Some Mechanical and Structural Aspects of the Smolensk Crash
By Dr. Gregory Szuladzinski, Ph.D., MSME
Appendix V. Aircraft Rotation around the Longitudinal Axis
The rolling began near critical point K and was due to loss of lift on the wing. The following calculation is approximate and simplified. The inertia data of the aircraft is assumed constant in spite of the accident, during which the lift force vanishes on the left wing together with the loss of part of that wing.
Takeoff mass: M0 = 110.5 t
mass at point K: M = 78.6 t
moment of inertia around the longitudinal axis, at startup: I0 = 1928 t-m2
moment of inertia around the longitudinal axis, point K: I = 1371 t-m2
(Assumed in proportion to the mass)
estimated lift force on the wing P ≈ 0.4 Mg = 0.4x78, 600x9.81 = 308,400 N
estimated distance of the center of aero of the wing from longitudinal axis: r = 8.2 m
Wing moving upwards (right) will develop less lift than the same wing in normal flight, because it reduces the effective angle of attack. Here we assume a loss of 20%.
torque M = 0.8Pr = 0.8x308,400x8.2 = 2.023 x106 Nm
angular acceleration: ε = M / I = 1.476 rad/s2
the time t it takes the turn by the angle θ: θ = εt2 / 2, i.e. for θ = 90°:
Over time, the state of affairs will change and it is unlikely that such acceleration could be continued. Therefore, we assume that the angle of 90° above the angular velocity is constant:
ω = ε t =1.476x1.459 = 2.153 rad/s
time to reach 1200 : (Δθ=30°), Δt = Δθ/ω = 0.243 s, t120 = 1.459 + 0.243 = 1.70 s
time to reach 200°: (Δθ=110°), Δt = Δθ/ω = 0.892 s, t200 = 1.459 + 0.892 = 2.35 s
One can obtain more precise results if, instead of the above intuitive assumptions, one uses more accurate data that can be obtained from the aerodynamic simulation.
According to the MAK report, the time to reach the turn of 2000 was 6 s, but it is unclear whether this time was calculated in a manner consistent with this calculation. Of course, if not all of the lift on the wing has lost, turning was slower than the above shows and the calculated time would be longer.