Vibration in a Cracked Machine Tool Spindle with Magnetic Bearings
|
|
- Sharlene Long
- 6 years ago
- Views:
Transcription
1 Te Open Mecanical Engineering Journal, 8,, Open Access Vibration in a Cracked Macine Tool Spindle wit Magnetic Bearings Huang-Kuang Kung and Bo-Wun Huang Department of Mecanical Engineering, Ceng Siu University, Taiwan Abstract: Due to manufacturing flaws or cyclic loading, cracks frequently appear in a rotating spindle system. Tese cracks markedly affect te dynamic caracteristics in iger modes of te rotating macinery. For faster rotational speeds, especially for super-ig-speed cutting, a spindle wit magnetic bearings is necessary. However, most investigations into spindle system dynamic caracteristics ave been confined to ball-bearing-type spindles. Te dynamic response of rotating cracked spindle systems wit magnetic bearings is examined in tis article. A Euler-Bernoulli beam of circular cross section is used to approximate te spindle and te Hamilton principle is employed to derive te equation of motion for te spindle system. Te effects of crack dept, rotation speed and bearing lengt on te dynamic response of a rotating magnetic bearing spindle system are studied. INTRODUCTION Cracks frequently appear in rotating macinery due to manufacturing flaws or cyclic fatigue during operation. Numerous cracks can be observed after severe operating conditions, especially in ig speed spindles [, ]. ocal structural irregularities caused by cracks in te spindle may significantly cange te dynamic beavior of a rotating macinery system. Te effects of cracks on te dynamic and static beaviors of structures ave been studied by a number of researcers [3-5]. Te effects of cracks on spindle dynamics, saft and rotor systems, were also studied by researcers [6-9]. Wen a spindle rotates, te vibrational response is altered by te crack opening and closing in eac cycle. Most investigations were motivated by te ypotesis tat only opening cracks markedly cange te spindle dynamics. Tis paper focuses on te dynamics of a spindle wit a transverse crack. Hig speed macining is one of te most modern manufacturing engineering tecnologies. In a macining system, te spindle is te most critical element tat affects te dynamic performance and capabilities of te system in te macining process. However focusing exclusively on te spindle system is insufficient because te bearings can cange te dynamics of a macining spindle system. Hence, te bearing effects on te spindle system must also be considered. Bearings are used in many rotating macines to brace te rotating spindles and rotors. In te past, te required rotor speed was low, allowing ball and roller bearings to be used in rotating macinery. Hig temperatures are generated wit ball-bearing spindle systems operating at ig speeds. Te ig temperatures often bring about macine failure. To attain greater complexity and accuracy, modern engineering tecnologies demand macinery tat can be run at ig speeds. To avoid te ig temperatures generated by te contact between te spindles and bearings, non-contact magnetic bearings are used for te spindle and rotor in ig speed rotating macinery. Traditionally, ball bearings ave been used to support te spindle systems wen te rotational speed was not ig. Previous investigations on bearing spindle systems were confined to spindles wit ball bearings. In some studies, te focus was on te dynamic response of a spindle supported by bearings [, ]. At iger speeds, tis bearing canges te stiffness of te entire spindle system and significantly alters te system properties [-5]. Precise macining requires iger spindle speeds, making te magnetic-bearing spindle necessary. Investigations as [6-] studied te performance and dynamic properties of magnetic bearings. Most studies deal wit a magnetic ring for a radiaagnetic bearing used as an unlimited one long magnetic bar for a permanent magnetic bearing. Investigation as [] studied te bearing capacity and stiffness of radiaagnetic bearings. Tus far, most investigations as [-4] on te dynamic caracteristics of a cracked spindle system were limited to ball-bearing-type spindles. Tis study examines te crack effects on te dynamic response of a rotating spindle system wit magnetic bearings. A Euler-Bernoulli beam of circular cross section was used to approximate te spindle model. Te equations of motion for te bearing-spindle system were derived using te Galerkin metod and Hamilton principle. A model te size of an actual spindle system was used. To simplify te calculations, massless springs were employed to model te stiffness of te magnetic bearings. Te effects of crack dept, rotational speed and bearing lengt on te dynamic response of a spindle system were investigated. Address correspondence to tis autor at te Department of Mecanical Engineering, Ceng Siu University, Taiwan; uangbw@csu.edu.tw a spindle supported by bearings X/8 8 Bentam Open
2 Vibration in a Cracked Macine Tool Spindle wit Magnetic Bearings Te Open Mecanical Engineering Journal, 8, Volume 33 k x : te bearing stiffness in u deflection at a position z, k y : te bearing stiffness in v deflection at a position z, k x : te bearing stiffness in u deflection at a position z, k y : te bearing stiffness in v deflection at a position z, : density, A : cross section area, : rotating speed, : Dirac delta fuction, z : te first located position of bearings, z : te second located position of bearings. For convenience, te dimensionless equations of motion for tis spindle are: a simple model of bearing spindle system Fig. (). A rotating spindle wit bearings sceme. η u A v + u + u 4 A 4 + k x u z z +k x u z z } = (3) R Fig. (). Geometry of a cracked spindle. Teory Tis paper considers a spindle supported by magnetic bearings, as sown in Fig. (a), to elucidate te dynamic response of a spindle system. Fig. (b) presents a simple model for tis bearing-spindle system. In tis model, massless springs are employed to simulate te stiffness of te magnetic bearing and support te spindle. Te rotational speed of te spindle cannot be ignored in te rotating macinery bearing application. In tis study, te deflection components (z,t), and u(z,t) denote te two transverse flexible deflections of te spindle system. E and I represent te Young s Modulus and area inertia of te spindle, respectively. Only te transverse flexible deflections are studied in tis article. According to [5], te governing equations of te spindle system are displayed as: A u A v A u + u + k x u ( z z )+ k x u ( z z )= A v + A u A v + v + k y v ( z z )+ k y v ( z z )= were a o dξ ξ () () v + A u + v + v 4 A 4 + k y v z z + k y v z z } = were te dimensionless parameters are given using: z = z, z = z, z = z, = u ( z )= u z k x = k x 3, v ( z )= v z, k y = k y 3, k x = k x 3, k y = k y 3 and te boundary conditions are: u = u = u = v = u = v = (4) A 4, (5), (6) (7) v =, at z = (8) v =, at z = (9) Te Galerkin metod is employed to derive te spindle equations of motion in matrix form. Terefore, te solutions for Eqs. (3) and (4) can be assumed to be: m u ( z,t)= i ( z )p i () t () i= m v ( z,t)= i ( z )q i () t () i= were i ( z ), i ( z ) are comparison functions for te spindle system, and p i (), t q i () t are te time coefficients to be determined for te system. Te exact solution for a beam wit free-free boundary conditions is considered, and five comparison function modes are used. i ( z )= i ( z )= ( i z ) ( cos i z cos i z ) ()
3 34 Te Open Mecanical Engineering Journal, 8, Volume Kung and Huang i = i 3 ( u i ), i =,, 3, (3) were u is te unit step function. Substituting Eqs. () and () into Eqs. (3) and (4) respectively, te equations of motion in matrix form for te spindle system can be derived as: () M pt G + p t M qt () G q() t K + e pt () K + K e qt () pt () K qt () K + s pt () K + K s qt () s pt () = K s qt () () (4) were = A. 4 Te elements of te matrices in te above equation are given as follows, ( m ij ) = i ( m ij ) = i ( g ij ) = i ( g ij ) = i ( k e ) ij ( k e ) ij ( k s ) ij ( k s ) ij ( k s ) ij ( k s ) ij ij j dz = k ij j dz = k (5) (6) j dz (7) j dz (8) = = i i = k x i z = k y i z = k x i z j dz (9) j dz () { } j ( z ) { } j ( z ) { } j ( z ) { } T () { } T () { } T (3) { } j ( z ) { } T (4) = k y i z For te sake of convenience, Eq. (4) can be rewritten as, M { X }+ G { X }+ K { X }= (5) were M = M M (6) G = G G K e K = K e + K K K s + K s + K s K s (7) (8) A space vector is introduced in Eq. (5) to solve te eigenvalue problem for te system. X { V }=. (9) X Substituting Eq. (9) into Eq. (5), te equation can be rearranged as; M { V }+ G K { K K V }= (3) Te non-dimensional frequency n in Eq. (3), i.e., te natural frequency of te spindle system, is defined as: n = n / for n =,,... (3) 4 A In industry, ball bearings are frequently used to support rotating spindles in rotating macinery. Recently, magnetic bearings ave been employed increasingly to support spindles because tey must rotate at iger speeds. Few investigations focused on te dynamic responses of defective spindle systems wit magnetic bearings. Terefore, tis investigation addresses te dynamic response of a cracked spindle supported by magnetic bearings. Crack Effect Considering a crack located at z = z on tis spindle, te strain energy of te defective spindle will include te released energy caused by te crack. Fig. () sows te geometry of a cracked spindle. Te released energy caused by a crack, as noted in [6], wit a dept of a may be expressed as: b μ U c = K E I d (3) b were b = R ( R a) and μ is te Poisson's ratio of te spindle, K I is te stress intensity factor under a mode I load and R is te radius of te spindle. In tis case, te stress intensity factors K I can be approximated as K I = 4M b R F R 4 ( ) (33) were, M b is te bending moment, and = R (34)
4 Vibration in a Cracked Macine Tool Spindle wit Magnetic Bearings Te Open Mecanical Engineering Journal, 8, Volume 35 = a + R R (35) F = tan sin cos 4 (36) Te notations a and R are te maximum crack dept and radius of te spindle, respectively. Based on te investigations in [6, 7], alterations of te elastic deformation energy caused by lateral bending moments are te only important canges in te case of slender beams wit a crack. Te released energy of te crack wit respect to due to te bending moment is obtained as: a b U c = E ( v μ ) z z b z ( R ) F d d dz (37) Similarly, te released energy of te crack wit respect to is derived as follows, U c = E ( μ ) were F F a b u z z z b (38) d d dz = tan sin cos 3 (39) For simplification, te dimensionless equations are employed as: U c = 4R ( μ u ) z z Q z R, R dz (4) U c = 4R ( μ v ) z z Q z R, R dz (4) were Q Q R, arbr R = R R F (4) d R d R d R d R R, ar br R = R R F (43) b R Te bearing-spindle wit a crack can be obtained as: u A v + u + u 4 A 4 8R μ ( )Q R, R u z z +k x u ( z z )+ k x u ( z z )} = v + A u + v + v 4 A 4 8R μ ( )Q R, R v ( z z ) + k y v ( z z )+ k y v ( z z )} = (44) (45) Similarly, te equations of motion for te defective spindle, i.e. Eq. 3, can be rearranged in matrix form using Galerkin s metod as follows: () M pt G + p t M qt () G q() t K + e pt () K + K e qt () K pt () K qt () c pt () K + K c qt () s K s pt () K + qt () s pt () = K s qt () were ( k c ) ij ( k c ) ij 8R μ = 8R μ = Q Q R, R R, R i i z () j ( z ) z j ( z ) (46) (47) z =z (48) z =z Supported by Magnetic Bearing Few investigations on radiaagnetic bearings were found, so tese bearings were selected for tis article. Te rotor is kept in te desired position by a magnetic bearing stator using a magnetic field induced by permanent magnets. According to [], te bearing force F r is derived as follows, F r = B B r r S (49) 4μ Were B r : remanence of te externaagnetic loop for te magnetic bearing; B r : remanence of te internaagnetic loop for te magnetic bearing; μ : permanence in vacuum and, (5) S = S 3 + S 4 S 3 S 4 + ( e + R 3 cos R cos ) (5) Rx = x + x x + R 3 sin R sin
5 36 Te Open Mecanical Engineering Journal, 8, Volume Kung and Huang + ( e + R 3 cos R cos ) (5) Rx = x + x x + R 3 sin R sin + ( e + R 4 cos R cos ) (53) Rx 3 = x + x x + R 4 sin R sin + ( e + R 4 cos R cos ) (54) Rx 4 = x + x x + R 4 sin R sin S 3 = R R 3 ( e + R 3 cos R cos )dddxdx Rx 3/ (55) size actually used in engineering applications are addressed and a magnetic bearing is considered in tis work. Te dimensions R=.m and =.m of a rotating spindle are assumed. Te bearings positions are assumed to be z = and z =. A spindle system braced by a magnetic bearing is important in engineering applications, especially for igspeed rotationaacinery. For te above-mentioned spindle dimensions, te important magnetic bearing parameters were selected as follows. Nd-Fe-B material was employed to model te elements of a permanent magnet in te radiaagnetic bearings. For tis material, te remanence B r = B r =.3 wb m can be sown. Corresponding to te spindle imension, te lengt of te magnetic = mm and te clearance mg =.m were assumed. Q S 3 = l l m m ( + ) R R e R cos R cos dddxdx 3 3 3/ Rx (56) Q S 4 = R R 4 ( e + R 4 cos R cos )dddxdx Rx 3 3/ (57).. S 4 = R R 4 ( e + R 4 cos R cos )dddxdx Rx 4 3/ (58) were R : external radius of te externaagnetic loop R+ g + mg R : internal radius of te externaagnetic loop R+ g + mg R 3 : external radius of te internaagnetic loop R+ R 4 : internal radius of te internaagnetic loop R : lengt of te magnetic loop mg : clearance between te internal and externaagnetic loops g : te tickness of te magnetic loop e : is eccentric of te magnetic bearing Consequently, te stiffness of a radiaagnetic bearing can be derived as follows: k m = F r e = B r B r 4μ S e g (59) ANAYSIS AND DISCUSSION In ultra-ig-speed macining, using magnetic bearings to support te spindle is necessary [8]. Te dynamic properties of a multi-mode spindle wit bearings of te Crcak dept a/r Fig. (3). Te variations in te crack flexibility wit te crack dept ratio. Te variation in te crack flexibility wit te crack dept ratio is plotted in Fig. (3). Te numerical analysis reveals tat te crack flexibility increases as te crack dept is increased. From te results, te crack dept markedly affects te saft stiffness. As a wole, tese results and tose from previous investigation [9] are identical. Fig. (4) presents te natural frequencies of a spindle bearing system wit and witout cracks. Te logaritmic scale was employed to study iger modes in tis figure. At lower modes, te natural frequencies of te spindle bearing system cange sligtly regardless of weter tere is a crack or not in te system. However, at iger modes, te natural frequencies of a spindle bearing system decrease wen tere is a crack in tis spindle system. Wit magnetic bearings, te crack effect on te dynamics of a spindle system as more influence at te iger modes tan at te lower modes. Te effect of te crack dept on te natural frequencies is considered in Fig. (5). In tis figure, te lower and iger mode natural frequencies, te st and 5 t modes, are studied togeter. Bot te st and 5 t natural frequencies decrease wit increasing crack dept. If te crack dept were., te crack size would ave little influence on te natural frequencies of a rotating blade system. However, te natural frequencies are depressed significantly wen te crack size is larger tan..
6 Vibration in a Cracked Macine Tool Spindle wit Magnetic Bearings Te Open Mecanical Engineering Journal, 8, Volume 37 no crack wit a crack ad= Mode Number Fig. (4). Te natural frequencies of a magnetic-bearing spindle wit and witout a crack, ( =.5, a D=.5, z =.5 ) natural frequencies of te system is te lowest wen te crack is located at te middle of te spindle. Te penomena illustrated in Fig. (6a) and (6b) are similar. Te effect of rotation speed on te dynamics of a cracked spindle wit magnetic bearings is plotted in Fig. (7). Double roots for te natural frequencies of a cracked spindle wit magnetic bearings are observed only if tis system as no rotation speed. Wit rotation speed, te natural frequencies of tis system are divided into two parts, te forward and backward frequencies. Only te st and nd natural frequencies are sown in tis figure. Te st natural frequency of a cracked spindle wit magnetic bearings increases as te rotational speed increases. However, it was found tat if te rotational speed increases, te nd natural frequency decreases Crack location z (a) te st mode Crack dept a/d (a) te st mode Crack dept a/d (b) te 5t mode Fig. (5). Te natural frequencies of a magnetic-bearing spindle wit different crack depts, ( =.5, z =.5 ). Crack location dramatically canges te dynamics of a spindle wit magnetic bearings. Fig. (6) sows te variation in te natural frequencies of a spindle wit different crack locations. As mentioned above, only te st and 5 t natural frequencies were examined. Wen te crack is located at bot ends, te natural frequencies are almost te same as a system witout a crack. Te value of te Crack location z (b) te 5t mode Fig. (6). Te natural frequencies of a magnetic-bearing spindle wit different crack locations, ( =.5, a D=.5 ). For radiaagnetic bearings, te lengt of te bearings remarkably alters te system stiffness. Fig. (8) illustrates te variations in natural frequencies for a cracked spindle wit different magnetic bearing lengts. Te results indicate tat te natural frequencies of a magnetic-bearing spindle increase as te magnetic bearing lengt increases. Finally, te frequency responses of a spindle system wit and witout cracks are illustrated in Fig. (9). Bot te lower and iger frequency domains were examined. In te lower frequency domain, te figure sows tat te frequency response of a bearing-spindle system witout a crack is almost te same as one wit a crack. Te peak frequency response values for a spindle wit magnetic bearings are depressed in te iger frequency
7 38 Te Open Mecanical Engineering Journal, 8, Volume Kung and Huang domain if tere is a crack in te spindle. As above, te effect of a crack may significantly influence te dynamics of a spindle system wit magnetic bearings at iger modes te st mode te nd mode 3 - wit a crack witout cracks Rotational speed Ω Fig. (7). Te natural frequencies of a magnetic-bearing spindle wit different rotational speeds, ( z =.5, a D=.5 ) engt of bearing m Fig. (8). Te natural frequencies of a rotating cracked spindle wit different lengts of magnetic bearing, ( z =.5, a D=.5 ). CONCUSIONS Te effect of cracks on te dynamics of a spindle supported by magnetic bearings was studied. Te most significant observations in tis study are summarized as follows:. Wit magnetic bearings, te natural frequencies of a spindle system decrease as te dept of te crack increases.. At iger modes, a crack wilarkedly affect te dynamics of a spindle wit magnetic bearings. However, at lower modes, te natural frequencies of te spindle bearing system cange just sligtly no matter weter tere is a crack or not. 3. Te rotational speed and magnetic bearing lengt will significantly influence te dynamics of a spindle wit magnetic bearings Frequency (b) a iger frequency domain Fig. (9). Te frequency response of a rotating spindle wit and witout a crack, ( =.5, a D=.5, z =.5 ) 4. Te crack will dramatically affect te dynamic caracteristics of a spindle wit magnetic bearings if te crack is located at te middle of te spindle. REFERENCES Frequency (a) a lower frequency domain [] S. C. Huang, Y. M. Huang and S. M.Sie, Vibration and Stability of a Rotating Saft Containing a Transverse Crack, Journal of Sound and Vibration, vol.6, pp , 993. [] S. K. Baumik, R. Rangaraju, M. A. Venkataswamy, T. A. Baskaran and M. A. Parameswara, Fatigue fracture of cranksaft of an aircraft engine, Engineering Failure Analysis, vol. 9(3), pp ,. [3] P. F. Rizos, N. Aspragatos, A. D. Dimarogonas, Dimarogonas Identification of Crack ocation and Magnitude in a Cantilever Beam from te Vibration Mode, Journal of Sound and Vibration, vol. 38, pp , 99. [4] D. Broek, Elementary Engineering Fracture mecanics, Martinus Nijoff Publisers, 986. [5] H. Tada, P. Paris and G. Irwin, Te Stress Analysis of Crack Handbook, Hellertown, Pennsylvania: Del Researc Corporation, 973. [6] B. Grabowski, Te Vibrational Beavior of a Turbine Rotor Containing a Transverse Crack, ASME, Journal of Mecanic Design, vol., pp. 4-46, 98. [7] A. S. Sekar, Effects of cracks on rotor system instability, Mecanism and Macine Teory, vol. 35(), pp ,. [8] N. Bacscmid, P. Pennacci, E. Tanzi and A. Vania, Identification of transverse crack position and dept in rotor systems, Meccanica, vol. 6, pp ,.
8 Vibration in a Cracked Macine Tool Spindle wit Magnetic Bearings Te Open Mecanical Engineering Journal, 8, Volume 39 [9] P. N. Savedra and.a. Cuitino, Vibration analysis of rotor for crack identification, Journal of Vibration and Control, vol. 8(), pp.5-67,. [] M. J. Goodwin, Dynamics of Rotor-Bearing System, Unwin Hyman, ondon, 989. [] J. A. Brandon and K. J. H. Al-Sareef, On Te Applicability of Modal and Response Representations in Te Dynamic Analysis of Macine Tool Spindle Bearing Systems, IMecE, Journal of Engineering Manufacture, vol. 5, pp , 99. [] B.R. Jorgensen and Y. C. Sin, Dynamics of Spindle-Bearing Systems at Hig Speeds Including Cutting oad Effects, ASME, Journal of manufacturing Science and Engineering, vol., pp , 998. [3] W. R. Wang and C. N. Cang, Dynamic Analysis and Design of a Macine Tool Spindle-Bearing System, ASME, Journal of Vibration and Acoustics, vol. 6, pp. 8-85, 994. [4] R.G. Parker and C. j. Mote, Vibration and Coupling Penomena in Asymmetric Disk-Spindle Systems, ASME, Journal of Applied Mecanics, vol. 63, pp , 996. [5] I.Y. Sen, Closed-Form Forced Response of a Damped, Rotating, Multiple Disks/Spindle System, ASME, Journal of Applied Mecanics, vol. 64, pp , 997. [6] F.T. Barwell, Bearing System, Principles and Practice, Oxford University Press, New York, 979. [7] M. Cinta and A. B. Palazzolo, Stability and Bifurcation of Rotor Motion in a Magnetic Bearing, Journal of sound and vibration, vol. 4(5), pp , 998. [8] C. Delprete, G. Genta and A. Repaci, Numerical Simulation of te Dynamic Beavior of Rotors on Active Magnetic Bearings, In: Proceedings of te 7 t International Symposium on Transport Penomena and Dynamics of Rotating Macinery, Honolulu, Hawaii, USA, 998. [9] M. Komori and T. Hamasaki, Improvement and Evaluation of Bearing Stiffness in Hig Tc Superconducting Magnetic Bearing, IEEE Transactions on Applied Superconductivity, vol. (), pp ,. [] H. Cang and S. C. Cung, Integrated Design of Radial Active Magnetic Bearing Systems Using Genetic Algoritms, Mecatronics, vol. (), pp.9-36,. [] Q. Tan, M. iu and H. Meng, Study on Bearing Capacity and Stiffness of Radial Magnetic Bearing, Journal of Tribology (in Cina), vol. 4(4), pp , 994. [] Y. He, D. Guo and F. Cu, Using genetic algoritms to detect and configure saft crack for rotor-bearing system, Computer Metods in Applied Mecanics and Engineering, vol. 9(45), pp ,. [3] M. A. Moiuddin and Y. A. Kulief, Dynamic response analysis of rotor-bearing systems wit cracked saft, ASME, Journal of Mecanical Design, vol. 4(4), pp ,. [4] S. Prabakar and A. S. Sekar and A R. Moanty, Detection and monitoring of cracks in a rotor-bearing system using wavelet transforms, Mecanical Systems and Signal Processing, vol. 5(), pp ,. [5] B. W. Huang and H. K. Kung, A Variation of Instability in a Rotating Spindle System wit Various Bearings, International Journal of Mecanical Science, vol. 45(), pp. 57-7, 3. [6] A. D. Dimarogonas and S. A. Paipetis, Analytical Metods in Rotor Dynamics, Applied Science Publisers, New York, 983. [7] M. Krawczuk and W. M. Ostacowicz, Transverse Natural Vibrations of a Cracked Beam oaded wit a Constant Axial Force, ASME, Journal of Vibration and Acoustics, vol. 5, pp , 993. [8] S. Herranz, F. J. Campa, N ópez de acalle, et al., Te milling of airframe components wit low rigidity: a general approac for te avoidance of static and dynamic problems, Procc. Int. Mecanical Inst. Part B Journal of Engineering Manufacture, vol. 9(B), pp , 5. [9] M. C. Wu and S. C. Huang, Vibration and Crack Detection of a Rotor wit Speed-Dependent Bearings, International Journal of Mecanical Science, vol. 4(6), pp , 998. Received: November, 7 Revised: April 8, 8 Accepted: May 3, 8 Kung and Huang; icensee Bentam Open. Tis is an open access article distributed under te terms of te Creative Commons Attribution icense (ttp://creativecommons.org/licenses/by/.5/), wic permits unrestrictive use, distribution, and reproduction in any medium, provided te original work is properly cited.
Stability of Smart Beams with Varying Properties Based on the First Order Shear Deformation Theory Located on a Continuous Elastic Foundation
Australian Journal of Basic and Applied Sciences, 5(7): 743-747, ISSN 99-878 Stability of Smart Beams wit Varying Properties Based on te First Order Sear Deformation Teory ocated on a Continuous Elastic
More informationAN ANALYSIS OF AMPLITUDE AND PERIOD OF ALTERNATING ICE LOADS ON CONICAL STRUCTURES
Ice in te Environment: Proceedings of te 1t IAHR International Symposium on Ice Dunedin, New Zealand, nd t December International Association of Hydraulic Engineering and Researc AN ANALYSIS OF AMPLITUDE
More informationAnalysis of Stress and Deflection about Steel-Concrete Composite Girders Considering Slippage and Shrink & Creep Under Bending
Send Orders for Reprints to reprints@bentamscience.ae Te Open Civil Engineering Journal 9 7-7 7 Open Access Analysis of Stress and Deflection about Steel-Concrete Composite Girders Considering Slippage
More informationFlapwise bending vibration analysis of double tapered rotating Euler Bernoulli beam by using the differential transform method
Meccanica 2006) 41:661 670 DOI 10.1007/s11012-006-9012-z Flapwise bending vibration analysis of double tapered rotating Euler Bernoulli beam by using te differential transform metod Ozge Ozdemir Ozgumus
More information3. Using your answers to the two previous questions, evaluate the Mratio
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 0219 2.002 MECHANICS AND MATERIALS II HOMEWORK NO. 4 Distributed: Friday, April 2, 2004 Due: Friday,
More informationAnalysis of Static and Dynamic Load on Hydrostatic Bearing with Variable Viscosity and Pressure
Indian Journal of Science and Tecnology Supplementary Article Analysis of Static and Dynamic Load on Hydrostatic Bearing wit Variable Viscosity and Pressure V. Srinivasan* Professor, Scool of Mecanical
More information158 Calculus and Structures
58 Calculus and Structures CHAPTER PROPERTIES OF DERIVATIVES AND DIFFERENTIATION BY THE EASY WAY. Calculus and Structures 59 Copyrigt Capter PROPERTIES OF DERIVATIVES. INTRODUCTION In te last capter you
More information6. Non-uniform bending
. Non-uniform bending Introduction Definition A non-uniform bending is te case were te cross-section is not only bent but also seared. It is known from te statics tat in suc a case, te bending moment in
More informationNonlinear correction to the bending stiffness of a damaged composite beam
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. Nonlinear correction to te bending stiffness of a damaged composite beam W.
More informationFinite Element Analysis of J-Integral for Surface Cracks in Round Bars under Combined Mode I Loading
nternational Journal of ntegrated Engineering, Vol. 9 No. 2 (207) p. -8 Finite Element Analysis of J-ntegral for Surface Cracks in Round Bars under Combined Mode Loading A.E smail, A.K Ariffin 2, S. Abdulla
More informationTheoretical Analysis of Flow Characteristics and Bearing Load for Mass-produced External Gear Pump
TECHNICAL PAPE Teoretical Analysis of Flow Caracteristics and Bearing Load for Mass-produced External Gear Pump N. YOSHIDA Tis paper presents teoretical equations for calculating pump flow rate and bearing
More informationModel development for the beveling of quartz crystal blanks
9t International Congress on Modelling and Simulation, Pert, Australia, 6 December 0 ttp://mssanz.org.au/modsim0 Model development for te beveling of quartz crystal blanks C. Dong a a Department of Mecanical
More information232 Calculus and Structures
3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE
More informationJournal of Mechanical Science and Technology 23 (2009) 2072~2084. M. M. Najafizadeh * and M. R. Isvandzibaei
Journal of Mecanical Science and Tecnology (9 7~84 Journal of Mecanical Science and Tecnology www.springerlink.com/content/78-494x DOI.7/s6-9-4- Vibration of functionally graded cylindrical sells based
More informationVibration Analysis Of Cantilever Shaft With Transverse Cracks
Vibration Analysis Of Cantilever Shaft With Transverse Cracks R.K Behera, D.R.K. Parhi, S.K. Pradhan, and Seelam Naveen Kumar Dept. of Mech Engg. N.I.T., Rourkela,7698 Dept. of Mech. Engg Dept. of Mech.
More informationNumerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
More informationCALCULATION OF COLLAPSE PRESSURE IN SHALE GAS FORMATION AND THE INFLUENCE OF FORMATION ANISOTROPY
CALCULATION OF COLLAPSE PRESSURE IN SHALE GAS FORMATION AND THE INFLUENCE OF FORMATION ANISOTROPY L.Hu, J.Deng, F.Deng, H.Lin, C.Yan, Y.Li, H.Liu, W.Cao (Cina University of Petroleum) Sale gas formations
More informationComputational Method of Structural Reliability Based on Integration Algorithms
Sensors & ransducers, Vol. 54, Issue 7, July 03, pp. 5-59 Sensors & ransducers 03 by IFSA ttp://www.sensorsportal.com Computational Metod of Structural Based on Integration Algoritms * Cong Cen, Yi Wan
More informationOptimization of the thin-walled rod with an open profile
(1) DOI: 1.151/ matecconf/181 IPICSE-1 Optimization of te tin-walled rod wit an open profile Vladimir Andreev 1,* Elena Barmenkova 1, 1 Moccow State University of Civil Engineering, Yaroslavskoye s., Moscow
More informationVibro-Acoustics of a Brake Rotor with Focus on Squeal Noise. Abstract
Te International Congress and Exposition on Noise Control Engineering Dearbor MI, USA. August 9-, Vibro-Acoustics of a Brake Rotor wit Focus on Sueal Noise H. Lee and R. Sing Acoustics and Dynamics Laboratory,
More informationA = h w (1) Error Analysis Physics 141
Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.
More informationThe entransy dissipation minimization principle under given heat duty and heat transfer area conditions
Article Engineering Termopysics July 2011 Vol.56 No.19: 2071 2076 doi: 10.1007/s11434-010-4189-x SPECIAL TOPICS: Te entransy dissipation minimization principle under given eat duty and eat transfer area
More informationWINKLER PLATES BY THE BOUNDARY KNOT METHOD
WINKLER PLATES BY THE BOUNARY KNOT ETHO Sofía Roble, sroble@fing.edu.uy Berardi Sensale, sensale@fing.edu.uy Facultad de Ingeniería, Julio Herrera y Reissig 565, ontevideo Abstract. Tis paper describes
More informationFirst-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns
First-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns J. A. Loya ; G. Vadillo 2 ; and J. Fernández-Sáez 3 Abstract: In this work, closed-form expressions for the buckling loads
More informationNCCI: Simple methods for second order effects in portal frames
NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames Tis NCC presents information
More information(4.2) -Richardson Extrapolation
(.) -Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Suppose tat lim G 0 and lim F L. Te function F is said to converge to L as
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 34, pp. 14-19, 2008. Copyrigt 2008,. ISSN 1068-9613. ETNA A NOTE ON NUMERICALLY CONSISTENT INITIAL VALUES FOR HIGH INDEX DIFFERENTIAL-ALGEBRAIC EQUATIONS
More informationDamage Identification of a Long-Span Suspension Bridge Using Temperature- Induced Strain Data. Southeast University, Nanjing, China, ABSTRACT
Damage Identification of a Long-Span Suspension Bridge Using Temperature- Induced Strain Data *Qi. Xia 1) and Jian. Zang 2) 1 Scool of civil engineering, Souteast University, Nanjing, Cina 2 Key Laboratory
More informationVIBRATION-BASED METHODS FOR DETECTING A CRACK IN A SIMPLY SUPPORTED BEAM
Journal of Theoretical and Applied Mechanics, Sofia, 2014, vol. 44, No. 4, pp. 69 82 DOI: 10.2478/jtam-2014-0023 VIBRATION-BASED METHODS FOR DETECTING A CRACK IN A SIMPLY SUPPORTED BEAM Dimitrina Kindova-Petrova
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc
More informationBending analysis of a functionally graded piezoelectric cantilever beam
Science in Cina Series G: Pysics Mecanics & Astronomy 7 Science in Cina Press Springer-Verlag Bending analysis of a functionally graded pieoelectric cantilever beam YU Tao & ZHONG Zeng Scool of Aerospace
More informationSection 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,
More informationEffect of L/D Ratio on the Performance of a Four-Lobe Pressure Dam Bearing
Vol:, No:8, 007 Effect of L/D Ratio on te Performance of a Four-Lobe Pressure Dam Bearin G Busan, S S Rattan, and N P Meta International Science Index, Mecanical and Mecatronics Enineerin Vol:, No:8, 007
More informationCombining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)
More informationStatic Response Analysis of a FGM Timoshenko s Beam Subjected to Uniformly Distributed Loading Condition
Static Response Analysis of a FGM Timoseno s Beam Subjected to Uniformly Distributed Loading Condition 8 Aas Roy Department of Mecanical Engineering National Institute of Tecnology Durgapur Durgapur, Maatma
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationFabric Evolution and Its Effect on Strain Localization in Sand
Fabric Evolution and Its Effect on Strain Localization in Sand Ziwei Gao and Jidong Zao Abstract Fabric anisotropy affects importantly te overall beaviour of sand including its strengt and deformation
More information2040. Damage modeling and simulation of vibrating pipe with part-through circumferential crack
24. Damage modeling and simulation of vibrating pipe with part-through circumferential crack Zhihong Yu 1, Laibin Zhang 2, Jinqiu Hu 3, Jiashun Hu 4 1, 2, 3 College of Mechanical and Transportation Engineering,
More informationPECULIARITIES OF THE WAVE FIELD LOCALIZATION IN THE FUNCTIONALLY GRADED LAYER
Materials Pysics and Mecanics (5) 5- Received: Marc 7, 5 PECULIARITIES OF THE WAVE FIELD LOCALIZATION IN THE FUNCTIONALLY GRADED LAYER Т.I. Belyankova *, V.V. Kalincuk Soutern Scientific Center of Russian
More informationNONLINEAR SYSTEMS IDENTIFICATION USING THE VOLTERRA MODEL. Georgeta Budura
NONLINEAR SYSTEMS IDENTIFICATION USING THE VOLTERRA MODEL Georgeta Budura Politenica University of Timisoara, Faculty of Electronics and Telecommunications, Comm. Dep., georgeta.budura@etc.utt.ro Abstract:
More informationDerivation Of The Schwarzschild Radius Without General Relativity
Derivation Of Te Scwarzscild Radius Witout General Relativity In tis paper I present an alternative metod of deriving te Scwarzscild radius of a black ole. Te metod uses tree of te Planck units formulas:
More informationA general articulation angle stability model for non-slewing articulated mobile cranes on slopes *
tecnical note 3 general articulation angle stability model for non-slewing articulated mobile cranes on slopes * J Wu, L uzzomi and M Hodkiewicz Scool of Mecanical and Cemical Engineering, University of
More informationTO STUDY THE EFFECT OF CRACK ON NATURAL FREQUENCY IN A CANTILEVER STRUCTURE BY USING EULER S BEAM THEORY.
TO STUDY THE EFFECT OF CRACK ON NATURAL FREQUENCY IN A CANTILEVER STRUCTURE BY USING EULER S BEAM THEORY. Mr Ganesh G. Gade PREC, Loni, Ahemadnagar, India, ggade7@gmail.com Mr.Amol L. Khatode SGOI, COE,
More informationAvailable online at ScienceDirect. Procedia Engineering 125 (2015 )
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 15 (015 ) 1065 1069 Te 5t International Conference of Euro Asia Civil Engineering Forum (EACEF-5) Identification of aerodynamic
More informationParameter Fitted Scheme for Singularly Perturbed Delay Differential Equations
International Journal of Applied Science and Engineering 2013. 11, 4: 361-373 Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations Awoke Andargiea* and Y. N. Reddyb a b Department
More informationDifferentiation in higher dimensions
Capter 2 Differentiation in iger dimensions 2.1 Te Total Derivative Recall tat if f : R R is a 1-variable function, and a R, we say tat f is differentiable at x = a if and only if te ratio f(a+) f(a) tends
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximating a function f(x, wose values at a set of distinct points x, x, x 2,,x n are known, by a polynomial P (x
More informationSAMCEF For ROTORS. Chapter 1 : Physical Aspects of rotor dynamics. This document is the property of SAMTECH S.A. MEF A, Page 1
SAMCEF For ROTORS Chapter 1 : Physical Aspects of rotor dynamics This document is the property of SAMTECH S.A. MEF 101-01-A, Page 1 Table of Contents rotor dynamics Introduction Rotating parts Gyroscopic
More information5 Ordinary Differential Equations: Finite Difference Methods for Boundary Problems
5 Ordinary Differential Equations: Finite Difference Metods for Boundary Problems Read sections 10.1, 10.2, 10.4 Review questions 10.1 10.4, 10.8 10.9, 10.13 5.1 Introduction In te previous capters we
More informationSample Problems for Exam II
Sample Problems for Exam 1. Te saft below as lengt L, Torsional stiffness GJ and torque T is applied at point C, wic is at a distance of 0.6L from te left (point ). Use Castigliano teorem to Calculate
More informationThe effects of shear stress on the lubrication performances of oil film of large-scale mill bearing
Universit of Wollongong Researc Online Facult of Engineering - Papers (Arcive) Facult of Engineering and Information Sciences 9 Te effects of sear stress on te lubrication performances of oil film of large-scale
More informationExercises for numerical differentiation. Øyvind Ryan
Exercises for numerical differentiation Øyvind Ryan February 25, 2013 1. Mark eac of te following statements as true or false. a. Wen we use te approximation f (a) (f (a +) f (a))/ on a computer, we can
More informationSqueeze film effect at elastohydrodynamic lubrication of plain journal bearings
Science, Juliana Engineering Javorova, & Education, Anelia Mazdrakova 1, (1), 16, 11- Squeeze film effect at elastoydrodynamic lubrication of plain journal bearings Juliana Javorova *, Anelia Mazdrakova
More informationThe Verlet Algorithm for Molecular Dynamics Simulations
Cemistry 380.37 Fall 2015 Dr. Jean M. Standard November 9, 2015 Te Verlet Algoritm for Molecular Dynamics Simulations Equations of motion For a many-body system consisting of N particles, Newton's classical
More information4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.
Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra
More informationDevelopment and experimental verification of a numerical model for optimizing the cleaning performance of the doctor blade press roll tribosystem
Development and experimental verification of a numerical model for optimizing te cleaning performance of te doctor blade press roll tribosystem M. Rodríguez Ripoll, B. Sceicl,, D. Bianci, B. Jakab, F.
More informationPath to static failure of machine components
Pat to static failure of macine components Load Stress Discussed last week (w) Ductile material Yield Strain Brittle material Fracture Fracture Dr. P. Buyung Kosasi,Spring 008 Name some of ductile and
More informationSimulation and verification of a plate heat exchanger with a built-in tap water accumulator
Simulation and verification of a plate eat excanger wit a built-in tap water accumulator Anders Eriksson Abstract In order to test and verify a compact brazed eat excanger (CBE wit a built-in accumulation
More information4.2 - Richardson Extrapolation
. - Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Definition Let x n n converge to a number x. Suppose tat n n is a sequence
More informationAN IMPROVED WEIGHTED TOTAL HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES
AN IMPROVED WEIGHTED TOTA HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES Tomas A. IPO University of Wisconsin, 45 Engineering Drive, Madison WI, USA P: -(608)-6-087, Fax: -(608)-6-5559, lipo@engr.wisc.edu
More informationApplication of numerical integration methods to continuously variable transmission dynamics models a
ttps://doi.org/10.1051/ssconf/2018440005 Application of numerical integration metods to continuously variable transmission dynamics models a Orlov Stepan 1 1 Peter te Great St. Petersburg Polytecnic University
More informationLIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION
LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LAURA EVANS.. Introduction Not all differential equations can be explicitly solved for y. Tis can be problematic if we need to know te value of y
More informationA Multiaxial Variable Amplitude Fatigue Life Prediction Method Based on a Plane Per Plane Damage Assessment
American Journal of Mecanical and Industrial Engineering 28; 3(4): 47-54 ttp://www.sciencepublisinggroup.com/j/ajmie doi:.648/j.ajmie.2834.2 ISSN: 2575-679 (Print); ISSN: 2575-66 (Online) A Multiaxial
More informationJournal of Engineering Science and Technology Review 7 (4) (2014) 40-45
Jestr Journal of Engineering Science and Tecnology Review 7 (4) (14) -45 JOURNAL OF Engineering Science and Tecnology Review www.jestr.org Mecanics Evolution Caracteristics Analysis of in Fully-mecanized
More informationShear Thinning and Hydrodynamic Friction of Viscosity Modifier- Containing Oils. Part II: Impact of Shear Thinning on Journal Bearing Friction
Tribology Letters 208 66:9 ttps://doi.org/0.007/s249-08-040-z ORIGINAL PAPER Sear Tinning and Hydrodynamic Friction of Viscosity Modifier- Containing Oils. Part II: Impact of Sear Tinning on Journal Bearing
More informationHIGH ORDER SHEAR DEFORMATION THEORY FOR ISOTROPIC BEAMS WITH USING MIXED FINITE ELEMENT METHOD
Held on 1-, Nov, 15, in Dubai, ISBN:978819313731 HIGH ORDER SHEAR DEFORATION THEORY FOR ISOTROPIC BEAS WITH USING IXED FINITE ELEENT ETHOD Emra adenci, Department of Civil Engineering, Necmettin Erbakan
More informationFinite Difference Methods Assignments
Finite Difference Metods Assignments Anders Söberg and Aay Saxena, Micael Tuné, and Maria Westermarck Revised: Jarmo Rantakokko June 6, 1999 Teknisk databeandling Assignment 1: A one-dimensional eat equation
More informationThe development of contact and noncontact technique to study the heat dissipation in metals under loading
Te development of contact and noncontact tecnique to study te eat dissipation in metals under loading More info about tis article: ttp://www.ndt.net/?id=73 Abstract * ICMM UB RAS, Ac. Koroleva Str., 63
More informationDynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact
Paper ID No: 23 Dynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact Dr. Magnus Karlberg 1, Dr. Martin Karlsson 2, Prof. Lennart Karlsson 3 and Ass. Prof. Mats Näsström 4 1 Department
More informationSimulation and Experimental Research on Dynamics of Low-Pressure Rotor System in Turbofan Engine
Simulation and Experimental Research on Dynamics of Low-Pressure Rotor System in Turbofan Engine Shengxiang Li 1, Chengxue Jin 2, Guang Zhao 1*, Zhiliang Xiong 1, Baopeng Xu 1 1. Collaborative Innovation
More informationLines, Conics, Tangents, Limits and the Derivative
Lines, Conics, Tangents, Limits and te Derivative Te Straigt Line An two points on te (,) plane wen joined form a line segment. If te line segment is etended beond te two points ten it is called a straigt
More informationA Reconsideration of Matter Waves
A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum,
More informationConsider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.
Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions
More informationA First-Order System Approach for Diffusion Equation. I. Second-Order Residual-Distribution Schemes
A First-Order System Approac for Diffusion Equation. I. Second-Order Residual-Distribution Scemes Hiroaki Nisikawa W. M. Keck Foundation Laboratory for Computational Fluid Dynamics, Department of Aerospace
More informationACCURATE SYNTHESIS FORMULAS OBTAINED BY USING A DIFFERENTIAL EVOLUTION ALGORITHM FOR CONDUCTOR-BACKED COPLANAR WAVEG- UIDES
Progress In Electromagnetics Researc M, Vol. 10, 71 81, 2009 ACCURATE SYNTHESIS FORMULAS OBTAINED BY USING A DIFFERENTIAL EVOLUTION ALGORITHM FOR CONDUCTOR-BACKED COPLANAR WAVEG- UIDES S. Kaya, K. Guney,
More informationQuasiperiodic phenomena in the Van der Pol - Mathieu equation
Quasiperiodic penomena in te Van der Pol - Matieu equation F. Veerman and F. Verulst Department of Matematics, Utrect University P.O. Box 80.010, 3508 TA Utrect Te Neterlands April 8, 009 Abstract Te Van
More information1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point
MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note
More informationLast lecture (#4): J vortex. J tr
Last lecture (#4): We completed te discussion of te B-T pase diagram of type- and type- superconductors. n contrast to type-, te type- state as finite resistance unless vortices are pinned by defects.
More informationSimulation of a Single and Double-Span Guideway under Action of Moving MAGLEV Vehicles with Constant Force and Constant Gap
Simulation of a Single and Doule-Span Guideway under Action of Moving MAGLEV Veicles wit Constant Force and Constant Gap Reinold Meisinger Mecanical Engineering Department Nuremerg University of Applied
More informationSection A 01. (12 M) (s 2 s 3 ) = 313 s 2 = s 1, h 3 = h 4 (s 1 s 3 ) = kj/kgk. = kj/kgk. 313 (s 3 s 4f ) = ln
0. (a) Sol: Section A A refrigerator macine uses R- as te working fluid. Te temperature of R- in te evaporator coil is 5C, and te gas leaves te compressor as dry saturated at a temperature of 40C. Te mean
More informationProblem Set 4 Solutions
University of Alabama Department of Pysics and Astronomy PH 253 / LeClair Spring 2010 Problem Set 4 Solutions 1. Group velocity of a wave. For a free relativistic quantum particle moving wit speed v, te
More informationCHAPTER 4 QUANTUM PHYSICS
CHAPTER 4 QUANTUM PHYSICS INTRODUCTION Newton s corpuscular teory of ligt fails to explain te penomena like interference, diffraction, polarization etc. Te wave teory of ligt wic was proposed by Huygen
More informationIN stability studies, the full dynamic model of a power
Simplified time-domain simulation of detailed long-term dynamic models Davide Fabozzi Tierry Van Cutsem, Fellow, IEEE Abstract Time-domain simulation of power system long-term dynamics involves te solution
More informationHOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS
HOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS Po-Ceng Cang National Standard Time & Frequency Lab., TL, Taiwan 1, Lane 551, Min-Tsu Road, Sec. 5, Yang-Mei, Taoyuan, Taiwan 36 Tel: 886 3
More informationInvestigating Euler s Method and Differential Equations to Approximate π. Lindsay Crowl August 2, 2001
Investigating Euler s Metod and Differential Equations to Approximate π Lindsa Crowl August 2, 2001 Tis researc paper focuses on finding a more efficient and accurate wa to approximate π. Suppose tat x
More informationCLOSED CONVEX SHELLS Meunargia T. Mixed forms of stress-strain relations are given in the form. λ + 2µ θ + 1
Seminar of I. Vekua Institute of Applied Matematics REPORTS, Vol. 43, 207 CLOSED CONVEX SHELLS Meunargia T. Abstract. If Ω is a closed convex sell, ten S : x 3 = 0 is an ovaloid. It is proved tat in tis
More informationLarge eddy simulation of turbulent flow downstream of a backward-facing step
Available online at www.sciencedirect.com Procedia Engineering 31 (01) 16 International Conference on Advances in Computational Modeling and Simulation Large eddy simulation of turbulent flow downstream
More informationDesalination by vacuum membrane distillation: sensitivity analysis
Separation and Purification Tecnology 33 (2003) 75/87 www.elsevier.com/locate/seppur Desalination by vacuum membrane distillation: sensitivity analysis Fawzi Banat *, Fami Abu Al-Rub, Kalid Bani-Melem
More informationArbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations
Arbitrary order exactly divergence-free central discontinuous Galerkin metods for ideal MHD equations Fengyan Li, Liwei Xu Department of Matematical Sciences, Rensselaer Polytecnic Institute, Troy, NY
More informationImplicit-explicit variational integration of highly oscillatory problems
Implicit-explicit variational integration of igly oscillatory problems Ari Stern Structured Integrators Worksop April 9, 9 Stern, A., and E. Grinspun. Multiscale Model. Simul., to appear. arxiv:88.39 [mat.na].
More informationUNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics
UNIVERSITY O SASKATCHEWAN Department of Pysics and Engineering Pysics Pysics 117.3 MIDTERM EXAM Regular Sitting NAME: (Last) Please Print (Given) Time: 90 minutes STUDENT NO.: LECTURE SECTION (please ceck):
More informationDevelopment of new and validation of existing convection correlations for rooms with displacement ventilation systems
Energy and Buildings 38 (2006) 163 173 www.elsevier.com/locate/enbuild Development of new and validation of existing convection correlations for rooms wit displacement ventilation systems Atila Novoselac
More informationReflection of electromagnetic waves from magnetic having the ferromagnetic spiral
Reflection of electromagnetic waves from magnetic aving te ferromagnetic spiral Igor V. Bycov 1a Dmitry A. Kuzmin 1b and Vladimir G. Savrov 3 1 Celyabins State University 51 Celyabins Br. Kasiriny Street
More informationResearch Article New Results on Multiple Solutions for Nth-Order Fuzzy Differential Equations under Generalized Differentiability
Hindawi Publising Corporation Boundary Value Problems Volume 009, Article ID 395714, 13 pages doi:10.1155/009/395714 Researc Article New Results on Multiple Solutions for Nt-Order Fuzzy Differential Equations
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More informationDistribution of reynolds shear stress in steady and unsteady flows
University of Wollongong Researc Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 13 Distribution of reynolds sear stress in steady
More informationEuler-Bernoulli Beam Theory in the Presence of Fiber Bending Stiffness
IOSR Journal of Matematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 13, Issue 3 Ver. V (May - June 017), PP 10-17 www.iosrjournals.org Euler-Bernoulli Beam Teory in te Presence of Fiber Bending
More informationBENDING VIBRATIONS OF ROTATING NON-UNIFORM COMPOSITE TIMOSHENKO BEAMS WITH AN ELASTICALLY RESTRAINED ROOT
BENDING VIBRATIONS OF ROTATING NON-UNIFORM COMPOSITE TIMOSHENKO BEAMS WITH AN EASTICAY RESTRAINED ROOT Sen Yung ee and Jin Tang Yang Department of Mechanical Engineering, National Cheng Kung University,
More informationChapter 5 FINITE DIFFERENCE METHOD (FDM)
MEE7 Computer Modeling Tecniques in Engineering Capter 5 FINITE DIFFERENCE METHOD (FDM) 5. Introduction to FDM Te finite difference tecniques are based upon approximations wic permit replacing differential
More informationMIMO decorrelation for visible light communication based on angle optimization
MIMO decorrelation for visible ligt communication based on angle optimization Haiyong Zang, and Yijun Zu Citation: AIP Conference Proceedings 80, 09005 (07); View online: ttps://doi.org/0.03/.4977399 View
More information