# MSU ME 424 - Marc.TaperedBeamVibration (36 pages)

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## Marc.TaperedBeamVibration

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## Marc.TaperedBeamVibration

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Pages:
36
School:
Michigan State University
Course:
Me 424 -

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Application of the Finite Element Method Using MARC and Mentat 6 1 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 4 0 6 x 0 03x 2 Geometry Length L 10 Width b 1 uniform Thickness 2h a function of x Material Steel Yield Strength 36 ksi Modulus of Elasticity 29 Msi Poisson s Ratio 0 3 Specific Weight 0 284 lbf in3 Loading Free vibration x 2h L Figure 6 1 Geometry material and loading specifications for a tapered beam 6 2 Analysis Assumptions Because the beam is thin in the width out of plane direction a state of plane stress can be assumed The length to thickness ratio of the beam is difficult to assess due to the severe taper By almost any measure however the length to thickness ratio of the beam is less than eight Hence it is unclear whether thin beam theory will accurately predict the vibratory response of Application of the Finite Element Method Using MARC and Mentat 6 2 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 two noded 1D thin beam finite elements having a uniform cross sectional shape and mass distribution within each element Thus the geometry is idealized as having a piecewise constant cross section as shown in Figure 6 2 The uniform thickness within each element is taken to

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