Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams

Gupta, Rekha; Babu, G. Jagadish; Janardhan, G. Ranga; Rao, G. Venkateswara

doi:10.1016/j.finel.2009.04.001

Cited by 36 publications

(15 citation statements)

References 21 publications

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“…A clamped-clamped homogeneous uniform beam is considered and the resulted backbone curve for fundamental mode is validated with the results published by Gupta et al [31] as shown in Fig. 3.…”

Section: Resultssupporting

confidence: 60%

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Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

Lohar¹,

Mitra

Sahoo³

2016

IFSL

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Abstract.A displacement based semi-analytical method is utilized to study non-linear free vibration and mode shapes of an exponential tapered axially functionally graded (AFG) beam resting on an elastic foundation. In the present study geometric nonlinearity induced through large displacement is taken care of by non-linear strain-displacement relations. The beam is considered to be slender to neglect the rotary inertia and shear deformation effects. In the present paper at first the static problem is solved through an iterative scheme using a relaxation parameter and later on the subsequent dynamic analysis is carried out as a standard eigen value problem. Energy principles are used for the formulation of both the problems. The static problem is solved by using minimum potential energy principle whereas in case of dynamic problem Hamilton's principle is employed. The free vibrational frequencies are tabulated for exponential taper profile subject to various boundary conditions and foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane and in some particular case the mode shape results are furnished.

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Section: Resultssupporting

confidence: 60%

“…3. It should be mentioned here that to comply with the system analyzed by Gupta et al [31] the spring stiffness is considered as 0 N/m. The two sets of results show similar trends and the values show satisfactory matching, hence, establishing the validity of the present method.…”

Section: Resultsmentioning

confidence: 99%

Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

Lohar¹,

Mitra

Sahoo³

2016

IFSL

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“…A material model where the elastic modulus and density vary along the axial direction is considered and the expressions for these two parameters as function of the normalized axial coordinate are given as follows, ( ) ( ) ( ) ( ). However, the present formulation [12] and the comparative plot is furnished in Figure 2. It can be seen from the figure that the matching of the two sets of results is satisfactory.…”

Section: Resultsmentioning

confidence: 99%

Free vibration analysis of axially functionally graded linearly taper beam on elastic foundation

Lohar¹,

Mitra

Sahoo³

2016

IOP Conf. Ser.: Mater. Sci. Eng.

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Abstract. In the present study non-linear free vibration analysis is performed on a linearly tapered Axially Functionally Graded (AFG) beam resting on an elastic foundation with different boundary conditions. Firstly the static problem is carried out through an iterative scheme using a relaxation parameter and later on the subsequent dynamic problem is solved as a standard eigen value problem. Minimum potential energy principle is used for the formulation of the static problem and for the dynamic problem Hamilton's principle is utilized. The free vibrational frequencies are tabulated for different taper parameter and different foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane.

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“…NL H / presented in Table 6 are obtained on the basis of SHM assumption and these expressions can be corrected by using the harmonic balance method (HBM) discussed in Refs. [16,17]. The non-linear frequency parameters .…”

Section: Z âZmentioning

confidence: 99%

“…Large amplitude free vibration behavior of plates [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] have received considerable attention by various researchers due to its relative importance to the field of structural engineering. Leung and Mao [7] applied an Hamilton's formulation and symplectic integration methods to investigate the non-linear vibration behavior of beams and plates.…”

Section: Introductionmentioning

confidence: 99%

Large Amplitude Free Vibration Analysis of Thin Rectangular Plates: Simple Closed-form Solutions

Gunda¹

2013

Nonlinear Engineering

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Large amplitude free vibration behavior of thin, isotropic rectangular plate configurations are expressed in the form of simple closed-form solutions by using an application of the Ritz method based on coupled displacement fields. Influence of plate aspect ratio a b and Poisson ratio . / on the behavior of back-bone curves is briefly discussed for various boundary configurations of the rectangular plate. Proposed closed-form solutions are corrected for the simple harmonic motion (SHM) assumption using the well established harmonic balance method which is applicable for the homogeneous form of cubic non-linear Duffing equation.Keywords. Large amplitude free vibration, Thin rectangular plates, Ritz method, Closed-form solutions, Coupled displacement fields, von-Kármán type of geometric nonlinearity, Non-linear frequency. Nomenclature a = length of the rectangular plate b = width of the rectangular plate C = axial rigidity of the plate E h .1 2 / Á D = flexural rigidity of the plate E h 3 12.1 2 / Á E = Young's modulus h= thickness of the plate N x , N y and N xy = in-plane stress resultants M x , M y and M xy = moment resultants u = displacement in x-direction v = displacement in y-direction U = strain energy of the plate w = transverse displacement T = kinetic energy of the plate x, y = in-plane co-ordinates z 1 = maximum reference amplitude of the rectangular plate z 1 h = maximum reference amplitude to thickness ratio x , y and xy = in-plane strain terms L = linear frequency parameter h! L 2 a 4 4 D Á NL H = non-linear frequency parameter obtained on the basis of SHM assumption Â h! NL H 2 a 4 4 D Ã NL = non-linear frequency parameter h! NL 2 a 4 4 D Á = Poisson ratiox , y and xy = curvature terms Subscripts L = linear NL = non-linear NL H = non-linear based on harmonic motion assumption x = partial derivative with respect to x xx = second partial derivative with respect to x y = partial derivative with respect to y yy = second partial derivative with respect to y xy = partial derivative with respect to x and y 1 Introduction

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Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams

Cited by 36 publications

References 21 publications

Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

Free vibration analysis of axially functionally graded linearly taper beam on elastic foundation

Large Amplitude Free Vibration Analysis of Thin Rectangular Plates: Simple Closed-form Solutions

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