Rotational Systems

Rotational Systems (rotation)

Purpose

Analyze rotational dynamics including angular acceleration, torque requirements, power, and kinetic energy for systems with inertia and speed profiles. Supports motor sizing and transient speed-up/down screening.

Physics & theory

Newton's law for rotation: $T=I\alpha$, where $I$ is mass moment of inertia and $\alpha$ is angular acceleration. Kinetic energy $E_k=\frac{1}{2}I{\omega }^2$. Power $P=T\omega$ relates torque and angular velocity.

Speed change from ${\omega }_1$ to ${\omega }_2$ requires work $\Delta E=\frac{1}{2}I({\omega }_2^2-{\omega }_1^2)$. Time to accelerate depends on available torque net of load and friction. Reflected inertia through gear ratio $i$: $I_{\mathrm{eq}}=I/i^2$ when referred to motor shaft.

Dynamic analysis requires careful identification of mass, stiffness, and damping distribution. Natural frequencies depend on boundary conditions — a cantilever beam has fundamentally different modes than a simply supported beam of the same dimensions.

Damping limits resonant amplification; lightly damped structures (( zeta < 0.05 )) can see transmissibility peaks exceeding 10 near resonance. Separation margin between operating excitation and natural frequency should typically exceed 15–20% for rotating machinery.

Governing equations

Numerical method

Closed-form rotational dynamics (engine). User supplies inertia, torque, speed range; outputs acceleration time, peak power, energy. Optional gear ratio for reflected inertia.

Inputs

Outputs

• Angular acceleration, acceleration time, kinetic energy change, power at speed, torque utilization.

Design codes & checks

• Indicative: Torque capacity utilization

Assumptions & limitations

• Rigid body rotation; no torsional compliance or backlash dynamics.
• Constant torque during transient unless torque profile specified.
• No gyroscopic effects on supported shafts.
• Motor thermal limits not evaluated.

Verification

• CI: rotation-indicative-01.json
• Engineer sign-off: validation-master-checklist.md

References

1. Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 15.
2. Norton, R. L. Design of Machinery, 6th ed.
3. Rao, S. S. Mechanical Vibrations, 6th ed.
4. IEC 60034-12. Rotating electrical machines (motor sizing context).
5. Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.