The Effect of Rotatory Inertia and of Shear Deformation on the Frequency and Normal Mode Equations of Uniform Beams With Simple End Conditions

The natural frequencies for bending vibrations of Timoshenko cracked beams with simple boundary conditions have been obtained. The beam is modelled as two segments connected by two massless springs (one extensional and another one rotational). This model promotes discontinuities in both vertical displacement and rotation due to bending, which are proportional to shear force and bending moment transmitted by the cracked section, respectively. The differential equations for the free bending vibrations are established and then solved individually for each segment with the corresponding boundary conditions and the appropriated compatibility conditions at the cracked section. The problem is also solved by the perturbation method and both procedures are applied to the case of a simply supported cracked beam. The results show that the perturbation method provides simple expressions for the natural frequencies of cracked beams and it gives good results for shallow cracks. E mail address: ppfer@ing.uc3m.es (J. Ferna´ndez Sa´ez).

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“…[33]. Consider now that the beam has a crack of length a located at a distance L Ã ðe ¼ L Ã =LÞ from the left support.…”

Section: Problem Formulationmentioning

confidence: 99%

Natural frequencies for bending vibrations of Timoshenko cracked beams

Loya

Rubio

Fernández-Sáez

2006

Journal of Sound and Vibration

177

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“…The "Timoshenko beam parameters," r and s, are defined as the rotational inertia parameter and the shear deformation parameter, respectively, via [12,13] (2a, b)…”

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confidence: 99%

Timoshenko Beam Model for Lateral Vibration of Liquid-Phase Microcantilever-Based Sensors

Schultz

Heinrich

Josse

et al. 2013

MEMS and Nanotechnology, Volume 5

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Abstract:Dynamic-mode microcantilever-based devices are potentially well suited to biological and chemical sensing applications. However, when these applications involve liquid-phase detection, fluid-induced dissipative forces can significantly impair device performance. Recent experimental and analytical research has shown that higher in-fluid quality factors (Q) are achieved by exciting microcantilevers in the lateral flexural mode. However, experimental results show that, for microcantilevers having larger width-tolength ratios, the behaviors predicted by current analytical models differ from measurements. To more accurately model microcantilever resonant behavior in viscous fluids and to improve understanding of lateral-mode sensor performance, a new analytical model is developed, incorporating both viscous fluid effects and "Timoshenko beam" effects (shear deformation and rotatory inertia). Beam response is examined for two harmonic load types that simulate current actuation methods: tip force and support rotation. Results are expressed in terms of total beam displacement and beam displacement due solely to bending deformation, which correspond to current detection methods used with microcantilever-based devices (optical and piezoresistive detection, respectively). The influences of the shear, rotatory inertia, and fluid parameters, as well as the load/detection scheme, are investigated. Results indicate that load/detection type can impact the measured resonant characteristics and, thus, sensor performance, especially at larger values of fluid resistance.

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“…(6a,b) through those terms that involve  . When this parameter is set equal to zero, the equations reduce to the well-known in-vacuum results (e.g., [42,44]). …”

Section: B Derivation Of Equations Of Motionmentioning

confidence: 99%

Lateral-Mode Vibration of Microcantilever-Based Sensors in Viscous Fluids Using Timoshenko Beam Theory

Schultz

Heinrich

Josse

et al. 2015

J. Microelectromech. Syst.

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To more accurately model microcantilever resonant behavior in liquids and to improve lateral-mode sensor performance, a new model is developed to incorporate viscous fluid effects and "Timoshenko beam" effects (shear deformation, rotatory inertia). The model is motivated by studies showing that the most promising geometries for lateral-mode sensing are those for which Timoshenko effects are most pronounced.Analytical solutions for beam response due to harmonic tip force and electrothermal loadings are expressed in terms of total and bending displacements, which correspond to laser and piezoresistive readouts, respectively. The influence of shear deformation, rotatory inertia, fluid properties, and actuation/detection schemes on resonant frequencies (fres) and quality factors (Q) are examined, showing that Timoshenko beam effects may reduce fres and Q by up to 40% and 23%, respectively, but are negligible for width-tolength ratios of 1/10 and lower. Comparisons with measurements (in water) indicate that the model predicts the qualitative data trends but underestimates the softening that occurs in stiffer specimens, indicating that support deformation becomes a factor. For thinner specimens the model estimates Q quite well, but exceeds the observed values for thicker specimens, showing that the Stokes resistance model employed should be extended to include pressure effects for these geometries.

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The Effect of Rotatory Inertia and of Shear Deformation on the Frequency and Normal Mode Equations of Uniform Beams With Simple End Conditions

Abstract: New frequency and normal mode equations for flexural vibrations of six common types of simple, finite beams are presented. The derivation includes the effect of rotatory inertia and transverse-shear deformation. A specific example is given.

Cited by 396 publications

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Natural frequencies for bending vibrations of Timoshenko cracked beams

Natural frequencies for bending vibrations of Timoshenko cracked beams

Timoshenko Beam Model for Lateral Vibration of Liquid-Phase Microcantilever-Based Sensors

Lateral-Mode Vibration of Microcantilever-Based Sensors in Viscous Fluids Using Timoshenko Beam Theory

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