Natural vibration of circular and annular thin plates by Hamiltonian approach

Zhou, Zhenhuan; Wong, Kwok-Wo; Xü, Xinsheng; Leung, A.Y.T.

doi:10.1016/j.jsv.2010.09.015

Cited by 90 publications

(50 citation statements)

References 31 publications

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“…[1,6,9,10,16]. The numerical results for plates with sliding supports are shown in Table 5 with comparison to Refs.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Yalcin et al [9] have studied free vibration of circular plates by using differential transformation method (DTM). Zhou et al [10] applied the Hamiltonian approach to a solution of the free vibration problem of circular and annular thin plates. Kukla and Szewczyk [11] applied the Green's functions (influence function) and Bessel functions to the solution eigenvalue problem of annular thin plates with discrete elements such as oscillators.…”

Section: Introductionmentioning

confidence: 99%

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Green’s function approach to frequency analysis of thin circular plates

Żur¹

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract.The free vibration analysis of homogeneous and isotropic circular thin plates by using the Green's functions is considered. The formulae for construction of the influence function for all nodal diameters are presented in a closed form. The limited independent solutions of differential Euler equations were expanded in the Neumann power series using the method of successive approximation. This approach allows to obtain the analytical frequency equations as power series rapidly convergent to exact eigenvalues for different number of nodal diameters.

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“…[1,6,9,10,16]. The numerical results for plates with sliding supports are shown in Table 5 with comparison to Refs.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Green’s function approach to frequency analysis of thin circular plates

Żur¹

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…The numerical results for free annular plates with different boundary conditions on the inner edge are shown in Tab.1 in comparison to Vera et al (1998), Vogel and Skinner (1965). The dimensionless frequencies of annular plates with different boundary conditions on the outer edge and free inner edge are presented in Tab.2 in comparison to Zhou et al (2011). The numerical results for annular plates with the same boundary conditions on both edges are shown in Tab.3 in comparison with Zhou et al (2011).…”

Section: Resultsmentioning

confidence: 99%

“…The dimensionless frequencies of annular plates with different boundary conditions on the outer edge and free inner edge are presented in Tab.2 in comparison to Zhou et al (2011). The numerical results for annular plates with the same boundary conditions on both edges are shown in Tab.3 in comparison with Zhou et al (2011). …”

Section: Resultsmentioning

confidence: 99%

“…Taher et al (2006) studied free vibration of circular and annular plates with variable thickness and different combinations of boundary conditions. Zhou et al (2011) analyzed natural vibration of thin circular and annular plates by Hamiltonian approach. Wang (2014) studied vibration modes of concentrically supported free circular and annular plates.…”

Section: Introductionmentioning

confidence: 99%

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Green’s Function In Free Axisymmetric Vibration Analysis Of Annular Thin Plates With Different Boundary Conditions

Żur¹

2015

International Journal of Applied Mechanics and Engineering

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Free vibration analysis of homogeneous and isotropic annular thin plates by using Green's functions is considered. The formula of the influence function for uniform thin circular and annular plates is presented in closed-form. The limited independent solutions of differential Euler equation were expanded in the Neumann power series based on properties of integral equations. The analytical frequency equations as power series were obtained using the method of successive approximations. The natural axisymmetric frequencies for singularities when the core radius approaches zero are calculated. The results are compared with selected results presented in the literature.

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A Hamiltonian‐based analytical method for fracture analysis of linear thermo‐viscoelastic media

et al. 2017

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A Hamiltonian-based methodology is presented to study the fracture behaviors of the thermo-viscoelastic materials based on the Laplace transform. The governing equations and associated boundary conditions are rebuilt in a Hamiltonian form by using the symplectic mathematics in the frequency domain (s-domain). The fundamental unknown vector composed of both displacements and stresses variables is expanded in terms of the symplectic eigensolutions. The corresponding unknown coefficients of the symplectic series are determined from the outer boundary conditions. Thus, the main unknowns are obtained and transformed into the time domain (t-domain). The fracture parameters including stress intensity factors (SIFs) and J-integrals are derived simultaneously. Numerical examples as well as convergence studies are given and are found to be in good agreement with the ANSYS results. A parametric study of thermo-viscoelastic parameters is included also. Nomenclatureu i component of displacements σ ij , ε ij component of stresses and strains s ij , e ij component of deviatoric stresses and deviatoric strains α T coefficient of thermal expansion η coefficient of viscosity G shear modulus K bulk modulus υ Poisson's ratio (x 1 , x 2 ) Cartesian coordinates (r, θ ) Polar coordinates H Hamiltonian function q, p mutually dual vectors H Hamiltonian operator matrix μ eigenvalue of Hamiltonian matrix ψ (α) n , ψ (β) n eigensolution of α and β group K I , K II Mode I and II stress intensity factors J

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Natural vibration of circular and annular thin plates by Hamiltonian approach

Cited by 90 publications

References 31 publications

Green’s function approach to frequency analysis of thin circular plates

Green’s function approach to frequency analysis of thin circular plates

Green’s Function In Free Axisymmetric Vibration Analysis Of Annular Thin Plates With Different Boundary Conditions

A Hamiltonian‐based analytical method for fracture analysis of linear thermo‐viscoelastic media

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