__Spectrum
Compatible Accelerograms__

It
is possible to synthesize an accelerogram, which has certain known
characteristics of ground motion, e.g. peak ground acceleration, power spectral
density, duration of strong motion, shape of the strong motion window function
and, above every thing else, a specified response spectrum. Design response
spectra (DRS) represent a hypothetical earthquake. No real accelerogram will,
therefore, posses the characteristics of a specified design response spectrum (SDRS).
Also, the accumulated data set of recorded accelerograms from strong earthquakes
is not large enough to allow the use of any real accelerogram in design. An
accelerogram to meet the requirements of design has to be compatible with the
design response spectrum. This means that the time history response spectrum (THRS)
should be sufficiently close to the SDRS at all frequencies. In fact, for the
sake of conservatism, the THRS ordinates are required to remain above those of
the SDRS at all frequency points, except for a few. At frequencies, where the
THRS go below the SDRS, the deviation should remain small (5-10%).

a(t)
= I(t). SA_{n}
Sin (nw_{n}t
+ j_{n})

Here
I(t) describes the windowing function. I(t) is equal to 0 before t=0 and after
t=t_{d} where t_{d}
is the duration of the accelerogram. The amplitudes A_{n} and
phases j_{n
}are then adjusted, in an iterative process, to match the THRS with the
SDRS.

1. Gasparini,A.D. and Vanmarcke, E.H.(1976). Simulated Earthquake Ground Motion Compatible with Prescribed Response Spectra. Report No. 2 (Publication No. R76-4), Deptt. Civil Engineering. MIT, Massachusettes, USA.

2. Levy, S. and Wilkinson,J.P.D.(1976). Generation of Artificial Time Histories Rich in all Frequencies from given Response Spectra. Nuclear Engineering. and design. Vol. 38, pp 241-251.

**Synthesizing
Spectrum
Compatible Accelerograms**

Spectrum compatible accelerograms are required for dynamic analysis of the structures and for generating response spectra for different floor levels. In the earlier days of aseismic design such accelerograms were obtained by modifying a recorded accelerogram (by suitably adjusting its time and amplitude scales). However, digital computers allow easy generation of synthetic accelerograms , and achieving a higher level of compatibility. A brief outline of the principle of generating spectrum compatible accelerograms is given below.

Any periodic function can be represented as a sum of sinusoids in the form:

X
(t) = S
A_{n} Sin (w_{n}t+j_{n})
...... (1)

Here A_{n} is the amplitude and j_{n}
is the phase of the n^{th} sinusoid. For fixed amplitudes A_{n}
different ground motion time histories with similar appearances, but different
details, can be generated. For X(t) the amplitudes An are related to the power
spectral density (see Frequency content of accelerograms 5.4
) of the time series at angular frequency wn
by the following relationship:

G(w_{n}).Dw
= A_{n}^{2}/2 ...
(2)

As n becomes large the total power in the series is given by:

₯

ςG
(w).
dw
= SG(w_{n}).Dw
= S
A_{n}^{2}/2 ...
(3)

0 n n

Equation (1) represents a stationary time series, in which the power does not vary with time. Earthquake signals are of transient nature. They have an onset, a rise time during which the signal builds up, duration of strong motion and, finally, a signal decay time in which the signal reaches the average ground level. The transient character of the strong motion signal can be simulated by multiplying X(t) with a window function I(t) describing the character of the transient. Thus the artificial ground motion time history can be written as:

a
(t) = I (t). S
An Sin (w_{n}t
+ j_{n})
.... (4)

n

Matching
the response spectrum of the accelerogram with a target spectrum is done in
several cycles by comparing the calculated response spectrum with the target
response spectrum at selected frequencies[1]
^{,[2]}.
In the computation procedure the power spectral density (PSD) at the i^{th}
angular frequency is modified as follows:

G(w)_{i+1}
= G(w)_{i}
. S_{v}(w)/S_{v}^{(i)}(w)
... (5)

where
S_{v}(w)
is the spectral velocity in the target response spectrum, S_{v}^{i}(w)
is the spectral velocity and G(w)_{i}
the PSD at frequency w in the i^{th}
iteration. G(w)_{i+1
is the modified PSD, }for (i+1)^{th} iteration, if another
iteration is required. The
following table lists out the frequency intervals in different frequency ranges,
which have been recommended for comparing spectra for matching[3].

**Table- : Suggested Frequencies for calculation of
Response Spectra.
**

**
**

FREQUENCY RANGE |
INCREMENT |

0.5 3.0 |
0.1 |

3.0
3.6 |
0.15 |

3.6 5.0 |
0.20 |

5.0 8.0 |
0.25 |

8.0 15.0 |
0.50 |

15.0 18.0 |
1.0 |

18.0 22.0 |
2.0 |

22.0 34.0 |
3.0 |

__Frequency
Content Of An Accelerogram__

For a given accelerogram a (t) its Finite Fourier Transform (FFT) at
angular frequency w
is written as: FS (w) =
[U^{2}+V^{2}] ^{½ }
..
(1)

T

U = ς
a(t) Sin (wt).dt
.
(1a)

0

T

U
= ςa(t) Cos (wt).dt
. (1b)

0

Here T is the duration of strong motion. The Fourier amplitude spectrum FS (w) is written as the square root of the sum of squares of the real and imaginary parts, U and V, of F (w). FS (w) has the units of velocity. For an undamped SDOF system subjected to the acceleration a(t) at the base the response of the SDOF system and the Fourier amplitude are closely related. The equation of motion of the undamped SDOF system is written as:

d^{2}/dt^{2}
+ w_{n}
x =-a(t)
. (2)

Here x (t) is the
relative displacement of is mass and w_{n
}is its
natural_{ }frequency. The steady state response of the system is given
by

t

x(t)
= (1/w_{n})
ς -a(t)Sin
wn (t-t)dt
..
(3)

0

The right hand
side is known as Duhamels integral.[1]

Its relative
velocity is given by

t

dx/dt
= ς a(t)Cos(wt)dt
(4)

0

The total energy
per unit mass of the system, E(w_{n}),
at the end of time T can be written as the sum of the kinetic and the strain
energies:

E (w_{n})
= (1/2)[(dx/dt)^{2}] +(1/2) w_{n}
^{2}x^{2}(T)
(5)

Substituting the
values of x(T) and (dx/dt) in
Equation (6) gives the relationship between E (w_{n})
and FS (w_{n}).

E (w_{n}) = (1/2)[FS (w_{n})]^{2}
(6)

Thus at the end
of the strong motion the total energy per unit mass is one half of the square of
the Fourier amplitude at that frequency. Expanding Equation (5), denoting the
maximum relative velocity (spectral velocity) of the system SV (w_{n})
and assuming that the maximum velocity occurs at time t_{v} gives

t_{v}
t_{v}

SV
(w_{n})
= [ ς a(t)
Sin w_{n}tdt}^{2}+{ς a(t) Cos w_{n}
tdt}2]^{1/2}
. (7)

0
0

A comparison
between equations (1) and (7) shows that, for zero damping the maximum relative
velocity is equal to the Fourier amplitude when t_{v} = T.
If the maximum relative displacement SD (w_{n})
occurs at time t = t_{d}, and a pseudo velocity PSV (w_{n})
is defined as the product of the natural frequency and the maximum relative
displacement, SD (w_{n}),

PSV (w_{n})
= w_{n.}
SD (w_{n)
. }(8)

It can, then, be
seen that the pseudo velocity is equal to the Fourier amplitude if td = T. While
SV (w_{n})
and FS (w_{n})
are closely related, in general, the Fourier amplitude is less than SV (w_{n})
and PSV (w_{n}).
If w_{0}
is the highest frequency in a(t)
then the inverse Fourier transform relationship is written as:

w_{0
}

a
(t) = (1/T)ς F(w)dw
(9)

0

The intensity or
total energy of a(t) is given by

T

I
= ς a^{2}
(t) dt
. (10)

0

Using
the Persevals theorem on Fourier transforms the intensity can also be
written as

I =(1/p)ς[F
(w)]^{2 }dw
. (11)

The intensity per
unit time is, then, written as

T

G = (1/T) ς a^{2} (t) dt
= =(1/pT) ς[F
(w)]^{2}.dw
.
(12)

0

The right hand
side represents the mean square vale of the acceleration, a(t).

In the frequency
domain the **power spectral density** (PSD)
is defined as

G (w) = (1/pT)[F(w)]^{2}
. (13)

This gives the
relationship between the PSD and the mean square acceleration r

w_{0}

r = ς
G (w) dw
(14)

0

PSD is, normally,
represented as the product of a normalized PSD function G^{<n>}(w), which has a unit area under the w, G(w)
curve, and a mean square acceleration. Thus

G (w) = r.
G^{<n>}(w)
. (15)

PSD is an
important characteristic determining the frequency content of the accelerogram.

* *