Showing papers on "Rotary inertia published in 2014"

Motion of spheroid particles in shear flow with inertia

Abstract: In this article, we investigate the motion of a solid spheroid particle in a simple shear flow. Using a lattice Boltzmann method, we examine individual effects of fluid inertia and particle rotary inertia as well as their combination on the dynamics and trajectory of spheroid particles at low and moderate Reynolds numbers. The motion of a single spheroid is shown to be dependent on the particle Reynolds number, particle aspect ratio, particle initial orientation and the Stokes number. Spheroids with random initial orientations are found to drift to stable orbits influenced by fluid inertia and/or particle inertia. Specifically, prolate spheroids drift towards the tumbling mode of motion, whereas oblate spheroids drift to the rolling mode. The rotation period and the variation of angular velocity of tumbling spheroids decrease as Stokes number increases. With increasing Reynolds number, both the maximum and minimum values of angular velocity decrease, whereas the particle rotation period increases. We show that particle inertia does not affect the hydrodynamic torque on the particle. We also demonstrate that superposition can be used to estimate the combined effect of fluid inertia and particle inertia on the dynamics of spheroid particles at sufficiently low Reynolds numbers.

Size-dependent nonlinear vibration and instability of embedded fluid-conveying SWBNNTs in thermal environment

Abstract: The size-dependent nonlinear free vibration and instability of fluid-conveying single-walled boron nitride nanotubes (SWBNNTs) embedded in thermal environment are studied in this paper. The fluid-conveying SWBNNT is modeled as a Timoshenko beam by which the effects of transverse shear deformation and rotary inertia is taken into consideration. The modified strain gradient theory is used to capture the size effect. To consider the nonlinear effect, the geometric nonlinearity, based on von Karman׳s assumption is introduced to develop the nonlinear governing equations of motion. By employing Hamilton׳s principle, the governing equations and associated boundary conditions are derived. Thereafter, a numerical solution procedure based on the generalized differential quadrature (GDQ) is introduced, according to which the nonlinear governing equations and the corresponding boundary conditions are discretized via the operational matrix of differentiation. The discretized equations are then solved analytically through the harmonic balance approach. Effects of different parameters including material length scale parameter, spring and damping constants of surrounding viscoelastic medium, and flow velocity on the nonlinear free vibration and instability of SWBNNTs are examined.

Non-linear static bending and forced vibrations of rectangular plates retaining non-linearities in rotations and thickness deformation

Abstract: Geometrically non-linear static bending and forced vibrations of rectangular plates are studied allowing full non-linear terms associated with Green–Lagrange strain–displacement relations, second-order thickness stretching, third-order shear deformation and rotary inertia by using seven independent parameters to describe the shell kinematics. In particular, in addition to non-linearities in membrane and transverse deflection, non-linear terms associated with rotations and thickness deformation parameters are also included. In order to obtain the governing equations of motion, the three-dimensional constitutive equations are used, removing the assumption of zero transverse normal strain. The boundary conditions of the plate are assumed to be simply supported immovable and the equations of motion are derived by using a Lagrangian approach. The numerical solutions are obtained by using pseudo arc-length continuation and collocation scheme. In order to compare the non-linear static response, another analysis has also been carried out by using the finite element code ANSYS and three-dimensional solid modeling. Results reveal that the new theory with full geometric non-linearities provides significant accuracy improvement for rotational and thickness deformation parameters, and, unlike other shear deformation theories, predicts the correct thickness stretching along the plate.

Steady-state dynamic behavior of an on-board rotor under combined base motions.

Abstract: In the transportation domain, on-board rotors in bending are subjected not only to rotating mass unbalance but also to several movements of their base. The main objective of this article is to predict the dynamic behavior of a rotor in the presence of base excitations. The proposed on-board rotor model is based on the Timoshenko beam finite element. It takes into account the effects corresponding to rotary inertia, gyroscopic inertia, and shear deformation of shaft as well as the geometric asymmetry of disk and/or shaft and considers six types of deterministic motions (rotations and translations) of the rotor's rigid base. The use of Lagrange's equations associated with the finite element method yields the linear second-order differential equations of vibratory motion of the rotating rotor in bending relative to the moving rigid base which forms a non-inertial frame of reference. The linear equations of motion highlight periodic parametric terms due to the geometric asymmetry of the rotor components and time-varying parametric terms due to the rotational motions of the rotor rigid base. These parametric terms are considered as sources of internal excitation and can lead to lateral dynamic instability. In the presented applications, the rotor is excited by a rotating mass unbalance combined with constant rotation and sinusoidal translation of the base. Quasi-analytical and numerical solutions for two different rotor configurations (symmetric and asymmetric) are analyzed by means of stability charts, Campbell diagrams, steady-state responses as well as orbits of the rotor.

On the free vibrations of grid-stiffened composite cylindrical shells

Abstract: A unified analytical approach is applied for investigating the vibrational behavior of grid-stiffened composite cylindrical shells considering the flexural behavior of the ribs. A smeared method is employed to superimpose the stiffness contribution of the stiffeners with those of the shell in order to obtain the equivalent stiffness parameters of the whole panel. The stiffeners are modeled as a beam and considered to support shear loads and bending moments in addition to the axial loads. Therefore, the corresponding stiffness terms are taken into consideration while obtaining the stiffness matrices due to the stiffeners. Theoretical formulations are based on first-order shear deformation shell theory, which includes the effects of transverse shear deformation and rotary inertia. The modal forms are assumed to have the axial dependency in the form of Fourier series whose derivatives are legitimized using Stokes’ transformation. In order to validate the obtained results, a 3-D finite element model is also built using ABAQUS CAE software. Results obtained from two types of analyses are compared with each other, and good agreement has been achieved. Furthermore, the influence of variations in the shell thickness and changes of the boundary conditions on the shell frequencies is studied. The results obtained are novel and can be used as a benchmark for further studies.

A non-linear higher-order thickness stretching and shear deformation theory for large-amplitude vibrations of laminated doubly curved shells

Abstract: A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness stretching, higher-order shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in the in-plane and transverse displacements and are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. The theory uses the three-dimensional constitutive equations and does not need the introduction of traction/compression free hypothesis at the shell inner and outer surfaces. The traction/compression free condition is introduced only to obtain a simplified model with six parameters instead of eight. The third-order thickness stretching theory is applied to cross-ply symmetrically laminated circular cylindrical shells complete around the circumference and simply supported at both ends. Geometrically non-linear forced vibrations are studied by using the present theory and results are compared to those obtained by using a refined higher-order shear deformation non-linear shell theory, which neglects thickness stretching, and to results obtained by using first-order and second-order thickness stretching theories. Results obtained by using the reduced third-order thickness stretching model with six parameters are also presented and compared.

Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials

Abstract: Vibration analysis of nonlocal nanobeams based on Euler–Bernoulli and Timoshenko beam theories is considered. Nonlocal nanobeams are important in the bending, buckling and vibration analyses of beam-like elements in microelectromechanical or nanoelectromechanical devices. Expressions for free vibration of Euler–Bernoulli and Timoshenko nanobeams are established within the framework of Eringen’s nonlocal elasticity theory. The problem has been solved previously using finite element method, Chebyshev polynomials in Rayleigh–Ritz method and using other numerical methods. In this study, numerical results for free vibration of nanobeams have been presented using simple polynomials and orthonormal polynomials in the Rayleigh–Ritz method. The advantage of the method is that one can easily handle the specified boundary conditions at the edges. To validate the present analysis, a comparison study is carried out with the results of the existing literature. The proposed method is also validated by convergence studies. Frequency parameters are found for different scaling effect parameters and boundary conditions. The study highlights that small scale effects considerably influence the free vibration of nanobeams. Nonlocal frequency parameters of nanobeams are smaller when compared to the corresponding local ones. Deflection shapes of nonlocal clamped Euler–Bernoulli nanobeams are also incorporated for different scaling effect parameters, which are affected by the small scale effect. Obtained numerical solutions provide a better representation of the vibration behavior of short and stubby micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.

Resonant-type inertia linear motor based on the harmonic vibration synthesis of piezoelectric bending actuator

Abstract: Traditional piezoelectric inertia motors are generally driven at the quasi-static frequency range, which results in a relatively slow moving speed. In this paper, a piezoelectric bending actuator was designed for a resonant-type inertia linear motor. The driving mechanism of the actuator was also studied. The actuator's movement in a periodical sawtooth-shaped waveform was generated by composing two sinusoidal resonant bending vibrations with a frequency ratio of 1:2. A prototype inertia motor was fabricated. Experimental results confirmed the effectiveness of the design. The no-load maximum speed was 28.2 mm/s with drive voltage of 300 Vp–p for a base frequency of 587 Hz. At a preload force of 9.6 N and a driving voltage of 400 Vp–p for the base frequency, the linear speed was 18.5 mm/s with 0.02 N drag load. The moving direction could be reversed by changing the driving voltage's phase.

Finite element based vibration analysis of functionally graded spinning shaft system

Abstract: The present work deals with the study of vibration and stability analysis of a functionally graded spinning shaft system using three-noded beam element based on the Timoshenko beam theory. Material properties are assumed to be graded in radial direction according to power law gradation. In the present analysis, the mixture of aluminum oxide (Al2O3) and stainless steel (SUS304) has been considered as functionally graded material where metal (SUS304) content decreases towards the outer diameter of the shaft. The functionally graded shafts has been modeled as a Timoshenko beam, which contains discrete isotropic rigid disks supported by flexible bearing. The functionally graded shaft has been modeled based on first-order shear deformation beam theory with transverse shear deformation, rotary inertia, gyroscopic effect, strain and kinetic energy of shafts by adopting three-dimensional constitutive relations. The derivation of governing equations of motion has been obtained using Hamilton’s principle. Three-nod...

Free vibration analysis of a magneto-electro-elastic doubly-curved shell resting on a Pasternak-type elastic foundation

Abstract: Free vibration of a simply-supported magneto-electro-elastic doubly-curved thin shell resting on a Pasternak foundation is investigated based on Donnell theory. The rotary inertia effect is considered in the formulation. Maxwell equations for electrostatics and magnetostatics are used to model the electric and magnetic behavior. The partial differential equations of motion are reduced to a single ordinary differential equation and an analytical relation is obtained for the natural frequency. After validation of the present study, several numerical studies is done to investigate the effects of the electric and magnetic potentials, spring and shear coefficients of the Pasternak foundation, and the geometry of the shell on the vibration frequency.

Dynamic behavior of thin and thick cracked nanobeams incorporating surface effects

Abstract: Free transverse vibration of cracked nanobeams is investigated in the presence of the surface effects. Two nanobeam types, thin and thick, are studied using two beam theories, Euler–Bernoulli and Timoshenko. The influences of crack severity and position, surface density, rotary inertia and shear deformation, nanobeam dimension, mode number, satisfying balance condition between the surface layers and the bulk, boundary conditions and satisfying compatibility and boundary conditions with appropriate resultant moment and shear force are studied in details. It is found out that satisfying compatibility and boundary conditions with the resultant moment and shear force in presence of the surface effects and considering surface density neglected in previous work have significant effects on the natural frequencies of cracked nanobeams. In addition, rotary inertia and shear deformation cause a reduction in the crack and surface effects on the natural frequencies.

Transient analysis of FGM and laminated composite structures using a refined 8-node ANS shell element

Abstract: In this paper, we investigate the vibration analysis of functionally graded material (FGM) and laminated composite structures, using a refined 8-node shell element that allows for the effects of transverse shear deformation and rotary inertia. The properties of FGM vary continuously through the thickness direction according to the volume fraction of constituents defined by sigmoid function, but in this method, their Poisson’s ratios of the FGM plates and shells are assumed to be constant. The finite element, based on a first-order shear deformation theory, is further improved by the combined use of assumed natural strains and different sets of collocation points for interpolation the different strain components. We analyze the influence of the shell element with the various location and number of enhanced membrane and shear interpolation. Using the assumed natural strain method with proper interpolation functions the present shell element generates neither membrane nor shear locking behavior even when full integration is used in the formulation. The natural frequencies of plates and shells are presented, and the forced vibration analysis of FGM and laminated composite plates and shells subjected to arbitrary loading is carried out. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To validate and compare the finite element numerical solutions, the reference solutions of plates based on the Navier’s method, the series solutions of sigmoid FGM (S-FGM) plates are obtained. Results of the present theory show good agreement with the reference solutions. In addition the effect of damping is investigated on the forced vibration analysis of FGM plates and shells.

Analysis of curved beams using a new differential transformation based curved beam element

Abstract: Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved beam element including the effects of shear deformation considering axially functionally graded material The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved beam element The static and free vibration of axially functionally graded tapered curved beams including shear deformation and rotary inertia are studied through solving several examples Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal beams (both forms—prime and quadratic) with hinged-hinged, hinged-clamped and clamped-clamped and clamped-free end restraints Three general taper types (depth taper, breadth taper and square taper) for rectangular cross section are studied Out of plane vibration is studied and the lowest natural frequencies are calculated and compared with the published results Out of plane buckling is investigated for circular beams due to radial load

Natural vibrations of thick circular plate based on the modified Mindlin theory

Abstract: Outline of the modified Mindlin theory is presented in which the Mindlin mathematical model with three differential equations of motion for total deflection and rotations is decomposed into a single equation for pure bending vibrations with transverse shear and rotary inertia effects and two differential equations for in-plane shear vibrations. The governing equations are transformed from orthogonal to polar coordinate system for the purpose of circular plate vibration analysis. The fourth order differential equation of flexural vibrations is split further into two second order equations of Bessel type. Also, the in-plane shear differential equations are transformed to Bessel equation by introducing displacement potential functions. The exact values of natural frequencies are listed and compared with FEM results.

Vibration band gap properties of periodic Mindlin plate structure using the spectral element method

Abstract: In this paper, vibration band gap properties of periodic Mindlin plate structure with two simply supported opposite edges are studied using the spectral element method (SEM). Based on the Mindlin plate theory (MPT), the spectral Mindlin plate stiffness matrix is deduced. The spectral equation of the whole periodic structure is further established. Frequency responses are calculated by the spectral equations to illustrate band gap properties. Compared with the results based on the classic plate theory (CPT), the effects of the shear and rotary inertia on the band gap properties are analyzed. The results are also compared with those calculated by the finite element method (FEM). It can be observed that the SEM is more accurate in high frequency ranges. Furthermore, the influences of the material properties and structure dimension on the band gap behaviors are investigated.

Dynamic analysis of a spinning functionally graded material shaft by the p - version of the finite element method

Abstract: This paper is concerned with the dynamic behavior of the spinning Functionally Graded Material (FGM) shaft on rigid bearings. A p- version, hierarchical finite element is employed to define the model. A theoretical study allows the establishment of the kinetic energy and the strain energy of the shaft, necessary to the result of the equations of motion. In this model the transverse shear deformation, rotary inertia and gyroscopic effects have been incorporated. A hierarchical beam finite element with six degrees of freedom per node is developed and used to model the shaft. A program is elaborate for the calculation of the natural frequencies of a spinning FGM shaft. To verify the present model, the results are compared with those available in the literature. The efficiency and accuracy of the methods employed are discussed.

Chatter instability analysis of spinning micro-end mill with process damping effect via semi-discretization approach

Abstract: In this paper, the stability of delay differential equations (DDEs), describing self-excited vibrations in a micro-milling process, is investigated based on semi-discretization (SD) method. Due to the stubby geometry of micro-tools, the shear deformation and rotary inertia effects are considered for modeling the structure. The extended Hamilton’s principle is used to derive a detailed dynamical model of the spinning micro-tool with the support of misalignment in which the gyroscopic effects cause coupling of equations. Considering the actual geometry of the micro-end mill, exact dynamic stiffness (DS) formulations are developed to investigate the tool’s free vibration characteristics. The extracted mode shapes obtained from DS method are utilized as base functions in a Galerkin approach. Having considered regenerative cutting force, imposing the Galerkin method reduces the governing PDEs of the system to a set of DDEs. The resulting equations are discretized by means of SD procedure. Finally, numerical Floquet theory is utilized to investigate the stability of the system. Also, the effects of process damping on the stability diagrams are explored. The results show the efficiency of the proposed model and delineate the considerable influence of process damping on the stability borders of the system especially at low spindle speed.