MEE 323 Su21 - Homework #5

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Jan 9, 2024

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Summer 2021B 1 MEE 323 Computer Aided Engineering II Homework #5 Assigned: August 02, 2021 Due Date: August 09, 2021 by 11:59PM Instructions: Cr eate a folder named “Homework_ 5 ” in your U: \ drive. Save all of your model data and a copy of your homework report in this folder. Use this document as a template for your homework report. Turn in the assignment on the course GradeScope before the deadline. Homework Objectives: Learn heat transfer, thermal analysis, and thermal-mechanical coupling using FEA. Learn modal analysis and the calculation of natural frequencies and mode shapes using ANSYS. Question 1: Temperature Condition using Static Structural An assembly of two identical bars 2.5 mm in width is shown below. One is made of structural steel and the other is made of copper alloy; they are bonded to form a bimetallic strip. The left end is completely fixed. The bar is initially at room temperature (22 deg C) then experiences a uniform 25 deg C decrease in temperature. (a) Show the figure of the mesh. Use a “Relevance Center” setting of “Fine”.
Summer 2021B 2 (b) Plot the maximum displacement contour plot with undeformed edge. Find the maximum vertical displacement of the tip of the bimetallic strip. (c) Plot the von Mises stress contour plot. Find the magnitude and location of the maximum von Mises stress. Question 2: Steady State Thermal Analysis A furnace wall consists of two layers: fire brick and insulating brick. The temperature inside the furnace is 3000 deg F and the inner surface convection coefficient is 3.333e-3 BTU/s-ft 2 -degF. The ambient temperature is 80 deg F and the outer surface convection coefficient is 5.556e-4 BTU/s-ft 2 -degF. Model a section of the wall as shown below. Use perfectly insulated boundary conditions on all faces except the 1 inch x 1 inch faces that represent the inside and outside surfaces of the walls.
Summer 2021B 3 Material Properties: Fire Brick Wall: K = 2.222e-4 BTU/s-ft-degF Insulating Wall: K = 2.778e-5 BTU/s-ft-degF Geometric properties: Cross section: 1 inch x 1 inch Fire Brick Wall thickness: 9 inches Insulating Wall thickness: 5 inches (a) Show the figure of the mesh. Use a “Relevance Center” setting of “Medium”. (b) Show the final steady state temperature distribution in the wall. What are the maximum and minimum temperatures of the wall material?
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Summer 2021B 4 Max temperatures for the wall material are 2957.2F and the minimum temperature is 336.69F. (c) Show a plot of the thermal error and comment. Given that the largest magnitude of error is in the range of 10^-5 I would personally take this analysis as being accurate should this not be accurate enough for some precision engineering, the mesh could be refined further still as I only performed minor refinement on the area where the two materials join together.
Summer 2021B 5 Question 3: Transient Heat Transfer in an Infinite Slab A semi-infinite slab is initially at a temperature of 0 deg C. The solid is then suddenly exposed to an environment having a temperature of 1000 deg C and a surface convection coefficient h = 50 W/m 2 -degC. Run a transient analysis and determine the temperature distribution through the solid after 2000 seconds. You can approximate the slab by modeling it as a 2-D rectangle of dimensions 1 meter by 0.1 meter. Use appropriate time step settings for the transient thermal analysis (hint: use smallest time step of 0.1s and largest time step of 10 seconds). Material Properties: Thermal Conductivity, K = 54 W/m-degC Density, ρ = 7833 kg/m 3 Specific Heat, c = 465 J/kg-decC (a) Show the final temperature distribution in the slab at 2000 seconds. What are the maximum and minimum temperatures of the slab material? Comment on the results. The maximum temperature of the slab material was 157.13C and the minimum temperature was 0.004588C. It does make sense that the slab would not be entirely heated at this point given the small contact patch with the environment that is at 1000C as well as the small-time frame that it is exposed to the area of only 30 min. With convection it would take more time for this small contact area to come up closer to the environment temperature.
Summer 2021B 6 Question 4: Two structural steel rods of diameter 25 mm are joined to form a “T” part as shown. A 10 mm diameter through hole and a 10 x 10 mm square through hole are placed as shown in the figure. (The 65 mm dimension locates the left edge of the 10 x 10 square through hole). The left end is completely fixed. (a) Find the three lowest natural frequencies of vibration of the structure. Show plots of the mode shapes and describe in your own words the type of motion that occurs in each. Present the frequencies in Rad/sec, Hz, and RPM. Use the medium mesh size setting in the “relevance center”. a. Mode 1 75.746Hz = 475.92Rad/sec = 4,544.76 RPM Mode 1 is the top half of the t-bar structure rotating around the y axis; oscillating at 75.746Hz. b. Mode 2 85.271Hz = 535.77 Rad/sec = 5116.26 RPM
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Summer 2021B 7 Mode 2 is the top half of the t-bar structure rotating about the x-axis at a frequency of 85.271Hz c. Mode 3 159.85Hz = 1004.36 Rad/sec = 9591 RPM Mode 3 is the very top of the T-Bar structure rotating about the center bar (x-axis) (b) Use three different mesh densities by choosing 3 different element sizes with a factor of 2 between each (for example, you may choose mesh sizes of 8 mm, 4 mm, and 2 mm). Plot the three calculated frequencies as a function of the number of elements in the mesh. What do you observe? Element Size First Frequency Second Frequency Third Frequency 3 75.414 85.218 159.46 6 75.682 85.429 160.03 12 76.029 85.405 160.33 The mesh density / element size does not impact the found frequencies enough to warrant very fine meshing. As was mentioned in lecture, a fine mesh would benefit a modal analysis if there were also pre-stresses present. 0 50 100 150 200 0 5 10 15 Chart Title 1st Frequency 2nd Frequency 3rd Frequency
Summer 2021B 8 (c) How do your results change if you remove the two through holes in the structure? Show plots of the mode shapes. Compare numerically (percentage difference) the change in the natural frequencies of the structure. Use the smallest mesh size setting from part (b) above for this. Element Size First Frequency Second Frequency Third Frequency 3 81.835 87.8 167.17 Mode 1 Mode 2 Mode 3 Element Size First Frequency % Difference Second Frequency % Difference Third Frequency % Difference 3 7.85 2.94 4.61
Summer 2021B 9 Question 5: Compute the lowest two non-zero natural frequencies of the free-free vibration of the aluminum alloy U-bar shown below. Present your results in Hz. (a) Use a full 3-D model. Show plots of the two mode shapes and carry out a convergence analysis until your frequencies converge to within 1% between iterations. Why do you see zero frequency modes? What kind of motion of the structure do these modes represent? We see zero frequency modes because there are no supports on the body, they are motions of the system that occur when there are no supports on the system. a. Mode 1 Converges at 286.96 Hz Mode 1 represents a bending motion b. Mode 2 Converges at 707.14Hz Mode 2 represents a twisting motion
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Summer 2021B 10 (b) Use the “mid - surface” tool to create a 2 -D shell model of the structure (2-D surface body but 3-D analysis). Show plots of the two mode shapes and carry out a convergence analysis until your frequencies converge to within 1% between iterations. a. Mode 1 Converges at 280.47 Hz Mode 1 represents a bending motion b. Mode 2 Converges at 697.55Hz Mode 2 represents a twisting motion (c) Tabulate the mesh statistics of the 3-D and 2-D final models (after convergence). Compute the percentage difference in the natural frequencies predicted by these two models and also the percentage reduction in mesh statistics achieved using the 2-D model. Comment. The mesh statistics show an almost 98% reduction in elements by using a 2-D model over a 3- D model, while still achieving a very accurate estimate for the natural frequency of the object in question, with a percent difference of 0.17 for the first mode and 1.36 for the second mode. Showing that the 2-D analysis is the better choice when seeking to utilize computational resources efficiently. Mode/Model Type # of elements for convergence Converged Frequency % Difference in Frequency % Reduction in elements Mode 1 - 3D Model 62584 280.96 0.17 97.8 Mode 1 - 2D Model 1400 280.47 Mode 2 - 3D Model 62584 707.14 1.36 97.8 Mode 2 - 2D Model 1400 697.55