ENGI9496 Modeling and Simulation of Dynamic Systems Mechanical Rotation

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From Close, Frederick and Newell

Mechanical Rotational Elements  Direct analogy with mechanical translation:

velocity v → angular velocity ω [rad/s, deg/s, rpm] force F → torque (or moment) T or M [N-m] mass m → moment of inertia I or J [kg-m2] momentum p → angular momentum H or h [kg-m2/s] displacement x → angular displacement θ [rad, deg] stiffness k → torsional stiffness kT [N-m/rad] damping → torsional damping [N-s-m]

Recall that the radian is not really a dimension, so sometimes it will be omitted from the units. Angular displacement, velocity and acceleration:

, Power:

TP

Rotational Inertia (Kinetic Energy)  Rotational inertia (any object that resists rotational acceleration)

analog to Newton’s law for translation: Newton’s law for rotation                Unlike mass, moment of inertia for an object is not constant, but must be defined relative to a point

normally moment of inertia about centre of gravity, or about a fixed pivot point to convert moment of inertia about centre of gravity to moment of inertia about another point, use

Parallel Axis Theorem

ENGI9496 Modeling and Simulation of Dynamic Systems Mechanical Rotation

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Compliance (potential energy storage) 

Torsion spring (e.g., compliant shaft):

Resistance (dissipation)  Rotational damper (rotary dashpot, friction torque in bearings, drag torque on propellers)

   

ENGI9496 Modeling and Simulation of Dynamic Systems Mechanical Rotation

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Interconnection Laws  d’Alembert’s Principle        Law of Displacements  Equal/Opposite Reactions 

Free Body Diagrams and Equilibrium Equations