A complete listing of the MathCad program is attached at the end of this document.

Transcription

1 Sensor Products Dvson (Europe) RB WK39/ Detectng Wheel Rotaton R H Brown September Summary: a smple mathematcal descrpton of a rotatng wheel has been created to allow the acceleraton experenced at the valve stem locaton to be modelled. The effect of ACcouplng ths acceleraton sgnal (as would result from the use of a pezoelectrc accelerometer) can then be nvestgated. A dscrete-tme model has been used, whch allows the effect of the hgh-pass flter to be appled usng FFT routne. The results show that a hgh-pass flter wth cut-off frequency at around 1 Hz should be suffcent to detect the onset of rotaton (wthn a few rotatons), and that a hgher-order flter (greater than sngle-order RC) may be advantageous n some crcumstances. A complete lstng of the MathCad program s attached at the end of ths document. Introducton There are two components of acceleraton present n a rotatng wheel: the oscllatng sgnal due to the relatve orentaton of the accelerometer to gravtatonal pull (+/- 1 g), and the centrpetal acceleraton gven by v /R. If a DC-coupled accelerometer technology (e.g. capactve, pezoresstve) s used as a detector, the rotatonal (centrpetal) component wll very quckly domnate, wth the gravtatonal component appearng only as a slght rpple supermposed on the much slower, but hgher magntude, centrpetal sgnal. To detect the onset of wheel rotaton, a DC-coupled devce could be used to detect when the centrpetal acceleraton exceeds a gven threshold (mplyng a fxed vehcle speed threshold). If an AC-coupled (e.g. pezoelectrc) accelerometer s used, however, the majorty of the centrpetal acceleraton sgnal may be lost (dependng upon the exact locaton of the 3 db cut-off frequency). The purpose of ths model s to nvestgate the approxmate condtons whch could result n a threshold-detecton of rotaton. Settng up the model The attached MathCad lstng annotates each step, but bascaly four tme perods are ncluded: an ntal perod wth no moton, an acceleraton phase from rest to a maxmum speed, a constant speed phase, and a deceleraton phase back to rest. The duraton of each phase can be vared, and f the sum of all perods s less than the total tme record, a further phase of zero speed s assumed. Both the acceleraton and deceleraton phases are assumed to be lnear. Ths smplfes the model, and any practcal level of acceleraton can be nvestgated. The shape of the acceleraton profle tself s not consdered to be partcularly sgnfcant n ths analyss. The tre crcumference can be altered, as can the radus at the valve stem, to sut any gven wheel geometry. Flattenng of the tre s not taken nto account n the "dstance travelled" calculatons agan, ths s not consdered essental to the nvestgaton. Four plots are generated ntally: vehcle speed versus tme, dstance travelled versus tme, gravtatonal acceleraton versus tme, and total acceleraton (centrpetal plus gravtatonal) versus tme. Measurement Specaltes, Incorporated, Sensor Products Dvson, PO Box 799, Valley Forge, PA , USA Tel: Fax:

2 Sensor Products Dvson (Europe) RB WK39/ Applyng the hgh-pass flter In the fnal secton of the model, a smple hgh-pass flter s created by modellng the effect of an R-C combnaton (resstance & capactance). Ths s partcularly approprate, snce any pezoelectrc sensor wll have a fxed capactance, and wll most lkely see a fxed value of nput resstance wthn the electronc nterface. By selectng approprate value of C (equvalent to typcal value for small area & thckness of pezo flm), and antcpated value of R, the model drectly predcts the effect on the fnal output sgnal. The nfluence of ths RC combnaton s computed for each frequency lne of the spectrum of the acceleraton sgnal, and the product of the acceleraton and flter spectra s taken. Fnally, nverse FFT s used to return to the tme doman. The resultng plot s thus the acceleraton sgnal as seen after passng through the hgh-pass flter. In further analyses, the HPF characterstc was "squared" or rased to even hgher powers (3 rd or th ) before applyng to the acceleraton sgnal. Ths s equvalent to the effect of a nd, 3 rd or th order HPF wth same cut-off frequency. Observatons Wth the centrpetal component of acceleraton largely removed, a sngle-order HP-fltered sgnal shows the +/- 1g gravtaton sgnal very clearly. Under steady acceleraton, a slght lnear ncrease can stll be seen, whch dsappears at constant speed. Fast deceleraton causes a sgnfcant negatve-gong "transent", where the gravtatonal sgnal would be vrtually lost. By settng other values wthn the model, t has been found that an approxmate +.5 g threshold would be approprate to detect onset of rotaton, even when the acceleraton rate s very low (e.g. from rest to 5 kph n 5 seconds) trgger occurs after ether one or two complete rotatons (after one, threshold s just crossed). Usng hgher-order HP characterstc, the lnear ncrease at fast acceleraton s almost completely removed, and the negatve-gong transent under fast brakng s reduced n ampltude and duraton. A nd order flter gves good response, wth 3 rd or hgher orders not mprovng the sgnal sgnfcantly. Whether the nd order response s actually requred n practce would depend upon the algorthm used to detect (and mantan "condton" of) rotaton. Alternatvely, a hgher cut-off frequency of 1 st order HPF could be used (to remove more of the centrpetal sgnal), although ths would ncrease the dstance or speed requred to trgger detecton. A 1 Hz cut-off frequency appears to be approxmately best for detectng the gravtatonal sgnal. Wth around.5 g trgger level, an accelerometer wth 1 mv/g senstvty (such as MSI's ACH-1) would gve only 5 mv trgger level, so may requre gan of 1 or so to brng sgnals up to usable level. Alternatve components such as LDTC wth around 1 V/g senstvty at low frequency may also be sutable, although n ths case the mass-loaded cantlever structure would be expected to "bottom out" at hgh centrpetal acceleraton. Sgnal may be lost entrely, untl speed reduces to lower levels. Agan, ths may stll be acceptable, dependng upon the detecton algorthm. Measurement Specaltes, Incorporated, Sensor Products Dvson, PO Box 799, Valley Forge, PA , USA Tel: Fax:

3 WHEEL6.MCD R H Brown MSI Sensor Products Dvson 7 March 1 Ths program calculates gravtatonal and rotatonal acceleraton for typcal wheel at varyng road speeds. It also allows a hgh-pass flter to be appled to the "deal" acceleraton sgnal (usng FFT). A dscrete-tme (sampled data) approach s used. N 96 # tme samples, must be nteger power of (for FFT) (Mathcad v 3.1 wll accept max 96 here) 1,.. N 1 k 1,.. N (range varables) samptme. samplng nterval, n seconds. samptme sampfreq 1 samptme k freq. k sampfreq = 5 N sampfreq maxtme ( N 1). samptme maxtme = 8.19 overall length of tme record t t s the "pre-trgger delay" pror to movng off t1 5 maxvel 5 t1 s tme to accelerate from to full speed maxvel s full speed, n kph pos maxvel pos = 1 pos s constant postve acceleraton t1 typcal practcal max = ( to 1 kph n 5 s) t.5 cvel maxvel t s dwell tme at max speed t3 1. mnvel t3 s tme to decelerate back to. Ths can be adjusted to leave wheel n correct "phase" for later FFT neg maxvel t3 neg = 9.9 neg s constant deceleraton value typcal practcal mn =? cyctme t t1 t t3 cyctme = 6.5 check that total cycle tme fts wthn the tme record (cyctme < maxtme)! Defne the speed vs tme relatonshps: c1 t c c1 t1 c3 c t c c3 t3 f tme. < c t, tme. c1 pos, f c, maxvel, f c3, c tme. neg, f c,, f < t,,

4 velm 3.6 convert speed n kph to m/s dst defne ntal dstance travelled as zero, then form runnng ntegral for each successve tme ncrement dst. ( > 1) dst. 1 velm samptme crc 1.61 defne crcumference of typcal tre (Wndstar), n m mant( x) x floor( x) dstrem mant dst crc a functon to extract fractonal porton of dstance dvded by crcumference grav1 sn dstrem.. π grav sn dstrem.. π grav3 cos dstrem.. π calculates the gravtatonal acceleraton "seen" by the sensor, accordng to the poston of the wheel. Only grav1 s used n followng analyss, but any "phase" of ntal startng pont s possble. grav cos dstrem.. π rps velm crc calculates the revs per sec (Hz), and dsplays max value max( rps) = radfreq.. rps π converts revs/s nto radan frequency (rad/s) radus.8 Note: radus at valve stem locaton (not outer tre radus) accm. radfreq radus calculates centrpetal acceleraton (n m/s/s) accg accm 9.81 converts to unts of "g" acc accg grav1 total nstantaneous acceleraton = sum of gravtatonal and centrpetal Next, add plots of varous parameters versus tme (see next page)

5 Speed n kph vs tme: Total dstance travelled vs tme: dst Gravtatonal acceleraton (n g) vs tme: 1 grav Total acceleraton (n g) vs tme: 6 acc

6 Ths secton apples an R-C hgh-pass flter to the total acceleraton sgnal: mag fft( acc) calculate spectrum of the acceleraton tme sgnal C set value of source capactance R set value of nput resstance RC. =.15 dsplay tme constant (n s) fc 1. π. RC. calculate & dsplay cut-off frequency (-3 db) fc = 1.61 ω. π. freq convert frequency values to from Hz to radans ω.1 nsert fnte (non-zero) freq for DC (requred because ths term appears n denomnator) X 1. ω C reactance of capactor over frequency Z ( j. X) mpedance of capactor Vout R R Z calculate output of R-C flter out ( mag. Vout) apply RC flter to the acceleraton spectrum (note: can change exponent of Vout to yeld hgher-order flter) facc fft( out) fnally, take nverse FFT to convert fltered spectrum back nto tme doman Acceleraton sgnal after hgh-pass flter: 5 facc

7 out mag. Vout apply RC flter to the acceleraton spectrum (note: can change exponent of Vout to yeld hgher-order flter) facc fft( out) fnally, take nverse FFT to convert fltered spectrum back nto tme doman Acceleraton sgnal after hgh-pass flter: facc

8 out mag. Vout 3 apply RC flter to the acceleraton spectrum (note: can change exponent of Vout to yeld hgher-order flter) facc fft( out) fnally, take nverse FFT to convert fltered spectrum back nto tme doman Acceleraton sgnal after hgh-pass flter: facc

9 out mag. Vout apply RC flter to the acceleraton spectrum (note: can change exponent of Vout to yeld hgher-order flter) facc fft( out) fnally, take nverse FFT to convert fltered spectrum back nto tme doman Acceleraton sgnal after hgh-pass flter: facc

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