ME 475 Modal Analysis of a Tapered Beam

Transcription

1 ME 475 Modal Analysis of a Tapered Beam Objectives: 1. To find the natural frequencies and mode shapes of a tapered beam using FEA.. To compare the FE solution to analytical solutions of the vibratory response of a uniform beam. Analysis Problem Statement: Find the natural frequencies and mode shapes of vibration for a cantilevered tapered beam. The geometrical, material, and loading specifications are given in Figure 1. The height of the beam is h inches, where h is described by the equation: h = 4 0.6x x. Figure 1. Geometry, material, and loading specifications for a tapered beam. The following assumptions will be made about the beam: 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 less than eight. Hence, it is 1

2 unclear whether thin beam theory will accurately predict the vibratory response of the beam. Therefore, both a D plane stress elasticity analysis and a thin elastic beam analysis will be performed. 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 planar beam finite elements having 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. 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.

3 (a) (b) (c) Figure. Piecewise continuous version of the tapered beam, for (a), 4 (b), and 6 elements (c). The second model is a D plane stress model of the geometry as shown in Figure 1. The D finite element model of this structure will be developed using D plane stress bilinear four-noded quadrilateral finite elements. In the present analysis, the geometry and material properties are 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 1. Note that the mass distribution is accurately represented in the D 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 3

4 harmonic motion of the structure. The eigenvectors are the mode shapes associated with each frequency. Analysis Procedure: The 1D beam analysis should be performed three times, each with a different mesh. Meshes consisting of, 4, and 6 elements should be developed. The D analysis should be performed only once, as described in the procedure below. 1D Beam Analyses: You may find it helpful to create a separate CAE database and job name for each analysis. 1. Create a D planar, deformable, wire part.. Create a horizontal wire of length After finishing the part, partition the line using the Partition Edge: Enter Parameter tool, shown in Figure 3. Choose the line, and then enter a number between 0 and 1. This number specifies the distance along the length of the edge to partition. Figure 3. Button for Partition Edge: Enter Parameter tool. o For the element analysis, enter 0.5. Now the line should be partitioned into two regions. o For the 4 element analysis, enter 0.5. Then choose the left region, and partition using a value of 0.5 again. Do the same for the right region. You should now have it partitioned into 4 equally spaced regions. o For the 6 element analysis, enter 0.5. Then choose the left region, and partition using a value of Choose the nd region from the left, and enter 0.5. Choose the right region, and enter Choose the right-most 4

5 region, and enter 0.5. You should now have it partitioned into 6 equally spaced regions. o To check this, choose Tools Query Point/Node. The partitions will appear, so you can verify that they were set up correctly. 4. Calculate the value of h at the x-value of the centroid for each of the elements in this analysis. Create a different profile for each element, a Rectangular section with a = 1 and b = h. 5. Create a different section for each element in this analysis. They should all be Beam sections, with Section Integration: Before Analysis. Enter Young s Modulus = , Shear Modulus = , a section Poisson s Ratio = 0.3, and the Section Material Density = The density here was converted from the specific weight given above by dividing by g (acceleration due to gravity, * 1 in/s/s). Apply each section its profile, as created in step 4. No output points are needed here since the stress values will not be utilized. 6. Visualize the beam profiles, as in the Kframe lab. Your model should look like Figure. 7. Instance the part in the assembly. 8. Create a Frequency step. Use the defaults except: Number of eigenvalues requested = value (10). 9. The only field outputs needed are S (stress) and U (displacement). 10. Apply a boundary condition constraining the left-hand point in all three DOF (U1, U, and UR3). Since this is free vibration, no loads will be applied. 11. Use a mesh seed of 10. This will ensure only one element per section. o Use B1 (shear-flexible) beam elements. Using B3 elements here will give poor results. This is because the shear stiffness of this tapered beam is not small enough compared to the bending stiffness to neglect, and Euler-Bernoulli beam elements do not allow for transverse shear deformation. 1. As noted earlier, create a job with a name that is different for each analysis. 13. Submit the job. D Analysis: 1. Create a D planar, deformable, shell part. 5

6 . Create a point at (0, 0). Fix it with a constraint. 3. Create horizontal and vertical construction lines through this point. 4. Create a vertical line, starting at the point created in step, of length Create a vertical line, starting at the horizontal construction line, but to the right of the line made in step 4. It should have a height of 1, and have a horizontal distance of 10 from the vertical construction line. 6. Create a point, assign it a horizontal distance from the vertical construction line of Perform step 6 four more times, but use distances of, 4, 6 and 8, respectively. 8. Calculate h at the x = 1,, 4, 6, and 8. Use these for the vertical distance for each point from the horizontal construction line. Now, there should be 7 points which lie on the top surface of the tapered beam shown in Figure 1 (the 5 points created in steps 6 and 7, along with the tops of the two vertical lines from steps 4 and 5). 9. Create a spline through the 7 points to define the top surface of the tapered beam. 10. Create a perpendicular constraint between the spline and the right-hand vertical line. 11. Choose the Mirror tool (shown in Figure 4). Figure 4. Button for the Mirror tool. 1. Choose a Copy mirror operation 13. Choose the horizontal construction line as the mirror line. 14. Select everything created in previous steps, then deselect (using the CTRL key) the two construction lines. 15. Finish the sketch to create the part. It should look like Figure 5 when finished. 6

7 Figure 5. D part model of tapered beam. 16. Create a material called Steel with properties E = , ν = 0.3, and ρ = (converted from the specific weight, as noted in the 1D Beam Analysis section). 17. Create and apply a plane stress section with a thickness of Use steps 7-9 from the 1D Beam Analysis procedure. 19. Apply a boundary condition constraining only displacement DOF (U1 and U) on the left-hand face. No loads are applied. 0. To mesh, use a mesh seed of 0.. Mesh with quads, using the Structured technique, and minimize the mesh transition. Also, be sure to use CPS4, 4-noded bilinear plane stress quadrilaterals (fully integrated). 1. As noted earlier, create a job with a name that is different for each analysis.. Submit the job. 7

8 Post-Processing: To view the results, open the output database for each analysis. View a contour plot of the model. By default, it will show the 1 st natural frequency and modes associated with it. The natural frequency is listed in Hz in the text shown on the plot ( Freq = XXXX (cycles/time) ). The deformation is the eigenvector, or mode shape of the structure for this mode of vibration. The magnitude of the displacements is arbitrary, meaning that one could multiply them by any factor and have the same mode shape (though displacement normalization was used, so the maximum displacement reported by Abaqus will be unity). Thus, the stress values reported are meaningless. However, the stress contours shown are accurate, to the degree of accuracy of the model. It should be clear which modes are bending modes and which are axial modes of vibration. You can view the next mode by clicking the Next button, up through the 1 st ten modes of vibration (as we requested in the step). 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 ( x t ) U ( x) F ( t ) u, = (1) 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: d U dx β + β U = ω m EA = 0 () 8

9 In the above equation, ω= πf 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: U = Asin βx + B cosβx (3) 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: U F ( 0) x= L = 0 du = EA dx x= L = 0 (4) The first boundary condition yields B = 0. The second boundary condition yields: Aβcos βl =0 (5) Since cos βl =0 0 and A = 0 is the trivial solution, we must have: (6) which is referred to as the characteristic equation. There are many solutions to this equation: nπ βn L = for n=1, 3, 5, 7, (7) The circular frequencies of the motion are then obtained as: ω = n nπ EA ml (8) where E is the Young s Modulus, A is the cross-sectional area, m is the mass/length, and L is the total length of the beam. The eigenfunctions, or mode shapes, are then found to be: U n = A n n x sin π (9) L 9

10 where A n are the amplitudes of the shapes, which cannot be determined uniquely. The lowest natural frequency associated with axial motion of the bar is found for n = 1: π EA ω 1 = ml (10) By approximating the shape and mass distribution of the tapered bar as being uniform, this formula can be used to estimate the lowest natural frequency associated with axial motion of the tapered bar. The results will not be exact, but they should be sufficiently close to indicate whether the finite element results are reasonable, which is the goal of validation. In a similar way, formulas for the first three flexural frequencies of a cantilevered uniform beam can be obtained as: ω = ω = ω = EI ml 4 EI ml EI ml 4 4 (11) Note that Abaqus presents the frequencies in Hz, so the circular frequencies calculated using the formulas above should be converted to Hz for comparison. The above formulations are valid for a beam that has a uniform distribution of shape and mass along the length. In order to use these results for validating the finite element results of the current tapered beam, a suitable uniform shape must be assumed. As a first-order approximation, the uniform beam shape may be taken as the shape at the mid-span of the tapered beam. In other words, the cross-sectional shape at the point x = 5.0 in the tapered beam may be used as the uniform shape in the validation calculations. The mass distribution is also assumed to be uniform, and follows directly from the assumed shape and the known material density. Note, however, that the overall mass of the uniform beam may be different than that of the tapered beam. Moreover, the distribution of the mass as well as the stiffness is very different. 10

11 It should be expected that the tapered beam under consideration would behave differently than a uniform beam, so comparisons between the finite element results of the tapered beam and those of the above hand calculations must be performed thoughtfully. For example, it may be expected that the tapered beam will have a higher frequency than the uniform beam for at least two reasons. First, the tapered beam is stiffer near its root, where the loads are highest. Second, the tapered beam has less mass near its free end than does the uniform beam, so the inertial loading on the tapered beam is less than that in the uniform beam. By using this kind of intuition, a meaningful comparison between the hand calculations and the finite element results can be made. Report Requirements: A full report is required for this lab (see ME 475 Lab Report Format Guidelines). 1. Complete the following table: Model ID 1D - two elements 1D - four elements 1D - six elements D - plane stress elements Validation hand calculation Frequency of First Axial Mode (Hz) Frequency of First Bending Mode (Hz) Frequency of Second Bending Mode (Hz) Frequency of Third Bending Mode (Hz). Comment on the convergence of frequencies and mode shapes in the 1D beam solutions to that of the D solution. 3. Comment on the validity of the solutions, with respect to the hand calculations performed. 4. For each model, include a deformed mesh plot of the first 5 modes of vibration (these may be included in the appendix). 11