AE/ME 631 _HW_3, Feb 25,2015Problem 3-1: Beam vibration by Ritz_super-element, monomial basis(a) Development a 10-dof beam Ritz-super-element for a uniform beam use the following data: L=100 in., E=10^7 psi. Rho=0.1/386.4 lb-s^2/in^4, A=4 in^2, I=16/12 in^4. Use monomials for the basis function and use the nodal dof given in Figure A.(b) Perform vibration analysis use the model develpoed in Part(a) and compare the first 3 natural frequencies with those computed by FEM for the following BC's (1) C-F, (2) C-S, (3) C-C and (4) S-S (C=clamped, F=Free, S=supported)(c) Compare the the first 3 mode shapes of C-S model computed by the 2 methods by plotting them in the same figure use different color and line stype and identify them by legends.(d) Solve the C-S beam use {c}-based model also. You should get same results as the C-S beam results of Part(b)

Problem 3-2: Beam vibration by Ritz-super-element, poly-cosine basis(a) Development a 10-dof beam Ritz-super-element for a uniform beam use the following data: L=100 in., E=10^7 psi. Rho=0.1/386.4 lb-s^2/in^4, A=4 in^2, I=16/12 in^4. Use poly-cosine basis (i.e., use (1,x,x^2 and x^3 plus 6 cosine functions: cos(r*pi/L), r=1 to 6 the special nodal dof given in Figure B.(b) Perform vibration analysis use the model develpoed in Part(a) and compare the first 3 natural frequencies with those computed by FEM for the following BC's (1) C-F, (2) C-S, (3) C-C and (4) S-S (C=clamped, F=Free, S=supported)(c) Compare the the first 3 mode shapes of C-S model computed by the 2 methods by plotting them in the same figure use different color and line stype and identify them by legends.(d) Solve the C-S beam use {c}-based model also. You should get same results as the C-S beam results of Part(b)

Problem 3-3: Optimal design of a cantilever beam(a) find the beam shape of a linearly tapered beam to maximize the lowest natural frequency,under total beam mass(volume) constraint.The beam is made of isotropic material with Young's modulus E,mass density Rho. The length of the beam is L. The tip mass is Mt=W1/g.The beam has rectangular cross section with a unit thickness.The beam height various linearly from X1 at the support to X2 at the tip.(See Figure C-2)The vibration analysis is to be carried out by Ritz-super-element using a 10-dof model. Use monomial basis if the first letter in your last name is A to M Use poly-cosine basis if if the first letter in your last name is N to Z.Use the following numerical data: L=60 in., E=10^7 psi. Rho=0.1/386.4 lb-s^2/in^4. The tip weight W1 is 50 lbs. The beam allowable volume for the beam material is 1000 in^3.Plot the optimal beam shppe.(b) Plot contours of objective function and constraint boundaries in the design space. Plot design histories in the contour plots and use symbols to identify the initial and final designs(