Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 321

- estimation of the probability density of the amplitude and the phase shift. In
our example, we can explain the joint probability density q (x, v) of the W(t) and
dX/dt(t) processes by applying formula [8.23] with the expression of the potential
corresponding to the behavior law in Figure 8.17:

1 1

22 2 2

Ux Ȧ x ȦȦ xeHxe ª xeH xe º

¬ ¼

2

1

2

1 2

with Z 1 2 = k/m and Z 2 2 = (k + K)/m.

From q (x,v) we can derive the p(x max ) probability density of the X max (t)
envelope process;
- obtaining the PSD of the response.

We can eventually express for the general case the PSD of the response X(t) of
the oscillator by;

2

Y ³ Sx H Ȧ, Ȧ x,İ Sx px dx

[8.68]

0 max max max max

0

which corresponds to the statistic average according to x max of relation [8.15].

In our example H represents the transfer function of the Z 0 (x max ) resonance
pulsation and H the damping 1-dof linear oscillator:

2 2

H Ȧ, Ȧ ,İ 1/ ȦȦ 2iİȦȦ

[8.69]

0 0 0

Figure 8.24 shows the good harmony between the PSDs obtained by numerical
simulation and those obtained using the equivalent linearization method.