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.
A new finite element model is developed and subsequently used for transverse vibrations of tapered Timoshenko beams with rectangular cross-section. The displacement functions of the finite element are derived from the coupled displacement field (the polynomial coefficients of transverse displacement and cross-sectional rotation are coupled through consideration of the differential equations of equilibrium) approach by considering the tapering functions of breadth and depth of the beam. This procedure reduces the number of nodal variables. The new model can also be used for uniform beams. The stiffness and mass matrices of the finite element model are expressed by using the energy equations. To confirm the accuracy, efficiency, and versatility of the new model, a semi-symbolic computer program in MATLAB® is developed. As illustrative examples, the bending natural frequencies of non-rotating/rotating uniform and tapered Timoshenko beams are obtained and compared with previously published results and the results obtained from the finite element models of solids created in ABAQUS. Excellent agreement is found between the results of new finite element model and the other results.
A new linearly pre-twisted Timoshenko beam finite element, which has two nodes and four-degrees-offreedom per node, is developed and subsequently used for coupled bending-bending vibration analysis of pre-twisted beams with uniform rectangular cross-section. First, displacement functions based on two coupled displacement fields (the polynomial coefficients are coupled through consideration of the differential equations of equilibrium) are derived for pre-twisted beams whose flexural displacements are coupled in two planes. This approach helps to reduce the number of nodal variables. Next, the stiffness and mass matrices of the finite element model are obtained by using the energy expressions. Finally, the natural frequencies of pre-twisted Timoshenko beams are obtained and compared with previously published theoretical and experimental results to confirm the accuracy and efficiency of the present model. Excellent agreement is found with the previous studies. Also, the new pre-twisted Timoshenko beam element has good convergence characteristics. r
Coupled bending-bending-torsion vibration of a pre-twisted beam with aerofoil cross-section by finite element method by Bulent Yardimoglu and Daniel J. Inman [Shock and Vibration 10(4) (2003), 223-230] p. 224: in Eq. (13), upper limit of fourth integration should be L instead of T. p. 225: the second term at the right-hand side in Eq. (18) should be r xθ z instead of r s θ z. p. 225: in caption of Fig. 2, change "Go-ordinate" to "Co-ordinate". p. 226: in Eq. (30), insert the missing term θ y between θ x and ψ x so that it reads:
Longitudinal natural vibration frequencies of rods (or bars) with variable cross-sections are obtained from the exact solutions of differential equation of motion based on transformation method. For the rods having cross-section variations as power of the sinusoidal functions ofax+b, the differential equation is reduced to associated Legendre equation by using the appropriate transformations. Frequency equations of rods with certain cross-section area variations are found from the general solution of this equation for different boundary conditions. The present solutions are benchmarked by the solutions available in the literature for the special case of present cross-sectional variations. Moreover, the effects of cross-sectional area variations of rods on natural characteristics are studied with numerical examples.
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