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nzur
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Join Date: Apr 2008
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3D Kinematics - Angular Velocity

01/08/2015 5:52 AM

Hey folks,

I got a question about 3D kinematics.

Dynamics book by Kane & Levinson says:...

====================================================

Let b1, b2, b3 form a right handed set of mutually perpendicular unit vectors fixed in a rigid body B moving in a reference frame A. The angular velocity of B in A is defined as

Angular Velocity ~= (b1)(b2dot)[dot product](b3) + (b2)(b3dot)[dot product](b1) + (b3)(b1dot)[dot product](b2)

====================================================

Do you guys have any idea of how it can be derived like above ?

Confused on the definition... it is quite abstract...
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Rixter
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#1

Re: 3D kinematics - angular velocity

01/08/2015 6:36 AM

r = [b1 b2 b3] is the displacement of the object B from the origin in A

v = [b1dot b2dot b3dot] is the velocity of object B in A

Angular velocity ω = (r X v)/|r|2 where "X" is the vector cross product
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Codemaster
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#2

Re: 3D Kinematics - Angular Velocity

01/08/2015 8:10 AM

I'm a bit rusty on vectors, but trying to compare your formula with Rixter's. As b1 etc are unit vectors |r|2 = 1, which simplifies things a little.

Can you explain what (b1)(b2dot)[dot product]b3 means? The dot product of vectors b and bdot

b.bdot = b1.b1dot + b2.b2dot + b3.b3dot if it's any help.

Cross product

b X bdot = (b2.b3dot - b3.b2dot),(b3.b1dot - b1.b3dot),(b1.b2dot - b2.b1dot). Doesn't seem to agree, but I don't suppose Kane & Levinson would get it wrong!

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Rixter
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#4
In reply to #2

Re: 3D Kinematics - Angular Velocity

01/08/2015 12:47 PM

"Angular Velocity ~= (b1)(b2dot)[dot product](b3) + (b2)(b3dot)[dot product](b1) + (b3)(b1dot)[dot product](b2)"

There sure seems to be an error. Dot product implies two vectors and returns a scalar value. The dot product of two perpendicular vectors is zero. I don't see how to interpret any of the terms above as not the dot product of two perpendicular vectors. It would be good to see this in context.
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Rixter
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#5
In reply to #4

Re: 3D Kinematics - Angular Velocity

01/08/2015 1:08 PM

I believe "[dot product]" should be "[cross product]".
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Tornado
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#3

Re: 3D Kinematics - Angular Velocity

01/08/2015 12:46 PM

Definitions are given, not derived.

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