Showing papers on "Rotary inertia published in 2018"

Stochastic dynamic stability analysis of composite curved panels subjected to non-uniform partial edge loading

Abstract: The stochastic dynamic stability analysis of laminated composite curved panels under non-uniform partial edge loading is studied using finite element analysis. The system input parameters are randomized to ascertain the stochastic first buckling load and zone of resonance. Considering the effects of transverse shear deformation and rotary inertia, first order shear deformation theory is used to model the composite doubly curved shells. The stochasticity is introduced in Love's and Donnell's theory considering dynamic and shear deformable theory according to the Sander's first approximation by tracers for doubly curved laminated shells. The moving least square method is employed as a surrogate of the actual finite element model to reduce the computational cost. The results are compared with those available in the literature. Statistical results are presented to show the effects of radius of curvatures, material properties, fibre parameters, and non-uniform load parameters on the stability boundaries.

Nonlinear vibration analysis of functionally graded beams considering the influences of the rotary inertia of the cross section and neutral surface position

Abstract: In the paper work, the nonlinear vibration response of functionally graded (FG) Euler–Bernoulli beam resting on elastic foundation is studied. Based on von Karman’s geometric nonlinearity, the partial differential governing equations describing the nonlinear vibration of FG Euler–Bernoulli beam are derived from Hamilton’s principle and are reduced to an ordinary nonlinear differential equation with quadratic and cubic nonlinear terms via Galerkin’s procedure. Due to unsymmetrical material variation along the thickness of FG beam, the neutral surface concept is proposed to remove the stretching and bending coupling effect and the rotary inertia of the cross section is incorporated to obtain an analytical solution. Numerical results are presented to show the effects of the nonlocal parameters and vibration amplitude on the frequency responses. This results may be useful in design and engineering applications.

Vibration Analysis of Material Size-Dependent CNTs Using Energy Equivalent Model

Abstract: This study presents a modified continuum model to investigate the vibration behavior of single and multi-carbon nanotubes (CNTs). Two parameters are exploited to consider size dependence; one derived from the energy equivalent model and the other from the modified couple stress theory. The energy equivalent model, derived from the basis of molecular mechanics, is exploited to describe size-dependent material properties such as Young and shear moduli for both zigzag and armchair CNT structures. A modified couple stress theory is proposed to capture the microstructure size effect by assisting material length scale. A modified kinematic Timoshenko nano-beam including shear deformation and rotary inertia effects is developed. The analytical solution is shown and verified with previously published works. Moreover, parametric studies are performed to illustrate the influence of the length scale parameter, translation indices of the chiral vector, and orientation of CNTs on the vibration behaviors. The effect of the number of tube layers on the fundamental frequency of CNTs is also presented. These findings are helpful in mechanical design of high-precision measurement nano-devices manufactured from CNTs.

Virtual inertia optimisation control of DFIG and assessment of equivalent inertia time constant of power grid

Abstract: Due to decoupling between mechanical part and electrical part of doubly fed induction generator (DFIG), the DFIG has no natural frequency response capability, which results in decreasing of total rotary inertia of power grid, and the frequency stability of power grid will face larger challenges. This study proposes an assessment method of equivalent inertia time constant, further gives the assessment value of equivalent inertia time constant of DFIG. Then, a virtual inertia optimisation control of DFIG using rotor current direct control based on status assessment value is proposed, which includes conventional function, status assessment and additional virtual inertia control. The assessment values of equivalent inertia time constant of wind farms, synchronous generator and power grid connected with multi-type generators are calculated, which are related to penetration of wind power and control strategies. The simulation results verify the efficacious of the proposed virtual inertia optimisation control and the accuracy of assessment method of equivalent inertia time constant, and the frequency stability of power grid is improved in the condition of different active power of wind farms and penetration of wind power.

Stress-Strain Assessment for Laminated Aircraft Cockpit Windows at Static and Dynamic Loads

Abstract: The method of strength calculations for laminated aircraft cockpit windows influenced by different operating factors (bird strike, pressurization) is devised. The method is based on embedding the initial uncanonical shell in the auxiliary one of canonical form in plan with the boundary conditions, which permit of a simple analytical problem solution as a trigonometric series. For satisfying the initial boundary conditions, the auxiliary shell is supplemented with compensating loads, which are continuously distributed over the contour of the initial shell. The compensating loads enter in the equations of motion for the auxiliary shell as integral relations. The system of motion equations is rearranged in the system of ordinary differential equations of second order, which is integrated by the solution expansion in the Taylor series. The windows are treated as a laminated open-ended cylindrical shell consisting of isotropic layers of constant thickness. The laminated window model is based on the modified theory of first order that accounts for transverse shear strains, thickness reduction, rotary inertia, and compression of the normal element in each layer. For the composition, the hypothesis of broken line is valid. The model of pressure pulse that apparently represents the effect of the bird impact on the windows was constructed on the basis of experimental studies. The stress-strain state of the window element in AN aircrafts was evaluated, set on the bird strike and cockpit pressurization. Five window alternatives are examined. Calculation results are in good agreement with experimental data. The results become theoretical and practical backgrounds for engineering calculations and optimum design of laminated aircraft window elements influenced by different operating factors. Thus, the advanced method can be applied to estimation of the lifetime of existing window elements and development of the new ones.

The isogeometric free vibration and transient response of functionally graded piezoelectric curved beam with elastic restraints

Abstract: This paper presents the natural vibration and transient response of functionally graded piezoelectric materials (FGPMs) curved beam with a numerical method. The unknown fields, geometry and material properties are described by the isogeometric method. The effects of axis extensibility, shear deformation, rotary inertia, general boundary condition and variable curvature are considered. The material with electrical properties are assumed varying continuously. The convergence and accuracy are validated in several numerical examples. Moreover, the effects of boundary conditions, material parameters and geometric proprieties are examined systematically. Finally, the transient response of curved beam subjected forces are fully reported.

Resonance frequencies of size dependent perforated nonlocal nanobeam

Abstract: Perforation is a common procedure in fabrication process of micro/nano-electromechanical systems (M/NEMS). Therefore, this paper presents an effort to study the resonance frequencies of size dependent regular square perforated nonlocal nanobeam, which not be discussed before. Equivalent characteristic parameters of perforated beam such as, bending stiffness, shear stiffness, mass, and rotary inertia are presented. The size-scale effect of long-range atomic interaction of nanobeam is described by using nonlocal differential form of Eringen model. Constitutive and governing equations of local and nonlocal perforated Timoshenko and Euler–Bernoulli nanobeam are derived. Analytical solution are exploited to solve the proposed model and derived closed form frequency equations as function of nanoscale and perforation parameters. The verification of current model is presented and compared with published works. Numerical results are illustrated to present the influences of length scale parameter, number of perforated holes, perforation size, and shear effects on the natural frequencies of both nanobeams theories. The obtained results are supportive in mechanical design of high-precision measurement nanobeams sensor and actuators.

The effect of initial geometric imperfection on the nonlinear resonance of functionally graded carbon nanotube-reinforced composite rectangular plates

Abstract: The purpose of the present study is to examine the impact of initial geometric imperfection on the nonlinear dynamical characteristics of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) rectangular plates under a harmonic excitation transverse load. The considered plate is assumed to be made of matrix and single-walled carbon nanotubes (SWCNTs). The rule of mixture is employed to calculate the effective material properties of the plate. Within the framework of the parabolic shear deformation plate theory with taking the influence of transverse shear deformation and rotary inertia into account, Hamilton’s principle is utilized to derive the geometrically nonlinear mathematical formulation including the governing equations and corresponding boundary conditions of initially imperfect FG-CNTRC plates. Afterwards, with the aid of an efficient multistep numerical solution methodology, the frequency-amplitude and forcing-amplitude curves of initially imperfect FG-CNTRC rectangular plates with various edge conditions are provided, demonstrating the influence of initial imperfection, geometrical parameters, and edge conditions. It is displayed that an increase in the initial geometric imperfection intensifies the softening-type behavior of system, while no softening behavior can be found in the frequency-amplitude curve of a perfect plate.

Free vibration characteristics of delaminated composite pretwisted stiffened cylindrical shell

Abstract: Effects of delamination on free vibration characteristics of laminated stiffened cylindrical shells with pretwist are analyzed by finite element method. The investigation is carried out using an eight-noded quadratic isoparametric shell element, which incorporates the transverse shear deformation and rotary inertia along with a three-noded beam element for the stiffener. The multipoint constraint algorithm has been included to guarantee the compatibility of deformation, equilibrium of resultant forces, and moments at delamination crack tip. The general dynamic equilibrium equation is derived from Lagrange’s equation of motion for moderate rotational speeds for which the Coriolis effect is neglected. The standard eigenvalue problem is solved utilizing QR iteration algorithm. The accuracy of the present formulation is validated with benchmark solutions is available in the literature. The present work concerns about the effects of delamination, fiber orientation, twist angle, stiffener depth-to-shell thickne...

Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams

Abstract: The dynamic stability of axially moving viscoelastic Rayleigh beams is presented. The governing equation and simple support boundary condition are derived with the extended Hamilton’s principle. The viscoelastic material of the beams is described as the Kelvin constitutive relationship involving the total time derivative. The axial tension is considered to vary longitudinally. The natural frequencies and solvability condition are obtained in the multi-scale process. It is of interest to investigate the summation parametric resonance and principal parametric resonance by using the Routh-Hurwitz criterion to obtain the stability condition. Numerical examples show the effects of viscosity coefficients, mean speed, beam stiffness, and rotary inertia factor on the summation parametric resonance and principle parametric resonance. The differential quadrature method (DQM) is used to validate the value of the stability boundary in the principle parametric resonance for the first two modes.

Analytical modeling of the coupled nonlinear free vibration response of a rotating blade in a gas turbine engine

Abstract: In this paper, we investigate the free vibration response of a rotating blade in a gas turbine engine. The blade is modeled as a tapered Timoshenko beam with nonlinear variations in its cross-section properties. The governing equations of motions are derived using Lagrangian mechanics and Rayleigh–Ritz method. These equations take into account centrifugal stiffening, axial and lateral coupling due to Coriolis effect, shear deformation, and rotary inertia. We examine the effect of the beam geometry upon its axial and lateral free vibration response. The effects of rotational speed, taper ratio, chord ratio, hub radius, and slenderness ratio on the natural frequencies are analyzed. The results of our analysis indicate that the taper ratio, slenderness ratio, and rotational speed of the beam govern its free lateral vibration response. The axial vibration of the beam is significantly affected by the slenderness ratio, but it is found to be independent of the hub radius.

Inertia Effect on the Load Capacity of Large Water-Lubricated Thrust Bearing

Abstract: With an assumption that the profile of the velocity is not affected by the inertia force, a numerical model based on an extended Reynolds equation, which takes the full inertia effect and turbulenc...

Nonstationary analysis of nonlinear rotating shafts passing through critical speed excited by a nonideal energy source

Abstract: In this paper, the effect of nonlinearity on vibration of a rotating shaft passing through critical speed excited by nonideal energy source is investigated. Here, the interaction between a nonlinear gyroscopic continuous system (i.e. rotating shaft) and the energy source is considered. In the shaft model, the rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The nonlinearity is due to large deflection of the shaft. Firstly, nonlinear equations of motion governing the flexural–flexural–extensional vibrations of the rotating shaft with nonconstant spin are derived by the Hamilton principle. Then, the equations are simplified using stretching assumption. To analyze the nonstationary vibration of the nonideal system, multiple-scale method is directly applied to the equations expressed in complex coordinates. Three analytical expressions that describe variation of amplitude, phase, and angular acceleration during passage through critical speed are derived. It is shown that...

Natural Frequencies of Rectangular Plate With- and Without-Rotary Inertia

Abstract: A nine-node isoparametric plate element, in conjunction with first-order shear deformation theory, was used for free vibration analysis of rectangular plates. Both thick and thin plate problems were solved for various aspect ratios and boundary conditions. In this work, the primary focus is on the effect of rotary inertia on the natural frequencies of rectangular plates. It is found that rotary inertia significantly affects thick plates, while it can be ignored for thin plates. The numerical convergence is very rapid and based on a comparison with data from the literature; it is proposed that the present formulation can yield highly accurate results. Finally, some numerical solutions are provided here, which may serve as benchmarks for future research on similar problems.

Effects of rotary inertia on sub- and super-critical free vibration of an axially moving beam

Abstract: The most important issue in the vibration study of an engineering system is dynamics modeling. Axially moving continua is often discussed without the inertia produced by the rotation of the continua section. The main goal of this paper is to discover the effects of rotary inertia on the free vibration characteristics of an axially moving beam in the sub-critical and super-critical regime. Specifically, an integro-partial-differential nonlinear equation is modeled for the transverse vibration of the moving beam based on the generalized Hamilton principle. Then the effects of rotary inertia on the natural frequencies, the critical speed, post-buckling vibration frequencies are presented. Two kinds of boundary conditions are also compared. In super-critical speed range, the straight configuration of the axially moving beam loses its stability. The buckling configurations are derived from the corresponding nonlinear static equilibrium equation. Then the natural frequencies of the post-buckling vibration of the super-critical moving beam are calculated by using local linearization theory. By comparing the critical speed and the vibration frequencies in the sub-critical and super-critical regime, the effects of the inertia moment due to beam section rotation are investigated. Several interesting phenomena are disclosed. For examples, without rotary inertia, the study overestimates the stability of the axially moving beam. Moreover, the relative differences between the super-critical fundamental frequencies of the two theories may increase with an increasing beam length.

Thouless-Valatin rotational moment of inertia from linear response theory

Abstract: Spontaneous breaking of continuous symmetries of a nuclear many-body system results in the appearance of zero-energy restoration modes. These so-called spurious Nambu-Goldstone modes represent a special case of collective motion and are sources of important information about the Thouless-Valatin inertia. The main purpose of this work is to study the Thouless-Valatin rotational moment of inertia as extracted from the Nambu-Goldstone restoration mode that results from the zero-frequency response to the total-angular-momentum operator. We examine the role and effects of the pairing correlations on the rotational characteristics of heavy deformed nuclei in order to extend our understanding of superfluidity in general. We use the finite-amplitude method of the quasiparticle random-phase approximation on top of the Skyrme energy density functional framework with the Hartree-Fock-Bogoliubov theory. We have successfully extended this formalism and established a practical method for extracting the Thouless-Valatin rotational moment of inertia from the strength function calculated in the symmetry-restoration regime. Our results reveal the relation between the pairing correlations and the moment of inertia of axially deformed nuclei of rare-earth and actinide regions of the nuclear chart. We have also demonstrated the feasibility of the method for obtaining the moment of inertia for collective Hamiltonian models. We conclude that from the numerical and theoretical perspective, the finite-amplitude method can be widely used to effectively study rotational properties of deformed nuclei within modern density functional approaches.

Nonlinear transient dynamic response of a blade subject to a pulsating load in a decaying centrifugal force field

Abstract: This study is concerned with an analytical solution for the nonlinear transient dynamic response of a rotating blade in a gas turbine engine experiencing dynamic unbalance due to a blade-off scenario leading to rotor unbalance. As a consequence of rotor unbalance, the blade experiences a pulsating load at its tip due to contact with the casing and a decaying centrifugal force field due to the rotor deceleration. The governing equations of motion consider the blade as an elastic Timoshenko beam of varying thickness subject to changing rotational speeds. The analysis takes into account the variation in the thickness of the beam, the coupled axial and lateral displacements of the beam as a result of Coriolis component of acceleration, shear deformation, rotary inertia, and friction resulting from the contact between the blade and the casing. Our findings indicate that the thickness profile of the beam plays a significant role in the transient response. It further reveals that the decaying centrifugal force field leads to a dramatic change in the dynamic response and the resulting forces on the blade.

Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

Abstract: Problems related to the transverse vibration of a rotating tapered cantilever beam with hollow circular cross-section are addressed, in which the inner radius of cross-section is constant and the outer radius changes linearly along the beam axis. First, considering the geometry parameters of the varying cross-sectional beam, rotary inertia, and the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. Next, dimensionless parameters and variables are introduced for the differential equation and boundary conditions, and the differential quadrature method (DQM) is employed to solve this differential equation with variable coefficients. Combining with discretization equations for the differential equation and boundary conditions, an eigen-equation of the system including some dimensionless parameters is formulated in implicit algebraic form, so it is easy to simulate the dynamical behaviors of rotating tapered beams. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.

Parametric instability of thick doubly curved CNT reinforced composite sandwich panels under in-plane periodic loads using higher-order shear deformation theory:

Abstract: This paper focuses on the problem of parametrically excited doubly curved sandwich shells with carbon nanotubes reinforced composite (CNTRC) facesheets subjected to in-plane periodic load. The panels consist of cylindrical and spherical shells modeled using QUAD-8 element which was developed using higher-order shear flexible theory. The formulation considers the secondary effects such as the influence of in-plane and rotary inertia terms, and the aerodynamic pressure when the panel is exposed to air flow. The governing equations developed are solved based on eigenvalue approach. The limits of the principal instability zone predicted here are graphically represented using excitation frequencies against the load amplitudes. The results of this study are tested against the available solutions in the literature. A detailed study considering various design parameters including structural theories on the dynamic instability boundaries and its associated origin of instability regions is conducted. These paramete...

Vibration of nonlinear bolted lap-jointed beams using Timoshenko theory

Abstract: This paper investigates the vibrational behavior of a system which consists of two free–free Timoshenko beams interconnected by a nonlinear joint. To model the bolted lap joint interface, a combination of the linear translational spring, linear and nonlinear torsional springs, and a linear torsional damper is used. The governing equations of motion are derived using the Euler–Lagrange equations. The reduced-order model equations are obtained based on Galerkin method. The set of coupled nonlinear equations are then analytically solved using the harmonic balance approach and numerical simulation. A parametric study is carried out to reveal the influence of different parameters such as linear and nonlinear torsional spring, linear translational spring, and linear torsional damper on the vibration and stability of the bolted lap joint structure. It is shown that the effect of the nonlinear torsional spring on the response of the system is significant. Interestingly, it is observed that in the presence of the nonlinear spring the softening behavior could be changed to hardening behavior. In addition, the effects of the different engineering beam theories on the modeling of the substructures are studied and it is observed that considering the effect of the rotary inertia and shear deformations is significant. In addition, it is observed that neglecting each of them can yield completely wrong interpretations of the system behavior and incorrect results.

Hydroelastic analysis of articulated floating elastic plate based on Timoshenko–Mindlin plate theory

Abstract: The wave interaction with articulated floating elastic plate is investigated considering the Timoshenko–Mindlin thick plate theory for both finite and shallow water depths. The elastic plates are m...