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Influence of in-plane loading on the vibrations of the fully symmetric mechanical systems via dynamic simulation and generalized differential quadrature framework

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Abstract

Honeycomb structures are one type of structure that has the geometry of a honeycomb to allow the minimization of the amount of used material to reach minimal material cost and minimal weight. In this regard, this article deals with the frequency analysis of imperfect honeycomb core sandwich disk with multi-scale hybrid nanocomposite (MHC) face sheets. The honeycomb core is made of aluminum due to its low weight and high stiffness. The rule of the mixture and modified Halpin–Tsai model are engaged to provide the effective material constant of the composite layers. By employing Hamilton’s principle, the governing equations of the structure are derived and solved with the aid of the generalized differential quadrature method. Afterward, a parametric study is carried out to investigate the effects of the thickness to length ratio of the honeycomb core, honeycomb core thickness to inner radius ratio, value fraction of carbon fibers, radius ration of the disk, the angle of honeycomb network, the weight fraction of CNTs, and tensile and compressive in-plane force on the frequency of the sandwich disk with honeycomb core and MHC. The results show that the critical fiber angle is \(\theta_{{\text{f}}} /\pi =\) 0.5 for C–C and C–S boundary conditions. Another consequence is that when the structure is fixed with S–S boundary conditions, for \(p =\) 500 and \(p = 1000\), as well as the critical dimensionless angle for fibers is 0.5, there are two more range for critical fiber angle in which they are \(0.275 \le \theta_{{\text{f}}} /\pi \le 0.375\) and \(0.23 \le \theta_{{\text{f}}} /\pi \le 0.39\), respectively. Additionally, the range of the critical dimensionless angle for fibers increases by increasing the applied load. Some new results related to dynamic behavior of an MHC are also presented, which can serve as benchmark solutions for future investigations.

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Abbreviations

h, R i, and R o :

Thickness, the inner and outer radius of the disk, respectively

CNTs:

Carbon-nanotubes

F and NCM:

Indicates fiber and nanocomposite matrix, respectively

\(\rho ,\,E,\,\nu \, {\rm {and}} \,G\) :

Illustrates the density, Young’s module, Poisson’s ratio, and shear parameter, respectively

V NCM, V F :

Volume fractions of nanocomposite matrix and fiber, respectively

l CNT, t CNT, d CNT, E CNT and V CNT :

Indicates the length, thickness, diameter, Young’s module, and volume fraction of carbon nanotubes, respectively

\({V}_{\mathrm{CNT}}^{*}\), W CNT :

Effective volume fraction and weight fraction of the CNTs, respectively

Nt, V CNT :

Layer number and volume fraction of CNTs

\(E_{1}^{*}\) and \(E_{2}^{*}\) :

Represents Young modulus in R and \(\theta\) directions, respectively

\(\nu_{12}^{*}\) and \(\nu_{21}^{*}\) :

Represents Poisson’s ratio in R and \(\theta\) directions, respectively

\(G_{12}^{*}\) :

Represent in-plane shear modulus

\(E_{S}\) and \(\rho_{S}\) :

Represent Young modulus and mass density of base material which is aluminum for honeycomb core, respectively

t m, h H, l m, and θ h :

Indicates the cell wall thickness, the sides of the hexagonal cell, and angle of honeycomb core, respectively

U, V, W :

Displacement fields of a disk

u, v, w, u 1, and v 1 :

Indicates the displacements of the mid-surface of the disk

\(\varepsilon_{RR}\) and \(\varepsilon_{\theta \theta }\) :

Indicates the corresponding normal strains in \(R\) and θ directions, respectively

\(\gamma_{RZ} ,\,\gamma_{R\theta } \, {\rm {and}} \,\gamma_{\theta Z}\) :

Represents the shear strain in the R–Z, R–\(\theta\) and \(\theta\)–Z plane

U *, T *, and W * :

Represents corresponding strain energy of the system, kinetic energy of the system, and the work is done, respectively

\({Q}_{ij}\) and \({\stackrel{-}{Q}}_{ij}\) :

Stiffness elements and stiffness elements that relate to orientation angle and the orientation angle, respectively

θ :

Represents the lamination angle with respect to the disk R axis

N r and N θ :

Represent, the number of grid points along the radial and circumferential directions, respectively

d, b, and \(\delta\) :

Indicates d as a subscript stand for the domain grid-points, b as subscript stands for boundary grid-points and the displacement vector, respectively

F ij and K ij :

Components of force and stiffness matrices, respectively

F ij * and K ij * :

Components of force and stiffness matrices in the GDQ method, respectively

ω n and \({\stackrel{-}{\omega }}_{\mathrm{n}}\) :

Represent dimensional and non-dimensional of natural frequency, respectively

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Funding

National Natural Science Foundation of China (51675148). The Outstanding Young Teachers Fund of Hangzhou Dianzi University (GK160203201002/003). National Natural Science Foundation of China (51805475).

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Authors and Affiliations

  1. School of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou, 310018, China

    M. S. H. Al-Furjan & Jing Ni

  2. School of Materials Science and Engineering, State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou, 310027, China

    M. S. H. Al-Furjan

  3. Faculty of Aerospace Engineering, K.N. Toosi University of Technology, Tehran, Iran

    Mahmoud Fereidouni

  4. Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam

    Mostafa Habibi

  5. Faculty of Electrical–Electronic Engineering, Duy Tan University, Da Nang, 550000, Vietnam

    Mostafa Habibi

  6. College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Raneen Abd Ali

  7. Department of Mechanical Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran

    Mehran Safarpour

Authors
  1. M. S. H. Al-Furjan
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  2. Mahmoud Fereidouni
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  3. Mostafa Habibi
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  4. Raneen Abd Ali
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  5. Jing Ni
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  6. Mehran Safarpour
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Correspondence to Mostafa Habibi.

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Al-Furjan, M.S.H., Fereidouni, M., Habibi, M. et al. Influence of in-plane loading on the vibrations of the fully symmetric mechanical systems via dynamic simulation and generalized differential quadrature framework. Engineering with Computers 38 (Suppl 5), 3675–3697 (2022). https://doi.org/10.1007/s00366-020-01177-7

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