# Tutorial 5

<h1>Performing modal analyses using CMISS</h1>

<p>In this laboratory we will use modal analysis to investigate the frequencies and shapes of vibration modes of various bodies. This is discussed in Chapter 6 of the <a href="/documentation/course_notes/fembem_notes/" target="new">FEM/BEM Notes</a>.</p>

<h1>Modal analysis of a plate</h1>

<p>The first body is the standard plate that you should now be familiar with. Run through <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/4/45/451/index.html" target="new">example_451</a>.</p>

<h1>Modal analysis of a clamped beam</h1>

<p>You should now be familiar with the steps involved with solving a modal analysis problem. We will now analyse a common engineering structure: that of a steel beam clamped at both ends. The beam considered here is a steel beam of length *l*=1m, breath *b*=1m and depth *d*=1m. Run through <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/4/45/452/index.html" target="new">example_452</a>. </p>

<p>Compare the first three modal frequencies you obtain with the frequencies obtained by analytically solving the vibrating beam equation. The analytic frequency for mode *n* is</p>

<p>f_n = (pi/2) sqrt{EI/(rho A *l*^4)} (beta_n *l*)^2</p>

<p>where *E* is the Young's Modulus, *I* the second moment of area for the beam - for the beam considered here *I*=(*b* *d*^3)/12 where *b* is the breath of the beam and *d* the depth of the beam - rho is the density of the beam, *l* the length of the beam, and beta_n is the *n*^th root of the equation:</p>

<p>cos(beta *l*) cosh(beta *l*) = 1</p>

<p>The first three roots of this equation are: (beta_1 *l*) = 4.730040745; (beta_2 *l*) = 7.853204624; (beta_3 *l*) = 10.99560784.

<p>Which modes are the most accurate in the finite element model?</p>

<h1>Modal analysis of a steel building</h1>

<p>Consider now the modal analysis of a steel building frame. Run through <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/4/45/453/index.html" target="new">example_453</a>. Before you draw each mode shape, have a think about what you expect the mode shape to be.</p>

<p>Now consider the case of using Mass Lumping. Under a mass lumping assumption all the mass of the building is concentrated at the nodes instead of being evenly distributed throughout the length of the beams and columns. This has the effect of diagonalising the global mass matrix. Rerun example_453 except answer 'Y' to the question: 'Do you want mass lumping' when you define the solution parameters. How have the modal frequencies changed?</p>

<hr> <p><b><a href="http://cmiss.bioeng.auckland.ac.nz/cgi-bin/lab-question-form.cgi?stage=1&laboratory=5" target="new">Tutorial 5 Quiz</a></b> | Tutorial 4 | Tutorial 6</p> <hr>