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Analyzes magnetic levitation systems using Earnshaw's theorem to check passive static stability, identifies diamagnetic, superconducting, active feedback, or spin-stabilized mechanisms for maglev, bearings, and Levitron devices.
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Evaluates and compares levitation mechanisms (magnetic, acoustic, aerodynamic, electrostatic) via structured trade studies for transport, sample handling, display, bearings, or precision measurement.
Analyzes spacecraft GNC/ADCS systems: sensor/actuator selection, pointing budgets, disturbance torque analysis, control modes, momentum management. For ADCS design and performance evaluation.
Provides patterns for physics-based simulations including numerical integration, rigid body dynamics, fluid simulation, finite element methods, and multi-physics coupling. Covers accuracy, stability, performance.
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Determine whether a given magnetic system can achieve stable levitation, identify which physical mechanism enables or forbids it, calculate the conditions for force balance and stability, and verify that the levitation is stable against perturbations in all spatial degrees of freedom including tilting modes.
Establish the complete physical description of the object and field source before any analysis:
## System Characterization
- **Object**: [mass, geometry, mu or chi_v, sigma]
- **Field source**: [type, configuration, key parameters]
- **Field profile**: [B(r) functional form or measured map]
- **Gradient**: [dB/dz at intended levitation point]
- **Environment**: [temperature, pressure, vibration]
Expected: A complete specification of the object and field source sufficient to determine forces and stability without further assumptions.
On failure: If the magnetic susceptibility or moment is unknown, measure or estimate it from material data tables. Without this quantity, force calculations are impossible. For composite objects, compute an effective susceptibility from the volume-weighted average.
Determine whether passive static levitation is possible for the given system:
## Earnshaw Analysis
- **Object magnetic classification**: [paramagnetic / diamagnetic / ferromagnetic / superconducting / permanent magnet]
- **Susceptibility**: chi_v = [value with units]
- **Earnshaw verdict**: [FORBIDDEN / PERMITTED]
- **Reasoning**: [which condition of the theorem applies or fails]
Expected: A definitive classification of whether the proposed levitation is Earnshaw-forbidden or Earnshaw-permitted, with the specific physical reasoning documented.
On failure: If the object has mixed magnetic character (e.g., a ferromagnetic core with a diamagnetic shell), analyze each component separately. The overall stability depends on the net energy landscape, which may require numerical field computation.
If Earnshaw's theorem forbids passive static levitation, identify which of the four standard circumvention mechanisms applies:
Diamagnetic levitation: The levitated object itself is diamagnetic (chi_v < 0). Examples: pyrolytic graphite over NdFeB magnets, water droplets and frogs in 16 T Bitter magnets. Requires strong field gradients; the condition is (chi_v / mu_0) * B * (dB/dz) >= rho * g, where rho is density.
Superconducting levitation: The object is a type-I or type-II superconductor below T_c.
Active electromagnetic feedback: Sensors measure the object's position, and a controller adjusts electromagnet currents to maintain the equilibrium. Examples: EMS maglev trains (Transrapid), active magnetic bearings. Requires power supply, sensors, and a control system with bandwidth exceeding the mechanical resonance frequency.
Spin-stabilized levitation: A spinning permanent magnet (Levitron) achieves a gyroscopic stabilization of the tilting mode that Earnshaw's theorem otherwise makes unstable. The spin must exceed a critical frequency omega_c for gyroscopic stiffness to overcome the magnetic torque. The object must also remain within a narrow mass window.
## Circumvention Mechanism
- **Mechanism**: [diamagnetic / superconducting (Meissner or flux pinning) / active feedback / spin-stabilized]
- **Physical basis**: [why this mechanism evades Earnshaw's theorem]
- **Key requirements**: [material property, field strength, temperature, spin rate, or control bandwidth]
- **Limitations**: [load capacity, power consumption, cryogenics, mass window]
Expected: Identification of the specific mechanism with its physical basis clearly explained, including quantitative requirements for the mechanism to function.
On failure: If the system does not clearly fit any of the four mechanisms, check for hybrid approaches (e.g., permanent magnets for the primary force with eddy current damping for stability, or diamagnetic stabilization of a paramagnetic system). Also consider whether the system uses electrodynamic levitation (moving conductors in a magnetic field), which is a distinct mechanism based on Lenz's law.
Compute the force balance and quantitative conditions for stable levitation:
Vertical force balance: The magnetic force must equal gravity.
Solve for levitation height: The force balance equation F_z(z) = m * g determines the equilibrium height z_0. For analytic field profiles, solve algebraically. For measured or numerically computed fields, solve graphically or numerically.
Restoring force gradient (stiffness): Compute k_z = -dF_z/dz evaluated at z_0. For stable levitation, k_z > 0 (force decreases with increasing height). The natural frequency of vertical oscillation is omega_z = sqrt(k_z / m).
Lateral stiffness: Compute the restoring force gradient in the horizontal plane, k_x = -dF_x/dx. For Earnshaw-permitted systems (diamagnetic, superconducting), this should be positive. For feedback systems, it depends on the sensor-actuator geometry.
Load capacity: Determine the maximum mass that can be levitated by finding the field gradient at which the equilibrium becomes marginally stable (k_z -> 0 at the maximum displacement).
## Levitation Conditions
- **Force balance equation**: [F_z(z) = m*g, explicit form]
- **Equilibrium height**: z_0 = [value]
- **Vertical stiffness**: k_z = [value, units N/m]
- **Vertical natural frequency**: omega_z = [value, units rad/s]
- **Lateral stiffness**: k_x = k_y = [value, units N/m]
- **Maximum load**: m_max = [value, units kg]
Expected: A complete force balance with the equilibrium position determined, stiffness values computed for vertical and lateral directions, and the load capacity estimated.
On failure: If the force balance has no solution (magnetic force too weak to overcome gravity), the system cannot levitate the specified object. Either increase the field gradient (stronger magnets, closer spacing), reduce the object mass, or switch to a material with higher susceptibility. If stiffness is negative in any direction, the equilibrium is unstable in that direction -- return to Step 3 to identify an appropriate stabilization mechanism.
Confirm that the levitation is stable against perturbations in all six rigid-body degrees of freedom (three translations, three rotations):
Translational stability: Verify k_z > 0, k_x > 0, k_y > 0. For axially symmetric systems, k_x = k_y by symmetry. Compute the restoring force for small displacements delta_x, delta_y, delta_z from equilibrium.
Tilting stability: Compute the restoring torque for small angular deflections theta_x, theta_y about the horizontal axes. For a magnetic dipole, the torque depends on the field curvature and the object's moment of inertia. Tilting instability is the primary failure mode of passive permanent magnet levitation (and the mode that spin stabilization in the Levitron addresses).
Spin stability (if applicable): For spin-stabilized systems, verify that the spin rate exceeds the critical frequency omega > omega_c. The critical frequency is determined by the ratio of magnetic torque to angular momentum. Below omega_c, precession leads to tilting instability.
Dynamic stability: For active feedback systems, verify that the control loop has sufficient phase margin (> 30 degrees) and gain margin (> 6 dB) at all resonance frequencies. Check that sensor noise does not excite instability.
Thermal and external perturbations: Assess the effect of temperature fluctuations (critical for superconductors near T_c), air currents (significant for diamagnetic levitation of light objects), and mechanical vibration (transmitted through the field source mounting).
## Stability Analysis
| Degree of Freedom | Stiffness / Restoring | Stable? | Notes |
|-------------------|----------------------|---------|-------|
| Vertical (z) | k_z = [value] | [Yes/No] | [primary levitation axis] |
| Lateral (x) | k_x = [value] | [Yes/No] | |
| Lateral (y) | k_y = [value] | [Yes/No] | |
| Tilt (theta_x) | tau_x = [value] | [Yes/No] | [most common failure mode] |
| Tilt (theta_y) | tau_y = [value] | [Yes/No] | |
| Spin (theta_z) | [N/A or value] | [Yes/No] | [only relevant for spin-stabilized] |
Expected: All six degrees of freedom are either inherently stable (positive restoring force/torque) or stabilized by an identified mechanism (feedback, gyroscopic, flux pinning). The system is confirmed viable for levitation.
On failure: If any degree of freedom is unstable and no stabilization mechanism is identified, the levitation design is not viable as specified. The most common fix is adding an active feedback loop for the unstable direction, adding diamagnetic material for passive stabilization of a lateral mode, or increasing spin rate for gyroscopic stabilization. Return to Step 3 to incorporate the additional mechanism.
evaluate-levitation-mechanism -- comparative analysis to select the best levitation approach for an applicationanalyze-magnetic-field -- detailed computation of magnetic field profiles needed as input to this skillformulate-maxwell-equations -- derive the electromagnetic field equations governing the levitation systemdesign-acoustic-levitation -- alternative non-magnetic levitation approach for comparisonformulate-quantum-problem -- quantum mechanical treatment for superconducting levitation (BCS theory, Ginzburg-Landau)