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: hxx=−+4060032.. 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 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) Poisson’s Ratio: 0.3 Specific Weight: 0.284 lbf / in3 Loading: Free vibration Figure 6.1 Geometry, material, and loading specifications for a tapered beam. 2h L xApplication 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 be equal to the actual thickness of the tapered beam at the x-coordinate 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 three-dimensional behavior, the beam model will predict three-dimensional 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. The second model is a 2D plane stress model of the geometry as shown in Figure 6.1. The 2D finite element model of this structure will be developed using 2D plane stress bilinear four-noded quadrilateral finite elements. In the present analysis, the geometry and material properties are Figure 6.2 Idealized geometry for a tapered beam. L xApplication of the Finite Element Method Using MARC and Mentat 6-3 symmetric about the mid-plane of the beam. However, the vibratory response is not symmetric about this plane. Hence, it is necessary to model the entire domain of the beam, as shown in Figure 6.1. Note that the mass distribution is accurately represented in the 2D model. In free vibration analysis, no loads are applied. The goal of the analysis is to determine at what frequencies a structure will vibrate if it is excited by a load that is applied suddenly and then removed. These frequencies are called natural frequencies, and they are a function of the shape, the material, and the boundary constraints of the structure. Mathematically, the natural frequencies are associated with the eigenvalues of an eigenvector problem that describes harmonic motion of the structure. The eigenvectors are the mode shapes associated with each frequency. 6.4 Finite Element Model The procedure for creating the finite element model and obtaining the finite element solution for each type of model is presented at the end of this chapter. The 1D beam analysis should be performed three times, each with a different mesh. Meshes consisting of 2, 4 and 6 elements should be developed. The 2D analysis should be performed only one time, using the mesh described within the procedure. 6.5 Model Validation Simple analytical formulas are available for predicting the axial and bending natural frequencies of a uniform cantilevered beam. These results can be used to estimate the natural frequencies in a tapered beam and thus to assess the validity of the finite element results (i.e., to make sure that the finite element results are reasonable and do not contain any large error due to a simple mistake in the model). For the case of axial vibratory motion, an analytical solution can be developed as follows. We assume that the axial displacement of the bar u(x,t) is separable in space and time, or )()(),( tFxUtxu= where F(t) is a harmonic function. When the material and geometric properties are uniform throughout the bar, the governing eigenvalue equation for axial vibratory motion is: 0222=+ UdxUdβ EAm22ωβ=Application of the Finite Element Method Using MARC and Mentat 6-4 In the above equation, fπω2=is the circular frequency measured in radians per second, f is the frequency measured in cycles per second, m is the mass per unit length of the bar, E is the modulus of elasticity, and A is the cross-sectional area. The solution to the above equation takes the form: xBxAUββcossin+= where A and B are constants of integration that must be determined from the boundary conditions. In the present problem, the boundary conditions are of the form: 0)0(=U 0====LxLxdxdUEAF The first boundary condition yields B = 0. The second