For the multi-degree-of-freedom (MDOF) system, at the time instant
\(\tau =t\), the structural response can be defined by the function
$$\begin{aligned} \varPhi = G \int _{0}^{t} \left[ q_{\alpha }({\mathbf {h}}, {\mathbf {b}}; \tau ), {\mathbf {h}} \right] \delta (t-\tau )\text {d}\tau , \quad \;\; t \in [0,T] \end{aligned}$$
(7)
where
t denotes the running terminal time and
\(\delta (t-\tau )\) is the Dirac delta distribution. The function (
7) is satisfying the equations of motion (
5) and being an explicit and implicit function of the vector of design variables
\({\mathbf {h}} = \{h^d\}\),
\(d=1,2,\ldots , D\), and the vector of random variables
\({\mathbf {b}} = \{b^r\}\),
\(r=1,2,\ldots , R\), i.e.,
$$\begin{aligned}&Q_{\alpha } ({\mathbf {h}}, {\mathbf {b}}; \tau ) - K_{\alpha \beta }({\mathbf {h}}, {\mathbf {b}})\, q_{\beta } ({\mathbf {h}}, {\mathbf {b}}; \tau ) - C_{\alpha \beta }({\mathbf {h}}, {\mathbf {b}})\, {\dot{q}}_{\beta } ({\mathbf {h}}, {\mathbf {b}}; \tau )\nonumber \\&\qquad - M_{\alpha \beta }({\mathbf {h}}, {\mathbf {b}})\, {\ddot{q}}_{\beta } ({\mathbf {h}}, {\mathbf {b}}; \tau ) = 0, \quad \;\; \alpha ,\beta =1,2,\ldots , N \nonumber \\&q_{\alpha } ({\mathbf {h}}, {\mathbf {b}}; 0) = 0, \quad {\dot{q}}_{\alpha } ({\mathbf {h}}, {\mathbf {b}}; 0) = 0 \end{aligned}$$
(8)
The first two probabilistic moments—expectations
\({\overline{b}}^{\,\,r}\) and cross-covariances
\(\text {Cov} (b^r, b^s)\) of the random variables
\(b^r\) are defined as
$$\begin{aligned}&{\overline{b}}^{\,\,r} = \text {E}\left[ b^r \right] = \int _{-\infty }^{+\infty } b^r p\,(b^r) \text {d}b^r\end{aligned}$$
(9)
$$\begin{aligned}&\text {Cov} (b^r, b^s) = \text {E}\left[ (b^r - b_0^r)(b^s - b_0^s) \right] = \text {R} (b^r, b^s) \sqrt{\text {Var} (b^r) \text {Var} (b^s)} \end{aligned}$$
(10)
with
$$\begin{aligned}&\text {R} \ (b^r, b^s) = \int _{-\infty }^{+\infty } \int _{-\infty }^{+\infty } b^r b^s p(b^r, b^s) \text {d}b^r \text {d}b^s\end{aligned}$$
(11)
$$\begin{aligned}&\text {Var} = \alpha ^2 \text {E}^2 \big [ b^r \big ] \end{aligned}$$
(12)
where
\(\text {R} (b^r, b^s)\),
\(\text {Var} (b^r, b^s)\),
\(p(b^r, b^s)\) and
\(\alpha \) denote functions of correlation, variance, joint probability density and the coefficient of variation, respectively.
The functions of random variables
\(K_{\alpha \beta }\),
\(C_{\alpha \beta }\),
\(M_{\alpha \beta }\),
\(q_{\alpha }\) and
\(Q_{\alpha }\) will be handled by the perturbation scheme. Assume that
\(K_{\alpha \beta }\),
\(C_{\alpha \beta }\),
\(M_{\alpha \beta }\),
\(q_{\beta }\) and
\(Q_{\alpha }\) are twice differentiable with respect to the design variables
\(h^d\) and the random variables
\(b^r\). Using the chain rule of differentiation, and since the running terminal time
t takes on some a priori selected value in the time interval [0,
T], leads to the expression for the derivative of
\(\varPhi \) with respect to
\(h^d\) in the form
$$\begin{aligned} \varPhi ^{;d}(t)= & {} \int _{0}^{t} \left[ G^{.d}(\tau ) \, \delta (t-\tau ) + G_{\! .\alpha }(\tau ) \, q_{\alpha }^{;d}(\tau )\, \delta (t-\tau ) \right] \text {d}\tau \nonumber \\= & {} G^{.d}(t) + \int _{0}^{t} G_{\! .\alpha }(\tau ) \, q_{\alpha }^{;d}(\tau )\, \delta (t-\tau ) \text {d}\tau \end{aligned}$$
(13)
where
\((\cdot )^{;d}\) is the first ordinary derivative with respect to the
dth design variable,
\((\cdot )^{.d}\) and
\((\cdot )_{.\alpha }\) are the first partial derivatives with respect to the
dth design variable and
\(\alpha \)th nodal displacement, respectively. The components
\(G^{.d}\) and
\(G_{.\alpha }\) in Eq. (
13) are known, because
G is an explicit function of its arguments. While
\(q_{\alpha }\) are implicit with respect to
\(h_d\), the derivatives
\(q_{\alpha }^{;d}\) must be determined. Differentiating the equation of motion (
8) with respect to the design variables
\(h_d\) yields
$$\begin{aligned} M_{\alpha \beta } {\ddot{q}}_{\beta }^{;d}(\tau ) + C_{\alpha \beta } {\dot{q}}_{\beta }^{;d}(\tau ) + K_{\alpha \beta } q_{\beta }^{;d}(\tau ) - R_{\alpha }^{d}(\tau ) = 0 \end{aligned}$$
(14)
where
$$\begin{aligned} R_{\alpha }^{d}(\tau ) = Q_{\alpha }^{.d}(\tau ) - M_{\alpha \beta }^{.d} {\ddot{q}}_{\beta }(\tau ) - C_{\alpha \beta }^{.d} {\dot{q}}_{\beta }(\tau ) - K_{\alpha \beta }^{.d} q_{\beta }(\tau ) \end{aligned}$$
(15)
To eliminate
\(q_{\alpha }^{;d}\) from (
13), the adjoint system method is employed. Pre-multiplying (
14) by the transpose of an adjoint vector
\(\lambda _{\alpha }(\tau )\), which is initially independent of random design variables, integrating by parts with respect to
\(\tau \) and equating the coefficients at
\(q_{\alpha }^{;d}\) in the resulting equation and (
13), we obtain the differential equations of motion for the adjoint system in the form
$$\begin{aligned}&M_{\alpha \beta } {\ddot{\lambda }}_{\beta }(\tau ) - C_{\alpha \beta } {\dot{\lambda }}_{\beta }(\tau ) + K_{\alpha \beta } \lambda _{\beta }(\tau ) = G_{\!.\alpha }(t) \, \delta (t-\tau ) \nonumber \\&\quad \lambda _{\alpha } (t) = 0, \quad {\dot{\lambda }}_{\alpha } (t) = 0, \quad \tau \in [0,t], \quad \;\; t \in [0,T] \end{aligned}$$
(16)
Substituting (
16) into (
13) and taking into account (
14) we obtain
$$\begin{aligned} \varPhi ^{;d}(t) = G^{.d}(t) + \int _{0}^{t}\! \lambda _{\alpha } (\tau )\, R_{\alpha }^{d}(\tau )\, \text {d}(\tau ) \end{aligned}$$
(17)
Now, the functions of random variables
\(M_{\alpha \beta }\),
\(C_{\alpha \beta }\),
\(K_{\alpha \beta }\),
\(Q_{\alpha }\),
\(G_{.\alpha }\),
\( q_{\beta }\),
\(\lambda _{\alpha }\),
\(M_{\alpha \beta }^{.d}\),
\(C_{\alpha \beta }^{.d}\),
\(K_{\alpha \beta }^{.d}\),
\(Q_{\alpha }^{.d}\) and
\(G^{.d}\) are expanded around the expectations
\({\overline{b}}^{\,\,r}\) via the second-order perturbation, with a small parameter
\(\theta \), generally expressed as
$$\begin{aligned} (\cdot ) ({\mathbf {h}}, {\mathbf {b}}) = (\cdot )^0 + \theta (\cdot )^{;r} \Delta b^r + {\frac{1}{2}} \ \theta ^2 (\cdot )^{;rs} \Delta b^r \Delta b^s, \quad \;\; r,s=1,2,\ldots , R \end{aligned}$$
(18)
where
\(\Delta b^r\) is the perturbational increment of
\(b^r\) with respect to
\(b_0^r\), and
\((\cdot )^0\),
\((\cdot )^{;r}\) and
\((\cdot )^{;rs}\) describe the zeroth, first and second ordinary derivatives with respect to
\(b^r\).
The expansions of
\(M_{\alpha \beta }\),
\(C_{\alpha \beta }\),
\(K_{\alpha \beta }\),
\(Q_{\alpha }\),
\(G_{\!.\alpha }\),
\(q_{\beta }\) and
\(\lambda _{\alpha }\) are substituted into (
8) and (
16). Equating the coefficients of the given parameter
\(\theta \) to zeroth, first and second power leads to
-
1 pair of the zero-order equations
$$\begin{aligned}&M_{\alpha \beta }^0 {\ddot{q}}_{\beta }^0(\tau ) + C_{\alpha \beta }^0 {\dot{q}}_{\beta }^0(\tau ) + K_{\alpha \beta }^0 q_{\beta }^0(\tau ) = Q_{\alpha }^0(\tau )\nonumber \\&q_{\alpha }^0(0)=0; \quad {\dot{q}}_{\alpha }^0(0)=0; \quad \tau \in [0,t]; \quad t\in [0,T] \nonumber \\&M_{\alpha \beta }^0 {\ddot{\lambda }}_{\beta }^0(\tau ) - C_{\alpha \beta }^0 {\dot{\lambda }}_{\beta }^0(\tau ) + K_{\alpha \beta }^0 \lambda _{\beta }^0(\tau ) = G_{\!.\alpha }^0(t) \delta (t-\tau )\nonumber \\&\lambda _{\alpha }^0(t)=0; \quad {\dot{\lambda }}_{\alpha }^0(t)=0; \quad \tau \in [0,t]; \quad t\in [0,T] \end{aligned}$$
(19)
-
r pairs of the first-order equations
$$\begin{aligned}&M_{\alpha \beta }^0 {\ddot{q}}_{\beta }^{;r}(\tau ) + C_{\alpha \beta }^0 {\dot{q}}_{\beta }^{;r}(\tau ) + K_{\alpha \beta }^0 q_{\beta }^{;r}(\tau ) = Q_{\alpha }^{r}(\tau )\nonumber \\&q_{\alpha }^{;r}(0)=0; \quad {\dot{q}}_{\alpha }^{;r}(0)=0; \quad \tau \in [0,t]; \quad t\in [0,T]\nonumber \\&M_{\alpha \beta }^0 {\ddot{\lambda }}_{\beta }^{;r}(\tau ) - C_{\alpha \beta }^0 {\dot{\lambda }}_{\beta }^{;r}(\tau ) + K_{\alpha \beta }^0 \lambda _{\beta }^{;r}(\tau ) = G_{\alpha }^{r}(\tau ,t) \nonumber \\&\lambda _{\alpha }^{;r}(t)=0; \quad {\dot{\lambda }}_{\alpha }^{;r}(t)=0; \quad \tau \in [0,t]; \quad t\in [0,T] \nonumber \\&r=1,2,\ldots , R \end{aligned}$$
(20)
-
1 pair of the second-order equations
$$\begin{aligned}&M_{\alpha \beta }^0 {\ddot{q}}_{\beta }^{(2)}(\tau ) + C_{\alpha \beta }^0 {\dot{q}}_{\beta }^{(2)}(\tau ) + K_{\alpha \beta }^0 q_{\beta }^{(2)}(\tau ) = Q_{\alpha }^{(2)}(\tau )\nonumber \\&q_{\alpha }^{(2)}(0)=0; \quad {\dot{q}}_{\alpha }^{(2)}(0)=0; \quad \tau \in [0,t]; \quad t\in [0,T] \nonumber \\&M_{\alpha \beta }^0 {\ddot{\lambda }}_{\beta }^{(2)}(\tau ) - C_{\alpha \beta }^0 {\dot{\lambda }}_{\beta }^{(2)}(\tau ) + K_{\alpha \beta }^0 \lambda _{\beta }^{(2)}(\tau ) = G_{\alpha }^{(2)}(\tau ,t) \nonumber \\&\lambda _{\alpha }^{(2)}(t)=0; \quad {\dot{\lambda }}_{\alpha }^{(2)}(t)=0; \quad \tau \in [0,t]; \quad t\in [0,T] \end{aligned}$$
(21)
where
$$\begin{aligned} \begin{array}{rcl} q_{\alpha }^{(2)} &{}=&{} {\frac{1}{2}} \, q_{\alpha }^{;rs} \, \text {Cov} (b^r, b^s)\\ \lambda _{\alpha }^{(2)} &{}=&{} {\frac{1}{2}} \, \lambda _{\alpha }^{;rs} \, \text {Cov} (b^r, b^s) \end{array}, \quad \;\; r,s=1,2,\ldots , R \end{aligned}$$
(22)
while the first- and second-order primary and adjoint generalized load vectors are defined by
$$\begin{aligned} Q_{\alpha }^{r}(\tau )= & {} Q_{\alpha }^{;r}(\tau ) - M_{\alpha \beta }^{;r}{\ddot{q}}_{\beta }^0(\tau ) - C_{\alpha \beta }^{;r}{\dot{q}}_{\beta }^0(\tau ) - K_{\alpha \beta }^{;r}q_{\beta }^0(\tau ) \nonumber \\ G_{\alpha }^{r}(\tau ,t)= & {} G_{.\alpha }^{;r}(t)\delta (t-\tau ) - M_{\alpha \beta }^{;r}{\ddot{\lambda }}_{\beta }^0(\tau ) + C_{\alpha \beta }^{;r}{\dot{\lambda }}_{\beta }^0(\tau ) - K_{\alpha \beta }^{;r}\lambda _{\beta }^0(\tau )\end{aligned}$$
(23)
$$\begin{aligned} Q_{\alpha }^{(2)}(\tau )= & {} \Big [{\frac{1}{2}} \,Q_{\alpha }^{;rs}(\tau ) - M_{\alpha \beta }^{;r}{\ddot{q}}_{\beta }^{;s}(\tau ) - C_{\alpha \beta }^{;r}{\dot{q}}_{\beta }^{;s}(\tau ) - K_{\alpha \beta }^{;r}q_{\beta }^{;s}(\tau ) \nonumber \\&-{\frac{1}{2}} \, M_{\alpha \beta }^{;rs}{\ddot{q}}_{\beta }^{\,0}(\tau ) -{\frac{1}{2}} \, C_{\alpha \beta }^{;rs}{\dot{q}}_{\beta }^0(\tau ) -{\frac{1}{2}} \, K_{\alpha \beta }^{;rs}q_{\beta }^0(\tau ) \Big ]\text {Cov} (b^r, b^s) \nonumber \\ G_{\alpha }^{(2)}(\tau ,t)= & {} \Big [ {\frac{1}{2}} \,G_{.\alpha }^{;rs}(t)\delta (t-\tau ) - M_{\alpha \beta }^{;r}{\ddot{\lambda }}_{\beta }^{;s}(\tau ) + C_{\alpha \beta }^{;r}{\dot{\lambda }}_{\beta }^{;s}(\tau ) - K_{\alpha \beta }^{;r}\lambda _{\beta }^{;s}(\tau ) \nonumber \\&-{\frac{1}{2}} \, M_{\alpha \beta }^{;rs}{\ddot{\lambda }}_{\beta }^{\,0}(\tau ) +{\frac{1}{2}} \, C_{\alpha \beta }^{;rs}{\dot{\lambda }}_{\beta }^0(\tau ) -{\frac{1}{2}} \, K_{\alpha \beta }^{;rs}\lambda _{\beta }^0(\tau ) \Big ]\text {Cov} (b^r, b^s) \end{aligned}$$
(24)
Having solved the initial-terminal systems (
19)–(
21) for the zero-, first- and second-order primary and adjoint displacements, velocities and accelerations, the solution to the time instant stochastic sensitivity problem can be received by setting
\(\theta = 1\) in the expansions of
\(M_{\alpha \beta }\),
\(C_{\alpha \beta }\),
\(K_{\alpha \beta }\),
\(Q_{\alpha }\),
\(G_{\!.\alpha }\),
\(q_{\beta }\),
\({\dot{q}}_{\beta }\),
\({\ddot{q}}_{\beta }\),
\(\lambda _{\alpha }\),
\({\dot{\lambda }}_{\alpha }\) and
\({\ddot{\lambda }}_{\alpha }\) via the second-order perturbation. In this way, the dynamic case of the second-order accurate expectations and cross-covariances for the time instant sensitivity gradient are written, respectively, as
$$\begin{aligned} {\text {E}} \left[ \varPhi ^{.d} (t) \right]= & {} G^{0.d}(t) + \int _0^t \Big \{ {{{\mathcal {A}}}}_{\alpha }^d(\tau ) \big [\lambda _{\alpha }^0(\tau ) + \lambda _{\alpha }^{(2)}(\tau )\big ]- {{{\mathcal {D}}}}_{\alpha }^{d(2)}(\tau ) \lambda _{\alpha }^0(\tau )\Big \} \text {d}\tau \nonumber \\&+\, \bigg \{ {{\frac{1}{2}}} G^{.d;rs}(t) + \int _0^t \big [{{{\mathcal {B}}}}_{\alpha }^{dr}(\tau ) \lambda _{\alpha }^{;s}(\tau ) + {{{\mathcal {C}}}}_{\alpha }^{drs}(\tau ) \lambda _{\alpha }^0(\tau ) \big ] \text {d}\tau \bigg \}\text {Cov} (b^r, b^s)\end{aligned}$$
(25)
$$\begin{aligned} \text {Cov} (\varPhi ^{.d}(t_1), \varPhi ^{.e}(t_2))= & {} \Bigg (G^{.d;r}(t_1) G^{.e;s}(t_2) \nonumber \\&+ \,G^{.d;r}(t_1) \int _0^{t_2} \big [{{{\mathcal {A}}}}_{\alpha }^e(\tau )\lambda _{\alpha }^{;s}(\tau )+ {{{\mathcal {B}}}}_{\alpha }^{es}(\tau ) \lambda _{\alpha }^0(\tau )\big ] \text {d}\tau \nonumber \\&+ \,G^{.e;r}(t_2) \int _0^{t_1} \big [{{{\mathcal {A}}}}_{\alpha }^d(\tau )\lambda _{\alpha }^{;s}(\tau )+ {{{\mathcal {B}}}}_{\alpha }^{ds}(\tau ) \lambda _{\alpha }^0(\tau )\big ] \text {d}\tau \nonumber \\&+ \int _0^{t_1} \int _0^{t_2} \Big \{ {{{\mathcal {A}}}}_{\alpha }^d(\tau ) {{{\mathcal {A}}}}_{\beta }^e(\nu ) \lambda _{\alpha }^{;r}(\tau ) \lambda _{\beta }^{;s}(\nu ) \nonumber \\&+ \, \big [{{{\mathcal {A}}}}_{\alpha }^d(\tau ) {{{\mathcal {B}}}}_{\beta }^{er}(\nu ) + {{{\mathcal {A}}}}_{\beta }^e(\nu ) {{{\mathcal {B}}}}_{\alpha }^{dr}(\tau )\big ] \lambda _{\alpha }^{;s}(\tau ) \lambda _{\beta }^0(\nu ) \nonumber \\&+ \, {{{\mathcal {B}}}}_{\alpha }^{dr}(\tau ) {{{\mathcal {B}}}}_{\beta }^{es}(\nu ) \lambda _{\alpha }^0 (\tau )\lambda _{\beta }^0(\nu )\Big \} \text {d}\tau \text {d}\nu \nonumber \\&- \, {\overline{\varPhi }}^{\,\, .d}(t_1) \bigg \{ {{\frac{1}{2}}} \, G^{.e;rs}(t_2) + \int _0^{t_2} \big [{{{\mathcal {B}}}}_{\alpha }^{er}(\tau ) \lambda _{\alpha }^{;s}(\tau )\nonumber \\&+ \, {{{\mathcal {C}}}}_{\alpha }^{ers}(\tau ) \lambda _{\alpha }^{0}(\tau ) \big ] \text {d}\tau \bigg \} - {\overline{\varPhi }}^{\,\, .e}(t_2) \bigg \{ {{\frac{1}{2}}} \, G^{.d;rs}(\tau ) \nonumber \\&+\int _0^{t_1} \big [{{{\mathcal {B}}}}_{\alpha }^{dr}(\tau ) \lambda _{\alpha }^{;s}(\tau ) + {{{\mathcal {C}}}}_{\alpha }^{drs}(\tau ) \lambda _{\alpha }^{0}(\tau ) \big ] \text {d}\tau \bigg \} \Bigg ) \text {Cov} (b^r, b^s)\nonumber \\&+ \, {\overline{\varPhi }}^{\,\, .d}(t_1) \int _0^{t_2} \big [{{{\mathcal {D}}}}_{\alpha }^{e(2)}(\tau ) \lambda _{\alpha } ^0(\tau ) -{{{\mathcal {A}}}}_{\alpha }^{e}(\tau ) \lambda _{\alpha }^{(2)} (\tau ) \big ] \text {d}\tau \nonumber \\&+ \, {\overline{\varPhi }}^{\,\, .e}(t_2) \int _0^{t_1} \big [{{{\mathcal {D}}}}_{\alpha }^{d(2)}(\tau ) \lambda _{\alpha }^0(\tau ) - {{{\mathcal {A}}}}_{\alpha }^{d}(\tau ) \lambda _{\alpha }^{(2)} (\tau ) \big ] \text {d}\tau \end{aligned}$$
(26)
where
$$\begin{aligned} {{{\mathcal {A}}}}_{\alpha }^d(\tau )= & {} Q_{\alpha }^{0.d}(\tau ) - K_{\alpha \beta }^{0.d} q_{\beta }^0(\tau ) - C_{\alpha \beta }^{0.d} {\dot{q}}_{\beta }^0(\tau ) - M_{\alpha \beta }^{0.d} {\ddot{q}}_{\beta }^0(\tau ) \nonumber \\ {{{\mathcal {B}}}}_{\alpha }^{dr}(\tau )= & {} Q_{\alpha }^{.d;r}(\tau ) - K_{\alpha \beta }^{0.d} q_{\beta }^{;r}(\tau ) - K_{\alpha \beta }^{.d;r} q_{\beta }^0(\tau ) - C_{\alpha \beta }^{0.d} {\dot{q}}_{\beta }^{;r}(\tau ) \nonumber \\&- \,\, C_{\alpha \beta }^{.d;r}(\tau ) {\dot{q}}_{\beta }^0(\tau ) - M_{\alpha \beta }^{0.d} {\ddot{q}}_{\beta }^{;r}(\tau ) - M_{\alpha \beta }^{.d;r} {\ddot{q}}_{\beta }^0(\tau ) \nonumber \\ {{{\mathcal {C}}}}_{\alpha }^{drs} (\tau )= & {} {\frac{1}{2}} Q_{\alpha }^{.d;rs}(\tau ) - K_{\alpha \beta }^{.d;r} q_{\beta }^{;s}(\tau ) - {\frac{1}{2}} K_{\alpha \beta }^{.d;rs} q_{\beta }^0(\tau ) - C_{\alpha \beta }^{.d;r} {\dot{q}}_{\beta }^{;s}(\tau ) \nonumber \\&- \,\, {\frac{1}{2}} C_{\alpha \beta }^{.d;rs} {\dot{q}}_{\beta }^0(\tau ) - M_{\alpha \beta }^{.d;r} {\ddot{q}}_{\beta }^{;s}(\tau ) - {\frac{1}{2}} M_{\alpha \beta }^{.d;rs} {\ddot{q}}_{\beta }^0(\tau ) \nonumber \\ {{{\mathcal {D}}}}_{\alpha }^{d(2)}(\tau )= & {} K_{\alpha \beta }^{0.d} q_{\beta }^{(2)}(\tau ) + C_{\alpha \beta }^{0.d} {\dot{q}}_{\beta }^{(2)}(\tau ) + M_{\alpha \beta }^{0.d} {\ddot{q}}_{\beta }^{(2)}(\tau ) \end{aligned}$$
(27)
with
\(t, t_1, t_2 \in [0,T]\);
\(d, e = 1,2,\ldots , D\);
\(r, s = 1,2,\ldots , R\);
\(\alpha , \beta = 1,2,\ldots , N\).