Expert-level dynamics covering kinematics and kinetics of particles and rigid bodies, Newton-Euler equations, energy methods, impulse-momentum, and vibrations.
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Particle kinematics: position, velocity, acceleration as functions of time. Projectile motion: independent horizontal and vertical motions under gravity. Relative motion: velocity and acceleration of one point relative to another. Rigid body rotation: v = omega cross r, a = alpha cross r minus omega squared r. General plane motion: translation of reference point plus rotation about it.
Newton second law: F = ma, vector equation in inertial frame. Work-energy: work done equals change in kinetic energy. Impulse-momentum: integral of F dt equals change in linear momentum. Impact: coefficient of restitution relates velocities before and after collision.
Translation: F = m times a of center of mass. Rotation about fixed axis: M = I alpha, moment equation about axis. General plane motion: F = m times a_cm and M_cm = I_cm times alpha. Angular momentum: H = I omega, conserved without external moments.
Free vibration: x double-dot plus omega_n squared x = 0, solution is sinusoidal. Natural frequency: omega_n = sqrt of k over m for spring-mass system. Damped vibration: damping ratio zeta determines response type. Forced vibration: resonance when excitation frequency equals natural frequency. Transmissibility: ratio of transmitted to applied force in vibration isolation.
| Pitfall | Fix |
|---|---|
| Non-inertial reference frame in Newton law | Transform to inertial frame first |
| Wrong mass moment of inertia | Use parallel axis theorem when rotating about non-centroidal axis |
| Sign error in impact problems | Define positive directions consistently before and after impact |
| Ignoring constraint forces in energy methods | Constraint forces do no work in ideal joints |
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Expert-level abstract algebra knowledge. Use when working with groups, rings, fields, modules, Galois theory, representation theory, or algebraic structures. Also use when the user mentions 'group', 'ring', 'field', 'homomorphism', 'isomorphism', 'normal subgroup', 'quotient group', 'Sylow theorem', 'Galois theory', 'polynomial ring', 'ideal', 'module', 'vector space', or 'representation theory'.
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Expert-level acoustics knowledge. Use when working with sound waves, acoustic propagation, resonance, standing waves, Doppler effect, room acoustics, ultrasound, noise control, musical acoustics, or architectural acoustics. Also use when the user mentions 'sound wave', 'decibel', 'frequency', 'resonance', 'standing wave', 'Doppler effect', 'acoustic impedance', 'reverberation', 'ultrasound', 'noise control', 'harmonics', or 'sound intensity'.
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Expert-level aerodynamics covering subsonic and supersonic flow, lift and drag, airfoil theory, boundary layers, compressible flow, and CFD methods.
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Expert-level air quality engineering covering pollutant sources, atmospheric dispersion, air quality standards, emission controls, and air quality modeling.
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Expert-level aircraft structures covering structural design philosophy, wing and fuselage analysis, fatigue and damage tolerance, composite structures, and certification.
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Expert-level algorithms and complexity theory covering sorting, searching, graph algorithms, dynamic programming, greedy algorithms, NP-completeness, and algorithm design paradigms.
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