Analytical solutions of the forced vibration of Timoshenko micro/nano-beam under axial tensions supported on Winkler–Pasternak foundation

Abstract

Axially loaded micro/nano-beams supported on foundations are extensively applied in micro/nano-medicine field. This paper aims to derive analytical solutions of the forced vibration of Timoshenko micro/nano-beam under axial tensions supported on Winkler–Pasternak foundation subjected to a harmonic load. Timoshenko beam theory and nonlocal strain gradient theory are employed to describe a micro/nano-beam model. Based on Hamilton’s principle, the vibration equations of the micro/nano-beam are derived. Directions of axial tensions during the deformation of the micro/nano-beam are described by transition parameters. Using the weighted residual method, the variational consistency boundary conditions (BCs) can be derived according to the vibration equations. Explicit expressions of steady-state dynamic responses of the micro/nano-beam are obtained by Green's function method and superposition principle. Numerical examples are performed to verify present solutions by some published articles. Influences of some important parameters, such as transition parameter, nonlocal parameter, and strain gradient parameter, on dynamic behaviors of the system are also investigated. Furthermore, numerical results reveal that the variational consistency BCs and the transition parameter produce significant effects on vibration characteristics of the system. More specially, the variational consistency BCs change the vibration shape of simply supported BCs into that of clamed-clamed BCs.

Appendix

According to Eqs. (43)–(44), the following forms are obtained by inverse Laplace transform:

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} b_{0} + s^{2} b_{1} + b_{2} } \right)\left( {a_{5} s^{2} + a_{6} } \right)e^{{ - s\xi_{0} }} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {H\left( {\xi - \xi_{0} } \right)} A_{i} \left( {\xi - \xi_{0} } \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)\left( {a_{5} s_{i}^{2} + a_{6} } \right)} \right], \\ \end{aligned} $$

(A1)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} b_{0} + s^{2} b_{1} + b_{2} } \right)\left( {s^{3} a_{0} + sa_{1} } \right) - \left( {s^{3} a_{3} + sa_{4} } \right)\left( {sb_{3} + b_{4} } \right)}}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ \begin{array}{llll} \left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)\left( {s_{i}^{3} a_{0} + s_{i} a_{1} } \right) \hfill \\ - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)\left( {s_{i} b_{3} + b_{4} } \right) \hfill \\ \end{array} \right]} , \\ \end{aligned} $$

(A2)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} b_{0} + s^{2} b_{1} + b_{2} } \right)\left( {s^{2} a_{0} + a_{1} } \right) - \left( {s^{3} a_{3} + sa_{4} } \right)sb_{3} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ \,& \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)\left( {s_{i}^{2} a_{0} + a_{1} } \right) - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)s_{i} b_{3} } \right]} , \\ \end{aligned} $$

(A3)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} b_{0} + s^{2} b_{1} + b_{2} } \right)sa_{0} - \left( {s^{3} a_{3} + sa_{4} } \right)b_{3} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)s_{i} a_{0} - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)b_{3} } \right]} , \\ \end{aligned} $$

(A4)

$$ L^{ - 1} \left( {\frac{{s^{4} b_{0} + s^{2} b_{1} + b_{2} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right){ = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)} , $$

(A5)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} b_{0} + s^{2} b_{1} + b_{2} } \right)\left( {s^{2} a_{3} + a_{4} } \right) - \left( {s^{3} a_{3} + sa_{4} } \right)\left( {s^{3} b_{0} + sb_{1} } \right)}}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)\left( {s_{i}^{2} a_{3} + a_{4} } \right) - \left( {s_{i}^{3} a_{3} + sa_{4} } \right)\left( {s_{i}^{3} b_{0} + s_{i} b_{1} } \right)} \right]} , \\ \end{aligned} $$

(A6)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} b_{0} + s^{2} b_{1} + b_{2} } \right)sa_{3} - \left( {s^{3} a_{3} + sa_{4} } \right)\left( {s^{2} b_{0} + b_{1} } \right)}}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)s_{i} a_{3} - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)\left( {s_{i}^{2} b_{0} + b_{1} } \right)} \right]} , \\ \end{aligned} $$

(A7)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} b_{0} + s^{2} b_{1} + b_{2} } \right)a_{3} - \left( {s^{3} a_{3} + sa_{4} } \right)sb_{0} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)a_{3} - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)s_{i} b_{0} } \right]} , \\ \end{aligned} $$

(A8)

$$ L^{ - 1} \left( {\frac{{ - \left( {s^{3} a_{3} + sa_{4} } \right)b_{0} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right){ = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ { - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)b_{0} } \right]} , $$

(A9)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{ - \left( {s^{3} b_{3} + sb_{4} } \right)\left( {a_{5} s^{2} + a_{6} } \right)e^{{ - s\xi_{0} }} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {H\left( {\xi - \xi_{0} } \right)} A_{i} \left( {\xi - \xi_{0} } \right)\left[ { - \left( {s_{i}^{3} b_{3} + sb_{4} } \right)\left( {a_{5} s_{i}^{2} + a_{6} } \right)} \right], \\ \end{aligned} $$

(A10)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} a_{0} + s^{2} a_{1} + a_{2} } \right)\left( {sb_{3} + b_{4} } \right) - \left( {s^{3} b_{3} + sb_{4} } \right)\left( {s^{3} a_{0} + sa_{1} } \right)}}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)\left( {s_{i} b_{3} + b_{4} } \right) - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)\left( {s_{i}^{3} a_{0} + s_{i} a_{1} } \right)} \right]} , \\ \end{aligned} $$

(A11)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} a_{0} + s^{2} a_{1} + a_{2} } \right)sb_{3} - \left( {s^{3} b_{3} + sb_{4} } \right)\left( {s^{2} a_{0} + a_{1} } \right)}}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)s_{i} b_{3} - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)\left( {s_{i}^{2} a_{0} + a_{1} } \right)} \right]} , \\ \end{aligned} $$

(A12)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} a_{0} + s^{2} a_{1} + a_{2} } \right)b_{3} - \left( {s^{3} b_{3} + sb_{4} } \right)sa_{0} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)b_{3} - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)s_{i} a_{0} } \right]} , \\ \end{aligned} $$

(A13)

$$ L^{ - 1} \left( {\frac{{ - \left( {s^{3} b_{3} + sb_{4} } \right)a_{0} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right){ = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ { - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)a_{0} } \right]} , $$

(A14)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} a_{0} + s^{2} a_{1} + a_{2} } \right)\left( {s^{3} b_{0} + sb_{1} } \right) - \left( {s^{3} b_{3} + sb_{4} } \right)\left( {s^{2} a_{3} + a_{4} } \right)}}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)\left( {s_{i}^{3} b_{0} + s_{i} b_{1} } \right) - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)\left( {s_{i}^{2} a_{3} + a_{4} } \right)} \right]} , \\ \end{aligned} $$

(A15)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} a_{0} + s^{2} a_{1} + a_{2} } \right)\left( {s^{2} b_{0} + b_{1} } \right) - \left( {s^{3} b_{3} + sb_{4} } \right)sa_{3} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)\left( {s_{i}^{2} b_{0} + b_{1} } \right) - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)s_{i} a_{3} } \right]} , \\ \end{aligned} $$

(A16)

$$ \begin{aligned} & L^{ - 1} \left( {\frac{{\left( {s^{4} a_{0} + s^{2} a_{1} + a_{2} } \right)sb_{0} - \left( {s^{3} b_{3} + sb_{4} } \right)a_{3} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right) \\ & \quad { = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)s_{i} b_{0} - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)a_{3} } \right]} , \\ \end{aligned} $$

(A17)

$$ L^{ - 1} \left( {\frac{{s^{4} a_{0} + s^{2} a_{1} + a_{2} }}{{\prod\nolimits_{i = 1}^{8} {\left( {s - s_{i} } \right)} }}} \right){ = }\sum\nolimits_{i = 1}^{8} {A_{i} \left( \xi \right)\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)} , $$

(A18)

where

$$ H\left( {\xi - \xi_{0} } \right){ = }\left\{ {\begin{array}{*{20}c} {1,} & {\xi \ge \xi_{0} } \\ {0,} & {\xi < \xi_{0} } \\ \end{array} } \right., $$

(A19)

$$ A_{i} \left( x \right) = \frac{{e^{{s_{i} x}} }}{{\left( {s_{i} - s_{1} } \right)...\left( {s_{i} - s_{i - 1} } \right)\left( {s_{i} - s_{i + 1} } \right)...\left( {s_{i} - s_{8} } \right)}} \, \left( {i = 1 \sim 8} \right). $$

(A20)

From Eqs. (A1)–(A20) and Eq. (43), Green’s functions can be derived

$$ \begin{aligned} W\left( {\xi ,\xi_{0} } \right) & = H\left( {\xi - \xi_{0} } \right)\varphi_{11} \left( {\xi - \xi_{0} } \right){ + }\varphi_{12} \left( \xi \right)W\left( 0 \right) \\ & \quad + \varphi_{13} \left( \xi \right)W^{\prime}\left( 0 \right) + \varphi_{14} \left( \xi \right)W^{\prime\prime}\left( 0 \right) \\ & \quad + \varphi_{15} \left( \xi \right)W^{\prime\prime\prime}\left( 0 \right) + \varphi_{16} \left( \xi \right)\Phi \left( 0 \right) + \varphi_{17} \left( \xi \right)\Phi^{\prime}\left( 0 \right) \\ & \quad + \varphi_{18} \left( \xi \right)\Phi^{\prime\prime}\left( 0 \right) + \varphi_{19} \left( \xi \right)\Phi^{\prime\prime\prime}\left( 0 \right), \\ \end{aligned} $$

(A21)

$$ \begin{aligned} \Phi \left( {\xi ,\xi_{0} } \right) & = H\left( {\xi - \xi_{0} } \right)\varphi_{21} \left( {\xi - \xi_{0} } \right) + \varphi_{22} \left( \xi \right)W\left( 0 \right) \\ & \quad + \varphi_{23} \left( \xi \right)W^{\prime}\left( 0 \right) + \varphi_{24} \left( \xi \right)W^{\prime\prime}\left( 0 \right) \\ & \quad + \varphi_{25} \left( \xi \right)W^{\prime\prime\prime}\left( 0 \right) + \varphi_{26} \left( \xi \right)\Phi \left( 0 \right) + \varphi_{27} \left( x \right)\Phi^{\prime}\left( 0 \right) \\ & \quad + \varphi_{28} \left( \xi \right)\Phi^{\prime\prime}\left( 0 \right) + \varphi_{29} \left( \xi \right)\Phi^{\prime\prime\prime}\left( 0 \right). \\ \end{aligned} $$

(A22)

The expression of φ_mn (m = 1, 2, n = 1 ~ 9) in Eqs. (A21) and (A22) is listed as follows:

$$ \begin{aligned} \varphi_{11} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)\left( {a_{5} s_{i}^{2} + a_{6} } \right)} \right]} , \\ \varphi_{12} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)\left( {s_{i}^{3} a_{0} + s_{i} a_{1} } \right) - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)\left( {s_{i} b_{3} + b_{4} } \right)} \right]} , \\ \varphi_{13} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)\left( {s_{i}^{2} a_{0} + a_{1} } \right) - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)s_{i} b_{3} } \right]} , \\ \varphi_{14} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)s_{i} a_{0} - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)b_{3} } \right]} , \\ \varphi_{15} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)} , \\ \varphi_{16} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)\left( {s_{i}^{2} a_{3} + a_{4} } \right) - \left( {s_{i}^{3} a_{3} + sa_{4} } \right)\left( {s_{i}^{3} b_{0} + s_{i} b_{1} } \right)} \right]} , \\ \varphi_{17} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)s_{i} a_{3} - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)\left( {s_{i}^{2} b_{0} + b_{1} } \right)} \right]} , \\ \varphi_{18} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} b_{0} + s_{i}^{2} b_{1} + b_{2} } \right)a_{3} - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)s_{i} b_{0} } \right]} , \\ \varphi_{19} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ { - \left( {s_{i}^{3} a_{3} + s_{i} a_{4} } \right)b_{0} } \right]} , \\ \end{aligned} $$

(A23)

$$ \begin{aligned} \varphi_{21} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ { - \left( {s_{i}^{3} b_{3} + sb_{4} } \right)\left( {a_{5} s_{i}^{2} + a_{6} } \right)} \right]} , \\ \varphi_{22} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)\left( {s_{i} b_{3} + b_{4} } \right) - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)\left( {s_{i}^{3} a_{0} + s_{i} a_{1} } \right)} \right]} , \\ \varphi_{23} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)s_{i} b_{3} - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)\left( {s_{i}^{2} a_{0} + a_{1} } \right)} \right]} , \\ \varphi_{24} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)b_{3} - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)s_{i} a_{0} } \right]} , \\ \varphi_{25} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ { - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)a_{0} } \right]} , \\ \varphi_{26} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)\left( {s_{i}^{3} b_{0} + s_{i} b_{1} } \right) - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)\left( {s_{i}^{2} a_{3} + a_{4} } \right)} \right]} , \\ \varphi_{27} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)\left( {s_{i}^{2} b_{0} + b_{1} } \right) - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)s_{i} a_{3} } \right]} , \\ \varphi_{28} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left[ {\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)s_{i} b_{0} - \left( {s_{i}^{3} b_{3} + s_{i} b_{4} } \right)a_{3} } \right]} , \\ \varphi_{29} & { = }\sum\limits_{i = 1}^{8} {A_{i} \left( \xi \right)\left( {s_{i}^{4} a_{0} + s_{i}^{2} a_{1} + a_{2} } \right)} . \\ \end{aligned} $$

(A24)

To determine the constants W(0), W′(0), W′′(0), W′′′(0), Φ(0), Φ′(0), Φ′′(0) and Φ′′′(0), the various order derivatives of φ_mn can be expressed with an unified form

$$ \varphi_{mn}^{\left( k \right)} { = }\sum\limits_{i = 1}^{8} {s_{i}^{k} A_{i} \left( \xi \right)g\left( {s_{i} } \right)} . $$

(A25)