MAE486_Fall11_L19_S

MAE486_Fall11_L19_S - MAE 486 Design of Mechanical Systems...

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Unformatted text preview: MAE 486 Design of Mechanical Systems Lecture 19 Fall 2011 Today s topics: Modeling (Chapter 10); Review Example of Dimensional Analysis •  List all the variables in the problem (k=?): •  •  Express each of the variables in terms of their basic dimensions: •  Determine the required number of pi terms •  Select the repea=ng variables. –  In this problem, we select three repea=ng variables, D, N, and p. •  Select the variable to be the dependent variable. –  The dependent variable is pf. The nonrepea=ng variables are pf, C and μ. •  Form a pi term by mul=plying one of the nonrepea=ng variables by the product of the repea=ng variables Review Geometric Similarity •  Consider the example of a cylindrical bar of length L with deforma=on δ under force P. What is the scale factor? •  What are the variables? •  Choose repea=ng variables: •  Find pi groups: •  How to use the above rela=on in model study? –  Suppose we wish to use a plas=c model, Em=0.4 x 10^6 lb/in2 to model a steel bar, Ep=30 x 10^6 lb/in2, loaded to 50000 lbs, if the model is built to a 1 to 10 scale (S=0.1), what is the load we should apply to the model? Review Finite ­Diﬀerence Method •  When numerical methods become necessary and powerful? •  Finite ­diﬀerence method: q – rate of energy genera=on per unit volume, k– thermal conduc=vity. •  Consider two ­dimensional case Review Finite ­Diﬀerence Method (cont’d) •  First, approximate the ﬁrst ­order deriva=ve: •  Secondly, approximate the second ­order deriva=ve: •  Do the same for y: •  Finite diﬀerence equa=on: •  Understanding the equa=on: •  One example: Steady ­state boundary temperatures on the four sides of the furnace wall are shown. Calculate temperatures at all nodes. Finite Element Analysis •  What are the limits of ﬁnite diﬀerence method (FDM)? •  Solu=on: Finite element method –  What is the main idea? Finite Element Analysis (cont’d) •  Originally for structure analysis. •  Bolts & nuts ﬁrst: •  Cons.tu.ve equa.ons: •  Elements behave: Finite Element Analysis (cont’d) •  What are F_1 and F_2 in the above 1 ­node model? •  Rewrite in matrix form: •  S=ﬀness matrix, and rela=on to elas=c modulus: Finite Element Method (cont’d) •  Extension to two linear elements (s=ll 1D) •  Combine together in one matrix form: Example •  Problem statement: –  Two bars of diﬀerent materials are welded together, end ­to ­end. –  P3= 10 kN. –  The materials: •  (1) Bar 1: mild steel, A1=70 mm2, L1=100 mm, E1=200GN/m2, •  (2) Bar 2: aluminum alloy, A2=70 mm2, L2=280 mm, E2=70 GN/m2. –  Ques=on: Find •  The stress in each bar; •  The total elonga=on of the structure; •  The reac=on force of the structure on the wall; Example (Con=nued) •  Find the spring constant for each element and construct the s=ﬀness matrix •  Find the force ­displacement equa=on •  Solve the equa=on Types of Elements •  (a) and (b) : •  (c): •  (d): triangle, (e): tetrahedron (tet), and (f): hexahedron (hex) are isoparame*c elements, when the boundaries are curved in 3D •  (g): •  (h): •  (i): FEM Errors •  For a linear axial element with displacements u=u1 at x=0 and u=u2 at x=L, how the displacement is varying along the element? Shape factor: •  How about two ­ and three ­dimensional elements with more nodes and displacements at a node? –  Example: Consider a 2D triangular element with three nodes and two displacements per node, the components of displacement u (along x ­axis) and v (along y ­axis) are •  What is the strain in x ­axis? FEM Errors (cont’d) •  Discre*za*on errors: •  Formula*on errors: (Lineariza=on error) •  Convergence error: –  H ­element method, –  P ­element method, FEM Errors (cont’d) h ­element method p ­element method Steps in FEA Process •  Perform a preliminary analysis to deﬁne the problem •  Pre ­processing phase –  The geometry of the part is imported from the CAD model –  Make decisions concerning the division of the geometry into elements (called meshing) –  Determine how the structure is loaded and supported, or determine the ini=al condi=on (e.g., temp. in thermal problem) –  Select the cons=tu=ve equa=on for describing the material that relates displacement to strain and then to stress •  Computa=on (FEA soiware) –  The FEA program renumbers the nodes in the mesh to minimize comp. resources by minimizing the size of global K –  It generates a s=ﬀness matrix k for each element and assembles the elements together to maintain a con=nuity in K. Apply boundary condi=ons –  Solve the massive matrix equa=on for the displacement vector and determine the constrain forces. •  Post processing (FEA soiware) –  Generate stress and strain ﬁeld values –  Interpret data, generate visual displacement of data etc. –  Increasingly, FEA soiware is being combined with an op=miza=on package and used for itera=ve op=miza=on design ...
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