UNIVERSITY OF CINCINNATI
August 20, 2004
Date:___________________
Daniel R. Lazor Jr.
I, _________________________________________________________,
hereby submit this work as part of the requirements for the degree of:
Masters of Science
in:
Mechanical Engineering
It is entitled:
Considerations For Using The Dynamic Inertia Method In
Estimating Rigid Body Inertia Property
This work and its defense approved by:
Dr. Randall J. Allemang
Chair: _______________________________
Dr. David L. Brown
_______________________________
Dr. Allyn W. Phillips
_______________________________
_______________________________
_______________________________
CONSIDERATIONS FOR USING THE DYNAMIC
INERTIA METHOD IN ESTIMATING RIGID BODY
INERTIA PROPERTY
A thesis submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE
in the Department of Mechanical Engineering of the College of
Engineering
2004
by
Daniel R. Lazor Jr.
B.S.M.E. University of Cincinnati 1997
Committee Chair: Dr. Randall J. Allemang
ABSTRACT
Recently, advances in sensor technology have opened fields of study in areas not
previously considered in structural dynamics. With the introduction of the six-degree of
freedom load cell, all translational and rotational force measurements can be utilized to
improve dynamic
models by providing previously unobservable reaction force
measurements. In particular, these sensors have allowed for the development of the
Dynamic Inertia Method (DIM), a means of determining rigid body properties using
dynamic data. Although DIM has been proven to provide accurate estimates for a
structure’s inertia properties in a finite frequency band, the results are not always
consistent throughout the measured frequency band nor are the inertia properties always
accurate for all structures.
This thesis examines some of the critical issues necessary for accurate and consistent
inertia properties for structures.
Since arrayed triaxial accelerometers are used to
approximate rotational accelerations, accurate placement and orientation of the
accelerometers is paramount in producing accurate estimates of angular accelerations. A
method of determining direction cosines for transducers utilizing sonic digitization is
discussed. Also, the choice of units can affect the accuracy of the estimated inertia
properties since this provides a method of weighting the estimated input and response
rotations against the measured translations.
Some inertia parameters can be determined relatively easily using traditional methods.
These properties can then be utilized in the DIM to eliminate unknowns in the estimation
process as well as to better condition the dynamic data. Additionally, frequency response
function scaling can be used to adjust the estimated inertia properties to better match the
known properties of the structure.
Problems with previous multiple axis load cell designs utilizing stacks of piezoelectric
crystals has led to the development of a six-degree of freedom load cell based on the use
of redundant piezoelectric strain gages. Experimental examples are presented utilizing
both multiple axis load cell types.
PREFACE
I would like to express my thanks to PCB Piezotronics for providing funding and
equipment allowing me to work on this topic as well as, any on and off the wall ideas that
Uncle Dave dreams up.
Also, The Modal Shop provided continuous support by
providing equipment to complete this work.
Thank you Dave for providing me with guidance and direction, not always the shortest
but always the most interesting and educational. Thank you Randy for the reminders that
I had to finish this work before I finish the PhD and for keeping me on the shortest
routes. Thank you Allyn for always simplifying the numerical process, once I understood
what you were talking about. Thank you Rhonda for making sure I did everything I was
supposed to do and for cleaning up the mess when I didn’t. And thanks to the SDRL
members - Matt Witter for introducing me to DIM and for providing the foundation of
this work - Bill Fladung for indirectly showing me how to take good, useful data. You
taught me more on one test trip than I could learn in any number of classes - Susan
Declercq for her work on the strain gage based load cell - And there are many others.
A very special thanks is entitled to my wife, Holly, for providing just the right amount of
“encouragement” to remind me that I can’t just do what I want and need to do “real
work”. You have always been understanding and patient regarding my floating schedule,
and I thank you.
Thanks everyone, you all have helped and educated me more than I can express here.
TABLE OF CONTENTS
Abstract
Preface
List of Figures.................................................................................................................... v
List of Tables .................................................................................................................... ix
Nomenclature .................................................................................................................... x
Abbreviations ................................................................................................................. xiii
1.0
Introduction....................................................................................................... 1
2.0
Automated Transducer Orientation ............................................................... 5
2.1
Direction Cosine Application ............................................................................. 9
2.1.1
Digitizer Array Registration ............................................................................. 11
2.1.2
Transducer Orientation Measurements ............................................................ 15
2.1.3
Excitation Line Of Action.................................................................................. 21
2.2
Analytical Direction Cosine Sensitivity Analysis.............................................. 22
2.2.1
Triaxial Response Direction Cosine Sensitivity Analysis ................................. 24
2.2.1.1
Analytical Example ........................................................................................... 24
2.2.2
Experimental Example – Bungee Supported Calibration Mass ....................... 33
2.2.3
Excitation Line Of Action Sensitivity Analysis.................................................. 37
ii
3.0
Numerical Modifications and Data Selection ............................................... 40
3.1
Perimeter Excitation Selection ......................................................................... 40
3.2
Known Property Elimination And Data Scaling............................................... 48
3.2.1
Known Property Elimination ............................................................................ 50
3.2.2
Data Scaling...................................................................................................... 53
3.2.2.1
FRF Scaling ...................................................................................................... 54
3.2.2.2
Unit Scaling ...................................................................................................... 55
4.0
Experimental Examples.................................................................................. 57
4.1
Calibration Mass .............................................................................................. 57
4.1.1
Hanging Calibration Mass By Bungee Cords .................................................. 61
4.2
X-38 Inertia Experimental Results.................................................................... 73
5.0
Conclusions and Future Work....................................................................... 86
5.1
Conclusions....................................................................................................... 86
5.2
Future Work ...................................................................................................... 89
Appendix A: Supporting Theory................................................................................... 97
Rigid Body Dynamics........................................................................................................ 97
Perimeter Response Selection........................................................................................... 99
The Dynamic Inertia Method .......................................................................................... 101
Appendix B: Experimental Examples......................................................................... 105
Inverted Calibration Mass Mounted On Air Ride........................................................... 105
iii
Calibration Mass Fixed To Floor ................................................................................... 110
X-38 DIM Results............................................................................................................ 114
iv
LIST OF FIGURES
Figure 1 - Effective Digitizer Volume ................................................................................ 6
Figure 2 - Single Axis Direction Cosine Representation.................................................... 9
Figure 3 - Example of Digitized Nodes ............................................................................ 13
Figure 4 - Transducer Mounting Base .............................................................................. 16
Figure 5 - Accelerometer Orientation Jig ......................................................................... 16
Figure 6 - Digitization of the Orientation Jig.................................................................... 17
Figure 7 - Mass Estimates for a Single Triax with Orientation Errors ............................. 25
Figure 8 - Single Sensor Rotation about the X and Y Axes Only, Mass and CG.............. 36
Figure 9 - Single Sensor Rotation about the X and Y Axes Only, Inertia Terms.............. 36
Figure 10 - Inertia Properties Jackknife Estimates ........................................................... 42
Figure 11 - Figure 10 - Inertia Properties Jackknife Error................................................ 43
Figure 12 - Analytical Model Jackknife Standard Deviation History .............................. 44
Figure 13 - Standard Deviations for Jackknife Process - Hanging Cal Mass................... 45
Figure 14 - Reaction Force Autopower and PCA for 17 Removed Inputs....................... 46
Figure 15 – Piezoelectric Strain Gage Based 6 DOF Load Cell....................................... 58
Figure 16 - Calibration Mass Model Dimensions............................................................. 59
Figure 17 – Calibration Mass Model and Typical Calibration Setup ............................... 60
Figure 18 - Calibration Mass Accelerometer Locations ................................................... 62
Figure 19 - Calibration Mass Load Cell Location ............................................................ 62
Figure 20 - Calibration Mass Lines of Action .................................................................. 63
Figure 21 - Calibration Mass Rigid Body Motion Error................................................... 64
v
Figure 22 - Calibration Mass Transducer Axes Rigid Body Errors.................................. 64
Figure 23 - Calibration Mass Jackknife Cycle Inertia Parameters ................................... 65
Figure 24 - Calibration Mass Jackknife Cycle Standard Deviation.................................. 66
Figure 25 - Calibration Mass DIM Parameters................................................................. 67
Figure 26 - Calibration Mass FRF Scaling to Match the Known Mass Parameter........... 68
Figure 27 - Calibration Mass Inertia Parameters with FRF Scaling................................. 69
Figure 28 - Calibration Mass Inertia Parameters with Known Mass Substitution ........... 70
Figure 29 - Calibration Mass Estimation of Mass Moment of Inertia Parameters ........... 71
Figure 30 - Calibration Mass Inertia Properties without the Reactions Forces ................ 72
Figure 31 - Calibration Mass Parameter Estimates Using All Input Excitations ............. 73
Figure 32 - X-38 Test Vehicle .......................................................................................... 74
Figure 33 - X-38 Jack Point Details, Nose, Port and Starboard ....................................... 75
Figure 34 - X-38 Accelerometer Locations and Orientations........................................... 76
Figure 35 - X-38 Load Cell Locations and Orientations .................................................. 76
Figure 36 - X-38 Input Lines of Action Locations and Orientations................................ 77
Figure 37 - X-38 Complex Mode Indicator Function....................................................... 78
Figure 38 - X-38 Solution – All Data Included in Estimation (Case 1) ........................... 79
Figure 39 - X-38 Rigid Body Motion Errors .................................................................... 80
Figure 40 - X-38 Filtered Rigid Body Motion Errors....................................................... 80
Figure 41 - X-38 Solution – Applied Rigid Body Motion Filtering (Case 2)................... 81
Figure 42 - X-38 Solution - Parameter Variations for 11 Cases....................................... 84
Figure 43 - X-38 Solution - Parameter Errors for 11 Cases ............................................. 85
vi
Figure 44 - Calibration Mass on Air Ride - Rigid Body Motion Errors......................... 105
Figure 45 - Calibration Mass on Air Ride - RB Motion Initial Channel Errors ............. 106
Figure 46 - Calibration Mass on Air Ride - RB Motion Final Channel Errors .............. 106
Figure 47 - Calibration Mass on Air Ride - Filtered RB Motion - DIM Solution.......... 107
Figure 48 - Calibration Mass on Air Ride - FRF Scaling ............................................... 107
Figure 49 - Calibration Mass on Air Ride – Scaled FRF - DIM Solution ...................... 108
Figure 50 - Calibration Mass on Air Ride - Known Mass - DIM Solution .................... 108
Figure 51 - Calibration Mass on Air Ride - Known Mass and CG - DIM Solution....... 109
Figure 52 - Calibration Mass on Ground - RB Motion Errors........................................ 110
Figure 53 - Calibration Mass on Ground - RB Motion Initial Channel Errors............... 111
Figure 54 - Calibration Mass on Ground - RB Motion Final Channel Errors ................ 111
Figure 55 - Calibration Mass on Ground - Filtered RB Motion - DIM Solution............ 112
Figure 56 - Calibration Mass on Ground – Scaled FRF - DIM Solution........................ 112
Figure 57 - Calibration Mass on Ground – Known Mass - DIM Solution ..................... 113
Figure 58 - X-38 – All Data Included in Estimation (Case 1) – DIM Solution.............. 114
Figure 59 - X-38– Applied Rigid Body Motion Filtering (Case 2) – DIM Solution...... 115
Figure 60 - X-38 Mass Scaling Convergence ................................................................. 115
Figure 61 – X-38 – Scaled FRF (Case 3) - DIM Solution .............................................. 116
Figure 62 - X-38 Jackknife Cycle Inertia Property Variations ....................................... 117
Figure 63 - X-38 Jackknife Cycle Standard Deviation of Inertia Properties .................. 117
Figure 64 – X-38 - Jackknife (7 Inputs Removed) (Case 4) - DIM Solution ................. 118
Figure 65 – X-38 - Jackknife – Scaled FRF (Case 5) - DIM Solution ........................... 119
vii
Figure 66 – X-38 - Jackknife – Known Mass (Case 6) - DIM Solution......................... 120
Figure 67 – X-38 - Jackknife – Known Mass and CG (Case 7) - DIM Solution............ 121
Figure 68 – X-38 – Input Excitation Only – Known Mass (Case 8) - DIM Solution..... 122
Figure 69 – X-38 – Input Excitation Only – Reaction Forces Not Considered.............. 123
Figure 70 – X-38 – Scaled FRF with Known Mass (Case 10) - DIM Solution.............. 124
Figure 71 – X-38 - Scaled FRF with Known Mass and CG (Case 11) - DIM Solution . 125
viii
LIST OF TABLES
Table 1 - X-38 Direction Cosine Orthogonality Check .................................................... 20
Table 2 - Single Triaxial Response Orientation Randomization – 0.02 m Cube.............. 26
Table 3 - Single Triaxial Response Orientation Randomization – 2.0 m Cube................ 27
Table 4 - Single Triaxial Response Orientation Randomization – 200 m Cube............... 28
Table 5 - Single Triaxial Response Orientation Mean Value Comparison ...................... 29
Table 6 - Four Triaxial Response Orientation Randomization – 2.0 m Cube .................. 30
Table 7 - Four Triaxes with Orientation Errors - Mean Value Comparison..................... 31
Table 8 - All Triaxial Response Orientation Randomization – 2.0 m Cube..................... 32
Table 9 - Four Triaxial Response Orientation Mean Value Comparison ......................... 33
Table 10 - All Triaxial Response Orientation Randomization, Calibration Mass............ 34
Table 11 - All Response Orientation Mean Value Comparison – Calibration Mass........ 35
Table 12 - Line of Action - Origin Sensitivity Comparison ............................................. 38
Table 13 - Line of Action - Off Origin Sensitivity Comparison....................................... 39
Table 14 - Jackknife 7 Remaining Inputs Excitation Verification ................................... 48
Table 15 - 2.0 m Cube Model – Property Changes Due to Unit Changes........................ 56
Table 16 - Calibration Mass DIM Parameters Summary.................................................. 71
Table 17 - X-38 DIM Solution Case List.......................................................................... 82
Table 18 - X-38 DIM Results for All Cases ..................................................................... 83
Table 19 - Calibration Mass on Air Ride – DIM Results ............................................... 109
Table 20 Calibration Mass Fixed to Ground - Dim Results ........................................... 113
ix
NOMENCLATURE
[ ]-1
matrix operator denoting inverse
[ ]+
matrix operator denoting pseudoinverse
[ ]T
matrix operator denoting transpose
m
mass
Ixx
mass moment of inertia about noted axis
Ixy
cross product of inertia between noted axes
Fx
force along noted axis
Mx
moment about noted axis
XCG
location of center of gravity with respect to measurement point P along noted
direction
{FP}
6-DOF force vector at measurement point P
{FP}S
vector containing force information from multiple forcing conditions at
measurement point P, multiple {FP}’s stacked vertically
[AMP]
6-DOF acceleration at measurement point P rearranged into matrix form
x
[AMP]S
matrix containing acceleration information from multiple forcing conditions at
measurement point P, multiple [AMP]’s stacked vertically
{F}i
6-DOF force as measured at point i on an object
[Φ]i
force resolution matrix which moves the 6-DOF force from any point i on an
object to the measurement point P
∆Xi
displacement of point i along noted axis
θx
rotational displacement about noted axis
Xi
physical dimension between point i and measurement point P along noted axis
{qm}
vector of perimeter response measurements
{KP}
6-DOF motion of point P based upon rigid body interpolation
[ΨRB]
rigid body transformation matrix
[W]
weighting matrix used to exclude perimeter response measurements
{qf}
vector of calculated perimeter response measurements based upon rigid body
fit
ε
rigid body fit error term
[C]
direction cosine matrix
xi
{S}
direction cosine scaling vector
[C’]
scaled direction cosine matrix
[I]
identity matrix
x
geometric location in the G - global, L - local or sensor, and DA –digitizer
y
array coordinate systems
z
G , L , DA
x
y
z
distance offset between coordinate systems
O
[T]
digitizer array transformation matrix
[OT]
orthogonal transformation to rotate a sensor about the global axes
xii
ABBREVIATIONS
3D
three dimensional
6 DOF
six degrees of freedom
CG
center of gravity
CS
coordinate system
CMIF
complex mode indicator function
DACS
digitizer array coordinate system
DIM
Dynamic Inertia Method
DOF
degree of freedom
FRF
frequency response function
GCS
global coordinate system
RB
rigid body
SVD
singular value decomposition
xiii
1.0 INTRODUCTION
The development of the procedure known as the Dynamic Inertia Method (DIM) was
conducted by the Structural Dynamics Research Laboratory at the University of
Cincinnati. The method resulted as an application related to the introduction of a force
sensor that measures the three orthogonal forces as well as the three rotational moment
degrees of freedom.
The rigid body inertia properties relate the specific force required to accelerate a structure
in all degrees of freedom. This means that the translational accelerations relate to the
force by the mass of the structure. Also, the force required to angularly accelerate the
structure depends on the mass moment of inertia properties. In general, the mass, center
of gravity and mass moment of inertia define the behavior of the structure’s motion to a
given input force and are known as the rigid body inertia properties.
By measuring all the forces acting on a structure, including the reaction forces resulting
from the structure support system, as well as the response of the structure to the input
forces, an estimate of the rigid body inertia properties can be computed.
Methods have been developed to estimate rigid body inertia properties utilizing dynamic
data as well as historical methods utilizing bifilar and trifilar pendulums. The pendulum
techniques have the disadvantage of costly fixturing to support the structure, and in
addition, the testing can be tedious and cumbersome. The advantage of the DIM is that it
can be easily integrated into a standard modal test, for example ground vibration test for
1
aircraft, with no additional data acquisition time since DIM utilizes frequency response
function data to estimate the inertia properties.
It can easily be concluded that the accuracy of the DIM can be limited by the quality of
the data, geometric sensor locations and orientations, excitation force locations, and
numerical conditioning of the data. As in all dynamics testing, efforts should be made to
acquire high quality data in order to increase the confidence and accuracy of the
estimated parameters.
However, depending on the requirements of the test, highly
controlled measurement parameters are not always required. For example, the precision
of modal analysis testing does not require an exact geometric location of a sensor or
excitation, and the orientation of a sensor can be considered good enough by a skilled
eye. This is the case because the apparent orientation of the sensor can often be affected
by the cross-axis sensitivities of a transducer, and the cross-axis sensitivities are rarely
considered. Thus the results of a modal model are not likely impacted by a slightly
skewed sensor or excitation. Since the DIM approximates rotational degrees of freedom
utilizing translational measurements and geometric positions, the location and orientation
of a sensor can cause large errors in the estimation of the rigid body inertia properties.
This thesis focuses on methods to reduce the errors in the inertia properties estimation
process and increase the confidence in the solution result. One such method involves
determining the geometric location of a sensor on the test structure as well as the
orientation of the transducer using a 3D sonic digitizer. The method can also be applied
2
to determine the line of action normal to a surface for use with an instrumented impact
hammer or electro-mechanical shaker.
The confidence of the data can be improved by two methods. As previously presented
[29][37], a set of redundant, arrayed accelerometers can be used to estimate the rigid
body motion of a structure. Response selection can be performed by estimating the
degree of rigid body error associated with each response measurement and eliminating
measurements with a high degree of error by using a weighting matrix. This thesis
presents a simple way of verifying the consistency of the excitation to the structure by
using a statistical jackknife approach.[15] Measurements corresponding to an excitation
that produces inconsistent parameters are rejected from the dataset used in the DIM
estimation procedure.
Also, this thesis presents the modification to the DIM equations needed to apply known
rigid body inertia properties to the estimation process resulting in a reduction of the
number of unknown parameters.
Any property that can be easily computed via
traditional means can be substituted to reduce the number of unknown parameters in the
solution equations and improve the numerical conditioning.
Another method of
incorporating known parameters into the final solution is to simply scale the FRF matrix
in a manner that results in the estimates matching the known parameters.
Finally, the prototype six axis load cell utilized in [37] was found to introduce errors into
the estimation process that could not always be explained. Therefore, a new six degree of
freedom force sensor was developed which utilized a redundant set of piezoelectric strain
3
gages on a metal structure in an effort to produce a simpler and more effective multiple
axis load cell.
Section Two of this thesis describes the procedure used to locate and orient transducers
using a 3D sonic digitizer.
Section Three discusses the numerical modifications,
perimeter excitation selection, and known parameter substitution. Section Four works
through the DIM for two experimental cases with additional cases in Appendix B.
Appendix A briefly presents the supporting theory behind rigid body dynamics, the
dynamic inertia method, and perimeter response selection.
4
2.0 AUTOMATED TRANSDUCER ORIENTATION
The motivation behind attempting to accurately describe the local orientation of a
transducer in global coordinate space is to improve the accuracy of the estimated angular
measurements from either multiple axis sensors or a skewed excitation line of action.
The estimated unmeasured degrees of freedom are derived from rigid body dynamics.
Thus it is necessary to determine, with a high degree of accuracy, the geometric location
and local orientation of the transducer.
Currently, there are many methods and instruments that can be used to accurately obtain
the locations and orientations of sensors. Methods using photogrammetry have proven to
provide reasonable accuracy. These methods include using several photos of a structure
with coded targets to produce a 3D digital image. The reverse process has also been
developed in which a camera mounted on a probe is placed at the node of interest and a
photo is taken of several coded targets. While these methods can provide high accuracy,
the acquisition time can be quite lengthy, generally require a significant amount of user
intervention, and can be very costly.
Another method of obtaining geometric locations and orientation requires the use of a
sonic digitizer.
The accuracy of the digitizer is generally less than that of the
photogrammetry methods, however the cost of the equipment is significantly less.
The sonic digitizer operates on the basic principle of acoustics that sound travels at a
known, constant rate in a consistent medium. An array of microphone receivers is placed
5
within sensing distance of a hand held probe. The hand held probe contains emitters that
generate ultrasonic pulses and are arranged linearly on the probe in order to describe a
line in 3D space. The equation x = cτ , where x represents the distance, c is the speed of
sound, and τ is the time delay from an emitter to a receiver, provides the linear distance
between each emitter on the hand held probe and each microphone receiver on the array.
By calculating the distance between each emitter on the hand held probe and each
microphone on the receiver array, the tip of the probe with a known length can be
computed using triangulation techniques. Figure 1 shows an example of the effective
volume that can be measured using the sonic digitizer. This volume is determined by the
strength of the emitted ultrasonic pulse, the noise floor of the microphones, and the
maximum value of the hardware counters used to determine the time delay. [5] [21]
Figure 1 - Effective Digitizer Volume
The accuracy of digitization requires an accurate value for the speed of sound of the
acoustic media within the effective digitizer volume. The speed of sound is determined
6
simultaneously with each digitization point using a calibration bar. The calibration bar
contains a single emitter and receiver separated by a known, fixed distance. Since the
distance is known and the time delay is measured, an accurate calibration of the speed of
sound can be determined.
Accurate transducer orientations and locations are vital to providing quality rigid body
approximations of the moments and angular accelerations. A 3D sonic digitizer can be
used to measure the direction cosine angles and origin of a sensor relative to the global
coordinate system of a structure. The direction cosine angles for the line of action for an
excitation can also be determined. Due to the simplicity of operation and the amount of
time needed to acquire data, the orientation of transducers can be determined quickly in
an ideal acoustic environment.
The limitations of using the sonic digitizer result primarily from environmental
conditions and the physical size of the test structure. Since the digitizer uses ultrasonic
pulses, any environmental condition causing interference of this signal can have a drastic
influence on the accuracy of the signal as well as limit the size of the effective digitizer
volume. Structures that cannot be acoustically isolated from surrounding structures can
also be problematic since reverberations from nearby objects can cause a time delay to
appear much larger and triangulate to a point farther away. Large structures that do not
fit into the effective digitizer volume require the digitization to be done in patches with
either the array or structure being moved to locate the nodes in the volume. Any relative
movement between the receiver array and the test structure creates offsets and rotations
7
relative to the global coordinate system and must be determined.
This requires an
additional step of registering the array using known geometric points either on the
structure or any non-moving reference frame relative to the global coordinate system.
This registration process can create additional errors associated with the digitized
transducer locations and orientations.
8
2.1 DIRECTION COSINE APPLICATION
The direction cosine transformation for a sensor can be thought of as the cosine values of
the angles between the local axes of the transducer and each global coordinate system
axis. There are three direction cosine values for each axis of a sensor. Figure 2 shows
the three angles measured to determine the direction cosine vector for the local X axis of
the transducer. To complete the direction cosine matrix, the cosine values of the angle
between both the transducer’s Y and Z axes to the global axes must also be determined.
Figure 2 - Single Axis Direction Cosine Representation
For a single axis accelerometer, the relationship between the local axis and the global
axes can be represented by
9
x
C11
y
z
= C 21 a L
C31
G
(1)
Note that the previous equation does not provide a means of measuring three axes of
acceleration while only using a single axis transducer. The global representation is only
the projection of the skewed axis onto each global axis and does not constitute the total
global acceleration. For the case when a single axis sensor is oriented primarily in the
global X direction while only slightly skewed in the Y and Z axes with the global
acceleration of the structure being measured were exactly in the global Y direction, the
results of applying the direction cosines to this measurement would appear to have a
larger acceleration component along the global X axis rather than the Y. Therefore,
direction cosines only describe the projection of a single axis onto the global axis and do
not represent the 3D global accelerations.
The direction cosines are the orthogonal projection of a transducer’s local axes onto the
global coordinate system. Extending the single axis equation for use with a multiple axis
transducer is quite simple. The influence of each axis of the transducer can be linearly
combined to form the total measurement participating along each global axis. Triaxial
accelerometers typically have three orthogonal measurement axes. This may not always
be the case if three single axis accelerometers are collocated to form three independent
axes. The method presented here does not assume orthogonality of the transducer axes.
10
Therefore the direction cosine matrix is not assumed symmetric.
The equations
representing the direction cosine vectors for a tri-axial accelerometer can be written as
x
y
z
C11
G
C12
C13
= C 21 a xL + C 22 a y L + C 23 a z L
C 31
C 32
C 33
(2)
Equation 2 can be rewritten in the linear algebra form that expresses the equivalent
accelerations along each global coordinate system axis.
x
y
z
G
C11 C12 C13
= C 21 C 22 C 23
C 31 C 32 C 33
x
y
z
(3)
L
Note that this equation does not have the coupling issue that is seen with the single axis
measurement. Since the transducer measures all three local, translational accelerations,
the equations become decoupled and are completely orthogonal.
2.1.1 DIGITIZER ARRAY REGISTRATION
The first step in measuring the orientation of a transducer in the global coordinate system
is to register the microphone receiver array with the global coordinate space. Generally,
when a sonic digitizer is used to define the orientations of transducers, there are three
coordinate systems present. The first is the structure’s coordinate system, or global
coordinate space (GCS), which defines the measurement node locations on the structure.
11
Secondly, there is the local coordinate system (LCS) representing the measurement axes
of the transducer. The third CS is the digitizer array coordinate system (DACS) which
defines the coordinates of a digitized point relative to the digitizer microphone array.
Generally, the DACS is defined to match the structure space so that the point locations
measured by the digitizer are directly related to the structure in the global coordinate
space.
However, this is often not a simple task considering three points must be
accurately defined to specify the DACS in terms of the GCS. These points typically
represent the origin, a point on the X-axis and a point in the XZ plane. For large
structures, the position of the transducers and the origin may not be located within the
usable range of the digitizer, or the origin and XZ axis of the structure may be difficult to
digitize precisely. In this situation, a geometric transformation relating the array space
and global space must be determined in order to describe the digitized nodes in the global
coordinate system.
Registering the sonic digitizer in global coordinate space can be accomplished in
different ways. The first method can be used when the geometry of the structure is not
previously defined. This method simply defines the array space coordinate system by
digitizing the origin, a point on the +X axis, and a point in the XZ plane. Thus, the array
space defines the global space, and the systems are identical. Once the global coordinate
system is defined for the structures, any point in the digitizer volume can be digitized to
create a measurement node or a reference node.
12
Measurement nodes are geometric points used to define an impact or shaker input and
transducer locations. Reference nodes are also geometric points but are only used to
assist in registering the array to the GCS by providing additional points with geometries
in the global coordinate system.
Reference nodes are only needed if the span of
measurement nodes is not encompassed by the effective digitizer volume requiring the
receiver array to be moved to a different position in order to complete the digitization and
transducer orientation process. The reference nodes are used to register the array back to
the original global coordinate system. During the relocation of the receiver array, the
DACS can either remain constant or change. The geometric transformation resulting
from the registration process contains the linear offset between the digitizer array
coordinate system origin and the GCS origin, rotations of the DACS about the GCS axes,
and scaling differences related to a difference in units between the two coordinate
systems.
Figure 3 - Example of Digitized Nodes
13
The transformation equation used to register the array contains nine unknowns relating
the rotation and scaling of the array, and the three unknowns for the linear offset.
Registering the digitizer array space to the global coordinate system requires a minimum
of four points with known 3D geometries in the GCS. The general equation used to
register the digitizer array is defined by
x
T11 T12 T13
x
x
y
z
= T21 T22 T23
T31 T32 T33
y
z
+ y
z
G
A
(4)
O
Equation 4 can be simplified to single matrix equation.
x
T11 T12 T13 xO
y
= T21T22T23 y O
z
x
y
T31T32 T33 z O
G
(5)
z
1
A
Since the equation for a single node is rank deficient, additional reference or
measurement nodes are needed to determine the registration unknowns, and the solution
becomes
T11 T12 T13 x0
T21 T22 T23 y0 =
T31 T32 T33 z0
x
y
x
y
x
y
z
1
z
1
z
1
1
2
+
n
x
x
x
y
z
y
z
y
z
1
2
(6)
n
G
14
Since errors can exist in both the known reference nodes’ GCS positions and in the
digitized 3D values, a least squares approach is utilized when solving for the unknowns
to reduce the overall errors. To reduce the influence of random errors, multiple digitized
measurements can be recorded for the same reference node. By doing so, the small
variance errors resulting from inaccuracies of the sonic digitizer can be minimized by
averaging.
A condition of the registration process is that the digitized reference nodes occupy a
volume in 3D global space. When only coplanar nodes are used, the registration will
only be valid in that plane. Following registration, any digitized nodes that do not lie in
the plane used to register the array may have large errors associated with their GCS
values.
2.1.2 TRANSDUCER ORIENTATION MEASUREMENTS
A new transducer mounting base was developed to assist in bonding the transducer to a
test structure and measure the orientation of the transducer. The mounting base has three
feet allowing the base to be mounted to a curved surface with a sensor axis in the
direction normal to the surface. The top of the base has a countersink in the shape of the
triaxial accelerometer so the sensor can be pressed into the base in only one orientation.
This was done to reduce the likelihood of placing the accelerometer into the base in a
different orientation than was used to determine the orientation of the sensor.
15
Figure 4 - Transducer Mounting Base
The orientation jig used to determine the direction cosines for the triaxial accelerometers
was built from an accelerometer matching the dimensions of the mounting base and three
machined, steel rods. The rods have eight indentations located at one inch intervals
providing a resting spot for the tip of the hand held digitizer probe and thread into the
three, orthogonal axes of the orientation transducer.
Figure 5 - Accelerometer Orientation Jig
16
In order to determine the orientation and location of the transducer, a minimum of four
points must be digitized with at least one point on each axis. A least squares solution can
be achieved by digitizing all eight indentations on the three axes.
Figure 6 - Digitization of the Orientation Jig
The procedure used to determine the origin and the direction cosines of the transducer is
similar to that of the digitizer registration. The orientation jig is calibrated by accurately
measuring the location of the 24 indentations used to digitize the three axes relative to the
origin of the orientation jig. These locations on the jig can be thought of as the known
reference nodes on the structure.
The same equation used to determine the
transformation matrix and offset between the digitizer array coordinate system and the
GCS is used to determine the direction cosines and location of the transducer relative to
the GCS of the structure.
xi
yi
zi
jig
C11 C12
= C21 C22
C13
C23
xi
yi
C31 C32
C33
zi
x
+ y
L
z
(7)
O
17
When a point on the orientation jig is digitized, it is immediately transformed to the
global coordinate system if the digitizer has been registered. After the orientation jig has
been digitized and all points have been converted into the GCS, Equation 8 can be used
to determine the direction cosines and location of the sensor origin in global coordinates.
C11 C12 C13 x0
C21 C22 C23 y0 =
x
y
C31 C32 C33 z0
z
x
y
1
z
x
y
z
x
y
2
z
n
1
jig
x
y
z
1
1
+
x
y
z
2
1
(8)
n
L
It is important to note that the previous equation does provide an adequate solution for
the orientation and location of a transducer. Since the axes of the triaxial accelerometer
are orthogonal, the direction cosine matrix should be orthogonal and the norm of each
row and column should be unity. This method does not provide any equations that
constrain the direction cosine matrix to be orthogonal when the transducer’s local axes
are orthogonal and contain rows of unity length allowing errors to exist in the direction
cosine matrix. A correction factor can be used to force the rows of the direction cosine
matrix to unity.
{S } =
C112 + C122 + C132
C 212 + C 222 + C 232
C +C +C
2
31
2
32
(10)
2
33
18
The scaling differences are then removed by applying the scaling to each row of the
matrix.
C' =
{C11 C12 C13 }
1
S1
{C21 C22 C23 }
1
S2
{C31 C32 C33 }
1
S3
(11)
The orthogonality of the scaled direction cosine matrix is checked to verify that it is an
orthonormal matrix.
[I ] =
C'
T
C'
(12)
This method of acquiring 6 DOF transducer location data from the 3D Sonic Digitizer
was used in a field test during a dynamic inertia method test on the NASA X-38 V131R
space vehicle at NASA Dryden Flight Research Center. For most of the sensors, the
orthogonality check provided results that were reasonably close to identity. For the
sensors that indicated low orthogonality, the 3D digitizer and/or orientation jig was
repositioned for more direct observability by the 3D Digitizer and thus more accurate
measurement of these locations. However, due to the structural surroundings and sensor
locations on the structure, difficulties were encountered in digitizing points accurately.
Table 1 shows the orthogonality check on the 17 triaxial accelerometers used to compute
19
the inertia properties for the X-38 vehicle. The node numbers are listed to the left of the
orthogonality check matrices.
Table 1 - X-38 Direction Cosine Orthogonality Check
1106
1107
1303
1405
1409
1320
1
0.0567
0.0071
0.0567
1
0.0149
0.0071
0.0149
1
1
0.1842
-0.0268
0.1842
1
0.0143
-0.0268
0.0143
1
1
-0.0191
0.0195
-0.0191
1
0.0086
0.0195
0.0086
1
1
0.1514
-0.1238
0.1514
1
-0.014
-0.1238
-0.014
1
1
-0.0806
-0.1155
-0.0806
1
0.0433
-0.1155
0.0433
1
1
-0.4795
-0.0446
-0.4795
1
0.0199
-0.0446
0.0199
1
1319
1302
1428
1401
1321
1425
1
-0.1219
-0.5066
-0.1219
1
0.0224
-0.5066
0.0224
1
1
-0.2315
-0.1008
-0.2315
1
-0.0073
-0.1008
-0.0073
1
1
0.037
-0.2496
0.037
1
-0.0025
-0.2496
-0.0025
1
1
-0.0417
0.021
-0.0417
1
0.0178
0.021
0.0178
1
1
0.073
0.2573
0.073
1
-0.0044
0.2573
-0.0044
1
1
-0.0005
0.1113
-0.0005
1
0.0143
0.1113
0.0143
1
2302
1310
1308
1318
1316
1
0.0385
-0.0108
0.0385
1
0.0095
-0.0108
0.0095
1
1
0.0524
0.0015
0.0524
1
-0.0017
0.0015
-0.0017
1
1
0.027
-0.0237
0.027
1
-0.018
-0.0237
-0.018
1
1
-0.005
-0.0514
-0.005
1
-0.008
-0.0514
-0.008
1
1
-0.1861
-0.0388
-0.1861
1
-0.0859
-0.0388
-0.0859
1
Due to spatial limitations, one of the axes on the accelerometer jig may need to be
removed so that the jig can be placed on the desired measurement point. In this case,
only two axes can be digitized. As with the registration process, digitizing only two axes
results in direction cosines for the axes measured.
Therefore, an additional step is
required to obtain a fully populated direction cosine matrix. Since the axes of the
accelerometers are orthogonal, the third axis can be determined by computing the cross
product between the two measured axes. This solution is less accurate than measuring all
three axes, but will produce a normal axis to the plane defined by the two measured axes.
20
2.1.3 EXCITATION LINE OF ACTION
Not only are the direction cosines for the response transducers important when making
accurate measurements, but the excitation line of action for a skewed input force to the
structure also needs to be determined. Whether the excitation of the structure is due to an
instrumented impact hammer or an electro-mechanical shaker, the direction of the
excitation can be determined using a sonic digitizer.
Since there is only a single axis in which the input force acts, the solution of the direction
cosine differs from that of the multiple-axis transducer. The first step is to locate the
array in a location to measure the input point and register the array into global space. The
transducer orientation jig is placed into the mounting base to produce a normal vector to
the surface at the input node. The eight indentations of the jig axis are digitized with
several averages at every location. By taking the standard deviation normalized by the
largest singular value of the standard deviation, the effective participation in each global
coordinate direction is determined.
The solution to the equation for the line of action does not directly yield the location of
the input in the GCS. This can be done by using the line of action and extrapolating back
to the center of the mounting base. However, it is much quicker to simply digitize the
location of the input node directly.
The signs of the direction cosines, or the direction of the line of action, can be determined
by taking the sign of the input node coordinates minus the mean of the data from the
21
digitized axes. Therefore the equation to determine the orientation of the line of action
can be described by Equation 13.
C1
C2 = sgn
x
y
C3
z
x
− y
O
z
σx
1
∗ σy ∗
σ x2 + σ y2 + σ z2
σz
(13)
x , y , z represent the mean and σ x , σ y , σ z represent the standard deviations of the
digitized axes data.
2.2 ANALYTICAL DIRECTION COSINE SENSITIVITY ANALYSIS
The limitations of the accuracy of the automated direction cosine determination process
using a 3D sonic digitizer became apparent during the NASA X-38 field test. Several
problems were encountered due to environmental interference and structural limitations,
such as, the vehicle size and location next to a crane, as well as, the vehicle being
elevated only three feet above the ground. When the measurements for the indentations
on the orientation jig were monitored, a random jump of up to eight feet from the
expected location could be seen.
The scaling and orthogonality checks had been
performed satisfactorily, but the accuracy of the orientations was inadequate. An analysis
of an analytical model was performed to determine the sensitivity of the DIM to random
orientation errors. The analysis included a wide range of rotation modifications for single
22
and multiple responses. The sensitivity to errors of the excitation line of action was
analyzed as well.
An analytical model was created to simulate a structure in a free-free boundary condition.
The boundary condition simplified the process by removing reaction forces from the
estimation process. The model was made to match the inertia property of the calibration
mass that is used to determine the sensitivities of multiple axis load cells and contained
eight triaxial accelerometer measurement points located equidistant from the GCS origin.
Two types of models were tested. The first model had three cases where the response
measurements were taken within different volumes. The accelerometer arrangement was
that of a square with each measurement locate 0.01m, 1.00m and 100m from each axis
for the three cases. Only 6 analytical forces were applied to the structure. These forces
were generated to excite the three translation forces and three rotational forces, and were
applied directly at the origin of the model. All the accelerometers were oriented to match
the global coordinate system.
The second model simulated a free-free boundary condition of the DIM tests performed
on the calibration mass in Section 4.1. The excitations of the structure were located at 24
points on the model. Eight excitations were applied to each of three surfaces in line with
the GCS. The locations and orientations of the inputs and response matched that of the
calibration mass DIM tests.
23
2.2.1 TRIAXIAL RESPONSE DIRECTION COSINE SENSITIVITY ANALYSIS
2.2.1.1 ANALYTICAL EXAMPLE
Each geometric case for the response orientation analysis consisted of nine variants of the
orientations. A single accelerometer was orthogonally rotated about the three local axes
using randomized θx, θy, θz angles with maximums values 0.1, 1, and 5 degrees. The
randomized angles were applied using Equation 14 and the transformation shown in
Equation 15. The inertia properties were then computed for 500 randomizations of the
angles using the DIM.
C = [OT ][C ]
[OT ] =
(14)
cosθ y cosθ z
cosθ y sin θz
− sin θ y
sin θx sin θ y cosθz
− cosθx sin θ z
sin θx sin θ y sin θ z
+ cosθx cosθz
sin θ x cosθ y
cosθ x sin θ y cosθ z
+ sin θ x sin θz
cosθx sin θ y sin θz
− sin θ x cosθz
cosθx cosθ y
(15)
Figure 7 shows the variation in the estimated mass with a single sensor randomly rotated
with a maximum of 0.1, 1.0, and 5.0 degrees about each of the sensor’s local axis. The
model used for this data has the sensor origins located at a distance of 0.10 meters from
each of the global axes.
24
DIM Estimates For Mass With Varying Orientation of a Single Sensor (0.1 Degrees)
Estimated Mass Value, Kg
18.185
18.18
18.175
18.17
18.165
18.16
DIM Estimates For Mass With Varying Orientation of a Single Sensor (1.0 Degrees)
Estimated Mass Value, Kg
18.2
18
17.8
17.6
17.4
17.2
17
DIM Estimates For Mass With Varying Orientation of a Single Sensor (5.0 Degrees)
Estimated Mass Value, Kg
20
18
16
14
12
10
8
6
0
50
100
150
200
250
Iteration Number
300
350
400
450
500
Figure 7 - Mass Estimates for a Single Triax with Orientation Errors
From 0.1 degrees to 1.0 degrees, a slight decrease in the mean of the mass estimates can
be seen. However, when the maximum error for the angles increases to 5.0 degrees, the
resulting mass estimates decrease significantly, and the deviation from the mean is much
larger. Table 2 provides the statistical data for all the inertia properties estimated for the
three maximum degrees conditions. It is easily seen, as well as logical, that as the
maximum degree of error on the direction cosine values increase, the resulting DIM
parameters have increasing errors.
25
Max Rotation of 5.0 Degrees
Max Rotation of 1.0 Degrees
Max Rotation of 0.1 Degrees
Table 2 - Single Triaxial Response Orientation Randomization – 0.02 m Cube
Minimum
Maximum
Mean
Standard
Deviation
Actual
Value
Mean
Error, %
Mass, kg
18.161677
18.181829
18.172661
0.004333
18.175000
0.012870
Xcg, m
0.035220
0.035253
0.035237
0.000009
0.035236
0.001285
Ycg, m
0.000089
0.000169
0.000130
0.000018
0.000130
0.240326
0.002191
Zcg, m
-0.085730
-0.085541
-0.085633
0.000037
-0.085632
Ixx, kg m^2
0.218164
0.218918
0.218547
0.000161
0.218554
0.003277
Ixy, kg m^2
-0.000425
0.000174
-0.000114
0.000128
-0.000109
4.402096
Ixz, kg m^2
0.077956
0.078352
0.078158
0.000085
0.078161
0.003897
Iyy, kg m^2
0.303944
0.304386
0.304155
0.000100
0.304172
0.005846
Iyz, kg m^2
0.000030
0.000385
0.000218
0.000075
0.000218
0.140292
Izz, kg m^2
0.158071
0.158236
0.158153
0.000037
0.158154
0.000504
Mass, kg
17.123981
18.182649
17.967635
0.200335
18.175000
1.140935
Xcg, m
0.035041
0.035393
0.035225
0.000091
0.035236
0.031837
Ycg, m
-0.000244
0.000510
0.000135
0.000171
0.000130
4.066245
0.023140
Zcg, m
-0.086555
-0.084747
-0.085612
0.000359
-0.085632
Ixx, kg m^2
0.210276
0.221280
0.216945
0.002120
0.218554
0.736156
Ixy, kg m^2
-0.003218
0.002731
-0.000140
0.001186
-0.000109
28.167346
0.894170
Ixz, kg m^2
0.074869
0.079797
0.077462
0.001044
0.078161
Iyy, kg m^2
0.294880
0.305187
0.302409
0.001992
0.304172
0.579753
Iyz, kg m^2
-0.001651
0.001979
0.000208
0.000738
0.000218
4.622580
Izz, kg m^2
0.156344
0.158830
0.157852
0.000454
0.158154
0.190782
Mass, kg
6.788984
18.107086
14.433371
2.426544
18.175000
20.586679
Xcg, m
0.033088
0.035947
0.035037
0.000502
0.035236
0.564396
Ycg, m
-0.001891
0.001837
0.000070
0.000876
0.000130
45.741520
Zcg, m
-0.090364
-0.081947
-0.085496
0.001819
-0.085632
0.158011
Ixx, kg m^2
0.132879
0.221517
0.191460
0.018857
0.218554
12.396937
Ixy, kg m^2
-0.013065
0.011084
0.000064
0.005440
-0.000109
158.714662
Ixz, kg m^2
0.039767
0.080830
0.066866
0.008556
0.078161
14.451059
Iyy, kg m^2
0.199393
0.304066
0.271354
0.021878
0.304172
10.789462
Iyz, kg m^2
-0.007850
0.006495
-0.000039
0.003323
0.000218
117.846011
Izz, kg m^2
0.140842
0.159047
0.153222
0.003718
0.158154
3.118428
The procedure described above to generate the statistical data was repeated for sensor
origins located farther away from the global axes to investigate the sensitivity to errors
associated with the approximated rigid body angular accelerations. Table 3 shows the
results of the 500 randomizations for an accelerometer located 1.0 meters from each
26
global axis. Table 4 shows the results of the 500 randomizations for an accelerometer
located 100 meters from each global axis.
Max Rotation of 5.0 Degrees
Max Rotation of 1.0 Degrees
Max Rotation of 0.1 Degrees
Table 3 - Single Triaxial Response Orientation Randomization – 2.0 m Cube
Standard
Deviation
Actual
Value
18.154737
0.015406
18.175000
0.035240
0.000085
0.035236
0.009937
0.000272
0.000129
0.000074
0.000130
0.610778
-0.085632
Minimum
Maximum
Mean
Mass, kg
18.082472
18.174932
Xcg, m
0.035048
0.035438
Ycg, m
-0.000010
Mean
Error, %
0.111488
Zcg, m
-0.085822
-0.085428
-0.085630
0.000099
Ixx, kg m^2
0.217446
0.219030
0.218404
0.000380
0.218554
Ixy, kg m^2
-0.000449
0.000210
-0.000106
0.000148
-0.000109
3.082053
Ixz, kg m^2
0.077688
0.078476
0.078105
0.000164
0.078161
0.071712
Iyy, kg m^2
0.303375
0.304362
0.303993
0.000227
0.304172
0.059084
Iyz, kg m^2
0.000013
0.000409
0.000217
0.000080
0.000218
0.299828
Izz, kg m^2
0.157871
0.158390
0.158133
0.000131
0.158154
0.013157
Mass, kg
11.854112
18.145692
16.419498
1.103552
18.175000
Xcg, m
0.033070
0.037169
0.035142
0.000843
0.035236
0.266352
Ycg, m
-0.001283
0.001518
0.000122
0.000718
0.000130
5.802512
-0.085632
0.001570
0.068673
9.658884
Zcg, m
-0.087471
-0.083446
-0.085579
0.000957
Ixx, kg m^2
0.173864
0.219122
0.205577
0.008594
0.218554
5.937503
Ixy, kg m^2
-0.002688
0.002282
-0.000041
0.001251
-0.000109
62.733052
Ixz, kg m^2
0.058475
0.078663
0.072691
0.003734
0.078161
6.999032
Iyy, kg m^2
0.248967
0.304363
0.288859
0.009788
0.304172
5.034458
Iyz, kg m^2
-0.001476
0.001775
0.000223
0.000698
0.000218
2.150347
Izz, kg m^2
0.149132
0.159015
0.155856
0.001909
0.158154
1.452843
Mass, kg
1.341097
16.732259
5.947561
3.055757
18.175000
Xcg, m
0.023580
0.043307
0.033886
0.004433
0.035236
3.833190
Ycg, m
-0.007049
0.007030
0.000046
0.003577
0.000130
64.595221
-0.085632
0.060787
67.276145
Zcg, m
-0.093080
-0.072608
-0.083748
0.004791
Ixx, kg m^2
0.094119
0.213223
0.127769
0.023188
0.218554
Ixy, kg m^2
-0.003256
0.002828
-0.000016
0.001657
-0.000109
84.923635
Ixz, kg m^2
0.025223
0.076002
0.040539
0.009767
0.078161
48.134496
Iyy, kg m^2
0.157209
0.294561
0.197577
0.026696
0.304172
35.044457
Iyz, kg m^2
-0.002521
0.002536
0.000171
0.001195
0.000218
21.603716
Izz, kg m^2
0.137102
0.158547
0.142716
0.004362
0.158154
9.761451
2.199306
41.539100
27
Max Rotation of 5.0 Degrees
Max Rotation of 1.0 Degrees
Max Rotation of 0.1 Degrees
Table 4 - Single Triaxial Response Orientation Randomization – 200 m Cube
Minimum
Maximum
Mean
Standard
Deviation
Actual
Value
Mean Error,
%
Mass, kg
0.355276
16.099852
2.383994
2.116082
18.175000
86.883113
Xcg, m
0.010359
0.050807
0.031058
0.008675
0.035236
11.858983
Ycg, m
-0.012667
0.013186
0.000177
0.006690
0.000130
36.282524
Zcg, m
-0.098870
-0.047062
-0.079792
0.010613
-0.085632
6.819398
Ixx, kg m^2
0.085723
0.202798
0.101916
0.016699
0.218554
53.367925
Ixy, kg m^2
-0.003348
0.002784
-0.000157
0.001228
-0.000109
43.721178
Ixz, kg m^2
0.023285
0.068314
0.029787
0.006747
0.078161
61.890541
Iyy, kg m^2
0.149022
0.286477
0.167031
0.018769
0.304172
45.086710
Iyz, kg m^2
-0.002363
0.002570
0.000153
0.000924
0.000218
29.969200
Izz, kg m^2
0.135345
0.153583
0.138311
0.003003
0.158154
12.546279
Mass, kg
0.002208
0.679220
0.031593
0.045178
18.175000
99.826172
Xcg, m
-2.013460
0.043207
-0.345983
0.308629
0.035236
1081.894601
Ycg, m
-0.129628
0.199997
0.011423
0.066416
0.000130
8711.753648
Zcg, m
-0.096587
3.156211
0.466204
0.451563
-0.085632
644.430317
Ixx, kg m^2
0.081956
0.092163
0.084446
0.001154
0.218554
61.361656
Ixy, kg m^2
-0.001893
0.001923
-0.000179
0.000846
-0.000109
63.840095
Ixz, kg m^2
0.021560
0.024990
0.022651
0.000633
0.078161
71.020285
Iyy, kg m^2
0.145538
0.154453
0.146605
0.000651
0.304172
51.801915
Iyz, kg m^2
-0.000816
0.001560
0.000199
0.000437
0.000218
8.883776
Izz, kg m^2
0.133524
0.137674
0.135501
0.000954
0.158154
14.323204
Mass, kg
-0.000398
0.035853
0.001451
0.002604
18.175000
99.992016
Xcg, m
-1065.1472
22865.9779
43.981544
1027.468388
0.035236
124719.0342
Ycg, m
-2191.1783
302.299032
-3.478937
101.368268
0.000130
2683754.024
Zcg, m
-32189.4572
1707.383930
-61.624847
1448.532908
-0.085632
71865.159656
Ixx, kg m^2
0.075344
0.095337
0.084552
0.004310
0.218554
61.312936
Ixy, kg m^2
-0.007002
0.005219
-0.000025
0.002778
-0.000109
77.397912
70.976320
Ixz, kg m^2
0.015679
0.032502
0.022685
0.002776
0.078161
Iyy, kg m^2
0.142548
0.151752
0.146829
0.001722
0.304172
51.728458
Iyz, kg m^2
-0.003870
0.004595
0.000429
0.001408
0.000218
96.882834
Izz, kg m^2
0.127036
0.145335
0.135474
0.003509
0.158154
14.340172
As would be expected, the estimates of the inertia properties become increasingly worse
as the distance of the accelerometer from the GCS origin increases. This is simply a
result of using rigid body properties to estimate angular accelerations. Since the rotations
resulting from rigid body dynamics are weighted by the distance from the 6 DOF location
point P to each measurement point i, large distances will weight the rotations more
28
causing larger RB rotation errors. This is why the two smaller cubes have similar values
since the distances are one meter or less from the global axes. Table 5 shows the
comparison for the estimated properties of the different sized volumes with varying
maximum angles.
Table 5 - Single Triaxial Response Orientation Mean Value Comparison
Max Angle
0.1 Degrees
1.0 Degrees
5.0 Degrees
Cube Size
0.02 m
2.0 m
200 m
0.02 m
2.0 m
200 m
0.02 m
2.0 m
200 m
Mass, kg
18.172661
18.154737
2.383994
17.967635
16.419498
0.031593
14.433371
5.947561
0.001451
Xcg, m
0.035237
0.035240
0.031058
0.035225
0.035142
-0.34598
0.035037
0.033886
43.981544
Ycg, m
0.000130
0.000129
0.000177
0.000135
0.000122
0.011423
0.000070
0.000046
-3.478937
Zcg, m
-0.085633
-0.085630
-0.07979
-0.085612
-0.085579
0.466204
-0.085496
-0.08375
-61.62485
Ixx, kg m^2
0.218547
0.218404
0.101916
0.216945
0.205577
0.084446
0.191460
0.127769
0.084552
Ixy, kg m^2
-0.000114
-0.000106
-0.00016
-0.000140
-0.000041
-0.00018
0.000064
-0.00002
-0.000025
0.022685
Ixz, kg m^2
0.078158
0.078105
0.029787
0.077462
0.072691
0.022651
0.066866
0.040539
Iyy, kg m^2
0.304155
0.303993
0.167031
0.302409
0.288859
0.146605
0.271354
0.197577
0.146829
Iyz, kg m^2
0.000218
0.000217
0.000153
0.000208
0.000223
0.000199
-0.000039
0.000171
0.000429
Izz, kg m^2
0.158153
0.158133
0.138311
0.157852
0.155856
0.135501
0.153222
0.142716
0.135474
The process described above was repeated on the same model increasing the number of
triaxial accelerometers with orientation errors from one to four. Table 6 shows the results
of the estimation process for the 2.0m cube, and Table 7 shows the comparison for all
three volumes and maximum angles.
29
Max Rotation of 5.0 Degrees
Max Rotation of 1.0 Degrees
Max Rotation of 0.1 Degrees
Table 6 - Four Triaxial Response Orientation Randomization – 2.0 m Cube
Standard
Deviation
Actual
Value
18.083871
0.070768
18.175000
0.501398
0.035233
0.000184
0.035236
0.010642
Minimum
Maximum
Mean
Mass, kg
17.647433
18.171326
Xcg, m
0.034708
0.035872
Mean
Error, %
Ycg, m
-0.000351
0.000722
0.000139
0.000203
0.000130
7.019616
Zcg, m
-0.086246
-0.085071
-0.085613
0.000217
-0.085632
0.021654
Ixx, kg m^2
0.215007
0.220520
0.217819
0.000934
0.218554
0.336080
Ixy, kg m^2
-0.001227
0.001014
-0.000096
0.000389
-0.000109
11.577813
Ixz, kg m^2
0.076561
0.079185
0.077870
0.000402
0.078161
0.373233
Iyy, kg m^2
0.299621
0.305787
0.303355
0.000899
0.304172
0.268783
Iyz, kg m^2
-0.000614
0.001071
0.000218
0.000351
0.000218
0.099840
Izz, kg m^2
0.157240
0.158940
0.158046
0.000299
0.158154
0.067897
Mass, kg
4.834684
17.254430
12.677715
2.487204
18.175000
Xcg, m
0.028797
0.040022
0.035020
0.001794
0.035236
0.614414
Ycg, m
-0.005292
0.005560
-0.000062
0.002052
0.000130
147.778979
-0.085632
30.246410
Zcg, m
-0.090620
-0.080426
-0.085459
0.002038
Ixx, kg m^2
0.119316
0.218761
0.178166
0.019345
0.218554
Ixy, kg m^2
-0.006249
0.006130
-0.000072
0.002423
-0.000109
34.054737
Ixz, kg m^2
0.037204
0.079610
0.061286
0.007972
0.078161
21.590585
Iyy, kg m^2
0.188826
0.299957
0.256536
0.022218
0.304172
15.661029
Iyz, kg m^2
-0.008337
0.006688
-0.000095
0.002445
0.000218
143.565697
Izz, kg m^2
0.140461
0.161730
0.151196
0.003800
Mass, kg
0.270840
12.220689
1.898271
1.266945
18.175000
Xcg, m
0.004654
0.058922
0.030341
0.009700
0.035236
13.894008
Ycg, m
-0.025361
0.026383
-0.000860
0.010397
0.000130
763.695847
-0.085632
0.158154
0.201507
18.479615
4.399238
89.555594
Zcg, m
-0.111312
-0.050498
-0.079419
0.010746
Ixx, kg m^2
0.084040
0.176840
0.098537
0.010512
0.218554
54.914238
Ixy, kg m^2
-0.011983
0.005723
0.000050
0.002102
-0.000109
146.281883
Ixz, kg m^2
0.020814
0.070475
0.028223
0.004650
0.078161
63.891161
Iyy, kg m^2
0.148468
0.251294
0.162905
0.011709
0.304172
46.443042
Iyz, kg m^2
-0.013346
0.019239
0.000032
0.002367
0.000218
85.243266
Izz, kg m^2
0.133340
0.166936
0.137956
0.002952
0.158154
12.770742
7.254653
30
Table 7 - Four Triaxes with Orientation Errors - Mean Value Comparison
Max Angle
0.1 Degrees
1.0 Degrees
5.0 Degrees
Cube Size
0.02 m
2.0 m
200 m
0.02 m
2.0 m
200 m
0.02 m
2.0 m
200 m
Mass, kg
18.166463
18.083871
0.570457
17.434105
12.677715
0.006051
10.084563
1.898271
0.000189
Xcg, m
0.035237
0.035233
0.016812
0.035214
0.035020
-1.67115
0.034331
0.030341
-6.96943
Ycg, m
0.000133
0.000139
0.000434
0.000141
-0.000062
0.024544
0.000085
-0.00086
-0.65679
Zcg, m
-0.085632
-0.085613
-0.06093
-0.085562
-0.085459
2.410368
-0.084415
-0.07942
9.24425
Ixx, kg m^2
0.218498
0.217819
0.088712
0.212904
0.178166
0.084384
0.158670
0.098537
0.085022
Ixy, kg m^2
-0.000129
-0.000096
0.000005
-0.000126
-0.000072
-0.00006
0.000145
0.000050
0.000033
Ixz, kg m^2
0.078137
0.077870
0.024168
0.075784
0.061286
0.022494
0.052908
0.028223
0.022907
Iyy, kg m^2
0.304097
0.303355
0.151405
0.297778
0.256536
0.146542
0.231776
0.162905
0.147133
Iyz, kg m^2
0.000221
0.000218
0.000035
0.000220
-0.000095
0.000024
0.000499
0.000032
0.000384
Izz, kg m^2
0.158146
0.158046
0.136154
0.157163
0.151196
0.135658
0.147383
0.137956
0.135487
By misaligning four sensors, it can be seen that for the small cube, the results are slightly
worse than the estimates for the single transducer case. A similar trend of deviation from
the actual values can be seen with the medium sized cube. For the large cube, some of
the estimated parameters are also worse than the estimates for the single transducer with
orientation errors. However, the CG terms get better with an increase in the maximum
angle as compared to the single sensor estimates. Since the CG terms are scaled by the
mass term when solving the DIM equation and the mass term for the larger cube is nearly
zero, it may be concluded that this is a coincidence and the data is not really better for a
larger volume when high degrees of error exist in the transducer orientations.
Table 8 and Table 9 tend to show the same parameter error trends as the previous two
cases. The most significant information that can be gathered from these tables is that
regardless of the total number of triaxial sensors containing orientation errors, the DIM
31
will produce reasonable results as long as the degree of the errors on each triaxial sensor
is minimized.
Max Rotation of 5.0 Degrees
Max Rotation of 1.0 Degrees
Max Rotation of 0.1 Degrees
Table 8 - All Triaxial Response Orientation Randomization – 2.0 m Cube
Standard
Deviation
Actual
Value
17.996429
0.138252
18.175000
0.982507
0.035223
0.000256
0.035236
0.038152
Minimum
Maximum
Mean
Mass, kg
17.136708
18.162563
Xcg, m
0.034447
0.035922
Mean
Error, %
Ycg, m
-0.000713
0.001094
0.000131
0.000300
0.000130
0.663489
Zcg, m
-0.086356
-0.084846
-0.085617
0.000296
-0.085632
0.016502
Ixx, kg m^2
0.211703
0.220223
0.217210
0.001500
0.218554
0.615074
Ixy, kg m^2
-0.001627
0.001360
-0.000125
0.000523
-0.000109
14.640126
Ixz, kg m^2
0.074928
0.079085
0.077601
0.000612
0.078161
0.716548
Iyy, kg m^2
0.295553
0.306085
0.302565
0.001524
0.304172
0.528453
Iyz, kg m^2
-0.001107
0.002009
0.000222
0.000526
0.000218
1.995069
Izz, kg m^2
0.156653
0.159215
0.157930
0.000440
0.158154
0.141161
Mass, kg
2.766647
17.089133
10.001365
2.924079
18.175000
Xcg, m
0.027148
0.042760
0.034996
0.002592
0.035236
0.681044
Ycg, m
-0.007121
0.008916
0.000099
0.003001
0.000130
23.876556
-0.085632
44.971855
Zcg, m
-0.095153
-0.077389
-0.085007
0.003010
Ixx, kg m^2
0.104658
0.225860
0.157999
0.022775
0.218554
Ixy, kg m^2
-0.006740
0.006456
-0.000032
0.002585
-0.000109
71.120229
Ixz, kg m^2
0.030723
0.084185
0.053160
0.009470
0.078161
31.987464
Iyy, kg m^2
0.169270
0.308985
0.233131
0.026299
0.304172
23.355674
Iyz, kg m^2
-0.006940
0.009225
0.000309
0.002866
0.000218
41.924716
Izz, kg m^2
0.138269
0.161536
0.148031
0.004454
0.158154
Mass, kg
0.120125
7.356258
1.273383
1.044933
18.175000
0.728903
27.706899
6.400524
92.993769
Xcg, m
-0.015662
0.070272
0.027714
0.014448
0.035236
21.348955
Ycg, m
-0.051282
0.046400
0.000752
0.014668
0.000130
479.748998
-0.085632
13.301032
Zcg, m
-0.123743
-0.000570
-0.074242
0.017165
Ixx, kg m^2
0.084309
0.201999
0.094062
0.010071
0.218554
Ixy, kg m^2
-0.006372
0.006289
-0.000047
0.002001
-0.000109
57.005365
Ixz, kg m^2
0.020870
0.056844
0.026330
0.004155
0.078161
66.313467
Iyy, kg m^2
0.147261
0.263677
0.157769
0.010979
0.304172
48.131624
Iyz, kg m^2
-0.010021
0.015201
0.000171
0.002138
0.000218
21.528510
Izz, kg m^2
0.132900
0.152946
0.137475
0.002724
0.158154
13.074722
56.961459
32
Table 9 - Four Triaxial Response Orientation Mean Value Comparison
Max Angle
0.1 Degrees
1.0 Degrees
5.0 Degrees
Cube Size
0.02 m
2.0 m
200 m
0.02 m
2.0 m
200 m
0.02 m
2.0 m
200 m
Mass, kg
18.156945
17.996429
0.320076
16.746627
10.001365
0.003403
7.553111
1.273383
0.000088
-128.299
Xcg, m
0.035235
0.035223
0.004390
0.035187
0.034996
-2.55453
0.033593
0.027714
Ycg, m
0.000127
0.000131
0.001094
0.000113
0.000099
-0.08441
-0.00012
0.000752
-83.1066
Zcg, m
-0.085629
-0.085617
-0.04073
-0.085613
-0.085007
3.872358
-0.08378
-0.07424
222.3874
0.084951
Ixx, kg m^2
0.218408
0.217210
0.086669
0.208332
0.157999
0.084551
0.141874
0.094062
Ixy, kg m^2
-0.000106
-0.000125
-0.00005
0.000070
-0.000032
-0.00004
0.000879
-0.00005
0.000033
Ixz, kg m^2
0.078085
0.077601
0.023437
0.073903
0.053160
0.022549
0.045803
0.026330
0.022883
Iyy, kg m^2
0.304019
0.302565
0.149208
0.291735
0.233131
0.146837
0.210835
0.157769
0.146802
Iyz, kg m^2
0.000196
0.000222
0.000062
0.000197
0.000309
-0.00007
0.000877
0.000171
0.000184
Izz, kg m^2
0.158122
0.157930
0.135988
0.156352
0.148031
0.135648
0.143956
0.137475
0.135461
2.2.2 EXPERIMENTAL EXAMPLE – BUNGEE SUPPORTED CALIBRATION MASS
An experimental was conducted on the load cell calibration mass to obtain data for a
sensitivity analysis. The same procedure used in the analytical example was repeated
with the test data.
A randomized sample was obtained by applying the orthogonal
transformations to a single accelerometer and to all eight accelerometers with a maximum
error of 0.1, 1.0 and 5.0 degrees. The interesting phenomenon that occurred with the real
data is that the parameters estimated for the different error conditions were far less
sensitive to orientation errors than were seen in the analytical case.
33
Max Rotation of 5.0 Degrees
Max Rotation of 1.0 Degrees
Max Rotation of 0.1 Degrees
Table 10 - All Triaxial Response Orientation Randomization, Calibration Mass
Minimum
Maximum
Mean
Standard
Deviation
Actual
Value
Mean Error,
%
Mass, kg
18.130835
18.135112
18.132996
0.000830
18.175000
0.231110
Xcg, m
0.034229
0.034353
0.034285
0.000019
0.035236
2.700737
Ycg, m
-0.000299
-0.000167
-0.000235
0.000024
0.000130
280.955222
3.979295
Zcg, m
-0.082310
-0.082144
-0.082224
0.000026
-0.085632
Ixx, kg m^2
0.219364
0.219700
0.219517
0.000055
0.218554
0.440613
Ixy, kg m^2
-0.001388
-0.001173
-0.001284
0.000039
-0.000109
1078.025760
0.539954
Ixz, kg m^2
0.078507
0.078678
0.078583
0.000027
0.078161
Iyy, kg m^2
0.310852
0.311283
0.311068
0.000076
0.304172
2.267085
Iyz, kg m^2
-0.002442
-0.002190
-0.002302
0.000044
0.000218
1156.266725
Izz, kg m^2
0.162735
0.162975
0.162843
0.000037
0.158154
2.964952
Mass, kg
18.105328
18.155663
18.133209
0.008597
18.175000
0.229935
Xcg, m
0.033810
0.034841
0.034296
0.000203
0.035236
2.667205
Ycg, m
-0.000994
0.000518
-0.000233
0.000224
0.000130
279.885408
3.989184
Zcg, m
-0.082918
-0.081421
-0.082216
0.000259
-0.085632
Ixx, kg m^2
0.217745
0.221146
0.219531
0.000586
0.218554
0.446995
Ixy, kg m^2
-0.002403
-0.000254
-0.001290
0.000411
-0.000109
1083.728336
0.564719
Ixz, kg m^2
0.077885
0.079432
0.078603
0.000265
0.078161
Iyy, kg m^2
0.308862
0.313000
0.311047
0.000761
0.304172
2.260008
Iyz, kg m^2
-0.003579
-0.001255
-0.002292
0.000400
0.000218
1151.465695
Izz, kg m^2
0.161828
0.163975
0.162870
0.000360
0.158154
2.981976
Mass, kg
17.965978
18.268425
18.141283
0.046865
18.175000
0.185514
Xcg, m
0.031549
0.037222
0.034247
0.001000
0.035236
2.807759
Ycg, m
-0.003955
0.003214
-0.000315
0.001219
0.000130
343.151482
3.829362
Zcg, m
-0.085571
-0.079248
-0.082352
0.001228
-0.085632
Ixx, kg m^2
0.212493
0.228718
0.219918
0.003001
0.218554
0.624174
Ixy, kg m^2
-0.006814
0.004084
-0.001284
0.001872
-0.000109
1077.882844
0.468947
Ixz, kg m^2
0.074506
0.082222
0.078528
0.001383
0.078161
Iyy, kg m^2
0.302015
0.320876
0.311900
0.003554
0.304172
2.540538
Iyz, kg m^2
-0.007813
0.004816
-0.002454
0.002190
0.000218
1225.718429
Izz, kg m^2
0.157361
0.167691
0.163030
0.001720
0.158154
3.083260
34
Table 11 - All Response Orientation Mean Value Comparison – Calibration Mass
Max Angle
1 Sensor
0.1
degrees
1 Sensor
1.0
degrees
8 Sensors
0.1
degrees
8 Sensors
1.0
degrees
8 Sensors
5.0
degrees
DIM
Solution
Actual
Values
Mass, kg
18.132970
18.133094
18.132996
18.133209
18.141283
18.175000
18.175000
Xcg, m
0.034284
0.034284
0.034285
0.034285
0.034247
0.032490
0.035236
Ycg, m
-0.000234
-0.000234
-0.000235
-0.000235
-0.000315
0.000289
0.000130
Zcg, m
-0.082223
-0.082221
-0.082224
-0.082224
-0.082352
-0.081580
-0.085632
Ixx, kg m^2
0.219514
0.219510
0.219517
0.219517
0.219918
0.220117
0.218554
Ixy, kg m^2
-0.001287
-0.001284
-0.001284
-0.001284
-0.001284
-0.001524
-0.000109
Ixz, kg m^2
0.078583
0.078581
0.078583
0.078583
0.078528
0.074788
0.078161
Iyy, kg m^2
0.311066
0.311060
0.311068
0.311068
0.311900
0.308555
0.304172
Iyz, kg m^2
-0.002302
-0.002302
-0.002302
-0.002302
-0.002454
-0.002898
0.000218
Izz, kg m^2
0.162844
0.162844
0.162843
0.162843
0.163030
0.162738
0.158154
It may be difficult to determine the direction cosines of all transducers as accurately as
would be needed to estimate accurate inertia properties. For this reason, it is important to
determine the degree of rigid body error caused by all of the response axes. The response
perimeter selection method in Appendix A covers the procedure developed to assist in the
removal of erroneous channels.
Due to the large change in estimated inertia parameters with the increasing orientation
errors, a single sensor was gradually rotated about the X and Y axes from -0.1 to +0.1
degrees with a constant step change in degrees. Figure 8 shows the dramatic change in
the estimated mass and CG properties with only a small change in transducer orientation.
Figure 9 shows the mass moment of inertia properties for the same incremental changes
in orientation.
35
0.035
Xcg, m
Mass, kg
15
10
0.03
0.025
5
0.02
0.1
0.1
0.05
0.1
0
0
-0.05
-0.1
Y-Axis Rotation, degrees
-0.1
0.05
0.05
0
-0.05
-0.05
-0.1
Y-Axis Rotation, degrees
X-Axis Rotation, degrees
-0.06
-0.07
Zcg, m
0.01
0.005
Ycg, m
0.1
0
0
-0.005
-0.1
0.05
-0.05
X-Axis Rotation, degrees
-0.08
-0.09
-0.01
0.1
0.1
0.05
0.1
0
0
-0.05
-0.1
Y-Axis Rotation, degrees
-0.1
0.05
0.05
0.1
0
0
-0.05
-0.05
-0.1
Y-Axis Rotation, degrees
X-Axis Rotation, degrees
-0.1
0.05
-0.05
X-Axis Rotation, degrees
Figure 8 - Single Sensor Rotation about the X and Y Axes Only, Mass and CG
x 10
2
Ixy, kg m2
Ixx, kg m2
0.2
0.15
0.1
0.1
0.05
0
-0.05
Y-Axis Rotation, degrees
-0.1
-0.1
-0.05
0
0.05
0
-2
0.1
0.1
0.05
0
-0.05
Y-Axis Rotation, degrees
X-Axis Rotation, degrees
-0.1
-0.1
-0.05
0
0.05
0.1
X-Axis Rotation, degrees
0.3
0.07
0.06
Iyy, kg m2
Ixz, kg m2
-3
0.05
0.04
0.25
0.2
0.03
0.1
0.05
0
-0.05
Y-Axis Rotation, degrees
-0.1
-0.1
-0.05
0
0.05
0.1
0.05
0
-0.05
Y-Axis Rotation, degrees
X-Axis Rotation, degrees
-3
0.1
-0.1
-0.1
-0.05
0
0.05
0.1
X-Axis Rotation, degrees
x 10
0.155
Izz, kg m2
Iyz, kg m2
2
0
-2
0.1
0.05
0
Y-Axis Rotation, degrees
-0.05
-0.1
-0.1
-0.05
0
0.05
X-Axis Rotation, degrees
0.1
0.15
0.145
0.14
0.1
0.05
0
Y-Axis Rotation, degrees
-0.05
-0.1
-0.1
-0.05
0
0.05
0.1
X-Axis Rotation, degrees
Figure 9 - Single Sensor Rotation about the X and Y Axes Only, Inertia Terms
36
2.2.3 EXCITATION LINE OF ACTION SENSITIVITY ANALYSIS
Due to complex geometries of test structures, providing excitation in the direction of
global axes may be impractical. Therefore, the direction cosine vector associated with
the input line of action must be determined. For electro-mechanical shaker excitation, the
line of action may only change slightly due to the structural dynamics during data
acquisition. For impact testing, the line of action and the point of force application may
vary significantly during the averaging process.
When large variations occur, this
inconsistency will typically be apparent in the coherence computations and the data can
be rejected. However, the impacts may actually be consistent along the same line of
action producing reasonable coherence functions, but the line of action of the data may
not be the same as the direction cosine values determined during test setup.
By
eliminating datasets that are inconsistent, more accurate inertia property estimates can be
achieved.
The first sensitivity model used the six inputs located at the origin of the model. By
doing so, the errors in the moment estimates associated with the rigid body
transformations were eliminated so the only errors applied are due to line action errors.
Table 12 shows the comparison of the results of the sensitivity analysis. For the inputs
colocated at the model’s origin, the results of the line of action errors seem to have very
little influence on the inertia property estimates.
37
Table 12 - Line of Action - Origin Sensitivity Comparison
Single LOA
Three LOA
All LOA
Max Angle
0.1
1.0
5.0
0.1
1.0
5.0
0.1
1.0
5.0
Mass, kg
18.174982
18.174980
18.174982
18.173099
18.173214
18.173161
18.128002
18.128856
18.129796
Xcg, m
0.035236
0.035236
0.035236
0.035236
0.035236
0.035236
0.035236
0.035236
0.035234
Ycg, m
0.000130
0.000130
0.000130
0.000130
0.000130
0.000130
0.000129
0.000129
0.000130
-0.085631
Zcg, m
-0.085632
-0.085632
-0.085632
-0.085631
-0.085632
-0.085631
-0.085632
-0.085632
Ixx, kg m^2
0.218554
0.218553
0.218554
0.218520
0.218535
0.218526
0.217946
0.217941
0.218025
Ixy, kg m^2
-0.000109
-0.000108
-0.000107
-0.000086
-0.000133
-0.000104
0.000123
-0.000034
-0.000164
Ixz, kg m^2
0.078161
0.078158
0.078163
0.078081
0.078171
0.078110
0.077834
0.077648
0.077894
Iyy, kg m^2
0.304172
0.304172
0.304172
0.304140
0.304141
0.304142
0.303388
0.303412
0.303399
Iyz, kg m^2
0.000215
0.000224
0.000217
0.000271
0.000208
0.000220
0.000117
0.000244
0.000109
Izz, kg m^2
0.158153
0.158152
0.158154
0.158116
0.158144
0.158123
0.157746
0.157665
0.157680
The model simulating the hanging calibration mass with 24 input locations not applied at
the GCS origin was used to verify the sensitivity of inaccurate line of action orientations
away from the origin. The analysis uses the same methodology that was used for the
accelerometer orientation sensitivity analysis. The three testing scenarios applied were
for a single input, half of the inputs, and all of the inputs to be randomly skewed at
maximum angles 0.1, 1.0, 5.0, and 10.0 degrees about each of the three global axes.
Table 13 compares the results of the estimated inertia properties for the varying degree of
line of action errors.
38
Table 13 - Line of Action - Off Origin Sensitivity Comparison
Single LOA Error
Half LOA Errors
All LOA Errors
Angle
0.1
1.0
5.0
10.0
0.1
1.0
5.0
10.0
0.1
1.0
5.0
10.0
Mass
18.17499
18.17496
18.17358
18.16818
18.17500
18.17405
18.15100
18.08139
18.17501
18.17310
18.12880
18.08139
Xcg
0.03524
0.03524
0.03522
0.03517
0.03524
0.03524
0.03523
0.03521
0.03524
0.03524
0.03527
0.03521
Ycg
0.00013
0.00013
0.00012
0.00009
0.00013
0.00013
0.00012
0.00008
0.00013
0.00013
0.00015
0.00008
Zcg
-0.0856
-0.0856
-0.0856
-0.0857
-0.0856
-0.0856
-0.0857
-0.0857
-0.0856
-0.0856
-0.0856
-0.0857
Ixx
0.21856
0.21855
0.21854
0.21858
0.21855
0.21854
0.21847
0.21791
0.21855
0.21850
0.21781
0.21791
Ixy
-0.0001
-0.0001
-0.0001
-0.0001
-0.0001
-0.0001
0.00001
0.00015
-0.0001
-0.0001
-0.0002
0.00015
Ixz
0.07816
0.07816
0.07815
0.07815
0.07816
0.07815
0.07807
0.07773
0.07816
0.07813
0.07802
0.07773
Iyy
0.30418
0.30416
0.30403
0.30391
0.30417
0.30416
0.30393
0.30262
0.30417
0.30415
0.30327
0.30262
Iyz
0.00022
0.00022
0.00019
0.00011
0.00022
0.00023
0.00021
0.00031
0.00022
0.00021
0.00016
0.00031
Izz
0.15815
0.15815
0.15814
0.15812
0.15815
0.15813
0.15802
0.15747
0.15815
0.15815
0.15778
0.15747
The results for the line of action sensitivity analysis indicate that the DIM estimation
process is not influenced by orientation errors as much as the method is for accelerometer
orientation errors. This is a beneficial effect for experimental testing scenarios that
utilize instrumented impact hammers to provide the input excitation. Impact testing can
become mundane and repetitive and errors in accurately providing excitation normal to
the surface of a structure are likely to occur. The lack of sensitivity to this error helps
reduce the overall errors in the inertia property estimations.
39
3.0 NUMERICAL MODIFICATIONS AND DATA SELECTION
There are no methods to compute perfect results from corrupt measurements.
The
purpose of this section is to provide three simple tools that will allow for the removal of
corrupt input data, manipulation of the solution equation to eliminate unknowns, and
scaling of the data by either directly scaling the FRF matrix by a scalar value or changing
the length unit. These simple tools along with the perimeter response selection method
discussed in Appendix B can yield a reduction in inertia property estimation errors by
providing optimal and consistent measurement data.
3.1 PERIMETER EXCITATION SELECTION
As previously discussed, response measurements can be selectively removed by directly
comparing the measurement to the approximated rigid body motion. Measurements that
yield high errors when compared to the least squares rigid body estimate can be removed
from the solution procedure by the application of a weighting matrix. Since this is an
inverse process in which the rigid body motion is described by a redundant set of
transducers, a single response measurement can be eliminated.
In the case of excitation, the rigid body motion resulting from excitation to a structure is a
forward process. This means that the motion is due to a linear combination of the
excitation and reaction forces acting on a structure.
A single excitation cannot be
removed from the structure without equivalently removing the response of the structure
due to that input. A method of simply removing all the measurements associated with the
40
corrupt input can be applied. This is achieved by eliminating the single column of the
FRF matrix which corresponds to the inconsistent input.
Determination of inconsistent inputs can be difficult. The jackknife procedure involves
an iterative approach were a single input is removed from the data group. The inertia
properties are then computed with the remaining data. The previously removed data is
returned to the data group and another input and resulting FRF column is removed from
the group. Again the inertia parameters are computed. This process is repeated until all
the inputs have been removed and added back to the data group. By inspecting the
estimated properties, any single input causing changes in the estimated parameters can be
considered an inconsistent excitation. This may be due to a number of data acquisition
factors, input channel overload during excitation, transducer failure, single conditioning
failure, and cable shorts. The inconsistency can also be due to transformation errors such
as incorrect geometric excitation location and inaccurate excitation line of action or
exciter orientations.
Once an erroneous excitation is detected, the input can be removed from the FRF matrix
and the jackknife process is repeated with the remaining inputs. This procedure is
repeated until the excitation set produces a consistent set of estimated parameters. This
means that once a consistent set is found, removing a single excitation has little to no
effect on the estimated properties.
The excitation should be checked to verify that all translational and rotational degrees of
freedom are still adequately excited for the remaining inputs. This can be verified by
41
computing the principle components on the remaining excitations.
Six curves
representing the six singular values computed at each frequency line should be clearly
indicated and well separated from the noise floor.[15]
A limitation of measurement optimization is that a reaction force cannot be removed
from the solution equation without negatively affecting the estimated inertia properties
since this will result in unmeasured forces acting on the structure.
In general, the
resulting estimates will not describe the isolated structure, but will also reflect the inertia
properties of the soft support system as well. By estimating the inertia properties in a
frequency band well away from the rigid body frequencies of the soft support system, a
more reasonable, but still contaminated, estimate can be achieved.
Min: 0.0350 Max: 0.0353 Mean: 0.0352 STD: 0.0001
0.0352
Xcg, m
Mass, kg
Min: 18.1409 Max: 18.1784 Mean: 18.1587 STD: 0.0080
18.17
18.16
18.15
0.035
-3
x 10
Min: -0.0856 Max: -0.0853 Mean: -0.0854 STD: 0.0000
Min: 0.0008 Max: 0.0016 Mean: 0.0015 STD: 0.0002
-0.0853
1.4
Zcg, m
Ycg, m
1.6
0.0351
1.2
1
-0.0854
-0.0854
-0.0855
-0.0855
0.8
x 10
0.215
15
0.214
10
Ixy, kg m2
Ixx, kg m2
Min: 0.2117 Max: 0.2151 Mean: 0.2128 STD: 0.0006
0.213
5
0
Min: 0.0764 Max: 0.0791 Mean: 0.0773 STD: 0.0005
Min: 0.3052 Max: 0.3061 Mean: 0.3057 STD: 0.0002
0.079
Iyy, kg m2
0.306
0.078
0.077
-3
x 10
0.3056
0.3054
Min: 0.1567 Max: 0.1584 Mean: 0.1571 STD: 0.0003
4
3
2
0
0.3058
Min: 0.0018 Max: 0.0054 Mean: 0.0047 STD: 0.0007
5
Izz, kg m2
Ixz, kg m2
Min: -0.0008 Max: 0.0017 Mean: 0.0008 STD: 0.0004
-5
0.212
Iyz, kg m2
-4
5
10
15
Removed Inputs ( )
20
0.158
0.1575
0.157
0
5
10
15
Removed Inputs ( )
20
Figure 10 - Inertia Properties Jackknife Estimates
42
1
Xcg : 0.0002
Mass : 0.0197
1
0.5
0
1
Zcg : 0.0001
Ycg : 0.0007
0
1
0.5
0.5
0.5
0
1
Iyy : 0.0004
Ixz : 0.0018
0
1
0.5
0.5
0
1
Izz : 0.0013
0
1
Iyz : 0.0029
0.5
0
1
Ixy : 0.0016
Ixx : 0.0023
0
1
0.5
0
0.5
0
5
10
Removed Input:
15
20
0.5
0
0
5
10
15
Max Err Input: 17
20
Figure 11 - Figure 10 - Inertia Properties Jackknife Error
In order to determine the input location causing the largest amount of error for all
properties, the estimates are scaled with the maximum set to unity. By summing the
scaled error of all the inertia properties for each jackknife case, the input causing the
largest overall deviation in the estimated parameters can be removed.
The difficulty in removing inputs involves deciding when a sufficient number has been
removed in order to produce the most consistent inertia property estimates. The standard
deviation of each jackknife process can be used to check the consistency of the remaining
inputs. In theory, as the standard deviation approaches zero, the remaining inputs are
consistent. Figure 12 shows an example of tracking the standard deviation as the inputs
43
with high degree of errors for the analytical model are successively removed. Each
successive jackknife cycle has an additional input removed.
-3
-5
Standard Deviations
x 10
6
4
2
4
2
-5
-5
x 10
10
Zcg
Ycg
15
5
5
4
3
2
1
-4
3
Ixy
Ixx
x 10
5
4
3
2
1
2
1
-4
-5
x 10
x 10
4
15
3
Iyy
Ixz
x 10
-4
x 10
2
10
5
1
-4
-4
x 10
3
6
4
Izz
Iyz
Standard Deviations
x 10
6
Xcg
Mass
8
2
1
2
3
4
5
6
Jack Knife Cycle
7
8
9
x 10
2
1
1
2
3
4
5
6
Jack Knife Cycle
7
8
9
Figure 12 - Analytical Model Jackknife Standard Deviation History
The use of the standard deviation value to determine the appropriate excitation set works
well for some field tests and fails for others. This is likely due to inconsistent excitation
errors on most of the inputs, error in the line of action, the local origin of the line of
action or the lack of sensitivity to the excitation errors. Since measurement errors exist in
experimental data, the errors associated with line of action orientation errors may be
masked causing a reduction in sensitivity to the line of action errors. Figure 13 shows the
deviation changes for the hanging calibration mass experimental data.
By visual
44
inspection of the standard deviation for each of the properties, it can be seen that there is
no obvious jackknife cycle that results in a consistent solution. It can be reasoned that
the errors due to each impact are not significant enough to cause a large change in the
estimated inertia properties. However, after removing eight impacts, an overall lowest
standard deviation is seen and may be considered to provide the most accurate inertia
property estimates.
-4
2.5
2
1.5
-4
-4
2.5
x 10
x 10
3
Zcg
Ycg
2
1.5
1
2
1
-3
-3
x 10
x 10
2
1.5
Ixy
Ixx
12
10
8
6
4
2
1
0.5
-3
-3
x 10
x 10
4
2
Iyy
3
Ixz
Standard Deviations
x 10
Xcg
Mass
Standard Deviations
0.024
0.022
0.02
0.018
0.016
0.014
0.012
2
1.5
1
1
0.5
-4
-4
x 10
x 10
12
8
Izz
Iyz
10
6
8
6
4
2
4
6
8
10
Jack Knife Cycle
12
14
16
18
4
2
4
6
8
10
Jack Knife Cycle
12
14
16
18
Figure 13 - Standard Deviations for Jackknife Process - Hanging Cal Mass
For the case above, an extremely high number of impacts has been removed in the final
jackknife iteration.
The original measurement set contained 24 inputs and the last
jackknife cycle has removed 17 of those impacts. The DIM estimates may have high
45
deviations due to the limited number of remaining impacts, meaning that there are not
enough unique inputs into the structure to excite all six degrees of freedom. Methods
exist to verify that the structure has been adequately excited. Computing an autopower
spectrum or principle component analysis of the reaction force measurements may seem
to indicate the quality of the six degree of freedom excitation. However, the reaction
forces are a result of the cross-axis dynamic coupling of a structure as well as the soft
support boundary, and it may appear that all six of the degrees of freedom are adequately
excited. Figure 14 indicates that some of the inputs are excited at higher levels than
others; however, they are all dynamically out of the noise floor and only separated by 3
orders of magnitude.
Reaction Force Autopow er
0
10
Autopower, Mag
-2
10
-4
10
-6
10
0
20
40
60
80
100
Frequency, Hz
120
140
160
180
200
140
160
180
200
Reaction Force Principle Component Analysis
0
10
-1
PCA, Mag
10
-2
10
-3
10
-4
10
0
20
40
60
80
100
Frequency, Hz
120
Figure 14 - Reaction Force Autopower and PCA for 17 Removed Inputs
46
A more realistic approach would be to compute the input autopower spectrum of the
combined inputs to the system.
But since FRFs are utilized in the DIM, the excitation
forces are considered to be unity since the FRF calculation scales the response data by the
input force levels. Therefore, a simple check of the remaining input line of action vectors
yields the degree of six axis excitation. The rank of the line of action matrix should be 6
when all degrees of freedom are being excited. A value less than 6 means that one or
more of the forces and moments are not excited. To determine the degree of excitation,
Equation 16 can be used to perform an autopower calculation on the line of action matrix.
Gφφ*
6x6
= {φ1}{φ2 } ...{φNi }
{φ1}{φ2 } ...{φNi }
T
(16)
The diagonal of the autopower matrix illustrates the level to which each degree of
freedom is being excited. For the same inputs used to display the reaction force data in
Figure 14, the rank, autopower, and singular values are listed in Table 14 along with
these values from the iteration cycle with 7 inputs removed. From this, it is clear to see
that the iteration cycle with only 7 inputs removed has adequate excitation as opposed to
the iteration cycle with the 17 inputs removed from the system. The rank gives a clear
indication that two degrees of freedom are not being excited. The singular values also
indicate that two of the axes are weakly excited, while the autopower shows that Fy and
Mx are the degrees of freedom with little to no excitation.
47
Table 14 - Jackknife 7 Remaining Inputs Excitation Verification
Rank
6
-
7 Removed Inputs
Autopower
SVD
8.0000000 2.8403300
7.0000000 2.6587660
2.0000000 1.4157163
0.0697039 0.2385172
0.0868963 0.1248926
0.0688710 0.1105373
Rank
4
-
17 Removed Inputs
Autopower
SVD
6.0000000 2.4597854
0.0000000 1.0001810
1.0000000 0.1195492
2.32E-08
0.0358427
0.0586602
4.33E-18
0.0078227
1.17E-18
3.2 KNOWN PROPERTY ELIMINATION AND DATA SCALING
In the case of the DIM, there are ten unknown parameters that must be estimated from the
measured data. These unknowns correspond to the mass, center of gravity, and mass
moment of inertia terms. It may be possible to determine any of these properties from
alternate methods. Mass may be the easiest term to establish by other means since this
only requires weighing the structure in a known gravitational field. Other terms, such as
the center of gravity may be approximated for symmetric structures, or computed from a
finite element model. However, modification to experimental structures may supercede
the integration of the changes into the finite element model thus making the inertia
property estimates inaccurate through the elimination of known parameters with incorrect
information.
48
For smaller structures, mass moments of inertia may be easily estimated using a
traditional swing test. For complicated, larger structures, this becomes increasingly more
difficult, expensive, and time consuming.
Therefore, mass may be a term that is consistently known and eliminated in the
estimation process. There may be situations where eliminating the known parameters
from the solution equation are not desired. For instance, fixing a known term throughout
the frequency band of estimation, thus eliminating the unknown, may not make physical
sense. If the reaction forces can not be removed due to transducer failure or inability to
instrument a structure, there will be frequencies at which the support system will be in
phase or out of phase with the structure changing the effective mass of the rigid body. It
may be more advisable to pick a higher or lower frequency band away from the
frequency region contaminated by the unmeasured support reactions and scale the FRF
matrix so that the resulting inertia property in that band matches the known quantity.
This can be done by iteratively estimating the inertia properties and scaling the FRF
matrix until the estimated property matches the known quantity in the frequency range
selected. The DIM estimation is then performed on the entire frequency band treating all
inertia properties as unknowns.
Another simple method of data scaling is achieved by changing the unit of length to a
more or less sensitive value. For example, using a metric length unit equal to meters will
not cause the rigid body approximated angular accelerations to be weighted as much as if
units of millimeters are selected. For a test setup with a high confidence of transducer
49
geometric and orientation accuracy, the length units can be scaled to provide a higher
sensitivity to rotations, thus increasing the numerical weighting of the mass moment
inertia property terms during the solution process.
3.2.1 KNOWN PROPERTY ELIMINATION
As previously mentioned, inertia properties of a structure that have previously been
determined by other means can be eliminated from the solution process by reformulating
the solution equation. Some terms can be directly substituted into the solution while
others cannot. Specifically, the mass moment of inertia terms can be directly replaced by
moving the known value to the left hand side of the solution equation. However, the
solution equation must be manipulated slightly in order to remove the mass and CG
terms. This is due to the coupling of the mass and CG terms along the off diagonals of
the mass matrix.
From the modified mass matrix, Equation 17, it is easily seen that the known inertia
terms can be removed by simply subtracting the known parameter from both sides of the
equation. Equation 18 shows that for a known Ixx term, the solution equation is easily
determined.
50
m
Fx
Fy
x
Fz
Mx
My
Mz
P
−θ z
0
θy
−θ x
y θz
z −θ y
=
0 0
θx
z
0
−y
−z
y
0
−x
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
θ y θz
0
0
θz
θx θ y θz
0
0
0
0
0
0
0
mX CG
mYCG
mZ CG
(17)
I xx
I xy
I xz
I yy
I yz
I zz
P
m
Fx
0
x
Fy
0
Fz
0
y θz
z −θ y
−
Mx
My
Mz
P
I xxθ x
=
0
−θ z
0
θy
−θ x
θx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
θy θz
0
0
0
θ y θz
0
0
0
z
−y
0
0
−z
0
x
0
0
0
0
y
−x
0
0
0
0
0
θz
mX CG
mYCG
mZ CG
(18)
I xy
I xz
I yy
I yz
I zz
P
This process of eliminating terms can be completed for each of the known mass moment
of inertia properties. In order to completely remove a known mass value from the
equation, it is not as simple as for the mass moment of inertia properties. The mass
column of the equation can be subtracted from both sides of the equation, but the scalar
value for the known mass must also be replaced on the CG terms. Equation 19 shows the
solution form when the mass term is known.
51
X CG
Fx
mx
0
−mθ z
mθ y
0
0
0
0
0
0
Fy
my
Fz
mz
−
=
0
θz
−mθ y
0
mθ x
−mθ x
0
0x
0
0
0
0
0
0
0
0
0
0
0
0
z
−my
θx θ y θz
0
0
0
0
− mz
0
mx
0
0
0
θy θz
0
0
my
−mx
0
0
0
0
Mx
My
Mz
P
0
0
YCG
Z CG
I xx
(19)
I xy
I xz
I yy
θz
I yz
I zz
P
In the case where a CG coordinate in known, either from a finite element model or other
means, the solution equation is again slightly modified. If the mass is also known, then
the terms can simply be eliminated by subtracting both sides of the equation. However,
for an unknown mass value with a known CG term, the term cannot simply be removed
by subtracting it from both sides of the equation due to the coupling with the mass term.
The terms can be eliminated by substituting the known value and reducing the number of
unknowns. Equation 20 is an example of the solution equation if only the XCG term is
known, and Equation 21 shows the solution equation when the CG is known in 3D space.
m
Fy
x
y + X CGθ z
−θ z
0
θy
−θ x
0
0
0
0
0
0
0
0
0
0
0
0
Fz
z − X CGθ y
θx
0
0
0
0
0
0
0
0
z
−y
θx θ y θz
0
0
0
− X CG z
X CG y
0
−x
x
0
θ y θz
0
Fx
=
Mx
My
Mz
P
0
0
0
0
0
0
0
0
θz
mYCG
mZ CG
I xx
(20)
I xy
I xz
I yy
I yz
I zz
P
52
Fx
Fy
x − YCGθ z + Z CGθ y
y + X CGθ z − Z CGθ x
0
0
0
0
0
0
0
0
0
0
0
0
Fz
z − X CGθ y + YCGθ x
0
0
0
0
0
0
θx θ y θz
0
0
0
θ y θz
0
=
Mx
My
Mz
P
YCG z − Z CG y
− X CG z + Z CG x
X CG y − YCG x
0
0
0
0
0
0
0
m
I xx
I xy
I yy
I yz
θz
0
(21)
I xz
I zz
P
From Equation 21, the mass moments of inertia terms can be solved directly if the mass
and CG terms are known. The interesting result that occurs is that the translational forces
no longer have an influence on the solution since mass moment of inertia properties do
not influence translational forces. Equation 22 shows the solution for the mass moment
of inertia properties and allows for a method of comparing the accuracy of the mass and
CG parameters substituted to the equivalent rigid body 6 DOF forces.
Fx
x − YCGθ z + Z CGθ y
0
0
0
0
0
0
I xx
Fy
Fz
y + X CGθ z − Z CGθ x
0
0
0
0
0
0
0
0
0
0
0
0
I xy
I xz
θx θ y θz
0
0
0
I yy
θ y θz
0
I yz
θz
I zz
−m
Mx
My
Mz
P
z − X CGθ y + YCGθ x
YCG z − Z CG y
=
− X CG z + Z CG x
0
0
0
X CG y − YCG x
0
0
0
0
0
(22)
P
3.2.2 DATA SCALING
Two approaches are presented to improve the quality of the DIM estimation results. The
first is to scale the FRF matrix forcing the solution of a parameter to match a known
property value within a specific frequency range. The second technique can be used to
53
weight the rotational degrees of freedom to better match the dynamic level of the
translational degrees of freedom. The procedure is a simple length unit change in order
to weight the approximated rigid body rotations.
3.2.2.1 FRF SCALING
Another method of forcing the estimated inertia properties to match a known property is
to scale the FRF matrix by a constant value. In general, this method can only be used to
scale a single term to exactly match the known value in a frequency band; otherwise, the
scalar value rarely converges for experimental test data. A scalar value can be used over
multiple estimated properties by determining a value that will minimize the errors across
the known properties, however, this method has not been investigated.
Since the CG terms are coupled with the mass, and mass is generally a property that can
be determined utilizing traditional techniques, the mass term is typically selected as the
parameter used to determine the scalar value.
An iterative estimation process is
performed on a selected frequency band. The mean value of the mass term is compared
to the known value, and the scaling of the FRF matrix is adjusted using a quadratic curve
fit to approximate the scalar value. The iteration process is repeated until the estimated
mass is within a specified tolerance of the known value or until the estimated parameter
diverges from the known value.
54
3.2.2.2 UNIT SCALING
Unit scaling provides a means of weighting the approximated rotational degrees of
freedom by changing the length unit associated with the geometry of the measurement
nodes and the length unit for the FRFs. In essence, the rotational degrees of freedom will
be more or less influential on the inertia property estimation process.
The results of scaling the length unit may have little to no influence on the estimated
properties. The sensitivity analysis for the transducer orientation shows that as the
volume of the cube increases, the errors resulting from inaccurate direction cosines
increases. The three volumes of 0.2m, 2.0m and 200m can be considered as a units
change from meters to decimeters to millimeters. Therefore, changing the length unit
from millimeters to meters results in a reduction in rotational influence by weighting the
translational measurements higher than the rotational measurements. For measurements
cases with accurate translational measurements, the length unit can be scaled down, for
example meters to kilometers, to decrease the rotational contamination due to direction
cosine errors increasing the accuracy of the mass and CG estimated terms. In cases of
accurate measurement locations and direction cosines, the length unit can be scaled up to
increase the rotational contributions increasing the accuracy of the estimated mass
moment of inertia parameters.
The analytical 2.0m cube was used to demonstrate the effectiveness of scaling unit length
values.
Errors were introduced in the model by offsetting all eight measurement
locations by 2.54cm along each GCS axis thus creating errors associated with the rigid
55
body transformations. The DIM solution results are listed in Table 15 for the different
length units.
Table 15 - 2.0 m Cube Model – Property Changes Due to Unit Changes
Units
millimeters
centimeters
meters
Actual
kilometers
Values
Error, %
Values
Error, %
Values
Error, %
Values
Error, %
Values
Mass, kg
17.73038
2.44635
17.73749
2.40720
18.17230
0.01485
18.17500
0.00000
18.175000
Xcg, m
0.03612
2.50811
0.03612
2.50766
0.03613
2.53965
0.03613
2.54000
0.035236
Xcg, m
0.00013
2.50812
0.00013
2.50870
0.00013
2.53982
0.00013
2.54000
0.000130
Xcg, m
-0.08778
2.50812
-0.08778
2.50865
-0.08781
2.53981
-0.08781
2.54000
-0.085632
Ixx, kg m^2
0.22411
2.54026
0.22416
2.56575
0.22757
4.12609
0.22759
4.13578
0.218554
Ixy, kg m^2
-0.00011
2.54032
-0.00011
2.57172
-0.00011
4.52331
-0.00011
4.53551
-0.000109
Ixz, kg m^2
0.08015
2.54030
0.08017
2.56925
0.08158
4.36843
0.08158
4.37968
0.078161
Iyy, kg m^2
0.31190
2.54022
0.31196
2.56167
0.31597
3.87701
0.31599
3.88519
0.304172
Iyz, kg m^2
0.00022
2.54040
0.00022
2.57922
0.00023
4.95380
0.00023
4.96855
0.000218
Izz, kg m^2
0.16217
2.54006
0.16218
2.54601
0.16276
2.91415
0.16277
2.91644
0.158154
A trend in the parameters can clearly be seen in the table. As the influence of the
rotations diminish, the estimated mass value approaches the actual value and the mass
moment of inertia terms deviate from the estimated values.
The other purpose for changing the length unit is to reduce the differences in dynamic
range between the rotational and the translational measurements.
In general, the
numerical conditioning of the solution equation in the DIM does not cause inversion
problems.
There may be situations when the direction cosine measurements are
considered accurate and an increase in rotational sensitivity is desired. The length unit
can be changed to weight the rigid body rotations increasing the resolution of the
rotational degrees of freedom.
56
4.0 EXPERIMENTAL EXAMPLES
Four experimental examples are given to illustrate the DIM process with caveats and
points of interest given throughout.
Three of the four are the same test structure
supported by different boundary conditions. The calibration mass was supported by
bungee cords, inverted and mounted to an air ride, and rigidly secured to ground.
The fourth example is a field test conducted at NASA Dryden Flight Research Center.
The NASA X-38 V-131R experimental crew return vehicle originally, intended as the
emergence escape vehicle for the International Space Station, was tested in order to
determine the inertia properties using the DIM. Due to the newness and somewhat
unproven technology of the DIM, NASA Dryden engineers determined the mass, CG,
and mass moments of inertia properties using traditional testing techniques in parallel.
The goal of the dynamics test was to estimate the ten inertia parameters experimentally
and compare the results with NASA determined parameters.
4.1 CALIBRATION MASS
The mass used to calibrate the 6-DOF load cell is the structure that was tested under three
different boundary conditions. Originally, a load cell was developed utilizing a stack of
piezoelectric crystals with eight voltage outputs. The load cell functioned well in some
cases, but was temperamental in others. This load cell was sensitive to the preload
mounting condition and a change in the preload would cause a change in the sensitivity
matrix. An alternate load cell was developed at the University of Cincinnati in an attempt
57
to create a more stable multiple axis load cell.[9] An array of 12 piezoelectric strain
gages was applied to a 4 spoke steel structure as shown in Figure 15. The arrangement
was such that all forces acting through the load cell passed through the arms containing
the strain gages. This isolates the test structure from the support system by measuring all
the translational and rotational forces acting on the structure.
Figure 15 – Piezoelectric Strain Gage Based 6 DOF Load Cell
The motivation for the three test cases was to validate the experimental load cell as well
as to provide test cases to evaluate the DIM using a structure that had well defined inertia
properties. The inertia properties were determined using AutoCAD and an accurate 3D
model that included the calibration mass, accelerometers, mounting plate, and any
mounting bolts. The density for each structural component was determined from the
weight of each component and the volumes computed from the AutoCAD model. Figure
16 and Figure 17 show the dimensions of the calibration mass used in the three
58
experimental DIM examples as well as a typical calibration setup using soft bungee cords
to support the calibration mass.
Another reason to use the calibration mass as the test structure was to create a nearly
circular test process. By testing the same structure used to calibrate the load cell in the
same configuration, a circular condition is utilized were the DIM solution should return
the exact inertia properties used in the calibration process for error free experimental
data.
Figure 16 - Calibration Mass Model Dimensions
59
Figure 17 – Calibration Mass Model and Typical Calibration Setup
The only difference in between the two test configurations is that in the calibration
procedure, the input forces are applied to the support side of the load cell, isolated from
the calibration mass by the load cell, and in the DIM procedure, the excitation was
applied directly to the calibration mass. In the calibration procedure, the input force does
not need to be considered since the load cell measures all the forces acting on the mass
causing the rigid body motion. In the DIM procedure, the input force needs to be
included as an additional force causing the rigid body motion.
The nearly circular process allowed for the ability to identify problems with the load cell.
If large deviations from the expected inertia parameters are estimated using the DIM, the
most likely problem would be a malfunctioning or inaccurate strain gage.
60
Additional test configurations were added to change the boundary condition. Since the
load cell isolates the calibration mass from the boundary support, the type of boundary
condition should not influence the results of the inertia estimation process. The results of
the air ride support and fix boundary conditions are presented in Appendix B:
Experimental Examples.
4.1.1 HANGING CALIBRATION MASS BY BUNGEE CORDS
The bungee cord support case exactly matched the configuration used to calibrate the
load cell.
Therefore, the estimated inertia parameters should identically match the
parameters used in the calibration process. The eight triaxial accelerometers were located
at the same locations used in the calibration process. The excitation was provided by 24
impacts from an instrumented impact hammer with eight impacts on three surfaces in line
with the global axes. Figure 18 thru Figure 20 shows the location and orientation of the
accelerometers, load cell, and impacts.
61
-0.1
3
1
2
-0.05
2
1
6
5
4
3
8
7
-0.05
0
Z X
Y
Z
Y
Z
X Y
0
4
-0.1
0.05
7
5
0.1
6
-0.1 -0.05
0
0.05
X
-0.15
8
0.1
0.15
0.2
-0.1
Z
Y X
0
6
2
Z
-0.1
-0.1
48
-0.05
0
0.1
0.05
X
0.1
0.15
2
4
1
-0.15
3
7
8
5
-0.05
-0.1
0.05
6
1
5
-0.15
0
Y
YZ X
0
-0.05
Z
-0.05
7
0.2
0.1
0.1
3
0.2
0
0
-0.1
Y
-0.1
X
Figure 18 - Calibration Mass Accelerometer Locations
0
-0.05
Z201
X
Z
Y
Z
X201
Y
0
-0.05
Y
-0.1
0.05
-0.15
-0.05
0
0.05
X
0.1
0.15
0.2
-0.05
Z
Y201
X
0
0
Z
0.05
X
201
Z
-0.05
Z
-0.05
Y
0
Y
-0.1
-0.1
0.2
-0.15
-0.15
-0.05
0
0.05
X
0.1
0.15
0.2
0.1
0.05
0
-0.05
0
Y
X
Figure 19 - Calibration Mass Load Cell Location
62
-0.1
28 34
27
12 13
11
14 15
16
17
0
Z X38
Y
23
35
-0.05
0
0.1
0.15
11
21
27
-0.05
24
-0.1
23
26
-0.15
22
28
17
25
-0.1
14
34
33
-0.05
12
0
0.05
X
37
36
0.1
35
0.15
32
31
0.2
0.05
15
31
11
17
24 27
13 35
23
36
26
37
38 14
2233 25
16
-0.1
-0.15
18
16
38
0
Y
21
-0.05
13
-0.05
YZ X
0
15
23
25
22
12 28
16
34 37 35
32
38 36 31
33
-0.2
-0.1
0.2
26
14
18
Z
Y X
0
-0.2
0.05
X
24
17
-0.1
-0.15
31
21
13
36
22 33
21
-0.1
11 27
15
-0.05
37
25
24
0.05
32
Z
Y
26
18
Z
-0.05
Z
Z
X Y
0
-0.2
0.2
2834 12
0.05
0
-0.05
Y
18
32
0.1
0
-0.1
-0.1
X
Figure 20 - Calibration Mass Lines of Action
Once the data has been acquired, the first step of the DIM process is to check the rigid
body response measurements. This includes selecting a frequency range, ideally the
same range that will be used to average the estimated inertia properties, and removing
any responses that exhibit large rigid body motion errors. Figure 21 shows that rigid
body motion errors for each response channel as a function of frequency.
63
Figure 21 - Calibration Mass Rigid Body Motion Error
The next step of the rigid body verification is to remove the transducer axes that do not
agree with the least squares rigid body motion.
Figure 22 - Calibration Mass Transducer Axes Rigid Body Errors
64
Ideally, the rigid body error should be minimized. It is important to note that the
minimum number of responses should not be exceeded. In order to determine the rigid
body motion, a minimum of 6 responses, two in each of the X, Y, and Z axes, is required.
Also, for each axis the two measurements cannot be located along the same axis as they
are measuring. For example, two sensors measuring X-axis acceleration cannot have
geometries with the same Y and Z values. This eliminates the ability to approximate the
rotation about the Y-axis. Therefore, care must be taken when eliminating response
measurements based on rigid body dynamics.
Following perimeter response selection, the jackknife process is utilized to select
perimeter excitations.
Initial Cycle Values
Initial Cycle Values
19.25
0.032
Xcg
Mass
19.2
19.15
19.1
0.0315
0.031
19.05
-4
x 10
-0.08
Zcg
Ycg
5
0
-0.082
-5
-0.084
Ixy
Ixx
-3
x 10
0.22
0.2
0.18
0.16
0.07
0.31
0.06
0.305
Iyy
Ixz
0.14
0.12
2
0
-2
-4
-6
-8
0.05
0.3
0.04
-3
x 10
0
0.164
Izz
Iyz
-1
-2
0.162
0.16
-3
2
4
6
8
10
Jack Knif e Cycle
12
14
16
18
2
4
6
8
10
Jack Knife Cycle
12
14
16
18
Figure 23 - Calibration Mass Jackknife Cycle Inertia Parameters
65
-4
2
1.5
-4
-4
2.5
x 10
x 10
3
Zcg
Ycg
2
1.5
1
2
1
-3
-3
x 10
2
12
10
8
6
4
2
1.5
Ixy
Ixx
x 10
1
0.5
-3
-3
x 10
x 10
4
2
Iyy
3
Ixz
Standard Deviations
x 10
2.5
Xcg
Mass
Standard Deviations
0.024
0.022
0.02
0.018
0.016
0.014
0.012
2
1.5
1
1
0.5
-4
-4
x 10
x 10
12
8
Izz
Iyz
10
6
8
6
4
2
4
6
8
10
Jack Knif e Cycle
12
14
16
18
4
2
4
6
8
10
Jack Knife Cycle
12
14
16
18
Figure 24 - Calibration Mass Jackknife Cycle Standard Deviation
By investigation of the standard deviations for the jackknife cycles, it would appear that
the 8th cycle has the least deviation throughout that cycle’s process. This means that the
inputs that had been removed through cycle seven may likely have errors causing
inaccuracies in the estimated inertia properties.
After the inputs have been removed from the data group, the inertia properties can be
estimated. The results of the estimate process are shown in Figure 25.
66
Figure 25 - Calibration Mass DIM Parameters
The results of the DIM solution show that the estimated parameters do not match the
values used in the calibration procedure. The solution curves show the values for the
parameters remain fairly constant throughout the 5-200 Hz frequency band. There are
inertia property errors due to the transducer cabling that are not considered. However,
this should not affect the estimated parameters since the DIM measurements were taken
in the same configuration as was the calibration. The calibration matrix for the load cell
contains the effects of the cabling, and the estimated parameters should exactly match the
mass matrix used to calibrate the load cell.
67
One consistent result is that the DIM parameters are all larger than the actual values.
This may indicate that a constant error exists in the processes. One source of error could
be an incorrect units conversion error since the data was taken using British units and
then converted to SI. A more likely error would be an incorrect calibration value for the
impact hammer.
Since the mass of the calibration mass is known, the next step in the DIM would be to
find a constant scaling term to force the estimated mass term to match the known value.
Figure 26 shows the convergence plot in the scaling process.
Figure 26 - Calibration Mass FRF Scaling to Match the Known Mass Parameter
Once the scaling term has been determined, the inertia properties can again be computed.
When the scaling is applied to the FRFs, the estimated parameters are worse than the
original prediction.
This would likely indicate that the errors associated with the
estimates are not due to an incorrect impact hammer calibration.
68
Figure 27 - Calibration Mass Inertia Parameters with FRF Scaling
Since the mass term is known, the estimation process can be supplemented by removing
the mass term and substituting the known parameter into the solution equation. Figure 28
shows the inertia property results with the mass parameter substitution.
69
Figure 28 - Calibration Mass Inertia Parameters with Known Mass Substitution
Some parameters show a slight increase in accuracy while others get worse. In an
attempt to solve for only the mass moment of inertia properties, the known values for the
mass and CG terms will be used in the solution. Figure 29 shows the results of the mass
moment of inertia estimations.
70
Figure 29 - Calibration Mass Estimation of Mass Moment of Inertia Parameters
Table 16 - Calibration Mass DIM Parameters Summary
General Solution
Known Mass
Substitution
FRF Scaling
Mass and CG
Substitution
Actual
Values
Values
Error, %
Values
Error, %
Values
Error, %
Values
Error, %
Mass, kg
19.191260
5.591510
18.175000
5.12E-07
18.175000
5.86E-14
18.175000
5.86E-14
18.175000
Xcg, m
0.031270
11.256700
0.031270
11.256700
0.032049
9.046060
0.035236
1.97E-14
0.035236
Ycg, m
0.000801
517.542000
0.000801
517.542000
0.000820
532.471000
0.000130
4.18E-14
0.000130
Zcg, m
-0.080649
5.818300
-0.080649
5.818300
-0.083163
2.883240
-0.085632
4.86E-14
-0.085632
Ixx, kg m^2
0.220392
0.840940
0.208721
4.499010
0.218695
0.064512
0.220824
1.038620
0.218554
Ixy, kg m^2
-0.003694
3289.390
-0.003498
3109.910
-0.003686
3282.740
-0.003758
3348.010
-0.000109
Ixz, kg m^2
0.073786
5.597860
0.069879
10.596900
0.073076
6.506370
0.074527
4.649410
0.078161
Iyy, kg m^2
0.303833
0.111511
0.287744
5.401030
0.302981
0.391709
0.304752
0.190558
0.304172
Iyz, kg m^2
-0.002418
1009.270
-0.002290
950.534
-0.002353
979.473
-0.002478
1036.990
0.000218
Izz, kg m^2
0.164289
3.879590
0.155589
1.621270
0.163983
3.685950
0.164947
4.295410
0.158154
Scaled Value
Substituted Value
71
The results previously listed in Table 16 generally show that the estimated inertia
parameters are close to the expected values but not the exact values. As a final check, the
DIM is used to estimate the parameters neglecting the reaction force measurements. This
is done to verify that the reaction forces are being compensated for as well as to
determine if including the reaction forces in the solution process improves the estimated
parameters or causes the estimated parameters to get worse.
Figure 30 - Calibration Mass Inertia Properties without the Reactions Forces
As a final check, since the jackknife procedure is somewhat arbitrary when the properties
are not known, the DIM solution is computed utilizing all of the input excitations. Figure
31 shows the estimated parameters using all of the input excitations to the calibration
72
mass. In general, the estimated parameters match the known parameters closer than any
of the DIM solutions previously computed. This may indicate that too many forces had
been removed causing some of the degrees of freedom to be inadequately excited. This
can be a common problem associated sigma based rejection techniques.
Figure 31 - Calibration Mass Parameter Estimates Using All Input Excitations
4.2 X-38 INERTIA EXPERIMENTAL RESULTS
The NASA X-38 V-131R crew return vehicle, Figure 32, was tested in August of 2001 at
NASA Dryden Flight Research Center.
The X-38 test differed from the test
73
configurations used during the calibration mass test. The size of the structure was much
larger than the calibration mass. The surface locations on the X-38 were curved and
required the use of the automated direction cosine process to provide the orientation of
the transducers and excitation lines of action.
Figure 32 - X-38 Test Vehicle
The vehicle was supported by three multiple axis load cells each aligned out of the global
coordinate systems. The soft support system was an active response system which used a
changing air pressure to the three air bladder supports to actively damp the vehicle’s rigid
body dynamics. Figure 33 shows the details of the mounting conditions for the three load
cells.
74
Figure 33 - X-38 Jack Point Details, Nose, Port and Starboard
A problem with the boundary condition was that the vehicle would occasionally become
unstable. This led to a situation that caused a bent support post that connected the port
side, rear load cell to the jack point of the vehicle. The question that arose due to the
damaged post was whether the load cell had also been damaged. Due to time limitations
during setup, a pre-calibration for all of the load cells could not be performed, so it was
not possible to determine if the sensor had been damaged by comparing pre and post test
calibration values.
Triaxial accelerometers were mounted at 21 locations on the vehicle. Three of the
locations were on the drogue chute which was believed to have a rigid body mode out of
phase with the vehicle around 5 Hz. Also, single axis accelerometers were mounted in
the Y axis direction, port to starboard, on the vertical tail to identify any low frequency
deformation modes of the tail section. The solution process used 18 of the 21 triaxial
sensors to determine the rigid body motion. The transducer locations and input lines of
action are shown in Figure 34 thru Figure 36.
75
-2
1405
1310
1106
1308
1428
1303
0 Z X 1401
Y
1302
2302
1
1107
1321
1316
1425
2
4
X
1401
1
1303
1409 1405
1320
1302
2302
14281425
1319
Z
6
1318
1310X
0
1319
2
0
1321
1107
1106
2
1320
1409
1318
Z
Y
-1
8
-2
-1
Y
1308
1316
0
Y
1
2
1321
1319
0
Z
Y X
1401
1302
1303
2302 1405
1428
1310
1308
0
2
1409
1425
1318
1316
4
X
1319
1320
6
1107
1425
1106
1316
1428
1409
1318
2302
1401 1308
1302
1405
1310
1303
2
Z
1
1321
1107
1106
1
0
8
1320
8
6
YZX
2
4
0
2
-2
Y
X
0
Figure 34 - X-38 Accelerometer Locations and Orientations
-2
2
7002
0 Z X
1.5
7001
Z
Y
-1
Y
1
Z
7001
0
0
2
4
X
7002
-1
-2
6
2
X
Z
Z
Y X
0
7001
2
4
X
7002
7003
6
7003
Y
0
Y
1
2
7003
2
1
0
1
0.5
7003
2
Z
Z
2
7002
1
6
0
2
Y
Z
X
7001
4
0
Y
2
-2
0
X
Figure 35 - X-38 Load Cell Locations and Orientations
76
-2
303
0 Z X 301
Y
1
1
6003
417
0
2
4
X
6
6008
303
320
315408
6001
6006
6002
0.5
6005
319
Z6003
301
6009
6010
313
310 X6011
312Y 308
311
0
6005
319
6008
2
6007
1.5
Z
Y
-1
417
2
6007
6001
6006
6002
320
315
408
6009
310 313
6011 312
308 311
6010
-0.5
8
-2
-1
0
Y
1
2
6008
417
319
6005
Z
Y X
303
301
0
408
315
6011
6009
6010
308 312
310
311
313
2
4
X
6
6005
319
320
6001
6006
6002
6003
8
2
6010
1
311
315
6009
308 312
408
313
6011310
Z
Z
1
0
6003
417
6008
6007
2
0
Y
2
Z
303
301
X
6007
320
6006
6002
6001
8
6
4
0
Y
2
-2
0
X
Figure 36 - X-38 Input Lines of Action Locations and Orientations
The vehicle was suspected to have deformation modes within the frequency band of the
collected data. A simple method of animating the vectors obtained from the singular
value decomposition of the FRF matrix was used to determine the frequencies at which
the X-38 exhibited deformation modes. Figure 37 shows the traditional CMIF plot
indicating the frequencies with suspected deformation modes. The 5 Hz region indicated
two closely spaced modes which appear to correspond to modes of the drogue chute.
Due to the location of the sensors, it was difficult to get accurate orientations, and the
spatial resolution was only to detect rigid body motion. The first deformation mode
appears around 23 Hz. Therefore, the inertia property estimation was performed in the 67 Hz frequency band.
77
X-38 Quadrature Complex Mode Indicator Function
-1
10
-2
10
CMIF
-3
10
-4
10
-5
10
0
5
10
15
20
25
30
Frequency, Hz
35
40
45
50
Figure 37 - X-38 Complex Mode Indicator Function
Figure 38 is the results of the DIM without performing any perimeter response of
excitation selection. When compared to the result obtained from NASA conventional
testing, the DIM results compare within 7% when the reaction forces are considered in
the parameter estimation.
78
Figure 38 - X-38 Solution – All Data Included in Estimation (Case 1)
Response measurements that contain a high degree of rigid body motion error are to be
removed from the data. Figure 39 shows that three to six responses exhibit a higher
degree of rigid body error than the bulk of the response measurements, and Figure 40
demonstrates that by removing the measurements with higher error, the overall error can
be reduced.
79
Figure 39 - X-38 Rigid Body Motion Errors
Figure 40 - X-38 Filtered Rigid Body Motion Errors
80
The DIM solution was used to estimate the inertia parameters after removing the
perimeter responses with high rigid body motion errors. The solution in Figure 41 shows
a slight improvement in most of the inertia properties compared to the solution without
the rigid body response filtering.
Figure 41 - X-38 Solution – Applied Rigid Body Motion Filtering (Case 2)
The iteration process using a variety of DIM solution options was performed on the
experimental data. Details of the 11 solution cases, Table 17, can be found in Appendix
B: Experimental Examples and the inertia properties’ results are listed in Table 18. A
81
graphical representation of the results is displayed in Figure 42 with the estimated
parameter errors shown graphically in Figure 43. Assuming the NASA determined
values for the inertia properties are correct, Case 3 provides the best overall solution
using the dynamic inertia method. This case only utilizes the filtered perimeter responses
and forces the mass term to match the NASA estimated mass using the constant scaling
term on the frequency response function.
Table 17 - X-38 DIM Solution Case List
Case
1
No rigid body response filtering applied
Case
2
Rigid body response filtered
Case
3
FRF scaling applied & Case 2
Case
4
Jackknife inputs & Case 2
Case
5
FRF scaling & Case 4
Case
6
Known mass parameter substitution & Case 4
Case
7
Known mass & CG parameters substitution & Case 4
Case
8
Impacts only - no reaction forces & Case 4
Case
9
Impacts only - no reaction forces & Case 4 with FRF scaling
Case
10
Known mass parameter substitution with FRF scaling & Case 4
Case
11
Known mass & center of gravity parameter substitution with FRF scaling & Case 4
Scaled Value
Substituted Value
Substituted with Scaled Value
82
Table 18 - X-38 DIM Results for All Cases
Case 1
Case 2
Case 3
Case 4
Value
Error, %
Value
Error, %
Value
Error, %
Value
Error, %
Mass, kg
8486.547
4.90991
8469.071
4.69388
8089.366
1.21E-09
8646.543
6.88778
Xcg, m
4.060316
1.98961
4.075128
1.63206
4.075128
1.63206
3.988496
3.72323
Ycg, m
-0.0098622
158.85
0.002500
34.373
0.002500
34.373
0.066552
1646.76
Zcg, m
0.7895187
7.18419
0.773482
5.007
0.773482
5.007
0.754465
2.42532
Ixx, kg m^2
5559.579
2.5412
5714.116
5.39149
5457.927
0.66633
5655.35
4.30761
Ixy, kg m^2
-41.16294
87.8529
-520.3749
53.5618
-497.0443
46.677
-730.7831
115.653
Ixz, kg m^2
1654.069
1.65407E+13
2284.452
2.28445E+13
2182.03
2.18203E+13
2576.27
2.57627E+13
Iyy, kg m^2
29178.69
7.63384
28304.97
4.41087
27035.94
0.270322
27763.91
2.41502
Iyz, kg m^2
1384.227
1.38423E+13
2837.061
2.83706E+13
2709.864
2.70986E+13
1320.583
1.32058E+13
Izz, kg m^2
26295.23
7.62155
26810.66
5.81077
25608.62
10.0337
32621.21
14.6024
Case 5
Case 6
Case 7
Case 8
Value
Error, %
Value
Error, %
Value
Error, %
Value
Error, %
Mass, kg
8089.366
9.02E-10
8089.366
1.12E-14
8089.366
1.12E-14
8677.2
7.26675
Xcg, m
3.988496
3.72323
4.175327
0.78661
4.14274
4.29E-14
3.959107
4.43266
Ycg, m
0.0665516
1646.76
0.0644545
1591.72
0.00381
1.14E-14
0.06654
1646.46
Zcg, m
0.7544649
2.42532
0.793638
7.74342
0.7366
3.01E-14
0.7567926
2.74133
Ixx, kg m^2
5290.923
2.4139
5336.043
1.58172
5741.736
5.90092
5525.588
1.91427
Ixy, kg m^2
-683.6919
101.756
-553.4353
63.3179
1101.076
224.926
-813.5586
140.08
Ixz, kg m^2
2410.257
2.41026E+13
1235.538
1.23554E+13
3007.32
3.00732E+13
2667.453
2.66745E+13
Iyy, kg m^2
25974.83
4.18453
21669.39
20.0663
24127.4
10.9993
28649.63
5.68224
Iyz, kg m^2
1235.486
1.23549E+13
1368.978
1.36898E+13
1877.212
1.87721E+13
1325.678
1.32568E+13
Izz, kg m^2
30519.12
7.21751
26104.76
8.29071
26986.45
5.19321
32081.59
12.7067
Case 9
Case 10
Case 11
Value
Error, %
Value
Error, %
Value
Error, %
Mass, kg
8089.366
1.12E-14
8089.366
1.12E-14
8089.366
1.12E-14
Xcg, m
4.154076
0.273625
3.988513
3.72282
4.14274
4.29E-14
Ycg, m
0.064335
1588.58
0.0665486
1646.68
0.00381
1.14E-14
Zcg, m
0.7982636
8.37138
0.7544071
2.41747
0.7366
3.01E-14
Ixx, kg m^2
5186.713
4.33595
5292.412
2.38645
5483.81
1.14371
Ixy, kg m^2
-627.277
85.1085
-683.2615
101.629
744.1076
119.585
Ixz, kg m^2
1257.176
1.25718E+13
2415.383
2.41538E+13
2978.581
2.97858E+13
Iyy, kg m^2
22301.86
17.7333
25974.64
4.18523
20877.44
22.9877
Iyz, kg m^2
1376.845
1.37685E+13
1236.059
1.23606E+13
1761.406
1.76141E+13
Izz, kg m^2
25291.32
11.1484
30515.17
7.20361
25623.25
9.9823
83
DIM Parameters
DIM Parameters
4.2
8500
Xcg
Mass
9000
4
8000
3.8
0.8
0.04
Zcg
Ycg
0.06
0.02
0.75
0
0.7
1000
500
5500
Ixy
Ixx
6000
0
-500
5000
4
x 10
3000
3
Iyy
Ixz
2000
2.5
1000
2
2500
2000
1500
1000
500
x 10
3.2
Izz
Iyz
4
3.4
3
2.8
2.6
1
2
3
4
5
6
DIM Case
7
8
9
10
11
1
2
3
4
5
6
DIM Case
7
8
9
10
11
Figure 42 - X-38 Solution - Parameter Variations for 11 Cases
The dotted lines in Figure 42 represent the NASA determined inertia properties for the
vehicle. For many of the parameters, the variations in the solution tend to drift around
the NASA values indicating that the various cases produce results close to the expected
values with some deviations.
84
DIM Parameter Errors
DIM Parameter Errors
4
Xcg, %
Mass, %
6
4
2
2
1
8
Zcg, %
1500
Ycg, %
3
1000
500
6
4
2
6
Ixy, %
Ixx, %
200
4
2
150
100
50
13
x 10
20
Iyy, %
Ixz, %
3
2.5
2
15
10
5
1.5
13
x 10
15
Izz, %
Iyz, %
2.5
2
10
1.5
1
2
3
4
5
6
DIM Case
7
8
9
10
11
5
1
2
3
4
5
6
DIM Case
7
8
9
10
11
Figure 43 - X-38 Solution - Parameter Errors for 11 Cases
85
5.0 CONCLUSIONS AND FUTURE WORK
5.1 CONCLUSIONS
Determining the inertia properties of a structure experimentally using dynamic data has
proven to be a feasible task with accurate measurements that include sensor locations and
orientations as well as calibrated dynamic force and acceleration data. Previous attempts
at using the DIM have generally been successful. However, the process rarely takes a
smooth, step by step approach and known inertia parameters of the test structure were
available during the DIM estimation process. Different test configuration and structure
types require attention to different details of the solution process. A clear, standardized
approach is still not apparent.
A common thread for all structural testing is the goal of acquiring accurate experimental
data. Multiple axis load cells and accelerometers are used to measure rigid body motion
and the forces causing that motion. This data can be used to approximate DOFs that
cannot be measured. Determining the direction cosine orientations of the transducers can
increase the accuracy of estimated inertia parameters when the orientations are accurate.
Currently the entire process of the DIM relies heavily on rigid body dynamics, meaning
that it is necessary to determine a structure’s rigid body motion in all translational and
rotational degrees of freedom. The direction cosine determination process can facilitate
the accurate approximation of a structure’s rigid body rotational degrees of freedom. The
86
use of a 3D sonic digitizer may have a cost advantage, but the limitations due to
environmental conditions can significantly impede its usefulness and accuracy.
The perimeter response selection technique can be utilized to remove measurements
generating erroneous rigid body motion, but there is too much flexibility in the response
selection process.
The assumption for the process is that most of the response
measurements contain only small rigid body errors.
When the rigid body error is
distributed evenly throughout the measurements with only a few transducers producing
relatively error free data, the large number of sensors with errors may skew the rigid body
participation to appear that the erroneous data fits better than the accurate measurements.
It is possible for a single sensor to provide an accurate estimation of a particular
translation, but the rigid body estimate may be skewed by measurements that are less
sensitive to that particular motion resulting in the removal of the accurate measurement.
The experimental results presented show that the DIM estimated parameters generally
agree with the expected values within five percent, but the confidence in the estimation
process is still low. The inertia properties were known but the DIM procedure clearly
indicated inaccurate inertia parameters. The tools presented in this thesis only provided
the means of better estimating parameters that are already known and for evaluating the
DIM methodology. These tools, in general, cannot be considered improvements to the
estimation method but simply provide ways to manipulate the data and solution process
to force the estimated parameters to match previously known values. The disadvantage
87
of the presented tools is that there currently is no way of determining whether to apply
them without knowing some of the tested structure’s inertia properties.
Note that for all experimental cases, the properties that were considered known, from a
computer model or produced using traditional estimation methods, may be incorrect due
to model inaccuracies or experimental errors. It is necessary to very accurately determine
the inertia properties of a structure so that the DIM and the variety of solution options can
be thoroughly evaluated. The use of possibly inaccurate geometry and inertia property
information has clouded the DIM evaluation process.
The dynamic inertia method does show promise as a tool for experimentally estimating
inertia properties, but the confidence in the estimated results is limiting practical
applications and user acceptance.
88
5.2 FUTURE WORK
The simplicity of the DIM method may have led to experimental evaluations prior to
gathering a complete understanding of the details leading to misconceptions about the
underlying theory. The analytical process needs to be evaluated from the beginning
ignoring the previous experimental work performed on the method.
Along with the theoretical evaluation, several other issues need addressed.
•
More accurate techniques, photogrammetry and optical methods, should be
investigated to improve the direction cosine computation process.
•
A method of calibrating 6 DOF accelerometers may provide a better method of
determining 6 DOF rigid body motion from arrayed single and triaxial
accelerometers.[8]
•
A mechanically optimized 6 DOF load cell needs to be designed to improve the
sensitivity of all six degrees of freedom with a minimum number of strain gages.
•
The development of a 6 DOF accelerometer may greatly improve the accuracy
and usability of the DIM by eliminating the need to approximate rotational DOFs
using rigid body dynamics.
•
Tools to confirm that the estimated inertia properties make physical sense and
agree with the experimental data.
89
90
References
[1] Allemang, R.J., “Vibrations: Experimental Modal Analysis” UC-SDRL-CN-20 263663/664, (1995)
[2] Andriulli, J.B. “A Simple Way to Measure Mass Moments of Inertia”, Sound and
Vibration, November, 1997
[3] Beer, F.P., Johnston, E.R. Jr., “Vector Mechanics for Engineers – Statics and
Dynamics”, Fifth Edition, McGraw-Hill, (1988).
[4] Bianchi, G., Fauda, A., Paolucci, F., “Guidelines for Experimental Identification of
Rigid Body Inertial Properties Using the Mass Lines Method”, Proceedings,
International Symposium on Modal Analysis (ISMA), K. U. Leuven, pp. 473-480,
(1998)
[5] Bono, Richard W., et al, Automated 3D Coordinate Digitizing for Modal/NVH
Testing, Sound And Vibration, January 1996
[6] Bretl, J., Conti, P., “Rigid Body Mass Properties From Test Data”, Proceedings,
International Modal Analysis Conference, pp. 655-659, (1987)
[7] Crowley, J., Brown, D.L., Rocklin, G.T., “Uses of Rigid Body Calculations in Test”,
Proceedings, International Modal Analysis Conference, pp. 487-493, (1986)
91
[8] Declercq, S.M., and Brown, D.L. “Feasibility Study of Multiple-Element
Accelerometers and Field Calibration Procedures”, Proceedings, International Modal
Analysis Conference, pp. 799-804, (2003)
[9] Declercq, S.M., Lazor, D.R., and Brown, D.L. “A Smart 6 DOF Load Cell
Development”, Proceedings, International Modal Analysis Conference, pp. 844-853,
(2002)
[10] Efron B. and Gong G., “A Leisurely Look at the Bootstrap, the Jackknife, and Cross
Validation”, The American Statistician, Vol. 37 No. 1, pp. 36-48, 1983
[11] Fladung, W.A., Phillips, A.W., Brown, D.L., Olsen, N., Lurie, R. “The Integration of
VXI Data Acquisition into MATLAB”, Proceedings of the 16th International Modal
Analysis Conference, (1998)
[12] Furusawa, M., Tominaga, T., “Rigid Body Mode Enhancement and Rotational DOF
Estimation for Experimental Modal Analysis”, Proceedings, International Modal
Analysis Conference, pp. 1149-1155, (1986)
[13] Furusawa, M., “A Method of Determining Rigid Body Inertia Properties”,
Proceedings, International Modal Analysis Conference, pp. 711-719, (1989)
[14] Gatzwiller, K.B., Witter, M.C., and Brown, D.L. “A New Method For Measuring
Inertia Properties”, Proceedings, International Modal Analysis Conference, pp.
1056-1062, (2000)
92
[15] Jolliffe, I.T., “Principle Component Analysis”, Springer Series in Statistics,
Springer-Verlag, 1986
[16] Lazor, D.R., and Brown, D.L. “Impedance Modeling with Multi Axis Load Cells”,
Proceedings, International Modal Analysis Conference, pp. 90-97, (2002)
[17] Lazor, D.R. Brown, D.L., and Dillon, M.J. “Automated Measurement of Direction
Cosines for Transducer Systems”, Proceedings, International Modal Analysis
Conference, pp. 799-804, (2001)
[18] Leurs, W., Gielen, L., Brughmans, M., Dierckx, B., “Calculation of Rigid Body
Properties From FRF Data: Practical Implementation and Test Cases”, Proceedings,
International Modal Analysis Conference - Japan, pp. 365-369, (1997)
[19] Li, S., Brown, D.L., Seitz, A., and Lally, M. “The Development of Six-Axis Arrayed
Transducer”, Proceedings, International Modal Analysis Conference (1994)
[20] Mangus, C., Passerello, C., VanKarsen, C., “Direct Estimation of Rigid Body
Properties From Frequency Response Functions”, Proceedings, International Modal
Analysis Conference, pp. 259-264, (1992)
[21] The Modal Shop, “3D Sonic Digitizer Technical Manual” Model 5230XL3B
[22] Nagy, C.J., “Weight, Balance, and Inertia Test Plan/Procedures Revision 3 XCRV024”, Technology Document, NASA Dryden Flight Research Center, Edwards Air
Force Base, CA, (1998)
93
[23] O’Callahan, J.C., Avitabile, P., Chou, C.M., Wu, C.H., “Consistent Scaling of Rigid
Body and Experimental Flexural Modes”, Proceedings, International Modal Analysis
Conference, pp. 1538-1544, (1987)
[24] Pandit, S. M., Hu, Z. Q., Yao, Y. X., “Experimental Technique for Accurate
Determination of Rigid Body Characteristics”, Proceedings, International Modal
Analysis Conference, pp. 307-311, (1992)
[25] Stebbins, M. A., Blough J. R., Shelley, S. J., Brown, D. L., “Measuring and
Including the Effects of Moments and Rotations for the Accurate Modeling of
Transmitted Forces”, Proceedings, International Modal Analysis Conference, pp.
429-436, (1996)
[26] Stebbins, M.A., Calibration and Application of Multi-Axis Load Cells, Masters
Thesis, 1997, University of Cincinnati.
[27] Stebbins, M.A., Blough, J.R., Shelley, S.J., and Brown, D.L. “Multi-Axis Load Cell
Calibration and Determination of Sensitivities to Forces and Moments”,
Proceedings, International Modal Analysis Conference, pp. 181-187, (1997)
[28] Stebbins, M. A., Brown, D. L., “Rigid Body Inertia Property Estimation Using A
Six-Axis Load Cell”, Proceedings, International Modal Analysis Conference, pp.
900-906, (1998)
94
[29] Stebbins, M.A and D.L. Brown, Rigid Body Intertia Property Estimation Using a
Six-axis Load Cell, Proc. of the 16th International Modal Analysis Conference, Santa
Barbara, CA, (1998)
[30] Thomson, D.J., and Chave, A.D., “Jackknife Error Estimates for Spectral Coherence,
and Transfer Function”, Advances in Spectrum Analysis and Array Processing, Vol.
1, Prentice Hall, pp.58-113.
[31] Urgueira, A. P. V., “On the Rigid Body Properties Estimation From Modal Testing”,
Proceedings, International Modal Analysis Conference, pp. 1479-1483, (1995)
[32] Urgueira, A. P. V., “Sensor Location for the Determination of Rigid Body
Properties”, Proceedings, International Modal Analysis Conference, pp. 442-445,
(1996)
[33] Witter, M. C., Brown, D. L., Dillon, M., “A New Method for RBP Estimation – The
Dynamic Inertia Method”, Society of Allied Weight Engineers, SAWE Paper No.
2461, 20 pp.
[34] Witter, M.C., Blough, J.R., and D.L. Brown, Measuring the 6 Degree of Freedom
Driving Point Frequency Response Function Using a 6 DOF Impedance Head and its
Validation, Proc. of the 23rd International Seminar on Modal Analysis, Leuven,
Belgium, September 1998.
95
[35] Witter, M. C., Brown, D. L., Gatzwiller, K. B., “A New Method for Measuring Rigid
Body Properties of Automotive Components”, Proceedings, International Congress
on Sound and Vibration, Technical University of Denmark, Lyngby, (1999)
[36] Witter, M.C., Phillips, A.W., Brown, D.L., “Extending the Utility of Rigid Body
Calculations in Test”, Proceedings, International Modal Analysis Conference, (2000)
[37] Witter, M.C., C Rigid Body Inertia Property Estimation using the Dynamic Inertia
Method, Masters Thesis, 2000, University of Cincinnati.
[38] Wolowicz, C.H., Yancey, R.B., “Experimental Determination of Airplane Mass and
Inertial Characteristics”, Technical Report NASA TR R-433, National Aeronautics
and Space Administration, Washington, D.C., (1974)
96
APPENDIX A: SUPPORTING THEORY
The information contained in this section is to familiarize the reader with the basic
concepts of the dynamic inertia method. More detail can be found in works previously
published. [23-27, 33-37]
RIGID BODY DYNAMICS
Rigid body dynamics is essential for the DIM as it is the fundamental method of
obtaining the rotational DOFs necessary for estimating the inertia parameters. The basic
principle is a translation and rotation located at point P on a structure can be represented
by an equivalent translation at any point i on the structure with a known geometric
distance from point P. The linearized form is shown in Equation 23 in 3D space.
∆X
∆X
1 0 0
∆Y
∆Z
0
= 0 1 0 − Zi
i
0 0 1
Yi
Zi
− Yi
0
Xi
− Xi
0
∆Y
∆Z
θX
θY
θZ
(23)
P
This rigid body transformation is rank deficient since a single 3D translation
measurement does not contain enough information to uniquely determine the 6 DOF rigid
body motion at point P. Therefore, an arrayed group of triaxial transducers can be used
97
to solve for the rigid body motion. A redundant set of measurements can be assembled to
produce a least squares estimate for the rigid body motion.
∆X i
1 0
∆Y i
0 1
∆Z i
0 0
∆X j = 1 0
∆Y j
0 1
∆Z j
0 0
0
0
0 − Zi
1 Yi
0
0
0 −Zj
1 Yj
Zi
0
− Xi
Zj
0
−Xj
− Yi
Xi
0
−Yj
Xj
0
∆X
∆Y
∆Z
θX
θY
θZ
(24)
P
The matrix in Equation 24 is commonly referred to as the rigid body transformation
matrix and is referred to in the following matrix form.
{q} = [ΨRB ]{K P }
(25)
This equation describes the 6 DOF motion at point P, vector K, as a result of the 3 DOF
translational motions on the structures, vector q. Solving for the 6 DOF rigid body
motion at point simply requires that the rigid body transformation matrix be invertible.
{K P } = [ Ψ RB ] {q}
+
(26)
The rigid body forces can be determined in much the same way as the rigid body motion.
The difference is the 6 DOF forces are a resultant of the force moment pair. In other
words, the resulting 6 DOF forces are due to a force at point q move a geometric distance
to point P producing a force-moment pair. Equation 27 shows the rigid body force
transformation.
98
Fx
1
0
0
0 0 0
Fx
Fy
0
1
0
0 0 0
Fy
0
0
1
0 0 0
Fz
0
− Zi
Yi
1 0 0
Mx
Zi
0
−Xi
0 1 0
My
−Yi
Xi
0
0 0 1
Mz
Fz
=
Mx
My
Mz
P
(27)
q
or in the matrix notation
{F }P = [Φ ]i {F }i
(28)
The force transformation equation differs from that of the rigid body motion equation in
that the system in not indeterminate. The equivalent 6 DOF forces at point P on a
structure due to multiple forces acting at several other location, i1, i2, i3…, is simply a
summation of all the force-moment pairs.
{F }i
{F }i
1
{F }P = [ Φ ]i [ Φ ]i
1
2
[ Φ ]i
2
N
(29)
{F }i
N
PERIMETER RESPONSE SELECTION
From rigid body dynamics, approximation of rotational degrees of freedom can be
estimated using the geometric rigid body transformation matrix. When a redundant set of
translational measurements is used to approximate the 6 DOF motion, measurements that
99
do not agree with the least squares approximation can be removed from the response data
by applying a weighting matrix. The weighting matrix is simply an identity matrix with a
size the same as the number of response measurements with zeros in the diagonal
positions of responses that are to be removed.
To determine which measurements do not agree with the least squares estimate of the
rigid body motion an error function is used to determine the amount of rigid body
deviation a particular measurement contains. The least squares 6 DOF rigid body motion
at point P is expanded using Equation 30 to determine the equivalent best fit translational
motion at each measurement point on the structure in the transducer’s axes direction.
{q }= [Ψ ][[ W ][Ψ ]] [W ]{q }
+
f
RB
RB
(30)
m
The vector qf is the rigid body fit equivalent motion resulting from the redundant
measurements, vector qm. If no rigid body motion error existed in the measurements then
the measurement vector qm would exactly match the fit vector qf.
The error term
describing the amount of rigid body error for each measurement i is described by
εi =
qmi − q f
qf
(31)
Eliminating the measurements with the largest rigid body error will improve the accuracy
of the estimated motion at point P. Care must be taken so that the error weighting matrix
does not remove measurements that will cause Equation 24 to become indeterminate. A
minimum of three measurements along each global axis direction must be used.
100
THE DYNAMIC INERTIA METHOD
The dynamic inertia method is primarily base on Newton’s second law.
F = mx
M = Iθ
The forces in the DIM equation contain the three translation forces and three rotational
moments. Also, the accelerations consist of the three translational and three angular
accelerations. The mass matrix is the rigid body mass and contains the mass, center of
gravity and mass moment of inertia terms for the structure. Therefore, the summation of
all the forces acting of the structure, both excitation and reaction, are a result of the mass
matrix multiplied by the accelerations. The rigid body equation of motion becomes
Fx
Fy
Fz
=
Mx
My
Mz
P
m
0
0
0
mZ CG
−mYCG
0
0
m
0
0
m
−mZ CG
mYCG
0
− mX CG
mX CG
0
0
− mZ CG
mYCG
I xx
I xy
I xz
mZ CG
−mYCG
0
mX CG
−mX CG
0
I yx
I zx
I yy
I zy
I yz
I zz
x
y
z
θ
θy
θz
(32)
x
P
where m is the mass of the structure, XCG, YCG, and ZCG are the location of the center of
gravity relative to the point P on the structure, and the lower right portion of the matrix is
the mass moment of inertia terms. It is assumed that the cross inertia terms, for example
Ixy and Iyx, are equal. This assumption reduces the amount of unknowns to 10, mass,
three CG terms, and 6 mass moment of inertia terms.
101
Equation 33 can be written in terms of the unknowns in order to solve for the inertia
properties.
m
Fx
Fy
x
Fz
Mx
My
Mz
P
0
−θ z
θy
−θ x
y θz
z −θ y
=
0 0
θx
z
0
−y
−z
y
0
−x
x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
θ y θz
0
0
θz
θx θ y θz
0
0
0
0
0
0
0
mX CG
mYCG
mZ CG
(33)
I xx
I xy
I xz
I yy
I yz
I zz
P
Since there are only six equations and 10 unknowns, the equation can be augmented. As
the name implies, dynamic measurements are used to estimate the inertia properties at
each frequency line. In general, the frequency response functions are utilized as the
fundamental datasets. This means that the forces and accelerations are made up of
complex values. By appending the real part and imaginary part of the solution equation,
the number of equations doubles, and the set of equations is no longer rank deficient.
Also, the forces are a result of all the excitations acting on the structure. Multiple inputs
can be stacked to produce a system of equations that is over-determined. Since multiple
inputs are used to excite the system, the likelihood of adequately exciting all translational
and rotational degrees of freedom increases. The solution equation for multiple inputs
can be written as
102
m
mX CG
{FP }1
{FP }2
=
[ AM P ]1
[ AM P ]2
[ AM P ]n
{FP }n
mYCG
mZ CG
Ixx
Ixy
Ixz
Iyy
Iyz
Izz
(34)
As mentioned, the complex parts of the forces and accelerations can be divided to
increase the number of equations. This also forces the inertia properties to be real values
which force a physical solution.
m
mX CG
real {FP }S
imag {FP }S
=
real [AM P ]S
imag [AM P ]S
mYCG
mZ CG
Ixx
(35)
Ixy
Ixz
Iyy
Iyz
Izz
{FP }S =
{FP }1
{FP }2
{FP }n
and
[ AM P ]S =
[ AM P ]1
[ AM P ]2
[ AM P ]n
103
The force vectors are the summation of all the forces acting on the structure to produce
the rigid body motion. The excitation forces used to input the dynamics forces to the
structure are measured, as well as the location and direction of the forces. The reaction
forces are measured at all locations where the structure is supported. By measuring all
the forces acting on the structure, the structure can be considered isolated from the
surroundings.
Thus, any accelerations of the structure are strictly a result of the
summation of all the forces acting on the structure and the mass matrix.
The acceleration of the structure is determined by mounting an array of accelerometers
on the structure in order to adequately measure all the degrees of freedom of the rigid
body motion. This motion is a result of all the forces acting on the structure and the rigid
body mass matrix of the structure. The array of accelerometers can be condensed to a
single point P colocated with the point P used to combine the force measurements to
produce a six degree of freedom acceleration vector.
104
APPENDIX B: EXPERIMENTAL EXAMPLES
The same procedure used to determine the inertia properties of the calibration mass
experimentally using the DIM was used for the inverted calibration mass mounted on an
air ride and rigidly secured to the floor.
INVERTED CALIBRATION MASS MOUNTED ON AIR RIDE
Figure 44 - Calibration Mass on Air Ride - Rigid Body Motion Errors
105
Figure 45 - Calibration Mass on Air Ride - RB Motion Initial Channel Errors
Figure 46 - Calibration Mass on Air Ride - RB Motion Final Channel Errors
106
Figure 47 - Calibration Mass on Air Ride - Filtered RB Motion - DIM Solution
Figure 48 - Calibration Mass on Air Ride - FRF Scaling
107
Figure 49 - Calibration Mass on Air Ride – Scaled FRF - DIM Solution
Figure 50 - Calibration Mass on Air Ride - Known Mass - DIM Solution
108
Figure 51 - Calibration Mass on Air Ride - Known Mass and CG - DIM Solution
Table 19 - Calibration Mass on Air Ride – DIM Results
General Solution
Mass, kg
FRF Scaling
Known Mass
Substitution
Mass and CG
Substitution
Actual
Values
Values
Error, %
Values
Error, %
Values
Error, %
Values
Error, %
19.20577
5.67137
18.17500
5.33E-07
18.17500
5.86E-14
18.175
5.86E-14
18.17500
0.03524
Xcg, m
0.03326
5.60374
0.03326
5.60374
0.03435
2.51889
0.03524
1.97E-14
Xcg, m
-0.00185
1330.510
-0.00185
1330.510
-0.00273
2006.490
0.00013
4.18E-14
0.00013
Xcg, m
-0.07971
6.91202
-0.07971
6.91202
-0.08206
4.17608
-0.08563
4.86E-14
-0.08563
Ixx, kg m^2
0.22669
3.72367
0.21453
1.84317
0.22512
3.00580
0.22822
4.42193
0.21855
Ixy, kg m^2
-0.01257
11431.50
-0.01189
10812.60
-0.01236
11242.20
-0.01260
11462.30
-0.00011
Ixz, kg m^2
0.07725
1.16062
0.07311
6.46532
0.07642
2.23300
0.07782
0.43664
0.07816
Iyy, kg m^2
0.30394
0.07493
0.28763
5.43790
0.30216
0.66184
0.30503
0.28253
0.30417
Iyz, kg m^2
-0.00401
1740.57
-0.00380
1641.79
-0.00447
1951.64
-0.00306
1302.81
0.00022
Izz, kg m^2
0.16176
2.27852
0.15308
3.21076
0.16115
1.89394
0.16242
2.69682
0.15815
109
CALIBRATION MASS FIXED TO FLOOR
Figure 52 - Calibration Mass on Ground - RB Motion Errors
110
Figure 53 - Calibration Mass on Ground - RB Motion Initial Channel Errors
Figure 54 - Calibration Mass on Ground - RB Motion Final Channel Errors
111
Figure 55 - Calibration Mass on Ground - Filtered RB Motion - DIM Solution
Figure 56 - Calibration Mass on Ground – Scaled FRF - DIM Solution
112
Figure 57 - Calibration Mass on Ground – Known Mass - DIM Solution
Table 20 Calibration Mass Fixed to Ground - Dim Results
General Solution
FRF Scaling
Known Mass Substitution
Actual
Values
Values
Error, %
Values
Error, %
Values
Error, %
Mass, kg
12.84594
29.32080
18.17500
4.62E-07
18.175
5.86E-14
18.17500
Xcg, m
0.04662
32.30810
0.04662
32.30810
0.03308
6.11014
0.03524
Xcg, m
0.00424
3170.22000
0.00424
3170.22000
0.00316
2335.36000
0.00013
Xcg, m
-0.12751
48.90940
-0.12751
48.90940
-0.08504
0.69145
-0.08563
Ixx, kg m^2
0.21016
3.83975
0.29735
36.05180
0.21151
3.22362
0.21855
Ixy, kg m^2
-0.00435
3890.870
-0.00615
5546.460
-0.00419
3748.960
-0.00011
Ixz, kg m^2
0.07644
2.19760
0.10816
38.37520
0.07639
2.26021
0.07816
Iyy, kg m^2
0.29940
1.56736
0.42361
39.26690
0.30061
1.17122
0.30417
Iyz, kg m^2
0.00164
654.25
0.00233
967.14
0.00155
613.33
0.00022
Izz, kg m^2
0.15716
0.62998
0.22235
40.59310
0.15718
0.61276
0.15815
113
X-38 DIM RESULTS
Case
1
No rigid body response filtering applied
Case
2
Rigid body response filtered
Case
3
FRF scaling applied & Case 2
Case
4
Jackknife inputs & Case 2
Case
5
FRF scaling & Case 4
Case
6
Known mass parameter substitution & Case 4
Case
7
Known mass & CG parameters substitution & Case 4
Case
8
Impacts only - no reaction forces & Case 4
Case
9
Impacts only - no reaction forces & Case 4 with FRF scaling
Case
10
Known mass parameter substitution with FRF scaling & Case 4
Case
11
Known mass & center of gravity parameter substitution with FRF scaling & Case 4
Figure 58 - X-38 – All Data Included in Estimation (Case 1) – DIM Solution
114
Figure 59 - X-38– Applied Rigid Body Motion Filtering (Case 2) – DIM Solution
Figure 60 - X-38 Mass Scaling Convergence
115
Figure 61 – X-38 – Scaled FRF (Case 3) - DIM Solution
116
Standard Deviations
0.02
50
Xcg
Mass
Standard Deviations
0.025
55
45
0.015
40
0.01
-3
x 10
0.016
5
0.012
0.01
Zcg
Ycg
0.014
4.5
4
0.008
0.006
250
Ixy
100
200
50
150
350
1100
300
1000
Iyy
Ixz
Ixx
150
250
900
800
200
700
150
2000
1000
Izz
Iyz
800
600
1500
400
1000
2
4
6
8
Jack Knif e Cycle
10
12
2
4
6
8
Jack Knife Cycle
10
12
Figure 62 - X-38 Jackknife Cycle Inertia Property Variations
Initial Cycle Values
Initial Cycle Values
4.05
Xcg
8600
4
8400
3.95
0.06
0.78
0.04
Zcg
Ycg
Mass
8800
0.02
0.77
0.76
0
4
1.08
x 10
0
1.04
Ixy
Ixx
1.06
1.02
-1000
-2000
1
4
5
x 10
x 10
-2.85
-2.9
Iyy
Ixz
1.74
-2.95
1.72
1.7
1.68
5
x 10
1.74
1.72
Izz
Iyz
-1000
-2000
1.7
1.68
1.66
-3000
2
4
6
8
Jack Knif e Cycle
10
12
2
4
6
8
Jack Knife Cycle
10
12
Figure 63 - X-38 Jackknife Cycle Standard Deviation of Inertia Properties
117
Figure 64 – X-38 - Jackknife (7 Inputs Removed) (Case 4) - DIM Solution
118
Figure 65 – X-38 - Jackknife – Scaled FRF (Case 5) - DIM Solution
119
Figure 66 – X-38 - Jackknife – Known Mass (Case 6) - DIM Solution
120
Figure 67 – X-38 - Jackknife – Known Mass and CG (Case 7) - DIM Solution
121
Figure 68 – X-38 – Input Excitation Only – Known Mass (Case 8) - DIM Solution
122
Figure 69 – X-38 – Input Excitation Only – Reaction Forces Not Considered
(Case 9) - DIM Solution
123
Figure 70 – X-38 – Scaled FRF with Known Mass (Case 10) - DIM Solution
124
Figure 71 – X-38 - Scaled FRF with Known Mass and CG (Case 11) - DIM Solution
125