Aerospace Fuel Tank Center of Mass and Slosh Baffle (Aerospace & Mechanical Engineering)

An aerospace structural dynamics engineer at the UNLV Howard R. Hughes College of Engineering is designing a lightweight titanium fuel cell integrated into the wingtip structure of an autonomous reconnaissance aircraft.

The interior reservoir of the fuel cell is bounded by the three coordinate planes $x = 0$, $y = 0$, $z = 0$ and the upper inclined wing-skin aerodynamic fairing plane:

Due to internal anti-slosh porous baffle foam and fuel pressurization, the effective mass density $\rho(x, y, z)$ within the cell increases with vertical height according to:

To evaluate wing aeroelastic flutter and provide precise mass properties for the flight-control autopilot:

1. State the general Type I boundary equations for the tetrahedral solid region $E$ in the order $dz \, dy \, dx$.
2. Formulate and evaluate the triple integral for the geometric volume $V(E) = \iiint_E 1 \, dV$ in cubic meters ($\text{m}^3$).
3. Formulate and evaluate the triple integral for the total fuel mass $M = \iiint_E \rho(x, y, z) \, dV$ in kilograms ($\text{kg}$).
4. Evaluate the first moment of mass about the $xy$-plane, $M_{xy} = \iiint_E z \rho(x, y, z) \, dV$, and determine the vertical center of mass coordinate $\bar{z} = \frac{M_{xy}}{M}$ in meters ($\text{m}$).
5. Explain why tracking shifts in the vertical center of mass $\bar{z}$ as fuel drains is critical for preventing wingtip stall and maintaining roll trim stability during high-$g$ maneuvers.