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== Key Constants and Physical Quantities ==
== Key Constants and Physical Quantities ==


* '''Planck's Constant (''h''):''' <math>\(6.62607015 \times 10^{-34}\) Js </math>
* '''Planck's Constant (''h''):''' <math>(6.62607015 \times 10^{-34}) Js</math>
* '''Reduced Planck's Constant (''<math>\hbar </math>''):''' \(\frac{h}{2\pi}\) </math>
* '''Reduced Planck's Constant (''<math>\hbar </math>''):''' <math>(\frac{h}{2\pi})</math>
* '''Speed of Light (''c''):''' <math>\(3.0 \times 10^8\) m/s </math>
* '''Speed of Light (''c''):''' <math>(3.0 \times 10^8) m/s </math>
* '''Elementary Charge (''e''):''' <math>\(1.602176634 \times 10^{-19}\) C </math>
* '''Elementary Charge (''e''):''' <math>\(1.602176634 \times 10^{-19}\) C </math>
* '''Vacuum Permittivity (''<math>\epsilon_0 </math>''):''' <math>\(8.854187817 \times 10^{-12}\) F/m </math>
* '''Vacuum Permittivity (''<math>\epsilon_0 </math>''):''' <math>(8.854187817 \times 10^{-12}) F/m </math>


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This document compiles all the mathematical equations, values, and data relevant to the MHD Core project, providing a comprehensive reference for team members and stakeholders.
This document compiles all the mathematical equations, values, and data relevant to the MHD Core project, providing a comprehensive reference for team members and stakeholders.

Revision as of 10:58, 10 November 2024

MHD Tech

Magnetohydrodynamics

Magneto Hydro Dynamic Core

A Levitation Power Core

Fundimental Technology for the operation of a Star Speeder and Magneto Speeder



https://github.com/Jthora/MHD-Core


MHD Core Project: Mathematical Equations and Data

This document compiles all the mathematical equations, values, and data discussed in the MHD Core project presentations.


Theoretical Foundations

Quantum Field Theorist's Equations

Zero-Point Energy of a Quantum Harmonic Oscillator:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \frac{1}{2} \hbar \omega}

  • E: Zero-point energy
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar} : Reduced Planck's constant
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} : Angular frequency

Casimir Effect Force Between Two Plates:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{\text{Casimir}} = \frac{\pi^2 \hbar c}{240} \frac{A}{L^4}}

  • FCasimir: Casimir force
  • c: Speed of light
  • A: Area of the plates
  • L: Separation between the plates

Dynamic Casimir Effect Photon Generation Rate:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma = \frac{\pi \omega_{\text{cavity}}^2}{3c^2} \left( \frac{\Delta L}{L} \right)^2}

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} : Photon generation rate
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} cavity: Resonant frequency of the cavity
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta L} : Modulation amplitude of cavity length
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} : Original cavity length

Expectation Value of the Energy-Momentum Tensor:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle T_{\mu\nu} \rangle = -\frac{\hbar c}{720 \pi^2} \frac{1}{L^4} g_{\mu\nu}}

  • Tμν: Energy-momentum tensor
  • gμν: Metric tensor of spacetime

Electromagnetic Field Specialist's Equations

Modified Wave Equation with Scalar Potential:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\epsilon_0}}

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} : Scalar potential
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} : Charge density
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_0} : Vacuum permittivity

Magnetic Flux Quantum:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_0 = \frac{h}{2e}}

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} 0: Magnetic flux quantum
  • h: Planck's constant
  • e: Elementary charge

Material Development

Superconducting Material Properties

Yttrium Barium Copper Oxide (YBCO):

  • Critical Temperature (Tc): Approximately 92 K
  • Critical Current Density (Jc): Exceeding \(1 \times 10^6\) A/cm² at 77 K

Barium Zirconate Nanoparticles Enhancement:

  • Increase in Critical Current Density: 30% under high magnetic fields

Quantum Behaviors in Superconducting Materials

Cooper Pair Formation:

  • Electrons form bound pairs enabling zero electrical resistance.

Flux Quantization Equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi = n \Phi_0}

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} : Magnetic flux through a superconducting loop
  • n: Integer (quantum number)
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} 0: Magnetic flux quantum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Phi_0 = \frac{h}{2e})}

Energy Gap in Superconductors:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E = 2\Delta}

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E} : Energy required to break a Cooper pair
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta} : Energy gap parameter

Engineering Design

Levitation System Equations

Magnetic Force Equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}_{\text{mag}} = \nabla (\mathbf{m} \cdot \mathbf{B})}

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} mag: Magnetic force
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{m}} : Magnetic moment
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{B}} : Magnetic field

Electrostatic Force Equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}_{\text{elec}} = Q \mathbf{E}}

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} elec: Electrostatic force
  • Q: Electric charge
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{E}} : Electric field

Control Systems and Simulations

Levitation Control Equations

State Equations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} \dot{\mathbf{x}} = \mathbf{v} \\ \dot{\mathbf{v}} = \frac{1}{m} \left( \mathbf{F}_{\text{mag}} + \mathbf{F}_{\text{elec}} + \mathbf{F}_{\text{dist}} \right) \end{cases} }

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}} : Position vector
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} : Velocity vector
  • m: Mass of the core
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}} dist: Disturbance force

Cost Function for Nonlinear Model Predictive Control (NMPC):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = \int_{t}^{t+T_p} \left[ \|\mathbf{x}_{\text{ref}}(t) - \mathbf{x}(t)\|_Q^2 + \|\mathbf{u}(t)\|_R^2 \right] dt}

  • J: Cost function
  • Tp: Prediction horizon
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}} ref: Reference position
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} : Control input
  • Q, R: Weighting matrices

Charge Regulation Equations

Sliding Surface Definition:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t) = e(t) + \lambda \int_{0}^{t} e(\tau) d\tau}

  • s(t): Sliding surface
  • e(t) = q_{\text{ref}}(t) - q(t): Charge error
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} : Positive constant

Control Law:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(t) = -k \cdot \text{sign}(s(t)) + \dot{q}_{\text{ref}}(t)}

  • u(t): Control input
  • k: Adaptive gain
  • sign(s(t)): Sign function

Acoustic Integration

Hypersound Frequencies and Phonon Interactions

  • Hypersound Frequency Range: Above 1 GHz
  • Phonon-Electron Coupling: Interaction mechanism between high-frequency phonons and electrons in materials.

Environmental Alignment

Schumann Resonance Frequencies

Schumann Resonance Modes
Mode Frequency (Hz) Wavelength (km)
1 ~7.83 ~38,300
2 ~14.3 ~21,000
3 ~20.8 ~14,400
4 ~27.3 ~11,000
5 ~33.8 ~8,900
  • Variability: Frequencies can shift by ±0.5 Hz due to ionospheric conditions.

Geomagnetic Pulsation Frequencies

Geomagnetic Pulsations
Category Frequency Range Associated Phenomena
Pc1 0.2–5.0 Hz Electromagnetic ion cyclotron waves
Pc2 5–10 mHz Field line resonances
Pc3 10–45 mHz Cavity modes in the magnetosphere
Pc4 45–150 mHz Large-scale magnetospheric oscillations
Pc5 1–7 mHz Solar wind coupling effects

Mathematical Modeling

System Dynamics Equations

Core Motion Equations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \ddot{\mathbf{x}} = \mathbf{F}_{\text{mag}}(\mathbf{x}, \dot{\mathbf{x}}, \mathbf{I}) + \mathbf{F}_{\text{elec}}(\mathbf{x}, \dot{\mathbf{x}}, Q) + \mathbf{F}_{\text{dist}}}

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddot{\mathbf{x}}} : Acceleration
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{I}} : Coil currents

State-Space Representation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} \dot{\mathbf{x}} = \mathbf{v} \\ \dot{\mathbf{v}} = \frac{1}{m} \left( \mathbf{F}_{\text{mag}} + \mathbf{F}_{\text{elec}} + \mathbf{F}_{\text{dist}} \right) \\ \dot{\boldsymbol{\theta}} = \boldsymbol{\omega} \\ \dot{\boldsymbol{\omega}} = \mathbf{I}^{-1} \left( \boldsymbol{\tau}_{\text{mag}} + \boldsymbol{\tau}_{\text{elec}} + \boldsymbol{\tau}_{\text{dist}} \right) \end{cases} }

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\theta}} : Orientation angles
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\omega}} : Angular velocities
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\tau}} mag, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\tau}} elec: Magnetic and electrostatic torques
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{I}} : Moment of inertia tensor

Sliding Surface for Adaptive Sliding Mode Control (ASMC):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t) = e(t) + \lambda \int_{0}^{t} e(\tau) d\tau}

Control Law for ASMC:


Control Algorithms Parameters

Parameters Definitions:

  • m: Mass of the core
  • , : Position and velocity vectors
  • , : Orientation and angular velocity vectors
  • mag, elec: Magnetic and electrostatic forces
  • dist: Disturbance forces
  • : Moment of inertia tensor
  • mag, elec: Magnetic and electrostatic torques
  • e(t): Error signal
  • : Positive constant for sliding surface
  • k: Adaptive gain for control law
  • : Control input vector

Key Constants and Physical Quantities

  • Planck's Constant (h):
  • Reduced Planck's Constant ():
  • Speed of Light (c):
  • Elementary Charge (e): Failed to parse (syntax error): {\displaystyle \(1.602176634 \times 10^{-19}\) C }
  • Vacuum Permittivity ():

This document compiles all the mathematical equations, values, and data relevant to the MHD Core project, providing a comprehensive reference for team members and stakeholders.