31 May, 1995 AUTHOR: Gordon B. Bowden grams... LJC 1.0 Hz GEOPHONE COIL RESISTANCE, OHMS

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1 I NLC ME NOTE I I TITLE: Mark L4C Geophone Design Constants lvo. l-94 Rev 2 DATE: 31 May, 1995 AUTHOR: Gordon B. Bowden Almost all vibration measurements made at SLAC have be made with several Mark 4LC geophones. The output response of these instruments continually comes in question. What do they measure and what are their calibration constants? These geophones contain a permanent magnet and a moving coil attached to a spring suspended mass. They are simple passive electro-mechanical sensors. Over a limited frequency range, they produce an output voltage proportional to the relative velocity of the mass, measured with respect to the permanent magnet which is fixed in the housing of the transducer. The dimensions of our transducer are given in the MARK PRODUCTS data sheets reproduced below TYPE... FREOUENCY... FREOUENCY CHANGE WITH TILT... FREOUENCY CHANGE WITH EXClTATlON.... SUSPENDED MASS:.... STANDARD COIL RESISTANCES... LEAKAGE TO CASE... TRANSDUCTION POWER... OPEN CIRCUIT DAMPING... CURRENT DAMPING... COIL INDUCTANCE... ELECTRIC ANALOG OF CAPACITY... ELECTRIC ANALdG OF INDUCTANCE... CASE HEIGHT.... CASE DIAMETER... TOTAL DENSITY... TOTAL WEIGHT... OPERATING TEMPERATURE... LIC 1.0 Hz QEOPHONE Movingdualcoil,humbuckwound * 0.05 Hz measured on 200 pound weight at 0.09 inches/second... Less than 0.05 Hz at 5 from vertical... Less than 0.05 Hz from 0 to 0.09 inches/second grams... 5OO, megohm minimum at 500 volts m3 watts/inch/second or 13.6 watts/meter/second... (bo) = 0.28 critical.... (bc) = Rc Rs+Rc Lc = Rc Lcinhenries... cc =,-, 73,500 (microfarads).... Rc Lm = 0.345Rc (henries).... 5% inches-13 cm... 3 inches-7.6 cm gramslcm % pounds-2.15 kilograms... Range: -2O to140*for -29*to6O C.. LJC 1.0 Hz GEOPHONE COIL RESISTANCE, OHMS TRANSDUCTION, VOLTS/IN/SEC COIL INDUCTANCE, HENRIES ANALOG CAPACITANCE, MICROFARADS ANALOG INDUCTANCE, HENRIES SHUNT FOR 0.70 DAMPING, OHM

2 Notes on Data Sheet Type The L4C has its coil moving in a magnet fixed to the housing of the geophone. The alternative arrangement which fixes the coil to the housing and uses a moving magnet for the proof mass has the disadvantage that the moving magnet has extra eddy current damping and sets up time varying magnetic fields which can be detected outside the housing by another geophone for instance. Standard Coil Resistance Our geophone has. the 5500 R coil. Transduction Power This power is the transduction constant squared divided by the coil resistance: G /R,. Open Circuit Damping Mechanical motion of the proof mass is damped even when the coil circuit is open and no current can flow. This damping comes from air resistance inside housing and mechanical hysteresis in the suspension springs. Damping is given in terms of damping ratio C. For a damped simple harmonic oscillator the damping force b = 2M&. L b = 2(1 Newton sec2/m)(.28)(2i; radians/set) = 3.52 Newton set/m. Coil Inductance For a given set of dimensions, coil inductance scales with resistance: L, = O.OOllR, =.0011(5500 0) = 6.05 Henries. Electric analog capacity and inductance These quantities are the electrical analogs of the mass and spring constant to be used in the dynamic analysis if the mechanical system is modeled by its electrical analog. Transduction Transduction constant G is the most critical parameter of the geophone. It specifys the voltage output of the coil for a given velocity. It depends on the magnetic field and the coil geometry and number of turns. Equations of Motion- a simple linear model Coil Voltage v,,il = G(k - i9) Coil Force FcO;l = -Gicoil Force and voltage equilibrium b(2, - 5) + k(x, - X) = MZ + Gi,,il (bj, + kxg) = (MZ + bj, + kx) + Gicoil G(i - js> = (.& + Rj)i,,;/ + Lc*

3 Laplace transformed equations. (bs + k)&(s) = (Ms2 + bs + k)x(s) + GI(s) kids) = WX(s)-X,(s)]= (R,+ R, + L,s)I(s) Solving for X(s) in the first equation and substituting this into the 2 nd, the transfer function between ground motion X0(s) and coil current I(s) can be written: I(4-1 -= x9 (4 L,s+R,+R, >( MGs3 MS" + (b + Lcs+y+R )s + k> C I At frequencies of 10 Hz or less, the coil impedance L,s is small compared to the coil resistance I?,: L,s/R, = (6.05 Henries)(lO * 2n rad/sec)/(5300 n> =.07. Neglecting the effect of the coil inductance and writing the transfer function for Kut(S) = R,](s) in terms of ground velocity X;,(s) = sx&) in standard form where k jill G ci: I/out(s) -R,G s* *g (4 = R, + R, s2 + 2woQ + w; > where TWO< = (b+ G2 Rc + R, JIM The conversion factor between Voui and ground velocity input X9 depends on the value chosen for the external damping resistance R,. If damping (C =.707) is chosen, then external shunt resistance RJ = is needed across the internal coil resistance R, = 5500R. The design transduction G = 7.02 volt set/in = volt set/meter so the conversion factor for a damped L4C geophone is: Vout -z= 8905 R (276.4 volt set/meter) 5500 l-l R A typical ground motion of 30 nanometers at 10 Hz should produce a 322 ~~011 signal at the geophone s output terminals if an 8905 R external damping resistor is used. To see the amplitude and phase variation of the signal with frequency, substitute s = iw and plot the amplitude and phase of the complex function e. 9 sw

4 ...I t.e B P 140 ' I il * 60- : 40-4

. The geophone has a 1 Hz resonance without the 8905 R shunt resistance which increases the damping to c z.7. The output signal is about 170 volt set/meter for frequencies above 2 Hz.

5 . The geophone has a 1 Hz resonance without the 8905 R shunt resistance which increases the damping to c z.7. The output signal is about 170 volt set/meter for frequencies above 2 Hz. The output voltage follows the input velocity for high frequencies but lags further behind as the frequency falls. b Calibration. The dimensions and constants come from the manufacturer s data sheets and the model of the geophone is a simple linear one. Our actual geophone may not have quite this response. Direct calibration of its output signal at 30 nanometers motion amplitude is difficult. It is hard to generate such small motions with the required accuracy and we as yet have no laboratory free from microseismic ground motion which is of this magnitude. A much larger amplitude calibration would be relatively simple but would presume that linear extrapolation down into the nanometer realm was valid. A sinusoidal motion of 100 microns is easy to generate with a motor and eccentric bearing. FFTB magnet positioners have proven that 1 micron motions are practical with standard roller bearing cams. Such a large motion is also easy to measure with an LVDT or Kaman proximity gage. At 10 Hz a 100 micron amplitude should generate about 1.1 volts signal which would require only a high input impedance follower between the geophone and the digitizer. Such a system should allow the amplitude and phase of the response to be measured over frequencies between.l Hz and 30 Hz. The upper frequency limit would be set by mechanical resonances in the shake table. Such a calibration scheme is sketched out below. Kaman Position Xducer

Electronics and Instrumentation Name ENGR-4220 Fall 1999 Section Modeling the Cantilever Beam Supplemental Info for Project 1.

Electronics and Instrumentation Name ENGR-4220 Fall 1999 Section Modeling the Cantilever Beam Supplemental Info for Project 1. Name ENGR-40 Fall 1999 Section Modeling the Cantilever Beam Supplemental Info for Project 1 The cantilever beam has a simple equation of motion. If we assume that the mass is located at the end of the