Exact Solution of Out-of-Plane Problems of an Arch with Varying Curvature and Cross Section

Tüfekçi, Ekrem; Doğruer, O. Yaşar

doi:10.1061/(asce)0733-9399(2006)132:6(600)

Cited by 19 publications

(13 citation statements)

References 18 publications

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“…Then, Huang et al [9] developed a dynamic stiffness matrix by using the Laplace transform technique for both the free vibration and forced vibration of non-uniform parabolic curved beams with various ratios of rise to span. Tufekci and Dogruer [10] obtained the exact solution of the differential equations for the static behavior of an arch with varying curvature and cross section including the shear deformation effect by using the initial value method.…”

Section: Introductionmentioning

confidence: 99%

Coupled out of plane vibrations of spiral beams for micro-scale applications

Karami

Yardimoglu

Inman

2010

Journal of Sound and Vibration

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a b s t r a c tAn analytical method is proposed to calculate the natural frequencies and the corresponding mode shape functions of an Archimedean spiral beam. The deflection of the beam is due to both bending and torsion, which makes the problem coupled in nature. The governing partial differential equations and the boundary conditions are derived using Hamilton's principle. Two factors make the vibrations of spirals different from oscillations of constant radius arcs. The first is the presence of terms with derivatives of the radius in the governing equations of spirals and the second is the fact that variations of radius of the beam causes the coefficients of the differential equations to be variable. It is demonstrated, using perturbation techniques that the derivative of the radius terms have negligible effect on structure's dynamics. The spiral is then approximated with many merging constant-radius curved sections joined together to approximate the slow change of radius along the spiral. The equations of motion are formulated in non-dimensional form and the effect of all the key parameters on natural frequencies is presented. Non-dimensional curves are used to summarize the results for clarity. We also solve the governing equations using Rayleigh's approximate method. The fundamental frequency results of the exact and Rayleigh's method are in close agreement. This to some extent verifies the exact solutions. The results show that the vibration of spirals is mostly torsional which complicates using the spiral beam as a host for a sensor or energy harvesting device.& 2010 Elsevier Ltd. All rights reserved. IntroductionWe are motivated to look at the vibrations of spiral shaped structures as a prelude to sensing and energy harvesting using the piezoelectric effect in micro-electro-mechanical systems (MEMS) devices. Vibrational energy harvesters convert vibrations available in the environment to electrical energy. The energy generated can be used to power sensor nodes [1]. These energy self-sufficient sensors can be placed in remote places to gather data and wirelessly transmit information. A key challenge in designing MEMS energy harvesters is to make them resonate with low frequency ambient vibrations. Spiral geometry has been suggested as a possible option for compact low frequency substrate in energy harvesting devices [2]. However the vibrational analysis of curved beam with varying radius (spirals) is missing in the literature [3]. Hu et al.[3] approximated the spiral by eccentric constant-radius arcs but only derived the governing equations and did not solve them for vibration characteristics. This paper attempts to solve the free vibrations of spiral beams and pave the way to the modeling of spiral MEMS harvesting devices.

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Section: Introductionmentioning

confidence: 99%

Coupled out of plane vibrations of spiral beams for micro-scale applications

Karami

Yardimoglu

Inman

2010

Journal of Sound and Vibration

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“…Then, Huang et al [8] developed a dynamic stiffness matrix by using the Laplace transform technique for both the free vibration and forced vibration of non-uniform parabolic curved beams with various ratios of rise to span. Tufekci and Dogruer [9] obtained the exact solution of the differential equations for the static behavior of an arch with varying curvature and cross section including the shear deformation effect by using the initial value method.…”

Section: Introductionmentioning

confidence: 99%

Coupled Out of Plane Vibrations of Spiral Beams

Karami

Yardimoglu

Inman

2009

50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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“…Studies for the out-of-plane buckling of non-circular arches have been carried out by Ojalvo et al (1969), Tokarz and Sandhu (1972), Tufekci and Dogruer (2006), and Wen and Lange (1981). Ojalvo et al (1969) derived a pair of equations which governs the out-of-plane buckling of planar curved members and buckling loads for ring segments that are subjected to pull and thrust loads.…”

Section: Introductionmentioning

confidence: 99%

“…Ojalvo et al (1969) derived a pair of equations which governs the out-of-plane buckling of planar curved members and buckling loads for ring segments that are subjected to pull and thrust loads. Tokarz and Sandhu (1972) obtained buckling loads for parabolic arches that are subjected to vertically distributed loads, while Tufekci and Dogruer (2006) proposed exact solutions of the equations for the out-of-plane deformation of arches with arbitrary axes and cross sections. Wen and Lange (1981) presented a finite element formulation for the buckling loads of curved beams with an arbitrary shape and constant cross section.…”

Section: Introductionmentioning

confidence: 99%

Out-of-plane buckling of arches with varying curvature

Moon¹,

Yoon

Lee

et al. 2009

KSCE Journal of Civil Engineering

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A number of theoretical solutions have been reported for the elastic out-of-plane buckling of arches with open thin-walled sections. However, most of theses studies are limited to arches with constant curvatures such as circular arches. This paper investigates the out-of-plane buckling of arches with varying curvature. Buckling equations for arches with varying curvature are derived and verified by comparing with those proposed earlier by different researchers. For this, the out-of-plane buckling of circular arches under pure axial compression and pure bending is studied. For an application of the derived buckling equations to arches with varying curvature, buckling loads of parabolic arches are determined. The solutions for parabolic arches are successfully verified by comparing with results of previous researcher and finite element analyses. Finally, comparative studies of circular and parabolic arches are conducted. The results show that the difference in buckling loads of circular and parabolic arches increases as the rise to span ratio increases.

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Exact Solution of Out-of-Plane Problems of an Arch with Varying Curvature and Cross Section

Cited by 19 publications

References 18 publications

Coupled out of plane vibrations of spiral beams for micro-scale applications

Coupled out of plane vibrations of spiral beams for micro-scale applications

Coupled Out of Plane Vibrations of Spiral Beams

Out-of-plane buckling of arches with varying curvature

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