ABSTRACT
We have obtained the equation of motion for the transverse vibration of a plate in Chapter 3 as D ∇ 4 w + ρ h ∂ 2 w ∂ t 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_1.tif"/> where
ω is the transverse displacement of the plate
ρ is the density of the material of the plate
h is the thickness of the plate
D = E h 3 12(1 − ν 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/inequ4_105_1.tif"/> is the flexural rigidity
v is the Poisson’s ratio
E is the Young’s modulus of elasticity
∇4 is the biharmonic operator
∇ 4 w = ∂ 4 w ∂ x 4 + 2 ∂ 4 w ∂ x 2 x y 2 + ∂ 4 w ∂ y 4 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_2.tif"/> Also, ∇4 w = ∇2(∇2 w) where ∇2 is the Laplacian operator, and this has been defined in Chapter 3 in the cases of rectangular, polar, and elliptical coordinates. For free vibration with circular (natural) frequency ω, we can write the motion of the plate in polar coordinates as w ( r , θ , t ) = W ( r , θ ) e i ω t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_3.tif"/> and in Cartesian coordinates w ( x , y , t ) = W ( x , y ) e i ω t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_4.tif"/> In general, this will be written in the form w = W e i ω t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_5.tif"/> 106By substituting Equation 4.5 in Equation 4.1, we get ( ∇ 4 − β 4 ) W = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_6.tif"/> β 4 = ρ h ω 2 D https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_7.tif"/> Equation 4.6 is then written as ( ∇ 2 + β 2 ) ( ∇ 2 − β 2 ) W = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_8.tif"/> whose solution may be obtained in the form of two linear differential equations: ( ∇ 2 + β 2 ) W 1 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_9.tif"/> ( ∇ 2 − β 2 ) W 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_10.tif"/> One can write the solution of Equation 4.8 as the superposition of the solutions of Equations 4.9 and 4.10. Let W 1 and W 2 be the corresponding solutions. Then, one may have the solution W of the original differential Equation 4.8 as W = W 1 + W 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429137099/7d497a6e-cf88-45c7-ae36-b06855672413/content/equ4_11.tif"/>