Problem $4.$ Twin Engine or Four Engine?

When the twin$-$engine Airbus A$330$ was developed, the company also designed a four$-$engine Airbus A$340$ for long$-$distance routes over the Pacific

Ocean. To save cost, the company decided to use the same fuselage and wing structures for these two aircraft, the only difference being the engines.

The design team wants to know the natural frequency difference between these two wing$-$engine assemblies.

Suppose $m_w$ is the total wing mass, $m_{330}=\frac{1}{3}{m}_w$ is the mass of the A$330$ engine, and $m_{340}=\frac{1}{5}{m}_w$ is the mass of the A$340$ engine.

$L$ is the total wing span, and the positions of these engines are shown in the schematic diagram.

What is the percentage difference in natural frequency between these two aircraft?

Reminder: As we discussed in Self$-$Learning Module $2,$ the cantilever beam's deformation is $w(t)=\frac{x^2}{2L^3}(3L-x)y_t$