• Structural Engineering and Mechanics
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• v.48 no.4
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• pp.519-545
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• 2013

An outline of the Timoshenko beam theory is presented. Two differential equations of motion in terms of deflection and rotation are comprised into single equation with deflection and analytical solutions of natural vibrations for different boundary conditions are given. Double frequency phenomenon for simply supported beam is investigated. The Timoshenko beam theory is modified by decomposition of total deflection into pure bending deflection and shear deflection, and total rotation into bending rotation and axial shear angle. The governing equations are condensed into two independent equations of motion, one for flexural and another for axial shear vibrations. Flexural vibrations of a simply supported, clamped and free beam are analysed by both theories and the same natural frequencies are obtained. That fact is proved in an analytical way. Axial shear vibrations are analogous to stretching vibrations on an axial elastic support, resulting in an additional response spectrum, as a novelty. Relationship between parameters in beam response functions of all type of vibrations is analysed.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.11 no.4
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• pp.27-35
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• 1991

This paper presents a numerical analysis procedure and a characteristics for vibration and buckling of the cylinderical panels. The panels with simply-simply or simply-clamped edge supports are subjectes to circumferential compressive or flexural stresses. The differential equations governing vibration and buckling for these panels are derived by using the fundamental differential equation of the Love-Timoshenko and are solved numerically via the Galerkin method. The panel with simply-clamped edge supports is used a trigonometric function or a eigen function of a beam as a trial function and the effects of trial functions on numerical solutions are displayed. Numerical results are presented to demonstrate the effects of the flexural parameters in natural frequencies and coefficients of critical buckling and some typical mode shapes of vibration and buckling are also presented.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2002.11b
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• pp.529-534
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• 2002

The receptance method was applied for the analysis of a cylindrical shell with a curvaturated plate attached at the top of the shell. The boundary conditions of the shell considered here were clamped at the bottom and free at the top of the shell. Before the analysis of the shell/plate combined structure, the natural frequencies of the plate and the shell were calculated separately and then they were used in the calculation of the frequencies of the combined structure by the receptance method. The frequency equation of the combined structure was derived from the continuity condition at the junction of the shell and the plate. The frequencies for various curvature factors of the plate were presented and compared with those from ANSYS to show its validity of the present method.

• Smart Structures and Systems
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• v.3 no.2
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• pp.147-172
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• 2007

The present paper is concerned with the design of distributed sensors and actuators. Strain type sensors and actuators are considered with their intensity continuously distributed throughout a continuous structure. The sensors measure a weighted average of the strain tensor. As a starting point for their design we introduce the concept of collocated sensors and actuators as well as the so-called natural output. Then we utilize the principle of virtual work for an auxiliary quasi-static problem to assign a mechanical interpretation to the natural output of the sensors to be designed. Therefore, we take the virtual displacements in the principle of virtual work as that part of the displacement in the original problem, which characterizes the deviation from a desired one. We introduce different kinds of distributed sensors, each of them with a mechanical interpretation other than a weighted average of the strain tensor. Additionally, we assign a mechanical interpretation to the collocated actuators as well; for that purpose we use an extended body force analogy. The sensors and actuators are applied to solve the displacement tracking problem for continuous structures; i.e., the problem of enforcing a desired displacement field. We discuss feed forward and feed back control. In the case of feed back control we show that a PD controller can stabilize the continuous system. Finally, a numerical example is presented. A desired deflection of a clamped-clamped beam is tracked by means of feed forward control, feed back control and a combination of the two.

• Journal of Korean Society of Steel Construction
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• v.12 no.6
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• pp.737-748
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• 2000

This paper presents a numerical analysis procedure and a characteristics for dynamic of cylindrical panels. The panels with simply-simply or simply-clamped edge supports are subjectes to circumferential compressive or flexural stresses. The differential equations governing vibration and dynamic for these panels are derived by using the fundamental differential equation of the Love-Timoshenko and are solved numerically by the Galerkin method. The panel with simply-clamped edge supports is used a trigonometric function or an eigen function of a beam as a trial function and the effects of trial functions on numerical solutions are displayed. Numerical results are presented to demonstrate the effects of the flexural parameters in natural frequencies and coefficients of critical buckling, and some typical mode shapes of vibration and buckling are also presented.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.19 no.1 s.71
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• pp.85-92
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• 2006

This paper investigates critical loads of the tapered Beck's columns with clamped and spring supports, subjected to a subtangential follower force. The linearly tapered columns with the solid rectangular cross-section is adopted as the column taper. The differential equation governing free vibrations of such Beck's columns is derived using the Bemoulli-Euler beam theory. Both divergence and flutter critical loads are calculated from the load-frequency curves which are obtained by solving the differential equation. The critical loads are presented as functions of various non-dimensional system parameters: the taper type, the subtangential parameter and the spring stiffness.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.18 no.11
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• pp.1185-1191
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• 2008

This paper presents an approach to define an optimal piezoactuator length to actively control structural vibration. The optimal ratio of the piezoactuator length against the beam length when a pair of piezoceramic actuator and accelerometer is used to suppress unwanted vibration with direct velocity feedback(DVFB) control strategy is not clearly defined so far. It is well known that DVFB control can be very useful when a pair of sensor and actuator is collocated on structures with a high gain and excellent stability. It is considered that three different collocated Pairs of piezoelectric actuators (20, 50 and 100 mm long) and accelerometers installed on three identical clamped-clamped beams($30{\times}20{\times}1mm$). The response of each sensor-actuator pair requires strictly positive real(SPR) property to apply a high feedback gain. However the length of the piezoactuator affects the SPR property of the sensor-actuator response. Intensive simulation and experiment show the effect of the actuator length variation is strongly related with the frequency range of the SPR property. Thus an optimal length ratio was suggested to obtain relevant performance with a good stability under the DVFB strategy.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.18 no.1
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• pp.123-130
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• 2008

An analytical method for the free vibration of a flexible rectangular plate in contact with water is developed by the Rayleigh-Ritz method. The plate clamped along the edges is partially contacted with water at both sides. It is assumed that the contained water is incompressible and inviscid. The wet mode shape of the plate is assumed as a combination of the dry mode shapes of a clamped beam. The liquid motion is described by using the liquid displacement potential and determined by using the compatibility conditions along the liquid interface with the plate. Minimizing the Rayleigh quotient based on the energy conservation gives an eigenvalue problem. It is found that the theoretical results can predict excellently the fluid-coupled natural frequencies comparing with the finite element analysis result.

• Advances in nano research
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• v.5 no.2
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• pp.141-169
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• 2017

In this paper, a nanobeam connected to a rotating molecular hub is considered. The vibration behavior of rotating functionally graded nanobeam based on Eringen's nonlocal theory and Euler-Bernoulli beam model is investigated. Furthermore, axial preload and porosity effect is studied. It is supposed that the material attributes of the functionally graded porous nanobeam, varies continuously in the thickness direction according to the power law model considering the even distribution of porosities. Porosity at the nanoscopic length scale can affect on the rotating functionally graded nanobeams dynamics. The equations of motion and the associated boundary conditions are derived through the Hamilton's principle and generalized differential quadrature method (GDQM) is utilized to solve the equations. In this paper, the influences of some parameters such as functionally graded power (FG-index), porosity parameter, axial preload, nonlocal parameter and angular velocity on natural frequencies of rotating nanobeams with pure ceramic, pure metal and functionally graded materials are examined and some comparisons about the influence of various parameters on the natural frequencies corresponding to the simply-simply, simplyclamped, clamped-clamped boundary conditions are carried out.

• Structural Engineering and Mechanics
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• v.70 no.6
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• pp.737-750
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• 2019

This paper investigates the static and dynamic behaviors of imperfect single walled carbon nanotube (SWCNT) modeled as a beam structure by using energy-equivalent model (EEM), for the first time. Based on EEM Young's modulus and Poisson's ratio for zigzag (n, 0), and armchair (n, n) carbon nanotubes (CNTs) are presented as functions of orientation and force constants. Nonlinear Euler-Bernoulli assumptions are proposed considering mid-plane stretching to exhibit a large deformation and a small strain. To simulate the interaction of CNTs with the surrounding elastic medium, nonlinear elastic foundation with cubic nonlinearity and shearing layer are employed. The equation governed the motion of curved CNTs is a nonlinear integropartial-differential equation. It is derived in terms of only the lateral displacement. The nonlinear integro-differential equation that governs the buckling of CNT is numerically solved using the differential integral quadrature method (DIQM) and Newton's method. The linear vibration problem around the static configurations is discretized using DIQM and then is solved as a linear eigenvalue problem. Numerical results are depicted to illustrate the influence of chirality angle and imperfection amplitude on static response, buckling load and dynamic behaviors of armchair and zigzag CNTs. Both, clamped-clamped (C-C) and simply supported (SS-SS) boundary conditions are examined. This model is helpful especially in mechanical design of NEMS manufactured from CNTs.