CAPÍTULO IV. METODOLOGÍA
4.7 Determinación de los sujetos de la investigación
Now, the relationship between the ground excitation and the absolute acceleration of mass m is considered. This relationship is a significant quantity in seismic isolation, and a more detailed deri-vation of the corresponding dynamic magnification is given.
1.2.2.2.1 Definition of Dynamic Magnification Factor of Absolute Acceleration
Using the method of complex response for the steady-state response (see Equations 1.45 and 1.46), results in
mx cx kx m x e+ + = ωf g2 j tωf (1.141)
Note that in Equation 1.141, x, x, and x are relative displacement, velocity, and acceleration, respectively. Denote
x x e= p0 j tωf (1.142)
where xp0 is the complex-valued amplitude of the relative displacement. Also,
x a e= p0 i tωf = −xp0ωf2ej tωf (1.143) Here, ap0 is the complex-valued amplitude of the relative acceleration, where
ap0= −ωf p2x 0 (1.144)
Also, the amplitude of the ground acceleration can be written as
ag= ωf g2x (1.145) Note that in Equations 1.144 and 1.145, ap0 and xp0 are complex valued, whereas ag and xg are real valued.
From Equation 1.130, the relationship between the real-valued amplitude of the relative displace-ment and the ground displacedisplace-ment is given by
x0= βdB gx (1.146)
Now, for the complex-valued base excitations,
xp0 = βCdB gx (1.147)
Here βdBC is a complex-valued term, instead of the real-valued dynamic magnification factor, βdB, expressed in the previous discussion; superscript C denotes that it is a complex-valued term. The result is
Consider the relationship between the relative acceleration and the ground acceleration using Equations 1.145 and 1.146:
Therefore, the dynamic magnification factor of the real-valued relative acceleration, denoted by βaB, can be written as
Furthermore, the complex-valued amplitude of the absolute acceleration can be written as ap0 + xg″. Therefore,
The real-valued amplitude of the absolute acceleration of mass m, denoted by xA″, can be obtained by taking the absolute value of (ap0 + xg″), that is,
xAʺ= ap0+xgʺ (1.152)
Therefore, the dynamic magnification factor of the absolute acceleration for the base excitation, denoted by βAB, can be obtained as follows:
where subscript A denotes absolute, whereas the lowercase subscripts stand for relative quantities.
Therefore,
By using the term βAB, the real-valued amplitude of the absolute acceleration can be written as
xAʺ= βAB gxʺ (1.155)
In Figure 1.18a, the dynamic magnification factor βAB is plotted vs. the frequency ratio for several different damping ratios. In Figure 1.18b, the phase angles are plotted vs. the frequency ratio.
From Figure 1.18a, it is seen that despite the damping ratios, the dynamic magnification factors all start from the value of unity when r = 0. After that, the magnitudes of the factors become larger and larger until they reach resonance points. Similar to all the various dynamic magnification fac-tors, when the damping ratio is small, the magnitude of the dynamic magnification factor of the absolute acceleration can be very large. As a difference from the dynamic magnification factors βa
and/or βdB, which will not have resonance when the damping ratio is >0.707, when the damping ratio reaches unity, βAB will still be greater than unity.
By continuously increasing the frequency ratio after the resonance point, the magnitude decreases. At a special frequency point, the value returns to unity. Interestingly, this frequency point, which is derived in the next subsection, is independent of the damping ratio. Consequently, all curves in Figure 1.18 pass through this unique point. Beyond that frequency point, the value of βAB will be less than unity. However, in contrast to all the dynamic magnification factors discussed above, which always reduce the magnitudes as the damping increases, the magnitude of βAB will have a different trend after this frequency point. That is, the larger the damping ratio, the smaller the reduction of the dynamic magnification factor.
Note that again, the phase angle should have a minus sign. Thus, by using Figure 1.18b, the plots of the phase angle vs. the frequency ratio show that as the frequency ratio increases, the phase difference between the mass and the ground will decrease from zero to –π. In this figure, the effect of the damping ratio is also realized. Generally, when the damping ratio is small, the phase will have a sharp turning point close to r = 1. However, when the damping ratio is large, the phase value will be increased more gradually with a less sharp turning frequency point. In
addition, when the damping is small, the phase difference will quickly reach –π or –180° as the frequency ratio becomes larger than unity. However, with a large damping ratio, the phase angle reaches a smaller maximum value in a more gradual manner. That is, the maximum value is also reduced. For example, when ξ = 1, the maximum phase difference is no longer –180°; instead, it becomes –90°.
1.2.2.2.2 Peak Value of the Dynamic Magnification Factor of Absolute Acceleration
In the preceding section, the nature of the dynamic magnification factor of absolute acceleration underground excitation was discussed in general terms. Now, a more detailed analysis is carried out at several frequency points. The first important point is when r = 0. It is easy to see that, in this case,
βAB|r=0=1 (1.157)
Next, the peak value of the dynamic magnification factor of absolute acceleration and the cor-responding frequency ratio are determined. The derivative of βAB is taken with respect to r and the result is set equal to zero. Thus,
drd d
dr r
r r
β ξ
AB= +
( )
ξ(
1−1 2)
+( )
2 =02
2 2 2 (1.158)
(a) 0
(b)
0.5 1 1.5 2 2.5 3
0 2 4 6 8 10
Frequency ratios
Magnitude
Damping ratio = 0.05 Damping ratio = 0.10 Damping ratio = 0.30 Damping ratio = 0.70 Damping ratio = 1.00
0 0.5 1 1.5 2 2.5 3
–3 –2.5 –2 –1.5 –1 –0.5 0
Frequency ratios
Phase angle (rad) Damping ratio = 0.05
Damping ratio = 0.10 Damping ratio = 0.30 Damping ratio = 0.70 Damping ratio = 1.00
FIGURE 1.18 Magnitude and phase of absolute acceleration due to ground excitation (a) dynamic magnifica-tion factor and (b) phase angle.
Solving Equation 1.158 to find the proper value of r, results in
From Equation 1.159, it is realized that since
ξ >0 (1.160)
the frequency ratio is always real valued. This fact implies that Equation 1.158 will always have a meaningful solution, so that regardless of the value of the damping ratio, the dynamic magnification factor of the absolute acceleration will always have a resonant peak value.
Substituting Equation 1.159 into Equation 1.153, the peak value of the dynamic magnification factor is given by
βAB
( )
ωA =2 2 8ξ2(
ξ4−4ξ2− +1 1 8+ ξ2)
−1 2/ (1.161)In Equation 1.161, ωA is the resonant frequency given by
It can be seen that when the damping ratio is small,
However, it is also observed that no matter how the damping ratio is chosen, the following is always the case:
Inequality Equation 1.164 indicates that the peak value of βAB(ωA) is always larger than the other types of dynamic magnification factors. When the damping ratio is sufficiently large, the difference becomes more significant. Furthermore, when r = 1,
β ξ
AB 1 1 4ξ
2
( )
= + 2 (1.165)which is 1 4+ ξ2 times the value of the other dynamic magnification factors, such as that of dis-placement, at r = 1.
1.2.2.2.3 Frequency Point of Dynamic Magnification Factor of Absolute Acceleration being Unity Consider the case when the value of the dynamic magnification factor of absolute acceleration reaches unity, besides the point when r = 0. Let
r = 2 (1.166)
Then, substituting Equation 1.165 into Equation 1.154 yields
βAB
( )
2 1= (1.167)Therefore, despite the value of the damping ratio, when Equation 1.165 is satisfied, the value of the dynamic magnification factor of absolute acceleration reaches unity again. Beyond this point,
βAB
(
ωf > 2ωf)
<1 (1.168)That is, the value of the dynamic magnification factor of absolute acceleration can be smaller than unity, only if Equation 1.168 is satisfied. Consequently, in base isolation design, the natural period, Tn, needs to be 2 times larger than the major driving period. Otherwise, the acceleration of the superstructure will not be reduced, but will instead be magnified.
1.2.2.2.4 Half-Power Points of Dynamic Magnification Factor of Absolute Acceleration To obtain the approximation of the half-power points, let
1 2
1 2
22 2 2 8 4 1 1 8
2
2 2 2 2 4 2 2 1 2
+
( )
(
−r)
+ξr( )
ξr = ⎡⎣⎢ ξ(
ξ − ξ − + + ξ)
−/ ⎤⎦⎥ (1.169)By solving Equation 1.169,
r b b ac
1 2 a
2 1 2
2 4
,
/
=⎧− −
⎨⎪
⎩⎪
⎫⎬
⎪
⎭⎪
(1.170)
where
a = 4ξ4 (1.171a)
b = −16ξ6+8ξ4+4ξ2−4ξ2 1 8+ ξ2 (1.171b)
c = −4ξ4+4ξ2+ −1 1 8+ ξ2 (1.171c)
When the damping ratio is sufficiently small, it can be proven that the frequency ratios calculated by Equation 1.170 can be used to determine the damping ratio with the help of Equation 1.122. In fact, when the actual damping ratio is <21%, there can be a <10% error in damping ratio overestima-tion. When the actual damping ratios are 30%, 40%, and 50%, the errors in damping ratio overesti-mation will be 18.82%, 31.11%, and 50.33%, respectively.
Example 1.10
A computer is mounted on a floor with total mass 25 (kg), which is subjected to ground harmonic excitation with an amplitude of 1 (g) and a driving frequency of 4 (Hz). The computer only allows 0.3 (g) of acceleration so that it is base isolated. The base isolator provides a stiffness of 3,000 (N/m) and a damping ratio of 10%. If the allowed relative displacement of the isolator is <1.6 (cm), the base isolation must be checked to determine if it can satisfy the required parameters.
With the given parameters, the natural frequency of the isolation system is 1.7 (Hz). The fre-quency ratio is then calculated to be 2.29, βdB = 1.23, and βAB = 0.257. Therefore, the amplitude of acceleration is 0.257 (g) < 0.3 (g), whereas the displacement of isolation is 1.9 > 1.6 (cm).
It is seen that although the acceleration satisfies the required level, the displacement does not.
Therefore, a different group of design data must be chosen. This time, k = 500 (N/m) is supposed.
The natural frequency of the isolation system is 0.712 (Hz). The frequency ratio is then calculated to be 5.62. When the damping ratio is chosen to be 0.7, βd = 1.001, βdB = 1.23, and βAB = 0.251.
Therefore, the amplitude of acceleration is 0.251 (g) < 0.3 (g), whereas the displacement of isola-tion is 1.55 < 1.6 (cm).
This example implies that when the level of acceleration is to be reduced, which is often the main goal of base isolation, the relative displacement of the isolator must be checked. In fact, in the design stage, it is best to consider the reduction of the acceleration and the regulation of the displacement simultaneously.