OQG MEGA: 35 kHz and beyond - V0.2
20 October 2026. Conditional inverse-load characterization, not a validated
prediction of MEGA thrust. The manuscript itself is not revised by this study.

1. PURPOSE AND WHAT CHANGED
The reference frequency is now 35 kHz, with mass and frequency sweeps from
2 to 200 g and 1 to 200 kHz. Vpp remains 100 V. No independent slider for bmin
or its spatial gradient is used. The hill rule below is held fixed. The earlier
linear-in-position stack approximation is replaced by an analytic distributed
axial rod, including internal kinetic energy, elastic storage, wave propagation,
Kelvin-Voigt dissipation, and the physical stack momentum. This is still a
one-dimensional linear surrogate; radial, mount, electrode, bond and nonlinear
material dynamics are not solved. The disk alone is assigned the OQG hill load.

2. MATERIAL AND SCENARIO PARAMETERS
All values are explicit in parameters.json. Ten parallel-electrically-connected
SM111 layers, 25.4 mm diameter, each 1 mm thick: total length 10 mm; stack mass
40.0299 g. Manufacturer values used: density 7900 kg/m^3, Young's modulus
73 GPa, d33=320 pm/V, relative free permittivity 1400, tan(delta)=0.004.
The tabulated dielectric values are at 1 kHz. Their extension over the full
drive spectrum is an assumption, not measured high-frequency device data.
The preload spring has 5% of the stack's static stiffness. Mechanical loss
uses eta/E=tau=1/(2*pi*35000*100); loss Q=100 at 35 kHz is a chosen continuum
loss parameter, not a measured assembled Q and not SM111's ceramic Qm=1800.
Driver efficiencies: 95% delivery, 90% recovered-energy return to the supply.
The baseline 50 N preload is tested and fails the 35 kHz reference tension test.
No component thermal, stress, current, voltage or frequency limit is certified.

3. KINEMATICS AND COUPLED HILL
Cycle phase s=f*t. The outward stroke occupies fraction D=0.25 (with D=0.50
as a sensitivity). For u=s/D during extension:
 x=X*[u-sin(2*pi*u)/(2*pi)]
 v=X*f*[1-cos(2*pi*u)]/D
 a=2*pi*X*f^2*sin(2*pi*u)/D^2
Thus vpk=2*X*f/D; outward apk=2*pi*X*f^2/D^2. The reported peak cycle
acceleration also checks the return. Return velocity has cosine ramps, each
of duration R=(1-D)/6 cycles, surrounding a constant-speed segment. Its
displacement, velocity and acceleration match continuously at the joins.

Compute the complete distributed kinetic energy at unit stroke:
 Ktotal(t)=m*v_tip^2/2 + rho*A/2 * integral_0^L[u_dot(z,t)^2 dz].
Set the hill H=lambda*max_cycle(Ktotal), with lambda=2, and define
 h=2*H/vpk^2 (h is a load coefficient with units of mass, not material mass).
During extension set Ug=-h*v_tip^2/2, so Ug reaches -H at peak outward speed
and returns to zero at the end of extension; Ug=0 throughout the return.
During release (0<s<D/2), prescribe zero spatial hill force and explicit-time
field exchange Pb=-h*v*a. During restoration (D/2<s<D), prescribe
 G=-h*a >=0, where the environmental force on the disk is -G and Pb=0.
Then integral_restore G dx=H. H and the gradient are the same load expressed
as work and force. This prescribed phase-resolved field is not inferred from
the voltage or from an independently solved stress/scattering calculation.

The manuscript's weak-depth, low-speed relations give
 1-bmin=H/(m*c0^2), and partial_x b=G/(m*c0^2) during restoration.
The explicit-time and convective contributions satisfy
 dUg/dt=Pb+G*v. In particular, b=1 is always the restoration endpoint.
During release partial_t b=Pb/(m*c0^2), with partial_x b=0 along the path;
during restoration partial_t b=0, with partial_x b as above. A full spatial
source configuration is not constructed by these along-path assignments.

A large acceleration alone does not specify a depth: the spatial distance and
work history are also needed. With fixed pulse shape and unchanged X, increasing
f multiplies v by f, a by f^2, and (in the quasistatic mode-shape limit) H and
1-bmin by f^2. At fixed voltage X is solved, so those scalings need not hold.
A very small dimensionless 1-b can accompany a substantial force across a
submicrometre climb because the conversion scale is m*c0^2. The inferred
fractional inertia change, about 3*(1-b), remains negligible in this study;
large modeled forces come from the prescribed gradient, not a large mass swing.

4. DISTRIBUTED PIEZOELECTRIC ACTUATOR
Let E*=E*(1+i*omega*tau) and kappa=omega*sqrt(rho/E*). For fixed root and
prescribed harmonic tip displacement U:
 u(z,omega)=U*sin(kappa*z)/sin(kappa*L)
 Zrod(omega)=A*E* * kappa*cot(kappa*L).
The static limit is EA/L. The required voltage in frequency space is
 theta*V=(Zrod+kl-m*omega^2)*U+G,
where theta=(EA/L)*N*d33, kl=0.05*EA/L. The uniform piezo eigenstrain appears
as the boundary force theta*V. It is reciprocal with the charge relation below.
The unit-stroke solution is scaled by X=100/peak_to_peak(Vunit). All elements
and H scale consistently: mechanical force and voltage linearly in X; energy
and depth quadratically. A DC bias shifts V to 0..100 V and the static
equilibrium by x_offset=theta*bias/(EA/L+kl). Storage and stress include this
bias shift. This is an inverse waveform calculation, not a fixed sine drive.
An ideal voltage source with sufficient current and bandwidth is assumed.

Internal velocities can persist when the tip is at rest; this is why all-cycle
distributed kinetic energy is used in H, rather than m_tip*vpk^2/2 alone.
At 35 kHz the hill exceeds all peak kinetic energy by 22.37 microjoules.
Elastic and electrical storage can also exchange energy within a cycle; they
are not declared unavailable, and their replenishment is included in power.
Disk restoration work is H - m*vpk^2/2 (37.62 microjoules at reference),
positive throughout the decelerating climb for this load. Net terminal work
during that interval is also positive in the reference (59.78 microjoules).

5. ENERGY, MOMENTUM AND TENSION
Cf=N*eps0*eps_r*A/t_layer; Cs=Cf-theta^2/(EA/L)>0.
Charge Q=Cs*V+theta*x_tip. Current includes dQ/dt plus harmonic dielectric
loss current: I_loss,n=omega_n*Cf*tan(delta)*V_n. This is a fixed loss-tangent
spectral model. Pterminal=V*I. The periodic cycle identity is
 Eterminal=H + Emechanical_loss + Edielectric_loss.
Emechanical_loss=integral_cycle integral_rod E*tau*A*u_zt^2 dz dt.
Stored energy is Cs*V^2/2 + integral_rod EA*u_z^2/2 dz + kl*x_tip^2/2.
Negative terminal power is returned energy, not negative consumed power.
 Pdriver=f*(Epositive/0.95 - 0.90*Ereturned).
The source's extra field-control power, cooling and auxiliaries are unknown;
Pdriver alone must not be called total installed-system input power.

Taking positive propulsion toward the fixed root, the ship reaction is
 Fship=m*a_tip + integral_rod rho*A*u_tt dz + G.
The momentum derivative integrates to zero in periodic steady state, hence
 Jmodel=integral G dt, Fmodel=f*Jmodel. Spring/damping forces are internal.
Whole-assembly source reactions must also be included: Fsystem=chi*Fmodel.
chi is unknown, and may be zero. The 0..1 sensitivity range is illustrative,
not a theorem bounding all possible source behavior. It is not a free tuning
knob. Source power and momentum transfer must come from the same field solution.
The passive no-field control has zero net average thrust at any resonance.

Axial compression at position z is
 C(z,t)=Ppreload+theta*V - EA*u_z - E*tau*A*u_zt.
The reported minimum preload is max(0,-min(C at zero added preload)), with
both ends and quadrature points tested. This is an incremental preload on the
chosen biased equilibrium. It holds the preload spring stiffness fixed; a
real preload mechanism changes mass, stiffness and losses. No contact loss,
buckling, fatigue, radial stress or bond failure is modeled.

6. FREQUENCY INTERPRETATION AND EXTREME-MOTION AUDIT
A 35 kHz repetition rate has a 28.57 microsecond cycle. D=0.25 demands a
7.14 microsecond extension, whose principal kinematic scale is 140 kHz.
The computed 95% capacitive-current spectral bandwidth is about 1.12 MHz.
It is based on cumulative (n*abs(V_n))^2, not a bandwidth specification for an
actual amplifier or a measure of all motional current. The illustrative radial
estimate Np/diameter=86.6 kHz lies inside this spectrum, so a 1D model cannot
certify this waveform. The D=0.5 waveform lowers bandwidth but still overlaps
unmodeled radial modes. A measured 35 kHz sinusoidal MEGA mode and this short
repetitive pulse are distinct driving conditions.

The ideal rod's first undamped axial resonance with a 10 g disk is calculated
by Re(Zrod)+kl-m*omega^2=0. A disk of 66.48 g places that root at 35 kHz for
this geometry; this is a parameter characterization, not an OTS recommendation.
The passive 100 Vpp sine model then gives vpk=3.495 m/s and apk=768594 m/s^2
for the chosen loss parameter. Its axial stress amplitude is about 126.8 MPa,
requiring approximately 64.25 kN oscillatory-force precompression to avoid
tension in that linear extrapolation. No safe operation is claimed. Its
mean thrust is zero because no environmental hill force is present. Loading
it with the stated uphill cycle does not preserve the passive resonant gain.

The imposed-stroke audit extends to 100 micrometres, deliberately outside the
100 Vpp budget. Required Vpp scales linearly with X. Only the orange marker
is the fixed-100-Vpp solution for that specified pulse. For instance 10 um at
35 kHz requires about 7.42 kVpp in the linear inverse model; this is neither
an available drive setting nor a reliable large-signal material prediction.

7. LIFT INTERPRETATION
The comparison is powered vertical thrust versus weight, not a calculated
change in Earth's gravity. Equivalent supported mass is Fsystem/g0. Bare
actuator mass m+ms omits preload hardware, mount, driver, power source,
cooling and payload. All must be counted in an installed lift claim.
At reference chi=1 gives 82.99 N and an equivalent 8.463 kg. A 50.03 g bare
actuator requires chi>=0.005912; a 1 kg total system requires chi>=0.1182.
These are algebraic thresholds conditional on the field solution and feasible
mechanics, not verified lift capabilities. System thrust/power is
 chi*Fmodel/(Pdriver+Pfield+Paux), not automatically the driver-only ratio.

8. NUMERICS AND REPRODUCIBILITY
Run python build_plots.py, then python build_report.py. Requires Python,
numpy, scipy, matplotlib, reportlab and pypdf. OPENBLAS_NUM_THREADS=1 is
recommended for these small repeated transforms. Reference: 4096 phase points,
96 Gauss-Legendre axial quadrature points. Sweeps: 1024 or 2048 phase points,
40 or 48 quadrature points. A reference refinement to 8192/160 changes mean
thrust by 0.000246% and driver power by 0.0207%. Reference cycle energy
closure error is 2.77e-6 relative. Maximum sweep closure is below 4.85e-5.
Fixed voltage, positive disk restoration work, hill greater than all peak
kinetic energy, and a zero-field zero-impulse control are checked. These
checks verify the stated surrogate, not its physical field source.

9. PRIMARY SOURCES AND MANUSCRIPT
[1] T. J. Desiato, Mach Effect Gravity Assist Warp Drive in Operational
    Quantum Gravity, V0.7, 20 October 2026. Equations (7)-(16), complete-cycle
    momentum and energy balances, and Appendix C.
[2] Fearn et al., Mach Effects for in Space Propulsion: Interstellar Mission,
    NIAC award NX17AJ78G, NASA NTRS 20190000915. Appendix B, printed B5,
    PDF page 64, identifies 35 kHz experimental operation. Its experimental
    results are not a calibration or validation of this OQG load model.
    https://ntrs.nasa.gov/api/citations/20190000915/downloads/20190000915.pdf
[3] STEMINC, Piezo Materials Properties, SM111 column.
    https://www.steminc.com/piezo/PZ_property.asp
[4] Physik Instrumente, Dynamic Operation.
    https://www.physikinstrumente.nl/en/expertise/technology/piezo-technology/properties-piezo-actuators/dynamic-operation
[5] Physik Instrumente, Electrical Operation of Piezo Actuators.
    https://www.physikinstrumente.com/en/expertise/technology/piezo-technology/properties-piezo-actuators/electrical-operation
Web sources accessed 20 October 2026. Source values and measured frequency
context are separated from the chosen geometry, damping and load assumptions.
