圓形振動(dòng)篩的設(shè)計(jì)

圓形振動(dòng)篩的設(shè)計(jì),圓形振動(dòng)篩的設(shè)計(jì),圓形,振動(dòng)篩,設(shè)計(jì) . Santus Emilia form 1 September to configurations C211 2007 Elsevier Ltd. All rights reserved. the drilling sites can be very costly and time consuming have been proposed. Miscow et al. 7 proposed a test rig based on four points bending scheme. The specimen is rotated at a frequency in the range 515 Hz and a constant loading employing a compressed rod inserted inside the kind of test is not suitable for a systematic assessment of the fatigue resistance, in particular when statistical evalua- tions are required. Smith et al. 9 employed a four point bending rig and a rotating cantilever beam rig to test innovative titanium drill pipe design. * Corresponding author. Tel.: +39 050 836607; fax: +39 050 836665. E-mail address: ciro.santusing.unipi.it (C. Santus). Available online at International Journal of Fatigue International for the recovery procedures. The working conditions of the drill string is described in Ref. 6. Drill strings rotating inside deviated wells experi- encerotating bending andthenfatigue damage,particularly attheconnectionswhicharedrillstringweakestpoints.Fati- gue failures usually are aggravated by corrosive environ- ment, improper equipment handling, excessive rotational speeds or loading. Coupling of various damage conditions reduces dramatically the fatigue life of the string. Full scale fatigue tests are therefore strategic for drilling contractors. Recently devices to test drill string connections hollow string under testing. With this solution the required axial load can be produced without any external frame. Moreover, tests in corrosive environment (NaCl solution) could be performed at low frequency (around 15 Hz, near to the actual frequency during drilling). These tests can be considered very representative of the operative conditions where fatigue acts in combination with mean stress and corrosive environment which is particularly eective at low rotating speed. However, this kind of test is very time consuming, indeed to produce a 10 10 6 cycles test, on a single specimen, four full months are necessary. Then this Keywords: Drill pipe connections; Full scale tests; Test rig design; Resonant testing machine; Fretting fatigue 1. Introduction In oil exploration long hollow drill strings are employed to reach the production area 4. Fatigue damage in drill string is a well known issue in oil drilling technology, recording more than 50% of failures 5, and failures at tensile axial load can also be superimposed. To produce the required high axial load the test structure is heavily loaded and a massive frame is necessary to this purpose. A similar four point bending test equipment has been employed by Grondin et al. 8 at a test frequency of around 7 Hz. They also developed an interesting solution for producing axial Resonant test rigs for fatigue string connections L. Bertini a , M. Beghini a , C a Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione b Eni S.p.A. Exploration and Production Division, Via Received 26 February 2007; received in revised Available online Abstract The paper presents two test rigs designed at the University of Pisa used for oil drilling. Two types of connection required dierent exploited in order to reduce the loads on the structure and the test duration. ratus and tests. Results of fatigue tests are reported and discussed. 0142-1123/$ - see front matter C211 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.08.013 full scale testing of oil drill a, * , A. Baryshnikov b , Universita di Pisa, Via Diotisalvi nC1762, 56126 Pisa, Italy n. 1, San Donato Milanese (MI) 20097, Italy 27 July 2007; accepted 21 August 2007 2007 perform bending fatigue tests on full scale drill pipe connections of test rig. In both cases, specimen resonance was This allowed a cost reduction in both the experimental appa- 30 (2008) 978988 Journalof Fatigue Nomenclature NC 26 type of connection according to API stan- dard 1,2 NC 50 type of connection according to API stan- dard 1,2 ADP-STJ 147 13 type of connection according to ISO stan- dard 3 F 1 eccentric rotating mass inertial force on bending arm 1 F 2 eccentric rotating mass inertial force on bending arm 2 m e eccentric rotating mass R e eccentricity of the rotating masses d displacement of the bending arm point where rotating masses are placed L. Bertini et al. / International Journal Veidt et al. 10 adopted a four point bending test facility similar to that employed by Miscow et al. 7, consisting in a very strong external frame able to produce tensile axial load in the specimen. In the present paper dierent schemes of test rigs for bending fatigue tests on drill string connections are pro- posed. The dynamic behavior near the resonance frequency is exploited to induce high bending moment in the connec- tion. The resonant condition is reached by means of rotat- ing eccentric masses. Through this technique the test frame just hold the specimen and no hydraulic actuator is employed, since the load is provided by inertia forces. As a consequence, both the complexity and the structural strength of the testing apparatus is much lower as com- pared to the four point test rig. Moreover, the resonance can be set by a proper choices of the masses and full scale tests can be run at a frequency up to 2530 Hz, thus reach- ing 10 10 6 fatigue cycles in about four full days test. A drawback of proposed test rigs is that mean axial load d e relative displacement of the rotating masses f rotational frequency of eccentric rotating masses x rotational speed of eccentric rotating masses f n natural frequency of the specimen acting as a dynamic system x n natural frequency of the specimen expressed as rotating speed n rotational frequency over natural frequency ratio D a dynamic amplification factor c phase angle between the two couples of rotational masses k b bending stiness of the specimen m a mass of the bending arm L a length of the bending arm I Q mass moment of inertia of the bending arm ID inner diameter OD outer diameter J area moment of inertia of the section W b section bending modulus L specimen length E Young modulus A i vibrating beam shape coecients (i =1,2,3,4) u(x,t) vibrating beam displacement as a function of position x and time t m mass of the vibrating beam m f fix mass to be placed at vibrating beam ends m C3 e rotating mass at one vibrating beam end R C3 e rotating mass eccentricity d C3 e relative displacement of the rotating mass of Fatigue 30 (2008) 978988 979 can not be applied (only alternating or rotating bending, i.e. cyclic stress at load ratio R = C01). On the contrary heavy test rig frame is able to exert high tensile axial load (as proposed in Refs. 10,7). Moreover, the choice of high frequency tests (faster than the working condition) reduces the possibility to test the eect of environment on fatigue. However, interesting comparisons between the basic fati- gue strength of dierent design solutions can be obtained in a relatively short time at a reasonable cost. 2. Connection types to test Two types of connections were considered: C15 high-strength steel connection (hereafter named as steel connection), related to standards Refs. 1,2; C15 aluminum light-weight pipe connection with steel tool joint (hereafter named as aluminum to steel connection), related to standard Ref. 3. q Specimen material density v length frequency of the vibrating beam A Cross section area of the specimen a 1 first harmonic amplitude of the strain gauge signal a 2 second harmonic amplitude of the strain gauge signal R load ratio r n nominal bending stress at the fatigue failure section M b bendingmomentatthefatiguefailuresection B Basquin equation constant b Basquin equation exponential r n,e nominal bending fatigue endurance limit j n nucleation slope of SN curve, in loglog coordinate j f fatigue failure slope of SN curve, in log log coordinate The steel connection is much more common in oil drilling and extensive technical literature can be found (papers 6,11,12 report the state of art about steel drill string connection fatigue). On the contrary, aluminum to steel connections have been recently developed by Russian drilling contractors and no systematic studies have been conducted yet. Aluminum drill pipes are spreading world- wide due to potential advantages, discussed in Ref. 13, based on the elevated strength-over-weight ratio and low stiness of the material as compared to quenched and tem- pered steels. Steel connections are composed of two conical threaded sides: Pin and Box attached to the pipe body by means of frictionwelding, asshowninFig.1a.Typicalfatiguecracks, leading to failure, usually nucleate at last engaged thread root either of the pin or of the box, as shown in Fig. 1b. For aluminum to steel connection, tool joints at pin and box sides feature conical thread and they are made of steel because they need to be engaged and disengaged very often, during the drilling operation. In this design, to connect the aluminum pipe body to the steel tool joints two other threaded connections are required; one for each side, Tool joint Pin Tool joint Box Friction weldin g Friction welding Body pipe Body pipe Stop face (shoulder) Conical threaded connection Box fatigue site (last engaged thread) Pinfatigue site (last engaged thread) time n n n Fig. 1. (a) Conical threaded connection between pin and box steel tool joints, attached to the body pipe by friction welding. (b) Fatigue nucleation sites either at the pin or the box sides. Tool joint Pin (Steel) Conical thread free portion Aluminum - steel conical threaded connection Body pipe (Aluminum) Conical threaded Steel Conical thread 980 L. Bertini et al. / International Journal of Fatigue 30 (2008) 978988 connection Tool joint Box (Steel) Body pipe (Aluminum) Aluminum - steel conical threaded connection Conical thread free surface Fig. 2. (a) Aluminum to steel threaded connectionis required (instead of friction the steel edge, is the failure mode of this type of connection. Steel edge, fretting on aluminum Aluminum free surface time n n welding) to connect the pipe body to the tool joint. (b) Fretting fatigue, at Fig. 2a. These other connections are assembled at the com- ponent manufacturing stage and they do not have to be dis- The eccentric counter-rotating masses generate two lon- Table 1 Main dimensions of the tested connection specimens Connection type Standard nomenclature Outer diameter (mm) Inner diameter (mm) Specimen length (m) Steel connection NC 26 88.9 38.1 1.2 Steel connection NC 50 168.8 71.4 1.2 Aluminum to steel connection ADP-STJ 147 13 147 107 3.7 Steel connection specimen Bending arm 2Bending arm 1 R e - m e F 1 F 2 Fig. 4. Two couple of counter-rotating masses, each hinged at the top of massive arms, generate cyclic bending on the specimen. L. Bertini et al. / International Journal of Fatigue 30 (2008) 978988 981 engaged for the whole life of the drill pipe. As depicted in Fig. 2a, the steel components feature a conical end without internal threads to shield the last engaged thread of the alu- minum pipe against fatigue due to bending. Fretting fati- gue, at the contact between the rounded edge of the steel component and the aluminum pipe body, generates fatigue crack nucleation, as illustrated in Fig. 2b. The main dimensions of the tested specimens are reported in Table 1. It is worth noting that the overall length of the alumi- num to steel connection is higher than the steel connec- tions, since two extra connections are required. As a consequence, two dierent test rigs were designed to pro- duce full scale fatigue testing on these two dierent connec- tion types. 3. Test rig design 3.1. Test rig for steel connections In Fig. 3 the test rig for steel connection is shown. Two couples of counter-rotating eccentric masses, at the top of two bending arms, induce inertial forces on the specimen. Force and displacements are in plane, then alternating (not rotating) bending is applied to the specimen, as sche- matically shown in Fig. 4. The system allows for shifting the phase between the two couples of rotating masses, then a phase angle c is introduced. Fig. 3. (a) Picture of the test rig. (b) gitudinal forces at the top of the two arms which, if the specimen is assumed to be rigid, are given by F 1 t2m e x 2 R e cosxt F 2 t2m e x 2 R e cosxt c 1 while the transversal components of the two forces are bal- anced for each couple of masses. If the forces F 1 and F 2 are in-phase (c = 0), no bending moment is induced in the specimen, since the specimen is supported by springs which allow in plane free rigid dis- placements. On the contrary, in the out-phase condition (c = p), the bending moment induced in the specimen is maximum. Moreover, the test rig operates at a frequency which is near (but lower) to the first resonance of the dynamic system in which the specimen is the spring and the two bending arms are the inertial bodies. Near to the resonance frequency the bending moment experienced by the specimen is much greater than that produced by forces F 1 and F 2 . On the basis of the following reasonable assumptions, a simple dynamic model of the system can be obtained: 1. out-phase condition, c = p; 2. bending arms are rigid as compared to the specimen; 3. specimen inertia is negligible in comparison to bending arms inertia; 4. bending deflection of the specimen is prevailing; 5. no damping eect is considered. Steel connection specimen. Q b n I k a m a L d cos( ee t)Rd 982 L. Bertini et al. / International Journal For assumption 1, half structure can be considered due to symmetry. As observed, by imposing dierent phase angle c, the bending moment can be continuously varied by a factor sin(c/2) ranging from the maximum value (c = p), to zero in the in-phase condition (c = 0). For assumptions 2 and 3 it follows that the dynamic system has one degree of freedom with the specimen as a spring and the arms as inertia. Moreover, by neglecting the spec- imen mass, the bending moment can be considered to be uniform along the specimen length. The here suggested model is depicted in Fig. 5. In order to estimate the natural frequency, f n = x n /(2p), model parameters can be evaluated as follows: C15 bending stiness: k b =2EJ/L, where E is the material Young modulus, J p 64 OD 4 C0ID 4 the section moment of inertia about the bending neutral axis and L the free bending specimen length; C15 the mass moment of inertia about the axis through point Q of the arm having mass m a is I Q 1 3 m a L 2 a , by assuming mass m a uniformly distributed over its length L a ; C15 the natural frequency is x n k b =I Q p . Let us consider the system loaded by a periodic force with a rotational speed of x as indicted in Fig. 5b. The displace- ment d can be obtained by solving the equation, neglecting any damping: k d I d 2x 2 R m cosxt2 Q I Q ID OD b k 2/L b k n Q Q I a L e 2m Fig. 5. (a) Natural frequency of the dynamic system, f n = x n /(2p). (b) Excited vibration of the system, at an imposed frequency f = x/(2p). Resonance is the condition: n = x/x n =1. b L 2 a Q L 2 a e e giving, the solution 1 : d 2x 2 R e m e k b C0I Q x 2 L 2 a cosxt3 The nominal bending stress amplitude r n is defined as bending moment divided by bending modulus of the pipe section. It can be related to the frequency ratio n = x/x n as follows: 1 Initial condition transient is neglected, since it vanishes very quickly, due to actual damping. r n 2R e m e L a W b x 2 1 1C0n 2 4 where W b = J/(OD/2) is the bending modulus. The form of Eq. 4 suggests the definition of a dynamic amplification factor: D a 1 1C0n 2 5 which is the amplification of the forces (F 1 ,F 2 ) due to the eccentric masses produced by the inertia forces at the arms. As the damping has been neglected, Eq. 4 indicates that the bending stress increases indefinitely when the frequency of the rotating masses approaches the natural frequency (n ! 1). In practice, it was observed that for the proposed test rig, Eq. 4 gives reasonable prediction up to n C25 0.95. For frequency near to the resonance the dynamic amplifi- cation depends strongly on damping, particularly when damping is a small quantity as in the present condition. In order to obtain a controllable behavior, the test rig was operated in sub-resonance (more details are given later) then Eq. 4 is accurate enough. It is worth noting that previous assumptions and model approximations were used for interpreting the phenome- non and defining the main quantities of the apparatus, however they produce no eect on the accuracy of the test. Indeed, the eective dynamic bending stress was continu- ously measured on the specimens by means of strain gauges, during tests. Set point was keep constant within a predetermined range (5% of the nominal value) by a closed loop system controlling the phase shift between the two couples of counter-rotating masses. 3.2. Test rig for aluminum to steel connections The rig for testing the aluminum to steel connections was designed for longer specimens as previously discussed. Its layout is shown in Fig. 6. The axial extension of the connections and the reduced bending stiness did not allow the previous testing scheme to be adopted. In this case, it was decided to give to the specimen both the elastic and inertia characteristics of the dynamic system. As previously, it was set to operate in the region of sub-resonance. The external load was pro- duced by rotating an eccentric mass located at one end of the specimen. In order to keep the symmetry of the struc- ture and to produce the maximum bending load in the cen- ter (where the connection is located), two masses were clamped at the ends of the specimen. This configuration can be modeled by assuming the specimen as a massive beam (with total mass m uniformly distributed on its length L) carrying two point-like masses m f at the ends, Fig. 7a. By spinning the eccentric mass m C3 e , rotating bending was induced in the specimen (with high amplification near the resonance) Fig. 7b. of Fatigue 30 (2008) 978988 The dynamic behavior of the system can be predicted by solving the 4th order partial dierential equation 14: specimens. (b) Details of the connection to test. L. Bertini et al. / International Journal of Fatigue 30 (2008) 978988 983 C0EJ 4 ux; t ox 4 m L o 2 ux; t ot 2 6 Fig. 6. (a) Picture of test rig for aluminum to steel connection f m x ),( txu Lm, f m ID OD Null displacement points f m * ef mm Lm, ),( txu m e * x )cos( * e * e tRd Fig. 7. (a) Dynamic model to find the natural frequency. (b) Eccentric rotating mass m C3 e to excite the dynamic system. where u(x,t) indicates the lateral displacement of the spec- imen axis as function of the position x and time t. The solu- tion is ux;tA 1 cosvxA 2 sinvxA 3 coshvx A 4 sinhvxC138cosxt7 where the time frequency x and the length frequency v are related by the relation: v 2 x qA EJ r 8 Natural frequencies x n are those values of x, solutions of the characteristic equation, that make null the determinant of the linear system (having the unknown coecients A i (i = 1,2,3,4) obtained by imposing boundary conditions. The vibration of a uniform beam (without masses m f )is a classic result 14, giving the following characteristic equation: cosvLcoshvLC01 0 9 For the considered condition (E = 73 GPa, m = 54.6 kg, OD = 147 mm, ID = 121 mm and L = 3.7 m) the first nat- ural frequency is 64.4 Hz. The characteristic equation for the vibrating beam with point-like masses at the two ends was obtained 2 : 2 Symbolic software MathematicaC228 ver. 5.1 was used to manipulate the algebra. coshvL cosvLC02 m f m vLsinvL C16C17 C02 m f m Lm f EJ EJ qA s xsinvLC0vLcosvL ! C2sinhvLC01 0 10 The external masses introduce a parameter m f /m in the characteristic equation. The solution of transcendental Eqs. (9) and (10) cannot be obtained in analytic form. In Fig. 8 the graphs of Eqs. (9) and (10) are plotted versus f, showing the graphical determination of the first two res- onant freq