Measuring the Inertia Tensor

Measuring the Inertia Tensor
Huw Williams
Visiting Professor of Mathematics, Coventry University
Presented at IMA Mathematics 2007 Conference
26 th April 2007

Introduction
• This talk is based on a real industrial problem that I am
currently involved in
• The problem is not really new but it is still an active field
in Engineering Science and Sports Science
• Software continues to deliver more simulation capability
but without accurate input data we may fall prey to the
“garbage in, garbage out”* syndrome or even as the
output becomes glossier “garbage in, gospel out”
• Hopefully this talk will support the view that there is
interesting mathematics to be found wherever we care to
look
* Coined by the late Stephen “Wilf” Hey, whose last published
article covered network analysis using ideas from graph theory, a
subject founded by Euler
Page 2

Agenda
• What is the inertia tensor?
• Why measure the inertia tensor?
• How is the inertia tensor measured?
• A novel solution to the inertia tensor
measurement problem*
• Some questions regarding the measurement of
the inertia tensor
*Due to Ian Kellaway of Jaguar Cars Ltd
Page 3

What is the inertia tensor?
• Basic concept
• Mathematical formulae
• Special case: fixed axis of rotation
Page 4

Inertia tensor – basic concept
Particle
m
v
p
Rigid Body
h
p = m v
h =
J ω
ω
A rigid body is a collection of particles that has angular
momentum h as well as linear momentum p. LM is a scalar
multiple of the linear velocity but AM is a tensor multiple of the
angular velocity and need not lie in the same direction.
Page 5

Page 6
Inertia tensor – mathematical formulae 1
p m v
=
J ω
h =
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
z
y
x
z
y
x
v
v
v
m
v
p
p
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
z
y
x
zz
zy
zx
yz
yy
yx
xz
xy
xx
z
y
x
J
J
J
J
J
J
J
J
J
h
h
h
ω
ω
ω

Inertia tensor – mathematical formulae 2
J
∫V
[( ⋅ ) − ⊗ ]dm
= r r I r r
J
( = ∫ y
2
+ z
2
)dm
xx
V
J
yz
=
J
zy
= −
∫
V
yz dm
J
( = ∫ z
2
+ x
2
)dm
yy
V
J
zx
=
J
xz
= −
∫
V
zx dm
J
( = ∫ x
2
+ y
2
)dm
zz
V
J
xy
=
J
yx
= −
∫
V
xy dm
I
Page 7
is the (3x3) identity tensor

Inertia tensor - special case: fixed axis of rotation
J
=
n ⋅
( J n )
is the moment of inertia,
where n is a unit vector along the fixed axis of rotation.
Some useful special cases are when the axis is aligned, in
turn, with the coordinate axes:
Page 8
( ) J
xx
J = i ⋅ J i =
( ) J
yy
J = j ⋅ J j =
( ) J
zz
J = k ⋅ J k =
which can be used to find the diagonal elements of J

Inertia tensor - special case: fixed axis of rotation
The off-diagonal elements can be arrived at through
considering
n =
1
( j + k )
2
n =
1
( k + i)
2
n =
1
( i + j)
2
Page 9
1
J = J + +
yz
yy
2
( ) J J
1
J = J +
zx
2
+
zz
1
J = J +
xy
2
+
xx
zz
( J J )
xx
( ) J J
These formulae are useful in measuring the inertia tensor
yy

Why measure the inertia tensor?
• It is fundamental to the dynamic properties of vehicles
(especially space vehicles) and their components
– e.g. the EU-sponsored INPROP project
• CAE (computer aided engineering) models need
accurate inertia properties if they are to make valid
predictions
– e.g. vehicle safety analysis (show movie)
• Sports governing bodies and sports equipment
manufacturers are using inertia tensor measurements
– e.g. the polar moment of inertia of golf clubs is restricted
because club manufacturers have been using it to reduce the
likelihood of poor shots due to off-centre contact with the ball
Page 10

How is the inertia tensor measured?
• Moments of inertia (diagonal elements) are typically
measured by attaching the body to a rotating table and
measuring the period of torsional oscillations due to
stored energy (can be elastic or gravitational)
– e.g. golf club head MOI machine, trifilar pendulum
• Products of inertia (off-diagonal elements) can be
measured by spinning the body and measuring the
torque produced or by measuring the moment of inertia
for different orientations of the body and using the
formulae already given
Page 11

A novel solution to the inertia tensor
measurement problem
Mounting the test piece in a rhombicuboctahedral frame
allows trifilar suspension (on the square faces) on the
coordinate axes and their 45 degree intermediates making it
possible to determine the inertia tensor using the
mathematical formulae
Page 12

Rombicuboctahedron
Graphics from Wikipedia
Page 13
Euler would
have been 300
on 15 th April and
this is one of
many topics that
bear his imprint

Some questions regarding the measurement of
the inertia tensor with a trifilar pendulum
• What are the assumptions implicit in the formula for the
period of oscillation and what happens if they are not fully
satisfied?
• What happens if the upper and lower triangles are
equilateral but of differing dimensions?
• How does the golf club regulatory device work?
Page 14

Trifilar pendulum assumptions
• The motion is purely torsional
– There is no swaying motion
• The combined test piece is rotating about a principal axis
of inertia
– i.e. there are no products of inertia that could generate torques
that could disturb the motion
• The suspending wires are sufficiently stiff to make the
contribution to the motion from any elastic modes of
vibration insignificant
• The centre of gravity of the combined test piece is at the
centre of the pendulum’s triangle
Page 15

Basic Trifilar pendulum theory
Assuming small angles
θ
L
ψ
r
L
r ψ = Lθ
ψ
r
h
1
h = Lθ
2
=
2
1
2
r
2
L
ψ
2
Page 16

Basic trifilar pendulum theory
Introducing the lagrangian
1
L = J &
2
Page 17
ψ
2
−
1
2
mgr
L
2
ψ
and using the Euler-Lagrange equations
d
dt
⎛
⎜⎜
⎝
∂L
∂
&
ψ
it follows that
J &
⎞
⎟⎟ −
⎠
mgr
2
∂L
∂
ψ
ψ + ψ
L
=
= 0
0
2

Basic trifilar pendulum theory
Hence the angular frequency of the oscillations is given by
ω 2 =
mgr 2
L J
and the period of oscillation is
T
=
2
π
L J
mgr
2
from which J can be found given the period that can be
measured directly
Page 18

Different sized triangles
R
In this case (for small angles)
h =
1
2
R r
L
ψ
2
L
T
= 2π
L J
mgR r
ψ
r
h
This idea is not recommended
because any error in the
alignment of the centre of gravity
would cause the test piece to tilt
Page 19

Advanced trifilar pendulum theory
This formula holds even if the
triangle is not equilateral and the
centre of gravity is noncentral
The derivation is quite interesting
and can be made rather elegant but
is too long to enter into here
T
= 2π
Page 20
mgr
L J
r
r
sinθ
+
sinθ
2 3 1 3 1 2 1 2
1r2
r3
r1
sinθ1
+ r2
sinθ
2 + r3
sin
r
r
+
r
r
sinθ
θ
3
3

Measurement robustness
• The formula given on the previous slide can be used to
analyse the experimental error and tolerance the
experiment to ensure the desired accuracy
• In practice the wires are long and the amplitude of
oscillation very small (of the order of 10 mrad) so the
assumptions hold true and the separation between sway
modes, the torsional mode and the modes due to the
wire elasticity is large enough to prevent non-principal
axes of rotation causing an issue
• The angular velocity is also kept very small thus avoiding
any untoward effects from gyroscopic torques
Page 21

Golf club inertia properties
The USGA has been conducting research into the effects of
moment of inertia. Moment of inertia in driver heads has
approximately tripled since 1990, about the time of the
introduction of the first oversized drivers. In simplest terms,
moment of inertia relates to a clubhead's resistance to twisting
on off-center hits. A club with a high moment of inertia can be
said to be more forgiving of hits away from the center of the
face.
"The USGA is concerned that any further increases in
clubhead moment of inertia may reduce the challenge of the
game," the memo reads. "The USGA is conducting this
research to determine whether a limit on moment of inertia
should be established."
Page 22

Golf club measurement
From the USGA and R&A measurement procedure
Page 23

Golf club measurement
The measurement is done on an air bearing torsion machine
rather than a trifilar rig but the principles are similar. The
measurement is done with the clubhead in nine different
locations so that the combined centre of gravity of the jig and
clubhead moves around. The moment of inertia is given by
J
=
J
+ J + +
jig jig
c lub
( ) m m r
2
c lub
where r is the distance from the combined cg to the axis of
rotation and the parallel axis theorem has been used.
A simple regression of J against r 2 gives the combined MOI as
the intercept and the combined mass as the slope.
Page 24

Golf club measurement
J
Page 25
r
2

Summary
• Measurement of inertia properties remains an
active research field
• The trifilar pendulum method is still a good
choice
• Interesting mathematics can be found
everywhere
Page 26

Conclusion
• Thank you for your participation
• Are there any questions?
Page 27

The INPROP project
Page 28

Trifilar Pendulum
Crede, Charles E., "Determining Moment of Inertia", Machine Design , August,
1948, pp. 138.
Page 29

Leonhard Euler
Now Euler was great in his own sort of way,
The results that he found were quite good for his day,
But among all his work lies one gem that excels,
That plucks at the heart strings,
the proud spirit swells,
To the prophet, the thinker, the mystic , the sage,
To the white-bearded hermit bent double with age,
Inexpressible solace one formula brought,
Namely, e to the pi i + 1= nought.
Page 30
Sources: Wikipedia and the Archimedean