Applied Geophysics : Seismics

Applet    ( in separatem Fenster, ca. 820 x 610 Pixel )

The applet simulates

an astatic horizontal seismometer / tiltmeter with a "vertical" axes of revolution.

Dimensions and technical specifications are taken from

a Sprengnether S-5100-H seismometer.  ( Sprengnether Instruments Inc., St. Louis, Mo. USA )

The initial state of the applet represents an instrument

completely assembled from factory-adjusted parts and ready for installation and adjustment.

Seismometer

Screenshot

Specifikations

Comments

Coordinate System

Equation of Motion

Additional Parameters

Simulation

Howto

Operation

Mechanical Adjustment

Electrical Adjustment

Position Marker

Period Measurement

Sensitivity to Tilt

"Cross Coupling"

Download

Screenshot :

The mechanical system of the instrument is displayed in the graphics area of the applet as a simplified horizontal section  ( scale approx. 1 [Pixel/mm] ) :

A pendulum  ( beam + seismic mass, blue )

is suspended by two hinges  ( crossed flexures, H, red ),  forming a vertcal axes of revolution.

The tilt of the instrument, i.e. the deviation of directions of hinge line and local gravity

is adjusted by three levelling screws  ( orange ),  and

can be observed at two bubble levels  ( orange )  of different sensitivity :

LONG LEVEL  -0.05...+2.0 [deg]  ( 36 [seca/pixel] ) => natural period ≥ approx. 6 [s] for tilt ≤ +2.0 [deg],

CROSS LEVEL  ±200 [seca]     ( 1.25 [seca/pixel] ) => static angular excursion of beam.

The angular excursion of the pendulum

is limited by two stops to approx. ±1.6 [deg]  ( = ±11 [mm] from zero of scale S )

and

can be obseved as a moving blue mark at an enlarged scale,

its value being displayed  ( S_SCL [mm] )  at the upper edge of the graphics area.

Electro-mechanical components are displayed as simplified electric circuits :

the damping coil  ( orange )  with adjustable external damping resistance R_EXT and

the calibration coil  ( green )  with adjustable DC-current I_CAL.

The output signals for

displacement  ( DSP OUT, blue )  and velocity  ( VEL OUT, magenta )

can be displayed on a "strip chart recorder" with selectable amplitude and time scale.

Program and recorder are controlled in a menue field above the graohics area of the applet.

Moving Mass

11.126

[Kg]

Moment of Inertia

1.333

[kg∗m^2]

Distance from Hinge line to :

Center of Oscillation

35.75

[cm]

Center of Mass

33.50

[cm]

Center line of Signal Coils

35.64

[cm]

Center line of Calibration Coil

17.15

[cm]

Scale

43.18

[cm]

2 Signal / Damping Coils :

Coil Constant

90.0

[Vs/m]

Coil Resistance

500

[Ohm]

Calibration Coil :

Coil Constant

5.0

[N/A]

Coil Resistance

68

[Ohm]

Screenshot    Table of Content    Top od Page

The angular excursion of the pendulum is described in a carthesian coordinate system fixed to the frame of the instrument.

Z = direction of axes of revolution  ( hinge line ),  "vertical", ( pos. downward ),

X = "along beam" from hinge line H to zero of scale S,  "horizontal",  ( in applet pos. to left )

Y = "across beam",  "horizontal",  ( in applet pos. upward )

"vertical" = Z-axes, evtl. tilted against local direction of gravity

"horizontal" = XY-plane, perpendicular to axes of revolution

The equilibrum of all internal and external forces acting on the pendulum leads to a differential equation of 2nd order

for the variables  ( time functions )

and with constant parameters of the instrument

adjustable quantities

and the external disturbance

contributing to the moment of gravitational acceleration and

neglected in the adjustment procedures performed in a very quiet environment.

Using the approximation for small angular excursions of the moment of gravity

and deviding the resulting differential equation by the moment of inertia leads to

the normalized equation of motion

with commonly used abbreviations

Some of the parameters used inthe calculations,

the torsion constant  τ  and the angular offset  ϕ₀  of the axes of revolution and

the mechanical damping  δ₀  of the instrument,

are not listed with the manufactorer's specifikations.

Their values have to be estimated / set arbitrary :

In the applet these parameters are derived from the static equilibrum

with an adjustment

leading to a natural frequency and a static angular excursion of the beam

For the calculation of the simulation

the corresponding natural period is set to  ` T₀ = 100 [s],  the angular offset tonbsp; ϕ₀ ca. 0.2 [deg]

and

the dimensionless mechaical damping  α₀( T₀ ) = 0.05

is set to an extremely small value, to allow the observation of several free oscillations at periods above 100 [s].

( realistic : 2 to 3 oscillations above 100 [s] )

The assumption  T₀ = 100 [s]  leads to

I.e. at an exactly levelled instrument ( G_X = G_Y = 0 ) we observe

a natural period  T_PER  of approx. 100 [s],

a mechanical damping  α₀  of approx. 0.05

and

a static excursion of the beam of approx. 0.31 [deg]  ( S_SCL approx. 2.15 [mm] ),

whereas for different values of the natural period

the mechanical damping increases / decreases as  T_PER, the static excursion as T_PER².

The angular excursion  ϕ(n∗δt)  of the pendulum is calculated recursively in constant time steps of δt = 80 [ms],

evaluating a difference equation derived from the equation of motion

with

coefficients, derived from the values of the mechanical and electrical specifications, the additional parameters and the adjustable quantities.

The real time behaviour of the system depends on the graphics power of the local computer, and

can be modified in the dialogue area of the applet  ( see below ).

The output signals for displacement ( DSP OUT ) and velocity ( VEL OUT )

are displayed on a "analogue" voltage meter / chart recorder,

scaled to correspond to the motion observed at the scale.

Additional Parameters    Equation of Motion    Coordinate System

Screenshot    Table of Content    Top od Page

Dialogue area :
( Screenshot )

HALT / RUN / RES

Stops and starts continuous calculations and graphic display

and

resets mechanical adjustments to initial values ( after assembly of the instrument ).

STEP

Initiates calculation and graphic display for one time step ( 80 [ms] ).

POS / NEG

Selects the polarity of the calibration dc-current, raising / lowering the pedulum from its actual zero position.

ON / OFF

Swiches the calibration current on / off.

T_REAL = ... Allows to adjust the real time behavior :

T_CAL displays the result of calculation of one time step ( 80 [ms] ) in real time,

T x 2, T x 4 reduces,

T / 2, T / 4, T / 10 increases the apparent speed by factors of 2, 4, 10,

and

AUTO selects the highest speed, compatible with given computing power.

DISPLAY / RECORDER Selects the the display mode of the output signals :

DISPLAY simulates an analog voltage meter,

RECORDER simulates a strip chart recorder,

both calibrated in [mm] and [mm/s] referred to scale, according to the selected respective full scale value F.S. for DISPLACEMENT and VELOCITY.

40 [sec] ... 800 [sec]

is enabled in RECORDER mode only and selects the display length ( => resolution ) in time.

HELP displays

hints to possible mouse button actions and

the actual position of both bubble levels  ( resolution 1/1000 [seca] ),

INFO

the horizontal components G_X and G_Y of gravity.

HELP + INFO lists

the actual values of the "counters" of the levelling screws, the resulting tilt of the instrument

and

approximate values of the static excursion S_SCL, of the natural period T_PER and of the damping α,

esimated from the coeffitients of the equation of motion,

at the screen / the JAVA-console.

Screenshot :

( adjustment s. screenshot applet )

The fields | + + | + | - | - - |  increment / decrement the setting of the corresponding levelling screw :

left mouse button by 1  ( | + | - | )  and by 100  ( | + + | - - | )

and

middle or right button by 10  ( | + | - | )  and by 1000  ( | + + | - - | ).

1000 counts correspond to

approx. 1.47 [mm] for the levelling screw at the hinge end of the instrument  ( -> LONG LEVEL -> natural period )

and

approx. 16.3 [µm] for both screws at the scale end  ( -> CROSS LEVEL -> zero position ).

The different "pitch" of the screws allows

to cover the range of approx. 0 to +2 [deg] for the LONG LEVEL  ( T_PER >100  to approx. 6 [s] ),

and

to adjust the CROSS LEVEL sufficiently exact at natural periods T_PER above 100 [s] 

( sensivity to cross tilt proportional to T_PER² ).

At the original instrument

only one of the scale end screws is replaced by a motor driven wedge for the fine adjustment of the cross tilt,

in the applet the scale end screws are both of equal sensitivity

to allow the adjustment of the cross tilt without change to the natural period  ( LONG LEVEL ).

R_EXT  ( external damping resistance, orange ) and

I_CAL   ( calibration current, green ) :

left mouse button increments ( + ) and decrements ( - ) the respective column,

middle or right button sets the respective parameter to max / min value

( R_EXT -> 9999.999 / 0000.000 [kOhm], I_CAL -> 9.9999 / 0.0000 [mA] ).

The actual position of the beam can be observed as a blue marker

moving between the limit stops ±11 [mm] of an enlarged scale left of the instrument.

Its value is displayed above the instrument ( S_SCL [mm], blue ).

Whenever the pedulum has come to rest

( observable at VELOCITY OUTPUT in DISPLAY / RECORDER mode at high resulution, i.e. F.S. < 100 [µm/s] ),

this postion can be marked as actual zero position ( red marker ) with the left mouse button

( middle or right mouse button removes the red marker ).

Its value is displayed above the instrument ( S_REF [mm], red ).

Pendulum excursions from the actual zero postion, normally caused by switching on a calibration current,

are displayed above the instrument ( S_AMP [mm], green ).

Setting the zero position marker starts the period measurement :

Postion adjustment and raising / lowering the pendulum by a calibration current

should be performed at relatively high damping values

( R_EXT = 0.5 ... 10 [kOhm], depending on the actual free period ).

Switching off the calibration current with R_EXT ≥ 1 [MOhm]

should cause very weakly damped oscillations about the actual zero postion.

The time difference between two adjacent zero crossings in the same direction ( passings of the red marker )

is displayed above the intrument ( T_PER [sec], red ).

This value represents the period of the weakly damped pendulum.

To facilitate the calculation of the actual damping value and the natural period of undamped system,

the period values and the pendulum velocities at zero crossings  ( => extrema of velocity )  are printed to the screen / JAVA Console  ( HELP = OFF ).

Postion Marker    Electrical Adjustment    Mechanical Adjustment    Operation

Comments    Screenshot    Table of Content    Top od Page

In the applet the adjustment procedures are performed in an idealized environment,

without seismic noise and without temperture / pressure induced long period tilt,

and threfore the the instrument can be adjusted to natural periods T_PER of up to approx. 1000 [sec] without stability problems  ( manufactorer's recommendation : T_PER ≤ 100 [s] ).

The magnitude of the influence of long period tilt to the performance of the instrument can be deduced

from the estimated approximate values of T_PER and S_SCL, listed with HELP+INFO ON

at a natural period  T_PER approx. 200 [s].

Reference adjustment :
( Screenshot of list )

X = -22.288 [seca] => T_PER  200.008 [s]
Y =  -0.160 [seca]  => S_SCL  -0.015 [mm]    ( compensation of the angular offset of the hinges )

Tilt across beam :
( Screenshot of list )

X = -22.288 [seca] => T_PER  200.008 [s]    ( unchanged,  see ref. )
Y =  -0.140 [seca] => S_SCL  +1.062 [mm]

ΔY = +0.020 [seca] => ΔS_SCL  +1.077 [mm]

Beam excursion of  +1.077 [mm]  resulting from a tilt across beam of  +0.020 [seca]  ( magnitude of earth tides )

=> sensitivity of tidal registrations.

Tilt along beam :
( Screenshot of list )

X = -22.304 [seca] => T_PER  200.223 [s]
Y =   -0.160 [seca]  => S_SCL  -0.015 [mm]   ( unchanged,  see ref. )

ΔX = -0.016 [seca] => ΔT_PER  +0.215 [s]

Change of natural period of  +0.215 [s]  resulting from a tilt along beam of  -0.016 [seca] 

=> stability of natural period for long period tilts along beam  ( here : magnitude of earth tides ).

Conclusion :

Since the sensitivity to tilt is proportional to T_PER²,

the effect of temperature / pressure induced and tidal long period tilt is strongly reduced and in most cases negligible

for use of the instrument as a horizontal seismometer in the period range below 100 [s].

Sensitivity to Tilt    Operation    Comments    Screenshot    Table of Content    Top od Page

The beam excursion S_SCL of up to  ±11 [mm]  ( approx. ±1.6 [deg] )  leads to errors, corresponding to the cross coupling effects known from gravity meters.

Reference adjustment :
( Screenshot of list )

X = -22.288 [seca] => T_PER  200.008 [s]
Y =   0.000 [seca]  => S_SCL  +8.601 [mm]    ( due to the angular offset of of the hinges )

Negative acceleration along beam :
( Screenshot of list )

X = -23.008 [seca] => T_PER  210.467 [s]
Y =    0.000 [seca]  ( unchanged;ndert )  => S_SCL  +9.524 [mm]

ΔX = -0.720 [seca]  ( approx. -34 [µm/s²] )  => ΔS_SCL  +0.923 [mm]

Positive acceleration along beam :
( Screenshot of list )

X = -21.568 [seca] => T_PER  190.967 [s]
Y =    0.000 [seca]  ( unchanged )  => S_SCL  +7.841 [mm]

ΔX = +0.720 [seca]  ( approx. +34 [µm/s²] )  => ΔS_SCL  -0.760 [mm]

Change of excursion S_SCL witout change of tilt across beam

caused by a change of the tilt along beam,

and

leadig to unsymmetrical excursions, depending on direction of tilt along beam.

The values listed above for static accelerations demonstrate the priciple of the cross coupling error.

Horizontal accelerations of comparable magnitude

i.g. correspond to seismic waves in a period range far  below 200 [s],

and the resulting error depends

on the signal period compared to the natural period of the instrument and

on the phase difference between accelerations along and across beam.

Class and html files for a local installation of the applets Long Period Horizontal and Verical Seismometer are available as a zip file and as a tar.gz file.

More applets : The author's homepage

"Cross Coupling"    Sensitivity to Tilt    Operation    Comments    Screenshot    Table of Content    Top of Page

Back to the Applet Index ( Geophysics Dept. TU Clausthal )