A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams

Adair, Desmond; Jaeger, Martin

doi:10.1177/1077546317714183

Cited by 21 publications

(14 citation statements)

References 37 publications

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“…6. It is exhibited from the results that the differences are no slight for K w0 = 50 EI L 2 and K w0 = 1000 EI L 2 .…”

Section: Mode Shapes Of the Cantilever Non-uniform Ebb On Winkler-typmentioning

confidence: 79%

“…The effect of the spring supports and the non-uniformity in the cross section on the natural frequencies are assessed in this paper. Also, the power series method is applied to determine the dynamic response of the rotating non-uniform cantilever EBBs by Yang et al [51] and Adair and Jaeger [2]. Unfortunately, it is not always possible to obtain the analytical solution for the free vibration of the non-uniform EBB with arbitrary varying cross section.…”

Section: Introductionmentioning

confidence: 99%

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Free vibrations of non-uniform beams on a non-uniform Winkler foundation using the Laguerre collocation method

Ghannadiasl

Zamiri

Borhanifar

2020

J Braz. Soc. Mech. Sci. Eng.

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Natural frequencies and free vibration are important characteristics of beams with non-uniform cross section. Hence, the solution for free vibrations of non-uniform beams is presented using a Laguerre collocation method. The elastically restrained beam model is based on the Euler-Bernoulli theory. Also, the non-uniform beam is rested on a non-uniform foundation (Winkler type). The Laguerre collocation method is introduced for solving the differential equation. This approach reduces the governing differential equation to a system of algebraic equations, and finally, the problem is greatly simplified. Properties of Laguerre polynomials and the operational matrix of derivation are first presented. Eventually, the proposed method is applied for solving the governing differential equation subject to initial conditions, and the results are compared with other results from the literature.

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“…6. It is exhibited from the results that the differences are no slight for K w0 = 50 EI L 2 and K w0 = 1000 EI L 2 .…”

Section: Mode Shapes Of the Cantilever Non-uniform Ebb On Winkler-typmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Free vibrations of non-uniform beams on a non-uniform Winkler foundation using the Laguerre collocation method

Ghannadiasl

Zamiri

Borhanifar

2020

J Braz. Soc. Mech. Sci. Eng.

View full text Add to dashboard Cite

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“…The aforementioned PVDF modal sensor study mainly focused on the static structural system, whereas a lot of structures in various engineering occasions, such as gas turbine blade, helicopter blade, and artificial satellite boom, are usually modeled as the rotating flexible beams, for which the external unbalanced force causing its resultant vibration cannot be absolutely eliminated. Many studies (Adair and Jaeger, 2018; Banerjee, 2000; Naguleswaran, 1994; Wang and Wereley, 2004) indicate that the structural dynamic characteristics will become more complicated because of the presence of centrifugal stiffening effect, which undoubtedly puts forward new challenges to the shaped modal sensor design.…”

Section: Introductionmentioning

confidence: 99%

Design of distributed piezoelectric modal sensor for a rotating beam with elastic boundary restraints

Chen

2020

Journal of Vibration and Control

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As an advanced sensing technique, modal sensors have been attracting a lot of research interest in modal filtering and active control fields. Most of the existing investigations are mainly focused on the static structure. In contrast, there is little effort made for its rotating counterpart, which is frequently encountered in various power machineries. Motivated by such limitation, a unified framework for the distributed piezoelectric modal sensor design of rotating beams with elastic boundary restraints is proposed using polyvinylidene fluoride piezoelectric integral equation and the second-order structural modal functions. A boundary smoothed Fourier series is used to obtain the modal information of rotating beams by solving the differential governing equation and elastic boundary conditions, simultaneously. Modal sensor shape of rotating beams can be determined for any boundary condition by simply setting the elastic restraining coefficients accordingly, instead of reformulating the equation or rewriting the codes like other approach usually does. Numerical examples are presented to demonstrate the correctness and effectiveness of the proposed framework. Modal sensitivity coefficient and charge output frequency response under external excitation are calculated to demonstrate the performance of the designed piezoelectric modal sensors. Influence of rotation speed and boundary restraining stiffness on the modal sensing accuracy of the shaped polyvinylidene fluoride sensor is analyzed and addressed. To our best knowledge, this work represents the first time that an analytical solution for the distributed piezoelectric modal sensor design of a rotating beam with general boundary conditions is derived, which can shed some new lights on further design and implementation of polyvinylidene fluoride modal sensing technique for rotating structures.

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“…Structure elements with a non-uniform cross section along the length of the beam elements have been used extensively to determine natural frequencies and mode shapes in engineering equipment such as turbine blades, helicopter rotor blades, and propellers. The dynamic behaviors of these structures are influenced by the cross-sectional shapes varying along the length of the elements, and many researchers have studied suitable shape functions for practical applications for a variety of shapes for non-uniform beam elements [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Free vibration analysis of tapered Rayleigh beams using the transfer matrix method

Lee

2020

J Braz. Soc. Mech. Sci. Eng.

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In this study, an exact transfer matrix method is developed to analyze the effect of rotatory inertia and tapering on the free vibration characteristics of tapered Rayleigh beams. The beam model developed addresses practical cross sections, including single-tapered and double-tapered beams. The governing differential equation is solved using the Frobenius method for power series solutions, and then, the transfer matrix is formulated to transform the state vectors defined from the displacements, bending moment, and shear force at the start and end points. To investigate the effect of rotatory inertia and taper ratio on the natural frequencies and mode shapes of beam structures, the length-to-height ratios of tapered beams with four classical boundary conditions and various taper ratios are examined. The accuracy of the developed method is demonstrated using the comparative results predicted in previous studies. Through a parametric study, the numerical results that are useful in practical applications are offered, and relevant conclusions not discussed in previous studies are proposed. Furthermore, the results discussed can serve as a benchmark solution.

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A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams

Cited by 21 publications

References 37 publications

Free vibrations of non-uniform beams on a non-uniform Winkler foundation using the Laguerre collocation method

Free vibrations of non-uniform beams on a non-uniform Winkler foundation using the Laguerre collocation method

Design of distributed piezoelectric modal sensor for a rotating beam with elastic boundary restraints

Free vibration analysis of tapered Rayleigh beams using the transfer matrix method

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