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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
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The Art of Practical andPrecise Str
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Special Credits:Mrs. Helen Pierson,
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Table of ContentsBy James G. Pierso
Page 8 and 9: ContentsApplications Note 4-10:Lead
Page 10: ContentsCHAPTER 8Electrical Conside
Page 13 and 14: CHAPTER 1Strain-BasedMeasurementInt
Page 15 and 16: Applications Note 1-1: Introduction
Page 17 and 18: Applications Note 1-1: Introduction
Page 19 and 20: Applications Note 1-2:The Art of Ap
Page 21 and 22: Applications Note 1-31. The presenc
Page 23 and 24: Applications Note 1-3result in a pe
Page 25 and 26: Applications Note 1-4:The measureme
Page 27 and 28: Applications Note 1-4:uncertainty i
Page 29 and 30: Applications Note 1-5:All strain-ba
Page 31 and 32: Applications Note 1-6:Applications
Page 33 and 34: Applications Note 1-6:ical size of
Page 35 and 36: Applications Note 1-6:and where:j i
Page 37 and 38: Applications Note 1-6:5. millivolts
Page 39: Applications Note 1-6:.Electrodynam
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