Problem #$2(75$ pts$)$:

The Concorde used $4$ Olympus $593$ turbofan engines. The intake system is comprised of movable ramps and spill doors that control the shock conditions within the intake and thus the airflow conditions that enter the engine. The high$-$speed turbulent air creates a harmonic axial load vector on the first intake ramp. The first ramp can be modeled as a cantilever beam with $2$ elements of different material properties and area. Element one is fixed on the left side and a harmonic load is applied to the right side of $F_3=P_{0}cos(t).$ For the system shown below find the following if $u_1=F_1=F_2=0.$

a$)$ Write the global equation of motion for the entire system. Define $(\frac{AL}{6}{)}_i$ as $m_i$ and $(\frac{AE}{L}{)}_i$ as $k_i.$ The global mass matrix $\{M_{gg}\}$ and $\{K_{gg}\}$ should be $33$ matrices.

b$)$ Reduce the system of equations to $2$ equations by eliminating row $1$ and column $1$ of $\{M_{gg}\}$ and $\{K_{gg}\}.$ If $m_1=2,m_2=1,k_1=4,$ and $k_2=2,$ then find the eigenvalues and eigenvectors of the reduced mass and stiffness matrix $\{M_{\text{r}\text{e}\text{d}}\}$ and $\{K_{\text{r}\text{e}\text{d}}\}.$

c$)$ Decouple the two equations of motion using the eigenvector expansion theorem where $\{M_{\text{r}\text{e}\text{d}}\}{}_i^T{}_i=G{M}_{\text{r}\text{e}\text{d}}^{(i)},\{K_{\text{r}\text{e}\text{d}}\}{}_i^T{}_i=G{K}_{\text{r}\text{e}\text{d}}^{(i)},$ and $\{F\}{}_i^T=G{F}_{\text{r}\text{e}\text{d}}^{(i)}.$

d$)$ Solve for the modal solution $q_2(t)$ and $q_3(t)$ based on the form of $G{F}_{\text{r}\text{e}\text{d}}^{(i)}.($step$,$ harmonic, polynomial, etc..$)$ if $P_0=5N,$ and $=2ra\frac{d}{s}.$

e$)$ Solve for the physical coordinates $u_2(t)$ and $u_3(t)$ using the mode shapes found in part b $,$ then find the unknown reaction force $R(t).$

f$)$ Finally, write the piecewise solution $u(x,t)$ for each bar element by combining $u_i(t)$ and the shape functions ${}_i(x).$

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Figure $2$ Concorde Intake Ramp Modeled as a Two Bar Element Finite Element Model