ORIGINAL_ARTICLE
Experimental and Numerical Study of Preform Design in Multi Stage Deep Drawing of High Strength Thin Steel Sheet
In this paper, experimental results of a deep drawing process to produce a cylinder of high strength steel with a spherical head were compared with it’s simulation results and three proposal design types. Meanwhile, the amount of limiting draw ratio in some stages was determined. Accuracy and precision of the results of a finite element software to predicting the multi stage deep drawing process of high-strength thin steel sheets was measured. ABAQUS version 6-9-3 was used to simulate the process.
In this research, the raw material was a circular blank of annealed steel AISI-4130 with 2 mm thickness that in experimental test, subjected to one drawing stages, three redrawing stage and two heat treatment stages. The uni-axial tensile test was performed to determine the mechanical properties of the steel sheet.
Numerical thickness distribution was compared with the experimental results in different stages of deep drawing and the accuracy was good (about 2.55%). Base on this results the proposal designs were simulated to introduce the properties of the most suitable design types.
http://jsme.iaukhsh.ac.ir/article_515281_0d9ac6b045d797443f1784bfd8b304ac.pdf
2012-12-21
1
16
simulation
Limiting Draw Ratio
Finite elements method
Deep drawing process
Steel AISI-4130
A.
Zamani Alishah
1
MSc Student, Mechanical Engineering, Research & Science Azad University, Hashtgerd, Iran
AUTHOR
M.
Tajdari
tajdari@yahoo.com
2
Professor, Mechanical Engineering, Research & Science Azad University, Kermanshah, Iran
LEAD_AUTHOR
J.
Eskandari Jam
3
- Associate Professor, Mechanical Engineering, Research & Science Azad University, Kermanshah, Iran
AUTHOR
J.
Seydi
4
Assistant Professor, Mechanical Engineering, Research & Science Azad University,Ilam, Iran
AUTHOR
[1] طالبی خورزوقی، احمد، طراحی و ساخت ابزار و قالب فـرآیند کششعمیق به کمک رایانه، پایاننامه کارشناسی ارشد، دانشگاه تربیت مدرس، 1381، صفحه 2.
1
[2] بابایی، بهروز، طـراحی پریفـرم بین مراحل در فـرآیند کشش عمیق قطعـات متقارن، پایاننامه کارشناسی ارشد، دانشگاه تربیت دبیر شهید رجائی، 1389، صفحات 11 تا 59.
2
[3] Moaveni S., Numerical experiments for a Mechanics of Materials course, International Journal Engineering, Vol. 14, No 2, 1998, pp. 122-129.
3
[4] Altan T., Khamitkar S., Kinzel G. L. , and Esche S. K., Process and die design for multi-step forming of round parts from sheet metal, Journal of Materials Processing Technology Vol. 59, 1996, pp. 24-33.
4
[5] Tisza M., Integration of numerical modelling and knowledge based systems in metal forming, 6th ICTP Conference, Nürnberg, 1999, pp.117-128.
5
[6] Choi T.H. , Choi S., Na K.H. , Bae H.S. , Chung W.J., Application of intelligent design support system for multi-step deep drawing process, Journal of Materials Processing Technology, Vol. 130–131, 2002, pp. 76–88.
6
[7] Sheng Z. Q. , Taylor R. Strazzanti M., FEM-based progressive drawing process design, Journal of Advanced Manufacturing Technology, Vol. 36, 2008, pp. 226–236.
7
[8] For bar products; plate, sheet and tubing may be slightly different, Central Steel & Wire Company Catalog, 2006-2008 ed, pp. 246.
8
[9] http://www.efunda.com.
9
[10] Schuler., Metal Forming Handbook, Springer-Verlag Berlin Heidelberg, 1998.
10
[11] Ivana Suchy., Handbook of die design, 2nd Edition, McGraw Hill, 2006.
11
[12] “ABAQUS/Standard example: forming a channel, ABAQUS, Getting Started with ABAQUS”, Version 6.8, 12.5.
12
[13] Huang Y-M., Tsai Y-W., Li C-L., Analysis of forming limits in metal forming processes, Journal of Materials Processing Technology, Vol. 201, 2008, pp. 385–389.
13
ORIGINAL_ARTICLE
Modeling and Sensitivity Analysis of Full Toroidal Continuously Variable Transmission
http://jsme.iaukhsh.ac.ir/article_515291_f888bc5a970213c730e816e43741ea2e.pdf
2012-12-21
17
26
Power transmission system
Continuously variable transmission
Full toroidal
Optimization
Sensitivity analyzing
M.
Delkhosh
1
PhD student of mechanical engineering, Sharif University of Technology, Tehran
AUTHOR
M.
Saadat Foumani
m_saadat@sharif.ir
2
Assistant professor of mechanical engineering, Sharif University of Technology, Tehran.
LEAD_AUTHOR
ORIGINAL_ARTICLE
Bending Analysis of Carbon Nanotubes with Small Initial Curvature Embedded on an Elastic Medium Based on Nonlocal Elasticity and Galerkin Method
Carbon nanotubes have an important role in reinforcing nanocomposits. Many experimental observations have shown that in the most nanostructures such as nanocomposites, carbon nanotubes (CNTs) are often characterized by a certain degree of waviness along their axial direction. In the present paper, the effects of initial curvature, influence of surrounding medium that is modeled as Winkler elastic foundation on behavior of slightly curved carbon nanotubes are investigated. To capture the small size effects, nonlocal elasticity theory is implemented. Bending governing equations are derived using the principle of minimum total potential energy and these nonlinear equations are solved by Newton Raphson method. It is shown that the larger the initial curvature, the higher deflection can occur. Furthermore, neglecting the effect of initial curvature of CNTs can lead to incorrect results.
http://jsme.iaukhsh.ac.ir/article_515297_735e3adc4621a5119382b458b62eb44c.pdf
2012-12-21
27
36
Bending Analysis
Carbon Nanotube with Initial Curvature
Nonlocal elasticity theory
Newton Raphson Method
Galerkin Method
A.
Arefi
1
MSc Student, Department of Mechanical Engineering, Isfahan University of Technology
AUTHOR
M.
Salimi
salimi@cc.iut.ac.ir
2
Professor, Department of Mechanical Engineering, Isfahan University of Technology
LEAD_AUTHOR
[1] Iijima S., “Helical microtubules of graphitic carbon”, Nature, 354, 1991, pp. 56-58.
1
[2] Thostenson E.T., Ren, Z., Chou, T.W., “Advances in the science and technology of carbon nanotubes and their composites: a review”, Composites Science and Technology, Vol. 61, 2001, pp. 1899–1912.
2
[3] Zhou S.J., Li Z.Q., “Length scales in the static and dynamic torsion of a circular cylindrical micro-bar”, Shandong University Technology, Vol. 31, 2001, pp. 401–407.
3
[4] Fleck N.A., Hutchinson J.W., “Strain gradient plasticity: theory and experiment”, Acta Metal Material, Vol. 42, 1994, pp. 475-487.
4
[5] Yang A.C.M., Chong D.C.C., Lam P., “Couple stress based strain gradient theory for elasticity”, Solids Structure, Vol. 39, 2002, pp. 2731–2743.
5
[6] Eringen A.C., “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves”, Journal of Applied Physics, Vol. 54, 1983, pp. 4703–4710,.
6
[7] Lu P., Lee H.P., Lu C., Zhang P.Q., “Dynamic properties of flexural beams using a nonlocal elasticity model”, Journal of Applied Physics, Vol. 99, 2006, No. 073510.
7
[8] Peddiseon P., Buchanan J.R., McNitt R.P., “Application of nonlocal continuum models to nanotechnology”, International Journal of Engineering Science, Vol. 41, 2003, pp. 305-312.
8
[9] Lu P., Lee H.P., Lu C., Zhang P.Q., “Application of nonlocal beam models for carbon nanotubes”, International Journal of Solids Structure, Vol. 44, 2007, pp. 5289-5300.
9
[10] Reddy J.N., “Nonlocal theories for bending, buckling and vibration of beams”, International Journal of Engineering Science, Vol. 45, 2007, pp. 288-307.
10
[11] Aydogdu M., “A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration”, Physica E, Vol. 41, 2009, pp. 1651-1655.
11
[12] Lee H.L., Chang W.J., “Surface and small scale effects on vibration analysis of a nonuniform nanocantilever beam”, Physica E, Vol. 43, 2010, pp. 466-469.
12
[13] Simsek M., “Forced vibration of an embedded single-walled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory”, Steel & Composite Structures, Vol. 11, 2011, pp. 59-76.
13
[14] Zhang Y.Q., Liu G.R., Han X., “Effect of Small Length Scale on Elastic Buckling of Multi-Walled Carbon Nanotubes Under Radial Pressure”, Physics Letters A, Vol. 69, 2006, pp. 370-376.
14
[15] Murmu T., Pradhan S.C., “Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM”, Physica E, Vol. 41, 2009, pp. 1232–1239.
15
[16] khademolhosseini F., “Application of Nonlocal continuum shell models for torsion of single-walled carbon nanotubes”, Department of Mechanical Engineering, Sharif University of Technology, 2009.
16
[17] Senthilkumar V., “Buckling analysis of a single-walled carbon nanotube with nonlocal continuum elasticity by using differential transform method”, Advanced science letter, Vol. 3, 2010, pp. 337-340.
17
[18] Wang Yi-Ze., Li F.M., Kishimoto K., “Scale effects on thermal buckling properties of carbon nanotube”, Physics Letters A, Vol. 374, 2010, pp. 4890-4893.
18
[19] Civalek O., Demir C., “Buckling and bending analysis of cantilever carbon nanotubes using the Euler-Bernoulli beam theory based on nonlocal continuum model”, Asian journal of Civil Engineering, Vol. 12, 2011, pp. 651-661.
19
[20] Wang C.M., Duan W.H., “Free vibration of nanorings/arches based on nonlocal elasticity”, Journal of Applied Physics, Vol. 104, 2008, 014303.
20
[21] Tepe A., “Nano-scale analysis of curved single walled carbon nanotubes for in-plane loading”, Journal of Computational and Theoretical Nanoscience, Vol. 7,2010, pp. 2405-2411.
21
[22] Duan W.H., Wang C.M., “Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory”, Nanotechnology, Vol. 18, 2007, No. 385704.
22
[23] Aghababaei R., Reddy J.N., “Nonlcal third order shear deformation plate theory with application to bending and vibration of plates”, Journal of Sound and Vibration, Vol. 326, 2009, pp. 227-289.
23
[24] Babaei H., Shahidi A.R., “Small scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method”, Archive Applied Mechanics, Vol. 81, 2010, pp.1051-1062.
24
[25] Ansari R., “Nonlocal finite element model for vibration of embedded multi layered graphene sheets”, Computational MaterialsScience, Vol. 49, 2010, pp. 831–838.
25
[26] Eringen A.C., “Nonlocal Continuum Field Theories”, Springer, NewYork, 2002.
26
[27] Zhang Y.Q., Liu G.R., Xie X.Y., “Free transverse vibrations of double walled carbon nanotubes using a theory of nonlocal elasticity”, Physics Review B, Vol. 71(19), 2005, No. 195404.
27
[28] Wang Q., “Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum model”, Journal of Applied Physics, Vol. 98, 2005, No. 124301.
28
[29] Phadikar J.K., Pradhan S.C., “Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates”, Computational materials science, Vol. 49, 2010, pp. 492-499.
29
[30] Fung Y.C., Kaplan A., “Buckling of low arches or curved beams of small curvature”, NACA TN 2840,1952.
30
[31] Fisher F.T., Bradshaw R.D., Brinson, L.C., “Fiber waviness in nanotube-reinforced polymer composites-I: modulus prediction using effective nanotube properties”, Composite Science Technology, Vol. 63, 2003, pp. 1689-2391.
31
[32] Qian D., Dickey E.C., Andrews, R., Rantell, T., “Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites”, Applied physics Letter, Vol. 76(20), 2003, pp. 2868-2938.
32
[33] Bradshaw R.D., Fisher F.T., Brinson L.C., “Fiber waviness in nanotube-reinforced polymer composites—II: modeling via numerical approximation of the dilute strain concentration tensor”, Composite Science Technology, Vol. 63, 2003, pp. 1705-1727.
33
[34] Aydogdu M., “A general nonlocal beam theory: its application to nanobeam bending”, buckling and vibration,Physica E, Vol. 41, 2009, pp. 1651-1655.
34
ORIGINAL_ARTICLE
Study of the Effects of Lacing Rods Location on Natural Frequencies in Last Stage Blades of a Steam Turbine
In this study, effects of lacing rods location on natural frequencies of last stage of a Steam Turbine Blades (STBs) have been investigated. STBs are critical and important elements in a turbine as well as power plants. One of the most important parameters in turbine blades is lacing rods location. In this paper, first, a three dimensional (3D) scan was used to produce the geometrical model of the blade. Then, last stage of a low pressure turbine have been produced and simulated by assembling of the one-blade model. There are two rows of lacing rods in the last stage of turbine. In this paper, effecting the lacing rods location on natural frequencies of system has been studied. According to the Campbell diagrams, the results showed there is no resonance in the last stage blades of the steam turbine for different locations of the lacing rods.
http://jsme.iaukhsh.ac.ir/article_515328_d66fffe8b8271a9e916c5b4dd814d81f.pdf
2012-12-21
37
46
Lacing rod
Turbine blades
Finite Element Method
resonance
M.
Nozarpour
1
- MSc Student, School of Mechanical Engineering, Islamic Azad University, Ahvaz Branch, Ahvaz, Iran
AUTHOR
A.
Rahi
a_rahi@sbu.ac.ir
2
Assistant Professor, School of Mechanical Engineering, Abbaspour College of Technology, Shahid Beheshti University, Tehran, Iran
LEAD_AUTHOR
[1] Stodola A., Steam and Gas Turbines, Vol. 1 and 2, McGraw-Hill, New York, 1927.
1
[2] Lamb H., Southwell R.V., The Vibration of a Spinning Disc, Process Royal Society of London, Vol. 99, pp. 272, 1922.
2
[3] Kroon R., Turbine Blade Vibration Due to Partial Admission, Transaction of ASME, InternationalJournal of Applied Mechanic, Vol. 7, pp. 161-165, 1940.
3
[4] Judge J., Pierre C., Mehmed O., Experimental Investigation of Mode Localization and Forced Response Amplitude Magnification for a Mistuned Bladed Disc, Journal of Engineering for GasTurbines and Power, Vol. 123, No. 4, pp. 940- 950, 2001.
4
[5] Periera J.C., Torres L.A.M., Rosa E., A Low Cycle Fatigue Analysis on a Steam Turbine Bladed Disk-case Study, 12th IFToMM World Congress, Besancon, Brazil, June 18-21, 2007.
5
[6] Moffatt S., He L., Blade Forced Response Prediction for Industrial Gas Turbines, International Gas Turbine & Aeroengine Congress & Exhibition, June 16-19, Atlanta, Georgia, USA, 2003.
6
[7] Christophe P., Jiang D., Finite-element-based Modal Reduction of a Rotating Blade With Large-amplitude-motion using Nonlinear Normal Modes, Department of Mechanical Engineering and Applied Mechanics, University of Michigan, 1997.
7
[8] Rao J.S., Turbomachine Blade Vibration, New age international publishers, 1987.
8
[9] وهابی، ح.، طراحی مکانیزمهای صنعتی با استفاده از روش المان محدود در نرمافزار ANSYS، چاپ اول، تهران، انتشارات اندیشهسرا، 1390.
9
]10] بهزاد، م.، حسینی، س.م.ر.، ابراهیمی، ع.ر.، تحلیل ارتعاشات آزاد پره توربینهای گازی به منظور جلوگیری از خستگی دور بالا، سومین کنفرانس ملی نگهداری و تعمیرات، تهران، 1384.
10
]11] فتحی، م.، تاثیر ارتعاشات در شکست پره های توربین بخار، اولین کنفرانس ملی شبیه سازی سیستمهای مکانیکی، اهواز، دانشگاه آزاد اسلامی،1390.
11
]12] آنتریسر، پ.م و دیگران، راهنمای کاربران عملیات حرارتی، ربیعی، بهناز و دیگران، چاپ اول، تهران، انتشارات جهان نو، 1379.
12
[13] Ansys Help, Release 13, Mechanical APDL, Advance Analysis Techniques Guidence, Cyclic Symmetry Analysis.
13
[14] Tsai G.C., Rotating Vibration Behavior of the Turbine Blades with Different Groups of Blades‚ Journal of Sound and Vibration, Vol. 271, 2004, pp. 547-575.
14
ORIGINAL_ARTICLE
Prediction of Mode II of Fracture Toughness in Laminate Composites
In this paper, effects of ply orientation of adjacent plies with (ϕ//θ) interfaces on mode II critical strain energy release rate (fracture toughness) of multidirectional (MD) laminate has been studied. Ply orientation of adjacent plies is one of the most important parameters affects the mode II critical strain energy release rate () in the initiation of the delamination. To study this parameter, End Notch Flexure (ENF) specimen has been used for measuring of laminated composites. Eventually, the purpose is to predict of of MD composite specimen, without direct experimental tests and finite element modeling using the results of unidirectional (UD) ply. First, of unidirectional composites will be studied and by the results obtained, the behavior of multidirectional laminated composites is predicted. In this context, a comprehensive method was proposed that combines prediction methods, and analytical modeling. The obtainedof multidirectional laminated composites with (ϕ//θ) interfaces can be used for design purposes. Results obtained using this method has been compared with the results of numerical and theoretical methods. This prediction method reduces the calculation costs of FE and analytical models, and also the costs of experiments significantly.
http://jsme.iaukhsh.ac.ir/article_515332_526826ba5537a90ea4c6c06bf5544221.pdf
2012-12-21
47
60
Composite
Initiation of delamination
Prediction
Mode II
strain energy release rate
A.
Zeinedini
zeinedini@iust.ac.ir
1
PhD Student, Mechanical Engineering, School of Mechanical Engineering, Iran University of Science and Technology
LEAD_AUTHOR
M.
Alizadeh
2
Assistance Professor, Mechanical Engineering, School of Mechanical Engineering, Iran University of Science and Technology
AUTHOR
[1] ASTM D5528. Standard test method for mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites, Annual book of ASTM standards, Vol. 15, 2007, pp. 1-12.
1
[2] Sriharan S., Delamination Behavior of composite, Published by Woodhead Publishing and Maney Publishing on behalf of The Institute of materials, Mainerals & Mining, CRC Press Boca Raton Boston New York Washington, (2008).
2
[3] Sheinman I., Kardomateas G.A., Energy release rate and stress intensity factors for delaminated composite laminates, International Journal Solids Structure, Vol. 34(4), 1997, pp. 451–9.
3
[4] Sela N., Ishai O., Interlaminar fracture toughness and toughening of laminated composite materials, a review Composites, Vol. 20(5), 1989, pp. 416.
4
[5] Barrett J.D., Foschi R.O., Mode II stress intensity factors for cracked wood beams, Engineering Fracture Mechanism, Vol. 9(3), 1977, pp. 371–387.
5
[6] O’Brien T.K., Characterization of delamination onset and growth in a composite laminate, .In: Reifsnider KL, editor. Damage in composite materials, American Society for Testing and Materials, ASTM STP, Vol. 775, 1982, pp. 140–167.
6
[7] Davies P., Casari P., Carlsson LA., Influence of fibre volume fraction on mode II interlaminar fracture toughness of glass/epoxy using the 4ENF specimen, Composite Science Technology, Vol. 65, 2005, pp. 295–300.
7
[8] Arrese A., Carbajal N, Vargas G., Mujika F., A new method for determining mode II R-curve by the End-Notched Flexure test, Engineering Fracture Mechanism, Vol. 77, 2010, pp. 51–70.
8
[9] Brunner AJ., Blackman BRK., Davies P., An status report on delamination resistance testing of polymer–matrix composites, Engineering Fracture Mechanism, Vol. 75, 2008, pp. 2779–2794.
9
[10] Blackman BRK., Kinloch AJ., Paraschi M., The determination of the mode II fracture resistance, GIIc, of structural adhesive joints: an effective crack length approach, Engineering Fracture Mechanism, Vol. 72, 2005, pp. 877–897.
10
[11] Miyagawa H., Chiaki S., Ikegami, K., Experimental Determination of Fracture Toughness of CFRP in Mode II by Raman Spectroscopy, Applied Composite Materials, Vol. 8, 2001, pp. 25–41.
11
[12] Jar P.Y.B., Dick T.M., Kuboki T., Comparison of testing methods for fibre-reinforced polymers (FRP) in resistance to in-plane sliding mode of delamination (Mode II), Journal Material Science, Vol. 40, 2005, pp. 1481–1484.
12
[13] Gallagher E., Kuboki T., Jar P.Y.B., Cheng J.J.R., in Proceedings CD of ANTEC, Society of Plastics Engineers, 2004.
13
[14] Gdoutos E.E., Pilakoutas K., Chris A., Rodopoulos., Failure Analysis of Industrial Composite Materials, McGraw-Hill Professional, 2000, pp. 553.
14
[15] Tsai S.W., Introduction to Composite Materials, Technomic Publishing Company, 1980.
15
[16] Davidson B.D., Kruger R., Konig M., Effect of stacking sequence on energy release rate distributions in multidirectional DCB and ENF specimens, Engineering Fracture Mechanism, Vol. 55, 1996, pp. 557–569.
16
[17] Sun C.T., Zheng S., Delamination characteristics of double-cantilever beam and end-notched flexure composite specimens, Composite Science and Technology, Vol. 56(4), 1996, pp. 451–459.
17
[18] Shokrieh M.M., Heidari-Rarani M., Ayatollahi M.R., Delamination R-curve as a material property of unidirectional glass/epoxy composites, Materials and Design, 2012.
18
[19] Chang. F.K., Chang. K.Y., A Progressive Damage Model for Laminated Composites Containing Stress Concentrations, Journal Composite Material, Vol. 21, 1987, pp. 834-855.
19
[20] Olsson R.A., “A simplified improved beam analysis of the DCB specimen”, Composites Science and Technology, Vol. 43, 1992, pp. 329-338.
20
[21] Rybicki E.F., Kanninen M.F., A Finite Element Calculation of Stress Intensity Factors by a Modified Crack Closure Integral, Engineering Fracture Mechanics, Vol. 9, 1997, pp. 931-938.
21
[22] Krueger R., Goetze D., Influence of Finite Element Software on Energy Release Rates Computed Using the Virtual Crack Closure Technique: History, Approach and Applications, NASA/CR-2006-214523.
22
[23] De Morais AB., Pereira AB., Application of the effective crack method to mode I and mode II interlaminar fracture of carbon/epoxy unidirectional laminates, Composites Part A Vol. 38, 2007, pp. 785–794.
23
ORIGINAL_ARTICLE
Nonlinear Vibration Analysis of an Euler-Bernoulli Beam Resting on a Nonlinear Elastic Foundation under Compressive Axial Force
This paper studies the nonlinear vibration analysis of a simply supported Euler-Bernoulli beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes concept in the case of three-to-one (3:1) internal resonance. The beam’s governing nonlinear PDE of motion and also its boundary conditions are derived and then solved using the method of Multiple Time Scales. Under three to-one-internal resonance condition applying nonlinear normal modes the steady state stability analysis of the beam’s vibrations is performed. Then the effect of changing the value of different parameters on the beam’s dynamic response and the steady state stability analysis is investigated.
http://jsme.iaukhsh.ac.ir/article_515336_47b02051df423c6405ebec28df3d0adb.pdf
2012-12-21
61
72
Nonlinear vibrations
Euler-Bernoulli beam
Nonlinear elastic foundation
The 3:1
A.
Mamandi
a.mamandi@piau.ac.ir
1
Assistant Professor, Mechanical and Aerospace Engineering, Department of Mechanical Engineering, Parand Branch, Islamic Azad University, Tehran, Iran.
LEAD_AUTHOR
M.H.
Kargarnovin
2
Professor, Mechanical Engineering, Department of Mechanical Engineering, Sharif university of Technology, Tehran, Iran
AUTHOR
[1] Nayfeh A.H., Lacarbonara W., Chin C.-M., Nonlinear normal modes of buckled beams: three-to-one and one-to-one internal resonances, Nonlinear Dynamics, Vol. 18, 1999, pp., 253-273.
1
[2] Santee D.M., Goncalves P.B., Oscillations of a beam on a non-linear elastic foundation under periodic loads, Shock and Vibration, Vol. 13, 2006, pp. 273-284.
2
[3] Tsiatas G.C., Nonlinear analysis of non-uniform beams on nonlinear elastic foundation, Acta Mechanical, Vol. 209, 2010, pp. 141-152.
3
[4] Kuo Y.H., Lee S.Y., Deflection of nonuniform beams resting on a nonlinear elastic foundation, Computers and Structures, Vol. 51, 1994, pp. 513-519.
4
[5] Hsu M.H., Mechanical analysis of non-uniform beams resting on nonlinear elastic foundation by the differential quadrature method, Structural Engineering and Mechanics, Vol. 22, 2006, pp. 279-292.
5
[6] Oz H.R., Pakdemirli M., Ozkaya E., Yilmaz M., Non-linear vibrations of a slightly curved beam resting on a non-linear elastic foundation, Journal of Sound and Vibration, Vol. 212, 1998, pp. 295-309.
6
[7] Pellicano F., Mastroddi F., Nonlinear dynamics of a beam on elastic foundation, Nonlinear Dynamics, Vol. 14, 1997, pp. 335-355.
7
[8] Balkaya M., Kaya M.O., Saglamer A., Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method, Achieve of Applied Mechanics, Vol. 79, 2009, pp. 135-146.
8
[9] Birman V., On the Effects of nonlinear elastic foundation on free vibration of beams, Journal of applied Mechanics, Vol. 53, 1986, pp. 471-474.
9
[10] King M.E., Vakakis A.F., An energy-based approach to computing resonant nonlinear normal modes, Journal of Applied Mechanics, Vol. 63, 1996, pp. 810-819.
10
[11] King M.E., Vakakis A.F., An energy-based formulation for computing nonlinear normal modes in undamped continuous systems, Journal of Vibration and Acoustics, Vol. 116, 1994, 332-340.
11
[12] Vakakis A.F., Nonlinear mode localization in systems governed by partial differential equations, Applied Mechanics Review, Vol. 49, 1996, pp. 87-99.
12
[13] Pellicano F., Vakakis A.F., Normal modes and boundary layers for a slender tensioned beam on a nonlinear foundation, Nonlinear Dynamics, Vol. 25, 2001, pp.79-93.
13
[14] Jiang D., Pierre C., Shaw S.W., The construction of non-linear normal modes for systems with internal resonance, International Journal of Non-Linear Mechanics, Vol. 40, 2005, pp. 729-746.
14
[15] Mazzilli C.E.N., Sanches C.T., Baracho O.G.P., Wiercigroch M., Keber M., Non-linear modal analysis for beams subjected to axial loads: analytical and finite-element solutions, International Journal of Non-Linear Mechanics, Vol. 43(6), 2008, pp. 551-561.
15
[16] Nayfeh A.H., Mook D.T., Nonlinear oscillations, Wiley-Interscience, New York, 1995.
16