Dynamics cheat sheet

Every time we take derivatives, we stop and look. For any vector that originates from the moving frame,
we must apply rule (1) to it. That is all. In the above, only r needs rule (1) applied to it, since that is the
only vector measure from the moving frame. Replacing ṙ p by V p and ṙ o by V o , meaning the velocity of P and o,
Hence the above becomes
V p = V o + ṙ
and now we apply rule (1) to expand ṙ
V p = V o + (V rel + ω × r) (3)
where V rel is just d dt r∣ ∣
relative
The above is the final expression for the velocity of the particle V p using its velocity as measured by the
moving frame in order to complete the expression.
So the above says that the absolute velocity of the particle is equal to the absolute velocity of the base of the
moving frame + something else and this something else was (V rel + ω × r)
Now we will find the absolute acceleration of P . Taking time derivatives of (3) gives
˙V p = ˙V
( )
o + ˙Vrel + ˙ω × r + ω × ṙ
(4)
As we said above, each time we take time derivatives, we stop and look for vectors which are based on the
moving frame, and apply rule (1) to them. In the above, ˙Vrel and ṙ qualify. Apply rule (1) to ˙V rel gives
˙V rel = a rel + ω × V rel (5)
where a rel just means the acceleration relative to moving frame. And applying rule (1) to ṙ gives
Replacing (5) and (6) into (4) gives
ṙ = V rel + ω × r (6)
a p = a o + (a rel + ω × V rel + ˙ω × r + ω × (V rel + ω × r))
= a o + a rel + (ω × V rel ) + ( ˙ω × r) + (ω × V rel ) + (ω × (ω × r))
= a o + a rel + 2 (ω × V rel ) + ( ˙ω × r) + (ω × (ω × r)) (7)
Eq (7) says that the absolute acceleration a p of P is the sum of the acceleration of the base a o of the
moving frame plus the relative acceleration a rel of the particle to the moving frame plus 2 (ω × V rel ) + ( ˙ω × r) +
(ω × (ω × r))
Hence, using Eq(3) and Eq(7) gives us the expressions we wanted for velocity and acceleration.
10 Miscellaneous hints
1. When finding the generalized force for the user with the Lagrangian method (the hardest step), using the
virtual work method, if the force (or virtual work by the force) ADDS energy to the system, then make
the sign of the force positive otherwise the sign is negative.
2. For damping force, the sign is always negative.
3. External forces such as linear forces applied, torque applied, in general, are positive.
4. Friction force is negative (in general) as friction takes energy from the system like damping.
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