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The Art of Practical andPrecise Strain BasedMeasurement- Second EditionBy James G. Pierson090103Pierson & Associates © LLCi

Revision Date: September 1, 1999Copyright © 1992 by Pierson & Associates LLC. All rightsreserved. Printed in the United States of America, Canada and theUnited Kingdom. Except as permitted under the United StatesCopyright Act of 1976, no part of this publication may bereproduced or distributed in any form or by any means, or stored ina database retrieval system, without the written permission of thepublisher.ISBN 1-895976-00-6Disclaimer: The enclosed information is given to the user as correctto the best of the authorÕs knowledge. The use of any or all of theinformation enclosed herein is entirely at the risk of the user. Inacceptance of this publication, the user hereby idemnifies PiersonAssociates with regards to the information and use of any and allinformation contained herein. This work has been prepared byJames G. Pierson of Pierson & Associates LLC. and is continuouslyrevised and updated. Any and all comments and suggestions aremost welcome and appreciated. With your input, the cause ofprecision measurement can continuously evolve and improve! Mr.Pierson may be contacted at the following address:Attention: James PiersonPierson & Associates LLC7A Sanders RoadRockaway, NJ07866-2008INTERNET: http://www.PiersonOnline.comDial: 800-565-6075 USA and Canadaii

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

Preface to Second Edition:Given a perfectionist disposition, a project like this handbook is anever-ending task, a source of happiness and a source of frustrationall wrapped up in 850 pages of type. Reviewing material that onewrote years prior can be a frightening experience, the usual reactionto a passage being ÒI actually wrote that?Ó ÒClunkyÓ is being far tookind a way of describing some passages. As time passes and wegrow in many different ways, maturity brings new insights that wesimply couldnÕt see during the first pass. The second edition rightsa bunch of wrongs, adds concepts that should have existed allalong, clarifies clunky wording, offers new insights and revisesother concepts to reflect current thinking in the measurementsciences.We have heard it stated that the world is changing to a knowledgebased economy. We have also heard it stated that knowledge ispower. Knowledge regarding the performance of the machines orproducts that our companies fabricate empowers us to make correctengineering and business decisions to best our competition. Thereal danger in physical measurement occurs when we mistakeperception for knowledge (truth) and make decisions based uponwhat we perceive to be true. Acting upon perceptions, not basedupon the truth; we chase ghosts with success always eluding us.When perceptions equate to the truth we have earned the power tosucceed.Without you the reader, of what value is any written work? I thankyou in advance for the time you spend with this material. I sincerelyhope 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 3rdEdition in 2005!) - Jim Pierson September 1, 1999iv

Table of ContentsBy James G. Pierson090103 Pierson & Associates LLCThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999i

ContentsCHAPTER 1Strain-Based Measurement Introduction:Applications Note 1-1: Introduction 1-1Applications Note 1-2:What is Applications Engineering? 1-6Applications Note 1-3The Measuring System Source of Information: The Sensor 1-8Applications Note 1-4:Linear versus Nonlinear Sensors 1-12Applications Note 1-5:The Strain-Based Sensor DeÞnition 1-16Applications Note 1-6:The Measurement System Transfer Function 1-19Applications Note 1-7:The Fourier Series 1-40Applications Note 1-8:Zeroth-, First-, and Second-Order System DeÞnitions 1-46Applications Note 1-9:The Second Order System 1-51Applications Note 1-10:Fluidic-, Gas-, and Structural-Damping of Second-Order Sensor Structures 1-58Applications Note 1-11:The Strain-Gaged Cantilevered Beam 1-67Applications Note 1-12:Pressure References 1-73Applications Note 1-13:The Diaphragm Strain-Gaged Pressure SensorGeneral Discussion: 1-80CHAPTER 2Data Quality, The Environment and Physical ConstraintsApplications Note 2-1:Introduction to The Statement of Objectives 2-1Applications Note 2-2:The Environment Assessment 2-17Applications Note 2-3:The ÒMicroÓ and ÒMacroÓ Perspectives 2-25Applications Note 2-4:The DeÞnition of, and Sources of, Noise 2-29Applications Note 2-5:The Statement of Physical Constraints 2-37Applications Note 2-6:Static and Dynamic Measurement Environments 2-46ii The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

ContentsApplications Note 4-10:Lead-Wire Conductors 4-71Applications Note 4-11:Lead-Wire Insulations 4-57Applications Note 4-12:Strain Bridge Wiring and Lead Wire Effects 4-80Applications Note 4-13:Fatigue and the Metal-Foil Strain Gage 4-95Applications Note 4-14:Fatigue and the Piezoresistive Strain Gage 4-97Applications Note 4-15:The Metallic-Foil Comparison To the Piezoresistive Strain Gage 4-101Applications Note 4-16:The Implications of Sensor Size and Mass 4-103Applications Note 4-17:Real Versus Pseudo Calibration 4-111Applications Note 4-18:Resistance Calibration General Discussion 4-112Applications Note 4-19:The Piezoresistive Strain-Gaged Sensor and Resistance Calibration 4-121CHAPTER 5Error ComponentsApplications Note 5-1:Zero or Null-Bias Stability 5-1Applications Note 5-2:Creep 5-12Applications Note 5-3:Hysteresis DeÞnition 5-18Applications Note 5-4:Linearity DeÞnition 5-27Applications Note 5-5:Combined Nonlinearity and Hysteresis 5-30Applications Note 5-6:Resolution 5-33Applications Note 5-7:Nonrepeatability and Reproducibility 5-34Applications Note 5-8:Acceleration Sensitivity of Strain-Based Pressure Sensors 5-38Applications Note 5-9:Transverse Sensitivity Considerations for the Cantilevered-Beam Accelerometer 5-40Applications Note 5-10:Uncertainty 5-47iv The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

ContentsApplications Note 5-11:Performance Parameter Distributions 5-87Applications Note 5-12:Cost Relationships 5-93CHAPTER 6Mechanical Considerations in Sensor DesignApplications Note 6-1:Sensor Spring Member Materials and Mechanical Design 6-1Applications Note 6-2:Mounting-Strain Effects on Transducers 6-49Applications Note 6-3:Loading Surface Geometry 6-53Applications Note 6-4:The Nature of Epoxies 6-56CHAPTER 7Thermal CompensationApplications Note 7-1:Piezoresistive Strain-Bridge Thermal-Sensitivity Compensation 7-1Applications Note 7-2:Piezoresistive Strain-Gage Active Thermal-Sensitivity Compensation 7-13Applications Note 7-3:Piezoresistive Semiconductor Strain-Gaged Wheatstone Bridge Thermal-Zero Shift 7-16Applications Note 7-4:Considerations for the Quarter, Half and Fully-Active Piezoresistive Bridge 7-28Applications Note 7-5:Constant-Current Excitation for Piezoresistive Strain-Gaged Sensors (Simple) 7-31Applications Note 7-6:Dual-Tracking Constant-Current Excitation for Piezoresistive Sensors 7-36Applications Note 7-7:General Notes Regarding Piezoresistive Strain-Gaged Sensor Thermal Performance 7-39Applications Note 7-8:Metallic Strain-Gage Thermal-Sensitivity Compensation 7-42Applications Note 7-9:Metallic-Strain-Gage Thermal-Zero Compensation 7-50Applications Note 7-10:Transient Thermal Compensation 7-54Applications Note 7-11:Thermal Isolation, Cooling and Control of Sensor Structures for Minimized Thermal Error 7-58Applications Note 7-12:Thermal Design Considerations 7-66The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999v

ContentsCHAPTER 8Electrical ConsiderationsApplications Note 8-1:AmpliÞcation 8-1Applications Note 8-2:Zero-Trim Methods for the Wheatstone Bridge 8-27Applications Note 8-3:Dynamic and Static Excitation 8-32Applications Note 8-4:Grounding and Shielding Techniques for Strain-Gaged Sensors 8-38Applications Note 8-5:Failure-Mode Analysis of the Wheatstone Bridge 8-54Applications Note 8-6:Aliasing 8-72Applications Note 8-7:The Electrical-, Mechanical-, and Thermal-Time Domains 8-75CHAPTER 9PerformanceApplications Note 9-1:Knowledge-Based Error Correction 9-1Applications Note 9-2:Sensor Performance SpeciÞcation 9-11Applications Note 9-3:SpeciÞcation Considerations 9-25CHAPTER 10Calibration and TestApplications Note 10-1:Strain Gaged Accelerometer Calibration 10-1Applications Note 10-2:Gravimetric Calibration Methods, Acceleration and Force: 10-17Applications Note 10-3:Impulse-Hammer Test Methods (Piezoelectric) 10-24Applications Note 10-4:QualiÞcation and Acceptance Testing 10-27Applications Note 10-5:Military SpeciÞcation Environmental Test 10-29Applications Note 10-6:QualiÞcation Test: Acceleration 10-31Applications Note 10-7:Strain-Gaged Pressure Transducer Calibration 10-39vi The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

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

CHAPTER 1Strain-BasedMeasurementIntroduction:Applications Note 1-1: IntroductionThe Wheatstone Bridge strain-gaged sensor type is, by a large margin,the most popular transducer element in use. Unfortunately, thestrain-based sensor is also the most misapplied sensor type. By someestimates, as much as 80% of all new sensor purchases are made inthe act of replacing a prior sensor technology or type. This representsa massive waste of human effort as well as financial resources. Historically,the costs of these errors in judgement have been buriedwithin research and development budgets. The objective of this workis to show how we can work smarter, rather than harder, by gaininginsight into the simple physics of the strain-based sensor so thaterrors in physical measurement are minimized. When a measurementis made we are attempting to understand our product or process underspecific operating conditions. Based upon our measurements we perceivespecific values when the true value of a parameter may be quitedifferent. The difference between the truth and the perception is error.A ten percent error in the magnitude of peak strain experienced by acomponent in use implies two consequences; the fatigue life expect-The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-1

Strain-Based Measurement Introduction:ancy of the component could be reduced by a factor of up to two times, possiblydoubling warranty costs when the strain is underestimated, or we areshipping much more material than is necessary to support the design lifeexpectancy when the peak strain is overestimated. When strain is overestimated,vehicles become much more massive than is necessary to meet orexceed the design life for an intended use profile increasing vehicle mass,decreasing vehicle fuel efficiency, increasing the cost of vehicle structures andbraking systems, requiring higher capacity propulsive systems and increasingthe cost of safety systems in the event of collision. It has been estimated thatgreater 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 happensat 15% or more error. We live in the so called Òinformation ageÓ. It is impliedby this statement that we make engineering and business decisions based uponcorrect information. The business of measurement is to gain understanding of aproduct or a process not simply to collect numbers. Our mission is to minimizethe difference between the perception and the truth.Strain gages are configured within the sensor to provide a differential outputthat is in proportion to the applied physical parameter, whether this parameteris force, pressure, acceleration, or strain. This work is a generalized treatmentof the thermal, mechanical, and electrical behavior of real-world sensors andrefers, in most cases, to the physical input as the input parameter without constraintto any particular sensor type.Many exhaustive works are available that mathematically model virtually anysensor in the frequency domain and time domain. It is the authorÕs opinionthat, as engineers, we are capable of mathematically complicating any issuethat we choose, and, if complex models are required, we are able to researchthe required theory and assemble these models. The models used herein areentirely based upon the two principal relationships of OhmÕs Law (V = IR) andNewtonÕs Law (F = MA). In many instances, these simple relationships can beused to create mathematical representations of surprising fidelity even whencompared to the more complex mathematical models.CHAPTER 1-2 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-1: IntroductionEvery effort has been made in this work to avoid math-intensive models of sensorperformance. When possible, simpler and more understandable mathematicalmodels have been used rather than the more rigorous models in the interest of clarity.The focus of this work is the thermomechanical behavior of the generalizedsensor. The material discussed in the following pages is an ÒApplications CompendiumÓdetailing many of the very simple phenomena that have frustrated many anengineer. To an instrument engineer, the embarrassment of not having consideredthis or that phenomenon in a particular measurement environment, is a familiarfeeling. We are spared this embarrassment when the ÒtruthÓ remains unknown.Indeed, in the mind of the purist measurement statistician, the ÒtruthÓ remains foreverunknowable as even national reference standards possess finite uncertaintylimitations.Much of the information contained in this work will appear to be of the simple ÒIknew thatÓ kind. It is written to be so. After having been frustrated by poor dataquality in the past, how many times have you slapped your forehead and exclaimedÒI should have known better!Ó? The parameter that very likely was the source offrustration, was either not properly specified, absent altogether, or specified in anambiguous way. In many cases, the manufacturers themselves truly have no ideahow their devices will or wonÕt function in specialized environments or in uniquecombinations of environments. In the end, as engineers, it is our responsibility todefine to the sensor manufacturer, our assessment of the measurement environment,and to work with the manufacturer to achieve mutual success.Much of the information contained in these pages has not previously been in printor has been buried within written works intended more to impress colleagues thanto instruct others. The material contained herein is common-sense physics. Youwill also note that some repetition of material exists within these pages. This repetitionis necessary to link associated concepts together and to provide a writtenwork 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 metallicfoil-basedstrain gage. The very high intrinsic strain sensitivity of the piezoresistivedevice makes visible phenomena that are typically low-level error sources forthe metallic-foil-gaged sensor but are, none-the-less, sources of error that mayThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-3

Strain-Based Measurement Introduction:make the difference between success and failure in measurement. Piezoresistivesensors, by a large margin, are manufactured in greater quantities for dedicatedvehicular use than all other strain-based sensors combined. For thisreason, the piezoresistive-based sensor is a primary focus of discussion.This book is organized as a reference manual containing a selection of ÒApplicationsNotesÓ that pertain to specific sensor parameters, considerations, mediaconcerns, lead-wire considerations, and a host of others. You are encouraged toreview the Introductory section to lay a firm foundation for the applicationnotes that follow.Sensors are epidemically misapplied to various measurements. In some situationsit is the fact that the engineer has achieved a Òcomfort levelÓ in the use ofa particular sensor technology and is unwilling to risk the use of a potentiallymore-appropriate technology. Knowledge of other technologies is the key thatunlocks the door to move beyond the comfort zone. In other instances, it issimply ignorance of the intricacies of the sensor that lead to the Òround peg inthe square holeÓ situations that occur frequently in measurement.Why are there hundreds of different sensor types commercially available? Simply,each sensor type is appropriate for use in specific environments and is lesswell-suited for use in others. Each measurement environment is unique andoften ÒgreyÓ areas are encountered where the performance of the sensor is notwell-defined by the manufacturerÕs specifications. This book breaks with traditionin attempting to define attributes of the sensor that the reader can utilize toqualify a sensor geometry and performance for the measurement environmentthat he or she faces.I firmly believe that any reader, who has been Òdown in the trenchesÓ makingreal-world measurements, will find some gems contained herein that have beenÒGotchasÓ in prior measurement efforts. As oneÕs experience level grows in thefield of measurement, it becomes clear that no ÒminusculeÓ errors exist. Thecomplete treatment of sensor error components may seem to dwell upon smallerror contributors yet, all too often, engineers find themselves in a court-of-lawarguing over ÒminuteÓ errors that, due to the nature of the measurement, havebecome major error drivers. In short, the Òminute errorÓ can, all too easily,CHAPTER 1-4 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-1: Introductionbecome the bone of contention when a major program has been delayed or expensiveretesting must be conducted due to poor-quality data.In closing, I would leave you with the thought that, the instrumentation engineer isperhaps one of the worldÕs great unsung heroes when one considers that all that weknow of materials, our bodies, our universe, subatomic as well as chemical andmacroscopic phenomena, we have learned through the science of observation andmeasurement. The human biological sensor system, including our senses of sight,smell, touch, temperature sensitivity, sound, and the like, functions as a group oftransducers providing information to us about our world. In instrument engineering,separate sensors, specifically fabricated to be sensitive to one or another phenomenon,are used to provide information to us concerning the inner workings ofour creations, whether they be machines or processes. In summary, the knowledgethat we possess about our world is provided to us by observation and measurementwhere the validity of this knowledge is dependent upon the quality of the observationor measurement.I sincerely hope that the you will enjoy this work as much as I have enjoyed creatingit.James PiersonPierson Associates IncorporatedFirst Edition: 19922nd Edition: September 1999The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-5

Strain-Based Measurement Introduction:Applications Note 1-2:What is Applications Engineering?The simple Funk and WagnallÕs definition of the meaning of ÒApplicationÓ isas follows: 1. ÒThe act of applyingÓ 2. ÒCapacity of being used; relevance, asof a theoryÓ 3. ÒClose AttentionÓ; as in application to oneÕs bookÓ.The definition of ÒEngineeringÓ: ÒThe art and science concerned with the practicalapplication of scientific knowledge, as in the design, construction, andoperation of roads, bridges, harbors, buildings, machinery, lighting, and communicationssystems etc.ÓThe additional definition of the meaning of the word ÒArtÓ is also appropriateas: ÒAny system of rules and principles that facilitates skilled human accomplishment:also the application of these rules and principles.ÓThe above definitions are not comprehensive but are a subset of the definitionsmost suited to this discussion. Applications Engineering is the Òdown in thetrenchesÓ real-world application of a technology for practical purposes.Applications Engineering is an art, consistent with the above definition, in thatit is the assimilation and distillation of a multitude of realities regarding thelimitations and physics of the world around us and the use of imperfect materialsand processes to effect the practical and useful implementation of a technology.All that we know of the machines and processes that we create is provided by meansof sensors that are subject to the imperfections and physical limitations of the worldwe live in. This knowledge is imperfect where the difference between the truth andperception is error. - JPThe objective of Applications Engineering, with respect to sensors, is to minimizethe difference between the perception and the truth, given the physicalconstraints and the environmental conditions within which the sensor mustoperate.CHAPTER 1-6 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-2:The Art of Applications Engineering, as described in the ensuing text, is the resultof a collection of experiences in measurement over many years and for many variedmeasurement environments. A preliminary review of the entire text will familiarizethe reader with many of the concepts contained herein, where the details canbe studied in depth on an as-required basis for various measurements and environmentsas they are encountered in practice.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-7

Strain-Based Measurement Introduction:Applications Note 1-3The Measuring System Source of Information: TheSensorMany component parts comprise the measurement system where all parts mustfunction in unison to produce valid information regarding the measured parameteror measurand. Of all of these subcomponent parts, the sensor is most criticalas the source of information upon which the balance of the system depends.The output of any sensor is organized energy where some deterministic relationshipexists between a quality or quantity of this energy and the present stateof the measurand. All sensor output signals are comprised of a random componentand a predictable component of signal. The ratio of the predictable componentto the random component is a measure of the signal-to-noise ratiowhich can be used to quantify the useful measurement range of the sensor.The Funk and WagnallÕs definition of the transducer, as ÒAny device wherebyenergy may be transmitted from one system to another system whether of thesame type or different typeÓ, is most appropriate when a sensor is viewed as aform of energy translation device. The definition of the sensor as ÒThat whichreceives and responds to a stimulus or signal; especially, an instrument ordevice, as an antenna, gyroscope, or photoelectric cell etc., designed to detectand respond to some force, change, or radiation for purposes of information orcontrolÓ, is equally appropriate. Some use the term ÒsensorÓ to refer to the Òasproduced but not yet finishedÓ transducer, as would be the case with an uncompensatedstrain-gaged device, in contrast with the term ÒtransducerÓ referringto the finished and fully specification-controlled device. Rather than attempt todelineate between the two terms, they are used herein interchangeably inaccordance with the above definitions.Several facts may be stated that are applicable to all of the many types of sensorsthat exist or, in fact, will exist. These facts derive from fundamental lawsof mechanics and thermodynamics and are stated as follows:CHAPTER 1-8 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-31. The presence of any physical sensor inserted into any mechanical or thermodynamic systemwill alter, to some extent, the system itself. This phenomenon can be visualized by consideringthat the addition of an accelerometer to a structure, for the purpose of determining the dynamicbehavior of the structure, implies the addition of the accelerometerÕs mass, thereby changing thestructureÕs dynamic characteristics. This fact is very similar to the Heisenberg Uncertainty Principleof nuclear physics, in that, the act of observation changes the system being observed. Asanother example, the flush-diaphragm pressure sensor must experience bending to produce outputas we shall see. The bending of such a diaphragm implies that the volume of the measuredcavity 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 measurandis static or very slowly changing with respect to time, must shed energy in order to provideinformation regarding the measurand. Sensors that do not dissipate heat or energy by someother mechanism, are not capable of providing information (organized energy) as an outputwhen the input is static and unchanging (invariant) with respect to time. It is important to notethat the piezoelectric sensor is of the nondissipative variety and cannot provide informationconcerning time invariant parameters. It is equally important to realize that energy must be subtractedfrom the measurand by the piezoelectric sensor, for any information in the form of anoutput to be realized, regardless of the time rate of change of the measurand. This situation isvisualized 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 productof mass and acceleration (F = ma), the accelerometer will therefore require a given forceinput 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 ofmass, acceleration and displacement (E =mad), where this energy is supplied to the accelerometerby the structure to which it is mounted.3. The presence of a physical sensor inserted into any mechanical or thermodynamic system willexchange energy in many different forms with the measured system and the measurement systemto which it communicates. This statement means that the mere presence of the sensor massin a small chemical reaction vessel will imply that calories will flow, either from the sensor intothe vessel or from the measurand to the sensor mass, and thus could influence the behavior ofthe measurand. In the case of the static-capable (zero hertz) sensor, a portion of the energyrequired to be shed, in the provision of information regarding the measurand, will be absorbedby the measurand thereby altering its state. Additionally, the energy state existing at the outputof a sensor is changed when the sensor is connected to any other element of the measurementsystem.Just as the product of force and distance equals mechanical energy, the product of charge ßowper unit time (current) and voltage equal energy per unit time or power. Since all elements of themeasurement system possess some Þnite input impedance, it follows that, when a voltage differenceexists between these elements, some current must ßow between them for each element toperform itÕs intended function. At close to zero hertz, impedance becomes simply resistance andThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-9

Strain-Based Measurement Introduction:the current that will ßow from a sensor into the balance of the measurement system willequal I=V/R in accordance with OhmÕs Law, where V is the voltage state at the input to themeasurement system and R is the input resistance of the measurement system. Clearly, thehighest value of R yields the lowest value of I for any voltage state existing at the measurementsystem input. Another implication of this energy exchange is realized when one considersthat the sensor can be modeled, in accordance with TheveninÕs Theorem, as a voltagesource (V signal ) in series with an equivalent output resistance (R output ) where V signal is theopen circuit output of the sensor.FIGURE 1-1. Energy must flow from the sensor into the MeasurementSystem:Energy Flow(Closed circuit)++R outputVV inputsignalR instrument--Since no measurement system possesses an inÞnite input resistance, but rather some Þnitevalue (R instrument ) of input resistance, then the voltage state at the input to the measurementsystem is calculated by the simple voltage divider:VV signal ´ R instrumentinput = --------------------------------------------------R o + R instrumentwhere the true sensor output is decreased by:V signal Ð V inputV-------------------------------------- ´ 100 1 inputR= Ð ---------------- ´ 100 = 1 Ð ---------------------------------------instrument´ 100V signalV signalR o + R instrument(EQ 1-1)Therefore, the input voltage to the measurement system must always be less than V signal .Only when R instrument equals an inÞnitely high value does V input equal V signal . In accordancewith equation 1-1, at close to zero hertz, a strain bridge possessing an output resistanceof 350 ohms operating into a voltmeter with an input resistance of 10 K ohm willCHAPTER 1-10 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-3result in a perceived signal 3.38% less than the true open circuit voltage the strain bridge wouldotherwise provide. Although the zero hertz case is useful to consider, most measurement systemsare required to measure dynamically changing inputs. In this case, equation 1-1 isextended to encompass the dynamic case by replacing resistances with impedances Z output andZ instrument .It is equally important to be aware that energy will be exchanged between the sensor and theenvironment in many different forms and via many different paths where all forms and paths ofenergy 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 goalis to design an accelerometer, we would, of course, set out to design the device so that it is notinfluenced by temperature, pressure, rate of temperature change, humidity, substrate strain, radiation,transverse acceleration inputs, and a host of other possible characteristics of the environment.In reality, this is not achievable. The physical constraints of having to fabricate sensorstructures with imperfect materials, having imperfectly-toleranced subcomponent parts and inusing less-than-perfect transduction methods, means that we must carefully design the sensorwith these shortcomings in mind.5. As the information provided by a sensor is conducted, conveyed, transformed into other energyforms, or interpreted, it is degraded at every opportunity. The random content of any organizedform of energy will increase at any and all opportunities. The act of amplifying a sensor signalwill undoubtedly increase the magnitude of the perceived signal but will likewise increase thepercentage of randomness or noise that is present in the amplified signal. In many ways, the tendencytowards randomness may be likened to the thermodynamic property of entropy which isthe 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 thepresence of the sensor within a system, to minimize the influence that the sensorwill have on the measured phenomena as well as being aware of the influence ofother aspects of the environment on the quality of the information that the sensorprovides.It is the sensor that is the source of information upon which aerodynamicists makedesign modifications, how an old bridge is buttressed, or whether the air bag in ourautomobile should initiate. It is the quality of this information that will establishour success, or in some cases, our very survivability. The strain-based sensor, themost prevalent sensor type in the world today, is the focus of the following dissertation.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-11

Strain-Based Measurement Introduction:Applications Note 1-4:Linear versus Nonlinear SensorsOur preference for sensors that produce a linear output with respect to a linearly-appliedinput arises out of expediency rather than necessity. Many sensortypes will produce nonlinear outputs similar to the output produced by a thermistor,for example (Figure 1-2). The highly nonlinear output of the commonthermistor is typically linearized by the addition of a low-TCR (thermal coefficientof resistance) resistor, installed in parallel with the thermistor.FIGURE 1-2. The Linearized Response.Figure 1-1Resistance(Ohms)TOhmsT Low-TCR ResistanceIdeal Linear ResponseActual ResponseTemperatureThermistor ResponseTemperatureLinearized Thermistor ResponseThe desire to work with linear output sensors is understandable as the need forcomputational linearization of the resultant data is eliminated. In the case ofsensors used for high-frequency measurement, inadequate time may existbetween data points to execute a linearization algorithm if the sampling rate isto support a useful bandwidth in close to real time. The methods used for outputlinearization require prior knowledge of the sensor output with respect tothe physical input.CHAPTER 1-12 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-4:The measurement of the output of a nonlinear sensor may be linearized by measuringthe output of the sensor at 0%, 20%, 40%, 60%, 80%, and 100% of the fullscaleinput of the sensor (Figure 1-3). This data may be used to form a piece-wiselinear curve that closely approximates the nonlinear output function but willrequire significant computational time for each data point collected and will, therefore,limit the maximum practical output bandwidth.FIGURE 1-3. Piece-wise LinearizationM4M5M2M3OutputY5Y4Y3Y2Y1M1Where:Output = Mi (Input) + Yi(For each of the appropriatecurve sections)0% 20% 40% 60% 80% 100%InputPiece-wise linearization of a nonlinear lineThe measurement of the output function used for piece-wise data linearization canalso be used as an input to a polynomial curve-fit program wherein the measurementsystem either computes the curve-fit in real time for each data point collectedor creates a Òlook-upÓ map of output versus input parameters in memory. The useof an internal look-up map of the sensor output with respect to its input will greatlyincrease the potential output frequency response, as simple decision algorithmsmay be used rather than complex real time computations. The implementation ofThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-13

Strain-Based Measurement Introduction:polynomial curve fit routines is easily accomplished with spreadsheet programssuch as MicrosoftÕs Excel by using the ÒTrendlineÓ menu selection.Curve fit routines make non-obvious underlying assumptions and typicallyprocess the supplied data to yield curve fit coefficients that minimize the leastsquares error between the computed curve and the data provided. The assumptionsmade with respect to a least squares fit are that the data is continuous inthe region of interest, the rate of change of data within the region of interest iscontinuous and the scatter of data within the region of interest is bounded andsymmetrically distributed. In some cases there can exist a wider scatter of dataat one end of the region of interest than at the other. In such cases, any leastsquares fit is of questionable value. Superior curve fits, providing uniformscatter on either side of the computed response, always result when the form ofthe curve fit relationship and the theoretical relationship defining the sensorbehavior are the same. If the sensor responds in an exponential fashion withrespect to the input, then the exponential relationship used in a curve fit to thedata will produce superior results. Strain based sensors tend to provide outputresponses that are closely approximated by polynomial relationships of theform:y = ax 2 + bx + c(EQ 1-2)Where a, b and c are quadratic coefficients, y is the output and x is the physicalinput to the sensor. Calculation of the physical input creating the output, y, isachieved by finding the real root of the quadratic equation given by computing:Ð b ± b 2 Ð 4acx = --------------------------------------2a(EQ 1-3)It is the nature of quadratic relationships that two roots will result with one rootyielding a nonsensical result. Although Excel curve fit routines allow the userto specify virtually any number of polynomial coefficients, it is seldom beneficialto specify more than three coefficients. Keep in mind that an 11-coefficientcurve fit to an 11-point data set will result in a perfect fit as the computedresponse is forced through each data point. This appears to be desirable, however,the computed curve fit will make some rather wild gyrations betweendata points producing sometimes large errors! When we discuss measurementCHAPTER 1-14 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-4:uncertainty in Chapter 5we learn that it is desirable to maximize the Òdegrees offreedomÓ to minimize measurement uncertainty. When an eleven-coefficient fit isapplied to an eleven-point data set, where each computed coefficient effectivelynegates one degree of freedom, zero degrees of freedom results! To produce superiorresults, the order of the curve fit used must be much less than the number ofdata 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 plentyof data points exist, higher order polynomial coefficients tend to become exceedinglysmall in magnitude and calculations utilizing more than three coefficientsbecome cumbersome and processor-intensive therefore limiting the rate at whichdata can be processed. In all cases, the computed curve fit should be printed overlyingthe collected data so that the quality of the fit as well as the scatter around thecomputed response can be assessed. Another measure of the quality of a curve fitutilizes the ÒStandard Estimate of ErrorÓ or ÒSEEÓ method computed as follows:SEE =å ( Y i Ð Y-------------------------------- ci( N Ð K) Where: N= Number of data pointsK= Number of curve fit coefficientsY i = ith measurementY ci = computed value of fit at the ith data point(EQ 1-4)The most rapid method of sensor output linearization involves the use of analograther than digital correction of the output data. By using amplifiers in the signalpath that possess nonlinear gains that effectively compensate for the nonlinear outputof the sensor, the effective bandwidth of the collected data is limited only bythe bandwidth of the linear and nonlinear amplifiers employed.With present computational capabilities, computational-corrected nonlinear outputsensors are best suited to lower-frequency applications of less than 1 KHz. Themajority of pressure sensor uses fall into this frequency band. The implementationof nonlinear analog amplifiers, or the computational methods described above,implies prior knowledge of the sensor output characteristic. The collection of thisknowledge, as well as the implementation of the required software implies thatadditional cost per data channel will be incurred.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-15

Strain-Based Measurement Introduction:The key to success in the use of nonlinear sensors is dependent upon themechanical and electrical repeatability of the nonlinear output with respect toits input from cycle to cycle and over time. For the most part, zero instabilitiescan be normalized out of the data as long as the zero output is updated withrespect to time, or Òautomatic-null-on-power-upÓ amplifiers are implemented,where the sensor is at some known physical input reference at the time thatpower is applied.Applications Note 1-5:The Strain-Based Sensor DeÞnitionThe strain-based sensor is any sensor structure that produces output directly asa result of the strain induced within the sensor spring member as result of aparameter input such as force, strain, pressure, acceleration, and many otherforms of input. The strain-based sensor family includes passive as well asactive or self-generating sensor types where the passive group is dominated bythe strain gage sensor type and the active group is dominated by the piezoelectricclass of sensors.FIGURE 1-4. The Cantilever Beam Accelerometer:Figure 1-3Strain gages2 on top2 underneathMassCantilever BeamMassAccelerationCHAPTER 1-16 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-5:All strain-based sensor structures possess an internal spring member upon whichthe parameter acts to produce output. In the case of the cantilevered-beam accelerometerof Figure 1-4, the seismic mass will produce an inertial force acting on thebeam as a function of the acceleration of the device. The bending strain createdmay be measured by a variety of means, the most common being by means ofstrain gages bonded to the bending beam. In the case of the cantilevered-beamaccelerometer, the sensor spring member is the beam itself, as it is the beam thatprovides the elastic restoring forces that act in opposition to the applied inertialloads. In the case of the flush-diaphragm pressure sensor of Figure 1-5, unequalpressures acting upon the diaphragm structure results in a force imbalance producingbending and resultant strain. In all cases, the strain-based sensor possesses aninternal spring member that is configured to respond mechanically to a specificparameter input. The shape of the spring member will vary widely as a function ofthe parameter to which the sensor has been designed to respond.FIGURE 1-5. The Strain-Based Pressure Sensor:Strain gagesPressure Loading (P applied )P referenceDiaphragm bending whereP applied > P referenceThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-17

Strain-Based Measurement Introduction:The piezoelectric class of sensors utilizes the charge-generating characteristicsof the piezoelectric class of materials as a function of applied stress and resultantstrain to produce charge outputs as a function of the applied parameterinput. The spring member in a piezoelectric sensor structure is generally thepiezoelectric material itself.The performance of any strain-based sensor technology relates directly to themechanical, thermal, and electrical qualities of the internal spring member andstrain-sensing mechanism utilized.Other selected non-strain-based sensor technologies are discussed with theobjective of providing you with a fuller and more complete perspective of theworld of measurement in general.The measurement environment and the parameter to be measured as well as thetime-varying nature of both will impose constraints upon the shape, position,mass, stiffness, structural support, thermal inertia, thermal impedance, allowablematerials, and processes that can be used with present technology to fabricatethe sensor. The Art of Applications Engineering is the study of theimplications of these constraints, relative to the many varied measurementenvironments, with the objective of maximizing the validity of the informationoutput that the sensor provides.CHAPTER 1-18 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-6:Applications Note 1-6:The Measurement System Transfer FunctionThe transfer function is the complete relationship defining the output of a sensorfor any set of measurand inputs. The transfer function is also the relationship bywhich the time-varying input measurand is related to the resulting time-varyingoutput of a sensor. The transfer function is generally accepted as:F outH(f) where:( f ) = H( f)F in ( f )F out = the output function of the sensor.F in = the input measurand function.Note that each element of the signal-processing system will also possess a uniquetransfer function as well.The Concept of Spatial Independence:The concept of spatial independence means that a variable measurand is spatiallyuniform and that the value of the measurand is constant and independent of position,relative to the size of the sensorÕs active member. The statically-pressurizedvessel meets the requirements of spatial independence. Tight turbulent flow fields,as may be encountered around an antenna mounted to a high-performance fighteraircraft, will show high-level pressure variations over small linear distances,implying that the value of the measurand may not be constant over the sensitivearea of the sensor. In cases such as this, the requirement for spatial independencemay not be met. Figure 1-6 graphically shows the concept of spatial independence.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-19

Strain-Based Measurement Introduction:FIGURE 1-6. Spatial Independence:Static PressureThe requirement for spatialindependence is satisfiedPressure SensorThe requirement for spatialindependence is NOT satisfiedConstant Pressure linePressure SensorTurbulent Flow FieldSensor active diameterConstant Pressure lineThe requirement for spatialindependence is satisfied Pressure Sensor Turbulent Flow FieldSensor active diameterThe concept of spatial independence implies that the transfer function of anygiven sensor is only valid within a specific frequency band related to the phys-CHAPTER 1-20 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-6:ical size of the sensorÕs active member. For the measurement of higher frequencies,spatial independence will require that smaller and smaller sensors beemployed 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 measurandsmay be vector quantities such as velocity, acceleration, and force, requiring adirection and magnitude to define the measurand.The effectiveness of the flush-diaphragm pressure sensor in sensing a pressureshock wave is a good example in that the wave has magnitude as well as directionand the sensor will not respond with equal outputs depending upon the angle of thewave relative to the sensor surface. In cases such as this, the sensor would alsorequire a Òfield responsivityÓ calibration in order to define the sensitivity of thedevice as a function of angle relative to a measurand input.Microphones are typically calibrated to show the field responsivity of this sensortype 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 thedefinition of the field responsivity of a sensor may also be necessary in order todefine the total response of the sensor, the transfer function of the measurementsystem is as shown in Figure 1-7.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-21

Strain-Based Measurement Introduction:FIGURE 1-7. The Transfer Function:Qw(t) X H(w) = Q o (w)InputQ i (t)H 1 (w)H 2 (w)H 3 (w)H n (w)Outputf 1+ f 2 + f 2 =f nMagnitude H(w) = H 1 (w) X H 2 (w)........H n (w)And where the Phase Response is given by:Phase j(w) = j 1 (w) + j 2 (w) +........... j n (w)The total output of the measuring system is therefore:F out ( w) = H( w)F in ( w)(EQ 1-5)and the total Phase Response is given by:j out ( w) = j( w) + j in ( w)(EQ 1-6)Although convolution defines a specific mathematical operation in time andfrequency domain analysis, equation 1-5 (Figure 1-8) can be thought of as theconvolution 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 thetransfer function, can be thought of as deconvolution and is expressed as:F in (w) = F out (w)/ H(w)CHAPTER 1-22 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-6:and where:j in (w) = j out (w) - j(w)FIGURE 1-8. Convolution and Deconvolution:F in(w)H(w)F out (w)CONVOLUTION:F out(w) = H(w) X F in (w)DECONVOLUTION:(Data Reduction)F in (w)=F out (w)H(w)Convolution occurs during the process of measurement and deconvolution is theprocess of the data reduction. Most data reduction difficulties come from thedeconvolution of collected data with the defined transfer function of the measurementsystem.The convolution function applied to spectral analysis of data is a very well-definedcomplex mathematical process where convolution in the time domain is equivalentto multiplication in the frequency domain. The relationship between convolutionand multiplication in the two domains allows the use of the convolution function tocompute transfer functions and time domain outputs of digitally-implemented signalfiltering routines.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-23

Strain-Based Measurement Introduction:The Sensor Transfer Function Measurement:The measurement of the sensor transfer function is the determination of themagnitude and phase response of the sensor when it is presented with a knownparameter input function. In the determination of the transfer function of anysensor, it is necessary that a dynamic parameter input be provided in order thatthe frequency-domain response of the device can be established. In the case ofboth load cells and strain-gaged pressure sensors, it is most common that thesedevices are calibrated using static or deadweight methods that will provideonly zero-hertz information and provide inadequate information to establishthe entire transfer function of the sensor. In measurement environments wheredynamic inputs are expected, choose a sensor having a flat frequency responsethat is at least equal to or greater than the highest expected frequency present inthe measurand. Dynamic calibration methods are implemented in the calibrationof the self-generating class of sensors, piezoelectrics for instance, due tothe difficulties and inaccuracies that arise when attempting high precisionstatic calibrations of inherently dynamic devices. In this case high precisionmeans calibrations performed to better than ±1% full scale output (FSO) nonlinearityand hysteresis.The objective in calibrating any sensor is to establish the ÒsensitivityÓ of thesensor to a physical input. The problem with this is that this sensitivity numb erchanges depending upon the frequency of the input. In the simplest case, wewould prefer to model the output of a sensor as a linear line or y=mx + b relationshipwhere y is the output, m is the sensitivity, x is the input and b is whatyou get out with no input. When the linear model would allow excessive uncertaintyto result, we are forced to use more complex and typically polynomialrelationships to model the sensor output. The sensitivity of any sensor may bequoted in a number of different ways as follows:1. millivolts per unit of input ie: mV/microstrain at some defined excitation (What you get outper 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 excitationper unit of input)4. millivolts per milliamp ie: mV/ma (Common with constant -current driven sensors)CHAPTER 1-24 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-6:5. millivolts per milliwatt ie: mV/mW (Common with optically-based sensors)Be aware that transducers with sensitivities quoted as full scale values in mV/Vmay meet all other specifications at only one calibrated excitation level. You donot necessarily have the latitude to select any arbitrary level of excitation. This isparticularly true with resistance based sensors where thermal specifications mayonly be valid at some defined level of input excitation. For instance, in the case ofstrain 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 voltage-excitedcase, is equal to V 2 /R and in the current-excited case, equals I 2 R.Additionally, sensitivities quoted in mV/V/unit of input can result in extremelysmall numbers where significant digits can mean a great deal. For example a50,000 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 sometechniques by which the transfer function of a sensor may be defined, where themost common methods in use are described in detail in following chapters:Accelerometers:The calibration of strain-based accelerometers is performed to determine the transferfunction of the device and is usually made by one of the following techniques.Sinusoidal Discrete Frequency:To perform discrete-frequency sinusoidal calibrations, the sensor ismounted to an electrodynamic shaker system and is driven to vibrate at aselection of discrete frequencies at a calibrated and known peak accelerationlevel. The output of the device is then compared to the output of a calibratedreference accelerometer mounted inside the shaker armature. Thephase difference between the reference accelerometer and the accelerometerbeing tested can also be measured and reported during this processThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-25

Strain-Based Measurement Introduction:FIGURE 1-9. The Electrodynamic Shaker System:Absolute Position Detector (for absolutecalibrations)Ch ACh BOscilloscopeAccelerometer undertestReference OutputNormalizingAmplifierElectrodynamicShakerVarious Methods:- Vibrating Wedge- Interferometry- Retroreflective- Ronchi RulingsShaker headReferenceAccelerometerShaker ArmatureShaker Drive PowerSinusoidal SweepGenerator:Discrete or SweepPower AmplifierFrequency ReferenceLogarithmic RatioAmplifierdB RatioPlotter/ RecorderdB versus FrequencySinusoidal Swept Frequency:The sinusoidal-swept frequency test is performed by mounting anaccelerometer to an electrodynamic shaker system and smoothlysweeping the input excitation from a low to high frequency at a constantpeak acceleration level. The output of the test accelerometer iscontinuously compared to the output of the calibrated standard accelerometermounted in the shaker armature with the difference usually plottedin decibels. The high end of the frequency sweep is generallyadequate to clearly identify the resonant response peak of the accelerometer.As in the sinusoidal-discrete frequency test method, the phaseinformation may also be measured.CHAPTER 1-26 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-6:.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 apseudo-random input (broad-band noise) is presented to the sensor and thesensor response to this input is recorded. The pseudo-random input is thesummation of a large number of sinusoids of arbitrary frequency, phase,and having a defined a RMS (root-mean-square) amplitude. This calibrationis performed by plotting the power spectral density of the output of thesensor as a function of frequency.Shock Machine and Hopkinson BarShort-duration transient inputs are usually made by mounting the accelerometerto a shock machine or structural Òwave-guideÓ (Hopkinson Bar)and applying inputs that approximate the unit-impulse function. The Hopkinsonbar is a freely-suspended, simple cylindrical bar that is impacted atone end where the compression wave travels through the bar to the free endwhere the wave is converted to motion of the end plane to which the accelerometeris mounted. A reference accelerometer is generally mounted tothe free end of the bar to monitor the end-plane motion. In most cases, astrain gage, affixed to the side of the bar, is used to provide a trigger signalto a digital storage oscilloscope to allow capture of the wave form. Theunit-impulse function is not easily physically approximated due to theimpossibility of generating a pulse of unit amplitude and areaCentrifuge:Static zero-hertz input calibrations are generally performed using a centrifugesystem where the accelerometer to be tested is mounted at a knownradius on the centrifuge disc and the disc is rotated at a known speed. Theknown radius and rotational speed produce a primary centripetal accelerationloading that is directly traceable and is given by:The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-27

Applications Note 1-6:Deadweight test apparatus possess the important inherent ability of applyingexactly the same loads to the sensor being calibrated regardless ofwhether loads are increasing or decreasing. Insight into the value of thisinherent advantage of deadweight calibration systems follows in the discussionof hydraulic test systems.HydraulicIn higher-load-range calibrations, a reference load cell is installed into theload path in series with the device to be calibrated where the assembly ishydraulically loaded and the output of the test load cell is compared to theoutput of the reference load cell. Calibrations of this type are known ascomparison calibrations as the sensor under test is being compared to thereference load cell. Moorehouse Instruments of Pennsylvania manufacturesload frames of this type. As one might suspect, the precision of this type ofcalibration is limited by the precision of the reference load cell otherwiseknown as the Òtransfer standardÓ. A calibration showing zero nonlinearityand hysteresis simply means that no substantial difference exists betweenthe load cell being calibrated and the reference load cell. Such results donot imply zero error. Typically, the reference load cell is chosen to showbetween three and ten times lower nonlinearity and hysteresis errors thanthe sensor being calibrated. Accordingly, to calibrate a load cell to ±.1% offull scale output (FSO) nonlinearity and hysteresis, the reference load cellshould 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 dueto seal leakage around the hydraulic cylinder piston seal. High pressurehydraulic lines provided with this type of system are often elastomeric innature and it is common to change to stainless steel tubing thereby reducingthe applied force decay rate due to hydraulic line expansion. Often,high speed digital data acquisition systems are used to collect many samplesof both the transfer standard and the sensor under test at each inputload level where this data is averaged. This has the effect of averagingnoise to zero and reducing the uncertainty due to load decay.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-31

Strain-Based Measurement Introduction:The hysteresis in the output of any sensor is the difference between theoutput of the sensor when inputs are monotonically increased and theoutput of the sensor when inputs are monotonically decreased. Industrystandard 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 determininghysteresis loss is to find the magnitude of input where the maximumdifference exists between the Òup-loadÓ response and the ÒdownloadÓresponse. This is not trivial in the case of high performance calibrationsperformed on comparison calibration systems due to theimpossibility of exactly repeating the up-load points on the unload portionof the test. To alleviate this problem it is recommended that the unloadresponse be characterized with a three-coefficient polynomialcurve fit and the outputs then computed for the average inputs achievedon the up-load line. In this way, true hysteresis can be measured.FIGURE 1-11b: The Hydraulic Tester, Transfer Calibration: or Transfer Standard) Changeable Spacerto accommodatevarious sensor styles Reference Sensor (Proving RingHydraulic PumpHydraulic RamLocation of Sensor forCompression LoadingLocation of Sensor forTension LoadingCHAPTER 1-32 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-6:Phase information is not available in this type of static testing. In the caseof the piezoelectric load cell, very long time constants of the output do permitstatic calibration within defined precision limits.GravimetricThe ÒdrivenÓ gravimetric method of calibration utilizes an inertial massmounted to the load cell and driven by a shaker system where the frequencyis either set to discrete points or swept over a defined frequencyband and the magnitude and phase data are recorded. By measuring theapplied acceleration with an internally-mounted reference accelerometer,and in knowing the mass of the inertial weight, the applied force may becalculated by the relationship F = MA. The free-fall gravimetric methoduses a load cell as the reference sensor to calibrate an accelerometermounted on the opposite side of the inertial mass. The gravimetric methodis reciprocal in nature, as will be discussed, but is normally limited to ± 1%precision which is generally adequate for transfer function determinationbut inadequate for high-precision calibration. Both the free-fall gravimetricand driven systems are depicted in Figure 1-13.FIGURE 1-13. The Gravimetric Test System: Load Cell M Test Mass Accelerometer Elastic Suspension:Manually Deflectedto create momentary Free-FallconditionsPostGuide TubeBeamDamperBaseTie rodsMassElectrodynamicShakerLoad CellReference AccelerometerF = MAFoamThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-33

Strain-Based Measurement Introduction:As in accelerometer calibration, the gravimetric method may be usedwith broadband noise inputs, sinusoidal or discrete-frequency inputsgenerated by an electrodynamic shaker to determine the load cell transferfunction.Pressure Transducers:Pressure sensors may be calibrated by many different methods for the purposeof determining the sensor transfer function where some of these methods arediscussed as follows:Deadweight PressurePressure transducers are statically calibrated by means of the deadweighttester where calibrated masses are sequentially added to a platformthat is supported by a precision piston. The pressure is increasedmanually or automatically, by either a ram or controlled-pressure regulator,to the point where the pressure, generated by the calibratedweights acting over the area of the precision piston, is balanced and thepiston is free-floating. The precision of this type of calibration systemrelates to the area difference between the piston and bore within whichit moves. Traceable uncertainties of as low as .0004% of applied pressureare achievable with deadweight calibrators where the piston andbore are precision lapped to minimize area differences. As the area differencebetween the piston and bore results in a leakage path, it is commonfor the piston to slowly settle where the pressure will ultimatelydecay to zero. Operating procedures for deadweight pressure calibratorswill often specify that the weights be manually rotated to distributethe fluid film and minimize friction while data is collected at each inputlevel. The sensor to be calibrated is in communication with the balancedpressure reservoir and the sensor output is recorded. As with allstatic-calibration methods, phase data is not available.CHAPTER 1-34 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-6:FIGURE 1-14. The Deadweight Pressure Test System:When the Free-floatingpiston is supported bythe hydraulic column.the pressure = W/APneumatic Shock Tube:CalibratedWeightsFree-floating pistonArea = A inches 2Sensor under testPressurizing Ram Pressure sensors may be dynamically calibrated by means of the shocktube where a pressurized reservoir of gas is maintained at a known pre-setlevel and is separated from a long tubular section maintained at anotherknown pressure by a rapid-acting valve or diaphragm. When the valve isopened or the diaphragm is deliberately punctured, the reservoir vents intothe shock tube tubular section very rapidly. A very high-frequency-capablereference piezoelectric pressure sensor is normally mounted close to thesensor under test and the outputs are compared to determine the responseof the sensor to the pressure shock wave. It is important that the referencesensor have a frequency response that is much higher than that of the testsensor and sufficiently high to prevent resonant ringing. The shock tubeapparatus can be used with various gases and Prichard pressure levels toachieve microsecond rise times and high pressure impulse loadings. Electrically-drivenshock tubes produce a high pressure shock wave resultingfrom an electrical spark discharged across a gap between electrodes. Pressureamplitudes of 1,000 psi or more are not uncommon. The impulseresponse of the sensor under such dynamic conditions may be used todetermine the transfer function of the sensor including phase data.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-35

Strain-Based Measurement Introduction:FIGURE 1-15. The High-Pressure Shock Tube:Driver SectionPiercing MechanismHigh PressureAluminum DiaphragmShock PulseTest SectionSensor Under TestTime-of-arrival sensorLow PressureHelium SupplyAcoustic Horn:The transfer function of low-pressure range sensors may also be determinedby use of the acoustic horn method where an acoustic generator(horn) is mounted into a procreated chamber and the horn output isswept over the frequency range of interest. The maximum pressuresgenerated by the acoustic horn system are generally less than 1 psi(approximately 170-dB sound pressure level).FIGURE 1-16. The Acoustic Horn:AcousticGeneratorPower AmplifierReference SensorSweep GeneratorOscillatorSensor under TestPressure SourceRapidly-Valved Systems:CHAPTER 1-36 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-6:Rapid-opening valve systems such as the Aronson shockless pressure calibratorin which two chambers, at different pressure pre-charge levels, areseparated by a low-mass poppet valve supported by a spring and providedwith a long slender actuating rod. The actuating force rod is provided witha platen, upon which weights are dropped, rapidly opening the poppetvalve. Once opened, the larger and higher pressure chamber rapidly equilibrateswith the smaller volume test chamber. To facilitate observation ofthe generated output of the sensor, a piezoelectric accelerometer or loadcell is normally used to detect the shock of impact of the weight against theplaten where this output is used for oscilloscope-triggering purposes. Thepressure sensor under test is mounted into the smaller of the two chambersin order that the larger chamber can pressurize the smaller volume rapidly.Chamber volume ratios of 2/1000 are common for very rapid rise-time testing.The rapid-acting valve systems are able to generate single positive- ornegative-going pressure steps that are useful in transfer function and phasemeasurement.FIGURE 1-17. The Aronson Shockless Pressure Step Generator:P PoppetSensor under testPoppet Volume(P Poppet )O-ring sealMassPoppet Valve HeadPoppet Valve StemCompression SpringHousing SupportPressure Reservoir(P reservoir )Poppet lifter andlockdown tabGuide TubeImpact PlateAccelerometer triggerThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-37

Strain-Based Measurement Introduction:Hydraulic Impulse:The hydraulic impulse systems in common use involve the dropping ofa calibrated weight from a controlled height onto a moveable precisionpiston which compresses a column of hydraulic fluid. Two forms ofthis calibrator are in use; one utilizes a reference pressure sensor usuallyof the tourmaline piezoelectric type; the other computes the generatedpressure based upon the mass of the weight, local gravity, velocity,and piston area. Peak pressure magnitudes of between 100 and 20,000psi are achievable with rise times on the order of 3 milliseconds andshowing pulse durations on the order of 6 milliseconds.FIGURE 1-18. The Hydraulic Impulse Test System:Drop Tube9-lb massImpact Pad:Modifies Pulse durationPistonTypical Range:0 to 20,000 psiRise Time:3 millisecondsPulse Duration:6-8 millisecondsTest sensor Reference Sensor(Piezoelectric tourmaline)Hydraulic ReservoirOther Methods:Other methods, not so commonly used for the determination of thedynamic response of sensors, are many and varied. Modified air com-CHAPTER 1-38 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-6:pressors (Figure 1-19) can be used to generate repeatable sinusoidal pressurewaves where the test pressure sensor output is compared with outputof a reference sensor. The modified air compressor method is typicallyused at a fixed frequency depending upon the design of the compressormotor.The vibrating fluidic-column method (Figure 1-20) is also based upon thecomparison of outputs between a reference sensor and the sensor undertest. This method involves the use of a fluidic column that supports amoveable piston/mass that is then mounted to an electrodynamic shakersystem and is driven to vibrate at a selection of frequencies.FIGURE 1-19. The Sinusoidal Pressure Source:Pressure sensorunder testPressure Relief ValvePressure Transfer StandardCompressorFIGURE 1-20. The Hydraulic-Column Pressure Source:Free MassSensor underTestHydraulic columnPressure TransferStandardShakerThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-39

Strain-Based Measurement Introduction:Applications Note 1-7:The Fourier SeriesThe Fourier series is a mathematical expression that is used to define any timevaryingfunction as an infinite sum of the multiples of the fundamental frequencythat comprise the time function. The Fourier series is given by:qt ()=¥a---- 0+ a2 å ( k coskw 1 t + b k sinkw 1 t)k = 1(EQ 1-7)The Fourier series is not just the mathematical expression of a time function, itis the time function. Another way of regarding the Fourier Series is to considerthat every frequency component required to define a specific time function canbe regarded as a physically real subcomponent of the time function. The valueof 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 signalis that the transfer function of the sensor and measurement system will acton each of the harmonic components as if these had been input separately. Theresult of this logic is that the resultant time-varying output is the summation, orsuperposition, of each of the outputs from the sensor or measurement systemcorresponding to each of the Fourier input components. Figure 1-21 shows thewaveform that will result if three simple Fourier components are summed. Itcan be seen intuitively that any complex waveform can be created by the successivesumming of the Fourier subcomponents. Correspondingly, each of theFourier components of the input will be phase-shifted by a different phaseangle depending upon the phase relationship of the measurement system transferfunction. The end result is that the output from the measurement system canbe both amplitude- and phase-distorted unless care is taken to assess the frequencyof the highest expected frequency component, and then to assess theCHAPTER 1-40 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-7:amplitude and phase characteristics of the measurement system transfer function.An infinitely-sharp discontinuity in the input waveform implies that the measurandcontains an infinite sum of Fourier components. The square-wave or triangularpulseinput wave forms, therefore, contain an infinite series of Fourier components.The existence of a large number of Fourier input components implies thatthere will exist, with a high probability, frequency inputs at or near the resonancefrequency of the sensor. In the undamped sensor case, these inputs can result inoverhanging of the sensor and/or resonant ringing of the output. In general, verylittle energy usually exists in the frequency components of the input that are at, orgreater than, the tenth harmonic or tenth Fourier component.FIGURE 1-21. A Complex Input Waveform:Magnitude Fourier Component 11Fourier Component 2.5Fourier Component 30Time-.5-121.5Fourier Sum = Component 1 + Component 2 + Component 31.5Net Waveform0TimeThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-41

Strain-Based Measurement Introduction:Figures 1-22 and 1-23 illustrate how even small parameter inputs, at or nearresonance, can produce high-level ringing of the sensor output.FIGURE 1-22. A Complex Input to the Sensor Transfer Function:Magnitude+1g0g-1g+10 gComponent 1: ± 10g at 100 HzComponent 2: ± 1g at 1000 HzTime-10 g+11 gPhysical Input toAccelerometerSummed WaveformTime-11 gMagnitudedB+40 dBCalibration FrequencyResonant Peak0 dB100 HzComponent 11000 HzComponent 2Sensor Transfer Function (Magnitude)Resonance Frequency = 1000HzCalibration at 100 Hz = 1mV/gSensor Range = 10 gSensor Overrange = 100gLog FrequencyCHAPTER 1-42 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-7: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 gearmeshingcomponents. In this case, the 10-g range accelerometer shows a sensitivityof 1 mV/g, as calibrated at 100 Hz, with an undamped resonant frequency of1,000 Hz showing a maximum response of +40 db (= 20 log V o /V cal , where V cal isthe calibrated sensitivity of the sensor at a much lower frequency within the ÒflatÓfrequency response range of the sensor). It would be reasonable for the accelerometerto show a maximum overhang tolerance of ±100 g before mechanical damageis 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 componentat 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 ofthe device, the accelerometer will mechanically amplify the ±1g input by the +40dB mechanical gain at resonance, which, when multiplied by the 1mV/g sensitivity,will result in a ±10 mV output component at 1,000 Hz as shown. The net outputwaveform shows massive ringing at 1000 Hz where the information of interestis marginally evident as the modulated envelope of the response shown. Frequencyfiltering of the output of the sensor would help to filter out the 1,000-Hz component,however, the 100-g maximum over-range of the accelerometer means that thesubject sensor is, in all likelihood, broken! If the maximum allowable overrange is100 g, this figure is given as being valid within the useful frequency range of thesensor and not at resonance. The ±1 g input at resonance could, therefore, be quitesufficient to cause overrange destruction of the device. The moral of this story isthat we must be aware of all components present within an input signal and assessthe potential impact of each component with respect to the transfer function of thesensor.The Fourier Series is not just a Òmathematically-elegantÓ method of expressing the inputwaveform but, more importantly, a valuable means of determining the power spectral density(the energy that exists within each of the Fourier components), and of visualizing thecomponents of the input and output separately. -JPThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-43

Strain-Based Measurement Introduction:FIGURE 1-23. The Net Accelerometer Output:mV+10 mV-10 mVAccelerometer Output Component at 100HzPeak Magnitude = 1mV/g X 10 g = ± 10 mVTimemV+100 mVAccelerometer Output Component at 1,000HzPeak Magnitude = 1mV/g X 1g X 100 = ± 100mV0 mVTime-100 mV+110 mVNet Accelerometer Output+100 mV+90 mV0 mVTime- 90 mV-100 mV-110 mVThe Fast Fourier Transform algorithm (FFT) is a method by which the numberof computations required to compute the Fourier series is dramatically reducedfrom N 2 calculations to Nlog 2 N calculations, allowing high-speed computersto rapidly compute the Fourier Spectrum and plot it as the power spectral density.The power spectral density is the plot of the relative magnitudes of each ofthe Fourier components as a function of frequency.CHAPTER 1-44 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-8:Applications Note 1-8:Zeroth-, First-, and Second-Order System DeÞnitionsThe definition of the order of a sensor is basically the generalized classification ofthe transfer function of the sensor.The Zero-Order Sensor:The zero-order sensor is sometimes referred to as the ÒZerothÓ order system,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. Thezero-order sensor structure is not realizable in practice; however, some sensorswill behave very much like zero-order sensors over limited bandwidthsin frequency. The zero-order system is shown in Figure 1-22 andcharacterized as follows:F out (t) = K F in (t) (EQ 1-8)Where: F out (t) = The output function with respect to time.F in (t) = The input function with respect to time.K = The proportionality constant.FIGURE 1-24. The Zero-Order Response:.Input/OutputMagnitudeInputOutput(LVDTResponse)TimeThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-45

Strain-Based Measurement Introduction:The Linear-Variable-Differential Transformer (LVDT)-based sensorclosely approximates the zeroth-order sensor at low-to-moderate frequenciesThe First-Order Sensor:A first-order system response is the solution to the first-order differential equationas follows:dF outk 1dT+ k 0 F out = b 0 F in(EQ 1-9)ordF outtæ ö + FèdTø out = KF in(EQ 1-10)Where: t = k 1 /k 0 , The sensor time constantK = b 0 /k 0 , The static sensitivity.If d /dT is replaced by the differential operator ÒDÓ, equation 1-10 becomes:KF in = ( 1 + tD)F outSince F out /F in is defined as the Transfer Function (H):HK= -------------------- =( 1 + tD)F--------- outF in(EQ 1-11)Implies:H( w)( jw)K= --------------------- = -----------------------( jw)( 1 + jwt)F outF in(EQ 1-12)Where the magnitude of this transfer function will equal:CHAPTER 1-46 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-8:H( w)=K------------------------1 + w 2 t 2(EQ 1-13)And where the phase response is given by:jw ( ) = atan( wt)(EQ 1-14)An example of the first-order response is the response of a thermocouple asshown in Figure 1-25. The maximum phase shift of the first-order sensoris -90 degrees as w approaches infinity. The first-order response is alsoreferred as the Òsingle-poleÓ response.FIGURE 1-25. The First-Order Step-Input Response:Input/OutputMagnitudeInputOutputt =Final Valuee(ThermocoupleResponse)1/eWhere e = 2.71828tTimeThe Second-Order Sensor:The second-order system response that is the solution to the second-order differentialequation:2dF out dF outk 2 + k2 1 + kd t dt0 F out = b 0 F in(EQ 1-15)If w 0 is defined as:The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-47

Strain-Based Measurement Introduction:k 0k 2---- and z=k 1-----------------2 k 0 k 2(EQ 1-16)where: w 0 is the natural frequency and z is the damping factor.Equation 1-15 becomes:1 dk----- 0 2z+ -------------- + k2 20 = KF inw 0 d t dk 0w 0dt(EQ 1-17)Where: K = The static sensitivity of the sensor.The second order transfer function in the frequency domain can be shown toequal:H( w)=K--------------------------------------------------------æ w1 Ð æ-----ö 2 + 2 jzæ-----w ööè èw 0ø èw 0øø(EQ 1-18)Where the magnitude of this function will equal:H( w)=K-----------------------------------------------------------w1 -----èæ Ð èæ øö2øö 2 + 2zw ----------èæ øö2w 0w 0(EQ 1-19)and where the phase response is given by:zjw ( ) = atan2------------------------æ w w----- Ð ----- 0 öè w øw 0(EQ 1-20)Examples of the second-order response, Figure 1-26, are abundant. Allspring-member-based sensors will show a second-order response, up tothe first resonance frequency, with varying degrees of damping. NoteCHAPTER 1-48 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-8:that the phase variation with frequency will equal -90 degrees at the naturalfrequency (w n ) and will approach -180 degrees as the frequencyapproaches infinity. The second-order response is sometimes also referredto as the Ò2-poleÓ response or the simple Òspring/mass/damperÓ systemresponse.FIGURE 1-26. The Second-Order System Step-Input Response:Input/OutputAmplitudeInputOutputw n decreasingDamping Factor ( z ) = .7TimeThe Second-Order Response to a Step-Input(The effect of Natural Frequency)Input/ OutputMagnitude21.5z = .1z = .3z = .5Damping Factor IncreasingOutput1Input.50 secondsTimeThe Second-Order Response to a Step-Input(The effect of Increasing Damping Factor)The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-49

Strain-Based Measurement Introduction:Applications Note 1-9:The Second-Order SystemAn accelerometer having an internal spring member that responded to accelerationinputs in two or more dimensions simultaneously would be of little use aswe 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 single-degree-of-freedomstructure. The single-degree-of-freedom spring-member-basedsensor class invariably show second-order responses.It should be noted that no mechanical system is able to perfectly imitatethe single-degree-of-freedom system and will simultaneously supportmotion, to some extent, in other dimensions. Even well-designed sensorstructures will show some degree of ÒtransverseÓ responsivity. Thesecond-order system is defined as a system where the transfer functionis the solution to a second-order differential equation. The second-orderresponse defines the response of any mechanical system possessingmass, where the restoring forces acting upon this mass are provided bya spring constant, and resistance to motion is provided by mechanicaldamping. The spring constant is quantified by the stiffness of the springmember. The damping forces acting upon the mass may result from theintrinsic structural damping of the material that comprises the springmember, or may derive from fluidic- or gas-damping provided in thesensor design. In the case of the piezoelectric element, the spring rateof the piezoelectric element is high and the internal structural dampingof the element is low yielding a close-to-zero-damped second-orderresponse. In the case of the seismometer, the spring restoring forces aregenerally provided by discrete spring elements of relatively low springrate, yielding fundamentally low resonance frequencies.CHAPTER 1-50 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-9:Seismic Instruments:The term ÒSeismic InstrumentsÓ is used to refer to the entire class of instruments,the dynamic behavior of which can be derived from the sum of theforces acting upon the internal seismic element, where the seismic elementis constrained to move primarily in one dimension (single-degree-of-freedomsystem). The seismic mass in low g range cantilever beam accelerometersmay consist of a concentrated mass at one end of the beam or can bethe distributed mass of the beam alone in very high range devices. In thecase of the pressure sensor diaphragm, the seismic mass is the distributedmass of the diaphragm, and the restoring spring force is provided by thespring constant of the diaphragm material. Load transducers are identical tothe diaphragm pressure sensor from the seismic viewpoint, with the springmember configured to produce bending in response to direct-force loading,as opposed, to the distributed-force loading that pressure environmentspresent to the diaphragm pressure sensor.NewtonÕs Law, F = MA, is the fundamental relationship which all seismicinstruments obey, in that, the net force exerted on a mass will result in theacceleration or deceleration of the mass with resulting displacement. Allseismic instruments show primarily second-order responses. It should alsobe noted that all sensor structures will show multiple higher-order resonancesas the various internal structures within the sensor possess differentmasses, geometries, and stiffness, and will support resonances at differentfrequencies. The objective of good sensor design is to ensure that theseinternal structures do not support resonant responses within the useful frequencyresponse range of the sensor.Referencing Figure 1-27, a seismic system is modeled showing the seismicmass and the spring as well as damping forces that will act upon this mass.This mechanical model is excited by the forcing function q i acting upon themass where the resulting displacement of the mass is defined as q o .The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-51

Strain-Based Measurement Introduction:FIGURE 1-27. The Second-Order System Mechanical Model:q i InputMassq o OutputSpringK Lbf/InRDamper Lbf/ft/secR proportional to velocityFoundationThe electrical output of the strain-based sensor is a direct and proportionalfunction of the spring member displacement. For this reason, thefunction q o (t) is taken as being proportional to the output that wouldresult from the second-order sensor structure. The velocity of the massis defined as dq o (t)/dt and the acceleration of the mass is defined asd 2 q o (t)/dt 2 . The force that the spring will exert upon on the mass will beopposite to the forcing input and will equal -K q o (t), where K is thespring constant having units of force-per-unit-displacement. Themechanical damper will also provide a force that is opposite to the forcinginput as a function of the mass velocity and will equal -Rdq o (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 massMA = Input force - Spring force - Damping forcedMA = q i () t Ð Kq 0 () t Ð R q0 () t dt(EQ 1-21)CHAPTER 1-52 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-9:Where:dMA = M q0 () t = F2d t2(EQ 1-22)Substituting (1-22) for MA in (1-21) and solving for q i (t) yields:2d dq i () t = M q0 () t + R q0 () t + Kq2d t dt0 () t(EQ 1-23)Converting from the time domain into the frequency domain using the relationshipq i (t) = e jwt as representative of a sinusoidal input, and knowingthat the output will equal:q 0 () t = H( w)q i () t(EQ 1-24)(where H(w) is the system transfer function)Then, the output must be of the form:q 0 () t = H( w)e jwt(EQ 1-25)H(w) also represents phase as well as magnitude of response. Havingestablished this equality, differentiation of equation (1-25) yields the furtherequalities:d q0 () t = H( w)jwe jwtdt(EQ 1-26)and2d q0 () t = ÐH( w)w 2 e jwtdt 2(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:The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-53

Strain-Based Measurement Introduction:H( w)=1--------------------------------------------------------------æ w1 -----èÐ èæ w 0øö2 ö 2 2z w ø+ -----èæ w 0øö2(EQ 1-28)andjw ( )=2zatan------------------------æ w w----- Ð ----- 0 öè w øw 0(EQ 1-29)Where:j(w) = The phase relationship between the output and the input forcingfunction.w 0=K----MRadians/SecondWhere: w 0 = 2p f 0(EQ 1-30)RMz = ----------------2 MK(EQ 1-31)The implications of equations (1-28) and (1-29) are shown graphicallyin Figure 1-28 and are stated as follows:1. As w approaches zero radians/second: The second-order system is stiffness-dominated orÒspring-controlledÓ where the transfer function H(w) approaches unity in value.2. As w approaches infinity radians/second: The second-order system is mass-dominatedwhere the reaction of the mass to the input approaches zero, or the value of the transferfunction value itself approaches zero.3. As w approaches infinity: The phase angle between the input and the output approaches180 degrees.4. When the phase angle between the input and the output equals 90 degrees, w = w 0 which istermed the natural frequency of the system.5. If this system possessed zero damping (an impossibility), the value of the transfer functionwill approach infinity at the natural frequency of the system.CHAPTER 1-54 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-9:6. When the phase angle between the input and the output equals 90 degrees, the natural frequencyof the system may be determined and from this quantity, the stiffness and mass-ratio of the systemcan be directly determined by means of equation (1-30)FIGURE 1-28. The Second-Order System Magnitude and Phase Response:4z = .1Magnitude ofResponse32z = .31z = .5z = 5z = 2z = 1z = .7.5.125.2.5 1 2 3w/w 0logThe Second-Order Magnitude ResponsePhaseAngle0-30-60-90-120z = 5-150z = .1z = .7z = 2-180If the second-order system possessed a damping factor that approacheszero in value, the flat frequency response range of the second-order sensorThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-55

Strain-Based Measurement Introduction:can be determined by finding the frequency, at which the value of thetransfer function H(w) is equal to unity plus 5%, or numerically equalto 1.05 as follows:If z = 0 (theoretically), then equation (1-28) simplifies to:1.05 = 1/[1-w 2 /w 0 2 )yielding:w = .218 w oInstrument designers and salespersons alike have, for many years,stated that the undamped second-order sensor has a flat frequencyresponse range of up to 20% of the primary resonance frequency of thesensor without really knowing why this is so. The above relationshipshows that the undamped second-order system will provide a theoreticaloutput that is within 5% of the low-frequency sensitivity of the sensorat up to 21.8% of the resonance frequency. Although all sensorstructures will possess intrinsic structural damping to some extent, renderingthe 21.8% figure as a theoretical maximum, the industryacceptedflat response range of 20% of resonance is a reasonable usefulflat bandwidth.CHAPTER 1-56 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-10:Applications Note 1-10:Fluidic-, Gas-, and Structural-Damping of Second-OrderSensor StructuresThis discussion relates to the damping mechanisms used in accelerometer structuresto absorb energy that would otherwise result in over-response or ringing ofthe accelerometer at its natural frequency.Damping of an accelerometer structure is the provision of an energyabsorptionsystem. Just as the shock absorbers of the family automobilewill become warm when driving over a rough road surface, the mediumused to absorb energy from the accelerometerÕs seismic system will alsoheat. With reference to the discussion of the second-order mechanical system,the resistance to motion that the damping medium exerts upon theseismic system is a function of the velocity of the seismic mass and, in general,relatively large displacements of the seismic system are required toachieve significant damping factors. The piezoelectric accelerometers arenot normally damped due to the extreme stiffness of the seismic systemresulting in low total deflections relative to the cantilevered-beam accelerometerstructures. Three basic damping mechanisms are discussed as follows:Fluidic Viscous Damping:Fluidic damping is generally achieved by filling the internal cavity of theaccelerometer with a silicone oil fill fluid. The silicone oil absorbs energyfrom the seismic system by one of two mechanisms. The first mechanismof energy absorption involves the bulk displacement of the silicone oil fillfluid from side to side as the seismic system moves. This bulk displacementof oil results in a damping factor contribution that is related to thebulk viscosity of the oil. The second mechanism relates to the film viscosityof the oil. Seismic systems are generally provided with overrange-limitingstops to impede the motion of the seismic assembly. The presence ofThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-57

Strain-Based Measurement Introduction:the stops effectively forms a damping gap or squeeze film that alsocontributes to the damping factor that results as shown in Figure 1-29.FIGURE 1-29. Fluidic Damping:BeamStrain Gages MassAccelerationFoundationFilm Damping ContributionBulk DampingCurvilinear Motion of Seismic MassDamping FluidThe damping factor ultimately achieved is highly controllable by precisionmixing of the damping fluids, in some cases even while the sensoris being dynamically excited. Usually a known high-viscosity value issolvent-diluted down to the desired damping factor. Due to the highvariation in damping fluid viscosity with changing temperature, theresulting sensor will show a strong damping-factor dependency withrespect to temperature. Typically the second-order response willchange by +5dB/100 degrees F due to this damping factor variation.Another implication of the damping-factor variation with temperatureis that the input and output phase response becomes a strong functionof temperature as well. Both effects are as shown in Figure 1-30.CHAPTER 1-58 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-10:FIGURE 1-30. Fluidic Damping, Thermal Effects:Magnitude ofResponse21.5.125DecreasingTemperaturez = 1z = .3z = .5z = .7Increasing Temperature.2 .5 1 2 3w/w 0The Second-Order Magnitude Response(Effect of Temperature on Fluidic Damping)logPhaseAngle0-30-60-90-120DecreasingTemperaturez = 5-150-180IncreasingTemperaturez = .1z = .7z = 2logIn fluid-damped systems, the presence of entrained air bubbles within theaccelerometer would cause large variations in damping factor as the bubblesmigrate around inside the accelerometer. The objective of fluidicdampingprocesses is to eliminate all internal air bubbles. These devicesare normally vacuum-filled by immersing the device to be damped in a volumeof silicone oil, pulling a vacuum over the oil and holding the vacuumuntil all of the air entrained within the device is removed and replaced bythe fill fluid. The vacuum-filling method also serves to effectively degasThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-59

Strain-Based Measurement Introduction:the fill fluid. Gases dissolved within the fill fluid can result in cavitationunder certain operating conditions. The implication of having acompletely oil-filled rigidly-confined volume is that the oil, whenheated, will tend to expand, but cannot. The internal pressure will risedramatically with increasing temperature producing large case strain,as shown in Figure 1-31.FIGURE 1-31. Fluidic Damping, Thermal Implications:Housing Distortion created by internal pressure risewith increasing temperature MassAcceleration FoundationSeals:Highly-Stressed atHigh TemperaureDamping FluidThe inclusion of an internal gas-filled expansion-absorbing bladder toaccommodate the fluid volumetric expansion is workable in the largeraccelerometer designs but is difficult to implement with the more miniaturizeddesigns. In completely fluid-filled designs, the pressure mayrise by as much as 2,000 psi per 100 degrees F. As a consequence, thisinternal pressure will act upon the available internal area producingforces that will stress the integrity of the sensor seals and will mechanicallystress the structure of the accelerometer. The loss of dampingfluid cannot be confirmed by visual inspection due to the small volumeof fluid present internally and the high possibility of the fluid beingexternally sourced. To detect the loss of damping fluid, an initial fre-CHAPTER 1-60 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

MagnitudeofResponseMagnitudeofResponseApplications Note 1-10:quency-sweep of the accelerometer is performed to establish the maximumresponse peak at a known and controlled temperature.FIGURE 1-32. Detection of Damping Fluid Leakage:Magnitude Response Plot BeforeThermal Cyling1z = .7ThermalCycling1w/w 0logHot/ ColdIncreased Peak is evidence of leakage1z = .51logw/w 0Magnitude Response Plot AfterThermal Cycling Showing Evidenceof Damping Fluid LeakageThe accelerometer is then thermally cycled several times from low to hightemperature and the frequency sweep is repeated. By comparing the initialfrequency sweep to the post thermal-cycling sweep, any increase in themaximum response peak is a direct indication of damping-fluid loss (Ref-The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-61

Strain-Based Measurement Introduction:erence Figure 1-32). It is critically important that the pre- and post-thermalcycling sweeps be performed at close to the same laboratorytemperature due to the strong dependency of damping factor withrespect to temperature.Regardless of the type of damping used, the flat frequency responsebandwidth of the accelerometer will be greater in the damped case thanin the undamped. In the undamped case the frequency response will beflat, ± 5%, to approximately 20% of the resonant frequency. In the critically-dampedcase (z =.7), the accelerometer will be flat to as much as60% of the resonant frequency at one temperature as shown in Figure1-33.FIGURE 1-33. The Effect of Damping on the Second-Order SystemResponse:Critically-Damped:Flat (± 5%) to 60%of Resonant FrequencyMagnitude ofResponse432Undamped:Flat (±5%) to 20%of ResonantFrequencyz = .11.5z = .7.125.2 .5 1 2 3w/w 0The Second-Order Magnitude ResponselogIn flight test environments it is not uncommon for the flight test accelerometersto experience very different temperatures at different pointson the air-frame. Some accelerometers may be mounted out in a coldCHAPTER 1-62 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-10:wing tip and others mounted next to hot propulsion systems. This variationin temperature implies that the damping factor will vary from unit-to-unitwith the associated loss of phase data from channel to channel. It is for thisreason that the piezoelectric-based undamped accelerometer is most oftenused for this type of modal analysis. If damped accelerometers are mountedinto thermal controller and heater assemblies, the accelerometers can beprecisely phased from unit-to-unit.Just as an automotive shock absorber will become warm when absorbingenergy, the damping fluid will also heat in measurement environments thatcontinuously present repetitive high-level inputs. The mechanical energyabsorbed will be converted into heat in the damping fluid causing thedamping fluid viscosity to decrease, reducing the fluidÕs ability to absorbadditional energy. It has been observed in this type of measurement environmentthat the damping factors can be reduced to such a low level thatthe accelerometer is often over-ranged after having operated successfullyfor several minutes under high-level input conditions.Gas damping:Gas damping is used to achieve the same end result as fluidic damping buttends to be much less sensitive to temperature variations. Gas damping isachieved by pre-charging the interior of the accelerometer with a dry puregas to a predetermined pressure. The technique of gas damping is mostoften used with the micro-machined silicon piezoresistive accelerometer.The relatively-low density of the silicon seismic mass results in a largeseismic-massarea in order to achieve significant damping factors. Gasdamping is achieved by forming the accelerometer structure such that precisiongaps and orifices are formed between the moving seismic elementand the outer housing of the device. The motion of the seismic elementcombined with the precision-damping gaps and orifices result in the gasbeing squeezed into a film as shown in Figure 1-34.The internal pressure will increase as temperature is increased but not tothe extent found in the fluid-damped versions. Although the internal pres-The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-63

Strain-Based Measurement Introduction:sure will increase by the ratio of P 1 /T 1 = P 2 /T 2 , assuming that the internalvolume is fixed, the gas density remains largely unchanged. Therelatively small changes in gas density with temperature change contributeto the small changes observed in damping factor with respect totemperature. The damping factor is predetermined to a large extent bythe dimensions of the gap between the seismic assembly and the outerhousing and is, therefore, a strong function of the fabrication tolerancesand is not easily modified as in the case of the fluid-damped devices.FIGURE 1-34. Gas Damping (Micro-machined Silicon):Piezo resistors (8)Bonding PadsSilicon Cap / Acceleration StopSilicon Seismic MassSilicon Base /Acceleration StopMallory BondsTop view of Seismic Element:Gaseous DampingPiezo ResistorsSeismic MassTabsCHAPTER 1-64 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-10:Structural or Solid Damping:It has been established that cyclically-stressed materials dissipate energywithin the material itself. This is the reason that the response of a cantileverbeam accelerometer or piezoelectric accelerometer is not infinite at the resonancefrequency due to the internal damping of the beam material itself.It has also been found that most structural materials such as steel and aluminumwill dissipate energy per stress cycle that is predominantly independentof frequency and is proportional to the square of the amplitude ofvibration. With the energy dissipated per cycle proportional to the square ofthe vibration amplitude, the loss coefficient is constant and the shape of thehysteresis curve remains independent of amplitude and strain rate. In concertwith the intrinsic structural damping afforded by the materials thatcomprise the sensor structure, external or extrinsic materials can be used tofurther aid in the structural damping of the sensor.FIGURE 1-35. Structural-Damping Model:AccelerationAccelerometerFoundationDamping MaterialKRFoundationSolid materials used for structural damping are generally mounted underthe accelerometer structure as shown in Figure 1-35 and therefore presentan additional spring and damper suspension which must be considered inthe determination of the total transfer function of the accelerometer. Soliddamping is not generally useful at low frequencies but can be of use in theexplosives and ordinance testing fields to mitigate the high-frequency componentsthat are induced in explosively-driven structures such as armorplating. It should be realized that structural damping is a form of frequencyThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-65

Strain-Based Measurement Introduction:filtering. It is often used for this purpose although great care must betaken in assessing the effects that the structural-damping materials mayhave on the frequency- and phase-response of the sensor.Applications Note 1-11:The Strain-Gaged Cantilevered BeamThe cantilevered beam is a fundamentally-simple mechanical structure asshown in Figures 1-36 and 1-37. Figure1-36 shows the metal-beam accelerometerstructure and Figure 1-37 shows the all-silicon design configuration.FIGURE 1-36. Cantilevered-Beam Accelerometer (Steel-Spring Member):Strain GagesAccelerationClampBeamMassThe Cantilever-Beam Accelerometer (Steel Beam)Center of Seismic MassCurvilinear Motion of Seismic MassDamping FluidIn the case of the silicon design, location of the strain gages as far as possiblefrom the bending axis of the beam results in significant strain multiplicationCHAPTER 1-66 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-11:and correspondingly high output sensitivities at low beam deflections. The highoutput,all-silicon designs are not typically damped as the low beam deflections donot appreciably displace a damping medium. The cantilevered beam, in general, isa high-strain low-force structure where the strain developed at the surface of thebeam is inversely proportional to the square of the beam thickness and is directlyproportional to the beam length. The strain developed in the beam surface is alsoinversely proportional to the beam width and the beam stiffness (YoungÕs Modulus).Most of the beam length serves only to generate moment loads at the straingage locations while significantly adding mass and contributing to beam deflection.The highest moments exist at the clamp, but continuously and linearly varyalong the entire length of the constant cross-section simple beam. Modified beamsections can be used to linearize the strain field at the strain gage locations butsuch modifications will contribute significantly to the cost of machining.FIGURE 1-37. Cantilevered-Beam Accelerometer (All-Silicon Discrete Gages):Strain GagesDisplacement from NeutralAxis Multiplies Strain at theStrain Gage Locations.ClampSeismicMass (Silicon)Acceleration-induced deflectionExtremely-high natural frequencydue to short effective beam lengthsand small deflections due to strainmulitiplication.This style of accelerometer is notnormally damped due to theextremely-small mass deflections.The cantilevered beam allows strain-sensitive elements to be bonded to both surfacesof the beam where the strain elements will be exposed to equal and oppositeThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-67

Strain-Based Measurement Introduction:strains. This characteristic allows high-output fully-active strain bridges to beimplemented.Other mechanical considerations of the cantilevered beam, used for eitherforce or acceleration measurement, relate to the resonant frequency of suchstructures. The cantilevered beam, used in the fabrication of the strain-gagedaccelerometer, will be designed to minimize the beam length. The resonant frequencyof the seismically-loaded beam, having negligible beam mass, is proportionalto the root of 1/L 3 . This means that the shorter beam lengths willyield much higher resonant frequencies than longer beam lengths. The piezoresistivestrain gages of the ÒUÓ type will provide minimum gage lengths thereforepermitting minimum beam lengths.The curvilinear motion of the loading point implies reduced perpendicularity atthe point of load application to the cantilevered beam at high acceleration inputlevels. Although small strains, on the order of 500 microstrain, are required bythe piezoresistive strain gage accelerometers, the curvilinear motion of the loadpoint will typically limit the nonlinearity performance of the device to between±.5 to ±1% of the full-rated output of the device. Additionally, the mechanicalstiffness and symmetry of the clamp structure will directly impact the symmetryof the strain field in positive- versus-negative bending of the beam. Theintegrally-machined clamp of Figure 1-36 will provide mechanical performancesuperior to that of the assembled clamp structure. The multi-componentclamp assemblies will contribute friction or hysteretic loss as well as assemblytolerances influencing the effective length dimension of the beam.CHAPTER 1-68 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-11:FIGURE 1-38. Integral versus Assembled Clamp Structures:Integrally-MachinedClampAccelerationAssembledClampStructureMechanical FrictionDeteriorates HysteresisPerformanceWith reference to Figure 1-37, the location of the strain-sensitive elements oneither side of the transverse neutral axis will result in low differential strains due totransverse bending of the beam, thereby minimizing the transverse sensitivity ofthe beam to transverse inputs. Additionally, the transverse strains that are inducedtend to be self-canceling with proper bridge wiring.Thermal Considerations:Mass MassIf the beam thickness is thin, the thermal tracking from strain gage to straingage is good. When the beam is exceptionally thin, the axial thermalimpedance of the beam is high and a mutual self-heating phenomenon canoccur. The operating heat shed by one strain gage will heat the adjacent andopposite strain gages, contributing to potential thermal instability. In thecase of the fluidically-damped cantilevered-beam accelerometers, thedamping fluid offers a well-coupled and low-thermal-impedance heat-sinkthat reduces self-heating. The self-heating phenomenon, shown in very thinbeam sections, most often occurs in piezoresistive strain-gage installations.The high operating power density and high thermal coefficients of gageresistance (TCR) tend to exacerbate the condition.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-69

Strain-Based Measurement Introduction:Beam Stress and Strain Relationships:With reference to Figure 1-39, a beam length of L to the strain gagelocations, having a width of d, thickness of b and where the load W isapplied at distance of L-a inches from the end of the beam, the maximumstress s is given as:s=Mb-------2I(EQ 1-32)Where M = MomentAnd where I = The Polar Moment of Inertia=b 3 d--------12(EQ 1-33)Substituting the Polar Moment of Inertia, I of (1-33) into (1-32) yields:s=6M--------b 2 d(EQ 1-34)And since e = s/E (where E = YoungÕs Modulus)and M = -W(L-a), therefore:6W( L Ð a)e = -------------------------Eb 2 d(EQ 1-35)The above calculations are valid for straight uniform-section beamsshowing small deflections where the width-to-thickness ratio is nominallybetween 8 and 15. The calculations given should be used asapproximations where the actual strains are determined by test 1 .1. Ref: Roarke, Warren Young, P100 case 1a.CHAPTER 1-70 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-11:FIGURE 1-39. Cantilevered-Beam Variable Definitions:LW = MaBeam Thickness (b)Acceleration MassLL-aaCenter of SeismicMass[Beam Width (d)]Beam Resonant Frequency Calculations:In the case where the beam mass is insignificant or is a small fraction of the seismicloading mass, the resonant frequency of the beam is given by:1.732F n = ------------2pEb 3 dg-------------------12WL 3(EQ 1-36)Where: F n = Hertzand where L refers to the total beam length from the point of the load applicationto the clamp.Additionally a simple correction for the strain-gage backing and bond-line thickness(t) is made as follows: (Reference Figure 1-40)The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-71

Strain-Based Measurement Introduction:eæb 2 -- + t öè øe i = -------------------b--2(EQ 1-37)Where: e i = The indicated strain which will be greater than the beam surfacestrain.The above correction is of particular importance when thin beams are beingconsidered.FIGURE 1-40. Adhesive Thickness Strain Correction:b / 2b / 2Strain Gage Backing /Adhesivee surfacee indicated (> e surface )BeamStrain GagesttApplications Note 1-12:Pressure ReferencesThe diaphragm-based pressure sensor bends in proportion to the pressure differencethat exists across the diaphragm. The pressure that exists ÒinsideÓ thepressure sensor may be fixed at some known value or it may be variable, as inthe case of the differential pressure sensor. Several different types of pressurereferences are used and will be discussed in turn:CHAPTER 1-72 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-12:PSIA (Absolute Reference):The psia-referenced sensor may be created in one of two ways. The moststraightforward method is to create an internal high-quality vacuum and tothen seal the reference cavity hermetically by welding a fused glass-tometalÒheaderÓ lead-wire feed-through onto the back of the sensor. Normallysuch headers are electron-beam welded in place, which is convenientsince the weld process occurs in a vacuum environment. The end result isthe creation of a Òclose-to-zero psiaÓ internal reference volume where thediaphragm will be prestressed due to normal atmospheric pressure toapproximately 14.7 psia. An absolute-referenced pressure sensor that producesan output of 1mV/psia and having a range of 15 psia will show 14.7mV ± zero-bias level output while sitting on your desk. Reference Figure1-39.FIGURE 1-41. True Absolute Pressure:WeldedFused Glass-to-Metal HeaderOutput14.7 mV@ 1mV/psia14.7 PsiaVacuum reference (Approximately 0 Psia)0Pressure14.7 psia(Zeroed Sensor)The second method of creating a psia-referenced pressure sensor is commonlyemployed in the miniature sensor fabrication industry and is termedthe ÒElectrically-ReferencedÓ method. Figure 1-42. In the electrically-referencedmethod, the internal cavity of the sensor is simply sealed with noattempt made to create an internal vacuum environment. Since the sensoroutput is idealized as a Y = mx+ b linear line, the zero-bias level of the sensoris electrically offset by means of the bridge-balance resistor to elevatethe output of the sensor to simulate the sea level output (nominally 14.7psia). Most miniature sensors are too small for the creation of a high-qual-The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-73

Strain-Based Measurement Introduction:ity internal vacuum sealed by hermetic methods, with the exception ofthe Mallory bonded capsule. As a result, most miniature sensors arefabricated using various epoxy materials that would outgas and deterioratean internal vacuum, even if one were initially provided.FIGURE 1-42. The Electrically-Referenced Absolute Pressure Sensor:AppliedPressureSeal (usually epoxy)Lead wiresAtmospheric Pressure Sealed InsideOutput14.7 mV0@ 1mV/PsiaPressure14.7 Psia( 0 Psig)Sensor Zero BeforeTrimmingThe zero level of the Electrically-Referencedpressure sensor is trimmed in the positivedirection.The use of the electrically-referenced method implies that, if the sensoris pressure-cycled from a high vacuum to a greater than 14.7 psia pressure,the bending diaphragm will progress through the stress-strainÒinflection pointÓ where the membrane transitions from negative topositive bending. Due to the asymmetry of the flush-diaphragm supportingclamp structure, the sensitivity of the diaphragm can be mismatchedby as much as ± 3% between positive and negative bending.This characteristic will not create error provided that the sensor hasbeen calibrated from a close-to-zero psia pressure to the maximumrange of the device. Properly calibrated, the inflection error becomes anintegral component of the nonlinearity error of the sensor.Another consideration in the Òtrue-psiaÓ versus the Òelectrically-referencedÓissue is that if no pressure difference exists between the insidereference cavity of the sensor and the outside world, then no tendencyCHAPTER 1-74 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-12:for leakage exists either. If the transducer is destined for terrestrial use atsea level, the differences between the true-psia and electrically-referenceddevice become academic. If, on the other hand, one erroneously assumesthat all psia sensors are of the Òtrue-psiaÓ sealed type, and the device purchasedis actually electrically-referenced, in a high-vacuum space environmentthe internal reference cavity will eventually leak out to the vacuum ofspace and this would be perceived as zero drift.The classical argument against the use of the electrical reference method isthat the internal gas will expand with increasing temperature, thus modifyingthe reference pressure. The statement is correct; the internal gas willthermally expand with increasing temperature in accordance with the relationship:P 1 /T 1 = P 2 /T 2Where pressure P 1 and pressure P 2 occur at temperature T 1 and temperatureT 2 respectively.Provided that the sensor is sealed prior to the thermal compensation process,the internal gas expansion with increasing temperature becomes a neteffect combined with the TCR mismatch shown by the bridge and is compensatedout as a net effect during the process of thermal-zero compensation.The end result is that there exists no way of determining whether a deviceis electrically-referenced or true-psia-referenced from the output terminalpoint of view. However, the difference can become very apparent dependingupon the pressure existing at the sensor lead exit.PSIS (Sealed-Gage Reference):The sealed-gage reference pressure is identical to the electrically-referencedabsolute method described above where the internal cavity is sealedat one atmosphere (14.7 psia) and the sensor zero output is trimmed to providea nominal null-bias output at 14.7 psia. This method is basically iden-The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-75

Strain-Based Measurement Introduction:tical to the electrically-referenced absolute method except that the zerooutput of the sensor is not artificially elevated. The same lead exit pressureconsiderations apply in the case of the psis device as apply in thepsia-electrically-referenced sensor. Reference Figure 1-43.FIGURE 1-43. The Sealed-Gage Reference:OutputAppliedPressureSeal (usually epoxy)Lead wiresAtmospheric Pressure Sealed Inside0mV-14.7 mV@ 1mV/Psis14.7 PsiaPressurePSIG (Gage Pressure):The internal cavity of the psig-referenced device is deliberately ventedto atmosphere. Bending of the sensor diaphragm will be in proportionto the difference in pressure between the measurand and the nominalatmospheric pressure present at the reference port. Reference Figure 1-42.FIGURE 1-44. The Gage Pressure Reference:Output@ 1 mV/PsigApplied PressureAtmospheric Pressure0 mV-14.7 mVPressureAtmospheric(Approximately 14.7 Psia)CHAPTER 1-76 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-12:PSID (Differential Pressure):The differential pressure sensor spring member bends in direct proportionto the difference in pressure applied across the spring member. ReferenceFigure 1-45. In some highly-symmetrical designs (ie: variable reluctance)the bending member can show high-symmetry of response in positive-versus-negativebending where either port of the sensor can be at a higherpressure than the other port. In other designs, particularly the flush-diaphragmtypes, the internal cavity of the device is generally defined as thelow pressure port. The low- or high-pressure ports may well have maximumline pressure (P line = (P 1 + P 2 )/2) tolerances due to strength and seallimitations. Generally, if positive bending only is to be supported, the sensorshould then be calibrated for positive bending. If positive and negativebending are to be supported, the sensor should then be bi-directional calibratedin each bending direction.Line Pressure:Differential pressure sensor specifications require the definition of severalpressures; P 1 , P 2 , P line and P ambient . In designs intended for high line pressureuse, where the housing of the sensor will be maintained at close toambient pressure, large pressure differences will exist between the insideof the sensor and external ambient pressure environment. In high line-pressureapplications, the high internal pressure acts not only across the springmember, but on all internal areas producing potentially large housingstrains. Since these strains will produce tensile stresses within the springmember, the differential pressure sensitivity of the sensor may be significantlyreduced at high line pressures. It should also be noted that high linepressures with the external case of the sensor maintained at ambient pressureusually implies that the lead wires exiting the sensor structure have anincreased tendency for leakage. To reduce line pressure effects and thepotential for lead exit leakage, high line pressure designs are sometimesconfigured so that the pressure surrounding the sensor is maintained ateither P 1 or P 2 rather than ambient pressure. When specifying the performanceof high line pressure differential sensors be careful to specify theThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-77

Strain-Based Measurement Introduction:maximum influence on differential pressure sensitivity at the highestexpected line pressure.FIGURE 1-45. The Differential Pressure Reference:OutputP 2 > P 1P 1 > P 2P 1 P 20 mVPressureP line = (P 1 + P 2 ) / 2Wet/Dry:Most differential pressure sensor designs are configured to tolerate ahostile or ÒwetÓ environment on one side of the spring member only. Inthese discussions, ÒwetÓ may refer to moisture-laden air, water in a liquidform, corrosive fluids or gases and just about any other media thatthe sensor manufacturer specifies as being compatible with the sensormaterials but not necessarily compatible with the inner workings of thesensor. The other side of the spring member may be exposed to P 2 ,however, this media must generally be a dry nonconducting and noncorrosivegas or fluid. Flush diaphragm strain-gaged differential pressuresensors are often wet/dry rated. In special cases, wet/dry designscan be vacuum-filled with silicone oil where the oil is retained bymeniscus forces effectively isolating the sensitive internal strain gagesfrom moisture exposure. Internal coatings are often used to desensitizethe internal circuitry to short term moist air exposure but rarely arecoatings a good long term solution.CHAPTER 1-78 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-12:Wet/Wet:Wet/wet differential pressure sensor designs are usually configured withthin flexible metal isolation diaphragms, ref Figure 1-48, used with anessentially incompressible silicone oil fill to communicate both P 1 and P 2to the spring member. Wet/wet differential pressure sensors are sometimesreferred to as Òhostile/hostileÓ designs and tend to be more costly thanother differential designs.Dry/Dry:Dry/Dry differential pressure sensor designs are the least expensive differentialdesigns available and will tolerate only dry nonconducting and noncorrosivefluid or gas exposure on either side of the sensor spring member.Often coatings are used to render these designs tolerant of short term exposureto moisture or condensate accounting for the popularity of this designfor heating, ventilating and air conditioning (HVAC) use.High Side:ÒHigh-side onlyÓ differential pressure sensors are designed such that one ofthe two pressure ports must always be maintained at a pressure that isgreater than the other port. Generally, the internal geometry of such designsis such that the spring member is well-supported for bending in one directionand poorly supported for bending in the other direction. In some cases,the sensor manufacturer will specify that the low side may be some smallpercentage of rated pressure greater than the high side. Generally, if a highside only design is subjected to full pressure reversals or the ports are confusedon installation, irreparable damage results.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-79

Strain-Based Measurement Introduction:Applications Note 1-13:The Diaphragm Strain-Gaged Pressure SensorThe general reference to the strain-gaged diaphragm pressure sensor refers toany sensor where the strain gages are directly bonded to, or are an intrinsic partof the diaphragm spring member of the sensor, and where pressure-inducedbending of the diaphragm produces strains within the diaphragm to which thestrain gages respond producing output. The bending diaphragm sensor structurepossesses zones of both tension and compression strains that are straingagedto produce the sensor output. The derivation of both radial and tangentialstrain components for the flat, circular, uniform-thickness, and fixedperimeterdiaphragm are provided.General Discussion:Recessed DiaphragmsFIGURE 1-46. The Recessed Diaphragm:Diaphragm strain-gaged pressure sensors fall into two general categories.The first is the recessed-diaphragm design shown as Figure 1-46,where the sensing diaphragm is recessed within the mounting port ofCHAPTER 1-80 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-13:the sensor. The significant dead volume produced by recessing of the sensordiaphragm severely limits the frequency response of such a sensor. Thecavity over the diaphragm will act as a Helmholtz cavity or pneumatic filter,attenuating dynamic pressure inputs significantly. The recessed-diaphragmdesign provides superior mechanical protection of the sensingdiaphragm and generally superior zero stability as the cavity will also addsignificant thermal mass to the sensor, rendering the design less sensitive tothermal-transient inputs. The added structural mass of this design also providesexcellent spring member support with the diaphragm removed fromthe generally high stresses found in threaded-mounting configurations. Inmeasurement environments where compacting particles could be present,the recessed-diaphragm design may not be the most suitable, as compactionof particles over the sensor diaphragm (Figure 1-47) will reduce thesensitivity of the sensor significantly. In cleaning of the pressure port anddiaphragm, only fluid cleaning solvents and solutions compatible with thewetted materials of the sensor should be used. The use of any mechanicalcleaning tools will undoubtedly result in damage to the sensor.FIGURE 1-47. Particulate Compaction (Recessed-Diaphragm):Compacted ParticlesParticulate-Laden FlowMany other variations of the recessed diaphragm design are commonlyavailable with many process-type designs being of the filled variety wherethe active sensor is isolated from the measurand (medium) by a flexibleThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-81

Strain-Based Measurement Introduction:fluid-filled bellows arrangement on one or both sides of the bendingdiaphragm as shown in Figure 1-48. In biomedical uses, the low-costdiffused semiconductor sensors are isolated from the medium by usingsilicone-gel materials that provide medium isolation by occupying thevolume that the moisture-laden media would otherwise occupy. Thisisolation method is sometimes referred to as isolation by the exclusionmethod and is depicted in Figure 1-49.FIGURE 1-48. Medium-Isolated, Filled Sensor Designs:LowPressureHighPressurePressure-SensingDiaphragm ModuleHeader AssemblyFill Fluid(Usually Silicon Oil)LeadwiresMedium-Isolating Diaphragms(Very Flexible)CHAPTER 1-82 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-13:FIGURE 1-49. Medium Isolation By ÒExclusionÓ:Silicone Gel FillSensorFlush Diaphragms:The second common type of strain-gaged diaphragm pressure sensor (Figure1-50) is the flush-diaphragm design where the sensing diaphragm isflush with the end of the sensor, eliminating all dead-volume cavities. Theadvantage of this style of sensor is that the elimination of the Helmholtzcavity over the sensor diaphragm allows much-improved frequencyresponses to be realized. The minimum mass of the flush-diaphragm sensorresults in high insensitivity to vibration and shock inputs. Since the sensitivemember is exposed, with no mechanical protection, care in handlingmust be exercised. The asymmetry of the structural diaphragm clamp andminimized excess structural material results in a sensor design that willshow different sensitivities for pressure-induced bending in the two bendingdirections. In the case of the differential sensor, it is highly recommendedthat the device be calibrated in both pressure directions assensitivity mismatches of as much as 3% are common in asymmetricallysupportedsensor structures such as the flush-diaphragm designs as shownin Figure 1-51.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-83

Strain-Based Measurement Introduction:FIGURE 1-50. Flush-Diaphragm Discrete-Bonded and Diffused Gages:MediaStrain GagesFlush Diaphragm (Discrete-Bonded Gages)Flush Diaphragm (Diffused Gages) Leads Diffused Silicon DieDiaphragm Support StructureThe direct exposure of the diaphragm to the measurand results inincreased thermal-transient sensitivities as the heat-dissipating straingages are thermally well-coupled to the measurand. The high heat densitiesof the miniaturized discrete-bonded piezoresistive strain gagedevices results in the warming of the measurand that is in direct contactwith the device in no-flow situations. When flow commences, thiswarmed layer is swept away and replaced with cooler medium, introducingsignificant thermal shock, particularly in the low-pressure andtherefore thin-diaphragm designs. The different coefficients of heatconduction between the sensor and the measurand, in the no-flow versusflow conditions, can cause deteriorated static data quality. In casessuch as this, the use of an essentially incompressible thermal barriermaterial, such as vacuum-degassed room-temperature-vulcanized rubber(RTV) or grease, will reach thermal equilibrium (equilibrate) withthe sensor diaphragm, reducing the thermal transient sensitivity of thesensor significantly (Figure 1-52). The vacuum-degassing of suchmaterials is essential in the removal of entrained gas bubbles that willCHAPTER 1-84 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-13:result in performance degradation due to the compressibility of theentrained gas within the pressure-communicating medium.FIGURE 1-51. Asymmetric Output Response:Outputm 1 m 2P ref > P measurandm 1P measurandm 2P measurand > P refP refFIGURE 1-52. Reduction of Thermal Transient Sensitivity:Vacuum-Degassed Thermal-Mitigating LayerMeasurand Diffused Silicon Diaphragms, Recessed and Flush Styles:Diffused silicon sensors are defined as single-crystal silicon structures intowhich piezoresistive strain gage elements are directly diffused as an integralpart of the spring member or diaphragm. Since the strain gages are anintegral part of the diaphragm, no strain gage bonding adhesive is presentor required. The diffused silicon piezoresistive sensor class are commonlyavailable in both the recessed- and flush-diaphragm styles. Silicon diaphragmsare commonly available in flat-plate and cup-diaphragm designs.The flat-plate design is as shown in Figure 1-50. The cup-diaphragmdevice of Figure 1-53 results in a superior mechanical clamp system whereThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-85

Strain-Based Measurement Introduction:the bending diaphragm is well-supported and no mechanical joints ordissimilar materials are present.FIGURE 1-53. The Cup Diaphragm Structure:Diffusion ZonesLeadsSilicon Substructure(N-Type typically)The silicon clamp and diaphragm must ultimately be sealed to an outerhousing, or made into a free-floating and hermetically-sealed pressurecapsule, to ensure that the internal pressure reference is maintained(Figure 1-54). The joining method commonly used to bond the sealingplate to the cup-diaphragm structure is known as the Mallory bond andis performed at elevated temperature by applying a high voltagebetween the two highly-polished surfaces. The electrostatic forces thatresult, due to the applied voltage potential between the two parts, producea high-quality nuclear-level seal between the two parts. Since thestrain gages are a diffused, and integral component of the bending diaphragmwith no adhesive bonding material required, the cup structureoffers a stable and low-hysteresis sensor design.FIGURE 1-54. The Cup-Diaphragm Mallory-Sealed Capsule:Diffusion ZonesLeadsP refMallory BondCHAPTER 1-86 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-13:The cup-diaphragm and capsule designs result in the active strain gagesgenerally being exposed to the pressure medium with the exception of thenew Sensonor isolated cup design. In water-laden environments or in thecase of blood pressure measurement, the silicon elements are either coatedwith parylene C or isolated from the medium by the principle of exclusionusing a silicone gel isolating material of Figure 1-49.The highly-miniaturized piezoresistive flush-membrane designs (Figure 1-50) provide extremely wide- and flat-frequency (± 5%) response ranges inexcess of 400 Kilohertz in higher pressure ranges. The combination of ahigh-stiffness, small-diameter, low-mass diaphragm and low operatingstrain requirements (due to the high gage factor shown by the piezoresistivestrain gage) all contribute to this extremely high-frequency capability.The small physical size of the diffused piezoresistive strain gage allows thefabrication of sensors as small as .030Ó in diameter, and the low mass of thediaphragm yields an exceptionally shock-tolerant sensor. It is not uncommonfor this style of sensor to survive exposure to 30,000-gravity (g) shockinputs.Diaphragm Strains:The following relationships may be used to approximate the induced strain in apressure-loaded membrane that is peripherally fixed and subject to the followingassumptions:¥ Uniform diaphragm thickness¥ The material is homogeneous and isotropic¥ InÞnitely-rigid perimeter clamp (For highest performance, the side-wall thickness is at least 10times the plate thickness)¥ Small, perfectly-elastic deßections of not greater than 30% of the diaphragm thickness.¥ No diaphragm stiffening contribution due to the presence of the strain gages.¥ The diaphragm thickness is less than 20% of the diaphragm diameter.¥ The diaphragm deßection is due to bending where no tensile forces exist in the median plane ofthe diaphragm.The quality of the above assumptions is often the reason for differences betweenthe computed and actual strains present. For example, the perimeter clamp will notThe Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-87

Strain-Based Measurement Introduction:be infinitely rigid and side-wall bending will occur especially in the case ofhigh-pressure sensor designs. For this reason, the calculations given are bestused as an initial estimation of the strains that will result for a given diaphragmthickness, diameter, and pressure loading.DeÞnitions: With reference to Figure 1-55.W = Total load in pounds forcew = Pounds per square inch (psi) where: W = wpa 2m = 1/n where n is PoissonÕs ratiot = Diaphragm thicknessE = YoungÕs Modulusa = Diaphragm radiusr = The distance from the diaphragm centere = s/E (strain)y c = Center of diaphragm deflection at maximum loadFIGURE 1-55. Variable Definitions for the Circular Flat Plate UnderUniform Pressure Loading:(W = total load in lbf)w psitrRadius aCLCHAPTER 1-88 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-13:Radial Stress:s r=3W æ--------------- ( 3m + 1) r2 ö---- Ð ( m + 1)8pmt 2 ç ÷èøa 2(EQ 1-38)Tangential Stress:s t=3W æ--------------- ( m + 3) r2 ö---- Ð ( m + 1)8pmt 2 çè a 2 ÷ø(EQ 1-39)At the center of the diaphragm: r = 0 and therefore:3W( m+1)s r = s t = Ð --------------------------èæ 8pmt 2 øö(EQ 1-40)The maximum deflection at the center of the diaphragm is given by:3wa 4 ( 1 Ð u 2 )y c = Ð--------------------------------( 16t 3 E)(EQ 1-41)The radial stress at the clamp occurs when r = a and is given by:3wa 2s ra = ------------4t 2(EQ 1-42)The tangential stress at the clamp also occurs when r = a and is given by:s t=3wa 2 uÐ----------------4t 2(EQ 1-43)By setting equation (1-38) to equal zero, the equation may be solved to find thevalue of r at which the radial stress is equal to zero with the result being:r = 0.63aWhen PoissonÔs Ratio = .3(EQ 1-44)The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-89

Strain-Based Measurement Introduction:Similarly, setting equation (1-39) equal to zero, the equation may be solved tofind the value of r at which the tangential stress is equal to zero with the resultbeing:r = 0.827aWhen PoissonÔs Ratio = .3(EQ 1-45)The radial and tangential stresses are shown graphically as Figure 1-54.FIGURE 1-56.Compression0r/ as rs t3wa 24t 23wa 2 n4t 2-3W(m+1)8p mt 2Tension1.001.0In the calculation of the first-order resonance frequency for the flat, circular,uniform section, plate 1 of thickness t, under uniformly-distributed loading conditions:¥ w = load per unit area including the intrinsic mass of the diaphragm.1. Roarke P716 case 10, 6th edition, Warren C. YoungCHAPTER 1-90 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999

Applications Note 1-13:¥ g = gravitational constant.¥ K 1 = Constant of 10.2 for the Þrst-order resonance (Higher-order resonances are given by constantsK 2 = 21.3, K 3 = 34.9 and K 4 = 39.8)¥ g = Plate density (Plate mass = pr 2 tg)K 1 Dgf n = ----- æ--------ö2pèwr 4 øWhere D =Et 3-------------------------12( 1 Ð u 2 )(EQ 1-46)Simplifies to:0.469t Egf n = -------------- æ----------------------ör 2 èg ( 1 Ð u 2 )ø(EQ 1-47)Other Considerations:1. As the diaphragm deflections become much larger than 30% of the diaphragm thickness, themedian-plane tensile stresses become dominant and diaphragm stretching will act to stiffen thediaphragm significantly. If nonlinearity performance of ±.2% is to be expected, the diaphragmmust not be deflected by more than 12% of the diaphragm thickness. If a nonlinearity performancelevel of 2% is acceptable, the diaphragm may deflect by as much as 30% of the diaphragmthickness. Additionally, with increasing diaphragm deflection and tensile loading, theinflection points on the diaphragm, for zero-radial and zero-tangential stress, will move outwardstoward the clamp, further deteriorating the nonlinearity performance of the sensor. Sincethe strain-gage class of sensors integrates the applied strain over a distance of the active diaphragm,the modification of the inflection points acts to distort the strain field that the straingages are positioned to measure.2. The ratio of the thickness-to-radius ratio of the diaphragm is an excellent indicator of diaphragmperformance. When this ratio is less than .15, the symmetrical bending theory,described, holds well. For values greater than .15, the loads normal to the plane of the diaphragm,and shear loads, must be taken into account. In the analysis of the performance of adiaphragm, both the deflection/diaphragm thickness ratio and the diaphragm thickness/radiusratio must be assessed for superior performance to result.The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999 CHAPTER 1-91

Strain-Based Measurement Introduction:3. In cases where diaphragms are of the recessed design, pressure loading of the internal cavitycan produce distortion of the sensor housing, resulting in direct tensile loading in theplane of the sensor diaphragm which acts to desensitize the diaphragm to bending. Thiseffect is most often experienced in low-pressure-range differential pressure sensors thatmust support high-level line pressures.CHAPTER 1-92 The Art of Practical and Precise Strain Based Measurement 2nd Edition © 1999