Try it! - Modal Truncation

Problem

Given is a simply supported Euler-Bernoulli beam:

 The eigenfrequencies $//<![CDATA[ \begin{array}{l}\omega_i\end{array} //]]>$ and mode shapes $//<![CDATA[ \begin{array}{l}\phi_i\end{array} //]]>$ can be found analytically with

Consider the two load cases

• A: $//<![CDATA[ \begin{array}{l}1\end{array} //]]>$$//<![CDATA[ \begin{array}{l}\mathrm{N}\end{array} //]]>$ at $//<![CDATA[ \begin{array}{l}\frac{l}{2}\end{array} //]]>$
• B: $//<![CDATA[ \begin{array}{l}0.5\end{array} //]]>$$//<![CDATA[ \begin{array}{l}\mathrm{N}\end{array} //]]>$ at $//<![CDATA[ \begin{array}{l}\frac{l}{4}\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\frac{3l}{4}\end{array} //]]>$

Which modes do not have a contribution to the beam's displacement following load case A, B, and the combination of A and B? Validate your result by calculating the response using the mode superposition method.