11-EFT.WP.Core.DrawingKinetics v1.0 | Chapter 5 Boundary Conditions and Operating Scenarios

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Chapter 5 Boundary Conditions and Operating Scenarios

I. Scope and Objectives

• Define the family of physical boundary conditions at the inlet ell = 0 and outlet ell = L(t): velocity/strain-rate drive, tension drive, impedance (Robin), and winding/unwinding boundaries.
• Unify clamping/slip/friction gauges and the coupling of temperature and environmental disturbances at the boundaries, so that geometry in Chapter 2 and conservation in Chapter 3 close consistently, and constitutive identification in Chapter 4 is well-posed.
• Deliver the Mx-10 operating-card template as the minimal reproducible set for experiments, simulations, and runtime, and bind it to I10-* implementations.

II. Terminology and Symbols

• Geometry & kinematics: lambda(ell,t), e = ln( lambda ), s = ( d/dt ) e, v(ell,t), A(ell,t), path gamma(ell), measure d ell.
• Conservation & power: rho_L = rho * A, J = rho_L * v, T_fil(ell,t), boundary power P_b = T_fil * v (evaluated at the boundary).
• Boundary-mode enumeration: BC.DIRICHLET(v|s), BC.NEUMANN(T), BC.ROBIN(Z_b), BC.WINDING, BC.UNWINDING, BC.CLAMP(mu,N).
• Environment & temperature: Theta_K(t) (boundary or ambient temperature); impacts on constitutive parameters per Chapter 4.

III. Postulates and Minimal Equations

1. P11-3 (boundary conservation and passivity)
   • For any boundary b ∈ {in,out}: J_out - J_in = ( d/dt ) ( ∫_0^{L(t)} rho_L d ell ).
   • Boundary power must not inject spurious negative non-physical energy: P_b >= - P_env, where P_env is declared external supply or dissipation.
2. Inlet/outlet basic gauges (aligned on ts)
   • Velocity drive (Dirichlet)
     1. v_in(ts) = v_cmd(ts) or s_in(ts) = s_cmd(ts).
     2. Corresponding mass flux: J_in = rho_L_in * v_in, with rho_L_in = rho_in * A_in.
   • Tension drive (Neumann)
     T_fil,in(ts) = T_cmd(ts); v_in is supplied by internal dynamics.
   • Impedance boundary (Robin)
     T_fil,b = Z_b * ( v_b - v_ref ) + T_0, with Z_b >= 0.
   • Winding / Unwinding
     v_out = omega_w * R_w; T_fil,out follows spool dynamics and friction.
3. Spool-radius evolution (layer-averaged, S12-5R)
   Capacity conservation gives dV/dt = A_out * v_out = 2 * pi * R_w * W_eff * ( dR_w/dt ), hence
   ( d/dt ) R_w = ( A_out * v_out ) / ( 2 * pi * R_w * W_eff ), with W_eff the effective laydown width.
4. Clamp and slip criterion (S12-5S)
   Static-friction cap: T_transfer <= mu_s * N. If T_req > mu_s * N, slip occurs and the boundary switches to
   T_fil = mu_k * N * sign( v_rel ) with v_rel > 0, v_rel = | v_jaw - v_b |.
5. Boundary-power ledger (consistent with Chapter 3)
   P_in = T_fil,in * v_in, P_out = T_fil,out * v_out; net mechanical power into the system P_net = P_in - P_out - P_loss,boundary.

IV. Data Gauges and Manifest (Mx-10 Operating Card)

1. Core fields
   • timebase.model : "linear", alpha : float, beta : float (so ts = alpha + beta * tau_mono).
   • length.L0 : float (initial effective length), A_in|out : float|series, rho_in : float|series.
   • mode.in : "v"|"s"|"T"|"Z"|"winding"|"unwinding"|"clamp"; mode.out : same options.
2. Inlet gauges (example keys)
   • in.v_cmd(t) or in.s_cmd(t) or in.T_cmd(t); in.Z_b; in.A; in.rho; in.Theta_K.
   • in.ramp : {"type":"S-curve","T_rise":float,"C2":true} (smooth start/stop).
3. Outlet gauges (winding/unwinding)
   • out.omega_ref(t) or out.v_cmd(t); R_core, W_eff, J_w (equivalent inertia), b_w (viscous damping).
   • out.friction : {"mu_s":float,"mu_k":float,"N":float}; out.backdrive.Z (backdrive impedance).
   • out.Theta_K (thermal boundary), out.h_conv (convective coefficient), Theta_amb.
4. Clamp/slip
   clamp.mu_s, clamp.mu_k, clamp.N, clamp.v_jaw(t), clamp.mode:"stick|auto".
5. Mass and power gates
   gate.mass : eps_mass_max, gate.power : eps_power_max, gate.bc.dim : true.
6. Traceability and versioning
   scenario.id, schema.version, seed, hash.config, hash.bc.

V. Algorithms and Implementation Bindings

1. I10-1 update_draw_state(state, bc:dict, dt:float) -> StepReport essentials
   • Parse boundary modes and commands; generate targets for v_in|s_in|T_in and v_out|T_out; apply C2 ramps to commands.
   • Clamp criterion: if T_req > mu_s * N, switch to slip; set T_fil,in = mu_k * N * sign( v_rel ) and release v_in.
   • Winding update: compute R_w^{n+1} via S12-5R; drive or constrain omega_w to enforce v_out = min( v_cmd, omega_w * R_w ).
   • Impedance boundary: evaluate T_fil,b = Z_b * ( v_b - v_ref ) + T_0 and close with constitutive tension.
   • Conservation alignment: write J_in = rho_L,in * v_in, J_out = rho_L,out * v_out, and synchronize energy/power terms.
   • Emit StepReport.bc: includes mode.switch, slip.events, R_w, P_in/out, violations.
2. I10-3 compute_instability_metrics(state) -> dict (boundary extensions)
   Return necking.proximity, slip.risk = T_req / ( mu_s * N ), spool.saturation = R_w / R_max.
3. I10-5 emit_metrics_drawing(state) -> dict must include
   TS.bc.slip_rate, TS.bc.switch_count, TS.bc.power_balance = ( P_in - P_out ) / max( |P_in|,|P_out| ).

VI. Metrology Workflow and Run Graph

1. Mx-10 operating-card creation and verification
   • Collect equipment geometry and drive capability: R_core, W_eff, J_w, b_w, mu_s/mu_k/N.
   • Choose boundary modes and trajectories: v_cmd(t) or T_cmd(t); configure ramping.
   • Declare thermal boundaries: Theta_amb(t), h_conv; if temperature coupling is enabled, align with Chapter 4 parameter gauges.
   • Time-base alignment: map raw tau_mono to ts = alpha + beta * tau_mono and unify windows.
   • Mass and power gates: compute eps_mass = | ( J_in - J_out - d/dt M_line ) | / scale and eps_power; both must be under thresholds.
   • Produce scenario.id and hash.bc as ingestion and reproduction anchors.
2. Alerts and rollback
   • A_BC_SLIP_EXCESS: slip.risk > 1.2; reduce speed and increase N, or switch to tension drive.
   • A_SPOOL_SAT: R_w approaching physical limit; enter a safe S-curve stop.
   • A_POWER_IMBAL: | TS.bc.power_balance | > thresh; audit sensors or impedance configuration.

VII. Verification and Test Matrix

1. Minimum required cases
   • Velocity-driven steady conservation: constant v_in = v_out = v0; verify J_in ≈ J_out and P_net ≈ 0 (no dissipative boundary terms).
   • Tension-driven step: step T_in to T1; verify inlet-region transient of v consistent with Chapter 4 constitutive laws.
   • Impedance match: given Z_b, sweep v_ref; measured T_fil matches H_sigma_s (Chapter 7 gauge).
   • Spool-radius growth: constant A_out, v_out; relative error of R_w(t) against S12-5R ≤ 1e-2.
   • Clamp slip: increase T_req until slip triggers; verify T_fil ≈ mu_k * N and v_rel > 0.
2. Boundary and extreme scenarios
   Low-temperature, high-viscosity drive saturation; high acceleration with significant J_w coupling; robustness to R_w jitter when W_eff exhibits gaps.

VIII. Cross-References and Dependencies

• Chapter 2: definitions and measure mapping for lambda, e, s, A.
• Chapter 3: conservation ledgers for J = rho_L * v and for P_in/out.
• Chapter 4: boundary compatibility with constitutive T_fil and passivity constraints.
• Chapter 7: spectral gauges and frequency-domain verification of impedance boundaries.
• Core.Metrology: time bases and thermal-boundary metrology. Core.DataSpec: operating-card schema. Core.Threads: alerts and TS.* metrics.

IX. Risks, Limits, and Open Questions

• The uniform-winding assumption (W_eff constant) fails under reciprocating laydown or edge reversal; a position-dependent W_eff(y,t) and layer-resolved models may be required.
• Rate and temperature dependencies of clamp friction are not explicit in S12-5S; under extreme conditions use mu( v_rel, Theta_K ).
• Linear approximation of boundary impedance can misfit strong back-drive servo systems; consider frequency-dependent Z_b( omega ) and delays.
• Sensor saturation or drift can corrupt P_net balance; coordinate with Chapter 8 error-budgeting.

X. Deliverables and Version Management

1. Artifacts
   • Mx-10 operating-card template and audit scripts (including gate.mass, gate.power, gate.bc.dim).
   • Reference implementations: boundary-update modules (winding, impedance, clamp-slip) and StepReport.bc field definitions.
   • Example scenarios and replay data: constant-speed winding, tension step, impedance sweep.
2. Version strategy
   New boundary types or energy-ledger changes are marked ADD/MOD; compatibility flag compat.bc.v1 accompanies migration notes in Appendix C.