Free Torsional Vibration Research Articles

Testing of wood shear modulus based on cantilevered square plate torsional vibration method

ABSTRACT The application of the energy method has proposed a dynamic testing method for the shear modulus in two mutually perpendicular directions on the principal surface of wood using a cantilevered square plate torsional vibration method. In the study, this method was applied to measure the shear modulus of larch wood in the LT and TL directions, as well as the longitudinal and transverse shear modulus of Laminated Veneer Lumber (LVL) sheets. The validity of the results was confirmed through experimental verification using the asymmetric four-point bending beam method, the free plate torsional vibration method, and the free square plate torsional mode method. Simulation calculations of the principal shear modulus for six species of trees with width-to-thickness ratios ranging from 10 to 25 were also conducted using this method. The conclusions drawn suggest that a width-to-thickness ratio of 15 for the cantilevered square plate provides sufficient accuracy for testing the principal shear modulus of wood. Furthermore, the difference in shear modulus between two mutually perpendicular directions on the principal surface of wood is dependent on the width-to-thickness ratio of the cantilevered square plate, indicating that this method outperforms other dynamic methods for testing wood shear modulus.

Elastic medium and torsional spring effects on the nonlocal dynamic of functionally graded porous nanotubes

In this study, Eringen’s nonlocal elasticity theory that applies the small size effects in functionally graded porous nanotubes embedded in an elastic matrix is discussed. The material properties of functionally graded porous nanotubes are taken into account to vary over the radius direction with a rule of mixture. The free torsional vibration relation according to nonlocal elasticity theory, via Hamilton’s principle, is obtained and an eigenvalue solution is constructed for the free torsional vibration response of the presented work. The presented analytical model is validated by comparing the calculated mathematical results for homogeneous nanotubes with rigid and non-rigid boundary conditions. Special attention is given to deformable boundary conditions, porosity coefficient, material grading coefficient and also to the influence of elastic medium on the free torsional vibration frequencies. In this paper, it has been proven that the influence of length, elastic medium, elastic torsional spring rigidities, material grading and porosity coefficients on the vary in the torsional vibration frequency of the functionally graded nanotube is not small.

STODOLA-VIANELLO ITERATION METHOD FOR FREE TORSIONAL VIBRATION ANALYSIS OF MONOSYMMETRIC BOX-BEAM BRIDGES

Box-beam bridges are used in large spans and wider decks due to their high strength and greater torsional and flexural stiffness. They are usually prone to vibration due to moving vehicular traffic. Their eigenfrequency analysis is a crucial aspect of their design in order to ensure that the natural frequency is not close to the excitation frequency to avert resonance failures. The free torsional vibration equation is a fourth order partial differential equation (PDE) with variable parameters. For prismatic cross-sections, homogeneous and isotropic materials the governing PDE have constant parameters. This paper explores the Stodola-Vianello iteration method (SVIM) for solving the PDE for isotropic, homogeneous prismatic box-beams. Harmonic response is assumed, decoupling the PDE to two equations, one in terms of time and the second an ordinary differential equation (ODE) in terms of space coordinates. The Stodola-Vianello method is used by the method of four successive integrations to express the ODE as an algebraic iteration problem with four constants of integration. The four boundary conditions are used to solve for the four constants of integration, thus making the problem determinate. Application of the boundary conditions results in the full determination of the iteration equations. For simply supported boundaries studied in the work, a trigonometric buckling shape function that satisfies all the boundary conditions is employed in the SVIM formula to obtain the next bucking modal shape function. The requirement for convergence is then used to establish the characteristic buckling equation from which the eigenvalues are obtained. The solution to the characteristic buckling equation gave the exact mathematical expression for the natural torsional frequency at the nth vibration mode. The frequencies are determined for the first eight vibration modes and compared with previously obtained values. It was found that the natural frequencies are identical with previously found values of the exact natural frequencies. The natural frequency obtained is exact because it satisfies the PDE and the boundary conditions at all points in the domain.

Torsional static and vibration analysis of a non-circular restrained micro/nanowire

This paper aims to study the static and free torsional vibration of non-circular micro/nanowires using the modified couple stress theory with several deformable and non-deformable support conditions. The geometrical properties of the micro/nanowires are considered to have triangular, elliptical and circular profiles. Equation of motion in torsional direction is derived based on torsional equilibrium and systems of linear equations are computed by adopting Fourier sine series with Stokes’ transformation. Moreover, deformable boundary conditions have been taken into account to formulate eigenvalue problem by employing the higher-order force boundary conditions. Material length scale parameter and the shape of cross-section have considerable impacts on torsional vibration frequencies of restrained microwire. The obtained frequency curves of restrained microwires are shown to be dependent on various parameters such as the shape of cross-section, deformable boundaries, material length scale parameter and also warping function.

Torsional static and free vibration analysis of noncircular short‐fiber‐reinforced microwires with arbitrary boundary conditions

AbstractThis study investigates the free torsional vibration response of short fiber‐reinforced noncircular microwires by considering the size effect. Moreover, the formulation of Eigen value problem based on forced boundary conditions that consider warping function and small‐scale parameter. The composite microwire has been considered to be composed of fiber reinforced by short particle distributed along the thickness direction based on different symmetric patterns. The partial differential equation with force boundary conditions of a microwire under the general boundary conditions are presented in a matrix form and are solved analytically via the modified couple stress theory. An analytical solution is obtained for a microwire with arbitrary boundary conditions using the Fourier sine series with Stokes' transformation and effect of different parameters including warping function and elastic springs at the ends, material length scale parameter, distribution of short fibers on the free torsional frequencies are investigated.

Determination of the critical velocity of a straight wing with a high aspect ratio

An aerodynamic problem on an air flow around a large aspect ratio rectangular wing is investigated in this study. According to the theory of Vlasov, the wing is considered to be a thin rod. External loads are assumed to be proportional to the airfoil angle of attack related to the dimensionless coefficient of the lift and the pitching moment coefficient. These coefficients depend on the airfoil parameters and the Mach number M and are determined by experimental measurements for subsonic and supersonic velocities. In this case, to define the unstable cases of the wing, one bases on the Lyapunov stability theory. Equations of bending and torsional free vibrations have resulted. Based on the analysis of natural frequencies (eigenfrequencies), it is possible to determine the changing positions of the real part and the imaginary part of the characteristic equation solution. These positions can cause instabilities for the wing such as torsional divergence and flutter.

Axial and torsional free vibrations of restrained single-walled boron nitride nanotube (SWBNNT) embedded in an elastic medium via nonlocal strain gradient theory

The effect of the surrounding elastic matrix on the axial and torsional vibrations of embedded single-walled boron nitride nanotube (SWBNNT) is studied in this work. The SWBNNT is modeled as a nanorod and the nonlocal strain gradient theory is utilized to derive the size-dependent equation of motion. Also, the surrounding elastic matrix is simulated by employing a one-parameter foundation model. Stokes’ transformation method as an accurate and efficient mathematical tool in conjunction with the Fourier trigonometric series solution is utilized to discretize the equation of motion. These two mathematical methods create two different eigenvalue problems, which can solve any boundary conditions for torsional and axial vibrations. Calculated Eigen frequencies are validated with earlier works in the literature. The effects of the surrounded elastic matrix parameter together with the other deformable boundary coefficients and also the small scale parameter on the axial and torsional vibration frequencies are researched and some conclusions are outlined.

Research progress on dynamic testing methods of wood shear modulus: A review

Wood is a non-homogeneous and orthotropic natural polymer material. It is important to test the wood shear modulus and elastic constants accurately and reliably using dynamic methods. Based on the introduction of the advantages of six common methods for dynamic testing of wood shear modulus, such as free plate torsional mode method, free bar torsional vibration method, and Timoshenko beam iterative method, issues associated with the applicability and accuracy of these methods are also pointed out. Recent methods, such as the free square plate torsional mode method and the square plate static torsional strain method, that were developed to dynamically test the shear modulus of wood and wood composite materials, are presented as effective ways to tackle these issues. These new approaches are expected to provide beneficial technical support for using small specimens, overcoming the size effect of specimens, simplifying the testing procedures, improving the test accuracy, and expanding the application range in the dynamic testing of wood shear modulus. These approaches have practical significance in promoting the industrialization and development of structural engineering, furniture and interior decoration, transportation, military, and musical instrument industries.

Free Torsional Vibration Analysis of Nanorods with Non-circular Cross-Sections Based on the Second-Order Strain Gradient Theory

Free Torsional Vibration Analysis of Nanorods with Non-circular Cross-Sections Based on the Second-Order Strain Gradient Theory

Porosity dependent torsional vibrations of restrained FG nanotubes using modified couple stress theory

In the present study, size-dependent static and free torsional vibration responses of functionally graded porous nanotubes are examined using Fourier sine series and Stokes’ transformation for the first time. The boundary conditions of functionally graded porous nano-sized tubes are defined by the two elastic torsional springs at the both ends. A power law rule is utilized to describe the distribution of functionally graded material and this distribution is considered through the radius of nanotube. The governing equations of the mechanical response of porous nanotubes with elastic boundary conditions and subjected to torsion are accomplished via the modified couple stress theory. The purpose of the presented work is to construct an eigen value solution including the small scale parameter based on the modified couple stress theory, torsional spring coefficients representing the boundaries of the porous nanotubes, functionally graded index caused by power law rule and porosity volume fraction.

An indirect method to determine nonlinearly elastic shear stress-strain constitutive relationships for nonlinear torsional vibration of nonlinearly elastic shafts

PurposeIn this paper, the authors aim to propose an effective method to indirectly determine nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials, and then study the nonlinear free torsional vibration of Al–1%Si shaft.Design/methodology/approachIn this study the authors use BoxLucas1 model to fit the determined-experimentally nonlinear elastic normal stress–strain constitutive relationship curve of Al–1%Si, a typical case of isotropic nonlinear elasticity materials, and then derive its nonlinear shear stress-strain constitutive relationships based on the fitting constitutive relationships and general equations of plane-stress and plane-strain transformation. Hamilton’s principle is utilized to gain nonlinear governing equation and boundary conditions for free torsional vibration of Al–1%Si shaft. Differential quadrature method and an iterative algorithm are employed to numerically solve the gained equations of motion.Findings The effect of four variables, namely dimensionless fundamental vibration amplitude , radius and length , and nonlinear-elasticity intensity factor , on frequencies and mode shapes of the shafts is obtained. Numerical results are in good agreement with reference solutions, and show that compared with linearly elastic shear stress-strain constitutive relationships of the shafts made of the nonlinear elasticity materials, its actual nonlinearly elastic shear stress-strain constitutive relationships have smaller torsion frequencies. In addition, but having opposite hardening effect, the rest of the four variables have softening effect on nonlinearly elastic torsion frequencies. Eventually, taking into account nonlinearly elastic shear stress-strain constitutive relationships, changes of the four factors, i.e. , , and , cause inflation and deflation behaviors of mode shapes in nonlinear free torsional vibration.Originality/valueThe study could provide a reference for indirectly determining nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials and for structure design of torsional shaft made of nonlinear elasticity materials.

Torsional and longitudinal vibration analysis of a porous nanorod with arbitrary boundaries

In the present study, free torsional and axial vibrations of porous nanorods with torsional, axial elastic boundary conditions are presented via Eringen’s nonlocal elasticity theory. This theory takes into account the small size effect into the differential formulation due to dealing with nano-sized structures. Even though free axial and torsional vibrations based analysis of nanorod is a widely investigated topic, there are only few researches which exist in the scientific literature pertaining to the vibration analysis of porous nanostructures with nonrigid (deformable) boundary conditions. This study brings a contribution to the literature by presenting both porosity and small size effects with deformable boundary conditions. It is also planned to examine the effects of porosity and small size in detail. A model that allows to examine all of these effects within the framework is created within the scope of this study. To this end two torsional and axial springs are attached to the porous nanorod at both ends separately. Present model bridges the gap between deformable and non-deformable (rigid) boundary conditions. An eigen-value problem is constructed including torsional and axial coefficients by using Stoke transformation and the Fourier sine series. The novelty of this work is that it seeks to construct a general eigen value algorithm for the torsional and axial vibration frequencies of the porous nanorods subjected to the deformable and rigid boundary conditions for the first time. Moreover, the effects of some parameters such as porosity index, axial and torsional restraints on the free vibration frequencies are investigated and some conclusions are drawn.

Free Torsional Vibration Analysis of Hollow and Solid Non-Uniform Rotating Shafts Using Distributed and Lumped Modeling Technique

In this paper, the torsional free vibration of solid and hollow rotating shafts with non-uniform tapered elements are investigated. To this end, the exact solution and also transfer matrix for the free torsional vibration of a hollow tapered shaft element with uniform thickness and also solid element are firstly obtained. Then, the natural frequencies are determined based on distributed and lumped modeling technique (DLMT). This technique is similar to transfer matrix method (TMM) but the exact solution is employed to obtain the transfer matrixes of the distributed element, therefore, there is no approximation and the natural frequencies and mode shapes are the exact values. To confirm the reliability of the presented method, the simulation results are compared with the results obtained from the other methods such as finite element method. It is shown that the proposed method provides highly accurate results and it can be simply applied to the complex torsional systems.

Characteristic sensitivity analysis of herringbone gear power-split transmission system

ABSTRACT The herringbone gear power-split transmission is widely used in high-speed and heavy-load transmission systems, and its dynamic analysis is becoming increasingly important. The characteristic sensitivity of a gear transmission system indicates the influence of the dynamic parameters on the natural frequency of the gear transmission system, and it functions as an important theoretical reference value in the dynamic design of a herringbone gear power-split transmission system. In this paper, the dynamic equations of a herringbone gear power-split transmission system for a ship are presented, and the natural frequencies and modal shapes are calculated based on the free torsional vibration differential equation. Subsequently, the modal shapes were classified into the coupling vibration model and the branch vibration model. The characteristic sensitivity was studied using modal analysis, and the characteristic sensitivity calculation equations for single and double root of natural frequency were deduced, and the characteristic sensitivity of the natural frequencies to the mesh stiffness was computed. Finally, the relationship between the modal strain energy and the characteristic sensitivity was analysed, and the physics essence of the characteristic sensitivity was revealed.

Torsional vibration test and analysis of an internal combustion engine power assembly by RLV-5500

The vibration characteristics of marine power plant and propulsion system shafting directly affect the navigation safety of the ship. Shafting torsional vibration is an important parameter to evaluate the stability and reliability of shafting. The demand of ship shafting torsional vibration measurement is gradually developing towards high resolution and high accuracy. According to the characteristics of ship shafting installation and operation, this paper selects an internal combustion engine powertrain as the test object, and uses rlv-5500 rotary laser vibration measurement system to test the shafting torsional vibration. The time-frequency analysis of the test results is carried out to calculate the comprehensive angular displacement and the angular displacement amplitude of each harmonic order. The maximum comprehensive amplitude of crankshaft free end torsional vibration is 0.33 °. The crankshaft will not be damaged by torsional vibration.

Torsional static and vibration analysis of functionally graded nanotube with bi-Helmholtz kernel based stress-driven nonlocal integral model

A torsional static and free vibration analysis of the functionally graded nanotube (FGNT) composed of two materials varying continuously according to the power-law along the radial direction is performed using the bi-Helmholtz kernel based stress-driven nonlocal integral model. The differential governing equation and boundary conditions are deduced on the basis of Hamilton’s principle, and the constitutive relationship is expressed as an integral equation with the bi-Helmholtz kernel. Several nominal variables are introduced to simplify the differential governing equation, integral constitutive equation, and boundary conditions. Rather than transforming the constitutive equation from integral to differential forms, the Laplace transformation is used directly to solve the integro-differential equations. The explicit expression for nominal torsional rotation and torque contains four unknown constants, which can be determined with the help of two boundary conditions and two extra constraints from the integral constitutive relation. A few benchmarked examples are solved to illustrate the nonlocal influence on the static torsion of a clamped-clamped (CC) FGNT under torsional constraints and a clamped-free (CF) FGNT under concentrated and uniformly distributed torques as well as the torsional free vibration of an FGNT under different boundary conditions.

Analytical study on free vertical and torsional vibrations of two- and three-pylon suspension bridges via d'Alembert's principle

This study derives the differential equations of free vertical bending and torsional vibrations for two- and three-pylon suspension bridges using d'Alembert's principle. The respective algorithms for natural vibration frequency and vibration mode are established through the separation of variables. In the case of the three-pylon suspension bridge, the effect of the along-bridge bending vibration of the middle pylon on the vertical bending vibration of the entire bridge is considered. The impact of torsional vibration of the middle pylon about the vertical axis on the torsional vibration of the entire bridge is also analyzed in detail. The feasibility of the proposed method is verified by two engineering examples. A comparative analysis of the results obtained via the proposed and more intricate finite element methods confirmed the former feasibility. Finally, the middle pylon stiffness effect on the vibration frequency of the three-pylon suspension bridge is discussed. It is found that the vibration frequencies of the first- and third-order vertical bending and torsional modes both increase with the middle pylon stiffness. However, the increase amplitudes of third-order bending and torsional modes are relatively small with the middle pylon stiffness increase. Moreover, the second-order bending and torsional frequencies do not change with the middle pylon stiffness.

Closed form solution for dynamic analysis of rectangular nanorod based on nonlocal strain gradient

ABSTRACT In this study, the free torsional vibration of the nanorod with noncircular cross-section based on the nonlocal strain gradient theory is investigated. The rectangular cross-section has been selected. The governing partial equation of the torsional vibration of the noncircular nanorod is derived using Hamilton’s principle. Using a Galerkin method, and by consideration of the rectangular shape function in series, the natural frequency of noncircular nanorod is obtained. The obtained results indicate the convergence of the nondimensional natural frequency of the nanorod by enhancement of the number of the rectangular shape function. Furthermore, the nondimensional natural frequency of the noncircular nanorod for different nondimensional nonlocal parameter and nondimensional material length scale parameter is studied. Eventually, the effects of both nonlocal parameter and material length scale parameter on the nondimensional natural frequency, which causes torsional stiffness softening and torsional stiffness hardening of the nanorod, is also illustrated.

Torsional Vibration Analysis of Carbon Nanotubes Using Maxwell and Kelvin-Voigt Type Viscoelastic Material Models

Torsional dynamic analysis of viscoelastic Carbon Nanotubes (CNT) has been carried out in the present work. Maxwell and Kelvin-Voigt type viscoelasticity are considered in the modeling of viscoelastic material. Nonlocal Elasticity Theory is used in the formulation of governing equation of motion and boundary conditions. Viscoelasticity and nonlocal effects of structure on the free torsional vibration of CNTs have been investigated. Clamped-clamped and clamped-free boundary conditions are considered. Present study results could be useful in design of nano-medicine delivery applications.

Analytical investigation on free torsional vibrations of noncircular nanorods

This paper is devoted to the free torsional behavior of the nanorods containing noncircular cross sections. The rectangular cross section is chosen to be the case of the study. Three various boundary conditions, namely the clamped–clamped (C–C), clamped–free (C–F), and clamped–torsional spring (C–T) boundary conditions, are used to model the nanorod. Hamilton’s principle is utilized to derive the equation of motion along with associated boundary conditions. The derived equation is reformulated by Eringen’s nonlocal elasticity approach to exhibit the small-scale effect. An analytical method is established to discretize and analyze the equation of motion. The novelty of this work is the analysis of the torsional vibration in rectangular nanorods, which are not observed in previous works. For the results, the influences of the horizontal and vertical aspect ratios ( $$a/b$$ and $$b/a$$ ) (for C–C and C–F boundary conditions) and the influences of the nonlocal parameter and stiffness of the boundary spring (for C–T boundary condition) are illustrated schematically and tabularly.