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8.5 references<br />
1. Vibration analysis by Robert K. Vierck<br />
2. Structural dynamics theory and computation, 5th edition by Mario Paz, William Leigh<br />
3. Dynamic of structures, Ray W. Clough and Joseph Penzien<br />
4. Theory of vibration,volume 1, by A.A.Shabana<br />
5. Notes on Diffy Qs, Differential equations for engineers, by Jiri Lebl, online PDF book, chapter 2.6, oct<br />
1,2012 http://www.jirka.org/diffyqs/<br />
9 Derivation of rotation formula<br />
This formula is very important. Will show its derivation now in details. It is how to express vectors in rotating<br />
frames.<br />
Consider this diagram<br />
P<br />
r<br />
Y<br />
X<br />
r p<br />
y<br />
r o<br />
o<br />
<br />
x<br />
Moving frame of<br />
reference, attached<br />
to body of interest<br />
Absolute (or inertial frame of reference)<br />
In the above, the small axis x, y is a frame attached to some body which rotate around this axis with angular<br />
velocity ω (measured by the inertial frame of course). All laws derived below are based on the following one rule<br />
d<br />
dt r ∣<br />
∣∣∣absolute<br />
= d dt r ∣<br />
∣∣∣relative<br />
+ ω × r (1)<br />
Lets us see how to apply this rule. Let us express the position vector of the particle r p . We can see by<br />
normal vector additions that the position vector of particle is<br />
r p = r o + r (2)<br />
Notice that nothing special is needed here, since we have not yet looked at rate of change with time. The<br />
complexity (i.e. using rule (1)) appears only when we want to look at velocities and accelerations. This is when<br />
we need to use the above rule (1). Let us now find the velocity of the particle. From above<br />
ṙ p = ṙ o + ṙ<br />
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