Chapter 3.2 Notes - Chapter 3 Measurement System Behavior...

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1 Chapter 3 Measurement System Behavior Part 2 1 st Order – Sine Function Input • Examples of Periodic: vibrating structure, vehicle suspension, reciprocating pumps, environmental conditions • The frequency of the input significantly affects measuring system time response. • Consider: t τω yy K A o += sin t yy = + 0 0 0 () General Solution yt Ce KA t ( )/[ ( )] s in ( tan ) / / =+ + τ ωτ ϖτ 1 2 12 1 Solve for C at t=0, y(0)=y 0 Output Transient Ce t / τ 0 With t going past 5 τ ωπ = 2f Note : amplitude and phase shifts are dependent on the frequency of input .
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2 Output Steady State • Amplitude and phase shift are functions of input frequency • Output frequency is the same as input frequency • Output remains as long as the forcing function Rewritten: yt Ce B t t ( ) () s i n [ ] / =+ + t ww j w Magnitude: BK A 1 () ( ) / [ ( )] / ωω τ 2 12 Phase Shift: ϕω ω τ t a n =− 1 Where τ is the time constant. Time Delay βϕωω 1 = (() ) / Value of β 1 is negative, indicating a time delay between input and output. Magnitude ratio of input/output: m( ω )=B/KA=1/(1+( ωτ ) 2 1/2 Note the frequency dependency
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3 • For value of w τ which gives m( ω ) near unity. The first order MS will transmit nearly all of the input signal with little delay or attenuation of signal. •B ( ω ) KA, ϕ ( ω ) ≈∅ o • If you want to monitor an input signal with high frequency, you will need τ to be very small in order to give m( ω ) 1 Freq response: magnitude ratio (Figliola, 3 rd ) Freq response: phase shift (Figliola, 3 rd ) Dynamic Error δ ( ω ) = m( ω ) – 1 – measures systems inability to adequately reconstruct the amplitude of input at a given frequency. • Want to minimize δ ( ω ) • Perfect reproduction is not possible Frequency Bandwidth • Frequency band over which m( ω ) 0.707 or where dB=20 log m( ω 1 ) does not drop more than –3 dB between ω n and ω m for a given τ • dB=20 log m( ω n ) + 20 log m( ω m ) • dB=20 log (1) + 20 log (0.4) •dB= -7 .95
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4 Determination of Frequency Response • The frequency response is found by dynamic calibration, by applying a simple periodic waveform input to the sensor stage and monitoring the output. However, this can be impractical in some physical systems.
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Chapter 3.2 Notes - Chapter 3 Measurement System Behavior...

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