Since finite element models (FEM) of rockets contain millions of degrees of freedom (DOF), direct simulation is computationally impossible for real-time control. Instead, engineers extract the lowest-frequency normal modes.
[ \mathbfw(\mathbfu, t) = \sum_i=1^n \boldsymbol\phi_i(\mathbfu) \eta_i(t) ]
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. dynamics and simulation of flexible rockets pdf
Liquid dynamics are notoriously difficult to model. In simulation, sloshing propellant is often represented as a mechanical analog—a "pendulum" or a "spring-mass-damper" system attached to the tank walls. This simple model predicts the forces the sloshing liquid exerts on the airframe.
Use standard missile equations (body axes). Include thrust, gravity, aerodynamics (lift, drag, pitch moment). Since finite element models (FEM) of rockets contain
Modern launch vehicles (e.g., SpaceX Starship, SLS, Ariane 6) are long, slender, and lightweight to maximize payload fraction. This structural flexibility introduces critical dynamics:
A rigid-body model is insufficient—flexible-body dynamics are essential for stability, payload comfort, and trajectory accuracy. The complete nonlinear equations for a flexible rocket
The complete nonlinear equations for a flexible rocket can be derived via Lagrange’s equations or Kane’s method. A simplified form of the constrained equations is:
To run a flexible simulation on flight hardware, use:
To simulate a flexible rocket, one cannot use Newton-Euler equations alone. The industry standard is the Lagrangian approach combined with Modal Analysis.