Engineering
Logic gates, bridge building, gears, and structural analysis
Euler Column Buckling
Elastic column stability with P_cr = π²EI/(KL)²: choose end conditions, effective length factor K, first mode shape, and load ratio P/P_cr.
Mohr Circle & Stress Transformation
Plane stress σx, σy, τxy: Mohr circle, transformed stresses on a rotated element, principal stresses, τmax, and θp.
Fracture Mechanics: Griffith / K_IC
Mode-I crack: K_I = Yσ√(πa), compare with K_IC, critical crack size, critical stress, and safe/unstable crack growth.
Thin-Walled Pressure Vessel Stress
Cylinder vs sphere membrane stresses: hoop σθ, longitudinal σz, von Mises stress, r/t thin-wall check, and yield safety factor.
Jeffcott Rotor Critical Speed
Single disk on a flexible shaft: ω_n = √(k/m), unbalance response, whirl orbit, phase lag, and critical-speed crossing.
Beam Deflection: Unit Load Method
Simply supported Euler-Bernoulli beam with point load P and UDL w: closed-form deflection vs virtual-work unit-load integral.
Hertzian Contact Stress
Sphere or cylinder on a flat: effective modulus, contact patch, peak pressure p0, elastic approach, and subsurface shear estimate.
Fatigue S-N Curve + Miner Rule
Basquin S-N curve with optional endurance limit, three cyclic load blocks, and Palmgren-Miner cumulative damage D = Σ n_i/N_i.
Vibration Isolation Transmissibility
SDOF base-excitation isolator: transmissibility T(r,ζ), resonance peak, phase lag, and the isolation region above r = √2.
Heat Exchanger ε-NTU
Parallel and counter-flow heat exchanger calculator: NTU = UA/Cmin, capacity ratio Cr, effectiveness, heat transfer, and outlet temperatures.
de Laval Nozzle Mach Number
Quasi-1D converging-diverging nozzle: area-Mach relation, choking pressure ratio, subsonic/supersonic branches, and a qualitative normal-shock mode.
Torsional Drivetrain Resonance
Two-inertia torsional drivetrain: shaft stiffness and damping, twist angle, first natural mode, resonance response, and optional backlash deadzone.
Cart–Pole: LQR vs MPC vs PID
Nonlinear cart inverted pendulum: linearize about upright φ = θ, discrete-time LQR from DARE (Q, R → K), finite-horizon LQ MPC with DARE terminal cost, and hand-tuned PID — same force limit, kicks, and plant as the mechanics cart-pole.
Bode & Nyquist from Pole–Zero Map
Click the s-plane to place real or complex-conjugate poles (●) and zeros (×); G(s)=K∏(s−z)/∏(s−p). Live Bode magnitude/phase and Nyquist plot with −1 point; phase and gain margins from principal crossings (teaching heuristics).
Root Locus: 1 + K·G(s) = 0
Same pole–zero plant editor as the Bode/Nyquist map: trace closed-loop poles as K sweeps log-spaced, asymptotes and real-axis locus sketch, Durand–Kerner roots at the current K; characteristic D(s)+K g₀ N₀(s)=0.
Particle Filter Localization (MCL)
1D or 2D toy robot with fixed range beacons: predict with noisy odometry, update with Gaussian range likelihood, systematic resample on low ESS; weighted mean vs ground truth.
EKF SLAM (toy)
Augmented state [x,y,θ, landmarks…]: noisy planar odometry predict, range & bearing updates with known IDs, map–robot covariance coupling; wrong map prior vs ground-truth Lissajous path.
MPC Pendulum Swing-Up (MPPI)
Sampling-based Model Predictive Control: K candidate torque rollouts over horizon H, MPPI cost-weighted update, bounded torque |u|≤u_max — swing up an inverted pendulum live and watch the planner replan.
A* / Dijkstra Pathfinding (Grid)
Interactive 40×28 grid pathfinder: A* (f=g+h), Dijkstra, or greedy best-first; Manhattan / octile / Euclidean heuristics, 4- vs 8-connectivity, paint walls + weighted cells, watch open / closed sets expand.
Minimum Spanning Tree (Prim & Kruskal)
Random planar points, complete Euclidean-weighted graph: step through Prim from a root or Kruskal with union–find; compare total MST weight.
Maze Generators + A* (Grid)
Perfect mazes on the same 40×28 cell lattice as the pathfinding lab: recursive backtracker, Wilson, Eller, or randomized Prim; then solve with Manhattan A* (4-neighbors). Paint walls or erase passages and move S/G.
RRT Path Planner (grid)
Same 40×28 wall map as A*: random samples, nearest-neighbor steer, goal bias, collision-checked edges; grow an RRT and compare summary stats with one-click Manhattan A* baseline.
Differential Drive Odometry
ω_L, ω_R → v, ω with two-wheel kinematics; integrate pose and watch dead-reckoning drift from biased wheel radii / track and noisy rate readings.
Holonomic 2D Hovercraft MPC (MPPI)
Planar double integrator with ‖u‖₂ thrust cap: sampling-based MPPI steers to a draggable goal while soft-penalizing circular obstacles — rollout fan and best predicted path drawn live.
3-Link 3D Arm Inverse Kinematics (CCD)
Continuation of two-link-arm-ik into 3D: 3 revolute joints (yaw + 2 pitches) solved with constrained Cyclic Coordinate Descent. Drag target in 3D or follow a helix / lemniscate / figure-8 trajectory.
Bicycle Model & Stanley Controller
Kinematic bicycle (rear-axle): δ = θ_e + atan2(k_e·e, v) Stanley path-following law. Pick oval, race-track, lemniscate, sine-road or S-curve and tune k_e, v, L; live cross-track e(t) and steering δ(t).
Stress–Strain & Hooke’s Law
Qualitative σ–ε curve: elastic Hooke region, yield, strain hardening, necking, and fracture. Drag strain and tune E, σ_y, σ_u.
Quadcopter 2D (Pitch)
Side view: two rotors, PD on thrust split vs pitch — whiteboard quad slice.
Watt Governor (schematic)
Flyballs and sleeve vs RPM; steam throttle gap — speed feedback cartoon.
Logic Gate Simulator
AND, OR, NOT, NAND, XOR. Drag, drop, connect, see output.
Truth Table Generator
Build a circuit, auto-generate the truth table.
Bridge Builder
Place beams and joints. Apply load. See stress distribution.
Gear Train
Connect gears, adjust teeth count, see speed and torque ratios.
Planetary Gear Set
Sun, planets, internal ring: Willis equation, hold Sun/ring/carrier and compare speed ratios.
Beam Q, M & N Diagrams
Simply supported beam: point load + UDL; shear, bending moment, and uniform axial diagrams.
AM / FM Modulation
Carrier + message: AM envelope vs FM phase; waveform and DFT spectrum snapshot.
Planar Truss (triangle)
Symmetric 3-bar truss: bar forces and reactions vs span, height, and apex load.
PID Controller (1D)
Cart on a track: Kp, Ki, Kd and random velocity impulses toward set-point x = 0.
Two-Link Arm IK (2R)
Planar 2R manipulator: mouse goal, elbow-up / elbow-down inverse kinematics; joint angles live.
Four-Bar Linkage
Crank–rocker geometry with coupler-curve trace; link lengths and crank speed.
Cam & Follower
Eccentric circular cam and knife-edge follower: lift and estimated velocity.
Finite State Machine
Traffic-light Moore machine: timed green–yellow–red cycle or manual step; state graph.
ADC / DAC (Sampling)
Sine → samples → quantization → ZOH; Nyquist and optional aliasing demo.
Stepper Motor (4-phase)
Full-step vs half-step commutation; rotor snaps as coils A–B–A′–B′ sequence.
Thermostat vs PID
First-order room: on/off hysteresis vs continuous PID heater power.