Dynamic Inflow Finite State — Robotics/Rotorcraft

Dynamic Inflow Finite State

Robotics / Rotorcraft

The Peters–He finite-state dynamic inflow model: the induced velocity normal to a rotor disc as a set of ordinary differential equations, one per inflow state, where each state is a harmonic m around the azimuth and a radial shape n along the blade. The inflow seen by blade b at radial element r is λ_i = Σ a_n^m·φ_n^m(r)·{cos|sin}(mψ_b), with ψ_b = ψ + 2πb/N_b and ψ' = Ω, and the states follow K·a' = |Ω|·(F − V·L(X)⁻¹·a): F is the blade load distribution projected onto each shape and harmonic, L the wake-skew matrix of the skew parameter X and V the mass-flow parameters.

• X = μ / (V_T + |λ_t|), signed, with λ_t = μ_z + (8/(3√3))·a₁⁰.
• V_T = √(μ² + λ_e²) on the first state and V = (μ² + λ_e(λ_e + ν))/V_T on the rest, with λ_e = μ_z + ν and ν = (2/√3)·[L⁻¹a]₁⁰.

Ports

• fz – the elemental force distribution, an [N_b,N_r] array of force per unit blade length (N/m, or lbf/ft in English units): one row per blade, one column per radial element. Its size sets the number of blades and elements.
• mu – the rotor advance ratio μ, a scalar [1,1], nondimensional.
• muz – the normal velocity ratio μ_z, a scalar [1,1], nondimensional.
• rho – the air density ρ > 0, a scalar [1,1] (kg/m³, or slug/ft³).
• lambda_i – the normal induced velocity ratio, the same [N_b,N_r] size as fz: the inflow at each blade's azimuth and each element's mid-radius, nondimensional.

Parameters

• Units – Metric (MKS) (the default) or English. The two are the same arithmetic; the choice only says which units the numbers are in.
• Rotor Rotational Speed – Ω in rad/s, a scalar. Defaults to 41.2596. A negative value turns the azimuth the other way; zero freezes the block.
• Rotor Radius – R, a scalar > 0 (m, or ft). Defaults to 5.08.
• Radial Locations Source – where the radial element edges come from:
  • Uniform distribution based on force input (the default) – N_r equal elements spanning the whole blade, edges 0, 1/N_r, …, 1.
  • Custom distribution – the edges come from the parameter below.
• Non-dimensional Radial Element Edges – the element edges under Custom distribution: N_r+1 strictly increasing values in [0,1]. Defaults to -1, which the uniform setting ignores.
• Highest Harmonic Index – M, a whole number from 0 to 6. Defaults to 2.
• Compute Spatial Modes – how many radial shapes each harmonic carries:
  • Consistently upgraded (the default) – harmonic m carries n = m+1, m+3, … up to M+1: (M+1)(M+2)/2 inflow states in all.
  • Constant – every harmonic carries N shapes, n = m+1, m+3, …, m+2N−1: (2M+1)·N inflow states.
• Number Of Spatial Modes – N, a whole number from 1 to 6, used under Constant. Defaults to 2.
• Integration Substeps – how many fourth-order Runge-Kutta steps a sample is integrated with on the discrete solver and in exported code, a whole number of 1 or more. Defaults to 100. No Simulink counterpart.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. A core holds the states, publishes the inflow from them, and integrates one sample with the inputs held and the substeps above. Everything the configuration fixes – the radial shapes, the Peters–He coefficients, the element weights – is folded to constants, and the wake-skew system is solved by straight-line elimination, so the emitted body grows with the number of states: exports are offered up to about 3000 elimination operations per derivative (every Consistently upgraded setting qualifies; the largest Constant ones do not, and asking for one logs why).

The three HDL targets are simulation-only real arithmetic, quantized at the port: square roots, divisions by signals and sines of a state have no Q16.16 form. The cores simulate correctly and are not offered as synthesizable.

Simulink bridge

Import and export, mapped to Aerospace Blockset's aerolibinflowmodels/Dynamic Inflow (Finite-State): Units → units, Rotor Rotational Speed → omega, Rotor Radius → radius, Radial Locations Source → radDistrSrc, Non-dimensional Radial Element Edges → radDistr, Highest Harmonic Index → indexHarmonics, Compute Spatial Modes → methodSpatialModes and Number Of Spatial Modes → numSpatialModes, the enumerations 1:1 and therefore lossless. Always emitted with omegaSrc = Dialog: taking the rotational speed from a port moves that block's port list, and this block has one. "Sampling Time (s)" does not cross: the Simulink block is continuous and defines no SampleTime parameter.

Notes

• Stateful, continuous and nonlinear: one continuous state per inflow state plus the azimuth, all starting at zero. The output reads the state alone, so the block has no direct feedthrough.
• The sign of μ matters: X is signed, unlike the 3-State block's. Measured against the Simulink block.
• The Simulink block limits μ and μ_z to [−1,4], refuses ρ ≤ 0 and a harmonic index or mode count above 6. This block refuses the last two at configuration and does not stop a run for the first three; it guards the divisions instead, so ρ ≤ 0 gives no forcing and a total flow of zero gives no damping.
• V is negative wherever μ² + λ_e(λ_e + ν) is, which is the model's own behaviour and makes the higher states unstable. Keep μ_z + ν clear of zero, as the Simulink block's own warning asks.