5ml AA312 2. The vibration in the vertical direction of an airplane and its wings can be modeled as a three-degree-of-freedom system with one mass corresponding to the right wing,

one mass for the left wing, and one mass for the fuselage. The stiffness connecting the three masses corresponds to that of the wing and is a function of the modulus E of the wing, moment of inertia I, and length L. Consider a gust of wind u(t) applied upwards onto the right wing. We are interested in the time response of the aircraft when experiencing such a wind disturbance. K= f(E, IL *(XR-Xf) mi ·xi = -K[xl-xf) xẻ |mxi + k [xl-xf) = Winter 2024 xe m Hm L XfA Figure 2: Simplified model of an aircraft and its wings. *(xe-xx) ↑ ↑k (xr-xf) 4m 5 hm Ilm (a) What are the equations of motion for the fuselage and each wing? (Ignore gravi- tational effects.) Homework 03 4m xf = K (xe-xf) +F (xr-xf) 4xt-k(xe-x)-k(x-XH) E0 kwxf +K (x-xe) †k (Xf-xr) =) Xr+ m u(t) риск) t q+ m √k (xr-x+) m x₂ = u(t) - K (xr-xf) mxr +k(xr-x+) = u(x) Instructor: Leung AA312 (more working out space) Winter 2024 6 Homework 03 Instructor: Leung AA312 (b) The equations of motion can be expressed in the following matrix-vector form: [100] m040 1 -1 0 xe -1 2 -1 −1 1 001 mass natrix 0 Stiffness matrix B (Feel free to verify this.) Suppose you can only observe xf. What is the equivalent state space dynamics of the system ☀ Ax+ Bu, y = Cx (D = 0). What is the state vector x, system matrix A, input matrix B, and observability matrix C? = = Mx + Kx = bu Mä = bu кх MY (M x ) = M² (by - Kx) my (bu-kx) x = M² 0 x = or 2² = M²² bu - M'kx Winter 2024 xe + x 3EI L³ M X + KX = bu [%] [1]. 0 x 7 x 1 муко O [#] муко ↓ system parrix A = X Homework 03 x=AxtBu y=cx 0 u. X |- |~ [min] ]+[~ State rector x ить ↓ input патх в Instructor: Leung и AA312 3000kg, - (c) With the programming language of your choice, simulate the impulse response for the fuselage for 20 seconds. Let E = 6.9 × 10⁹Nm-2, L 2m, m = I 5.2 × 10-6m². Include your plots and briefly comment on the response. Include a copy of your code. Winter 2024 = 8 Homework 03 = Instructor: Leung