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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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This note was uploaded on 02/26/2012 for the course 650 486 taught by Professor Zou during the Fall '11 term at Rutgers.
 Fall '11
 Zou

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