On the resonant component of the response of single degreeoffreedom systems under random loading M. Ashraf Ali and P. L. Gould
Department of Civil Engineering, l~ashington University, St. Louis, MO, USA (Received February 1984) A simple expression for the resonant component of the response variance of a single degreeoffreedom system under wind pressure loading is derived. For slowly varying excitation spectral densities, low structural damping and relatively wide excitation frequency bands, the expression gives a good estimate of the response variance. The derivation substantiates a frequently used approximation in the analysis of singledegreeoffreedom system under wind loading. Keywords: random excitation, resonance, complex frequency response function, spectral density, bandlimited white noise
Beginning in 1961, Davenport presented the first of a remarkable series of papers 1 which provide the basic framework for modern wind engineering. Within this framework, the mathematical sciences of statistics, probability and harmonic analysis are tightly interwoven with the engineering sciences of structural mechanics and dynamics. This required many engineers to expand their horizons in both mathematics and engineering and upgraded the level of wind engineering appreciably. By characterizing the response of structures in the frequency domain, the importance of resonance may be observed. A relatively simple formula for the variance of the resonant component of the response is stated in reference 2 but its origin has proved elusive. It is the purpose of this paper to present the derivation for this basic relationship. To introduce the background in a coherent fashion, the notation of reference 3 will be adopted.
Now, the spectral density of the response may be expressed in terms of the spectral density of the random loading p (e.g. wind pressure or ground acceleration loading) by:
1
Sx(cO) = ~IH(co)12 Sp (co)
(2)
where k is the stiffness of the system and H(oo) is the complex frequency response function, with the complex frequency response given by: IH(co) [2 =
=
1.0
{ [1  (wl63)2]2 + 4P (co/63)2} 1.0 [1 + (w/63) 4 + (4~2  2)(6o/63) 2]
(3)
Using equation (2) in equation (1) gives:
Background
'f
oo
The variance of a random variable X, Ox, representing the response of a structure, can be related to the spectral density of the variable Ss(cO) by:
a2X= i Sx(cO) d ~
(1)
0
where the response is defined in the frequency (w) domain.
280
EngngStruct., 1985. Vol. 7, October
~x =~"
IH(~)lesp(co) dee
(4)
0
For wind pressure loading, the response spectrum Sx(w) given by equation (2) can be divided into three parts: a (1) (2) (3)
quasistatic region resonant region inertial region
O 63 + A63+
01410296/85/0428003/$03.00 © 1985Butterworth& Co. (Publishers)Ltd
Resonant component o f response: M. Ashraf A l i and P. L. Gould
(I)=
I_. ~
(2)
..d 1
It should be noted that the two approximations are reasonable only when they are considered together. Nevertheless, it is convenient to develop them separately. ~ (3)
Approximation (1) If the spectral density of the input excitation Sp(6o) varies slowly and the structure has low damping, then the spectral density of the response is given by: s
Sx(6o) = Sp(6o)I//(6o)12/k2 ~ Sp(Co) la(6o)12/k 2
It should be pointed out that IH(6o)[Z/k 2 has a sharp peak at 6o = Goat moderate damping ratios/3; the peak gets sharper at lower damping ratios; and IH(6o)12/k~ = oo for /3 = 0 at 6o = ¢5. As a result, IH(6o)12/k 2 acts somewhat like a Dirac delta function, forcing the significant portion of the response to be concentrated at and around the natural frequency.
tlJ
Figure I Responsespeotrum
The response spectrum Sx(w) is shown qualitatively in Figure 1. The contribution of region (3) is small and is neglected. For region (1): tO
Approximation (2)
A(D 4
1
c?xO) = o~ = ~ 7
f
In(6o)12Sp(6o) d6o
(5)
For simplicity, let:  A~ = COx
o
~ + A~* = 6o~
(13a), (13b)
Substituting Sp(6o)  Sp(d) and noting that IH(6o)l 2 is an even function in 6o, one can write equation (7) as:
In this region, IH(6o)[2 ~ 1.0 and:
o~= St'(a~) IH(6o)l 2 d6o+ 2k21 J
o
L   (~11
For region (2):
i
]
IH(6o)l2 d6o (14)
('~Z
which is the variance of the response due to an excitation having bandlimited white noise spectral density, An indefinite integral can be obtained for equation (14) by using a partial fraction expansion. In this way one finds: 6
o~ has been approximated by: 24 /r
(12)
o~
(~7)
7ra)St'(~) [I(6o2/a~,/3)  I(6ol/C0,/3)1 4/3k2
(15)
for the variance of the response, where the integral factor I is given by:
where ~ is the natural frequency of the system and/3 is the damping coefficient. The objective of this paper is to show how the expression for the resonant part of the response variance, equation (8), can be derived. For this purpose, equation (7) is approximated as:
1
I(6o/Co,/3) =   tan t Tf
X In
2/3(6o/~) + 1  (6o/a))2
/3 2nX/i~
{1+ (6o/ )2 + 2 /1 1 + (6o/a~)2
}
2~/'1~ (16)
o~ = ~  S p ( a ) )
IH(6o)l2 d6o
(9)
o
with regard to equations (13a) and (13b), the integral factor in equation (15) may be rewritten as: I(w2/60,/3)  I(6ox/Co, /3) = I(1 + 6o3,/3)   I ( 1  wa,/3)
in the resonance region.
(17a) where, for the sake of convenience, it is assumed that A~= A~* = A~, and:
Approximations for resonant response Statement
A~ 6o3 = '  
With regard to equation (9), it can be seen that this equation incorporates two approximations: (1) Sp(6o)" Sp(¢2) = constant
(IO)
and: cb+&~* (2)
I f(6o) d6o~if(6o;c°)d6o ~AcD0
(11)
(17b)
The values of the factor in equation (17a) for different values of 6o3 and/3 are given in Table 1. Approximation (2) assumes that the ~xcitation has an ideal white noise rather than a bandlimited white noise spectral density. For the former case with 6oz = 0 and 6o2 ~ 0% the integral factor becomes: I(oo/co,/3)I(0/co,/3)= 1.00.0
= 1.0
Engng Struct., 1985, Vol. 7, October
(18)
281
Resonant component o f response: M. Ashraf A l i and P. L, Gould Table 1 Values factor in equation (17a) for different values of ~
of obtaining a close approximation would certainly be justified in this case. 6
and
Conclusion 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.05 0.05 0.05 0,05 0.05 0.05 0.05 0.05 0.05 0.05
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1:0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.937023 0.969154 0.980237 0.986041 0.989743 0.992410 0.994502 0.996255 0.997808 0.999253 0.875265 0.938457 0.960514 0.972097 0.979494 0.984823 0.989005 0.992511 0.995617 0.998506 0.815805 0.908053 0.940874 0.958185 0.969258 0.977244 0.983513 0.988769 0.993427 0.997760 0.759487 0.878078 0.921354 0.944318 0.959043 0.969674 0.978025 0.985031 0.991240 0.997016 0.706874 0.848655 0.901992 0.930513 0.948856 0.962118 0.972545 0.981296 0.989054 0.996273
0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.08 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.658257 0.819894 0.882825 0.916783 0.938702 0.954579 0.967073 0.977567 0.986872 0.995533 0.613696 0.791887 0.863886 0,903142 0.928590 0.947060 0.961613 0.973843 0.984694 0.994795 0.573088 0,764710 0.845205 0.889604 0.918523 0.939564 0.956164 0.970127 0.982520 0.994060 0.536214 0.738422 0.826809 0.876180 0.908509 0.932094 0.950730 0.966418 0.980350 0.993328 0.502797 0.713064 0.808723 0.862883 0.898553 0.924653 0.945311 0.962718 0.978187 0.992600
It has been demonstrated that the resonant component of the variance of the response of single degreeoffreedom systems under wind pressure loading, can be approximated by equation (8). The total variance of the response is then: o~. = ok + o~
(20)
Table 1 shows that the white noise approximation is very sound for relatively wide frequency bands and low damping ratios. Also, the socalled approximation (I) is reasonable if the system has low damping. Therefore, equation (8) works better for low values of damping ratios. References
1 Davenport, A. G. 'The application of statistical concepts to the wind loading of structures', Proc. Inst. Civil Engrs, 1961, 19, pp. 449472 2 Davenport, A. G. 'The response of slender, linelike structures to a gusty wind', Proc. Inst. Civil Engrs, 1962, 23, pp. 389408 3 Gould, P. L. and AbuSitta, S. H. 'Dynamic response of structures to wind and earthquake loading', Pentech Press, 1980 4 Lawson, T. V. 'Wind effects on buildings', Vol. 1, Design Applications, Applied Science Publishers Ltd, London, 1980, p. 81 5 Clough, R. W. and Penzien, J. 'Dynamics of structures', McGrawHill, New York, 1975, Art. 24, 5 6 Cranda11,S. H. and Mark, W. D. 'Random vibration in mechanical systems', Academic Press, New York, 1973, pp. 7880
Appendix With the help of equation (3), equation (9) may be written as:
Sp(63) 7 a~ 
2k 2
_J
dw
1 + (~o/63) 4 + (4/32


since the integrand is an even function in the excitation frequency, or: !2
s.(63) i
632
i
*/=/(1
2k 2
+ cJ3,~3)/(1cQ3,~)
/r63
H(w) =
(8)
where the second equality is obtained by a substitution of: Sp(63) = 4~2 k 2 S x ( 63)
Engng S t r u c t . , 1 9 8 5 , V o l . 7, O c t o b e r
 w 2 + 2i~Cow + Co21
(A1)
i6oBl + Bo w2A2 + iwA1 + Ao
(h2)
then:
i
(19)
A more direct evaluation of ok due to an ideal white noise excitation (equation (9)) is given in the Appendix. It is clear from Table 1 that for a bandlimited excitation, the integral factor in equation (1 5) is always less than unity. But it can be seen from the table that the factor is close to unity if the damping is low, and provided the band in question includes the natural frequency of the system, and is wide in comparison with the bandwidth (A63 = 2/~63) of the system. Therefore, the use of the ideal white noise hypothesis (equation (1 8)) for the purpose
282
a
!L dw
It may be shown that 6 if:
Therefore, equations (15) and (18) yield: 02R  4[3k 2 Sp(63) = rr1363S x ( 63)
2)(60/63) 2
IH(co)l 2 d ~ = 7r
(B:o/Ao)A2+ BI
(h3)
AIA2  
o a
comparing the integrand in equation (A1) with the righthand side of equation (A2), one finds: 7r63 o~ = 4  ~ Sp(63)
(A4)
which is the same expression given in equation (8) and in reference 3 (p. 33, equation (3.13).