Seismometer – The basic Principles Seismometer – The basic Principles
u
x x
0u
gu
mx
mx x
0x
ru ground displacement
x
rdisplacement of seismometer mass
x
0mass equilibrium position
The motion of the seismometer mass as a function of the ground displacement is given through a differential equation resulting from the equilibrium of forces (in rest):
F
spring+ F
friction+ F
gravity= 0
for example
F
sprin=-k x, k spring constant
F
friction=-D x, D friction coefficient
F
gravity=-mu, m seismometer mass
Seismometer – The basic Principles Seismometer – The basic Principles
u
gx x
0x
r.
..
Seismometer – The basic Principles Seismometer – The basic Principles
u
gx x
0x
rusing the notation introduced the equation of
motion for the mass is
m h k
m D
t u t
x t
x t
x r r r g
2 0 0
2 0
2 ,
) ( )
( )
( 2
) (
From this we learn that:
- for slow movements the acceleration and velocity becomes negligible, the
seismometer records ground acceleration
- for fast movements the acceleration of the
mass dominates and the seismometer
Seismometer – examples Seismometer – examples
u
g
x
x0 x
r
Seismometer – examples Seismometer – examples
u
g
x
x0 x
r
Seismometer – examples Seismometer – examples
u
g
x
x0 x
r
Seismometer – examples Seismometer – examples
u
g
x
x0 x
r
Seismometer – examples Seismometer – examples
u
g
x
x0 x
r
Seismometer – examples Seismometer – examples
u
g
x
x0 x
r
Seismometer – Questions Seismometer – Questions
u
g
x
x0 x
1. How can we determine the damping
rproperties from the observed behavior of the seismometer?
2. How does the seismometer amplify the ground motion? Is this amplification
frequency dependent?
We need to answer these question in order to determine what we really want to know:
The ground motion.
Seismometer – Release Test Seismometer – Release Test
u
g
x
x0 x
1. How can we determine the damping
rproperties from the observed behavior of the seismometer?
0 )
0 ( ,
) 0 (
0 )
( )
( )
(
0
2 0 0
r r
r r
r
x x
x
t x t
x h
t x
we release the seismometer mass from a given initial
position and let is swing. The behavior depends on the
relation between the frequency of the spring and the
damping parameter. If the seismometers oscillates, we
can determine the damping coefficient h.
Seismometer – Release Test Seismometer – Release Test
u
g
x
x0 x
r
0 1 2 3 4 5
-1 -0.5 0 0.5 1
F0=1Hz, h=0
D is pl ac em en t
0 1 2 3 4 5
-1 -0.5 0 0.5 1
F0=1Hz, h=0.2
0 1 2 3 4 5
-1 -0.5 0 0.5 1
F0=1Hz, h=0.7
Time (s)
D is pl ac em en t
0 1 2 3 4 5
-1 -0.5 0 0.5 1
F0=1Hz, h=2.5
Time (s)
Seismometer – Release Test Seismometer – Release Test
u
g
x
x0 x
The damping coefficients
rcan be determined from the amplitudes of
consecutive extrema a
kand a
k+1We need the logarithmic decrement
a
ka
k+1
1
ln 2
k k
a a
The damping constant h can then be determined through:
2
4 2
h
Seismometer – Frequency Seismometer – Frequency
u
g
x
x0 x
r
The period T with which the seismometer mass oscillates depends on h and (for h<1) is always
larger than the period of the spring T
0: 2
0
1 h T T
a
ka
k+1T
Seismometer – Response Function Seismometer – Response Function
u
g
x
x0 x
r
t i r
r
r t h x t x t A e
x ( ) 0 ( ) 0 2 ( ) 2 0
2. How does the seismometer amplify the ground motion? Is this amplification frequency dependent?
To answer this question we excite our seismometer with a monofrequent signal and record the response of the seismometer:
the amplitude response A
rof the seismometer depends on the frequency of the seismometer w
0, the
frequency of the excitation w and the damping constant h:
2 2 2
2 2
0 2
4 1
1
h T A T
A r
Seismometer – Response Function Seismometer – Response Function
u
g
x
x0 x
r