Problem Statement This lab seeks a FEA vibration normal mode analysis of the L-shaped membrane shown below wherein all edge displacement BCs are fixed at zero displacement.
Recall the Galerkin weak statement formulation for DP yields
The normal mode solution process assumes yh(x,y,t) = Y(x,y)eiwt hence
Since this GWSh
matrix statement is homogeneous, the determinant of w2 Þ wi2, i = 1,2,3,….,n,... Choosing the positive root, the natural vibration frequencies are w Þ wih = Ö(eigenvalues of [MASS]-1[STIFF]) The corresponding normal modes are characterized by the nodal distributions {Q}Þ {Qi (wi)} The first analysis
step is to estimate the accuracy of the base mesh solution, which of
course leads to a mesh refinement study. Thereafter, the design
component seeks to study the effect of filleting the convex corner on
the eigensolution, as one might seek to reduce the high stresses that
occur at sharp corners. The fillet is created by fitting a circle of
radius r. The following sequence
of FEMLAB commands creates the required geometry, generates the m-file
to solve for the eigenvalues and thereafter plots the variation of the
first eigenvalue for the filleted membrane with increasing radius.
a = 0.5; sq1 = square2(0,0,1); sq2 = move(sq1,0,-1); sq3 = move(sq1,-1,-1); sq4 = square2(-a,0,a); c1 = circ2(-a,a,a); geom = sq1+sq2+sq3+(sq4-c1); geomplot(geom) To solve the eigenvalue problem on this
geometry, start by defining the FEMLAB data clear fem fem.shape=2; fem.sshape=2; fem.dim=1; fem.bnd.h=1; fem.equ.c=1; fem.equ.da=1; Next solve the GWSh normal
mode matrix statement for each r from 0.1 to 1 with a step of
0.05 and save the smallest eigenvalue for each r r = 0.1:0.05:1; evalue = []; for a = r sq4 = square2(-a,0,a); c1 = circ2(-a,a,a); fem.geom = sq1+sq2+sq3+(sq4-c1); fem.mesh = meshinit(fem); fem.mesh = meshrefine(fem); fem.xmesh = meshextend(fem); fem.sol =
femeig(fem,’eigfun’,’fleig’,’eigpar’,[0 10]); evalue = [evalue fem.sol.lambda(1)]; end plot(r,evalue)
In case if you are not able to get matlab running in the background then try to make the fillet directly using the femlab GUI. |