Geophysics Department TU Clausthal

deutsch english Applied Geophysics : Seismics

Long Period Vertical Seismometer

Applet    ( in separate window, ca. 820 x 610 Pixel )

The applet simulates
an astatic vertical-seismometer / gravitymeter with Lacoste suspension
( zero-length spring on the hypotenuse of a right triangle ).
Dimensions and technical specifications correspond to
a Sprengnether S-5100-V 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 final adjustment.

- Table of Content -


Coordinate System
LaCoste - Geometry
"Zero-Length" Spring
Equation of Motion
Additional Parameters

Mechanical Adjustment
Electrical Adjustment
Position Marker
Period Measurment

Adjustment Procedures
Initial Setup
Period Adjustment
Period Stability


- Seismometer -

Screenshot :

The mechanical system of the instrument is displayed in a simplified vertical section ( scale approx. 1 [Pixel/mm] ).

A pedulum  ( beam + seismic mass, blue )
is suspended by two hinges  ( crossed flexures, H, red ),  forming a horizontal axes of revolution.

A spring with negative preload ( magenta ) is fixed
to the pendulum ( B ) by a suspension wire of fixed length and
to the frame of the instrument ( A ) by a wire of adjustable length.

The postion of the upper suspension point A is factory adjusted,
but can be moved along the center line of the spring, if necessary or for experimantal purposes.

The tilt of the instrument about the direction of the hinge line is
adjusted by a levelling screw ( orange, lower right ) and observed at a bubble level ( orange ).

The excursion of the pendulum, limited to ±11 [mm] by two stops,
can be observed as a moving blue mark at an enlarged scale, and
its value is displayed at the upper edge of the graphic area.

Electro-mechanical components are displayed as simplified electric cicuits :
the damping coil with adjustable external damping resistance ( orange ) and
the calibration coil with adjustable DC-current ( green ).

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 recoder are contolled in a menue field above the graphic area of the applet.


( from ASSEMBLY AND OPERATING INSTRUCTIONS, Sprengnether Instruments, Inc. )

Moving Mass 10.92 [Kg]
Spring Constant 1030 [N/m]
Distance from Hinge line to :
Center of Oscillation 35.8 [cm]
Center of Mass 32.2 [cm]
Center line of Signal Coils 35.7 [cm]
Center line of Calibration Coil 24.2 [cm]
Point of Suspension 18.7 [cm]
Scale 43.2 [cm]
2 Signal / Damping Coils :
Coil Constant 89.0 [Vs/m]
Coil Resistance 500 [ohms]
Calibration Coil :
Coil Constant 5.0 [N/A]
Coil Resistance 68 [ohms]

Screenshot    Table of Content    Top of Page

- Comments -

Coordinate System

X = direction of axes of revolution ( hingei line ), "horizontal
Y = coordinate "along beam", "horizontal", from hinge line to zero of scale ( in applet pos. to left )
Z = "vertical", pos. upward

"vertical"     = Z-axes, evtl. tilted against the local direction of gravity
"horizontal" = XY-plane
mechanical system symmetrical to YZ-plane

LaCoste - Geometry

leads to
for the magnitude and the vector of the force acting on suspension point B
with an "effective spring constant" κeff,  in general a function of the actual length L of the spring.

"Zero-Length" spring

This results in a momentum of the spring acting on the beam

"Zero-Length" Spring    LaCoste - Geometry    Coordinate System

Screenshot    Table of Content    Top of Page

Equation of Motion

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

for the varaiables  ( time functions )

and with constant parameters of the instrument

adjustable quantities

and the external disturbance

contributing to the momentum of gravitational acceleration and
neglected in the adjustment procedures performed in a very quiet environment.

Using the approximations for small angular excursions of the contributions of the forces

acting on the suspension point B of the spring


acting on the effective center of gravity

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

the normalized equation of motion

with commonly used abbreviations

Additional Parameters

Some of the parameters used in the calculations,
the additional mass  mW  and its distance  sW  from the hinge line,
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 specifications.

Their values have to be estimated / set arbitrary :

In the applet the values
for the additional mass  mW = 300 [g]  and for its distance reange  sW = 95 ± 80 [mm]
are assumed to be somewhat greater than in the original instrument.

The remaining 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 preriod is set to  T0 = 100 [s],  the angular offset to  ϕ0 ca. 0.2 [deg],
the dimensionless mechanical damping  α0( T0 ) = 0.05
is set to an extremely small value, to allow the observation of several free oscilations at periods above 100 [s].
( realistic : max. 2 to 3 oscilations above 100 [s] )

The assumption  T0 = 100 [s]  leads to


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

! For  L0 ≠ 0  the  "effective spring constant  κeff  is a funktion of the actual beam position  ϕ,  and therefore, the coefficients depending on  κeff  are recalculated for every time step !

The output signals
for displacement ( DSP OUT ) and velocity ( VEL OUT )
are scaled to correspond
to the motion observed at the scale ( distance to hinge line approx. 400 [mm] ).

In order to simplify the parameter adjustment,
presumably resulting approximate values of the natural period  T_PER,  the damping  α (T_PER)  and the static beam excursion  S_SCL (T_PER)
can be estimated from the coefficients of the equation of motion and listed to the display / JAVA-console.

Screenshot :
( adjustment s. screenshot applet )

Additional Parameters    Equation of Motion    Coordinate System

Screenshot    Table of Content    Top of Page

- HowTo -


Dialogue area :
( Screenshot )

Stops and starts continuous calculations and graphic display
resets mechanical adjustments to initial values ( after assembly of the instrument ).

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

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

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,
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,
some internal parameter values, not visible to the operator of a real instrument, and greatly simplifies the adjustment procedures.

Parameter values displayed :
Effective spring length,
"horizontal" component of gravity and
"horizontal" distance of upper suspension point from hinge line.
( "horizontal" = hinge line -> scale zero, here = parallel to base plate )

HELP + INFO lists
the actual "counters" of the boxes  |+++| ...  and
the resulting estimated values for the zero position S_SCL, the natural period T_PER and the damping α
at the screen / the JAVA-console.

Mechanical Adjustments

Boxes   | + + + | + +| + | - | - - | - - - | :

Left mouse button increments / decrements
by 1          ( | + | - | ),
by 100      ( | + + | - - | ),
by 10000  ( | + + + | = - - | )

and middle or right mouse button
by 10          ( | + | - | ),
by 1000      ( | + + | - - | ),
by 100000  ( | + + + | = - - | )

The value 1000 corresponds
to approx. 0.1 [mm] for the displacement of the additional mass,
to approx. 1.0 [mm] for the leveling screw,
to approx. 1.4 [mm] for the displacement of the upper suspension point  A  and
                                for the adjusment of the spring length L0.

The levelling screw ( bottom right, orange )
tilts the whole instrument about an axes parallel to the hinge line, thus varying the "horizontal" component of gravity.
( Tilt is not reflected by the graphics display, but may be observed at the bubble level. )

The upper spring suspension point ( upper right, red and orange )
is "factory pre-adjusted", and should remain unchanged during initial setup and adjustment procedures.
( Evtl. adjustments necessary for extreme long natural periods should be performed very carefully because of possible instability of the instrument. )

The spring length adjustment ( upper right, magenta ) shortens / lengthens the upper suspension wire :

During initial setup
the pendulum is raised / lowered util it oscilates between its limit stops ( ±11 [mm] at Scale ),
during period stability adjustments
postion changes are compesated by by moving the additional mass ( see below ) and the the spring assembly is adjusted to react as a so called zero length spring.

Zero length spring :
Force F prop. total length L of spring assembly, i.e. F = 0 for L = 0.
Technical realization :
Fixed-length lower suspension wire + negative preloaded spring ( = negative length ) + adjustable-length upper suspension wire => effective spring length = 0.

The additional Mass ( movable along the beam, orange )
allows to adjust the zero postion of the pedulum by shifting the effective center of gravity to and from the hinge line.

Electrical Adjustments

R_EXT  ( external damping resistance, orange ) and
I_CAL   ( calibration current, green ) :
Left mouse button increments ( + ) and decrements ( - ) the respective column,
middle or right mouse button sets the respective parameter value to max / min
( R_EXT -> 9999.999 / 0000.000 [kOhm], I_CAL -> 9.9999 / 0.0000 [mA] ).

Postion Marker

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 ).

Period Measurement

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.

Position Marker    Electrical Adjustments    Mechanical Adjustments    Operation

Comments    Screenshot    Table of Content    Top of Page

Adjustment Procedures

( taken from ASSEMBLY AND OPERATING INSTRUCTIONS, Sprengnether Instruments, Inc. )

The final result of the adjustment should be a seismometer with
a stable natural period ≥ 20 [sec],
uniform ( ±3 % ) over the whole range between the limit stops.

Initial Setup

1.  Adjust the spring length
( upper suspension wire, | + + | + | - | - - |, magenta,  ! not upper suspension point ! )
until the pedulum will oscillate in the range of its limit stops.
2.  Move the small additional mass
for fine adjustments of the pendulum position.

If the pendulum is unstable, that is, it tends to move to one of the stops,
tilt the insrument by raising its hinge end with the leveleling screw and
Repeat steps 1 and 2.

Period Adjustment

The natural period is adjusted by tilting the intrument :

Raising the hinge end
increases the "horizontal" component of gravity ( pointing from hinge to scale ) and thus decreases the natural period,
lowering the hinge end
decreases the "horizontal" component of gravity and thus increases the period.

Period Stability

1.  Move the additional mass away from the hinge until
the pedulum oscillates about a point near the lower limit stop ( see above, S_REF [mm] approx. -6 ... -8 ) and measure the period.
2.  Move the additional mass toward the hinge until
the pedulum oscillates about a point near the center point ( see above, S_REF [mm] approx. -1 ... +1 ) and measure the period.
3.  Move the additional mass toward the hinge until
the pedulum oscillates about a point near the upper limit stop ( see above, S_REF [mm] approx. +6 ... +9 ) and measure the period.

If the period is not uniform within ±3 %, especially if the periods measured near the limit stops are not shorter than measured near the center by approx. the same amount,
the spring assemly is not "zero length" ( see above )
the spring length needs adjustment.

Period longer at the upper position ( = spring assembly is "negative length" ) :
Increase spring length ( | + + | + |, magenta )
compensate by moving the additional mass towards hinge until the pedulum again oscillates in the range of its limit stops.

Period shorter at the Upper Position ( = spring assembly is "positive length" ) :
Decrease spring length ( | - - | - | magenta )
compensate by moving the additional mass away from hinge until the pedulum again oscillates in the range of its limit stops.

Repeat steps 1. to 3.

Period Adjustment    Initial Setup

HowTo    Comments    Screenshot    Table of Content    Top of Page


This is a simulation ( fairly realistic, as I hope ),
but there is no ground noise and for instance the period stability adjustment loop in reality implies some steps not implemented here :

  1. Remove the instrument cover.
  2. Lock the pendulum, to avoid excessive bumps to the limit stops.
  3. Unclamp the upper suspension wire.
  4. Adjust the spring length.
  5. Retighten carefully the upper suspension wire clamp, to avoid variations of the effective spring length if bending the wire upward / downward during oscillations of the pendulum.
  6. Unlock the pedulum and gently guide it to the limit stop, it tends to travel to.
  7. Reinstall the istrument cover ( if possible, without exciting weakly damped fre oscilations of the spring ).

( The leveling screw is mounted on the base plate outside the instrument cover and the additional mass on the beam is motor driven.
Electrical adjustments and observing / recording of output signals are of course performed outside the cover and at some distance from the instrument to avoid ground noise caused by the operator. )

- Download -

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

More applets at : the author's Homepage

Rev. 07-nov-2007

Comments to Fritz Keller
( ned gschempfd isch globd gnueg )

Adjustment Procedures    HowTo    Comments    Screenshot    Table of Content    Top of Page

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