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1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 10 NON LINEAR DYNAMIC AND STABILITY ANALYSIS OF BEAM USING FINITE ELEMENT IN TIME Prabhat Kumar Sinha1 , Ishan Om Bhargava2 , Saifuldeen Abed Jebur3 Department of Mechanical Engineering (SSET) Sam Higginbottom Institute of Agriculture Technology and Sciences, Allahabad ABSTRACT In this article the main focus is to predict nonlinear dynamic response of a beam using finite element in time under given condition. To do the nonlinear dynamic analysis of a beam, a distributed load is being applied and the beam is experiencing bending. The given beam is homogeneous in composition and isotropic in nature. Here, considering the stiffness of the beam and its effect on the deflection, under the distributed load conditions. The use of this method is in determining the variation of the beam under the given load and the corresponding load conditions. The result of this entire analysis will be appropriate and this will facilitate the complete knowledge about the nonlinear dynamic analysis. INTROUCTION The need of accurate prediction of nonlinear dynamic response of beam, isotropic in nature subjected to distributed loads is carried on through graphical analysis of various parameters applicable to the beam. The load applied is uniformly distributed, has changing point of application. The case discussed here is about transverse deflection and its effect on the dynamic response. Many researches have been conducted using different techniques taking into consideration the finite element method and its application to the solution of the problem acceptable results have been obtained. This is achieved by using Finite Element in Time, though the importance of the paper is unanimous and not under any discussion even though we have to consider some studies. Chen [1] have studied the instability behavior of beams with variable cross section subjected to sub-tangential non conservative follower forces, and their solution is numerically attained by using a Runge-Kutta method based on the above work . The dynamic stability of buckled beam considering the snap through motion under sinusoidal loading has been investigated by Poon et al [2]. Non-linear steady state response of beams, frames and shallow arches has been analyzed in the frequency domain using the h-version straight beam finite element by Chen et al [3]. The geometrically non-linear thermo elastic vibration analysis of straight and curved beams has been carried out using p-version INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 5, Issue 3, March (2014), pp. 10-19 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2014): 7.5377 (Calculated by GISI) www.jifactor.com IJMET © I A E M E
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 11 hierarchical finite elements by Ribeiro and Manoach [4]. The influence of temperature variation and curvature on the non-linear dynamics of curved beam has been studied. Periodic as well as a periodic motions were observed. The vibration of a curved beam in anti-symmetric mode, due to auto- parametric excitation, has been studied by Lee et al [5]. Chen and Yang [6] have studied, both theoretically and experimentally, the non- linear vibration of shallow arch under the harmonic excitation at one end. Based on the above formulation Mata et al.[7] have developed fully geometric and constitutive non-linear model for the dynamic behavior of beam structures considering an inter- mediate curved reference configuration. The model has been used to study the transient response of straight beams, frames and curved beams and [8, 9]. The solution of the problems was achieved using the analog equation method of Katsikadelis [10] as it was developed for the non-linear dynamic analysis of beams. It may be noted that the solution of banded system of equations is computationally more efficient. It can also be noted from the literature that the finite element based system of equations for beams/plates/shells are solved by the traditional shooting method in which the second order governing equations of motion are transformed to first order equations. This results in doubling the number of equations and the banded nature of the system of equations is destroyed. In this article, this issues has taken care of by directly applying the shooting method to the second order governing equations [10,11] . In spite of the importance of non-linear dynamic analysis of curved beams, non-linear forced response studies are relatively few. Sheinmann [11] has studied the dynamic buckling behavior of shallow and deep circular arches. The solution methodology used could not capture the converged solutions adequately for the deep arches.[12,14] .Vast literatures for the dynamics of axially moving continua have been comprehensively reviewed by Wickert and Mote, Abrate and Chen [15] The system of semi-discrete dynamic equations of motion is derived from the modified Hamilton principle in which only the strain variables are interpolated. Such a choice of the interpolated variables is an advantage over approaches, in which the displacements and rotations are interpolated, since the field consistency problem and related locking phenomena do not arise. Finite element dynamic analysis of geometrically exact planar beams has been done by Gams et al.[17] A new finite-element-based approach along with an iterative incremental method is developed to study the dynamic response of sandwich beam with hybrid composite face sheets and flexible core by Shariya and others. As a point of fact the literature suggest that a few response analysis have been carried out on assuming the solution which is a priori. The finite element in time is applied to carry the distributed load analysis. It has been observed that properly applied transient application of finite element method has been carried on partially fixed or supported beams under distributed load conditions. The further interpretation of various literatures suggest effectiveness of Finite Element Method in analyzing partially partially fixed or supported beams. Though solving a large number of problems is cumbersome this issue can be sorted out by using a small bandwidth of equations resulting in smaller matrix formation in large dynamical systems. In this paper repetition of equations is solution of the problem discussed Finite Element in Time methodology has been utilized. Its efficacy has been demonstrated for the non linear dynamic analysis of isotropic and composite curved beams. POTENTIAL ENERGY APPROACH Beams with cross sections that are symmetric with respect to plane of loading are considered here. A general horizontal beam has been shown with the cross section and the bending stress distribution. For small deflections,
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 12 L P Pm m Mk k x 0 (a) Y LOADED BEAM x v v' v' v (b) DEFORMRED NEUTRAL AXIS ,v is the deflection of the centroidal axis at x axis, and I is the moment of inertia of the section about the neutral axis (z- axis passing through the centroid).
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 13 M Ne utra l a xi s y x V y Centroid z dA y y SECTIONS OF THE BEAM AND ITS STRESS DISTRIBUTION FINITE ELEMNT ANALYSIS The strain energy in an element of length dx is = The total strain energy in the beam is given by: U =
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 14 The potential energy of the beam is: = Where p is the distributed load per unit length, is the point load at point m, is the moment of the couple applied at point k, is the deflection at point m, and is the slope at point k 1 2 3 4 51 2 3 4 Q1 Q3 Q5 Q7 Q9 Q2 Q4 Q6 Q8 Q1 0 Q2 i– 1 Q2 i DISCRETIZED BEAM ELEMENT 1 2 e 1 v1 v'1 v2 v'2 q1 q2 q3 q4 AN ELEMENT BEING ANALYZED THE FINITE ELEMENT FORMULATION The beam is divided into four elements. Each node has two degree of freedom. Typically the degrees of freedom of node i are . The degree of freedom is the transverse displacement and is slope or rotation. Q =
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 15 The vector represents the global displacement vector. For a single element, the local degrees of freedom are represented by The local and global displacement are in correspondence with each other. The local vector q is same as . The shape functions for interpolating v on an element are defined in terms of on -1 to +1. The shape functions for beam elements here are different as we are using a third degree polynomial. Since nodal values and nodal slopes are involved, Hermite shape functions have been taken, which satisfy both the nodal value and the slope continuity. Each of the shape functions is of a cubic order represented by: Slope = 0 1 = –1 = 0 = +1 Slope = 0 H1 1 –1 0 +1 Slope = 0 H3 1 1 2 Slope = 0 HERMITE SHAPE FUNCTIONS –1 0 +1 Slope = 0 H2 1 Slope = 1 –1 0 +1 H4 1 Slope = 0 Slope = 1 2 NON LINEAR HERMITE SHAPE FUCTIONS
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 16 The conditions given in the following table are satisfied: 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 = -1 = 1 The coefficient and can be easily obtained by imposing those conditions on these The Hermite shape functions can be used to write v in form Hence the coordinate transform is given by the relationship As The chain rule Noting that evaluated at nodes 1 and 2 are , respectively, thus
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 17 This is denoted as Where In the total potential energy of the system, the integrals are considered as summations over the integrals over the elements. The element strain energy has been given by: Then, substituting v = Hq following equation is obtained On substituting we get Each term in the matrix has been integrated. Knowing that This result in the form of element strain energy is given by
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 18 Where the element stiffness matrix as From the matrix the following has been observed Where which is the set of generalized virtual displacement on the element v = Hq and RESULT In this paper finite element in time develop as a analyzing tool for the supported beam element which has been discretized into finite elements and this element presents a dynamic model for an isotropic beam considering the bending of beam when the load has been applied .The formulation of the dynamic model has been carried and analyzed by taking into consideration the potential energy approach and the relevant energy formulations for the stress analysis of the beam which has been described and validated through the finite element in time. The formulation has been done using the Hermite shape functions which result in the most approximate stiffness matrix of the beam considering the degree of freedom and their orientation with respect to the differential equations of deflection. These equations have been considered over the boundary conditions and taken into consideration by the Hermite third degree polynomial. Though the curve so obtained are nonlinear they result in an augmented stiffness matrix. During the initial modeling and experimental work, emphasis has been made on dynamic analysis of beam and has been improved by increasing the nodal points and by analyzing the degree of freedom. REFRENCE [1] Chen, Y.Z., “Interaction between compressive force and vibration frequency for a varyingcross-section cantilever under action on generalized follower force,” Journal of Sound andVibration, 259, 991-999, (2003). [2] W.Y. Poon, C.F. Ng, Y.Y. Lee, Dynamic stability of a curved beam under sinusoidal loading, Proceedings of Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 216 (2002) 209–217. [3] S.H. Chen, Y.K. Cheung, H.X. Xing, Nonlinear vibration of plane structures by finite element and incremental harmonic balance method, Nonlinear Dynamics 26 (2001) 87–104. [4] P. Ribeiro, E. Manoach, The effect of temperature on the large amplitude vibrations of curved beams, Journal of Sound and Vibration 285 (2005) 1093–1107. [5] Y.Y. Lee, W.Y. Poon, C.F. Ng, Anti-symmetric mode vibration of a curved beam subject to auto parametric excitation, Journal of Sound and Vibration 290 (2006) 48–64.
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 10-19, © IAEME 19 [6] J.S. Chen, C.H. Yang, Experiment and theory on the nonlinear vibration of a shallow arch under harmonic excitation at the end, ASME Journal of Applied Mechanics 74 (2007) 1061– 1070. [7] P. Mata, S. Oller, A.H. Barbat, Dynamic analysis of beam structures considering geometric and constitutive nonlinearity, Computer Methods in Applied Mechanics and Engineering 197 (2008) 857–878. [8] P. Ribeiro, Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methods, Computers and Structures 82 (2004) 1413–1423. [9] P. Ribeiro, Forced large amplitude periodic vibrations of cylindrical shallow shells, Finite Elements in Analysis and Design 44 (2008) 657–674 . [10] J.T. Katsikadelis, The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies, Appl. Mech. 27 (2002) 13–38. [11] I. Sheinman, Dynamic large displacement analysis of curved beams involving shear deformation, International Journal of Solids and Structures 16 (1980) 1037–1049. [12] U.Lee, H.Oh, Dynamics of an axially moving viscoelastic beam subject to axial tension, InternationalJournalofSolidsandStructures42 (2005)2381–2398. [13] U.Lee, J.Kim, H.Oh, Spectral analysis for the lateral vibration of an axially moving Timoshenkobeam, JournalofSoundandVibration271 (2004) 685–703. [14] F.Fung,H.C.Chang, Dynamic and energetic analyses of a string/slidernon- linear coupling system by variable grid finite difference, Journal of Sound and Vibration 239(3)(2001) 505– 514. [15] L.Q. Chen, D.Hu, Natural frequencies of nonlinear vibration of axially moving beams, Nonlinear Dynamics 62(2011)125–134. [16] Finite element dynamic analysis of geometrically exact planar beams Original Research Article Computers & Structures, Volume 85, Issues 17–18, September 2007, Pages 1409- 1419M. Gams, M. Saje, S. Srpčič. [17] Non-linear dynamic analysis of a sandwich beam with pseudoelastic SMA hybrid composite faces based on higher order finite element theory Original Research Article Composite Structures, Volume 96, February 2013, Pages 243-255 S.M.R. Khalili, M. Botshekanan Dehkordi, E. Carrera, M. Shariya. [18] Prabhat Kumar Sinha, Vijay Kumar, Piyush Pandey and Manas Tiwari, “Static Analysis of Thin Beams by Interpolation Method Approach to MATLAB”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 254 - 271, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [19] Jn Mahto, Sc Roy, J Kushwaha and Rs Prasad, “Displacement Analysis of Cantilever Beam using FEM Package”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 3, 2013, pp. 75 - 78, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [20] Prabhat Kumar Sinha, Chandan Prasad, Mohdkaleem and Raisul Islam, “Analysis and Simulation of Chip Formation & Thermal Effects on Tool Life using FEM”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 53 - 78, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [21] Prabhat Kumar Sinha and Rohit, “Analysis of Complex Composite Beam by using Timoshenko Beam Theory & Finite Element Method”, International Journal of Design and Manufacturing Technology (IJDMT), Volume 4, Issue 1, 2013, pp. 43 - 50, ISSN Print: 0976 – 6995, ISSN Online: 0976 – 7002.
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