[ \mathbf{M}(\boldsymbol{\eta}) \ddot{\mathbf{q}} + \mathbf{D} \dot{\mathbf{q}} + \mathbf{K} \mathbf{q} = \mathbf{F} {aero} + \mathbf{F} {thrust} + \mathbf{F}_{control} ]
Here, (\boldsymbol{\phi}_i) is the mode shape (eigenvector) and (\eta_i(t)) is the modal coordinate (amplitude). A standard PDF will show that only the first 5 to 10 bending modes matter for flight control, as higher modes have high natural frequencies and are damped by structural damping. The holy grail of flexible rocket simulation is the nonlinear coupled ODE: dynamics and simulation of flexible rockets pdf
Modern rockets—such as the SpaceX Starship, NASA’s SLS, or the European Ariane 6—are marvels of structural efficiency. They are, essentially, oversized soda cans filled with propellant. Their high slenderness ratio (length-to-diameter) makes them prone to bending, sloshing, and vibration. They are, essentially, oversized soda cans filled with
% Load FEM results (e.g., from NASTRAN output) modes = load('rocket_modes.mat'); % Contains freq, damping, shape vectors f_flex = modes.freq(1:5); % First 5 bending modes (Hz) zeta_flex = [0.005, 0.01, 0.02, 0.03, 0.04]; % Structural damping ratios The state vector x has 12 rigid states (6DOF pos/vel) + 10 flexible states (modal displacement/velocity for 5 modes). from NASTRAN output) modes = load('rocket_modes.mat')