• Advances in nano research
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• v.8 no.1
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• pp.25-36
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• 2020

Rotating systems concern with torsional vibration, and it should be considered in vibration analysis. To do this, the time-dependent torsional vibrations in a single-walled carbon nanotube (SWCNT) under the linear and harmonic external torque, are investigated in this paper. Eringen's nonlocal elasticity theory is considered to demonstrate the nonlocality and constitutive relations. Hamilton's principle is established to derive the governing equation of motion and consequently related boundary conditions. An analytical method, called the Galerkin method, is utilized to discretize the driven differential equations. Linear and harmonic torsional loads, along with determined amplitude, are applied to the SWCNT as the external torques. SWCNT is considered under the clamped-clamped end supports. In free vibration, analysis of small scale effect reveals the capability of natural frequencies in different modes, and this results desirably are in coincidence with another study. The forced torsional vibration in the time domain, especially for carbon nanotubes, has not been done before in the previous works. The previous forced studies were devoted to the transverse vibrations. It should be emphasized that the dynamical analysis of torsion is novel, workable, and at the beginning of the path. The variations of nonlocal parameter, CNT's thickness, and the influence of excitation frequency on time-dependent angular displacement and nondimensional angular displacement are investigated in the context.

• Structural Engineering and Mechanics
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• v.89 no.2
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• pp.103-119
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• 2024

The present research investigates the thermodynamically bending behavior of FG sandwich plates, laying on the Winkler/Pasternak/Kerr foundation with various boundary conditions, subjected to harmonic thermal load varying through thickness. The supposed FG sandwich plate has three layers with a ceramic core. The constituents' volume fractions of the lower and upper faces vary gradually in the direction of the FG sandwich plate thickness. This variation is performed according to various models: a Power law, Trigonometric, Viola-Tornabene, and the Exponential model, while the core is constantly homogeneous. The displacement field considered in the current work contains integral terms and fewer unknowns than other theories in the literature. The corresponding equations of motion are derived based on Hamilton's principle. The impact of the distribution model, scheme, aspect ratio, side-to-thickness ratio, boundary conditions, and elastic foundations on thermodynamic bending are examined in this study. The deflections obtained for the sandwich plate without elastic foundations have the lowest values for all boundary conditions. In addition, the minimum deflection values are obtained for the exponential volume fraction law model. The sandwich plate's non-dimensional deflection increases as the aspect ratio increases for all distribution models.

• Steel and Composite Structures
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• v.42 no.5
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• pp.699-710
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• 2022

This work presents a comprehensive study on the surface energy effect on the axial frequency analyses of AFGM nanorods in cylindrical coordinates. The AFGM nanorods are considered to be thin, relatively thick, and thick. In thin nanorods, effects of the inertia of lateral motions and the shear stiffness are ignored; in relatively thick nanorods, only the first one is considered; and in thick nanorods, both of them are considered in the kinetic energy and the strain energy of the nanorod, respectively. The surface elasticity theory which includes three surface parameters called surface density, surface stress, and surface Lame constants, is implemented to consider the size effect. The power-law form is considered for variation of the material properties through the axial direction. Hamilton's principle is used to derive the governing equations and boundary conditions. Due to considering the surface stress, the governing equation and boundary condition become inhomogeneous. After homogenization of them using an appropriate change of variable, axial natural frequencies are calculated implementing harmonic differential quadrature (HDQ) method. Comprehensive results including effects of geometric parameters and various material properties are presented for a wide range of boundary condition types. It is believed that this study is a comprehensive one that can help posterities for design and manufacturing of nano-electro-mechanical systems.

• Structural Engineering and Mechanics
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• v.76 no.5
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• pp.619-629
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• 2020

In current study, surface/interface effects for pull-in voltage and viscous fluid velocity effects on dimensionless natural frequency (DNF) of fluid-conveying piezoelectric nanosensor (FCPENS) subjected to direct electrostatic voltage DC with nonlinear excitation, harmonic force and also viscoelastic foundation (visco-pasternak medium and structural damping) are investigated using Gurtin-Murdoch surface/interface (GMSIT) theory. For this analysis, Hamilton's principles, the assumed mode method combined with Lagrange-Euler's are used for the governing equations and boundary conditions. The effects of surface/interface parameters of FCPENS such as Lame's constants (λI,S, μI,S), residual stress (τ0I,S), piezoelectric constants (e31psk,e32psk) and mass density (ρI,S) are considered for analysis of dimensionless natural frequency respect to viscous fluid velocity u̅f and pull-in voltage V̅DC.

• Wind and Structures
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• v.25 no.2
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• pp.131-156
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• 2017

This paper deals with the dynamic stability of embedded functionally graded (FG)-carbon nanotubes (CNTs)-reinforced micro cylindrical shells. The structure is subjected to harmonic non-uniform temperature distribution and 2D magnetic field. The CNT reinforcement is either uniformly distributed or FG along the thickness direction where the effective properties of nano-composite structure are estimated through Mixture low. The viscoelastic properties of structure are captured based on the Kelvin-Voigt theory. The surrounding viscoelastic medium is considered nonhomogeneous with the spring, orthotropic shear and damper constants. The material properties of cylindrical shell and the viscoelastic medium constants are assumed temperature-dependent. The first order shear deformation theory (FSDT) or Mindlin theory in conjunction with Hamilton's principle is utilized for deriving the motion equations where the size effects are considered based on Eringen's nonlocal theory. Based on differential quadrature (DQ) and Bolotin methods, the dynamic instability region (DIR) of structure is obtained for different boundary conditions. The effects of different parameters such as volume percent and distribution type of CNTs, mode number, viscoelastic medium type, temperature, boundary conditions, magnetic field, nonlocal parameter and structural damping constant are shown on the DIR of system. Numerical results indicate that the FGX distribution of CNTs is better than other considered cases. In addition, considering structural damping of system reduces the resonance frequency.

• Honam Mathematical Journal
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• v.38 no.2
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• pp.385-391
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• 2016

We consider Jacobi filds as the first derivatives for ${\varepsilon}$, the energy of harmonic extensions, in a given manifold. In this paper we see that the Jacobi fild is bounded by the given boundary map. Here we give no restriction concerned with the curvature for the given manifold.

• Coupled systems mechanics
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• v.1 no.2
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• pp.155-163
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• 2012

This paper presents analytical approximate solutions for the initial post-buckling deformation of the pipes in oil and gas wells. The governing differential equation with sinusoidal nonlinearity can be reduced to form a third-order-polynomial nonlinear equation, by coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. The linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations, unlike the classical method of harmonic balance. We are hence able to establish analytical approximate solutions. The approximate formulae for load along axis, and periodic solution are established for derivative of the helix angle at the end of the pipe. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.

• Structural Engineering and Mechanics
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• v.88 no.5
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• pp.405-417
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• 2023

This paper reveals theoretical research to the nonlinear dynamic response and initial geometric imperfections sensitivity of the spinning graphene platelets reinforced metal foams (GPLRMF) cylindrical shell under different boundary conditions in thermal environment. For the theoretical research, with the framework of von-Karman geometric nonlinearity, the GPLRMF cylindrical shell model which involves Coriolis acceleration and centrifugal acceleration caused by spinning motion is assumed to undergo large deformations. The coupled governing equations of motion are deduced using Euler-Lagrange principle and then solved by a combination of Galerkin's technique and modified Lindstedt Poincare (MLP) model. Furthermore, the impacts of a set of parameters including spinning velocity, initial geometric imperfections, temperature variation, weight fraction of GPLs, GPLs distribution pattern, porosity distribution pattern, porosity coefficient and external excitation amplitude on the nonlinear harmonic resonances of the spinning GPLRMF cylindrical shells are presented.

• Steel and Composite Structures
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• v.34 no.4
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• pp.615-623
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• 2020

In this paper, free vibration analysis of a functionally graded cylindrical nanoshell resting on Pasternak foundation is presented based on the nonlocal elasticity theory. A two-dimensional formulation along the axial and radial directions is presented based on the first-order shear deformation shell theory. Hamilton's principle is employed for derivation of the governing equations of motion. The solution to formulated boundary value problem is obtained based on a harmonic solution and trigonometric functions for various boundary conditions. The numerical results show influence of significant parameters such as small scale parameter, stiffness of Pasternak foundation, mode number, various boundary conditions, and selected dimensionless geometric parameters on natural frequencies of nanoshell.

• Bulletin of the Society of Naval Architects of Korea
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• v.23 no.2
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• pp.1-13
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• 1986

The nonlinear hydrodynamic forces acting on a two-dimensional circular cylinder, oscillating with large amplitude in the free surface, are calculated by using the Semi-Lagrangian Time-Step-ping Method used by O.M. Faltinsen. In present calculation the position and the potential value of free surface are calculated using the exact kinematic and dynamic free surface boundary condition. At each time step an integral equation is solved to obtain the value of potential and normal velocity along the boundaries, consisting of both the body surface and the free surface. Some effort was devoted to the elimination of instability arising in the range of high frequency. Numerical simulations were performed up to the 3rd or 4th period which seems to be enough for the transient effect to die out. Each harmonic component and time-mean force are obtained by the Fourier transform of forces in time domain. The results are compared with others' experimental and theoretical results. Particularly, the calculation shows the tendency that the acceleration-phase 1st-harmonic component(added mass) increases as the motion amplitude increases and a reverse tendency in the velocity-phase 1st-harmonic component(damping coefficient). The Yamashita's experimental result also shows the same tendency. In general, the present result show relatively good agreement with the Yamashita's experimental result except for the time-mean force.