yp kA sin t 1 2 n 2 2 2 n 2 2 n Arc tan 2 1 2 n Desired system behavior sometimes 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 under damped system occurs at D n 1 2 2 p phase shift occurs at resonance for undamped system characteristic of resonance 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 A strain gage measurement system is mounted on an airplane wing to measure wing oscillation and strain during wind gusts The strain system has a 90 rise time of 100 ms a ringing frequency of 1200 Hz and a damping ratio of 0 8 Estimate the dynamic error in the measurement of a 1Hz oscillation Also estimate any time lag Dynamic error M 1 What kind of system is this 2 1 n We are given the ringing frequency d Natural frequency n Drive frequency 2p d 1 2 2 p 1200 0 36 4000p We can plug in and find out M 1 2 n 2 2p 5 x10 4 n 4000p M 1 2 2 n 1 1 5x10 2 0 80 5x10 2 4 2 4 2 1 2 There is virtually no dynamic error in this system 2p How do we find the time delay td 2 4 2 0 8 5 x 10 Arc tan n Arc tan 2 4 2 1 5 x10 1 2 n 8x10 4 rad td 0 127ms Select one set of appropriate values for damping ratio and natural frequency for a second order instrument used to measure frequencies up to 100 rad s with no more than 10 dynamic error A catalog offers models with damping ratios of 0 4 1 and 2 and natural frequencies of 200 and 500 rad sec In the end we have to check these M 1 1 2 n 2 2 2 n 2 n n M M 1 200 4 5 1 18 0 18 200 1 0 5 64 0 36 200 2 0 5 47 0 53 500 4 2 1 03 0 03 500 1 0 2 96 0 04 500 2 0 2 80 0 20 1 The system is under damped 2 We have a static sensitivity of 1 0 mm mm and an input magnitude of 1 0 Therefore that magnitude ratio is M 5 The dynamic error is M 1 5 0 1 0 4 0 3 The time lag is 10 ms the phase lag is thus 2p 10 p rad 90deg rees 40 2 4 Assume we are operating close to the natural frequency M n 5 M n 0 1 1 5 2 EXAMPLE In order to measure the vibration of a test part the output of an air coupled transducer is sent into a spectrum analyzer The spectrum analyzer display is shown in the figure to the right Note that the spectrum analyzer displays the magnitude of the signal versus frequency an output analogous to the FFT of a time domain signal The measurement is made using a transducer second order system with a resonance frequency of 15kHz a damping ratio of 0 8 and a static sensitivity of 10V mm You note that the part seems to be vibrating strongly at two resonance frequencies and you are interested in quantifying the magnitude of the displacement at each frequency a Estimate the dynamic error at each frequency b Estimate the actual magnitude of the displacement in mm of the part at each frequency c Estimate the time delay at the lowest frequency WOW WE NEED TO DECIDE HOW TO ATTACK THIS PROBLEM y t A sin t System 1 K1 M 1 Frequencies 1 2p f1 2p 4 9 x103 30 78 x103 rad s 2 2p f 2 2p 20 x103 125 66 x103 rad s System Parameters n 2p f n 2p 15 x103 94 24 x103 rad s 0 8 K 10V m m y t AK1M 1 sin t NOW WE CAN CALCULATE THE DYNAMIC ERROR The magnitude responses are M 1 M 2 1 2 3 2 3 30 78 x10 2 0 8 30 78 x10 1 3 94 24 x103 2 94 24 x10 1 2 2 3 2 3 125 66 x10 2 0 8 125 66 x10 1 3 3 2 94 24 x 10 94 24 x10 2 What is the dynamic error at each frequency 1 M 1 1 0 034 2 M 2 1 0 562 1 0 966 0 7980 0 2730 1 0 438 0 6502 4 551 HOW ABOUT THE INPUT AMPLITUDE y t A sin t System 1 K1 M 1 AK1M 1 0 32V 1 0 25V 2 A 1 0 32V 1 0 033mm 10V mm 0 96 A 2 0 25V 1 0 058mm 10V mm 0 43 y t AK1M 1 sin t FINALLY WE HAVE THE TIME DELAY AT THE LOWEST FREQUENCY What do we need first 3 2 0 8 30 78 x10 2 3 94 24 x 10 5225 n Arc tan Arc tan 0 5292rad Arc tan 2 2 8933 1 30 78 x103 1 2 3 2 n 94 24 x10 What is the time delay 5292 17 2m s 3 30 78 x10