OQG MEGA REPEATED-PULSE CHARACTERIZATION V0.1
20 October 2026 | Prepared for Todd J. Desiato

PURPOSE AND STATUS
Characterize a powered climb from b_min to b=1 at a fixed voltage swing,
including the disk and PZT kinetic energy, elastic restraint, preload,
return impulses, electrical storage, losses, and driver recovery.
This is a conditional INVERSE-DYNAMICS load study. A desired periodic motion
and supporting-field load are specified; the voltage needed for them is
calculated and scaled to 100 V peak-to-peak. The waveform shape is allowed to
change across the sweeps. It is not a test of one unchanged voltage waveform.
The field response b(x,t) is required by the load, not predicted from voltage.
Full net external impulse retention and zero extra field-control power are
the reference boundary conditions, not measured efficiencies. Their effects
are displayed separately. No catalog device has been selected or optimized.

INPUTS: WHAT IS SOURCED AND WHAT IS A REFERENCE CHOICE
SM111 manufacturer small-signal constants [1]: d33=320 pm/V; Y33=73 GPa;
density=7900 kg/m^3; free relative permittivity=1400 at 1 kHz; tan(delta)=0.4%
at 1 kHz, i.e. 0.004, NOT 0.4. The listed ceramic Qm is 1800; it is NOT used
as the quality factor of an assembled actuator. Radial frequency constant
Np=2200 Hz m is used only for a coarse bandwidth screen.
Illustrative geometry: ten 25.4 mm diameter, 1 mm thick elements, mechanically
in series, electrically in parallel. This is not a verified part assembly.
Reference drive: unipolar 0-100 V, 1 kHz, 10 g disk; 25% extension and 75%
return. Each return velocity ramp lasts 12.5% of the full cycle.
Chosen preload spring stiffness=5% of stack stiffness; illustrated preload
50 N. Both are adjustable screening inputs, not a recommendation or rating.
Chosen assembled Q=100 at the 10 g reference; corresponding viscous damping
is held fixed in mass sweeps. Delivery efficiency=95%, recovery=90%; no
quiescent or extra field-control power in the reference. These are scenarios.
Constant dielectric loss across the spectrum is an extrapolation of the
1 kHz small-signal value. High-field hysteresis and quadratic electrostriction
(including 2f), amplitude-dependent capacitance and stiffness, adhesive and
electrode masses, transverse and distributed modes are not fitted here.

MECHANICAL MODEL AND MOVING STACK
Let m be disk mass, ms stack mass, L stack length, A area, k=Y*A/L,
alpha=N*d33, theta=k*alpha, kl preload stiffness, and d viscous damping.
Assume axial stack displacement u(z)=x*z/L. Its kinetic energy is
K_stack=ms*v^2/6 and its momentum is ms*v/2. Therefore the generalized
inertial mass me=m+ms/3 differs from the momentum coefficient mp=m+ms/2.
K_total=me*v^2/2. These distinct coefficients avoid confusing kinetic energy
with the force at the ship end of a distributed stack.

For cycle phase s=f*t, extension fraction D=0.25, and 0<=u=s/D<=1:
x=X*[u-sin(2*pi*u)/(2*pi)]
v=(X*f/D)*[1-cos(2*pi*u)]
a=(2*pi*X*f^2/D^2)*sin(2*pi*u)
The disk reaches peak velocity at s=D/2 and stops at maximum extension s=D.
The ambient return has smooth cosine velocity ramps and a constant-velocity
middle. Position, velocity, and acceleration are continuous and periodic.

REQUIRED HILL AND ENERGY CLOSURE
Let lambda=DeltaUg/K_total,peak. The reference lambda=2 ensures that even
ALL the disk-plus-stack kinetic energy cannot finish the climb.
Let G be the inward environmental force magnitude on the disk:
  release half:       G=0
  restoration half:   G=-lambda*me*a  (a<0)
  ambient return:     G=0
Set Ug=-lambda*K_total during extension and Ug=0 during return.
This is a selected trajectory through the supporting state, not a universal
relation between velocity and b. During release, the state changes explicitly
in time; during restoration the disk climbs a locally stationary gradient.
The required explicit-time field power is Pb=-lambda*me*v*a in release,
and Pb=0 in restoration and return. It removes DeltaUg in the release phase.
The corresponding local field derivatives, to leading order near b=1, are:
  partial_x b=G/(m*c0^2),   partial_t b=Pb/(m*c0^2).
They satisfy db/dt=partial_t b+v*partial_x b along the selected path.
This construction supplies the force and energy requirement to a future
scattering/stress calculation; it does not establish that source solution.

Disk pushing force Sdisk=m*a+G. Its restoration work is
Wdisk,2=DeltaUg-K_disk,peak, strictly positive in the reference.
Beyond all moving kinetic energy, the actuator must supply
Wadded=DeltaUg-K_total,peak=(lambda-1)*K_total,peak > 0.
Preload elastic storage and dissipation add further requirements. Electrical
work during restoration is calculated directly and is positive in the
reference. It is not equated to the smaller net consumed energy per cycle.
For positive ship force throughout restoration, require lambda*me>mp.
The reference lambda=2 and sensitivity range 1.6..6 satisfy that condition.

FIXED VOLTAGE SWING, LOADED STROKE, AND PZT TENSION
Generalized inverse dynamics gives:
Vraw=[(k+kl)*x+d*v+me*a+G]/theta.
First evaluate this for unit stroke, then set X=Vpp/ptp(Vraw_per_unit_stroke).
Thus free stroke is not imposed as the loaded stroke. Add a DC shift to put
the applied voltage at 0..Vpp, and shift the static equilibrium by
x_offset=theta*voltage_bias/(k+kl). The geometry is referenced to minimum
cycle extension; this static offset does not affect cycle velocity or impulse.
The reported preload is the force at minimum stroke in this shifted frame.
The piezo compression forces are
Ctop=P0+kl*x+d*v+m*a+G,
Croot=Ctop+(ms/2)*a.
The assumed linear rod mode gives intermediate forces between these ends.
Minimum preload to avoid axial tension is the negative of the smallest
dynamic end force, or zero if both remain compressive. PZT restraint is a
real elastic load and is not counted as external propulsion reaction.
No allowable ceramic tension, preload maximum, bond strength, or fatigue
life has been assumed. The full mechanical design requires those ratings.

SHIP FORCE, RETURN, AND AVERAGE THRUST
After subtracting the onboard preload spring and damping reaction, the
ship force in the propulsion direction is Fship=mp*a+G.
Every cycle starts and ends at zero internal velocity. Therefore
J=integral(Fship dt)=integral(G dt)=lambda*me*v_peak.
Average conditional thrust is Fmean=f*J, with the entire return included.
Multiplying by rJ gives the net assembly value when impulse retention differs
from the ideal boundary condition. Peak force is not average thrust.
With G=0 and no supporting-state change, the cycle impulse is zero to
numerical precision. An internal piezo/preloader/oscillating mass alone is
not assigned a propulsion effect by this calculation.

ELECTRICAL MODEL AND POWER
Free capacitance Cf=N*eps0*epsr*A/t. The reciprocal linear two-port uses
Cs=Cf-theta^2/k > 0, and charge Q=Cs*V+theta*x.
Its ideal current is Cs*dV/dt+theta*v. A harmonic dielectric-loss current
adds real dissipation using Cf and tan(delta). Fourier harmonics are retained.
The electrical-plus-elastic storage used for the complete balance is
Estored=Cs*V^2/2+(k+kl)*x_abs^2/2. It returns to its initial value each cycle.
Static preload contributes constant-force work that cancels against its
preload reaction; it does not provide cycle energy.
Terminal instantaneous power is V*I, including motional current.
Eplus is its positive integral; Eminus is the magnitude of its negative
integral. Net terminal energy=Eplus-Eminus=DeltaUg+Eviscous+Edielectric.
Source power=f*(Eplus/eta_delivery-eta_recovery*Eminus)+Padditional.
Additional power includes amplifier idle power and any separate field-control
power; setting it to zero is an optimistic reference, not a measured value.
The lossless-cycle ratio is F/P=2/v_peak for this chosen trajectory.
The photon comparison is (F/P)*c0. Figures primarily show engineering N/W
and do not imply experimental verification of their conditional numerators.

NEAR-AMBIENT OQG MASS CHANGE
DeltaUg=m*c0^2*(1-bmin). The reference 1-bmin is about 1.54e-22 and the
fractional axial inertia increase bmin^-3-1 is about 4.63e-22. Throughout
the displayed base map the deficit remains below about 1.65e-19.
The low-speed, near-ambient mass approximation is therefore far more accurate
than the mechanical model. The model does not generate a large heavy/light
inertia swing; its conditional thrust comes from the required spatial field
gradient. Prescribing a much lower bmin at the same voltage would require
checking the vastly larger restoration energy, not simply changing mass.
Compute delta=1-b directly from energy; subtracting it from 1 in float64
would round away the effect. CSV columns retain delta and 3*delta explicitly.

MODEL BANDWIDTH SCREEN
The harmonic frequency containing 95% of capacitive-current weight is
compared with 0.2 times the smaller of the estimated loaded axial frequency,
unloaded longitudinal rod frequency, and radial ceramic frequency.
This 20% criterion is an analyst-selected conservative screen, not a product
rating. Dotted/hatch regions are extrapolations. Solid curves also retain
all source-response assumptions. Exact poles, resonance enhancement,
stability, ringing, contact loss, driver bandwidth and current saturation,
and thermal behavior require a forward distributed model or measurements.
The waveforms here are periodic desired cycles, not startup transients.

VALIDATION AND REPRODUCTION
Reference traces use 16384 samples/cycle; curves use 2048 and maps 1024.
Energy and momentum integrals are independently compared with closed forms.
A zero-field passive control gives zero cycle impulse. A 4x reference-grid
refinement checks net driver power. See data/verification.json.
All physics parameters are exposed in parameters.json. Edit that file, run
python build_plots.py, then python build_report.py. Requirements: Python 3,
numpy, scipy, matplotlib, reportlab, pypdf. Models and CSVs use SI units.
The report uses the currently generated results. Some plot captions show the
default reference case and should be updated if default parameters change.

WHAT TO DETERMINE BEFORE PARTS OPTIMIZATION
1. Driven source response: map actual voltage, strain/stress, acceleration
   history, and surrounding field to b, partial_x b, partial_t b and external
   boundary stress, including reactions on all onboard hardware.
2. Actual stack dimensions/wiring, disk mass, preload, stroke-versus-voltage,
   complex impedance, assembled damping, nonlinear strain/2f coefficient,
   admissible tension/compression and frequency-dependent behavior.
3. Driver voltage bias, current/bandwidth limits, electrical recovery,
   quiescent losses and additional field-control energy.
These plots identify the mechanical and energy targets for that work. They
do not choose an optimum by maximizing an assumed field load.

SOURCES (accessed 20 October 2026)
[1] STEMINC, Piezo Material Properties, SM111 column.
https://www.steminc.com/piezo/PZ_property.asp
[2] Physik Instrumente, Forces & Stiffness.
https://www.physikinstrumente.nl/en/expertise/technology/piezo-technology/properties-piezo-actuators/forces-stiffnesses
[3] Physik Instrumente, Dynamic Operation.
https://www.physikinstrumente.nl/en/expertise/technology/piezo-technology/properties-piezo-actuators/dynamic-operation
[4] Physik Instrumente, Electrical Operation of Piezo Actuators.
https://www.physikinstrumente.com/en/expertise/technology/piezo-technology/properties-piezo-actuators/electrical-operation
[5] Todd J. Desiato, Mach Effect Gravity Assist Warp Drive in Operational
Quantum Gravity, manuscript V0.7, 20 October 2026: Eqs. (6), (9)-(23), (26)-(28).
