Inverse Dynamic Analysis of an Inclined FGM Beam Due to Moving Load for Estimating the Mass of Moving Load Based on a CGM

Abstract

In this paper, the use of the inverse solution method for estimating the mass of moving load on an inclined functionally graded material (FGM) Timoshenko beam is discussed based on the measured displacements. Fletcher–Reeves (FR) method is used to solve the inverse problem. Also, in the process of solving the inverse problem and in order to solve the direct problem, Newmark method is applied for discretization of the time domain and finite element method (FEM) is used for discretization of the space domain. According to many other similar types of research, instead of conducting an experiment and measuring displacements of the beam as inputs of the inverse solution, those inputs are obtained from the direct problem and then random errors are added. Furthermore, an inclined functionally graded (FG) beam with different boundary conditions and power-law exponents due to different speeds of the moving mass has been studied to evaluate the performance of the proposed method. The accuracy of results is very high in all cases.

Appendices

Appendix 1

$$\begin{aligned} \left[ {M^{\text{e}} } \right] & = \left[ {\begin{array}{*{20}c} {M_{ij}^{11} } & {M_{iJ}^{12} } & {M_{ij}^{13} } \\ {M_{Ij}^{21} } & {M_{IJ}^{22} } & {M_{Ij}^{23} } \\ {M_{ij}^{31} } & {M_{iJ}^{32} } & {M_{ij}^{33} } \\ \end{array} } \right],\;\left[ {K^{\text{e}} } \right] = \left[ {\begin{array}{*{20}c} {K_{ij}^{11} } & {K_{iJ}^{12} } & {K_{ij}^{13} } \\ {K_{Ij}^{21} } & {K_{IJ}^{22} } & {K_{Ij}^{23} } \\ {K_{ij}^{31} } & {K_{iJ}^{32} } & {K_{ij}^{33} } \\ \end{array} } \right] \\ M_{ij}^{11} & = \int_{0}^{l} {\rho_{0} \phi_{i}^{(1)} \phi_{j}^{(1)} } dx,\;M_{iJ}^{12} = 0,\;M_{ij}^{13} = \int_{0}^{l} {\rho_{1} \phi_{i}^{(1)} \phi_{j}^{(3)} } {\text{d}}x,\;M_{Ij}^{21} = 0,\;M_{IJ}^{22} = \int_{0}^{l} {\rho_{0} \phi_{I}^{(2)} \phi_{J}^{(2)} } {\text{d}}x,\;M_{Ij}^{23} = 0, \\ M_{ij}^{31} & = (M_{ij}^{13} )^{\text{T}} ,\;M_{iJ}^{32} = 0,\;M_{ij}^{33} = \int_{0}^{l} {\rho_{2} \phi_{i}^{(3)} \phi_{j}^{(3)} } {\text{d}}x, \\ K_{ij}^{11} & = \int_{0}^{l} {A_{11} \frac{{\partial \phi_{i}^{(1)} }}{\partial x}\frac{{\partial \phi_{j}^{(1)} }}{\partial x}} {\text{d}}x,\;K_{iJ}^{12} = 0,\;K_{ij}^{13} = \int_{0}^{l} {B_{11} \frac{{\partial \phi_{i}^{(1)} }}{\partial x}\frac{{\partial \phi_{j}^{(3)} }}{\partial x}} {\text{d}}x,\;K_{Ij}^{21} = 0,\;K_{IJ}^{22} = \int_{0}^{l} {k_{s} A_{55} \frac{{\partial \phi_{I}^{(2)} }}{\partial x}\frac{{\partial \phi_{J}^{(2)} }}{\partial x}} {\text{d}}x, \\ K_{Ij}^{23} & = \int_{0}^{l} {k_{s} A_{55} \frac{{\partial \phi_{I}^{(2)} }}{\partial x}\phi_{j}^{(3)} } {\text{d}}x,\;K_{ij}^{31} = (K_{ij}^{13} )^{\text{T}} ,\;K_{iJ}^{32} = (K_{Ij}^{23} )^{T} ,\;K_{ij}^{33} = \int_{0}^{l} {D_{11} \frac{{\partial \phi_{i}^{(3)} }}{\partial x}\frac{{\partial \phi_{j}^{(3)} }}{\partial x} + k_{s} A_{55} \phi_{i}^{(3)} \phi_{j}^{(3)} } {\text{d}}x, \\ \left[ {\rho_{0} ,\rho_{1} ,\rho_{2} ,A_{11} ,B_{11} ,D_{11} ,A_{55} } \right] & = \int\limits_{ - h/2}^{h/2} {\int\limits_{ - b/2}^{b/2} {\left[ {\rho (z),z\rho (z),z^{2} \rho (z),E(z),zE(z),z^{2} E(z),G(z)} \right] \, } } {\text{d}}y{\text{d}}z. \\ \end{aligned}$$

(A1–20)

where $i,j = 1,2$ and $I,J = 1,2,3$.

And $Q_{i}^{\text{e}} \, \left( {i = 1, \ldots ,7} \right)$ are the nodal forces of an element of the inclined beam that in this study according to boundary conditions for two end nodes are zero or unknown, and for other nodes are zero.

Appendix 2

$$\begin{aligned} \left\{ f \right\} & = \left[ {\begin{array}{*{20}c} {f_{1} } & {f_{2} } & {f_{3} } & {f_{4} } & {f_{5} } & {f_{6} } & {f_{7} } \\ \end{array} } \right]^{T} ,\;\left\{ q \right\} = \left[ {\begin{array}{*{20}c} {q_{1} } & {q_{2} } & {q_{3} } & {q_{4} } & {q_{5} } & {q_{6} } & {q_{7} } \\ \end{array} } \right]^{T} , \\ \left\{ {\dot{q}} \right\} & = \left[ {\begin{array}{*{20}c} {\dot{q}_{1} } & {\dot{q}_{2} } & {\dot{q}_{3} } & {\dot{q}_{4} } & {\dot{q}_{5} } & {\dot{q}_{6} } & {\dot{q}_{7} } \\ \end{array} } \right]^{T} ,\;\left\{ {\ddot{q}} \right\} = \left[ {\begin{array}{*{20}c} {\ddot{q}_{1} } & {\ddot{q}_{2} } & {\ddot{q}_{3} } & {\ddot{q}_{4} } & {\ddot{q}_{5} } & {\ddot{q}_{6} } & {\ddot{q}_{7} } \\ \end{array} } \right]^{T} , \\ \left[ m \right] & = m_{\text{c}} \left[ {\begin{array}{*{20}c} {m_{ij}^{11} } & {m_{iJ}^{12} } & {m_{ij}^{13} } \\ {m_{Ij}^{21} } & {m_{IJ}^{22} } & {m_{Ij}^{23} } \\ {m_{ij}^{31} } & {m_{iJ}^{32} } & {m_{ij}^{33} } \\ \end{array} } \right],\left[ c \right] = 2m_{c} v\left[ {\begin{array}{*{20}c} {c_{ij}^{11} } & {c_{iJ}^{12} } & {c_{ij}^{13} } \\ {c_{Ij}^{21} } & {c_{IJ}^{22} } & {c_{Ij}^{23} } \\ {c_{ij}^{31} } & {c_{iJ}^{32} } & {c_{ij}^{33} } \\ \end{array} } \right], \\ \left[ k \right] & = m_{\text{c}} v^{2} \left[ {\begin{array}{*{20}c} {k_{ij}^{11} } & {k_{iJ}^{12} } & {k_{ij}^{13} } \\ {k_{Ij}^{21} } & {k_{IJ}^{22} } & {k_{Ij}^{23} } \\ {k_{ij}^{31} } & {k_{iJ}^{32} } & {k_{ij}^{33} } \\ \end{array} } \right],\left\{ F \right\} = m_{\text{c}} g\left\{ {\begin{array}{*{20}c} {F_{i}^{1} } \\ {F_{I}^{2} } \\ {F_{i}^{3} } \\ \end{array} } \right\}. \\ \end{aligned}$$

(B1–8)

and

$$\begin{aligned} m_{ij}^{11} & = \phi_{i}^{(1)} \phi_{j}^{(1)} ,\;m_{iJ}^{12} = - \mu \phi_{i}^{(1)} \phi_{J}^{(2)} ,\;m_{ij}^{13} = 0,\;m_{Ij}^{21} = 0,\;m_{IJ}^{22} = \phi_{I}^{(2)} \phi_{J}^{(2)} ,\;m_{Ij}^{23} = 0,\;m_{ij}^{31} = 0,\;m_{iJ}^{32} = 0,\;m_{ij}^{33} = 0, \\ c_{ij}^{11} & = 0,\;c_{iJ}^{12} = - \mu \phi_{i}^{(1)} \phi_{J}^{{{\prime (2)}}} ,\;c_{ij}^{13} = 0,\;c_{Ij}^{21} = 0,\;c_{IJ}^{22} = \phi_{I}^{(2)} \phi_{J}^{{{\prime }(2)}} ,\;c_{Ij}^{23} = 0,\;c_{ij}^{31} = 0,\;c_{iJ}^{32} = 0,\;c_{ij}^{33} = 0, \\ k_{ij}^{11} & = 0,\;k_{iJ}^{12} = - \mu \phi_{i}^{(1)} \phi_{J}^{{{\prime }(2)}} ,\;k_{ij}^{13} = 0,\;k_{Ij}^{21} = 0,\;k_{IJ}^{22} = \phi_{I}^{(2)} \phi_{J}^{(2)} ,\;k_{Ij}^{23} = 0,\;k_{ij}^{31} = 0,\;k_{iJ}^{32} = 0,\;k_{ij}^{33} = 0, \\ F_{i}^{1} & = - \phi_{i}^{(1)} \left( {\mu \cos \theta + \sin \theta } \right),\;F_{I}^{2} = \phi_{I}^{(2)} \cos \theta ,\;F_{i}^{3} = 0 , { ( }i,j = 1,2; \, I,\;J = 1,2,3{ )} .\\ \end{aligned}$$

(B9–38)