Experimental Determination of
Rigid Body Parameters
Prof. Raul G. Longoria
October 20, 2000
Version 1.0
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Overview
• This lecture reviews the design of experiments
for measuring inertia properties of rigid bodies,
mainly for rotational dynamics.
• A knowledge of basic pendulum dynamics is
required.
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
1
Functional Types of Engineering Experiments
• Determination of material properties and object dimensions
• Determination of component parameters, variables, and
performance indices
• Determination of system parameters, variables and performance
indices
• Evaluation and improvement of theoretical models
• Product/process improvement by testing
• Exploratory experimentation
• Acceptance testing
• Use of physical models and analogues
• Teaching/learning through experimentation
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Specific Tasks
• Modeling of a rotational pendulum with either
two or three suspension files.
• Design of experiments using “bifilar” or
“trifilar” configuration.
• Perform experiments with simple rigid bodies
to confirm model.
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
2
Need for Inertia Properties
Suspension
Puma 560
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Moment of Inertia
• The moment of inertia, J, of a rigid body about
an axis is defined by,
2
J = ∫ r dm
• You can interpret J as a measure of a body’s
refusal to be angularly accelerated.
• Parallel axes theorem
J o = J CG + mlc2
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
3
Moments of Inertia
From Ogata, “System Dynamics”.
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Compound Pendulum
• Equation of motion for rotation of a rigid body
about a fixed axis.
• Only gravity is external force (neglect any
damping due to air, pivot, etc.)
• Apply Newton’s law,
Jθ!! = − mglc sin θ
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
4
Radius of gyration
• The radius of gyration, k, is the distance from
the point of suspension of the pendulum at
which we must concentrate the total mass, m, in
order to obtain the moment of inertia, J, of the
actual mass distribution.
2
J = mk
J
• Equivalent simple pendulum has leq =
mlc
• k is the geometric mean, 2
k = lc ⋅ leq
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Experiment Design - Bifilar
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
5
Experiment Design - Bifilar
Jθ!! = ∑ T = −
mg
sin φ × R × 2
2
L sin φ = R sin θ
L ⋅ φ = R ⋅θ
mg R
Jθ!! +
⋅ sin θ × R × 2 = 0
2 L
 mgR 2 
θ = 0
 JL 
Moment of inertia
θ!! + 
2
2
 Tn  mgR
J = 
L
 2π 
mgR 2
2π
ωn =
=
Tn
JL
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
“Adjustable” Bifilar
The torques on each file will be different here at
the point of attachment, so the period depends on x.
T = 2π
J
mgx (d − x) / l
Predict a
minimum period:
∂T
=0
 x
∂ 
d
ME 244L
Dynamic Systems and Controls Laboratory
⇒
x 1
=
d 2
Department of Mechanical Engineering
The University of Texas at Austin
6
Trifilar - More practical in the lab?
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Typical Drivetrain Values
Ref. SAE Universal Joint and Driveshaft Manual, 1979.
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
7
Other Methods (1)
Basic torsional pendulum
d 2G
Jd =
128π f n2 L
d = wire diameter
G=shear modulus of wire
f n = measured natural frequency of
motion in θ
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
Other Methods (2)
What is the model that governs this experiment design?
 t 2 2  WR 2
J =  − ⋅
H g 2
t = time to fall height, H
H = height weight falls
W = weight
ME 244L
Dynamic Systems and Controls Laboratory
Department of Mechanical Engineering
The University of Texas at Austin
8