coordinate invariant forms for axially symmetric...

B ∈ Bx

y

z

Coordinate Invariant Forms for Axially Symmetric Bodies

Let x-y-z be a set of principal Body fixed Cartesian coordinates (origin at body point B ∈ B), Ixy = Iyz = Izx = 0,

for whichIz = Ia & Ix = Iy = Ib

As previously discussed, this state of affairs establishes the z-axis through B as a Body axis of inertial/rotational symmetry, the unit vector as its associated axial unit vector

a=k ,and the constant non-negative values Ia & Ib respectively as the Body’s primary and secondary mass moments of inertia at B ∈ B. Moreover, for any argument vector A ,

I B A = I xA x i + I yA y j + I zA zk= IbA x i + IbA y j + IaA zk= Ib A x i + A y j + Ia A zk

= Ib A � + Ia A // ; // & � to a=k

I B A = IaA // + IbA �

= IaA // + Ib A <A //

= IbA + Ia < Ib A // ; A // = a •A a = A•a a

I B A = IbA + Ia < Ib A•a a

Thus, in terms of this axial unit vector , the Body’s primary mass moment

Ia ~ mass moment of inertia w.r.t. the Body axis thru BDB which is parallel to a ,and secondary mass moment

Ib ~ mass moment of inertia w.r.t. a Body axis thru BDB which is perpendicular to a ,it follows that:

I B A =IaA // + IbA � =

IaA ; whenever A // a

IbA ; whenever A � a

IbA + Ia < Ib A•a a

The primary advantage to this sort of ‘coordinate-invariant’ formulation is that the analyst is free to choose any set of ON-basis vectors to represent the vectors involved and to perform the required algebraic manipulations.Specifically, for such an axially symmetric rigid Body having the angular velocity and acceleration vectors tt&__ .

I B tt =Iatt// + Ibtt� =

Iatt ; whenever tt // a

Ibtt ; whenever tt � a

Ibtt + Ia < Ib tt•a a

B ∈ Bx

y

z

P.A. Dashner © 2014 page 1 of 6 Rough Draft: 6/3/16


and

I B __ =Ia__ // + Ib__� =

Ia__ ; whenever __ // a

Ib__ ; whenever __ � a

Ib__ + Ia < Ib __•a a.

Moreover,tt× I B tt =tt× Iatt// + Ibtt� ; tt = tt// + tt�

= Ia tt//+tt� ×tt// + Ib tt//+tt� ×tt�

= Ia zero + tt�×tt// + Ib tt//×tt� + zero

= Ia tt�×tt// + Ib tt//×tt�

tt× I B tt = Ia < Ib tt�×tt// = Ib < Ia tt//×tt�

or, alternativelytt× I B tt = tt× Ibtt + Ia < Ib tt•a a

= zero + Ia < Ib tt•a tt×a

tt× I B tt = Ia < Ib tt•a tt×a

so that

I B __ + tt× I B tt =Ia__ // + Ib__� + Ia < Ib tt�×tt//

Ib__ + Ia < Ib __•a a + tt•a tt×a .

For the special case of a slender rod with an ‘on-line’ Body point B, a unit vector parallel to the rod serves as the axially symmetric rod’s axial unit vector corresponding to a negligible (zero-valued) primary mass moment of inertia

Ia = 0and a positive secondary mass moment

Ib = IB = {mass moment of rod w.r.t. an axis thru B which is perpendicular to the rod}Thus, for a slender rod with an ‘on-line’ Body point B, it is a simple matter to confirm the specialized forms:

I B tt =IBtt�

IB tt < a•tt a

and

I B __ + tt× I B tt =IB __� + tt//×tt�

IB __ < a•__ a + a•tt a×tt .

The reader should check to confirm that these are precisely the relations previously derived and used during our initial focus on the dynamics of slender rod structures.


Precessional Analysis for Axially Symmetric BodiesLet B represent an axially symmetric body having an axis of rotational symmetry through B ∈B associated with the axial unit vector . In addition, suppose that there exists a ‘Body-following’ intermediate reference frame F relative to which the Body’s axial unit vector appears (over time) to maintain a fixed orientation, i.e.

dFdt a = 00 .

Such a Body-following Frame is said to be a Precessional Frame (or P-frame) for the axially symmetric Body B. The angular velocity of such a Body-following P-frame is notationally designated as

11 > ttF ~ Precession Rate vector � __F = 11•

and commonly referred to as the Body’s Precession Rate vector.Using the established notation ttB/F for the angular velocity of the Body relative to this P-frame, it immediately follows from the fundamental Galilean vector time-rate relation

dFdt a =

dBdt a + ttB/F×a

that00 = 00 + ttB/F×a � ttB/F×a = 00

owing to the (above) defining characteristic of a P-frame, and the obvious fact that the rigid Body’s own axis of inertial symmetry is most certainly Body-fixed. This, in turn, guarantees that

ttB/F // a � ttB/F| a � ttB/F = pa

so that the Body’s so-called (relative) Spin Rate vector

p > ttB/F ~ axial Spin Rate vector

is expressible in the formp = ttB/F = pa

in terms of some scalar “spin-rate” p = p(t) over the time period in question. From the compound rotation formula

tt > ttB = ttB/F + ttF ,

it then follows that the Body’s angular velocity tt is then given by the relation

tt = 11 + p

in terms of the above defined Precession & Spin rate vectors11 > ttF & p > ttB/F = pa

expressed in terms of the scalar-valued spin rate p = p(t), and the Body’s own axial unit vector .


A further implication of the defining characteristic of a P-frame and the Galilean time-rate relation is the following ground (absolute) time-rate relation for the Body’s axial unit vector, viz.

a• = dd ta =

dFdt a + ttF×a = 00 + 11×a

a• = 11×a .

From this, and the above relation for the Body’s angular velocity, the following expression for its angular acceleration vector, i.e.

dd ttt =

dd t 11 + pa

tt•= 11

•

+ p• a + pa•

__ = 11•

+ p• a + p 11×a

is easily obtained. Of particular interest is the special case of Simple-Steady-State-Precessional (SSSP) motion - a motion of this type for which both the Precession Rate vector (11) and the Body’s axial spin rate (p) are constant, i.e.

11•

= 00 & p• = 0 ; (SSSP) ,and for which

__ = 11×p ; p > pa

in terms of the Body’s (axial) Spin-Rate Vector .The important dynamical relations for Precessional motion are easily derived by substituting the above (boxed) relations for tt&__ .into the axially symmetric forms of the previous section. As this is just an algebraic exercise, we proceed to derive & present the important results without further commentary.

I B tt = I B 11 + pa = I B 11 + I B pa ; pa // a

= Ib11 + Ia < Ib 11•a a + Ia pa

I B tt = Ib 11 + Ia p + Ia < Ib 11•a a

tt× I B tt = 11 + pa × I B tt -----------------------------------------------------------------------------------------------

= 11× I B tt + p a× I B tt -----------------------------------------------------------------------------------------------

=11× Ib11 + Ia p + Ia < Ib 11•a a

+p a× Ib11 + Ia p + Ia < Ib 11•a a

-----------------------------------------------------------------------------------------------

=zero + Ia p + Ia < Ib 11•a 11×a

+p Ib a×11 + zero

-----------------------------------------------------------------------------------------------


-----------------------------------------------------------------------------------------------

=Ia p + Ia < Ib 11•a 11×a

+< Ib p 11×a

-----------------------------------------------------------------------------------------------

tt× I B tt = Ia p + Ia < Ib 11•a 11×a < Ib p 11×a

I B __ = I B 11•

+ p• a + p 11×a = I B 11

•

+ I B p• a + I B p 11×a ; p• a // ap 11×a � a

I B __ = I B 11•

+ Ia p• a + Ib p 11×a

I B __ + tt× I B tt = I B 11•

+ Ia p• a + Ia p + Ia < Ib 11•a 11×a

For the important special case of SSSP (constant 11 & p ) this reduces to

I B __ + tt× I B tt =SSSP Ia p + Ia < Ib 11•a 11×a

For the very special case of SSSP motion for which

11�a � 11•a = 0 ,

it would then follow that

I B __ + tt× I B tt =** Ia__ ; __ = 11×p ; p = pa

which is reminiscent the simple moment equation used in your 1st 2D planar dynamics class.


B ∈ Bx

y

z, z׳

x׳y׳

ββ

EndNote:If a Body is confirmed to have an axis of inertial symmetry through a point B ∈B by virtue of having identified a principal (x-y-z) cartesian system ‘originating’ at B ∈B having the requisite structure (i.e. Ix = Iy = Ib & Iz = Ia), it is easily verified that any other B-based cartesian system (x׳-y׳-z׳) for which

z׳ = z ⇒ & kv = k = a is also a set of principal coordinates, relative to which the Body has exactly the same diagonal mass moment of inertia matrix, i.e.

I B xvyvzv = I B xyz =Ib 0 00 Ib 00 0 Ia

.

This since

iv � a = kv � I B iv = Ib ivjv � a = kv � I B jv = Ib jvkv = a // a � I B kv = Iakv

from which it follows that

< I xvyv = iv• I B jv = iv• Ib jv = Ib iv• jv = 0< I yvzv = jv• I B kv = jv• Iakv = Ia jv• kv = 0< I zvxv = iv• I B kv = iv• Iakv = Ia iv• kv = 0

� I xvyv = I yvzv = I zvxv = I xy = I yz = I zx = 0

and

I xv = iv• I B iv = iv• Ib iv = Ib iv• iv = Ib � I xv = I x = IbI yv = jv• I B jv = jv• Ib jv = Ib jv• jv = Ib � I yv = I y = IbI zv = kv• I B kv = kv• Iakv = Ia kv• kv = Ia � I zv = I z = Ia

An immediate corollary to this result is that the secondary mass moment value Ib is the mass moment of inertia of the Body with respect to any and all axes passing through the point B ∈B which are perpendicular to (such as x׳ and y׳) the Body’s a-axis of inertial/rotational symmetry.

B ∈ Bx

y

z, z׳

x׳y׳

ββ

coordinate invariant forms for axially symmetric...

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