AERO4630-Project 3 Solved

Problem 1: Clamped beam with initial force
Let’s look at the same problem of clamped beam we did in the last project. We are looking at a beam of length

The beam is clamped at endsL = 1m, height H = 0.2m and width Wand= 0.2m. The properties are given by. E = 200GPa, ν = 0.3 and ρ = 7800kgm−3.

(1a)         The governing equation isx = 0         x = L                 div       + f = ρu                                                                               (1a.1)

σ                                                                                                                                 ¨

where we are solving for displacement vectorgiven through our usual linear elasticity laws u at every point as a function of time. The stress tensor is  tr (1a.2)

                  where     and        are the Lame’ parameters given in terms of         and     . The strain tensor       is defined as

                               λ          µ                                                                                      E          ν                                     ε

                                                                                                  grad u + (grad u)T                                                                    (1a.3)

xFirst, re-write the equations in the non-dimensional form. Just like we did in class, you can non-dimensionalize, y and z as

x˜ = x L, y˜ =  yL, z˜ = L z                (1a.4) and time as

 ˜t = t/tchar,          tchar = L/c,               c = sE      (1a.5) ρ

Derive the non-dimensional equation and obtain its weak form like we did in class.

(1b)              Now as a first step, we are imposing a force, . We are allowing the beam to deform quasi-statically (meaning, no timeF = 100N at the top face y = W on a small patch 0.48L <

dependence for this one). Obtain the deflection of the beam and include the deformed profile (use warpby vector filter).x < 0.52L 0.3H < z < 0.7H

(1c) Now we are going to let go of this force.obtained earlier as the initial condition and obtain the vertical vibrations of the beam. What’s the timeWe know that the beam should vibrate. Use the deflection period of the oscillation?

(1d)    Change the width. Plot the natural frequency of the system as a function offixed.   W and height H of the beam to Wnew = αW and H, keepingnew = βH, wherefixed. Repeat this for0 < α < 1 and,

              0keeping<β< 1                                                                                                               α                   β                                           β

α