Build your first Djed
Design for phase-true resonance (r ≈ 1, small zeta), keep alignment high (alpha_align → 1), couple to rotation with a tuned beta and flywheel inertia I, and avoid extremes that either bleed energy (too much damping, friction, misalignment) or choke the translator (over-heavy wheel, over-strong coupling). In these limits, geometry becomes behavior—and the Ankh-Djed turns timing into power.
Principle of operation: Mass–spring oscillation + tension alignment → resonance and phase-locking. Same math, any size.
Geometry: Tri-radial layouts, rope-to-flywheel linkage, spiral/arc paths—proportionate dimensions preserve behavior.
Materials palette: Rope/cable, wood/stone frames, compliant members (springs, rubber, stacked leaf-springs). Larger = thicker/longer, not different in kind.
Measurement & tuning workflow: Find TTT, estimate ω0=k/m\omega_0=\sqrt{k/m}ω0=k/m, adjust k, mass, and damping, then re-test—identical procedure at all scales.
Energy storage logic: Rotational storage Erot=12Iω2E_{\text{rot}}=\tfrac12 I\omega^2Erot=21Iω2 grows with III; bigger radii naturally favor flywheels.
Springing / compliance:
Stiffness doesn’t scale linearly. Target ω0=k/m\omega_0=\sqrt{k/m}ω0=k/m means k must rise with mmm.
Real large springs add mass and hysteresis; consider leaf packs, elastomer stacks, or cable-spring hybrids.
Damping & loss paths:
Structural joints, rope internal friction, ground coupling, air drag—all grow with size.
Keep damping ratio ζ = c / ( 2 × √(k × m) ) low enough for build-up; isolate bases and use low-loss bearings.
Ropes/Cables:
Creep, stretch, and safety factors dominate. Choose low-creep cable (e.g., HMPE, steel), manage bend radii, and design anchor redundancy.
Tension monitoring becomes essential at high loads.
Flywheel & torque transfer:
I_f scales approximately as M × R^2 : stresses and rim speeds matter. Use conservative rim speed limits, keyed hubs, and guarded enclosures.
Engagement mechanisms (pawls, one-way clutches, eccentric links) need fatigue-rated parts.
Foundations & anchorage:
Ground and frame compliance can “steal” energy. Design stiffer pedestals or tuned isolation so motion stays in the oscillator.
True-north alignment structures must resist wind and thermal drift.
Metrology & control:
At large mass, small phase error wastes cycles. Use simple timing marks or low-cost sensors to keep forcing near resonance.
Track amplitude ceilings to avoid non-linear snap, material yield, or unintended mode switching.
Safety & operations:
Stored energy and moving mass scale quickly. Add guards, brakes, catch-ropes, and safe-to-fail restraints.
Set clear test envelopes (max stroke, max RPM, abort conditions).
Keep ratios; change sizes. Preserve the small-model proportions (arm length : flywheel radius : stroke) when scaling.
Tune k/m, not just k or m. If mass goes up 10×, aim spring stiffness ~10× to hold ω0\omega_0ω0.
Design for low ζ first, add friction later. Build amplitude and coherence before introducing work loads (grinding, lifting).
Instrument early. A tape mark and stopwatch are enough to see if you’re drifting off resonance.
The concept scales; the losses and safety are what change. If you preserve geometry, tune k/mk/mk/m for the target period, manage damping, and engineer anchors and flywheel safely, the monolith-class Djed behaves like the tabletop one—just with forces big enough to matter.
As m → 0
• Natural frequency omega = sqrt(k/m) → ∞ (hyper-stiff, twitchy)
• Momentum storage collapses; hard to phase-lock; system flickers instead of builds
• Useful for sensing, not for work
As m → ∞
• omega → 0 (very slow)
• Huge momentum storage; excellent smoothing—but hard to start/stop
• Structural loads and anchor integrity dominate
As k → 0
• omega = sqrt(k/m) → 0 (sluggish)
• Minimal restoring force; large drifts; resonance hard to target
• More “pendulum-like,” less controllable
As k → ∞
• omega → ∞ (ultra fast)
• Tiny stroke; high shock loads; energy transfer inefficient
• Risk of structural ringing and fatigue
As c (or zeta) → 0
• Q → ∞; great energy retention; easy amplitude growth
• Also amplifies noise; control becomes critical
• Good for steady drives; bad for disturbed environments
As c (or zeta) → ∞
• Motion dies; resonance vanishes; safe but useless
• Heat dissipation ↑; mechanical work ↓
As r → 0 (far below resonance)
• Quasi-static push; little amplification
• Energy mostly dissipates each cycle
As r → 1 (on resonance)
• Maximum amplification; phase-locking easiest
• Nonlinearities and limits must be managed
As r → ∞ (far above resonance)
• System can’t follow; force “skips” across mass
• Heat and noise ↑; useful work ↓
As A_d → 0
• Only coherence/solar-tidal contributions remain
• Good for sensing; slow for work
As A_d → ∞
• Rapid growth to mechanical stops; nonlinearity dominates
• Risk of burst, clipping, damage; coupling to flywheel becomes essential
As alpha → 0 (purely linear)
• Predictable resonance; narrowband gain; clean control
• Limited soft-start options; abrupt stall under heavy loads
As alpha → ∞ (strongly nonlinear)
• Amplitude-dependent frequency; shell jumps; hysteresis
• Enables soft-starts and “burst” transfer, but harder to model
As alpha_align → 0 (orthogonal to inertial axis)
• Coherent forcing ≈ 0; only local drive matters
• Phase stability suffers over long runs
As alpha_align → 1 (aligned with inertial axis)
• Maximum coherent bias; long-term phase stability ↑
• Best for low-drive, long-duration amplification
As beta → 0
• Flywheel decoupled; oscillation builds but isn’t harvested
• Great for measurement; poor for work output
As beta → ∞
• Strong back-action; oscillation throttled by rotational load
• Can overdamp the translator; tune for sweet spot
As I (or R_f) → 0
• Very responsive; poor smoothing; energy bleeds each cycle
• Output feels “choppy”
As I (or R_f) → ∞
• Smoothing and energy carry-over ↑
• Harder to spin up; start-torque and bearing loss dominate
Practical cue: choose I so that 2–5 cycles can fill it meaningfully without stalling the translator.
As rho → 0 (tiny arm or huge wheel)
• Great smoothing, weak leverage; stroke feels “geared down”
• Demands more cycles to move the wheel
As rho → ∞ (long arm or tiny wheel)
• High leverage, jerky torque; risk of stalling mid-stroke
• Sensitive to geometry error; needs damping control
As b → 0 (nearly circular wrap)
• Constant lever arm; simple but shocky under heavy load starts
• Best when loads are already moving
As b → ∞ (steep spiral)
• Strongly varying leverage; excellent soft-start/ramp
• Control becomes timing-dependent; avoid over-ramping into slack
As Q → 0
• Flat response; little amplification; forgiving but weak
As Q → ∞
• Tall, narrow peak; huge amplification; fragile to detuning
• Needs stable alignment and clean drive
As phi_err → 0
• Inputs stack perfectly; amplitude climbs predictably
• Long coherence times; ideal for low-power build-up
As phi_err → ∞
• Inputs cancel; no net build; heating and noise dominate
• Indicates mistuning, misalignment, or load mismatch
As asymmetry → 0
• Planar motion; clean energy transfer; low drift
As asymmetry → ∞
• Precession/wobble; energy spilled into unwanted modes
• Use for special effects only (e.g., directional pumping)
As mu → 0
• Minimal heat; mechanics dominate
As mu → ∞
• Heat generation ↑ but steals mechanical Q
• Use clutched or duty-cycled coupling to avoid killing resonance
Green: r ≈ 1, zeta small but not zero, alpha_align → 1, moderate I, moderate beta, modest alpha (nonlinearity), small phi_err
Red: r ≪ 1 or ≫ 1, zeta too large, alpha_align ≈ 0 for long runs, I or beta extreme, large asymmetry, uncontrolled friction
Natural frequency: omega = sqrt(k / m)
Damped ratio: zeta = c / (2 * sqrt(k * m))
Driven response (magnitude):
A = F0 / (m * sqrt( (omega0^2 − omega_d^2)^2 + (2zetaomega0*omega_d)^2 ))
Stored energy (spring): E = 0.5 * k * x^2
Rotational energy: E_rot = 0.5 * I * omega_rot^2
Alignment factor: alpha_align = cos(theta)
Spiral radius: r(theta) = a + b * theta
Phase accumulation error (approx): phi_error_per_cycle ≈ Drift_rate / omega0, with Drift_rate ≈ omega_earth * sin(latitude)