Question

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Table of Contents

I. Problem 1: Modal Analysis of a Cantilever Beam………………………….03

A. ANSYS Solution……………………………………………………….03 B. Manual Solution………………………………………………………..06 C. Analytical Solution……………………………………………………..08

II. Problem 2: Flow Analysis in 90 Degree Planar Branch…………………...10

III. References…………………………………………………………………….15

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I. Problem 1: Modal Analysis of a Cantilever Beam

A. ANSYS Solution

Before creating a CAD model of the cantilever, a new material with properties given in the problem was added to engineering data.

i. Geometry:

The CAD model of the cantilever beam was designed in Design Modeler of ANSYS. A 0.5 mm x 10 mm rectangle was sketched and extruded by 0.2 m.

ii. Mesh:

Figure 1 : Cantilever Mesh

a

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Following settings were set to create the mesh:

Mesh Relevance 84 Relevance Centre Fine Element Size 1e-3m

iii. Boundary Conditions:

Figure 2:Boundary Conditions A fixed support was applied to one side of the beam to replicate cantilever behavior.

iv. Results:

The following results for natural frequencies were obtained:

Table 1: Modal Data

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To check for convergence of results, solutions for total deformations were generated, one for each mode with a convergence setting of 10 %.

Figure 3 Total Deformation Results: Mode 1 Figure 4 Mode 1: Converged Results

Figure 6: Total Deformation Results: Mode 2 Figure 5 : Mode 2: Converged Results

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B. Manual Solution

Figure 9: Standard Cantilever Beam

To determine the first three natural frequencies, energy method was used to first determine the equation relating the dynamic and static behavior of the beam. This relation is given by:

To find the non-trivial solution of the above equation, the following condition was solved using MATLAB.

Figure 8: Total Deformation Results: Mode 3 Figure 7: Mode 3: Converged Results

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i. MATLAB Program:

A MATLAB program was created to generate global K and M matrices for the cantilever beam with 500 elements.

Figure 10: MATLAB Code Part 1

Figure 11: MATLAB Code Part 2

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ii. Results:

Figure 12: Command Prompt Results Display

C. Analytic Solution

To find the analytic solution, the natural frequency calculator at

http://www.amesweb.info/Vibration/Vibration-Calculators.aspx

was used. This website uses the following formula to determine accurate values for natural frequencies.

Figure 13: Input Parameters

m

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Figure 14: Analytical Results The results of all three solution methods are consolidated in the tables below:

Table 2: Summary of Modal Data

Mode 1

(Hz) Mode 2

(Hz) Mode 3

(Hz) ANSYS 10.317 64.649 181.05

Manual 10.28 64.433 180.415 Analytic 10.302 64.387 180.575

Owing to the small discrepancies in the results, it can be concluded that all three methods yielded precise results. The % deviations (assuming analytic solution to be the theoretical solution) are given in Table 3.

Table 3: % Deviations of Modal Data

% Deviation

Mode 1

(Hz) Mode 2

(Hz) Mode 3

(Hz) Ansys 0.1456028 0.406914 0.263049

Manual 0.21355077 0.071443 0.088606

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II. Problem 2: Flow Analysis in 90 Degree Planar Branch

i. Geometry:

The plane cross section of the 3-D channel was sketched in Design Modeler with dimensions specified in the problem. Furthermore, a surface body was generated with the 2-D cross-section sketch. This sketch was then exported to generate the Mesh.

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Figure 15: Geometry of Channel

ii. Mesh:

To replicate the no-slip condition at the boundaries of the channel, inflation method was used to generate very fine mesh along the boundary edges of the channel. The mesh settings used are tabulated below:

Relevance 80 Element Size 5e-2 m Inflation Layers 50

Figure 16: Mesh of the Cross Section

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Figure 17: Close-up of Mesh showing Inflation Layers on Sides

The mesh file was then opened in Fluent for flow analysis.

iii. Fluent: Boundary Conditions

Physical data for air was added to the materials list. The following boundary conditions were set:

a. Velocity inlet: parabolic velocity profile was imported and applied to the inlet b. Pressure outlets: both outlets were set to pressure outlets with 0 pressure

A laminar flow model was assumed with the ‘Energy’ option on. The results were generated until convergence or 350 iterations, whichever came first.

iv. Results

The pressure and velocity contours for the laminar flow of air through the 90 degree channel were determined. These are shown in the figures below.

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Figure 18: Convergence of Residuals

The vertical red region indicates that maximum velocity is within the straight channel and relatively small amounts of air enters the branching channel with a very small velocity.

Figure 19: Velocity Contours

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Figure 20: Pressure Contour

Using the ‘Fluxes’ option in Fluent, the mass flow rates at the inlet and outlets were determined. The fluent results are shown in Figure 21.

Figure 21: Mass Flow Rates for Inlet/Outlets The fraction of mass flow rate for the main channel (outlet – y) was determined to be 0.8943 which is tabulated below.

Table 2: Mass flow Rates and Fraction for Main Channel

The experimental result of 0.89433 kg/s has a % deviation of 0.816281% compared to the reference value of 0.887 kg/s which indicates that the analysis was considerably accurate.

Mass flow rate, kg/s outlet x -0.071045168 outlet y -0.60124082

inlet 0.67228093 Fraction, F 0.89433

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References http://www.amesweb.info/Vibration/Vibration-Calculators.aspx

HAYES, R.E., et al. STEADY LAMINAR FLOW IN A 90 DEGREE PLANAR BRANCH. ac.els-cdn.com/0045793089900273/1-s2.0-0045793089900273-main.pdf?_tid=ae7e0468-85f8- 4a18-afa5-a06a3b631fec&acdnat=1524076072_ec1e95eaf0084ba15b23249d3d4ec4b2.