Dynamical responses and stabilities of axially moving nanoscale beams with time-dependent velocity using a nonlocal stress gradient theory

2016 ◽  
Vol 23 (20) ◽  
pp. 3327-3344 ◽  
Author(s):  
Jinjian Liu ◽  
Cheng Li ◽  
Changjin Yang ◽  
Jiping Shen ◽  
Feng Xie
Keyword(s):  
Boundary Conditions ◽  
Vibration Frequency ◽  
Stress Gradient ◽  
Equation Of Motion ◽  
Time Dependent ◽  
Gradient Theory ◽  
Strengthening Effect ◽  
Vibration Frequencies ◽  
Nonlocal Stress ◽  
Axially Moving

A higher-order mechanical model of axially moving nanoscale beams with time-dependent velocity was developed in the framework of nonlocal stress gradient theory. Based on the correlation between effective and common nonlocal bending moments, a sixth-order partial differential equation of motion with respect to the transverse displacement was derived. Unlike some previous work which assumed the velocity of axially moving nanoscale beam to be a constant, a time-dependent axial velocity was considered for the nanoscale beams. The resonance vibration frequencies were obtained according to the governing equation of motion and corresponding boundary conditions. It was concluded a nonlocal nanoscale strengthening effect that the vibration frequencies of such axially moving nanostructure increase with stronger nonlocal effects, or a larger dimensionless nonlocal nanoscale parameter causes a higher vibration frequency. A jumping phenomenon in frequency field was observed, and the vibration frequency may decrease or increase with an increase in the axial average velocity. Critical speeds of the axially non-uniformly moving nanoscale beams were defined and determined, and the critical speed versus nonlocal nanoscale revealed step and strengthening effects. The theoretical results obtained were compared with some experimental data and good agreement was achieved. Subsequently, the steady-state and stability of such moving nanostructures including the principal parametric and combination resonances were analyzed using a multiple-scale method. Some beneficial analytical procedures and theoretical formulations at nanoscale were provided. Based on specific boundary conditions, the stability boundaries of the axially accelerating nanoscale beams were determined and the unstable regions were influenced by nonlocal nanoscale significantly.

Vibration of a Segmented Rod

2020 ◽  
pp. 2071011
Author(s):  
C. Y. Wang ◽  
H. Zhang ◽  
C. M. Wang
Keyword(s):  
Boundary Conditions ◽  
Equation Of Motion ◽  
Chain Model ◽  
Vibration Frequencies ◽  
Difference Model ◽  
Euler Beam ◽  
Constant Stiffness ◽  
Continuous Rod ◽  
Number Of Segments ◽  
Motion Boundary

This paper presents the governing equation of motion, boundary conditions and exact vibration frequencies of a segmented rod where the segments are connected by hinges with elastic rotational springs of constant stiffness. The mass of each segment is assumed to be evenly distributed along the length of the rod. Another discrete model called Hencky bar-chain model (short for HBM; which is equivalent to the finite difference model for discretizing continuous rod) assumes the rod mass to be lumped at the ends instead and a different set of boundary conditions are adopted clamped end. The vibration results of a clamped–clamped segment rod are compared with those of the HBM. It is shown that the HBM underestimates the vibration frequencies when compared to the segmented rod model for a finite number of segments while both models furnish vibration solutions that converge to the solutions of Euler beam for infinitely large number of segments.

Buckling and frequency analysis of the nonlocal strain–stress gradient shell reinforced with graphene nanoplatelets

2019 ◽  
Vol 25 (19-20) ◽  
pp. 2627-2640 ◽  
Author(s):  
Masoud Mohammadgholiha ◽  
Ali Shokrgozar ◽  
Mostafa Habibi ◽  
Hamed Safarpour
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Stress Gradient ◽  
Graphene Nanoplatelets ◽  
Strain Gradient Theory ◽  
Small Scale ◽  
Gradient Theory ◽  
Various Boundary Conditions ◽  
Pattern Length

In this study, buckling and vibrational characteristics of a nanoshell reinforced with graphene nanoplatelets under uniform axial load are investigated. The material properties of the piece-wise graphene-reinforced composites (GPLRCs) are assumed to be graded in the thickness direction of a nanoshell and are estimated using a nanomechanical model. The effects of the small scale are analyzed based on nonlocal stress–strain gradient theory (NSGT). The governing equations and boundary conditions (BCs) are developed using Hamilton’s principle and are solved with assistance of the generalized differential quadrature method. The novelty of the current study is the consideration of GPLRC and size effect as well as satisfying various boundary conditions implemented on the proposed model using NSGT. The results show that, nonlocal parameter, graphene platelet (GPL) distribution pattern, length scale parameter, number of layers, and GPL weight function have significant influence on the buckling and natural frequency of the GPLRC nanoshell. Another significant result is that nonlocal parameter does not have any effect on the buckling load for each BC. The results of the current study are useful for design of the nanoactuators and nanosensors.

scholarly journals Vibration of Roller Chain Drives at Low, Medium and High Operating Speeds

1993 ◽  
Author(s):  
Woosuk Choi ◽  
Glen E. Johnson
Keyword(s):  
Boundary Conditions ◽  
Transverse Vibration ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Partial Differential ◽  
Periodic Load ◽  
Roller Chain ◽  
Axially Moving ◽  
The Impact ◽  
Chain Drives

Abstract A model based on axially moving material is developed to study transverse vibration in roller chain drives. A unique feature of the work presented in this study is that impact, polygonal action and external periodic load have been included through chain tension and boundary conditions and periodic length change is also considered. The impact between the engaging roller and sprocket surface is modeled as a single impact between two elastic bodies and the modeling of the polygonal action is based on a four bar mechanism (rigid four bar at low speeds, elastic four bar at moderate and high speeds). At low and medium operating speeds, the system equation of motion for the chain span is expressed as a mixed type partial differential equation with time-dependent coefficients and time-dependent boundary conditions. At high operating speeds, the system equations of motion are two partial differential equations for transverse and longitudinal vibrations respectively and they are nonlinearly coupled The effects on transverse vibration of center distance, the moment of inertia of the driven sprocket system, static tension, and external periodic load are presented and discussed. Solutions are obtained by a finite difference method and Galerkin’s method.

scholarly journals Hygro-Thermal Vibration of Porous FG Nano-Beams Based on Local/Nonlocal Stress Gradient Theory of Elasticity

Nanomaterials ◽  
2021 ◽  
Vol 11 (4) ◽  
pp. 910
Author(s):  
Rosa Penna ◽  
Luciano Feo ◽  
Giuseppe Lovisi ◽  
Francesco Fabbrocino
Keyword(s):  
Volume Fraction ◽  
Nonlocal Parameter ◽  
Stress Gradient ◽  
Structural Response ◽  
Theory Of Elasticity ◽  
Functionally Graded ◽  
Gradient Theory ◽  
Thermal Environments ◽  
Structural Problems ◽  
Nonlocal Stress

In this manuscript the dynamic response of porous functionally-graded (FG) Bernoulli–Euler nano-beams subjected to hygro-thermal environments is investigated by the local/nonlocal stress gradient theory of elasticity. In particular, the influence of several parameters on both the thermo-elastic material properties and the structural response of the FG nano-beams, such as material gradient index, porosity volume fraction, nonlocal parameter, gradient length parameter, mixture parameter is examined. It is shown how the proposed approach is able to capture the dynamic behavior of porous functionally graded Bernoulli–Euler nano-beams under hygro-thermal loads and leads to well-posed structural problems of nano-mechanics.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

scholarly journals Geometries with mismatched branes

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Andreas Karch ◽  
Lisa Randall
Keyword(s):  
Boundary Conditions ◽  
Effective Action ◽  
Cosmological Models ◽  
Time Dependent ◽  
De Sitter ◽  
Stabilization Mechanism ◽  
New Class ◽  
Vacuum Solutions ◽  
Single Coordinate

Abstract We study Randall-Sundrum two brane setups with mismatched brane tensions. For the vacuum solutions, boundary conditions demand that the induced metric on each of the branes is either de Sitter, Anti-de Sitter, or Minkowski. For incompatible boundary conditions, the bulk metric is necessarily time-dependent. This introduces a new class of time-dependent solutions with the potential to address cosmological issues and provide alternatives to conventional inflationary (or contracting) scenarios. We take a first step in this paper toward such solutions. One important finding is that the resulting solutions can be very succinctly described in terms of an effective action involving only the induced metric on either one of the branes and the radion field. But the full geometry cannot necessarily be simply described with a single coordinate patch. We concentrate here on the time- dependent solutions but argue that supplemented with a brane stabilization mechanism one can potentially construct interesting cosmological models this way. This is true both with and without a brane stabilization mechanism.

Rayleigh-Ritz Analysis of Vibrating Plates Based on a Class of Eigenfunctions

2009 ◽  
Author(s):  
Giuseppe Catania ◽  
Silvio Sorrentino
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Boundary Conditions ◽  
Linear Combination ◽  
Equation Of Motion ◽  
Extensive Literature ◽  
Shape Functions ◽  
Element Analysis ◽  
Vibrating Plates ◽  
General Basis

In the Rayleigh-Ritz condensation method the solution of the equation of motion is approximated by a linear combination of shape-functions selected among appropriate sets. Extensive literature dealing with the choice of appropriate basis of shape functions exists, the selection depending on the particular boundary conditions of the structure considered. This paper is aimed at investigating the possibility of adopting a set of eigenfunctions evaluated from a simple stucture as a general basis for the analysis of arbitrary-shaped plates. The results are compared to those available in the literature and using standard finite element analysis.

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