smart technologies for safety engineering

246 Smart Technologies for Safety Engineering
which, substituted into the simplified system of Equation (45) differentiated with respect to
both variables, yield the two following linear systems:
F ω 1
⎡
∂d
⎢
⎣
0 ⎤
K
∂
⎥
ˆαL ⎥
⎦ =
∂φ 0 k
∂ ˆαL
−δLN MNMuM
0
, F ω 1
⎡
∂d
⎢
⎣
0 ⎤
K
∂ ˆλl
⎥
⎦ =
∂φ 0 k
∂ ˆλl
0
−δliSiiεi
These systems can be used to obtain the corresponding derivatives of the nonvanishing virtual
distortions and, by Equations (44), of the response.
6.4.3.2 Redistribution of Material
In this case, the problem is to redistribute the material between the elements while preserving at
the same time the damping characteristics of the structure. The damping coefficients are hence
assumed to be constant, ˆαN = αN and ˆλi = λi, and the only variables are the cross-sectional
areas Âl of the elements of the modified structure. Since the element masses and stiffnesses
are modified, the damping-related virtual distortions also change, even though the damping
coefficients remain constant. Consequently, all four virtual distortions are nonvanishing and
the general form of the system of Equation (43) is retained, besides the minor computational
simplifications related to the fact that (αM)NM = αN MNM and (λS)ii = λiSii. The
response of the modified structure can be computed by solving the general system and by
substituting the resulting virtual distortions into Equations (42).
As the first step of the corresponding sensitivity analysis, Equations (41) are differentiated
with respect to the modified cross-sectional areas Âl of elements,
∂MNM
∂ Âl
∂ (αM) NM
∂ Âl
= Ml
NM
,
Al
M
= αN
l
NM
Al
,
∂ (λS) ii
∂ Âl
∂Sii
∂ Âl
= δilλi Ei
= δil Ei
Differentiation of the original system of Equation (43) yields the following linear system:
F ω
⎡
∂ f
⎢
⎣
0 K
∂ Âl
∂d 0 K
∂ Âl
∂φ0 k
∂ Âl
∂ε0 ⎤
⎡
⎥
2 ∂MNM
ω u M
⎥ ⎢
⎥ ⎢ ∂ Âl
⎥ ⎢
⎥ ⎢
⎥ ⎢ −iω
⎥ = ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ ⎣
⎦
k
∂ (αM) NM
u M
∂ Âl
−iω ∂ (λS) ⎤
⎥
ii ⎥
εi ⎥
∂
⎥
Âl ⎦
∂ Âl
− ∂Sii
εi
∂ Âl
After substitution of Equations (46), this system can be used to obtain the derivatives of all
virtual distortions and, by Equations (44), the derivatives of the response.
(46)