Free Torsional Vibration Research Articles

Torsional dynamics of bi-directional functionally graded small-scale tubes possessing a variable length scale parameter

The aim of the current study is to investigate torsional vibrations of bi-directional functionally graded (FG) small-scale tubes. The size effect is captured using the modified couple stress theory (MCST). Through-the-thickness distribution profile of material properties are characterized by employing a power-law function, which incorporates a gradation index, β, and longitudinal varation in material properties is featured by an exponential function which employs a gradient parameter, α. Similar to other material properties, continuous spatial variation of the length scale parameter is also incorporated into the analysis of two-dimensionally FG tubes. The model, including governing equations of motion and boundary conditions, is developed through the utilization of Hamilton’s principle. By employing differential quadrature method (DQM) numerical results regarding torsional vibrations are provided. Accuracy of the proposed model and procedures are verified through verification comparisons made to limiting cases available in the literature. Numerical results reveal effects of various material and geometric parameters on natural frequencies and indicate that torsional free vibration characteristics of small-sized tubes are sensitive to distribution profile of constituents, geometric aspect ratios and effective length scale parameter. Results also demonstrate that length scale parameter variation has significant influence on natural frequencies which justifies the necessity of its consideration in analyses.

Free torsional vibration of triangle microwire based on modified couple stress theory

In this article, the free torsional vibration of the noncircular microwire is obtained via modified couple stress theory. The noncircular cross section is considered to be equilateral triangle. The governing equations are derived using Hamilton’s principle, and the natural frequencies of the clamped-clamped and clamped-free microwires are obtained based on a Galerkin method. The natural frequency for analyzing of the results is considered to be dimensionless. The effects of the different parameters including the dimensionless material length scale parameter, the dimensionless edge length of the triangle, and mode number on the dimensionless natural frequency of the microwire are evaluated. Ultimately, the influence of the Poisson’s ratio on the natural frequency of the microwire under various boundary conditions is investigated.

Nonlocal Torsional Vibration of Elliptical Nanorods with Different Boundary Conditions

This work aims at investigating the free torsional vibration of one-directional nanostructures with an elliptical shape, under different boundary conditions. The equation of motion is derived from Hamilton’s principle, where Eringen’s nonlocal theory is applied to analyze the small-scale effects. The analytical Galerkin method is employed to rewrite the equation of motion as an ordinary differential equation (ODE). After a preliminary validation check of the proposed formulation, a systematic study investigates the influence of the nonlocal parameters, boundary conditions, geometrical and mechanical parameters on the natural frequency of nanorods; the objective is to provide useful findings for design and optimization purposes of many nanotechnology applications, such as, nanodevices, actuators, sensors, rods, nanocables, and nanostructured aerospace systems.

Dynamic identification and model updating of full-scale concrete box girders based on the experimental torsional response

The authors examined the dynamic behaviour of seven concrete box-girders of the A24 Italian motorway. The bridge spans derive from the same design and are nominally identical. However, they manifest a varied dynamic response, imputable to several factors. Some discrepancies may originate from the construction time, due to geometrical irregularities and unlike concrete curing. Others may descend from the time-degradation, soil-substructures interactions and other modifications in the boundary restraints, like the substitution of the original rack and roller bearings with multi-directional pots. The first bending mode shape is negligibly affected by the specific bearing, and the corresponding natural frequency has a marked dependence on the values of the Elastic Modulus, obtained from static load tests. The Least-Squares fit between the Elastic Modulus from static load tests and the first natural frequency matches with the corresponding expression of a simply-supported beam model. Conversely, the second mode shape, which is mainly torsional, is likely to be more sensitive to the bearings. The authors endeavored to estimate the values of two equivalent torsional springs, representatives of the actual restraint conditions by maximizing the Modal Assurance Criterion (MAC) between analytical and experimental mode shapes. The analytical mode shape is the solution of the equation of motion for free torsional vibration, with warping stiffness and warping inertia both neglected. The obtained values reported as the ratio between torsional stiffness of the boundary spring and the torsional stiffness of the beam cross-section yield the estimate of the vertical restraint offered by the two different typologies of bearings: the rack and roller and the multi-directional pot. The study encompasses the analysis of the outcomes of the dynamic identification and an elementary model updating.

On the viscoelastic carbon nanotube mass nanosensor using torsional forced vibration and Eringen’s nonlocal model

This article aims to investigate the forced and free dynamic torsional vibrations of single-walled carbon nanotube (SWCNT) embedded in a viscoelastic medium under a harmonic external torque. Eringen’s nonlocal elasticity is established. It takes the effect of the small size into the formulation due to dealing with nanostructures. The governing equation and corresponding boundary condition are derived using Hamilton’s principle. The boundary condition is considered clamped-disk. In order to reduce the derived partial differential equation (PDE) to ordinary differential equation (ODE) one and discretizing, a Galerkin method is employed in the time domain. The novelty of the present work is that it seeks to investigate the dynamic angular displacement in a viscoelastic SWCNT subjected to an external torque and mass nanosensor over time for the first time. A harmonic load along with determined amplitude is applied to SWCNT. Subsequently, the effects of the nonlocal parameter, damping ratio, damping coefficient, stiffness of the viscoelastic medium, excitation frequency, geometry, and disk’s mass moment of inertia on the angular displacement, in the time domain, are demonstrated. For free torsional vibration, the undamped natural frequency based on the mass moment of inertia of the disk is calculated for the first three mode numbers. Afterward, the effects of the damping coefficient on the real and imaginary parts of the eigenfrequency are investigated. Also, the effects of the stiffness of the viscoelastic medium, nonlocal parameter, and the length on the real part of the eigenfrequency are reported. Furthermore, the effects of the nonlocal parameter and damping coefficient on the damping ratio are illustrated schematically. A finite difference method (FDM) is established to assess the exact solution. Moreover, the results are validated with another study. The results are compatible in comparison with each other.

Torsional Vibration of nanowire with equilateral triangle cross section based on nonlocal strain gradient for various boundary conditions: comparison with hollow elliptical cross section

The torsional vibration is an unavoidable part of the mechanical behavior of the nanoscale materials. The nanostructures always are not circular. On the contrary, they have more often noncircular cross sections. In this paper, the size-dependent free torsional vibration of a nanowire with a triangular (equilateral triangle) cross section based on the nonlocal strain gradient theory is investigated for the first time. Three different boundary conditions such as clamped–clamped (C–C), clamped-free (C-F), and clamped–torsional spring (C-T) boundary conditions are utilized. Moreover, the first three natural frequencies of a hollow circular nanowire along with the nanowire with horizontal and vertical hollow elliptical cross sections are evaluated. The equation of motion, as well as the boundary conditions, is extracted by using Hamilton’s principle. An analytical method is employed to reduce the order of the equation of motion and ultimately solve it. The nanowire model contains the material length scale parameter and nonlocal parameter, simultaneously to take into account both stiffness-hardening and stiffness-softening effects on the vibration of the system. The effects of the nondimensional material length scale parameter and nondimensional nonlocal parameter on the first four nondimensional natural frequencies of the nanowire are assessed. The influences of ea/l and nondimensional material length scale parameter, and the effects of l/ea and nondimensional nonlocal parameter on the nondimensional responses are examined. Also, the effects of the nonlocal parameter on the first three natural frequencies of the model having circular and hollow elliptical cross sections are illustrated. The results are proved to be valid compared to another study.

Exponential and harmonic forced torsional vibration of single-walled carbon nanotube in an elastic medium

In this study, free torsional vibration and forced torsional vibration analysis under the time-dependent exponential and harmonic torsional loadings in single-walled carbon nanotube are investigated. The SWCNT is embedded in an elastic medium. Eringen's theory among the small-scale theories is selected. The nonlocal differential constitutive relation and corresponding boundary condition are derived via Hamilton's principle. Clamped–clamped boundary condition is utilized. The assumed modes method is employed for the dynamic torsional vibration in order to discretize the derived governing equations. The novelty of this work is devoted to the analysis of forced torsional vibration of a carbon nanotube embedded in an elastic medium under the various loadings. The angular displacement for the resonance frequency neglecting the elastic medium is illustrated. For the free analysis, the first three nondimensional natural frequencies with various small-scale parameters and stiffness of the elastic medium are calculated. The results are compared with another study for the first 10 mode numbers. The effects of the nonlocal parameter, length of carbon nanotube, stiffness of the elastic medium, thickness, time constant, and excitation frequency on the nondimensional and dimensional angular displacements are investigated, dynamically. For the greater values of the stiffness of the medium, the nonlocal parameter becomes negligible. When a time-dependent exponential torque is applied to the model, the angular displacement becomes greater and then lower by an increase in the value of the length, but the nondimensional angular displacement decreases continuously by increasing the value of the length under the time-dependent harmonic loading. Moreover, the angular displacement for a determined time becomes lower first and then becomes greater by increasing the time constant.

Effect of External Moving Torque on Dynamic Stability of Carbon Nanotube

In nano-dimension, the strength of the material is considerable, and the failure is unavoidable in a torsional mode. Because of this reason, the free and forced torsional vibrations of single-walled carbon nanotube (SWCNT) are investigated in this paper. For dynamic analysis, the moving harmonic torsional load is exerted to SWCNT. The related boundary condition and equation of motion are derived by Hamilton’s principle, and the equation is discretized by the Galerkin method. In order to demonstrate the nonlocality and small–scale effect, Eringen’s theory based on nonlocal elasticity theory is applied. A clamped-clamped (C-C) boundary condition is fitted for the end supports. The influences of the aspect ratio and mode number on the free natural frequency are investigated. Furthermore, the dynamic effects of nonlocal parameter, velocity, thickness, length, and excitation-to-natural frequencies on dimensional and nondimensional angular displacements are indicated. Moreover, the natural frequency was investigated due to the variation of the aspect ratio.

Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions

Nonlocal strain gradient continuum mechanics is a methodology widely employed in literature to assess size effects in nanostructures. Notwithstanding this, improper higher-order boundary conditions (HOBC) are prescribed to close the corresponding elastostatic problems. In the present study, it is proven that HOBC have to be replaced with univocally determined boundary conditions of constitutive type, established by a consistent variational formulation. The treatment, developed in the framework of torsion of elastic beams, provides an effective approach to evaluate scale phenomena in smaller and smaller devices of engineering interest. Both elastostatic torsional responses and torsional free vibrations of nano-beams are investigated by applying a simple analytical method. It is also underlined that the nonlocal strain gradient model, if equipped with the inappropriate HOBC, can lead to torsional structural responses which unacceptably do not exhibit nonlocality. The presented variational strategy is instead able to characterize significantly peculiar softening and stiffening behaviours of structures involved in modern Nano-Electro-Mechanical-Systems (NEMS).

Sensitivity analysis of free torsional vibration frequencies of thin-walled laminated beams under axial load

The paper addresses sensitivity analysis of free torsional vibration frequencies of thin-walled beams of bisymmetric open cross section made of unidirectional fibre-reinforced laminate. The warping effect and the axial end load are taken into account. The consideration is based upon the classical theory of thin-walled beams of non-deformable cross section. The first-order sensitivity variation of the frequencies is derived with respect to the design variable variations. The beam cross-sectional dimensions and the material properties are assumed the design variables undergoing variations. The paper includes a numerical example related to simply supported I-beams and the distributions of sensitivity functions of frequencies along the beam axis. Accuracy is discussed of the first-order sensitivity analysis in the assessment of frequency changes due to the fibre volume fraction variable variations, and the effect of axial loads is discussed too.

Nonlinear torsional vibrations of electromechanical coupling of diesel engine gear system and electric generator

New approach for analysis and detuning of nonlinear torsional vibrations of diesel electric generator gear system is suggested. Nonlinear magnetic field of the generator is calculated by finite element method to obtain the disturbing moments, which act on the gear system. Moreover, others disturbing moments, which arise due to operation in diesel engine cylinders, are accounted. Essential nonlinear model of torsional vibrations with sixteen degrees-of- freedom is derived. This model accounts the nonlinear moments of two elastic clutches and the backlashes of gear pairs. The comparison of the numerical simulations data with experimental results is performed to verify the obtained model. The approach for essential nonlinear free torsional vibrations calculations is suggested. The harmonic balanced method is the basis of this approach. Sensitivity theory is used to detune the nonlinear forced torsional vibrations from resonances. The results of nonlinear torsional vibrations analysis are discussed.

Longitudinal and Torsional Vibration Characteristics of Boron Nitride Nanotubes

Boron nitride nanotubes (BNNTs) are new cousins of carbon nanotubes (CNTs), which alternately substitute boron and nitrogen atoms for carbon atoms entirely. Its excellent physical and chemical properties, such as excellent thermal stability, chemical inertness and strong acid corrosion resistance, make BNNTs products be favorable to machine, thus scientific activities focused on them have profound practical significance and great application prospect. However, the vibration characteristics of BNNTs is still the least understood aspects. Longitudinal and torsional free vibration of single-walled boron nitride nanotubes are investigated through a coupling atomistic-continuum model with Euler beam model and atomistic simulations. Euler beam model with the involved atomistic-continuum multiscale model is capable of capturing the size effect since physical meanings of the integration for involved higher order gradient tensors are corresponding to size effect. The analytical solutions obtained by the coupling method are compared with atomistic simulation and are found to be in a good agreement. Computational results demonstrate that fundamental frequencies are in an order of 1 THz for both cantilevered and clamped boron nitride nanotubes, independent of tubular chirality. The effect of tubular radius on fundamental frequencies of both longitudinal and torsional free vibrations are large when it is small, especially the latter. The error between coupling method and atomistic simulation generally decreases as the length/diameter ratio increases. For a given length/diameter ratio, tubular radius have a distinctly decreasing effect on the error no matter what the kind of constraint is. However, a small difference observed in the study of torsional free vibration is that the clamped constraint gives rise to a slight increasing effect on the error as the tubular radius increases, but limited to a small degree. The vibrational modes confirm that the difference is originated from the involved breathing vibration in both torsional and longitudinal free vibration.

Axial and Torsional Free Vibrations of Elastic Nano-Beams by Stress-Driven Two-Phase Elasticity

Size-dependent longitudinal and torsional vibrations of nano-beams are examined by two-phase mixture integral elasticity. A new and efficient elastodynamic model is conceived by convexly combining the local phase with strain- and stress-driven purely nonlocal phases. The proposed stress-driven nonlocal integral mixture leads to well-posed structural problems for any value of the scale parameter. Effectiveness of stress-driven mixture is illustrated by analyzing axial and torsional free vibrations of cantilever and doubly clamped nano-beams. The local/nonlocal integral mixture is conveniently replaced with an equivalent differential law equipped with higher-order constitutive boundary conditions. Exact solutions of fundamental natural frequencies associated with strain- and stress-driven mixtures are evaluated and compared with counterpart results obtained by strain gradient elasticity theory. The provided new numerical benchmarks can be effectively employed for modelling and design of Nano-Electro-Mechanical-Systems (NEMS).

Formation of Surface Layers on Fe70Si15B15 Alloy upon Heating in an Inert Atmosphere with Air Impurities

The composition of surface layers formed during high-temperature interaction between Fe70Si15B15 alloy and air impurities in an inert atmosphere is determined using equilibrium thermodynamic means. It is shown that at T = 1473 K, Fe3O4, FeSiO3, and B2O3 are at the interface between the alloy and the gas. Intermediate layers of internal oxidation contain FeSiO3, B2O3, SiO2, and BN. The iron-based region adjacent to the alloy contains silicides Fe3Si and FeSi; borides Fe2B and FeB; and nitride BN. At a relatively low concentration of air impurities in the inert gas, the destruction of a surface film consisting mainly of SiO2 is detected in the temperature range of 1600–1800 K as an anomaly on the damping decrement polytherm of free torsional vibrations of the crucible with the melt.

Coupled Bi-Flexural–Torsional Vibration of Fluid-Conveying Pipes Spinning About an Eccentric Axis

This paper presents a dynamical model of a fluid-conveying pipe spinning about an eccentric axis. The coupled bi-flexural–torsional free vibration and stability are analyzed for such a doubly gyroscopic system. The partial differential equations of motions are derived by the extended Hamilton principle, and are then truncated by a 4-term Galerkin technique. The frequency and energy evolutions and representative mode shapes in the two transverse directions and torsional direction are investigated to unveil the essential dynamical attributes of the system. It is indicated that the stability of the present system mainly depends on spinning speed, flow velocity and eccentricity, while the torsional frequencies are almost immune to the flow velocity. The pipe reveals ‘traveling-wave’ modal vibrations in both transverse directions, and a general ‘standing-wave’ modal vibration in the torsional direction.

Coupled Bending–Bending–Axial–Torsional Vibrations of Rotating Blades

In this paper, the coupled bending–bending–axial–torsional free vibrations of rotating blades are investigated based on the Euler–Bernoulli beam model. The coupled partial differential equations governing flapwise, edgewise, axial and torsional motions are derived by the Hamilton’s principle, wherein three types of velocity-dependent terms, namely static centrifugal terms, dynamic centrifugal terms and gyroscopic coupling terms, are focused. The ordinary differential equations are acquired by the Galerkin truncation, and the natural frequencies in all directions and complex mode shapes of the rotating blades are analyzed in detail. It is revealed that the three types of velocity-dependent terms have different effects on the natural frequencies. The natural frequencies are noticeably dependent on the rotating speed and preset angle, except for the axial vibration, which is almost immune to the preset angle. The complex modal motions are displayed by a series of positions of the central line and free-end cross section for different time instants, showing the coupled vibrations among different directions.

실베스터-전달강성계수법에 의한 실습선 새동백호 추진축계의 비틀림 자유진동 해석

실베스터-전달강성계수법에 의한 실습선 새동백호 추진축계의 비틀림 자유진동 해석

Torsional vibration of cracked carbon nanotubes with torsional restraints using Eringen’s nonlocal differential model

Free torsional vibration of cracked carbon nanotubes with elastic torsional boundary conditions is studied. Eringen’s nonlocal elasticity theory is used in the analysis. Two similar rotation functions are represented by two Fourier sine series. A coefficient matrix including torsional springs and crack parameter is derived by using Stokes’ transformation and nonlocal boundary conditions. This useful coefficient matrix can be used to obtain the torsional vibration frequencies of cracked nanotubes with restrained boundary conditions. Free torsional vibration frequencies are calculated by using Fourier sine series and compared with the finite element method and analytical solutions available in the literature. The effects of various parameters such as crack parameter, geometry of nanotubes, and deformable boundary conditions are discussed in detail.

Analytical procedures for torsional vibration analysis of ship power transmission system

In this paper two relatively simple analytical procedures for free and forced torsional vibration analysis of ship power transmission systems are developed. In the first, approximate procedure, the shaft line is modelled as a two-mass system and analytical solution of the differential equations of motion is given. In the second one, a multi degree of freedom (d.o.f.) problem of the complete propulsion system is solved by the Rayleigh-Ritz method. A special attention is paid to the determination of the contribution of each cylinder to the primary and secondary engine torques by taking into account the firing order. The application of the two procedures is illustrated in the case of a typical propulsion system of a merchant ship with a slow-speed main engine connected directly to the propeller by a relatively short shaft line. The obtained results are verified by a comparison with measurements. All classification societies require calculation of the propulsion system operating parameters, but they do not provide simplified formulae for vibration analysis. The outlined analytical procedures can be used for the estimation of torsional vibration of the shaft line in the preliminary ship design stage as well as for ships in service.

Spectral Dynamic Analysis of Torsional Vibrations of Thin-Walled Open Section Beams Restrained Against Warping at One End and Transversely Restrained at the Other End

The present paper deals with spectral dynamic analysis of free torsional vibration of doubly symmetric thin-walled beams of open section. Spectral frequency equation is derived in this paper for the case of rotationally restrained doubly-symmetric thin-walled beam with one end rotationally restrained and transversely restrained at the other end. The resulting transcendental frequency equation with appropriate boundary conditions is derived and is solved for varying values of warping parameter and the rotational and transverse restraint parameter. The influence of rotational restraint parameter, transverse restraint parameter and warping parameter on the free torsional vibration frequencies is investigated in detail. A MATLAB computer program is developed to solve the spectral frequency equation derived in this paper. Numerical results for natural frequencies for various values of rotational and transverse restraint parameters for various values of warping parameter are obtained and presented in both tabular as well as graphical form showing the influence of these parameters on the first fundamental torsional frequency parameter.