INPUT sensor transducer Signal A MEASUREMENT SYSTEM Signal B OUTPUT Signal C Zero order responds instantly to measurand no instrument is truly zero order but many can be approximated as zero order First order show capacitive type energy storage effects mechanical analogs to capacitors are springs and devices that store thermal energy thermometer is a first order system Second order have inertial effects of inductance or accelerated mass as well as capacitive effects common spring mass systems bathroom scale Second order systems include a characteristic called damping or energy loss Covers systems that possess inertia accelerometers loudspeakers pressure transducers d2y dy Here s the equation a2 2 a1 a0 y F t dt dt We ll see this more often in the standard form 1 n a0 a2 2 y n 2 n y y kF t a1 2 a0 a2 1 k a0 Let s examine the solution to this equation 1 2 y y y kF t 2 n n In the absence of a forcing function we have the homogeneous equation y 2 n y n2 y 0 This has the solution yh t Ce t If we plug this back in we get the characteristic equation 2 2 n 2n 0 This quadratic equation has two roots given by 1 2 n n 1 2 The nature of the solution depends on this 1 2 n n 1 2 yh t C1e 1t C2e 2t Example Under damped Case 0 1 1 2 n i n 1 2 1 2 n n 1 2 We can plug this in above y h t C1 e i 1 n n 2 t C2e i 1 n n 2 t We can rewrite this as yh t e nt B1 sin d t B2 cos d t We see that the homogeneous transient solution is a damped sine wave d n 1 2 Damped natural frequency In the absence of damping the response oscillates at n yh t e nt C1 C2 n t Critically Damped Case 1 Over damped Case 1 y h C1 e 2 1 t n C2e 2 1 t n 1 2 y n 2 n y y kF t Lets consider a step function input F t AH t We have for t 0 1 2 n y 2 n y y kA homogeneous particular The total solution is given by yT yh y p Here y p kA Let s take the case of a critically damped system as an example yh t e n t C1 C2 n t yT kA e nt C1 C2 nt Taking the initial conditions as y t 0 dy t 0 0 dt We get yT kA kA 1 nt e nt Critically damped Under damped Case 0 1 n t yT kA kAe sin d t cos d t 2 1 Over damped Case 1 2 1 y t kA kA e 2 2 1 2 1 n t 2 1 2 1 2 e 2 1 t n An under damped system oscillates before reaching steady state T yT kA kAe n t sin d t cos d t 1 2 d n 1 2 1 2 T f d 1 n Ringing frequency Period Controls the duration of the transient response analogous to the time constant t in first order systems Ringing frequency period Oscillations reduce to 10 Reach 90 systems with damping ratio 0 7 gives good balance between speed and ringing system parameters may be chosen to fall in this range 1 2 n y 2 n y y kA sin t The total solution is given by yT yh y p Recall that the homogeneous part of the solution describes the transient behavior of the system here we can focus on the particular solution Note that the homogeneous solution depends on the damping ratio and has already been given How do we find the particular solution Method of undetermined coefficients y p a cos t b sin t yp kA sin t 1 2 n 2 2 2 n 2 2 n Arc tan 2 1 2 n Desired system behavior magnitude ratio 1 phase shift 0 Note from the forcing function n property of the system Magnitude ratio 1 around the resonance frequency for under damped cases Resonance frequency of underdamped system occurs at D n 1 2 2 phase shift occurs at resonance for undamped system characteristic of resonance For many sensors operating in the resonance band not desirable Systems with a damping ratio of greater than 707 do not resonate Transmission band 3dB M 3dB Filter band M 3dB eliminate high frequency signals Number of Students 15 Mean 74 Standard Deviation 12 10 5 0 40 60 80 Grade 100
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