A Novel Solution Method for Free Vibration Analysis of Functionally Graded Arbitrary Quadrilateral Plates with Hole

Abstract

Purpose

In this paper, a new meshfree moving least squares-Tchebychev (MLST) shape function is proposed to analyze the free vibration characteristics of functionally graded arbitrary quadrilateral plate with hole.

Method

The plate and hole have an arbitrary quadrilateral shape. The whole plate structure is separated into the segments with arbitrary quadrilateral shape by the domain discompose method and these segments are modeled to a square plate through the coordinate mapping. The fourth-order polynomial mapping approach is used as a mapping function for the coordinate mapping. The first-order shear deformation theory (FSDT) is adopted in theoretical formulation for the free vibration analysis of functionally graded arbitrary quadrilateral plate. The boundary and continuation conditions are generalized by the artificial spring technique. All the displacement functions containing the boundary and continuation conditions are expressed by the meshfree MLST shape function, on the base of this, the governing equation of arbitrary plate with hole are obtained. Thus, the natural frequency and mode shape of the functionally graded arbitrary quadrilateral plate with hole are obtained by solving the governing equation.

Results

The accuracy and reliability of the proposed method are verified by comparison with the results of literature and finite element method (FEM). The free vibration characteristics (i.e. natural frequency and mode shape) of the functionally graded arbitrary quadrilateral plate with arbitrary quadrilateral hole under different boundary conditions are proposed through the parameter research and some examples.

Appendices

Appendix A

${a}_{11}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}^{2}({x}_{i}){T}_{0}^{2}({y}_{i})$	${a}_{12}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}^{2}({x}_{i}){T}_{0}({y}_{i}){T}_{1}({y}_{i})$
${a}_{13}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}^{2}({x}_{i}){T}_{0}({y}_{i}){T}_{2}({y}_{i})$	${a}_{14}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{1}({x}_{i}){T}_{0}^{2}({y}_{i})$
${a}_{15}={a}_{24}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{1}({x}_{i}){T}_{0}({y}_{i}){T}_{1}({y}_{i})$	${a}_{16}={a}_{34}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{1}({x}_{i}){T}_{0}({y}_{i}){T}_{2}({y}_{i})$
${a}_{17}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{2}({x}_{i}){T}_{0}^{2}({y}_{i})$	${a}_{18}={a}_{27}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{2}({x}_{i}){T}_{0}({y}_{i}){T}_{1}({y}_{i})$
${a}_{19}={a}_{37}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{2}({x}_{i}){T}_{0}({y}_{i}){T}_{2}({y}_{i})$	${a}_{22}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}^{2}({x}_{i}){T}_{1}^{2}({y}_{i})$
${a}_{23}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}^{2}({x}_{i}){T}_{1}({y}_{i}){T}_{2}({y}_{i})$	${a}_{25}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{1}({x}_{i}){T}_{1}^{2}({y}_{i})$
${a}_{26}={a}_{35}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{1}({x}_{i}){T}_{1}({y}_{i}){T}_{2}({y}_{i})$	${a}_{28}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{2}({x}_{i}){T}_{1}^{2}({y}_{i})$
${a}_{29}={a}_{38}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{2}({x}_{i}){T}_{1}({y}_{i}){T}_{2}({y}_{i})$	${a}_{33}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}^{2}({x}_{i}){T}_{2}^{2}({y}_{i})$
${a}_{36}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{1}({x}_{i}){T}_{2}^{2}({y}_{i})$	${a}_{39}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{0}({x}_{i}){T}_{2}({x}_{i}){T}_{2}^{2}({y}_{i})$
${a}_{44}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}^{2}({x}_{i}){T}_{0}^{2}({y}_{i})$	${a}_{45}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}^{2}({x}_{i}){T}_{0}({y}_{i}){T}_{1}({y}_{i})$
${a}_{46}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}^{2}({x}_{i}){T}_{0}({y}_{i}){T}_{2}({y}_{i})$	${a}_{47}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}({x}_{i}){T}_{2}({x}_{i}){T}_{0}^{2}({y}_{i})$
${a}_{48}={a}_{57}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}({x}_{i}){T}_{2}({x}_{i}){T}_{0}({y}_{i}){T}_{1}({y}_{i})$	${a}_{49}={a}_{67}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}({x}_{i}){T}_{2}({x}_{i}){T}_{0}({y}_{i}){T}_{2}({y}_{i})$
${a}_{55}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}^{2}({x}_{i}){T}_{1}^{2}({y}_{i})$	${a}_{56}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}^{2}({x}_{i}){T}_{1}({y}_{i}){T}_{2}({y}_{i})$
${a}_{58}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}({x}_{i}){T}_{2}({x}_{i}){T}_{1}^{2}({y}_{i})$	${a}_{59}={a}_{68}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}({x}_{i}){T}_{2}({x}_{i}){T}_{1}({y}_{i}){T}_{2}({y}_{i})$
${a}_{66}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}^{2}({x}_{i}){T}_{2}^{2}({y}_{i})$	${a}_{69}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{1}({x}_{i}){T}_{2}({x}_{i}){T}_{2}^{2}({y}_{i})$
${a}_{77}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{2}^{2}({x}_{i}){T}_{0}^{2}({y}_{i})$	${a}_{78}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{2}^{2}({x}_{i}){T}_{0}({y}_{i}){T}_{1}({y}_{i})$
${a}_{79}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{2}^{2}({x}_{i}){T}_{0}({y}_{i}){T}_{2}({y}_{i})$	${a}_{88}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{2}^{2}({x}_{i}){T}_{1}^{2}({y}_{i})$
${a}_{89}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{2}^{2}({x}_{i}){T}_{1}({y}_{i}){T}_{2}({y}_{i})$	${a}_{99}={\sum }_{i=1}^{n}{\overset{\frown}{W}}_{i}{T}_{2}^{2}({x}_{i}){T}_{2}^{2}({y}_{i})$

Appendix B:

${L}_{11}^{i}={A}_{11}\frac{{\partial }^{2}{\phi }_{i}}{\partial {x}^{2}}+{A}_{66}\frac{{\partial }^{2}{\phi }_{i}}{\partial {y}^{2}}$	${L}_{12}^{i}={L}_{21}^{i}=({A}_{12}+{A}_{66})\frac{{\partial }^{2}{\phi }_{i}}{\partial x\partial y}$
${L}_{13}^{i}={L}_{31}^{i}={L}_{23}^{i}={L}_{32}^{i}=0$	${L}_{14}^{i}={L}_{41}^{i}={B}_{11}\frac{{\partial }^{2}{\phi }_{i}}{\partial {x}^{2}}+{B}_{66}\frac{{\partial }^{2}{\phi }_{i}}{\partial {y}^{2}}$
${L}_{15}^{i}={L}_{51}^{i}={L}_{24}^{i}={L}_{42}^{i}=({B}_{12}+{B}_{66})\frac{{\partial }^{2}{\phi }_{i}}{\partial x\partial y}$	${L}_{22}^{i}={A}_{66}\frac{{\partial }^{2}{\phi }_{i}}{\partial {x}^{2}}+{A}_{11}\frac{{\partial }^{2}{\phi }_{i}}{\partial {y}^{2}}$
${L}_{25}^{i}={L}_{52}^{i}={B}_{66}\frac{{\partial }^{2}{\phi }_{i}}{\partial {x}^{2}}+{B}_{11}\frac{{\partial }^{2}{\phi }_{i}}{\partial {y}^{2}}$	${L}_{33}^{i}={A}_{55}\frac{{\partial }^{2}{\phi }_{i}}{\partial {x}^{2}}+{A}_{44}\frac{{\partial }^{2}{\phi }_{i}}{\partial {y}^{2}}$
${L}_{34}^{i}=-{L}_{43}^{i}={A}_{55}\frac{\partial {\phi }_{i}}{\partial x}$	${L}_{35}^{i}=-{L}_{53}^{i}={A}_{44}\frac{\partial {\phi }_{i}}{\partial y}$
${L}_{44}^{i}={D}_{11}\frac{{\partial }^{2}{\phi }_{i}}{\partial {x}^{2}}+{D}_{66}\frac{{\partial }^{2}{\phi }_{i}}{\partial {y}^{2}}-{A}_{55}{\phi }_{i}$	${L}_{45}^{i}={L}_{54}^{i}=({D}_{12}+{D}_{66})\frac{{\partial }^{2}{\phi }_{i}}{\partial x\partial y}$
${L}_{55}^{i}={D}_{66}\frac{{\partial }^{2}{\phi }_{i}}{\partial {x}^{2}}+{D}_{11}\frac{{\partial }^{2}{\phi }_{i}}{\partial {y}^{2}}-{A}_{44}{\phi }_{i}$

Appendix C:

${C}_{11}^{i}={A}_{11}\frac{\partial {\phi }_{i}}{\partial x}-{k}_{x0}^{u}{\phi }_{i}$	${C}_{12}^{i}={A}_{12}\frac{\partial {\phi }_{i}}{\partial y}$
${C}_{13}^{i}={C}_{23}^{i}={C}_{43}^{i}={C}_{53}^{i}={C}_{31}^{i}={C}_{32}^{i}={C}_{35}^{i}=0$
${C}_{14}^{i}={C}_{41}^{i}={B}_{11}\frac{\partial {\phi }_{i}}{\partial x}$	${C}_{15}^{i}={C}_{42}^{i}={B}_{12}\frac{\partial {\phi }_{i}}{\partial y}$
${C}_{21}^{i}={A}_{66}\frac{\partial {\phi }_{i}}{\partial y}$	${C}_{22}^{i}={A}_{66}\frac{\partial {\phi }_{i}}{\partial x}-{k}_{x0}^{v}{\phi }_{i}$
${C}_{24}^{i}={C}_{51}^{i}={B}_{66}\frac{\partial {\phi }_{i}}{\partial y}$	${C}_{25}^{i}={C}_{52}^{i}={B}_{66}\frac{\partial {\phi }_{i}}{\partial x}$
${C}_{33}^{i}={A}_{55}\frac{\partial {\phi }_{i}}{\partial x}-{k}_{x0}^{w}{\phi }_{i}$	${C}_{34}^{i}={A}_{55}{\phi }_{i}$
${C}_{44}^{i}={D}_{11}\frac{\partial {\phi }_{i}}{\partial x}-{k}_{x0}^{x}{\phi }_{i}$	${C}_{45}^{i}={D}_{12}\frac{\partial {\phi }_{i}}{\partial y}$
${C}_{54}^{i}={D}_{66}\frac{\partial {\phi }_{i}}{\partial y}$	${C}_{55}^{i}={D}_{66}\frac{\partial {\phi }_{i}}{\partial x}-{k}_{x0}^{y}{\phi }_{i}$

Appendix D:

${C}_{11}^{i}={A}_{66}\frac{\partial {\phi }_{i}}{\partial y}-{k}_{y0}^{u}{\phi }_{i}$	${C}_{12}^{i}={A}_{66}\frac{\partial {\phi }_{i}}{\partial x}$
${C}_{13}^{i}={C}_{23}^{i}={C}_{43}^{i}={C}_{53}^{i}={C}_{31}^{i}={C}_{32}^{i}={C}_{34}^{i}=0$
${C}_{14}^{i}={C}_{41}^{i}={B}_{66}\frac{\partial {\phi }_{i}}{\partial y}$	${C}_{15}^{i}={C}_{42}^{i}={B}_{66}\frac{\partial {\phi }_{i}}{\partial x}$
${C}_{21}^{i}={A}_{12}\frac{\partial {\phi }_{i}}{\partial x}$	${C}_{22}^{i}={A}_{11}\frac{\partial {\phi }_{i}}{\partial y}-{k}_{y0}^{v}{\phi }_{i}$
${C}_{24}^{i}={C}_{51}^{i}={B}_{12}\frac{\partial {\phi }_{i}}{\partial x}$	${C}_{25}^{i}={C}_{52}^{i}={B}_{11}\frac{\partial {\phi }_{i}}{\partial y}$
${C}_{33}^{i}={A}_{44}\frac{\partial {\phi }_{i}}{\partial y}-{k}_{y0}^{w}{\phi }_{i}$	${C}_{35}^{i}={A}_{44}{\phi }_{i}$
${C}_{44}^{i}={D}_{66}\frac{\partial {\phi }_{i}}{\partial y}-{k}_{y0}^{x}{\phi }_{i}$	${C}_{45}^{i}={D}_{66}\frac{\partial {\phi }_{i}}{\partial x}$
${C}_{54}^{i}={D}_{12}\frac{\partial {\phi }_{i}}{\partial x}$	${C}_{55}^{i}={D}_{11}\frac{\partial {\phi }_{i}}{\partial y}-{k}_{y0}^{y}{\phi }_{i}$