Instructors:
Dr. Ismail H. Tuncer
Dr. Yusuf Ozyoruk
Dr. Nilay Sezer Uzol
Assistants: H.C. Onel, I. Tandogan and C. Erdogan
AE305 - NUMERICAL METHODS
HOMEWORK-II
October 25, 2019
Due on: Wednesday, November 6 @ 9.00
SOLUTION OF A SYSTEM OF ODE’S
A simplified MD3-160 aircraft wing model mounted on translational and rotational elastic
supports is shown in the figure.
The following second order ordinary differential equations approximate the vertical and angular
displacements of the wing, zand α, respectively :
Md2z
dt2+cz
dz
dt +kzz=1
2ρ∞U∞SCLsU2
∞+dz
dt
2
Id2α
dt2+cα
dα
dt +kαα=1
2ρ∞U∞SCLasU2
∞+dz
dt
2
The wing has a projected area, S= 15 m2, mass, M= 150 kg, mass moment of inertia,
I= 50 kg ·m2,a= 0.15 m, and assume that
Vertical stiffness, kz= 20 kN/m
Vertical damping, cz= 300N/m/s
Angular stiffness, kα= 15 kN m/rad
Angular damping, cα= 150 Nm/rad/s
CL= 2πα when α < 12 deg
The wing is subjected to a steady flow with a freestream velocity, U∞= 40 m/s, and at t= 0 it
is set to an angle of attack α(0) = 5 deg at z(0) = 0 p osition. Integrate the governing equations
in time to determine the dynamics of the wing. Use the incomplete 4th order classical
Runge-Kutta solver provided.
◦Plot the time variations of vertical displacement and angle of attack
◦Plot the time variations of vertical and angular velocities
◦Plot vertical velocity vs vertical displacement.
◦Plot angular velocity vs angle of attack.
◦Take cα= 90 N·m/rad/s, and discuss the results
◦Experiment with 20 < U∞<60m/s
◦Experiment with the step size.
◦Apply adaptive stepping for a bonus.
