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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]

Vector Product Definition

(ω × r ) Velocity (‘r’ is the position vector, ‘ω’ is the angular velocity.

Angular momentum/moment of momentum (‘r’ is the position

(mr × V) vector, ‘mV’ is linear momentum) – widely used in turbomachines,
rigid body dynamics, fluid mechanics, etc.…

r × F = τ Torque (‘r’ is the position vector)

Coriolis force (‘v’ is the linear velocity of the element subjected to

−2m(ω × V) angular velocity ‘ω’) – widely used in mechanisms, rotor dynamics,
fluid mechanics, cyclones…

Gyroscopic couple (‘I’ is the moment of inertia about ‘Ω’, ‘ω’ is the

(ω × IΩ) velocity of precession) – widely used in rigid body dynamics, vehicle
dynamics, rotor dynamics, etc.

ω × (ω × r ) Centripetal acceleration (negative of which gives the centrifugal
acceleration)

Scalar Product Definition

Work done (where (F) is the applied force and (S) is the

(F. S) displacement.
(F. V) Power (where (F) is the applied force and (V) is the velocity.

(T. ω ) Power (where (T) is the torque and (ω ) is the angular velocity.

(T. θ) Work done, where (θ) is the angular displacement.

1 (L. ω ) Rotational kinetic energy (‘L’ is the angular momentum) – Note that
the direction of angular momentum and angular velocity need not

2 necessarily co-inside.

1 Kinetic energy, where (p ) is the linear momentum (mV) and (V) is
2 (p . V) the velocity vector.

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 43