MHD Core: Difference between revisions
| Line 126: | Line 126: | ||
<math>\mathbf{F}_{\text{mag}} = \nabla (\mathbf{m} \cdot \mathbf{B})</math> | <math>\mathbf{F}_{\text{mag}} = \nabla (\mathbf{m} \cdot \mathbf{B})</math> | ||
* ''\mathbf{F}''<sub>mag</sub>: Magnetic force | * ''<math>\mathbf{F}</math>''<sub>mag</sub>: Magnetic force | ||
* ''\mathbf{m}'': Magnetic moment | * ''<math>\mathbf{m}</math>'': Magnetic moment | ||
* ''\mathbf{B}'': Magnetic field | * ''<math>\mathbf{B}</math>'': Magnetic field | ||
'''Electrostatic Force Equation:''' | '''Electrostatic Force Equation:''' | ||
| Line 134: | Line 134: | ||
<math>\mathbf{F}_{\text{elec}} = Q \mathbf{E}</math> | <math>\mathbf{F}_{\text{elec}} = Q \mathbf{E}</math> | ||
* ''\mathbf{F}''<sub>elec</sub>: Electrostatic force | * ''<math>\mathbf{F}</math>''<sub>elec</sub>: Electrostatic force | ||
* ''Q'': Electric charge | * ''Q'': Electric charge | ||
* ''\mathbf{E}'': Electric field | * ''<math>\mathbf{E}</math>'': Electric field | ||
---- | ---- | ||
Revision as of 10:53, 10 November 2024
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} }
- \mathbf{x}: Position vector
- \mathbf{v}: Velocity vector
- m: Mass of the core
- \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
- \mathbf{x}ref: Reference position
- \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
- \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
| 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
| 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}}}
- \ddot{\mathbf{x}}: Acceleration
- \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} }
- \boldsymbol{\theta}: Orientation angles
- \boldsymbol{\omega}: Angular velocities
- \boldsymbol{\tau}mag, \boldsymbol{\tau}elec: Magnetic and electrostatic torques
- \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:
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)}
Control Algorithms Parameters
Parameters Definitions:
- m: Mass of the core
- \mathbf{x}, \mathbf{v}: Position and velocity vectors
- \boldsymbol{\theta}, \boldsymbol{\omega}: Orientation and angular velocity vectors
- \mathbf{F}mag, \mathbf{F}elec: Magnetic and electrostatic forces
- \mathbf{F}dist: Disturbance forces
- \mathbf{I}: Moment of inertia tensor
- \boldsymbol{\tau}mag, \boldsymbol{\tau}elec: Magnetic and electrostatic torques
- e(t): Error signal
- \lambda: Positive constant for sliding surface
- k: Adaptive gain for control law
- \mathbf{u}(t): Control input vector
Key Constants and Physical Quantities
- Planck's Constant (h): \(6.62607015 \times 10^{-34}\) Js
- Reduced Planck's Constant (\hbar): \(\frac{h}{2\pi}\)
- Speed of Light (c): \(3.0 \times 10^8\) m/s
- Elementary Charge (e): \(1.602176634 \times 10^{-19}\) C
- Vacuum Permittivity (\epsilon_0): \(8.854187817 \times 10^{-12}\) F/m
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.
