Showing papers on "Rotary inertia published in 2009"

Timoshenko beam model for buckling and vibration of nanowires with surface effects

Abstract: In this paper, surface effects on the axial buckling and the transverse vibration of nanowires are examined by using the refined Timoshenko beam theory. The critical compression force of axial buckling and the natural frequency of nanowires are obtained analytically, in which the impacts of surface elasticity, residual surface stress, transverse shear deformation and rotary inertia have been included. The buckling and vibration behaviour of a nanowire is demonstrated to be size dependent, especially when its cross-sectional dimension reduces to nanometres. The surface effects with positive elastic constants tend to increase the critical compression force and the natural frequency, especially for slender nanowires, while the shear deformation lowers these values for stubby nanowires. This study may be helpful to accurately measure the mechanical properties of nanowires and to design nanowire-based devices and systems.

Vibrational modes of Timoshenko beams at small scales

Abstract: This letter presents a theoretical treatment of Timoshenko [S. Timoshenko, Philos. Mag. 41, 744 (1921)] beams, in which the influences of shear deformation, rotary inertia, and scale coefficient are taken into account. Based on the nonlocal elasticity theory, coupled equations for transverse deflection and rotation of cross section are derived. Free vibration of several typical beams is analyzed. Explicit expressions for modal shapes of vibration are presented. Natural frequencies are evaluated for free vibration of simply supported beams, clamped beams, cantilever beams, and clamped-hinged beams. The effects of the nonlocal parameter on natural frequencies and modal shapes are discussed in detail.

Free vibration of microscaled Timoshenko beams

Abstract: In this paper, a comprehensive model is presented to investigate the influence of surface elasticity and residual surface tension on the natural frequency of flexural vibrations of microbeams in the presence of rotary inertia and shear deformation effects. An explicit solution is derived for the natural oscillations of microscaled Timoshenko beams considering surface effects. The analytical results are illustrated with numerical examples in which two types of microbeams are configured based on Euler–Bernoulli and Timoshenko beam theory considering surface elasticity and residual surface tension. The natural frequencies of vibration are calculated for selected beam length on the order of nanometer to microns and the results are compared with those corresponding to the classical beam models, emphasizing the differences occurring when the surface effects are significant. It is found that the nondimensional natural frequency of the vibration of micro and nanoscaled beams is size dependent and for limiting case in which the beam length increases, the results tends to the results obtained by classical beam models. This study might be helpful for the design of high-precision measurement devices such as chemical and biological sensors.

A closed-form solution for critical buckling temperature of a single-walled carbon nanotube

Abstract: In this paper, Timoshenko beam model, including the effects of transverse shear deformation and rotary inertia, is employed to study the critical buckling temperature of armchair and zigzag single-walled carbon nanotubes (SWCNTs) subjected to a uniform temperature rise. A closed-form solution for the determination of critical buckling temperature is derived. The solution can be further reduced to obtain the results of the Euler beam model. The results show that the Euler beam model overpredicts the critical buckling temperature of SWCNTs, especially at relatively small length-to-diameter ratios and higher-order modes. In addition, the critical buckling temperature of armchair and zigzag SWCNTs for the first ten modes with different length-to-diameter ratios is compared.

An Equation Both More Consistent and Simpler Than the Bresse-Timoshenko Equation

Abstract: A simple equation is discussed which takes into account both shear deformation and rotary inertia in vibrating beams. This equation is both more consistent and simpler than the widely used one of Bresse-Timoshenko.

Optimized Measurements of Unmanned-Air-Vehicle Mass Moment of Inertia with a Bifilar Pendulum

Abstract: A bifilar (two-wire) pendulum is a torsional pendulum consisting of a test object suspended by two thin parallel wires. The pendulum oscillates about the vertical axis. The restoring torque of the bifilar pendulum is provided by the gravitational force as rotations from the rest state cause the test object to raise slightly. The mass moment of inertia is computed using dynamic modeling, measurements of the oscillation period, and the physical dimensions of the bifilar pendulum such as the length and separation displacement of the pendulum wires. A simulation technique is described that improves estimates of the mass moment of inertia by considering the nonlinear effects of damping and large angular displacements. An analysis of the error variance of mass moment of inertia measurements is also described. The resulting expression for the error variance is used to optimize the physical parameters of the bifilar pendulum to obtain the moment of inertia measurement with the minimum error variance. Monte Carlo simulations were used to validate the parameter optimization technique. Experimental results are presented for a uniform-density test object for which the moment of inertia is straightforward to compute from geometric considerations. Results are also presented for a small unmanned air vehicle, which was the intended application for this moment of inertia measurement technique.

Free vibration analysis of third-order shear deformable composite beams using dynamic stiffness method

Abstract: The dynamic stiffness method is introduced to investigate the free vibration of laminated composite beams based on a third-order shear deformation theory which accounts for parabolic distribution of the transverse shear strain through the thickness of the beam. The exact dynamic stiffness matrix is found directly from the analytical solutions of the basic governing differential equations of motion. The Poisson effect, shear deformation, rotary inertia, in-plane deformation are considered in the analysis. Application of the derived dynamic stiffness matrix to several particular laminated beams is discussed. The influences of Poisson effect, material anisotropy, slenderness and end condition on the natural frequencies of the beams are investigated. The numerical results are compared with the existing solutions in literature whenever possible to demonstrate and validate the present method.

Swinging motion control of suspended structures: Principles and applications

Abstract: This paper discusses the development of non-linear differential equation modeling and characteristic analysis of control force for motion control of suspended structures, and proposes an innovative passive control system for suppressing swinging motion of similar structures. Based on the classical Lagrangian principle, the system EOM can be established with respect to two basic motion modes: planar motion and swinging motion. The analytical results indicate that different control systems should be considered owing to the diverse motion characteristics of the suspended system. Furthermore, theoretical results show that the tuned mass damper (subsequently abbreviated as TMD) cannot be used for swinging motion control, due to the coupling effect between linear stroke of TMD mass and angular velocity of suspended structure. Then, based on thorough numerical analysis, the concept of tuned rotary inertia damper (subsequently abbreviated as TRID) control system is proposed, and the differential equations of motion, denoting the motion law of the whole system, are studied. In addition, optimization issues of TRID control are equivalent to the classical optimization problem of TMD control. At last, conclusions were extended to vibration or motion control of typical civil engineering structures, such as high-rising tower structures with prior bending deformation characteristics and long-span bridges with rotation vibration characteristics. Once the structural motion or vibration is similar to the single pendulum or inverted pendulum, the planar TMD control system will lose its effectiveness and the innovative TRID system should be considered for suppressing swinging motion of such structures. Copyright © 2009 John Wiley & Sons, Ltd.

Shear deformation effect in flexural–torsional vibrations of beams by BEM

Abstract: In this paper, a boundary element method is developed for the general flexural–torsional vibration problem of Timoshenko beams of arbitrarily shaped cross section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.

Thermal buckling of double-walled carbon nanotubes

Abstract: The thermal buckling of an armchair double-walled carbon nanotube (DWCNT), which is subjected to axial compression due to temperature rise, is derived and analyzed based on Timoshenko beam model, including transverse shear deformation and rotary inertia. According to the analysis, the effect of van der Waals force between the nanotubes on the critical buckling temperature of mode 1 of armchair DWCNT is significant, especially for smaller diameter nanotubes. The van der Waals force makes the DWCNT stiffer and increases the buckling temperature. In addition, the effect of shear deformation and rotary inertia on the buckling temperature is more obvious for the higher-order modes. The critical buckling temperature ratio of a Timoshenko beam to a Euler beam for the armchair DWCNT significantly decreases with increasing the diameter and mode number. Therefore, for the higher-order modes, the Timoshenko beam model is able to predict the critical buckling temperature of larger diameter DWCNT.

Inertia Shaping For Humanoid Fall Direction Change

Abstract: A system and method is disclosed for controlling a robot that is falling down from an upright posture. Inertia shaping is performed on the robot to avoid an object during the fall. A desired overall toppling angular velocity of the robot is determined. The direction of this velocity is based on the direction from the center of pressure of the robot to the object. A desired composite rigid body inertia of the robot is determined based on the desired overall toppling angular velocity. A desired joint velocity of the robot is determined based on the desired composite rigid body inertia. The desired joint velocity is also determined based on a composite rigid body inertia Jacobian of the robot. An actuator at a joint of the robot is then controlled to implement the desired joint velocity.

Natural frequencies of composite beams with a variable fiber volume fraction including rotary inertia and shear deformation

Abstract: The present study is concerned with the vibration analysis of symmetric composite beams with a variable fiber volume fraction through thickness First-order shear deformation and rotary inertia have been included in the analysis The solution procedure is applicable to arbitrary boundary conditions Continuous gradation of the fiber volume fraction is modeled in the form of an m-th power polynomial of the coordinate axis in the thickness direction of the beam By varying the fiber volume fraction within the symmetric composite beam to create a functionally graded material (FGM), certain vibration characteristics are affected Results are presented to demonstrate the effects of shear deformation, fiber volume fraction, and boundary conditions on the natural frequencies and mode shapes of composite beams

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part II: applications and validation

Abstract: In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. In Part I the governing equations of the aforementioned problem have been derived, leading to the formulation of five boundary value problems with respect to the transverse displacements, to the axial displacement and to two stress functions. These problems are numerically solved using the Analog Equation Method, a BEM based method. In this Part II, numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. Thus, the results obtained from the proposed method are presented as compared with those from both analytical and numerical research efforts from the literature. More specifically, the shear deformation effect in nonlinear free vibration analysis, the influence of geometric nonlinearities in forced vibration analysis, the shear deformation effect in nonlinear forced vibration analysis, the nonlinear dynamic analysis of Timoshenko beams subjected to arbitrary axial and transverse in both directions loading, the free vibration analysis of Timoshenko beams with very flexible boundary conditions and the stability under axial loading (Mathieu problem) are presented and discussed through examples of practical interest.

Inertia Match of a 3-RRR Reconfigurable Planar Parallel Manipulator

Abstract: Inertia match of the parallel manipulator means the ratio of the inertial load of the parallel manipulator converted to each actuator shaft and the moment of inertia of the actuator is kept within a reasonable range. Currently there are many studies on parallel manipulators, but few mention inertia parameters and inertia match of parallel manipulators. This paper focuses on the inertia characteristics of the 3-RRR reconfigurable planar parallel manipulator. On the basis of the inverse dynamic formulations deduced with the principle of virtual work, the inertia matrix of the 3-RRR planar parallel manipulator in the actuator space is obtained in algebraic form. Then, by unifying the dimension and averaging diagonal elements of the inertia matrix, the equivalent inertia of the parallel manipulator, which is the inertial load of the parallel manipulator converted to each actuator shaft, is determined. By transforming the inertia problem of the 3-RRR parallel manipulator into that of the serial multi-bar manipulator, the practicality of the equivalent inertia deduced by inverse dynamics is demonstrated. According to the physical meaning of the inertia equation, the manipulator is divided in to three parts. Further analysis is carried out on the contribution of each part to the equivalent inertia and their distributions in the required workspace, revealing that the passive links cannot ignored in calculating the equivalent inertia of the parallel manipulator. Finally, the inertia match for the 3-RRR reconfigurable parallel manipulator under three configurations is accomplished, and reducers are selected. The equivalent inertia calculation and the inertial match results illustrate that the inertia math is a necessary step to the design of the parallel manipulator, and inertia parameters dramatically affect dynamic performances of parallel manipulators. Besides, the equivalent inertia and inertial match principles, proposed in the paper, can be widely applied in the dynamic analysis and servomotors selecting for the parallel manipulator.

Molecular Dynamics Simulation of Double-Walled Carbon Nanotube Vibrations: Comparison With Continuum Elastic Theories

Abstract: Double-walled carbon nanotubes (DWNTs) are expected to be useful as elements in improving con- ventional polymer-based fibers and films. An extensive molecular dynamics simulation and continuum analyses are carried out to estimate the influence of matrix stiffness and the intertube radial displacements on free vibration of an individual DWNT. The effects of nanotube length and chirality are also taken into account. The continuum analyses are based on both Euler-Bernoulli and Timoshenko beam theories which considers shear deformation and rotary inertia and for both concentric and non-concentric assump- tions considering intertube radial displacements and the related internal degrees of freedom. New inter- tube resonant frequencies are calculated. Detailed results are demonstrated for the dependence of reso- nant frequencies on the matrix stiffness. The results indicate that internal radial displacement and sur- rounding matrix stiffness could substantially affect resonant frequencies especially for longer double- walled carbon nanotubes of larger innermost radius at higher resonant frequencies, and thus the latter does not keep the otherwise concentric structure at ultrahigh frequencies.

Effects of Damping on the Vibration Frequency of Atomic Force Microscope Cantilevers Using the Timoshenko Beam Model

Abstract: The modal frequencies of flexural vibration for an atomic force microscope (AFM) cantilever immersed in fluids have been derived based on Timoshenko beam theory, including the effects of rotary inertia and shear deformation, and a closed-form expression for the resonant frequencies of vibration modes has been obtained. The effects of quality factor and contact stiffness on the modal frequency are analyzed. The results show that the damping effect on the vibration frequency of AFM cantilever is obvious, especially for high-order modes and low contact stiffness. The vibration frequency of AFM cantilever shows a marked decrease with low quality factors. In addition, the effects of rotary inertia and shear deformation on the ratio of vibration frequency of a Timoshenko beam to that of an Euler beam are significant, especially for high-order modes and low quality factors. It indicates that the Timoshenko beam theory is able to evaluate the frequencies of flexural vibration of the higher modes for the AFM cantilever immersed in liquids.

Rotation of a slender particle in a shear flow: influence of the rotary inertia and stability analysis

Abstract: The rotation of an inertialess ellipsoidal particle in a shear flow of a Newtonian fluid has been firstly analyzed by Jeffery [17]. He found that the particle rotates such that the end of its axis of symmetry describes a closed periodic orbit. In the special case of a slender particle the Jeffery solution predicts the particle alignment parallel to the streamlines. In a recent publication [3] it was shown that the orbits are no longer observable if the rotary inertia is taken into account. Furthermore, in the case of a slender particle the inertia may cause the jump over the equilibrium alignment. In this paper we address a detailed analysis of the slender particle behavior in the shear flow. We recall the constitutive equation for the hydrodynamic moment and formulate equations of rotary motion. For a special initial condition we reduce the problem to a single second-order ordinary differential equation with respect to the angle of rotation about a fixed axis. The phase portrait of this equation illustrates different cases of the particle behavior depending on the initial conditions and the "inertia" parameter. They include the particle alignment to a semi-stable equilibrium position, the non-uniform rotation about a fixed axis as well as the quantization effect (the particle locates in the neighborhood of the first equilibrium point over a relatively long time and then rotates towards the next equilibrium point).

Dynamic Response of a Pre-Stressed Transversely Isotropic Plate to Impact by an Elastic Rod

Abstract: This article considers the problem of normal impact of a long thin elastic cylindrical rod upon an infinite pre-stressed elastic transversely isotropic plate possessing cylindrical anisotropy. The impact takes place at the center of the plate, whose equations of motion take both rotary inertia and shear deformations into account. During the shock interaction of the rod with the plate, waves with strong discontinuities are generated in the plate and begin to propagate. Behind the fronts of these waves, the solution is constructed in terms of ray series, the coefficients of which are the different order discontinuities in partial-time derivatives of the desired functions, and the time elapsed after the wave arrival at the point (of the plate) under consideration is a variable. The ray series coefficients are determined from recurrent equations within the accuracy of arbitrary constants, which are determined from the conditions of the dynamic contact interaction between the impactor and the target. These arb...