An advanced numerical simulation, convergence testing, and optimization platform developed to evaluate the frequency response of a lunar rover suspension system subjected to rough terrain kinematics. This framework transitions a traditional linear mass-spring-damper assumption into realistic nonlinear component models, mapping safe frequency operating ranges to prevent structural or payload damage from displacement and force transmissibility.
To protect onboard sensitive electronics and camera arrays during high-velocity traversal over unstructured lunar terrains, the payload suspension must maintain rigid tracking constraints:
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Payload Mass (
$m$ ): 0.5 kg -
Maximum Allowable Displacement (
$X_{max}$ ): 0.0875 m (Displacement Transmissibility limit$T_{d,max} = 1.75$ ) -
Maximum Allowable Transmitted Force (
$F_{max}$ ): 1.0 N -
Target Operational Bandwidth: 0.5 to 20.0 rad/s under a baseline peak terrain amplitude (
$Y$ ) of 0.05 m.
This pipeline evaluates a benchmark Linear System (
The mathematical simulation engine completely bypasses native solvers to execute raw, high-performance numerical analysis algorithms:
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Coupled Classical 4th-Order Runge-Kutta Solver ($O(h^4)$): Solves the coupled differential state equations for position (
$x$ ) and velocity ($\dot{x}$ ). Because acceleration depends nonlinearly on displacement differentials outside the base motion ($y(t) = Y\sin(\omega t)$), the state parameters are interlinked sequentially across alternating$k$ -step calculations. -
Self-Correcting Mesh Convergence Testing: Dynamically evaluates discretization errors. The function progressively halves the integration time step (
$\Delta t$ ) until the maximum global error tracking differential satisfies a strict displacement tolerance threshold ($E_{s,x} = 0.001\text{ m}$ ). -
Golden Section Search Optimization: Implements a localized Golden Section Search to locate the absolute maximum of the displacement transmissibility curve, pinpointing the damped natural frequency (
$\omega_d$ ) within a tolerance boundary of$E_{s,\omega} = 0.01\text{ rad/s}$ . - Automated Root-Finding Operands: Utilizes a custom boundary scanning algorithm paired with a secondary bisection loop to determine the exact frequencies where the non-linear displacement and force outputs cross safe thresholds, mapping the system's operational windows.
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Displacement vs. Force Operational Trade-offs: The optimization script successfully isolates the safe operational zones. For the nonlinear architecture, safe tracking occurs between
$[0, 2.9635]\text{ rad/s}$ and$[6.4592, 20.0]\text{ rad/s}$ , beyond which resonant frequencies violate structural force boundaries. -
Nonlinear Stiffening Effects: Numerical verification shows that the cubic stiffening elements shift the non-linear damped natural frequency upward to
$4.8455\text{ rad/s}$ , compared to the linear baseline resonance of$4.4187\text{ rad/s}$ . This highlights the severe underestimation risks associated with purely linear assumptions in aerospace engineering tracking.
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Main_Lunar_Simulation.m: Principal execution script managing data preallocation loops, plotting routines, and global orchestration. -
calculateSystemResponse.m: Core embedded$O(h^4)$ Runge-Kutta integration pipeline. -
calculateConvergedSystemResponse.m: Self-correcting iterative time-step refinement loop. -
w_d_Optimization.m: Golden Section Search optimization maximizer. -
UpperAndLowerBoundFinder.m/t_range.m: Boundary checking and bisection-driven root solvers. -
CombinedOperatingRange.m: Logical matrix comparison script resolving intersecting multi-variable boundary sets.
To compile the architecture and regenerate the structural response curves, execute the main routine within a native MATLAB workspace:
run('Main_Lunar_Simulation.m')