Free vibration of microscale frameworks using modified couple stress and a combination of Rayleigh–Love and Timoshenko theories

Mustapha, K.B.

doi:10.1177/1077546319892470

Cited by 6 publications

(12 citation statements)

References 78 publications

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“…However, it should be recognised that for a Timoshenko-Ehrenfest micro beam in bending or flexural motion, there are three displacement components v, θ and v 0 that arise will lead to a sixth order system, whereas in classical Timoshenko-Ehrenfest beam theory, the displacement field consists of v and θ only (without the v 0 ) resulting in a fourth order system. This difference was observed by earlier investigators (Ma et al, 2008;Mustapha, 2020;Reddy, 2011), but apparently, these investigators did not pay much attention to the additional displacement parameter v 0 , particularly when obtaining the results. One of the purposes of this investigation is to consider the new displacement component v 0 both in the theory as well as in the results.…”

Section: Derivation Of the Governing Differential Equationsmentioning

confidence: 71%

“…The general procedure for dynamic stiffness theory development and its application can be found in the work of Banerjee (1997Banerjee ( , 2015, amongst others. In the current work, bending or flexural vibration is given precedence in the development of the dynamic stiffness theory because axial vibration is not affected by the small length parameter when the MCST is used (Ma et al, 2008;Mustapha, 2020;Reddy, 2011). The investigation is carried out in following steps.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

Banerjee

Papkov

et al. 2021

Journal of Vibration and Control

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Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

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Section: Derivation Of the Governing Differential Equationsmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

Banerjee

Papkov

et al. 2021

Journal of Vibration and Control

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“…The reduced displacement field of this takes the form 72 :

u_{ax} ((), x, y, z, t) = ({, \begin{matrix} u_{x} = & u \\ u_{y} = & - italicυy \frac{\partial u ((), x, t)}{\partial x} \\ u_{z} = & - italicυz \frac{\partial u ((), x, t)}{\partial x} \end{matrix})

which yields the following vector of strain and symmetric curvature tensors:

({}, \begin{matrix} ε_{xx} \\ ε_{yy} \\ ε_{zz} \\ γ_{xz} \\ γ_{xy} \end{matrix}) = ({}, \begin{matrix} \frac{\partial u}{\partial x} \\ - υ \frac{\partial u}{\partial x} \\ - υ \frac{\partial u}{\partial x} \\ - \frac{1}{2} italicyυ \frac{\partial^{2} u}{\partial x^{2}} \\ - \frac{1}{2} italiczυ \frac{\partial^{2} u}{\partial x^{2}} \end{matrix})

({}, \begin{matrix} χ_{xy} \\ χ_{zx} \end{matrix}) = ({}, \begin{matrix} \frac{1}{4} italiczυ \frac{\partial^{3} u}{\partial x^{3}} \\ - \frac{1}{4} italicyυ \frac{\partial^{3} u}{\partial x^{3}} \end{matrix})

where

υ

denotes the Poisson's ratio. Besides, Equation (17) indicates the presence of three normal strains and two shear strains 69 . With the Rayleigh–Love theory, the practice is to include the effects of the lateral motions in the kinetic energy to account for the lateral inertia effects, 72 …”

Section: Theoretical Developmentmentioning

confidence: 99%

“…In conjunction with the aforesaid modeling details, a two‐node finite element with three degrees of freedom (axial, transverse and rotational displacements) depicted in Figure 3C will be employed in the solution phase 69 . As will be noticed, a global reference frame

italicXY

is attached at the base of the finite element, and a local coordinate system

italicxy

is attached to the element with its

x

‐axis aligning with the principal axis of the structure.…”

Section: Theoretical Developmentmentioning

confidence: 99%

“…Besides, Equation ( 17) indicates the presence of three normal strains and two shear strains. 69 With the Rayleigh-Love theory, the practice is to include the effects of the lateral motions in the kinetic energy to account for the lateral inertia effects, 72 and consider only the effect of the normal strain along the principal axis (ε xx ) in the strain energy expression. Bearing this in mind, along with Equation ( 18), the axial strain and kinetic energy expressions are obtained as:…”

Section: Axial Motionmentioning

confidence: 99%

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Modeling and analysis of nature‐inspired branched micropillars for enhanced dynamic bio‐sensing

Mustapha

Hawwa

Abakr

2021

Numer Methods Biomed Eng

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Research evidence abounds on the effectiveness of micropillar‐based microelectromechanical systems for the detection of a wide variety of ultrasmall biological objects for clinical and non‐clinical applications. However, the standard micropillar‐based sensing platforms rely on a single‐column micropillar with a spot at the tip for binding of objects. Although this long‐standing form has shown immense potential, performance improvement is hindered by the fundamental limits enforced by physical laws. Moreover, the single‐column micropillar has a lower sensing area and is ill‐suited for a simultaneous differential sensing of chemical/biological objects of different mass. Here, we report a new set of nature‐inspired, branched micropillar‐based sensing resonators to address the highlighted issues. The characteristics of the newly proposed branched micropillars are comprehensively examined with three payloads (Bartonella Bacilliformis, Escherichia coli, and Micro magnetic beads). Anchored on the capability of continuum theoretical framework, the mathematical model of the micropillar is formulated through the synthesis of the modified couple stress, the Rayleigh‐Love, and the Timoshenko theories. The finite element method is employed to shed light on the variability of the structures' resonant response under performance reduction factors (payload's rotary inertia, damaged substrate, and density of a surrounding fluid). The results obtained indicate superior performance indicators for the triply‐branched micropillar: enhanced response sensitivity for multiple payloads and less susceptibility to deterioration in resonant frequencies due to fluid immersion.

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Free Vibration of Micro‐Beams and Frameworks Using the Dynamic Stiffness Method and Modified Couple Stress Theory

Banerjee¹

2021

Modern Trends in Structural and Solid Mechanics 2

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Free vibration of microscale frameworks using modified couple stress and a combination of Rayleigh–Love and Timoshenko theories

Cited by 6 publications

References 78 publications

Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

Modeling and analysis of nature‐inspired branched micropillars for enhanced dynamic bio‐sensing

Free Vibration of Micro‐Beams and Frameworks Using the Dynamic Stiffness Method and Modified Couple Stress Theory

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