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Unformatted text preview: Application of the Finite Element Method Using MARC and Mentat 61 Chapter 6: Modal Analysis of a Cantilevered Tapered Beam Keywords: elastic beam, 2D elasticity, plane stress, convergence, modal analysis Modeling Procedures: ruled surface, convert 6.1 Problem Statement and Objectives It is required to determine the natural frequencies and mode shapes of vibration for a cantilevered tapered beam. The geometrical, material, and loading specifications for the beam are given in Figure 6.1. The geometry of the beam is the same as the structure in Chapter 3. The thickness of the beam is 2h inches, where h is described by the equation: h x x = + 4 0 6 0 03 2 . . 6.2 Analysis Assumptions Because the beam is thin in the width (outofplane) direction, a state of plane stress can be assumed. The lengthtothickness ratio of the beam is difficult to assess due to the severe taper. By almost any measure, however, the lengthtothickness ratio of the beam is less than eight. Hence, it is unclear whether thin beam theory will accurately predict the vibratory response of Geometry: Material: Steel Length: L=10 Yield Strength: 36 ksi Width: b=1 (uniform) Modulus of Elasticity: 29 Msi Thickness: 2h (a function of x) Poissons Ratio: 0.3 Specific Weight: 0.284 lbf / in 3 Loading: Free vibration Figure 6.1 Geometry, material, and loading specifications for a tapered beam. 2h L x Application of the Finite Element Method Using MARC and Mentat 62 the beam. Therefore, both a 2D plane stress elasticity analysis and a thin elastic beam analysis will be performed. 6.3 Mathematical Idealization Based on the assumptions above, two different models will be developed and compared. The first model is a beam analysis. In this model, the main axis of the beam is discretized using straight twonoded 1D thin beam finite elements having a uniform crosssectional shape and mass distribution within each element. Thus, the geometry is idealized as having a piecewise constant crosssection, as shown in Figure 6.2. The uniform thickness within each element is taken to be equal to the actual thickness of the tapered beam at the xcoordinate corresponding to the centroid of that element. Note that this type of geometry approximation also leads to an approximation of the overall mass as well as its distribution. Since the mass distribution plays a strong role in vibratory motion, the effect of this approximation should be considered carefully. As the mesh is refined, the error associated with this approximation will be reduced. Because beam elements are designed to capture threedimensional behavior, the beam model will predict threedimensional modes of vibration unless additional constraints are imposed. In the present case, we are most interested in the modes of vibration that occur within the plane shown in Figures 6.1 and 6.2. Therefore, we should apply constraints to the model such that the beam cannot translate or rotate out of the plane....
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 Spring '05
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