found that the current and voltage gradually decrease along the direction from the converter station to the earth electrode. An active arc extinguishing method based on variable resistance for arcing horn was proposed. An electromagnetic transient simulation model of the grounding electrode line is established based on the electrical parameters of a ±800 kV UHVDC transmission system. It is found that the absorbed energy of the arc extinguishing unit needs to be at least 31.2 and 49.6 kJ when the lightning current is 20 and 50kA, respectively. According to the calculation results, the arcing horn with arc extinguishing unit was developed and tested. The test results verifies that the designed arcing horn can meet the requirements of engineering application.

Modelling Wave Transmission for Transient Flow and Amplitude-Frequency Characteristics of Tubular String in a Water Injection Well

Fluid wave code communication is used in layered water injection intelligent monitoring systems, but a model of fluid transient flow wave signal transmission is still unknown. Impedance and transfer coefficient in power transmission theory were used to describe transient flow waves in the transmission process of a tubular string in a water injection well and a transient flow wave model was built based on the transfer matrix method. The relationship between pressure and discharge was analyzed when the transient flow waves moved along the tubular string, and the influence of terminal impedance and dip angle of the tubular string on the wave transmission was studied. Simulations showed that the transient flow waves were with standing wave distribution when the transient flow wave signals transmitted in the tubular string. Moreover, the transmission volatility under different terminal impedances was analyzed. The communication frequency was selected according to the wave amplitude ratio between the two ends of the water injection tubular string. The relationship between the influence of tubular string parameters and fluid characteristics on the wave velocity and wave amplitude in the signal transmission process was obtained by simulation analysis. The wave velocity tended to decrease as the gas content increased. As the tube diameter–thickness ratio increased, the wave velocity decreased. Taking data from a water injection well in Daqing Oilfield as an example, a two-layer water injection test platform was built to study the fluctuation of discharge and pressure at monitoring points in the tubular string. The experiment condition was that the depth of the injection well was 1400 m. It was verified by the experiments that the pressure and flow changes in the downhole and wellhead had good consistency during the transmission of transient flow waves. Comparing the experimental results with the numerical results, the errors of the wave velocity and wave amplitude were 0.69% and 3.85%, respectively, indicating the verification of the simulation model. This study provides a theoretical support for the transmission of transient flow wave signals in a water injection tubular string.

Optimal Design of Corona Ring for 132 kV Insulator at High Voltage Transmission Lines Based on Optimisation Techniques

The installation of a corona ring on an insulator string on a transmission line is one of the solutions to reduce the electric field stress surrounding the energised end of the insulator string. However, installing a corona ring with an optimum design to reduce the electric field magnitude on an insulator string is a challenging task. Therefore, in this work, a method to achieve the optimum design of a corona ring for 132 kV composite non-ceramic insulator string was proposed using two optimisation methods: the Imperialist Competitive Algorithm (ICA) and Grey Wolf Optimisation (GWO). A composite non-ceramic insulator string geometry with and without a corona ring was modelled in finite element analysis and used to obtain the electric field distribution in the model geometry. The electric field distribution was evaluated using a variation in the corona ring’s dimensions, i.e., the ring diameter, the ring tube diameter and the vertical position of the ring along the insulator string. From the results achieved, a comparison of the minimum electric field magnitude along the insulator string with a corona ring design shows that the minimum electric field magnitude is found to be lower using optimisation techniques compared to without using optimisation techniques by between 3.724% and 3.827%. Hence, this indicates the capability and effectiveness of the proposed methods in achieving the optimum design of a corona ring on an insulator string.

Design and Evaluation of a 180 nm Powerline Communication ASIC for Harsh Environment

Modern complex drilling systems contain communication nodes like sensors, actuators, and controllers, spread along the lower end of a drill string. Here, temperatures of more than 150°C and pressure levels up to 200MPa are present. These environmental conditions and mechanical shocks, are extremely challenging for the reliable use of electronic components. A powerline communication system is designed and evaluated to establish a robust communication channel with low amounts of wiring. This system can operate on highly distorted physical transmission channels by adding redundancy at the sender that can then be used to correct errors at the receiver. In order to synchronize the real-time clocks among different powerline stations, a new preamble extension approach that enables precise time synchronization between multiple bus nodes is added. After design and verification, this system was manufactured in XFAB 180nm Silicon-On-Insulator (SOI) technology allowing operating temperatures of up to 175°C. The die size is 5.25 mm × 5.25 mm and contains a complete HomePlug 1.0 communication stack with an environment for boot, interfacing, and debugging. Its data rate reaches 6.1Mbit/s using the fastest transmission mode and the theoretical maximum of 0.55Mbit/s in the robust OFDM (ROBO) mode, which is of particular interest for harsh environment applications. After verifying the fabricated die, a Printed Circuit Board (PCB) for climate chamber evaluation was designed and fitted. Measurements in this chamber carried out a maximum ambient temperature of 190°C for communication with a minimum self-heating of the die of 20°C measured at room temperature. In combination, this is 35°C above the specification of the technology process. The timing synchronization evaluation showed a precision of 55.6ns over the temperature range from -30°C to 185°C, which is as low as 1.5 clock cycles. Power measurements of up to 190°C have shown an average power consumption increase of only 63μW/K below 150°C and a maximum increase of 394μW/K above 150°C. To the best of the authors’ knowledge, this is the first high-temperature evaluation of a powerline communication ASIC, which is particularly designed for a drilling system’s harsh environment.

Analiza statystyczna uszkodzeń rur wiertniczych z grup wytrzymałości S-135 i G-105

The characteristic types of operational defects that can form on the inner or outer surface of drill pipes of strength groups S-135 and G-105 (according to API Spec 5DP) are described using the results of technical diagnostics from drilling wells in the Dnipro-Donetsk gas and oil region. In 2018 and 2019, the Ukrburgaz Drilling Department rejected 81 drill pipes of strength group S-135 and 89 drill pipes of strength group G-105 when drilling wells to a depth of 4000 to 6000 m. A statistical evaluation of the operational defects detected during deep drilling of wells (4000–6000 m) was carried out. Potentially dangerous areas were identified: in the drilling pipe upset zone and along the length of the drill string end drill pipes lifetime has been taken into account. It is recommended during defectoscopy of drill string pipes of the selected strength groups to pay close attention to the sections of pipes of strength group S-135 from the end of the coupling or nipple, in the range of 0.45 m to 0.57 m, and for sections of pipes of strength group G-105, in the range of 0.55 m to 0.63 m. In addition, given the depth of drilling (Lmax), when performing diagnostics on drill pipes, special attention should be paid to sections with the most likely defect (Lf ) along the length of the drill string. In particular, taking into account the relative length (Lf /Lmax) of the drill string, for pipes of strength groups S-135 and G-105, segments in the ranges of 0.34 to 0.47 and 0.43 to 0.52, respectively, were identified as having the highest probability of operational defects. The peculiarities of the influence of a drill pipe operating lifespan depending on the strength group were established. In particular, during the long-term deepening of drill pipes in strength group S-135, three stages of drilling were distinguished: Stage I – running-in (from start-up to 2000 hours); Stage II – stable work (2000 to 7000 hours); and Stage III – accelerated destruction (7000 hours and longer). It was found that during defectoscopy of the pipe, special attention should be paid to the drill pipe, the service life of which is 602–998 hours in the first stage, from 3348 to 5344 hours in the second stage, and from 8942 to 10 584 hours in the third stage, because these periods carry the greatest probability of originating an inadmissible defect. For long-term drilling works with pipes of strength group G-105, two stages of drilling were distinguished: the first stage, of stable work (up to 6000 hours), and the second stage, of accelerated destruction (6000 hours and longer). It was found that during defectoscopy of the pipe, special attention should be paid to the drill pipe, the service life of which is from 2692 to 3736 hours in the first stage and from 8744 to 10 983 hours in the second stage, because these periods demonstrate the greatest probability of an inadmissible defect.

Experimental study on flow-induced vibration response characteristics of leaching tubing in salt cavern gas storage

To address flow-induced vibration failure problems of water solution leaching tubing string in salt cavern gas storage, a laboratory experimental device for flow-induced vibration of leaching tubing string was developed based on the similarity principle, and the corresponding experimental bench was built, according to the on-site parameters of the M gas storage well in Jianghan Oilfield, China. The vibration simulation experiments were carried out, and the displacement and amplitude-frequency response data at the bottom end of the tubing string were measured. The influences of different operation parameters and structure modes, such as water injection flow rate, two-port distance, screen structure and circulation mode, on the vibration characteristics of the tubing string were explored, the failure mechanism of the leaching tubing string was revealed, and the corresponding vibration failure control methods were proposed. The results obtained demonstrate that, first, at the moment of pump start-up (corresponding to well opening on-site), the tubing string produced obvious water hammer phenomenon. With the increase of the water injection flow rate, the vibration displacement and frequency of the center tube first increase and then decrease, when the water injection flow rate is less than 1.55 L/min (corresponding to the on-site flow rate of 40 m3/h), the risk of vibration failure of leaching tubing can be reduced. Second, when the two-port distance is greater than 0.16 m (corresponding to the on-site two-port distance of 45 m), the amplitude and frequency of lateral vibration of the center tube increase significantly, and the risk of lateral vibration deformation of leaching tubing increases. Third, adding a symmetrical hole screen at the bottom of the center tube can significantly reduce the vibration displacement and frequency of leaching tubing, and improve the safety of the operation tubing string. Fourth, the vibration displacement and amplitude of the leaching tubing in the reverse circulation mode are smaller than those in the positive circulation mode. When the cavern shape requirements are met on-site, the vibration of the tubing string can be effectively reduced by using the reverse circulation cavern leaching.

Cost Management and Strings of Increasing Earnings

Barth et al. (1999) document that firms sustaining a string of increasing earnings have higher price-earnings ratios and Myers et al. (2007) find such firms receive higher abnormal stock market returns during the string periods compared to firms not exhibiting this earnings pattern. The primary objective of the current study is to improve our understanding of how firms can maintain a consecutive string of increasing earnings. Most firms that have a string on increasing earnings do so by increasing revenues (Ghosh et al., 2005). Firms that maintain a string of earnings during periods of decreasing revenues must effectively manage costs to decrease their expenses sufficiently to increase earnings during those periods. Prior literature does not provide insights into the characteristics of firms and their market environments that allow them to maintain a string of consecutive earnings over periods that include both increasing and decreasing revenues. This paper contributes to filling that gap in the literature using large data and shows that cost flexibility is an important factor associated with a firm that can maintain a string of increasing earnings and finds that demand uncertainty for a firm’s products is inversely related to a firm’s ability to maintain a string of increasing earnings. In addition, an examination of the asymmetric cost behavior of the firms that experience at least one sales decrease during a consecutive string of increasing earnings indicates that these firms are associated with anti-sticky cost behavior during the string periods and exhibit sticky costs when the string ends.

Cost Management and Strings of Increasing Earnings

Barth et al. (1999) document that firms sustaining a string of increasing earnings have higher price-earnings ratios and Myers et al. (2007) find such firms receive higher abnormal stock market returns during the string periods compared to firms not exhibiting this earnings pattern. The primary objective of the current study is to improve our understanding of how firms can maintain a consecutive string of increasing earnings. Most firms that have a string on increasing earnings do so by increasing revenues (Ghosh et al., 2005). Firms that maintain a string of earnings during periods of decreasing revenues must effectively manage costs to decrease their expenses sufficiently to increase earnings during those periods. Prior literature does not provide insights into the characteristics of firms and their market environments that allow them to maintain a string of consecutive earnings over periods that include both increasing and decreasing revenues. This paper contributes to filling that gap in the literature using large data and shows that cost flexibility is an important factor associated with a firm that can maintain a string of increasing earnings and finds that demand uncertainty for a firm’s products is inversely related to a firm’s ability to maintain a string of increasing earnings. In addition, an examination of the asymmetric cost behavior of the firms that experience at least one sales decrease during a consecutive string of increasing earnings indicates that these firms are associated with anti-sticky cost behavior during the string periods and exhibit sticky costs when the string ends.

Study on Dynamic Characteristics of Drill String Considering Rock-Breaking Process

Using various tools to obtain downhole data to reach a precise pore pressure model is an important means to predict overpressure. Most downhole tools are connected to the lower end of drill string and move with it. It is necessary to understand the motion state and dynamic characteristics of drill string, which will affect the use of downhole tools. In this paper, a drilling process considering rock-breaking process in vertical wells is simulated using finite element method. In the simulation, gravity is applied to the whole drill string. The contact force between PDC bit and formation is the weight on bit (WOB). And a rotation speed is applied to the upper end of drill string. Analysis of the results shows that the vibration amplitude of bottom hole WOB (contact force between PDC bit and formation, which is the real WOB in drilling process) is bigger than the amplitude of wellhead WOB (acquired through conversion using Hook load, which is on behalf of the WOB obtained on drilling site). Both wellhead WOB and bottom hole WOB decline with a fluctuation in drilling process. In small initial WOB and low rotation speed conditions, the fluctuation of wellhead WOB focuses on low frequency, the fluctuation of bottom hole WOB focus on high frequency, and the phase of them are not identical. In large initial WOB and high rotation speed conditions, the fluctuation of wellhead WOB and bottom hole WOB both become more irregular. As for wellhead torque and bottom hole torque, the fluctuation of them mainly focuses on low frequency. And in high rotation speed conditions, wellhead torque may become negative. The research results are beneficial to the usage of downhole tools.

Automatic machine for feeding liquid equal to the frequency of the pulsator-vibrator

The main task of the drilling fluid circulation during drilling is to remove the remnants of the drilled rocks to avoid additional wear of the rock-cutting equipment. Bottom hole cleaning efficiency depends on both the rate of flow of the drilling fluid and its performance. During backwashing, the jet enters the well and, having passed under the end of the drill string, comes out to the surface along the string of rods.

On the Law of Motion of the End of a Semi-Bounded String, Which Extinguishes Reflected Waves

The problem of oscillation of a semi-bounded string at a given initial perturbation '(x), when the left end of the string moves according to a given law f(t), is considered (t time). The solution of the problem is obtained in the final form for arbitrary functions '(x) and f(t). The direct, inverse and reflected from the end of the string wave components that make up the resulting waves are constructed. The influence of the function f(t) on the magnitude of the reflected waves is investigated. For an arbitrary initial perturbation of a string, the law of motion of its end is found, at which the reflected waves disappear. The relevance of the article is determined by a wide range of practical problems related to the vibrations of extended objects (bridges, beams, antennas), when the source of motion is an initial perturbation and a given external force applied to the end of the object.

Analysis of initiation of instability zones of drilling string and intensification of drilling process at well boring by rotating an d rotating-bit methods

Instable operation of a drilling string can be caused by design flaws, unreasonably selected drilling modes, as well as specific features of the drilling by bit, fixed on a long flexible rod with initial irregularities. It was established that a drilling string under the influence of constant torque and longitudinal feed load is deformed along a screw line. It was shown that the source of the dynamic state initiation of a rotating drilling string is an alternative friction between rock and the instrument, as well as variation of potential energy of the drilling string at its twisting and untwisting. At the string twisting the potential anergy is accumulating, at the untwisting it is released. In this case the drilling sting length begins to increase and the work of feed force is imposed to this shifting resulting in the instrument beat over the rock. The torsion oscillating process causes longitudinal periodic shifting of the string end, which lead to joint longitudinal and bending oscillations of the whole drilling column. Systems of differential equations obtained describing the joint bending-longitudinal oscillations of a drilling string, caused by torsion vibrations, initiating at the contact of a drilling instrument with rock. Approximate solutions of the systems presented, which take place with the basic natural frequency of the torsion oscillations of a two-mass system instrument - string - rotator drive. In this case the amplitudes of the drilling string bending oscillations are associated with characteristics variation of the friction between the instrument and the rock as well as with the angular frequency of the drilling string rotation. Zones of dynamic instability at rotation of the deformed drilling string due to acting force factors, as well as initial irregularities and centrifugal forces revealed. It was shown that they are initiated at critical speeds of the drilling string rotation, which coincides with any natural frequency of the bending oscillations of the stave. To intensify the drilling process, recommendation elaborated for equipping the rotary drilling rig with an additional pneumatic hammer. A calculated diagram of a drilling string, equipped by an additional pneumatic hammer considered. Parameters of the hammer in terms of frequency and amplitude are matched with the translational speed of well drilling. The total longitudinal and transverse shifts can be obtained by superposition of self-exciting and induces shifts.

Analysis of Serious Deformation of Cavitation Tubing in Salt Cavern Gas Storage in Jianghan Oilfield

Jianghan Oilfield salt cavern gas storage is characterized by depth, high temperature and high pressure. At present, the main problem of Jianghan Oilfield salt cavern gas storage is that in the process of cavitation, the frequent bending deformation at the end of the cavitation tubing string leads to the difficulty of operating the tubing string, which extends the cavitation period and affects the production of the gas storage. The problem of tubing string in the cavitation of salt cavern gas storage in Jianghan Oilfield was counted, and the causes of tubing string bending were analyzed from the aspects of salt layer geological characteristics, tubing string mechanics, fluid mechanics, etc. It was concluded that the injection rate was higher than the critical liquid carrying flow rate, leading to tubing string instability and deformation, which provided theoretical reference for the construction of salt cavern gas storage.

Composite Belt Ornaments with Bear Claw Pieces in Medieval Men’s Costume of the Perm Region, Western Urals

Longstanding excavations at the Boyanovo and Rozhdestvenskoye medieval cemeteries in the Perm Territory revealed a new type of belt ornament—pendants with arch-shaped pieces carved from dorsal plates of bear claws. Each piece has two drilled holes in the central third, and they were strung on two cords in a “rope ladder” fashion. Pieces made of bear claws were interchanged with bronze beads or pipes. At the ends of strings, bells or pendants were attached. Such ornaments were worn exclusively by boys and men of all ages (from two to sixty). Silver artifacts and other “elite” items, suggesting that they were markers of high social status, accompanied the ornaments. The use of bear claws might indicate an apotropaic function. The available facts point to the use in funerary costume only, but the difficulty of manufacturing such ornaments obviates the possibility of a one-off use. Previously, such an ornament was found only at Zagarye, a cemetery dating to the final stage of the Lomovatovka culture. The pendants, then, were used during the late 9th to the late 11th centuries.

When a dollar makes a BWT

The Burrows-Wheeler-Transform (BWT) is a reversible string transformation which plays a central role in text compression and is fundamental in many modern bioinformatics applications. The BWT is a permutation of the characters, which is in general better compressible and allows to answer several different query types more efficiently than the original string.It is easy to see that not every string is a BWT image, and exact characterizations of BWT images are known. We investigate a related combinatorial question. In many applications, a sentinel character $ is added to mark the end of the string, and thus the BWT of a string ending with $ contains exactly one $-character. Given a string w, we ask in which positions, if any, the $-character can be inserted to turn w into the BWT image of a word ending with $. We show that this depends only on the standard permutation of w and present a O(nlog⁡n)-time algorithm for identifying all such positions, improving on the naive quadratic time algorithm. We also give a combinatorial characterization of such positions and develop bounds on their number and value. This is an extended version of [Giuliani et al. ICTCS 2019].

Effect of spring mid-support condition on the vibrations of the axially moving string

In this study, the axially moving string with spring-loaded middle support is discussed. The supports assumed as simple support on the string both ends. The intermediate support shows the characteristics of the spring. The string velocity is accepted as harmonically varying around a mean value. The Hamiltonian principle is used to find the equations of motion. The equations of motion become nonlinear, considering the nonlinear effects caused by string extensions. The equations of motion and boundary conditions are become dimensionless by nondimensionalization. Approximate solutions were found by using multiple time scales which is one of the perturbation methods. By solving the linear problem that is obtained by the first terms of the perturbation series, the exact natural frequencies were calculated for the different locations of the mid-support, various spring coefficients, and various axial velocity values. The second-order nonlinear terms reveal the correction terms for the linear problem. Stability analysis is carried out for cases where the velocity change frequency is away from zero and two times the natural frequency. Stability boundaries are determined for the principal parametric resonance case.