Book File09_05_03

i
Pierson & Associates
©
LLC
The Art of Practical and
Precise Strain Based
Measurement- Second Edition
By James G. Pierson
090103

ii
Revision Date: September 1, 1999
Copyright © 1992 by Pierson & Associates LLC. All rights
reserved. Printed in the United States of America, Canada and the
United Kingdom. Except as permitted under the United States
Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in
a database retrieval system, without the written permission of the
publisher.
ISBN 1-895976-00-6
Disclaimer: The enclosed information is given to the user as correct
to the best of the authorÕs knowledge. The use of any or all of the
information enclosed herein is entirely at the risk of the user. In
acceptance of this publication, the user hereby idemnifies Pierson
Associates with regards to the information and use of any and all
information contained herein. This work has been prepared by
James G. Pierson of Pierson & Associates LLC. and is continuously
revised and updated. Any and all comments and suggestions are
most welcome and appreciated. With your input, the cause of
precision measurement can continuously evolve and improve! Mr.
Pierson may be contacted at the following address:
Attention: James Pierson
Pierson & Associates LLC
7A Sanders Road
Rockaway, NJ
07866-2008
INTERNET: http://www.PiersonOnline.com
Dial: 800-565-6075 USA and Canada

iii
Special Credits:
Mrs. Helen Pierson, my wife, cheerleader and best friend.
Mr. Michael Coope, President Copidate, U.K. (For encouragement
and support above and beyond the call of duty!)
Mr. Jim Lally, President PCB, Piezotronics. (No request for
information was too much to ask!)
Mr. Craig Rockafellow, General Motors Proving Grounds, Milford
Michigan. (For encouraging me to undertake this effort)
Mr. Terry Smith, former President of Sentech Systems Inc. Mount
Joy PA.
Mr. Richard Talmadge, Chief Engineer of the Structural Dynamics
Research Branch, Wright Patterson Air Force Base, Dayton, Ohio.
(For agreeing to the arduous task of editing!)
Mr. Andrejs Zeltkalns, a brilliant expatriated Latvian load-cell
designer and friend (For discussions, insight, and support that I
could never have found elsewhere!)
+ Several hundred others from whom I have learned and I wish that
I had the space to separately thank. This is no slight!

iv
Preface to Second Edition:
Given a perfectionist disposition, a project like this handbook is a
never-ending task, a source of happiness and a source of frustration
all wrapped up in 850 pages of type. Reviewing material that one
wrote years prior can be a frightening experience, the usual reaction
to a passage being ÒI actually wrote that?Ó ÒClunkyÓ is being far too
kind a way of describing some passages. As time passes and we
grow in many different ways, maturity brings new insights that we
simply couldnÕt see during the first pass. The second edition rights
a bunch of wrongs, adds concepts that should have existed all
along, clarifies clunky wording, offers new insights and revises
other concepts to reflect current thinking in the measurement
sciences.
We have heard it stated that the world is changing to a knowledge
based economy. We have also heard it stated that knowledge is
power. Knowledge regarding the performance of the machines or
products that our companies fabricate empowers us to make correct
engineering and business decisions to best our competition. The
real danger in physical measurement occurs when we mistake
perception for knowledge (truth) and make decisions based upon
what we perceive to be true. Acting upon perceptions, not based
upon the truth; we chase ghosts with success always eluding us.
When perceptions equate to the truth we have earned the power to
succeed.
Without you the reader, of what value is any written work? I thank
you in advance for the time you spend with this material. I sincerely
hope that you will gain a deeper understanding of the necessity,
value and process of precision physical measurement.
(PS: Did I mention this effort is never ending? Look for the 3rd
Edition in 2005!) - Jim Pierson September 1, 1999

The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 i
By James G. Pierson
090103 Pierson & Associates LLC
Table of Contents

Contents
ii The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999
CHAPTER 1 Strain-Based Measurement Introduction:
Applications Note 1-1: Introduction 1-1
Applications Note 1-2:
What is Applications Engineering? 1-6
Applications Note 1-3
The Measuring System Source of Information: The Sensor 1-8
Linear versus Nonlinear Sensors 1-12
The Strain-Based Sensor DeÞnition 1-16
The Measurement System Transfer Function 1-19
The Fourier Series 1-40
Zeroth-, First-, and Second-Order System DeÞnitions 1-46
The Second Order System 1-51
Fluidic-, Gas-, and Structural-Damping of Second-Order Sensor Structures 1-58
The Strain-Gaged Cantilevered Beam 1-67
Pressure References 1-73
The Diaphragm Strain-Gaged Pressure Sensor
General Discussion: 1-80
CHAPTER 2 Data Quality, The Environment and Physical Constraints
Introduction to The Statement of Objectives 2-1
The Environment Assessment 2-17
The ÒMicroÓ and ÒMacroÓ Perspectives 2-25
The DeÞnition of, and Sources of, Noise 2-29
The Statement of Physical Constraints 2-37
Static and Dynamic Measurement Environments 2-46

Contents
The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 iii
CHAPTER 3 Sensor Types
The Capacitive Sensor 3-1
The Force-Balance-Based Sensor 3-7
Force-Summing Strain Gaged Sensors 3-11
The Linear Variable Differential Transformer (LVDT): 3-14
The Potentiometric Sensor 3-21
The Variable Reluctance Sensor 3-26
The Resonant Cylinder- and Resonant-Beam Sensor 3-29
The Self-Generating Piezoelectric Sensor 3-35
The Metal Strain Gages 3-58
The Wheatstone Bridge 3-68
CHAPTER 4 Strain Gages
Wheatstone Bridge Gain Factor and Nonlinearities 4-1
Gage Factor and Sensitivity Relationships of Metal Strain Gages 4-12
Piezoresistivity 4-18
Strain-Gage Backing Material 4-43
Surface Preparation for Strain-Gaging 4-47
Strain-Gage Bonding Adhesives 4-49
Strain Gage Release Films, Clamping and Bonding 4-54
Soldering and Interbridge Wiring of Strain Gages 4-61
Electrical Connector Considerations 4-64

Contents
iv The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999
Lead-Wire Conductors 4-71
Lead-Wire Insulations 4-57
Strain Bridge Wiring and Lead Wire Effects 4-80
Fatigue and the Metal-Foil Strain Gage 4-95
Fatigue and the Piezoresistive Strain Gage 4-97
The Metallic-Foil Comparison To the Piezoresistive Strain Gage 4-101
The Implications of Sensor Size and Mass 4-103
Real Versus Pseudo Calibration 4-111
Resistance Calibration General Discussion 4-112
The Piezoresistive Strain-Gaged Sensor and Resistance Calibration 4-121
CHAPTER 5 Error Components
Zero or Null-Bias Stability 5-1
Creep 5-12
Hysteresis DeÞnition 5-18
Linearity DeÞnition 5-27
Combined Nonlinearity and Hysteresis 5-30
Resolution 5-33
Nonrepeatability and Reproducibility 5-34
Acceleration Sensitivity of Strain-Based Pressure Sensors 5-38
Transverse Sensitivity Considerations for the Cantilevered-Beam Accelerometer 5-40
Uncertainty 5-47

Contents
The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 v
Performance Parameter Distributions 5-87
Cost Relationships 5-93
CHAPTER 6 Mechanical Considerations in Sensor Design
Sensor Spring Member Materials and Mechanical Design 6-1
Mounting-Strain Effects on Transducers 6-49
Loading Surface Geometry 6-53
The Nature of Epoxies 6-56
CHAPTER 7 Thermal Compensation
Piezoresistive Strain-Bridge Thermal-Sensitivity Compensation 7-1
Piezoresistive Strain-Gage Active Thermal-Sensitivity Compensation 7-13
Piezoresistive Semiconductor Strain-Gaged Wheatstone Bridge Thermal-Zero Shift 7-16
Considerations for the Quarter, Half and Fully-Active Piezoresistive Bridge 7-28
Constant-Current Excitation for Piezoresistive Strain-Gaged Sensors (Simple) 7-31
Dual-Tracking Constant-Current Excitation for Piezoresistive Sensors 7-36
General Notes Regarding Piezoresistive Strain-Gaged Sensor Thermal Performance 7-39
Metallic Strain-Gage Thermal-Sensitivity Compensation 7-42
Metallic-Strain-Gage Thermal-Zero Compensation 7-50
Transient Thermal Compensation 7-54
Thermal Isolation, Cooling and Control of Sensor Structures for Minimized Thermal Error 7-58
Thermal Design Considerations 7-66

Contents
vi The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999
CHAPTER 8 Electrical Considerations
AmpliÞcation 8-1
Zero-Trim Methods for the Wheatstone Bridge 8-27
Dynamic and Static Excitation 8-32
Grounding and Shielding Techniques for Strain-Gaged Sensors 8-38
Failure-Mode Analysis of the Wheatstone Bridge 8-54
Aliasing 8-72
The Electrical-, Mechanical-, and Thermal-Time Domains 8-75
CHAPTER 9 Performance
Knowledge-Based Error Correction 9-1
Sensor Performance SpeciÞcation 9-11
SpeciÞcation Considerations 9-25
CHAPTER 10 Calibration and Test
Strain Gaged Accelerometer Calibration 10-1
Gravimetric Calibration Methods, Acceleration and Force: 10-17
Impulse-Hammer Test Methods (Piezoelectric) 10-24
QualiÞcation and Acceptance Testing 10-27
Military SpeciÞcation Environmental Test 10-29
QualiÞcation Test: Acceleration 10-31
Strain-Gaged Pressure Transducer Calibration 10-39

Contents
The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 vii
QualiÞcation Test: Pressure 10-48
Load Cell Calibration 10-56
QualiÞcation Test: Load 10-66
CHAPTER 11 Other Considerations and Special Environments
Leakage Path Analysis 11-1
Moisture 11-9
Corrosive Media Considerations 11-12
High-Resolution Unsteady- and Steady-Pressure Measurement 11-15
High Vacuum Environments 11-19
The Effects of Optical Radiation and Ionizing Radiation 11-21
Sensors for use in Abrasive Flows 11-32
Blast Pressure Measurement 11-34
Flight Test Acceleration Measurement 11-37
Application Note 11-10:
Robotics 11-39
Proper Application of Flat-Pack Style Sensors 11-40
CHAPTER 12 The Future 12-1
The Future of Sensing 12-1

Contents
viii The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-1
CHAPTER 1 Strain-Based
Measurement
Introduction:
Applications Note 1-1: Introduction
The Wheatstone Bridge strain-gaged sensor type is, by a large mar-
gin, the most popular transducer element in use. Unfortunately, the
strain-based sensor is also the most misapplied sensor type. By some
estimates, as much as 80% of all new sensor purchases are made in
the act of replacing a prior sensor technology or type. This represents
a massive waste of human effort as well as financial resources. His-
torically, the costs of these errors in judgement have been buried
within research and development budgets. The objective of this work
is to show how we can work smarter, rather than harder, by gaining
insight into the simple physics of the strain-based sensor so that
errors in physical measurement are minimized. When a measurement
is made we are attempting to understand our product or process under
specific operating conditions. Based upon our measurements we per-
ceive specific values when the true value of a parameter may be quite
different. The difference between the truth and the perception is error.
A ten percent error in the magnitude of peak strain experienced by a
component in use implies two consequences; the fatigue life expect-

Strain-Based Measurement Introduction:
CHAPTER 1-2 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999
ancy of the component could be reduced by a factor of up to two times, possi-
bly doubling warranty costs when the strain is underestimated, or we are
shipping much more material than is necessary to support the design life
expectancy when the peak strain is overestimated. When strain is overesti-
mated, vehicles become much more massive than is necessary to meet or
exceed the design life for an intended use profile increasing vehicle mass,
decreasing vehicle fuel efficiency, increasing the cost of vehicle structures and
braking systems, requiring higher capacity propulsive systems and increasing
the cost of safety systems in the event of collision. It has been estimated that
greater than half of all measurements made show peak errors greater than 15%.
If warranty costs can double at a peak error of 10%, just imagine what happens
at 15% or more error. We live in the so called Òinformation ageÓ. It is implied
by this statement that we make engineering and business decisions based upon
correct information. The business of measurement is to gain understanding of a
product or a process not simply to collect numbers. Our mission is to minimize
the difference between the perception and the truth.
Strain gages are configured within the sensor to provide a differential output
that is in proportion to the applied physical parameter, whether this parameter
is force, pressure, acceleration, or strain. This work is a generalized treatment
of the thermal, mechanical, and electrical behavior of real-world sensors and
refers, in most cases, to the physical input as the input parameter without con-
straint to any particular sensor type.
Many exhaustive works are available that mathematically model virtually any
sensor in the frequency domain and time domain. It is the authorÕs opinion
that, as engineers, we are capable of mathematically complicating any issue
that we choose, and, if complex models are required, we are able to research
the required theory and assemble these models. The models used herein are
entirely based upon the two principal relationships of OhmÕs Law (V = IR) and
NewtonÕs Law (F = MA). In many instances, these simple relationships can be
used to create mathematical representations of surprising fidelity even when
compared to the more complex mathematical models.

Every effort has been made in this work to avoid math-intensive models of sensor
performance. When possible, simpler and more understandable mathematical
models have been used rather than the more rigorous models in the interest of clar-
ity. The focus of this work is the thermomechanical behavior of the generalized
sensor. The material discussed in the following pages is an ÒApplications Compen-
diumÓ detailing many of the very simple phenomena that have frustrated many an
engineer. To an instrument engineer, the embarrassment of not having considered
this or that phenomenon in a particular measurement environment, is a familiar
feeling. We are spared this embarrassment when the ÒtruthÓ remains unknown.
Indeed, in the mind of the purist measurement statistician, the ÒtruthÓ remains for-
ever unknowable as even national reference standards possess finite uncertainty
limitations.
Much of the information contained in this work will appear to be of the simple ÒI
knew thatÓ kind. It is written to be so. After having been frustrated by poor data
quality in the past, how many times have you slapped your forehead and exclaimed
ÒI should have known better!Ó? The parameter that very likely was the source of
frustration, was either not properly specified, absent altogether, or specified in an
ambiguous way. In many cases, the manufacturers themselves truly have no idea
how their devices will or wonÕt function in specialized environments or in unique
combinations of environments. In the end, as engineers, it is our responsibility to
define to the sensor manufacturer, our assessment of the measurement environ-
ment, and to work with the manufacturer to achieve mutual success.
Much of the information contained in these pages has not previously been in print
or has been buried within written works intended more to impress colleagues than
to instruct others. The material contained herein is common-sense physics. You
will also note that some repetition of material exists within these pages. This repe-
tition is necessary to link associated concepts together and to provide a written
work that, in the end, can be used as a quick, stand alone reference manual.
The two primary strain gage types discussed are the piezoresistive and metallic-
foil-based strain gage. The very high intrinsic strain sensitivity of the piezoresis-
tive device makes visible phenomena that are typically low-level error sources for
the metallic-foil-gaged sensor but are, none-the-less, sources of error that may

make the difference between success and failure in measurement. Piezoresis-
tive sensors, by a large margin, are manufactured in greater quantities for dedi-
cated vehicular use than all other strain-based sensors combined. For this
reason, the piezoresistive-based sensor is a primary focus of discussion.
This book is organized as a reference manual containing a selection of ÒAppli-
cations NotesÓ that pertain to specific sensor parameters, considerations, media
concerns, lead-wire considerations, and a host of others. You are encouraged to
review the Introductory section to lay a firm foundation for the application
notes that follow.
Sensors are epidemically misapplied to various measurements. In some situa-
tions it is the fact that the engineer has achieved a Òcomfort levelÓ in the use of
a particular sensor technology and is unwilling to risk the use of a potentially
more-appropriate technology. Knowledge of other technologies is the key that
unlocks the door to move beyond the comfort zone. In other instances, it is
simply ignorance of the intricacies of the sensor that lead to the Òround peg in
the square holeÓ situations that occur frequently in measurement.
Why are there hundreds of different sensor types commercially available? Sim-
ply, each sensor type is appropriate for use in specific environments and is less
well-suited for use in others. Each measurement environment is unique and
often ÒgreyÓ areas are encountered where the performance of the sensor is not
well-defined by the manufacturerÕs specifications. This book breaks with tradi-
tion in attempting to define attributes of the sensor that the reader can utilize to
qualify a sensor geometry and performance for the measurement environment
that he or she faces.
I firmly believe that any reader, who has been Òdown in the trenchesÓ making
real-world measurements, will find some gems contained herein that have been
ÒGotchasÓ in prior measurement efforts. As oneÕs experience level grows in the
field of measurement, it becomes clear that no ÒminusculeÓ errors exist. The
complete treatment of sensor error components may seem to dwell upon small
error contributors yet, all too often, engineers find themselves in a court-of-law
arguing over ÒminuteÓ errors that, due to the nature of the measurement, have
become major error drivers. In short, the Òminute errorÓ can, all too easily,

become the bone of contention when a major program has been delayed or expen-
sive retesting must be conducted due to poor-quality data.
In closing, I would leave you with the thought that, the instrumentation engineer is
perhaps one of the worldÕs great unsung heroes when one considers that all that we
know of materials, our bodies, our universe, subatomic as well as chemical and
macroscopic phenomena, we have learned through the science of observation and
measurement. The human biological sensor system, including our senses of sight,
smell, touch, temperature sensitivity, sound, and the like, functions as a group of
transducers providing information to us about our world. In instrument engineer-
ing, separate sensors, specifically fabricated to be sensitive to one or another phe-
nomenon, are used to provide information to us concerning the inner workings of
our creations, whether they be machines or processes. In summary, the knowledge
that we possess about our world is provided to us by observation and measurement
where the validity of this knowledge is dependent upon the quality of the observa-
tion or measurement.
I sincerely hope that the you will enjoy this work as much as I have enjoyed creat-
ing it.
James Pierson
Pierson Associates Incorporated
First Edition: 1992
2nd Edition: September 1999

What is Applications Engineering?
The simple Funk and WagnallÕs definition of the meaning of ÒApplicationÓ is
as follows: 1. ÒThe act of applyingÓ 2. ÒCapacity of being used; relevance, as
of a theoryÓ 3. ÒClose AttentionÓ; as in application to oneÕs bookÓ.
The definition of ÒEngineeringÓ: ÒThe art and science concerned with the prac-
tical application of scientific knowledge, as in the design, construction, and
operation of roads, bridges, harbors, buildings, machinery, lighting, and com-
munications systems etc.Ó
The additional definition of the meaning of the word ÒArtÓ is also appropriate
as: ÒAny system of rules and principles that facilitates skilled human accom-
plishment: also the application of these rules and principles.Ó
The above definitions are not comprehensive but are a subset of the definitions
most suited to this discussion. Applications Engineering is the Òdown in the
trenchesÓ real-world application of a technology for practical purposes.
Applications Engineering is an art, consistent with the above definition, in that
it is the assimilation and distillation of a multitude of realities regarding the
limitations and physics of the world around us and the use of imperfect materi-
als and processes to effect the practical and useful implementation of a tech-
nology.
All that we know of the machines and processes that we create is provided by means
of sensors that are subject to the imperfections and physical limitations of the world
we live in. This knowledge is imperfect where the difference between the truth and
perception is error. - JP
The objective of Applications Engineering, with respect to sensors, is to mini-
mize the difference between the perception and the truth, given the physical
constraints and the environmental conditions within which the sensor must
operate.

The Art of Applications Engineering, as described in the ensuing text, is the result
of a collection of experiences in measurement over many years and for many var-
ied measurement environments. A preliminary review of the entire text will famil-
iarize the reader with many of the concepts contained herein, where the details can
be studied in depth on an as-required basis for various measurements and environ-
ments as they are encountered in practice.

The Measuring System Source of Information: The
Sensor
Many component parts comprise the measurement system where all parts must
function in unison to produce valid information regarding the measured param-
eter or measurand. Of all of these subcomponent parts, the sensor is most criti-
cal as the source of information upon which the balance of the system depends.
The output of any sensor is organized energy where some deterministic rela-
tionship exists between a quality or quantity of this energy and the present state
of the measurand. All sensor output signals are comprised of a random compo-
nent and a predictable component of signal. The ratio of the predictable com-
ponent to the random component is a measure of the signal-to-noise ratio
which can be used to quantify the useful measurement range of the sensor.
The Funk and WagnallÕs definition of the transducer, as ÒAny device whereby
energy may be transmitted from one system to another system whether of the
same type or different typeÓ, is most appropriate when a sensor is viewed as a
form of energy translation device. The definition of the sensor as ÒThat which
receives and responds to a stimulus or signal; especially, an instrument or
device, as an antenna, gyroscope, or photoelectric cell etc., designed to detect
and respond to some force, change, or radiation for purposes of information or
controlÓ, is equally appropriate. Some use the term ÒsensorÓ to refer to the Òas
produced but not yet finishedÓ transducer, as would be the case with an uncom-
pensated strain-gaged device, in contrast with the term ÒtransducerÓ referring
to the finished and fully specification-controlled device. Rather than attempt to
delineate between the two terms, they are used herein interchangeably in
accordance with the above definitions.
Several facts may be stated that are applicable to all of the many types of sen-
sors that exist or, in fact, will exist. These facts derive from fundamental laws
of mechanics and thermodynamics and are stated as follows:

1. The presence of any physical sensor inserted into any mechanical or thermodynamic system
will alter, to some extent, the system itself. This phenomenon can be visualized by considering
that the addition of an accelerometer to a structure, for the purpose of determining the dynamic
behavior of the structure, implies the addition of the accelerometerÕs mass, thereby changing the
structureÕs dynamic characteristics. This fact is very similar to the Heisenberg Uncertainty Prin-
ciple of nuclear physics, in that, the act of observation changes the system being observed. As
another example, the flush-diaphragm pressure sensor must experience bending to produce out-
put as we shall see. The bending of such a diaphragm implies that the volume of the measured
cavity must increase with increasing pressure, thereby altering the measurand.
2. All sensors that are capable of providing information at zero hertz or ÒDCÓ, when the measur-
and is static or very slowly changing with respect to time, must shed energy in order to provide
information regarding the measurand. Sensors that do not dissipate heat or energy by some
other mechanism, are not capable of providing information (organized energy) as an output
when the input is static and unchanging (invariant) with respect to time. It is important to note
that the piezoelectric sensor is of the nondissipative variety and cannot provide information
concerning time invariant parameters. It is equally important to realize that energy must be sub-
tracted from the measurand by the piezoelectric sensor, for any information in the form of an
output to be realized, regardless of the time rate of change of the measurand. This situation is
visualized best by considering the piezoelectric accelerometer mounted to a vibrating structure.
Since the accelerometer possesses mass and Newtons Law states that force will equal the prod-
uct of mass and acceleration (F = ma), the accelerometer will therefore require a given force
input to be displaced. Since work energy is equal to the product of force and displacement (E=
Fd), the energy required to cause the cyclic displacement of the sensor is equal to the product of
mass, acceleration and displacement (E =mad), where this energy is supplied to the accelerome-
ter by the structure to which it is mounted.
3. The presence of a physical sensor inserted into any mechanical or thermodynamic system will
exchange energy in many different forms with the measured system and the measurement sys-
tem to which it communicates. This statement means that the mere presence of the sensor mass
in a small chemical reaction vessel will imply that calories will flow, either from the sensor into
the vessel or from the measurand to the sensor mass, and thus could influence the behavior of
the measurand. In the case of the static-capable (zero hertz) sensor, a portion of the energy
required to be shed, in the provision of information regarding the measurand, will be absorbed
by the measurand thereby altering its state. Additionally, the energy state existing at the output
of a sensor is changed when the sensor is connected to any other element of the measurement
system.
Just as the product of force and distance equals mechanical energy, the product of charge ßow
per unit time (current) and voltage equal energy per unit time or power. Since all elements of the
measurement system possess some Þnite input impedance, it follows that, when a voltage differ-
ence exists between these elements, some current must ßow between them for each element to
perform itÕs intended function. At close to zero hertz, impedance becomes simply resistance and

the current that will ßow from a sensor into the balance of the measurement system will
equal I=V/R in accordance with OhmÕs Law, where V is the voltage state at the input to the
measurement system and R is the input resistance of the measurement system. Clearly, the
highest value of R yields the lowest value of I for any voltage state existing at the measure-
ment system input. Another implication of this energy exchange is realized when one con-
siders that the sensor can be modeled, in accordance with TheveninÕs Theorem, as a voltage
source (V signal) in series with an equivalent output resistance (Routput) where Vsignal is the
open circuit output of the sensor.
FIGURE 1-1. Energy must flow from the sensor into the Measurement
System:
Since no measurement system possesses an inÞnite input resistance, but rather some Þnite
value (Rinstrument) of input resistance, then the voltage state at the input to the measurement
system is calculated by the simple voltage divider:
(EQ 1-1)
Therefore, the input voltage to the measurement system must always be less than Vsignal.
Only when Rinstrument equals an inÞnitely high value does Vinput equal V signal. In accor-
dance with equation 1-1, at close to zero hertz, a strain bridge possessing an output resis-
tance of 350 ohms operating into a voltmeter with an input resistance of 10 K ohm will
Vsignal
Routput
Rinstrument
Vinput
+
-
+
-
Energy Flow
(Closed circuit)
Vinput
Vsignal Rinstrument´
Ro Rinstrument+
---------------------------------------------------=
where the true sensor output is decreased by:
Vsignal VinputÐ
Vsignal
--------------------------------------- 100´ 1
Vinput
Vsignal
-----------------Ð 100´ 1
Rinstrument
Ro Rinstrument+
---------------------------------------Ð 100´= =

result in a perceived signal 3.38% less than the true open circuit voltage the strain bridge would
otherwise provide. Although the zero hertz case is useful to consider, most measurement sys-
tems are required to measure dynamically changing inputs. In this case, equation 1-1 is
extended to encompass the dynamic case by replacing resistances with impedances Zoutput and
Z instrument.
It is equally important to be aware that energy will be exchanged between the sensor and the
environment in many different forms and via many different paths where all forms and paths of
energy ßow must be considered for valid information to result.
4. All sensors respond to all inputs; it is simply a matter of the degree of the response. If our goal
is to design an accelerometer, we would, of course, set out to design the device so that it is not
influenced by temperature, pressure, rate of temperature change, humidity, substrate strain, radi-
ation, transverse acceleration inputs, and a host of other possible characteristics of the environ-
ment. In reality, this is not achievable. The physical constraints of having to fabricate sensor
structures with imperfect materials, having imperfectly-toleranced subcomponent parts and in
using less-than-perfect transduction methods, means that we must carefully design the sensor
with these shortcomings in mind.
5. As the information provided by a sensor is conducted, conveyed, transformed into other energy
forms, or interpreted, it is degraded at every opportunity. The random content of any organized
form of energy will increase at any and all opportunities. The act of amplifying a sensor signal
will undoubtedly increase the magnitude of the perceived signal but will likewise increase the
percentage of randomness or noise that is present in the amplified signal. In many ways, the ten-
dency towards randomness may be likened to the thermodynamic property of entropy which is
the irreversible tendency of a system or the universe toward increasing disorder.
The focus of the instrument engineer is to be aware of the consequences of the
presence of the sensor within a system, to minimize the influence that the sensor
will have on the measured phenomena as well as being aware of the influence of
other aspects of the environment on the quality of the information that the sensor
provides.
It is the sensor that is the source of information upon which aerodynamicists make
design modifications, how an old bridge is buttressed, or whether the air bag in our
automobile should initiate. It is the quality of this information that will establish
our success, or in some cases, our very survivability. The strain-based sensor, the
most prevalent sensor type in the world today, is the focus of the following disser-
tation.

Linear versus Nonlinear Sensors
Our preference for sensors that produce a linear output with respect to a lin-
early-applied input arises out of expediency rather than necessity. Many sensor
types will produce nonlinear outputs similar to the output produced by a ther-
mistor, for example (Figure 1-2). The highly nonlinear output of the common
thermistor is typically linearized by the addition of a low-TCR (thermal coeffi-
cient of resistance) resistor, installed in parallel with the thermistor.
FIGURE 1-2. The Linearized Response.
The desire to work with linear output sensors is understandable as the need for
computational linearization of the resultant data is eliminated. In the case of
sensors used for high-frequency measurement, inadequate time may exist
between data points to execute a linearization algorithm if the sampling rate is
to support a useful bandwidth in close to real time. The methods used for out-
put linearization require prior knowledge of the sensor output with respect to
the physical input.
Figure 1-1
Resistance
(Ohms)
Temperature
Thermistor Response
Ideal Linear Response
Temperature
Ohms
Low-TCR Resistance
Linearized Thermistor Response
Actual Response
T
T

The measurement of the output of a nonlinear sensor may be linearized by measur-
ing the output of the sensor at 0%, 20%, 40%, 60%, 80%, and 100% of the full-
scale input of the sensor (Figure 1-3). This data may be used to form a piece-wise
linear curve that closely approximates the nonlinear output function but will
require significant computational time for each data point collected and will, there-
fore, limit the maximum practical output bandwidth.
FIGURE 1-3. Piece-wise Linearization
The measurement of the output function used for piece-wise data linearization can
also be used as an input to a polynomial curve-fit program wherein the measure-
ment system either computes the curve-fit in real time for each data point collected
or creates a Òlook-upÓ map of output versus input parameters in memory. The use
of an internal look-up map of the sensor output with respect to its input will greatly
increase the potential output frequency response, as simple decision algorithms
may be used rather than complex real time computations. The implementation of
Output
Input0% 20% 40% 60% 80% 100%
M1
M2 M3
M4 M5
Y1
Y2
Y3
Y4
Y5
Piece-wise linearization of a nonlinear line
Where:
Output = Mi (Input) + Yi
(For each of the appropriate
curve sections)

polynomial curve fit routines is easily accomplished with spreadsheet pro-
grams such as MicrosoftÕs Excel by using the ÒTrendlineÓ menu selection.
Curve fit routines make non-obvious underlying assumptions and typically
process the supplied data to yield curve fit coefficients that minimize the least
squares error between the computed curve and the data provided. The assump-
tions made with respect to a least squares fit are that the data is continuous in
the region of interest, the rate of change of data within the region of interest is
continuous and the scatter of data within the region of interest is bounded and
symmetrically distributed. In some cases there can exist a wider scatter of data
at one end of the region of interest than at the other. In such cases, any least
squares fit is of questionable value. Superior curve fits, providing uniform
scatter on either side of the computed response, always result when the form of
the curve fit relationship and the theoretical relationship defining the sensor
behavior are the same. If the sensor responds in an exponential fashion with
respect to the input, then the exponential relationship used in a curve fit to the
data will produce superior results. Strain based sensors tend to provide output
responses that are closely approximated by polynomial relationships of the
form:
(EQ 1-2)
Where a, b and c are quadratic coefficients, y is the output and x is the physical
input to the sensor. Calculation of the physical input creating the output, y, is
achieved by finding the real root of the quadratic equation given by computing:
(EQ 1-3)
It is the nature of quadratic relationships that two roots will result with one root
yielding a nonsensical result. Although Excel curve fit routines allow the user
to specify virtually any number of polynomial coefficients, it is seldom benefi-
cial to specify more than three coefficients. Keep in mind that an 11-coefficient
curve fit to an 11-point data set will result in a perfect fit as the computed
response is forced through each data point. This appears to be desirable, how-
ever, the computed curve fit will make some rather wild gyrations between
data points producing sometimes large errors! When we discuss measurement
y ax
2
bx c+ +=
x
bÐ b
2
4acÐ±
2a
--------------------------------------=

uncertainty in Chapter 5we learn that it is desirable to maximize the Òdegrees of
freedomÓ to minimize measurement uncertainty. When an eleven-coefficient fit is
applied to an eleven-point data set, where each computed coefficient effectively
negates one degree of freedom, zero degrees of freedom results! To produce supe-
rior results, the order of the curve fit used must be much less than the number of
data points to which the curve fit is being applied. With reference to equation 1-4,
N-K equals the number of degrees of freedom for the curve fit. Even when plenty
of data points exist, higher order polynomial coefficients tend to become exceed-
ingly small in magnitude and calculations utilizing more than three coefficients
become cumbersome and processor-intensive therefore limiting the rate at which
data can be processed. In all cases, the computed curve fit should be printed over-
lying the collected data so that the quality of the fit as well as the scatter around the
computed response can be assessed. Another measure of the quality of a curve fit
utilizes the ÒStandard Estimate of ErrorÓ or ÒSEEÓ method computed as follows:
(EQ 1-4)
The most rapid method of sensor output linearization involves the use of analog
rather than digital correction of the output data. By using amplifiers in the signal
path that possess nonlinear gains that effectively compensate for the nonlinear out-
put of the sensor, the effective bandwidth of the collected data is limited only by
the bandwidth of the linear and nonlinear amplifiers employed.
With present computational capabilities, computational-corrected nonlinear output
sensors are best suited to lower-frequency applications of less than 1 KHz. The
majority of pressure sensor uses fall into this frequency band. The implementation
of nonlinear analog amplifiers, or the computational methods described above,
implies prior knowledge of the sensor output characteristic. The collection of this
knowledge, as well as the implementation of the required software implies that
additional cost per data channel will be incurred.
SEE
Yi YciÐ( )
2
å
N KÐ( )
1 2¤
---------------------------------=
Where: N= Number of data points
K= Number of curve fit coefficients
Yi ith measurement=
Yci computed value of fit at the ith data point=

The key to success in the use of nonlinear sensors is dependent upon the
mechanical and electrical repeatability of the nonlinear output with respect to
its input from cycle to cycle and over time. For the most part, zero instabilities
can be normalized out of the data as long as the zero output is updated with
respect to time, or Òautomatic-null-on-power-upÓ amplifiers are implemented,
where the sensor is at some known physical input reference at the time that
power is applied.
The Strain-Based Sensor DeÞnition
The strain-based sensor is any sensor structure that produces output directly as
a result of the strain induced within the sensor spring member as result of a
parameter input such as force, strain, pressure, acceleration, and many other
forms of input. The strain-based sensor family includes passive as well as
active or self-generating sensor types where the passive group is dominated by
the strain gage sensor type and the active group is dominated by the piezoelec-
tric class of sensors.
FIGURE 1-4. The Cantilever Beam Accelerometer:
Figure 1-3
M
ass
M
ass
Acceleration
Strain gages
2 on top
2 underneath
Cantilever Beam

All strain-based sensor structures possess an internal spring member upon which
the parameter acts to produce output. In the case of the cantilevered-beam acceler-
ometer of Figure 1-4, the seismic mass will produce an inertial force acting on the
beam as a function of the acceleration of the device. The bending strain created
may be measured by a variety of means, the most common being by means of
strain gages bonded to the bending beam. In the case of the cantilevered-beam
accelerometer, the sensor spring member is the beam itself, as it is the beam that
provides the elastic restoring forces that act in opposition to the applied inertial
loads. In the case of the flush-diaphragm pressure sensor of Figure 1-5, unequal
pressures acting upon the diaphragm structure results in a force imbalance produc-
ing bending and resultant strain. In all cases, the strain-based sensor possesses an
internal spring member that is configured to respond mechanically to a specific
parameter input. The shape of the spring member will vary widely as a function of
the parameter to which the sensor has been designed to respond.
FIGURE 1-5. The Strain-Based Pressure Sensor:
Pressure Loading
Strain gages
(Papplied)
Preference
Diaphragm bending where
Papplied > Preference

The piezoelectric class of sensors utilizes the charge-generating characteristics
of the piezoelectric class of materials as a function of applied stress and result-
ant strain to produce charge outputs as a function of the applied parameter
input. The spring member in a piezoelectric sensor structure is generally the
piezoelectric material itself.
The performance of any strain-based sensor technology relates directly to the
mechanical, thermal, and electrical qualities of the internal spring member and
strain-sensing mechanism utilized.
Other selected non-strain-based sensor technologies are discussed with the
objective of providing you with a fuller and more complete perspective of the
world of measurement in general.
The measurement environment and the parameter to be measured as well as the
time-varying nature of both will impose constraints upon the shape, position,
mass, stiffness, structural support, thermal inertia, thermal impedance, allow-
able materials, and processes that can be used with present technology to fabri-
cate the sensor. The Art of Applications Engineering is the study of the
implications of these constraints, relative to the many varied measurement
environments, with the objective of maximizing the validity of the information
output that the sensor provides.

The Measurement System Transfer Function
The transfer function is the complete relationship defining the output of a sensor
for any set of measurand inputs. The transfer function is also the relationship by
which the time-varying input measurand is related to the resulting time-varying
output of a sensor. The transfer function is generally accepted as:
H(f) where:
Fout = the output function of the sensor.
Fin = the input measurand function.
Note that each element of the signal-processing system will also possess a unique
transfer function as well.
The Concept of Spatial Independence:
The concept of spatial independence means that a variable measurand is spatially
uniform and that the value of the measurand is constant and independent of posi-
tion, relative to the size of the sensorÕs active member. The statically-pressurized
vessel meets the requirements of spatial independence. Tight turbulent flow fields,
as may be encountered around an antenna mounted to a high-performance fighter
aircraft, will show high-level pressure variations over small linear distances,
implying that the value of the measurand may not be constant over the sensitive
area of the sensor. In cases such as this, the requirement for spatial independence
may not be met. Figure 1-6 graphically shows the concept of spatial independence.
Fout f( ) H f( )Fin f( )=

FIGURE 1-6. Spatial Independence:
The concept of spatial independence implies that the transfer function of any
given sensor is only valid within a specific frequency band related to the phys-
Static Pressure
Pressure Sensor
The requirement for spatial
independence is satisfied
Turbulent Flow Field
Constant Pressure line
independence is NOT satisfied
Pressure Sensor
Pressure Sensor
independence is satisfied
Constant Pressure line
Turbulent Flow Field
Sensor active diameter
Sensor active diameter

ical size of the sensorÕs active member. For the measurement of higher frequen-
cies, spatial independence will require that smaller and smaller sensors be
employed in any given medium.
The Nature of the Measurand:
Measurands may be scalar quantities such as voltage, current, or temperature,
meaning that they can be completely specified by magnitude alone. Other measur-
ands may be vector quantities such as velocity, acceleration, and force, requiring a
direction and magnitude to define the measurand.
The effectiveness of the flush-diaphragm pressure sensor in sensing a pressure
shock wave is a good example in that the wave has magnitude as well as direction
and the sensor will not respond with equal outputs depending upon the angle of the
wave relative to the sensor surface. In cases such as this, the sensor would also
require a Òfield responsivityÓ calibration in order to define the sensitivity of the
device as a function of angle relative to a measurand input.
Microphones are typically calibrated to show the field responsivity of this sensor
type to measurand inputs at various angles to the sensor active member.
The System Transfer Function:
Bearing in mind the requirements of spatial independence, and the fact that the
definition of the field responsivity of a sensor may also be necessary in order to
define the total response of the sensor, the transfer function of the measurement
system is as shown in Figure 1-7.

FIGURE 1-7. The Transfer Function:
Magnitude H(w) = H1(w) X H2(w)........Hn(w)
And where the Phase Response is given by:
Phase j(w) = j1(w) + j2(w) +........... jn(w)
The total output of the measuring system is therefore:
(EQ 1-5)
and the total Phase Response is given by:
(EQ 1-6)
Although convolution defines a specific mathematical operation in time and
frequency domain analysis, equation 1-5 (Figure 1-8) can be thought of as the
convolution of the input signal with the transfer function of the sensor system.
The reverse process, where the output function of the system is divided by the
transfer function, can be thought of as deconvolution and is expressed as:
Fin(w) = Fout(w)/ H(w)
H1(w) Hn(w)H3(w)H2(w)
Input
Qi(t)
f1
Output
Qw(t) X H(w) = Qo(w)
f2 f2 fn+ + =
Fout w( ) H w( )Fin w( )=
jout w( ) j w( ) jin w( )+=

and where:
jin(w) = jout(w) - j(w)
FIGURE 1-8. Convolution and Deconvolution:
Convolution occurs during the process of measurement and deconvolution is the
process of the data reduction. Most data reduction difficulties come from the
deconvolution of collected data with the defined transfer function of the measure-
ment system.
The convolution function applied to spectral analysis of data is a very well-defined
complex mathematical process where convolution in the time domain is equivalent
to multiplication in the frequency domain. The relationship between convolution
and multiplication in the two domains allows the use of the convolution function to
compute transfer functions and time domain outputs of digitally-implemented sig-
nal filtering routines.
CONVOLUTION:
DECONVOLUTION:
Fin(w) Fout(w)
H(w)
Fout(w) H(w) Fin(w)
Fin(w)
Fout(w)
H(w)
=
= X
(Data Reduction)

The Sensor Transfer Function Measurement:
The measurement of the sensor transfer function is the determination of the
magnitude and phase response of the sensor when it is presented with a known
parameter input function. In the determination of the transfer function of any
sensor, it is necessary that a dynamic parameter input be provided in order that
the frequency-domain response of the device can be established. In the case of
both load cells and strain-gaged pressure sensors, it is most common that these
devices are calibrated using static or deadweight methods that will provide
only zero-hertz information and provide inadequate information to establish
the entire transfer function of the sensor. In measurement environments where
dynamic inputs are expected, choose a sensor having a flat frequency response
that is at least equal to or greater than the highest expected frequency present in
the measurand. Dynamic calibration methods are implemented in the calibra-
tion of the self-generating class of sensors, piezoelectrics for instance, due to
the difficulties and inaccuracies that arise when attempting high precision
static calibrations of inherently dynamic devices. In this case high precision
means calibrations performed to better than ±1% full scale output (FSO) non-
linearity and hysteresis.
The objective in calibrating any sensor is to establish the ÒsensitivityÓ of the
sensor to a physical input. The problem with this is that this sensitivity numb er
changes depending upon the frequency of the input. In the simplest case, we
would prefer to model the output of a sensor as a linear line or y=mx + b rela-
tionship where y is the output, m is the sensitivity, x is the input and b is what
you get out with no input. When the linear model would allow excessive uncer-
tainty to result, we are forced to use more complex and typically polynomial
relationships to model the sensor output. The sensitivity of any sensor may be
quoted in a number of different ways as follows:
1. millivolts per unit of input ie: mV/microstrain at some defined excitation (What you get out
per unit of input)
2. millivolts per Volt of input ie: mV/V (What you get as a full scale per unit of excitation)
3. millivolts per Volt of excitation per unit of input ie: mV/V/lbf (output per unit of excitation
per unit of input)
4. millivolts per milliamp ie: mV/ma (Common with constant -current driven sensors)

5. millivolts per milliwatt ie: mV/mW (Common with optically-based sensors)
Be aware that transducers with sensitivities quoted as full scale values in mV/V
may meet all other specifications at only one calibrated excitation level. You do
not necessarily have the latitude to select any arbitrary level of excitation. This is
particularly true with resistance based sensors where thermal specifications may
only be valid at some defined level of input excitation. For instance, in the case of
strain gage based sensors, a doubling of the input excitation implies a 400%
increase in the power dissipation. This results from the fact that power, in the volt-
age-excited case, is equal to V2
/R and in the current-excited case, equals I2
R.
Additionally, sensitivities quoted in mV/V/unit of input can result in extremely
small numbers where significant digits can mean a great deal. For example a
50,000 lbf load cell at 2mV/V of input results in .00004 mV/V/lbf as a sensitivity.
The various test methods that are listed in the following text briefly describe some
techniques by which the transfer function of a sensor may be defined, where the
most common methods in use are described in detail in following chapters:
Accelerometers:
The calibration of strain-based accelerometers is performed to determine the trans-
fer function of the device and is usually made by one of the following techniques.
Sinusoidal Discrete Frequency:
To perform discrete-frequency sinusoidal calibrations, the sensor is
mounted to an electrodynamic shaker system and is driven to vibrate at a
selection of discrete frequencies at a calibrated and known peak accelera-
tion level. The output of the device is then compared to the output of a cal-
ibrated reference accelerometer mounted inside the shaker armature. The
phase difference between the reference accelerometer and the accelerome-
ter being tested can also be measured and reported during this process

FIGURE 1-9. The Electrodynamic Shaker System:
Sinusoidal Swept Frequency:
The sinusoidal-swept frequency test is performed by mounting an
accelerometer to an electrodynamic shaker system and smoothly
sweeping the input excitation from a low to high frequency at a con-
stant peak acceleration level. The output of the test accelerometer is
continuously compared to the output of the calibrated standard acceler-
ometer mounted in the shaker armature with the difference usually plot-
ted in decibels. The high end of the frequency sweep is generally
adequate to clearly identify the resonant response peak of the acceler-
ometer. As in the sinusoidal-discrete frequency test method, the phase
information may also be measured.
Ch A
Ch B
Electrodynamic
Shaker
Shaker head
Reference
Accelerometer
Shaker Armature
Accelerometer under
test
Sinusoidal Sweep
Generator:
Discrete or Sweep
Power Amplifier
Shaker Drive Power
Reference Output
Normalizing
Amplifier
Logarithmic Ratio
Amplifier
dB Ratio Plotter/ Recorder
Frequency Reference
dB versus Frequency
Oscilloscope
Absolute Position Detector (for absolute
calibrations)
Various Methods:
- Vibrating Wedge
- Interferometry
- Retroreflective
- Ronchi Rulings

.Electrodynamic Shaker Pseudo-Random Input:
In the pseudo-random shaker method of transfer function determination,
the accelerometer is mounted to an electrodynamic shaker system and a
pseudo-random input (broad-band noise) is presented to the sensor and the
sensor response to this input is recorded. The pseudo-random input is the
summation of a large number of sinusoids of arbitrary frequency, phase,
and having a defined a RMS (root-mean-square) amplitude. This calibra-
tion is performed by plotting the power spectral density of the output of the
sensor as a function of frequency.
Shock Machine and Hopkinson Bar
Short-duration transient inputs are usually made by mounting the acceler-
ometer to a shock machine or structural Òwave-guideÓ (Hopkinson Bar)
and applying inputs that approximate the unit-impulse function. The Hop-
kinson bar is a freely-suspended, simple cylindrical bar that is impacted at
one end where the compression wave travels through the bar to the free end
where the wave is converted to motion of the end plane to which the accel-
erometer is mounted. A reference accelerometer is generally mounted to
the free end of the bar to monitor the end-plane motion. In most cases, a
strain gage, affixed to the side of the bar, is used to provide a trigger signal
to a digital storage oscilloscope to allow capture of the wave form. The
unit-impulse function is not easily physically approximated due to the
impossibility of generating a pulse of unit amplitude and area
Centrifuge:
Static zero-hertz input calibrations are generally performed using a centri-
fuge system where the accelerometer to be tested is mounted at a known
radius on the centrifuge disc and the disc is rotated at a known speed. The
known radius and rotational speed produce a primary centripetal accelera-
tion loading that is directly traceable and is given by:

G = 2.842 X 10-5 X Radius(inches) X RPM2
Where G is acceleration in gravities.
FIGURE 1-10. The Hopkinson Bar:
It should be noted that two styles of centrifuges are in general use,
those that are designed to merely expose a product to a known acceler-
ation level and those that are of calibration quality. The calibration-
quality centrifuges possess a rotating disc to which the sensor is
mounted. At speed, the relatively-high mass of the disc builds a consid-
erable angular momentum vector, and, properly balanced, will show
small out-of-plane accelerations. The lower-cost centrifuge style has a
spin arm as opposed to a rotating disc and, due to aerodynamic wobble,
is not as stable as the disc-style centrifuge and is, therefore, not gener-
Pressure Source
Rapid-Acting Valve or Bursting Diaphragm
Projectile (or
Manually-Impacted)
Suspension
Axial Strain Gage (used to
trigger Oscilloscope or for absolute
measurement of compression wave
strain)
Test AccelerometerReference Accelerometer
Ch A
Ch B
Strain Gage
Conditioner
Trigger
Oscilloscope

ally suitable for calibration purposes. Sensor phase response is not measur-
able when static inputs are applied to the sensor.
FIGURE 1-11. The Centrifuge:
Load Cells:
The transfer function of a strain-based load cell may be determined by one of sev-
eral methods as follows:
Spin Axis
Radius
Accelerometer
Slip rings
Brushes
Applied Acceleration
Sensor Power
Voltmeter
Variable-Speed Drive Motor
Optical Encoder or Tachogenerator
Frequency counter
(RPM)
Drive Motor Controller

Deadweight:
The most common method of calibrating the low-range strain gage-
based load cell involves the application of a series of calibrated weights
to the sensor where the output is recorded and the sensitivity deter-
mined. The simple deadweight test system is shown as Figure 1-12.
The calibration accuracy of this type of test system is related directly to
the uncertainty associated with the weights used to apply load and to
the inherent friction of the load mechanism. The ÒguideÓ shown in Fig-
ure 1-12 is a bearing designed to allow minimum geometry variations
in the fixturing of the load cell relative to the test stand. This bearing
also presents frictional loss that will appear in the calibration results as
hysteresis. High performance dead-weight test systems typically utilize
low-friction knife edges in V-grooves, bell cranks and self-aligning
spherical bearings to reduce fixture friction, to accommodate small
nonparallelisms in test apparatus and to minimize geometry variations
in the application of loads from sensor-to-sensor and test-to-test. It
should be recognized that geometry variations of the test system in
response to increasing loads resulting from structural bending of the
test system can limit the calibration accuracy of the test system.
FIGURE 1-12. Deadweight Load Testing Apparatus:
Weights
Load Cell
Guide

Deadweight test apparatus possess the important inherent ability of apply-
ing exactly the same loads to the sensor being calibrated regardless of
whether loads are increasing or decreasing. Insight into the value of this
inherent advantage of deadweight calibration systems follows in the dis-
cussion of hydraulic test systems.
Hydraulic
In higher-load-range calibrations, a reference load cell is installed into the
load path in series with the device to be calibrated where the assembly is
hydraulically loaded and the output of the test load cell is compared to the
output of the reference load cell. Calibrations of this type are known as
comparison calibrations as the sensor under test is being compared to the
reference load cell. Moorehouse Instruments of Pennsylvania manufactures
load frames of this type. As one might suspect, the precision of this type of
calibration is limited by the precision of the reference load cell otherwise
known as the Òtransfer standardÓ. A calibration showing zero nonlinearity
and hysteresis simply means that no substantial difference exists between
the load cell being calibrated and the reference load cell. Such results do
not imply zero error. Typically, the reference load cell is chosen to show
between three and ten times lower nonlinearity and hysteresis errors than
the sensor being calibrated. Accordingly, to calibrate a load cell to ±.1% of
full scale output (FSO) nonlinearity and hysteresis, the reference load cell
should show between ±.03% and ±.01% FSO nonlinearity and hysteresis.
To achieve traceable high performance results with this type of apparatus,
one must consider that the loads applied will decay with respect to time due
to seal leakage around the hydraulic cylinder piston seal. High pressure
hydraulic lines provided with this type of system are often elastomeric in
nature and it is common to change to stainless steel tubing thereby reduc-
ing the applied force decay rate due to hydraulic line expansion. Often,
high speed digital data acquisition systems are used to collect many sam-
ples of both the transfer standard and the sensor under test at each input
load level where this data is averaged. This has the effect of averaging
noise to zero and reducing the uncertainty due to load decay.

The hysteresis in the output of any sensor is the difference between the
output of the sensor when inputs are monotonically increased and the
output of the sensor when inputs are monotonically decreased. Industry
standard procedures typically require that loads be applied in 11, 20%
of full scale input increments/decrements ie: 0%, 20%, 40%, 60%,
80%, 100%, 80%, 60%, 40%, 20%, 0% of input. The objective in deter-
mining hysteresis loss is to find the magnitude of input where the max-
imum difference exists between the Òup-loadÓ response and the Òdown-
loadÓ response. This is not trivial in the case of high performance cali-
brations performed on comparison calibration systems due to the
impossibility of exactly repeating the up-load points on the unload por-
tion of the test. To alleviate this problem it is recommended that the un-
load response be characterized with a three-coefficient polynomial
curve fit and the outputs then computed for the average inputs achieved
on the up-load line. In this way, true hysteresis can be measured.
FIGURE 1-11b: The Hydraulic Tester, Transfer Calibration:
Reference Sensor (Proving Ring
or Transfer Standard)
Hydraulic Pump
Hydraulic Ram
Location of Sensor for
Compression Loading
Location of Sensor for
Tension Loading
Changeable Spacer
to accommodate
various sensor styles

Phase information is not available in this type of static testing. In the case
of the piezoelectric load cell, very long time constants of the output do per-
mit static calibration within defined precision limits.
Gravimetric
The ÒdrivenÓ gravimetric method of calibration utilizes an inertial mass
mounted to the load cell and driven by a shaker system where the fre-
quency is either set to discrete points or swept over a defined frequency
band and the magnitude and phase data are recorded. By measuring the
applied acceleration with an internally-mounted reference accelerometer,
and in knowing the mass of the inertial weight, the applied force may be
calculated by the relationship F = MA. The free-fall gravimetric method
uses a load cell as the reference sensor to calibrate an accelerometer
mounted on the opposite side of the inertial mass. The gravimetric method
is reciprocal in nature, as will be discussed, but is normally limited to ± 1%
precision which is generally adequate for transfer function determination
but inadequate for high-precision calibration. Both the free-fall gravimetric
and driven systems are depicted in Figure 1-13.
FIGURE 1-13. The Gravimetric Test System:
Electrodynamic
Shaker
Mass
Load Cell
Reference Accelerometer
F = MA
Elastic Suspension:
Manually Deflected
to create momentary Free-Fall
conditions
Post
Beam
Damper
Base
Foam
Load Cell
Test Mass
Accelerometer
M
Tie rods
Guide Tube

As in accelerometer calibration, the gravimetric method may be used
with broadband noise inputs, sinusoidal or discrete-frequency inputs
generated by an electrodynamic shaker to determine the load cell trans-
fer function.
Pressure Transducers:
Pressure sensors may be calibrated by many different methods for the purpose
of determining the sensor transfer function where some of these methods are
discussed as follows:
Deadweight Pressure
Pressure transducers are statically calibrated by means of the dead-
weight tester where calibrated masses are sequentially added to a plat-
form that is supported by a precision piston. The pressure is increased
manually or automatically, by either a ram or controlled-pressure regu-
lator, to the point where the pressure, generated by the calibrated
weights acting over the area of the precision piston, is balanced and the
piston is free-floating. The precision of this type of calibration system
relates to the area difference between the piston and bore within which
it moves. Traceable uncertainties of as low as .0004% of applied pres-
sure are achievable with deadweight calibrators where the piston and
bore are precision lapped to minimize area differences. As the area dif-
ference between the piston and bore results in a leakage path, it is com-
mon for the piston to slowly settle where the pressure will ultimately
decay to zero. Operating procedures for deadweight pressure calibra-
tors will often specify that the weights be manually rotated to distribute
the fluid film and minimize friction while data is collected at each input
level. The sensor to be calibrated is in communication with the bal-
anced pressure reservoir and the sensor output is recorded. As with all
static-calibration methods, phase data is not available.

FIGURE 1-14. The Deadweight Pressure Test System:
Pneumatic Shock Tube:
Pressure sensors may be dynamically calibrated by means of the shock
tube where a pressurized reservoir of gas is maintained at a known pre-set
level and is separated from a long tubular section maintained at another
known pressure by a rapid-acting valve or diaphragm. When the valve is
opened or the diaphragm is deliberately punctured, the reservoir vents into
the shock tube tubular section very rapidly. A very high-frequency-capable
reference piezoelectric pressure sensor is normally mounted close to the
sensor under test and the outputs are compared to determine the response
of the sensor to the pressure shock wave. It is important that the reference
sensor have a frequency response that is much higher than that of the test
sensor and sufficiently high to prevent resonant ringing. The shock tube
apparatus can be used with various gases and Prichard pressure levels to
achieve microsecond rise times and high pressure impulse loadings. Elec-
trically-driven shock tubes produce a high pressure shock wave resulting
from an electrical spark discharged across a gap between electrodes. Pres-
sure amplitudes of 1,000 psi or more are not uncommon. The impulse
response of the sensor under such dynamic conditions may be used to
determine the transfer function of the sensor including phase data.
Calibrated
Weights
Sensor under test
Free-floating piston
Area = A inches2
Pressurizing Ram
When the Free-floating
piston is supported by
the hydraulic column.
the pressure = W/A

FIGURE 1-15. The High-Pressure Shock Tube:
Acoustic Horn:
The transfer function of low-pressure range sensors may also be deter-
mined by use of the acoustic horn method where an acoustic generator
(horn) is mounted into a procreated chamber and the horn output is
swept over the frequency range of interest. The maximum pressures
generated by the acoustic horn system are generally less than 1 psi
(approximately 170-dB sound pressure level).
FIGURE 1-16. The Acoustic Horn:
Rapidly-Valved Systems:
Driver Section
Aluminum Diaphragm
Test Section
Helium Supply
Sensor Under Test
Time-of-arrival sensor
Shock Pulse
Piercing Mechanism
High Pressure Low Pressure
Acoustic
Generator Sensor under Test
Power Amplifier
Sweep Generator
Oscillator
Reference Sensor
Pressure Source

Rapid-opening valve systems such as the Aronson shockless pressure cali-
brator in which two chambers, at different pressure pre-charge levels, are
separated by a low-mass poppet valve supported by a spring and provided
with a long slender actuating rod. The actuating force rod is provided with
a platen, upon which weights are dropped, rapidly opening the poppet
valve. Once opened, the larger and higher pressure chamber rapidly equili-
brates with the smaller volume test chamber. To facilitate observation of
the generated output of the sensor, a piezoelectric accelerometer or load
cell is normally used to detect the shock of impact of the weight against the
platen where this output is used for oscilloscope-triggering purposes. The
pressure sensor under test is mounted into the smaller of the two chambers
in order that the larger chamber can pressurize the smaller volume rapidly.
Chamber volume ratios of 2/1000 are common for very rapid rise-time test-
ing. The rapid-acting valve systems are able to generate single positive- or
negative-going pressure steps that are useful in transfer function and phase
measurement.
FIGURE 1-17. The Aronson Shockless Pressure Step Generator:
Accelerometer trigger
Mass
Sensor under test
Poppet Valve Head
Poppet Volume
(PPoppet)
Compression Spring
Housing Support
Pressure Reservoir
(Preservoir)
Poppet lifter and
lockdown tab
Guide Tube
Impact Plate
Poppet Valve Stem
O-ring seal
PPoppet

Hydraulic Impulse:
The hydraulic impulse systems in common use involve the dropping of
a calibrated weight from a controlled height onto a moveable precision
piston which compresses a column of hydraulic fluid. Two forms of
this calibrator are in use; one utilizes a reference pressure sensor usu-
ally of the tourmaline piezoelectric type; the other computes the gener-
ated pressure based upon the mass of the weight, local gravity, velocity,
and piston area. Peak pressure magnitudes of between 100 and 20,000
psi are achievable with rise times on the order of 3 milliseconds and
showing pulse durations on the order of 6 milliseconds.
FIGURE 1-18. The Hydraulic Impulse Test System:
Other Methods:
Other methods, not so commonly used for the determination of the
dynamic response of sensors, are many and varied. Modified air com-
9-lb mass
Piston
Hydraulic Reservoir
Reference Sensor
(Piezoelectric tourmaline)
Test sensor
Typical Range:
0 to 20,000 psi
Rise Time:
3 milliseconds
Pulse Duration:
6-8 millisecondsImpact Pad:
Modifies Pulse duration
Drop Tube

pressors (Figure 1-19) can be used to generate repeatable sinusoidal pres-
sure waves where the test pressure sensor output is compared with output
of a reference sensor. The modified air compressor method is typically
used at a fixed frequency depending upon the design of the compressor
motor.
The vibrating fluidic-column method (Figure 1-20) is also based upon the
comparison of outputs between a reference sensor and the sensor under
test. This method involves the use of a fluidic column that supports a
moveable piston/mass that is then mounted to an electrodynamic shaker
system and is driven to vibrate at a selection of frequencies.
FIGURE 1-19. The Sinusoidal Pressure Source:
FIGURE 1-20. The Hydraulic-Column Pressure Source:
Pressure Relief Valve
Pressure Transfer StandardPressure sensor
under test
Compressor
Shaker
Free Mass
Pressure Transfer
Standard
Sensor under
Test
Hydraulic column

The Fourier Series
The Fourier series is a mathematical expression that is used to define any time-
varying function as an infinite sum of the multiples of the fundamental fre-
quency that comprise the time function. The Fourier series is given by:
(EQ 1-7)
The Fourier series is not just the mathematical expression of a time function, it
is the time function. Another way of regarding the Fourier Series is to consider
that every frequency component required to define a specific time function can
be regarded as a physically real subcomponent of the time function. The value
of k in the above expression is equal to the number of sinusoids, or harmonics,
summed together to approximate the time function that is being modeled,
where higher values of k yield closer approximations to the time function.
The key issue concerning the Fourier Series expression of a time-varying sig-
nal is that the transfer function of the sensor and measurement system will act
on each of the harmonic components as if these had been input separately. The
result of this logic is that the resultant time-varying output is the summation, or
superposition, of each of the outputs from the sensor or measurement system
corresponding to each of the Fourier input components. Figure 1-21 shows the
waveform that will result if three simple Fourier components are summed. It
can be seen intuitively that any complex waveform can be created by the suc-
cessive summing of the Fourier subcomponents. Correspondingly, each of the
Fourier components of the input will be phase-shifted by a different phase
angle depending upon the phase relationship of the measurement system trans-
fer function. The end result is that the output from the measurement system can
be both amplitude- and phase-distorted unless care is taken to assess the fre-
quency of the highest expected frequency component, and then to assess the
q t( )
a0
2
----- ak kw1cos t bk ksin w1t+( )
k 1=
¥
å+=

amplitude and phase characteristics of the measurement system transfer function.
An infinitely-sharp discontinuity in the input waveform implies that the measurand
contains an infinite sum of Fourier components. The square-wave or triangular-
pulse input wave forms, therefore, contain an infinite series of Fourier compo-
nents. The existence of a large number of Fourier input components implies that
there will exist, with a high probability, frequency inputs at or near the resonance
frequency of the sensor. In the undamped sensor case, these inputs can result in
overhanging of the sensor and/or resonant ringing of the output. In general, very
little energy usually exists in the frequency components of the input that are at, or
greater than, the tenth harmonic or tenth Fourier component.
FIGURE 1-21. A Complex Input Waveform:
0
1
.5
-.5
-1
Time
Magnitude Fourier Component 1
Fourier Component 3
0
.5
1
1.5
2
Fourier Sum = Component 1 + Component 2 + Component 3
Fourier Component 2
Time
Net Waveform

Figures 1-22 and 1-23 illustrate how even small parameter inputs, at or near
resonance, can produce high-level ringing of the sensor output.
FIGURE 1-22. A Complex Input to the Sensor Transfer Function:
Time
Magnitude
0g
+1g
-1g
+10 g
-10 g
+11 g
-11 g
TimeSummed Waveform
Component 1: ± 10g at 100 Hz
Component 2: ± 1g at 1000 Hz
Magnitude
dB
+40 dB
0 dB
Log Frequency
1000 Hz
Component 2
100 Hz
Component 1
Sensor Transfer Function (Magnitude)
Resonance Frequency = 1000Hz
Calibration at 100 Hz = 1mV/g
Sensor Range = 10 g
Sensor Overrange = 100g
Calibration Frequency
Resonant Peak
Physical Input to
Accelerometer

The 100-Hz (±10 g peak) input may be the input of interest where the 1,000-Hz
(±1g peak) input is of secondary interest and may be due to bearing chatter or gear-
meshing components. In this case, the 10-g range accelerometer shows a sensitiv-
ity of 1 mV/g, as calibrated at 100 Hz, with an undamped resonant frequency of
1,000 Hz showing a maximum response of +40 db (= 20 log Vo/Vcal, where Vcal is
the calibrated sensitivity of the sensor at a much lower frequency within the ÒflatÓ
frequency response range of the sensor). It would be reasonable for the accelerom-
eter to show a maximum overhang tolerance of ±100 g before mechanical damage
is suffered. The net accelerometer output is as shown in Figure 1-23.
As shown, the 10-g input component will produce a ±10 millivolt output compo-
nent at 100 Hz, however, the situation is dramatically different for the ±1g, 1000-
Hz component. Since the 1,000-Hz component exists at the resonant frequency of
the device, the accelerometer will mechanically amplify the ±1g input by the +40
dB mechanical gain at resonance, which, when multiplied by the 1mV/g sensitiv-
ity, will result in a ±10 mV output component at 1,000 Hz as shown. The net out-
put waveform shows massive ringing at 1000 Hz where the information of interest
is marginally evident as the modulated envelope of the response shown. Frequency
filtering of the output of the sensor would help to filter out the 1,000-Hz compo-
nent, however, the 100-g maximum over-range of the accelerometer means that the
subject sensor is, in all likelihood, broken! If the maximum allowable overrange is
100 g, this figure is given as being valid within the useful frequency range of the
sensor and not at resonance. The ±1 g input at resonance could, therefore, be quite
sufficient to cause overrange destruction of the device. The moral of this story is
that we must be aware of all components present within an input signal and assess
the potential impact of each component with respect to the transfer function of the
sensor.
The Fourier Series is not just a Òmathematically-elegantÓ method of expressing the input
waveform but, more importantly, a valuable means of determining the power spectral den-
sity (the energy that exists within each of the Fourier components), and of visualizing the
components of the input and output separately. -JP

FIGURE 1-23. The Net Accelerometer Output:
The Fast Fourier Transform algorithm (FFT) is a method by which the number
of computations required to compute the Fourier series is dramatically reduced
from N2
calculations to Nlog2N calculations, allowing high-speed computers
to rapidly compute the Fourier Spectrum and plot it as the power spectral den-
sity. The power spectral density is the plot of the relative magnitudes of each of
the Fourier components as a function of frequency.
+10 mV
mV
-10 mV
Accelerometer Output Component at 100Hz
Peak Magnitude = 1mV/g X 10 g = ± 10 mV
Time
+100 mV
mV
-100 mV
Time
Accelerometer Output Component at 1,000Hz
Peak Magnitude = 1mV/g X 1g X 100 = ± 100mV
+110 mV
- 90 mV
-110 mV
+100 mV
+90 mV
0 mV
-100 mV
Net Accelerometer Output
Time
0 mV

Zeroth-, First-, and Second-Order System DeÞnitions
The definition of the order of a sensor is basically the generalized classification of
the transfer function of the sensor.
The Zero-Order Sensor:
The zero-order sensor is sometimes referred to as the ÒZerothÓ order sys-
tem, where the output of the sensor is a linear function of the input where,
in theory at least, this relationship remains valid at all frequencies. The
zero-order sensor structure is not realizable in practice; however, some sen-
sors will behave very much like zero-order sensors over limited band-
widths in frequency. The zero-order system is shown in Figure 1-22 and
characterized as follows:
Fout(t) = K Fin(t) (EQ 1-8)
Where: Fout(t) = The output function with respect to time.
Fin(t) = The input function with respect to time.
K = The proportionality constant.
FIGURE 1-24. The Zero-Order Response:.
(LVDT
Response)
Input
Output
Time
Input/Output
Magnitude

The Linear-Variable-Differential Transformer (LVDT)-based sensor
closely approximates the zeroth-order sensor at low-to-moderate fre-
quencies
The First-Order Sensor:
A first-order system response is the solution to the first-order differential equa-
tion as follows:
(EQ 1-9)
or
(EQ 1-10)
Where: t = k1/k0 , The sensor time constant
K = b0/k0, The static sensitivity.
If d /dT is replaced by the differential operator ÒDÓ, equation 1-10 becomes:
Since Fout/Fin is defined as the Transfer Function (H):
(EQ 1-11)
Implies:
(EQ 1-12)
Where the magnitude of this transfer function will equal:
k1
Td
dFout
k0Fout+ b0Fin=
t
Td
dFout
è ø
æ ö Fout+ KFin=
KFin 1 tD+( )Fout=
H
K
1 tD+( )
---------------------
Fout
Fin
----------= =
H w( )
Fout jw( )
Fin jw( )
----------------------
K
1 jwt+( )
------------------------= =

(EQ 1-13)
And where the phase response is given by:
(EQ 1-14)
An example of the first-order response is the response of a thermocouple as
shown in Figure 1-25. The maximum phase shift of the first-order sensor
is -90 degrees as w approaches infinity. The first-order response is also
referred as the Òsingle-poleÓ response.
FIGURE 1-25. The First-Order Step-Input Response:
The Second-Order Sensor:
The second-order system response that is the solution to the second-order differen-
tial equation:
(EQ 1-15)
If w0 is defined as:
H w( )
K
1 w
2
t
2
+
-------------------------=
j w( ) wt( )atan=
Input Output
Time
(Thermocouple
Response)
t
t =
Final Value
e
Where e = 2.718281/e
Input/Output
Magnitude
k2
t
2
2
d
d Fout
k1
td
dFout
k0Fout+ + b0Fin=

(EQ 1-16)
where: w0 is the natural frequency and z is the damping factor.
Equation 1-15 becomes:
(EQ 1-17)
Where: K = The static sensitivity of the sensor.
The second order transfer function in the frequency domain can be shown to
equal:
(EQ 1-18)
Where the magnitude of this function will equal:
(EQ 1-19)
and where the phase response is given by:
(EQ 1-20)
Examples of the second-order response, Figure 1-26, are abundant. All
spring-member-based sensors will show a second-order response, up to
the first resonance frequency, with varying degrees of damping. Note
k0
k2
----- and z
k1
2 k0k2
------------------=
1
w0
2
------
t
2
d
dk0 2z
w0
td
dk0
-------------- k0+ + KFin=
H w( )
K
1
w
w0
------
è ø
æ ö
2
2 jz
w
w0
------
è ø
æ ö+Ð
è ø
æ ö
--------------------------------------------------------=
H w( )
K
1
w
w0
------
è ø
æ ö
2
Ð
è ø
æ ö
2 2zw
w0
----------
è ø
æ ö
2
+
------------------------------------------------------------=
j w( ) 2
z
w
w0
------
w0
w
------Ð
è ø
æ ö
------------------------atan=

that the phase variation with frequency will equal -90 degrees at the natural
frequency (wn) and will approach -180 degrees as the frequency
approaches infinity. The second-order response is sometimes also referred
to as the Ò2-poleÓ response or the simple Òspring/mass/damperÓ system
response.
FIGURE 1-26. The Second-Order System Step-Input Response:
Input/Output
Amplitude
wn decreasing
Time
Damping Factor ( z ) = .7
The Second-Order Response to a Step-Input
(The effect of Natural Frequency)
Input
Output
Input/ Output
Magnitude
2
1.5
1
.5
Input
Output
Time0 seconds
Damping Factor Increasing
The Second-Order Response to a Step-Input
(The effect of Increasing Damping Factor)
z = .1
z = .3
z = .5

The Second-Order System
An accelerometer having an internal spring member that responded to acceler-
ation inputs in two or more dimensions simultaneously would be of little use as
we would be unable to define a dimension that produced any given response.
This is why an accelerometer spring member is designed to be flexible
(responsive) in one dimension and stiff (unresponsive) in all other dimensions.
Our goal would be to define the measurement axis precisely to produce a sin-
gle-degree-of-freedom structure. The single-degree-of-freedom spring-mem-
ber-based sensor class invariably show second-order responses.
It should be noted that no mechanical system is able to perfectly imitate
the single-degree-of-freedom system and will simultaneously support
motion, to some extent, in other dimensions. Even well-designed sen-
sor structures will show some degree of ÒtransverseÓ responsivity. The
second-order system is defined as a system where the transfer function
is the solution to a second-order differential equation. The second-order
response defines the response of any mechanical system possessing
mass, where the restoring forces acting upon this mass are provided by
a spring constant, and resistance to motion is provided by mechanical
damping. The spring constant is quantified by the stiffness of the spring
member. The damping forces acting upon the mass may result from the
intrinsic structural damping of the material that comprises the spring
member, or may derive from fluidic- or gas-damping provided in the
sensor design. In the case of the piezoelectric element, the spring rate
of the piezoelectric element is high and the internal structural damping
of the element is low yielding a close-to-zero-damped second-order
response. In the case of the seismometer, the spring restoring forces are
generally provided by discrete spring elements of relatively low spring
rate, yielding fundamentally low resonance frequencies.

Seismic Instruments:
The term ÒSeismic InstrumentsÓ is used to refer to the entire class of instru-
ments, the dynamic behavior of which can be derived from the sum of the
forces acting upon the internal seismic element, where the seismic element
is constrained to move primarily in one dimension (single-degree-of-free-
dom system). The seismic mass in low g range cantilever beam accelerom-
eters may consist of a concentrated mass at one end of the beam or can be
the distributed mass of the beam alone in very high range devices. In the
case of the pressure sensor diaphragm, the seismic mass is the distributed
mass of the diaphragm, and the restoring spring force is provided by the
spring constant of the diaphragm material. Load transducers are identical to
the diaphragm pressure sensor from the seismic viewpoint, with the spring
member configured to produce bending in response to direct-force loading,
as opposed, to the distributed-force loading that pressure environments
present to the diaphragm pressure sensor.
NewtonÕs Law, F = MA, is the fundamental relationship which all seismic
instruments obey, in that, the net force exerted on a mass will result in the
acceleration or deceleration of the mass with resulting displacement. All
seismic instruments show primarily second-order responses. It should also
be noted that all sensor structures will show multiple higher-order reso-
nances as the various internal structures within the sensor possess different
masses, geometries, and stiffness, and will support resonances at different
frequencies. The objective of good sensor design is to ensure that these
internal structures do not support resonant responses within the useful fre-
quency response range of the sensor.
Referencing Figure 1-27, a seismic system is modeled showing the seismic
mass and the spring as well as damping forces that will act upon this mass.
This mechanical model is excited by the forcing function qi acting upon the
mass where the resulting displacement of the mass is defined as qo.

FIGURE 1-27. The Second-Order System Mechanical Model:
The electrical output of the strain-based sensor is a direct and propor-
tional function of the spring member displacement. For this reason, the
function qo(t) is taken as being proportional to the output that would
result from the second-order sensor structure. The velocity of the mass
is defined as dqo(t)/dt and the acceleration of the mass is defined as
d2
qo(t)/dt2
. The force that the spring will exert upon on the mass will be
opposite to the forcing input and will equal -K qo(t), where K is the
spring constant having units of force-per-unit-displacement. The
mechanical damper will also provide a force that is opposite to the forc-
ing input as a function of the mass velocity and will equal -Rdqo(t)/dt,
where, R is in units of force-per-unit of mass velocity.
Since Force = mass X acceleration or F = MA, then:
MA = S of all forces acting upon the mass
MA = Input force - Spring force - Damping force
(EQ 1-21)
Mass
Foundation
qi Input
RK Lbf/In Lbf/ft/sec
R proportional to velocity
qo Output
Spring Damper
MA qi t( ) Kq0 t( ) R
td
d
q0 t( )ÐÐ=

Where:
(EQ 1-22)
Substituting (1-22) for MA in (1-21) and solving for qi(t) yields:
(EQ 1-23)
Converting from the time domain into the frequency domain using the rela-
tionship qi(t) = ejwt as representative of a sinusoidal input, and knowing
that the output will equal:
(EQ 1-24)
(where H(w) is the system transfer function)
Then, the output must be of the form:
(EQ 1-25)
H(w) also represents phase as well as magnitude of response. Having
established this equality, differentiation of equation (1-25) yields the fur-
ther equalities:
(EQ 1-26)
and
(EQ 1-27)
Substitution of equations (1-24), (1-25), (1-26), and (1-27) into equation
(1-23) and solving for H(w) yields:
MA M
t
2
2
d
d
q0 t( ) F= =
qi t( ) M
t
2
2
d
d
q0 t( ) R
td
d
q0 t( ) Kq0 t( )+ +=
q0 t( ) H w( )qi t( )=
q0 t( ) H w( )e
jwt
=
td
d
q0 t( ) H w( ) jwe
jwt
=
t
2
2
d
d
q0 t( ) H w( )w
2
e
jwt
Ð=

(EQ 1-28)
and
(EQ 1-29)
Where:
j(w) = The phase relationship between the output and the input forcing
function.
(EQ 1-30)
(EQ 1-31)
The implications of equations (1-28) and (1-29) are shown graphically
in Figure 1-28 and are stated as follows:
1. As w approaches zero radians/second: The second-order system is stiffness-dominated or
Òspring-controlledÓ where the transfer function H(w) approaches unity in value.
2. As w approaches infinity radians/second: The second-order system is mass-dominated
where the reaction of the mass to the input approaches zero, or the value of the transfer
function value itself approaches zero.
3. As w approaches infinity: The phase angle between the input and the output approaches
180 degrees.
4. When the phase angle between the input and the output equals 90 degrees, w = w0 which is
termed the natural frequency of the system.
5. If this system possessed zero damping (an impossibility), the value of the transfer function
will approach infinity at the natural frequency of the system.
H w( )
1
1
w
w0
------
è ø
æ ö
2
Ð
è ø
æ ö
2
2z
w
w0
------
è ø
æ ö
2
+
---------------------------------------------------------------=
j w( )
2z
w
w0
------
w0
w
------Ð
è ø
æ ö
------------------------atan=
w0
K
M
-----=
Radians/Second
Where: w0 2p f 0=
z
RM
2 MK
-----------------=

6. When the phase angle between the input and the output equals 90 degrees, the natural frequency
of the system may be determined and from this quantity, the stiffness and mass-ratio of the sys-
tem can be directly determined by means of equation (1-30)
FIGURE 1-28. The Second-Order System Magnitude and Phase Response:
If the second-order system possessed a damping factor that approaches
zero in value, the flat frequency response range of the second-order sensor
1
2
3
4
.5
.125
.5 1 2 3.2
Magnitude of
Response
The Second-Order Magnitude Response
z = .1
z = .3
z = .5
z = .7z = 1z = 2z = 5
w/w0
log
0
-30
-60
-90
-120
-150
-180
Phase
Angle
z = 5
z = 2z = .7z = .1

Book File09_05_03

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