Non-uniform Rods Research Articles

Stiffness parameter prediction for elastic supports of non-uniform rods

The present research focuses on establishing the stiffness parameter of elastic springs placed at the ends of non-uniform rods. The governing equation for the longitudinal vibrations of the rod was solved using the Haar wavelet integration method. The calculated natural frequency parameters closely aligned with those available in the literature. The normalised values of the first ten natural frequency parameters were used in the feature vector to predict the stiffness parameter of the springs. A feedforward neural network with two hidden layers made accurate predictions when the range of each natural frequency parameterwithin its domain exceeded one. The insights garnered from this study contribute to the design, optimisation and assessment of diverse engineering applications.

Exact general solutions for the mode shapes of longitudinally vibrating non-uniform rods via Lie symmetries

A Lie symmetry method-based approach is proposed for systematically computing general solutions in closed-form for the mode shape equation of non-uniform and unconventional vibrating rods. The mode shape equation is modeled by the elementary rod theory, addressing polynomial, exponential, trigonometric, and hyperbolic cross-section variations. The method provides algorithmic order-reduction steps for solving the investigated mode shape equation, producing a first-order Riccati equation whose integration reveals the aimed solutions for the problem. Illustrative examples are presented, including original solutions in closed-form as well as solutions previously obtained in the literature by other approaches. Mode shapes from general solutions with appropriate rod boundary conditions are also considered for different examples. • Original general solutions establish new exact mode shapes of non-uniform rods. • The Lie symmetry method allows addressing new cross-section variations systematically. • Mode shapes derive from the general solutions with rod boundary conditions. • Lie symmetry applications have been highlighted for further problems of interest.

Numerical investigation of geometrically nonlinear clamped uniform rods and rods with sections varying exponentially free vibration

The present paper is intended to investigate the problem of linear and non-linear longitudinal free vibration of uniform rods and rods whose cross-sections vary exponentially at large vibration amplitudes. The method adopted consists in discretizing the energy term on linear kij and non-linear rigidity tensor bijkl, as well as the mass tensor mij. Therefore, the formulation of this structure is based on Lagrange equations and the harmonic balance method so as to obtain the nonlinear algebraic equations. These latter are solved numerically and analytically through the explicit and linearized method. The response of Clamped-Clamped uniform and non-uniform rods on our structure are highlighted in the amplitude frequency and associated first three mode shapes. Moreover, this research leads to study the influence of the exponential slope on the maximum displacement, thus emphasizing the non-uniform bars usefulness. The obtained results are then compared with the available literature with a view to validating this theory. As a perspective, the method used in this paper would be pushed to study the FDM material, taking into account other parameters related to additive manufacturing, and later to be validated experimentally. Longitudinal vibrations are important in mechanical structures; therefore, the determination of their dynamic behaviour needs to be understood. In the present study, the effect of the displacement amplitude on the exponential slope of the structure was analysed, which led to the determination of the reduction range of the vibration amplitude under resonance. However, this should be taken into account in the design process. Besides, the usefulness of the non-linearity geometric effects was demonstrated to examine these structures by considering all the parameters involved. A linearized procedure is used to solve a nonlinear algebra equation. The use of this method leads to reduce calculation time contrary to iterative methods.

Comparative study of nickel selenide, iron selenide and platinum on triiodide reduction for dye-sensitized solar cells

In this work, nickel selenide (NiSe2) and iron selenide (FeSe2) were synthesized by a simple and mild hydrothermal method. The selenides were characterized by X-ray diffraction (XRD), energy dispersive X-ray spectroscopy (EDX), field-emission scanning electron microscopy (FESEM) and high-resolution transmission electron microscopy (HRTEM). XRD and EDX studies show the compositions of the products. The NiSe2 presents an irregular flaky-like structure whereas the FeSe2 forms a non-uniform rods structure. The catalytic effects of NiSe2 and FeSe2 in comparison to Pt were examined by cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS). The FeSe2 exhibits the highest catalytic activity and the lowest charge transfer resistance, followed by Pt and NiSe2. The photovoltaic and long-run performance of dye-sensitized solar cells (DSSCs) with poly(ε-caprolactone) (PCL)-dimethylformamide (DMF)-potassium iodide (KI) gel polymer electrolytes were investigated with respect to NiSe2, FeSe2 and Pt counter electrodes. The efficiencies attained for NiSe2, FeSe2 and Pt counter electrodes are 6.05, 7.20 and 6.99%, respectively. The FeSe2 demonstrates better corrosion resistance to the iodide/triiodide (I−/I3−) electrolyte than Pt.

Cooperative parameter estimation of a nonuniform payload by multiple quadrotors

AbstractThispaper addresses the problem of carrying an unknown nonuniform payload by multiple quadrotor agents. The load is modeled as a rigid body with unknown weight and position of Center of Gravity (CG) for the agents, and is included in their dynamic equations of motion. The agents and the load are assumed to be connected to each other by taut ropes. The Udwadia–Kalaba equation is used to calculate the constraint forces on the ropes acting on each quadrotor. Inner-loop and outer-loop controllers for quadrotors position and attitude control are presented. For the outer loop, an estimation algorithm based on the invariance and immersion adaptive control is utilized to estimate the unknown physical parameters of the payload including mass and CG position without using multi-axes force/torque sensors. The inner-loop controller employs an adaptive controller. Simulation results, for two and four agents carrying a nonuniform rod and cubic payload, show the effectiveness of the proposed algorithm. A case study is also performed to investigate the effect of quadrotors positioning on flight endurance of the cooperative aerial team carrying a nonuniform payload.

Large-mode-area all-solid anti-resonant fiber with single-mode operation for high-power fiber lasers.

We investigate the feasibility of applying an anti-resonant guiding mechanism in an all-solid anti-resonant fiber (AS-ARF) to achieve a large mode area (LMA) and single mode for high-power fiber laser applications. A novel, to the best of our knowledge, AS-ARF with nonuniform rods is proposed to enhance the single-mode property and enlarge the mode area. The numerical results show that the core diameter can expand to 57, 80, and 100 µm at the wavelengths of 1.064, 1.55, and 2 µm, respectively. The loss ratio of the lowest loss of higher-order modes to the loss of the fundamental mode can exceed 1000, 550, and 860 at the wavelength of 1.064, 1.55, and 2 µm; thus, robust single-mode operation can be ensured. Besides, the fiber can also be adapted to bent condition under certain heat load. These indicate that the proposed AS-ARF with nonuniform rods is a great candidate as an LMA fiber for high-power fiber lasers.

Research of Stability and Seismic Resistance of Non-Uniform Two-Layer Rods in Elastic Medium

Simulation considered after the main seismic impacts are removed from the structure, the two-layer rod, which is in an elastic environment, is subjected to periodic longitudinal forces. In the investigated work the Bubnov”s-Galerkin method was used, numerical calculations were made. The main area of dynamic instability is built and shown in the figure, where the dotted line marks the solution of a homogeneous problem.

An accurate and efficient series solution for the longitudinal vibration of elastically restrained rods with arbitrarily variable cross sections

Longitudinal vibration of non-uniform rod has been of great significance in various engineering occasions. The existing works are usually limited to the certain area variation and/or classical boundary condition. Motivated by this limitation, an efficient accurate solution is developed for the longitudinal vibration of a general variable cross-section rod with arbitrary boundary condition. Displacement function is invariantly expressed as the summation of standard Fourier series and supplementary polynomials, with an aim to make the calculation of all derivatives more straightforwardly in the whole solving region [0, L]. Energy principle is employed for system dynamics formulation, with the elastic boundaries considered as potential energy stored in the restraining spring. Arbitrary cross-section area variation is uniformly expanded into Fourier series. Numerical examples are presented for the natural frequency and mode shapes of non-uniform rod of free and clamped boundary conditions and compared with literature data. Results show good agreement with the previous analytical solutions. Influence of cross section area variation on vibration characteristics of non-uniform rods is then studied and discussed. One of the most advantages of the proposed model is that there is no need to reformulate the problem or rewrite the codes when the cross-section area distribution and/or boundary conditions change arbitrarily.

Investigation of nonuniform rod elements stability by direct integration method

The calculation for the equilibrium stability of a rod system with variable stiffness by the direct integration method is proposed. The method is based on the exact solutions of the corresponding differential equations. Using this method, the rod stability problem with arbitrary continuous variable flexural stiffness is solved. The formulas for the parameters of the frame structure stability are expressed in the analytical form and the method of their numerical implementation is provided. The frame structures with binomial distribution of stiffness are considered.

Elastic wave propagation in non-uniform rational B-spline rods under mechanical impact loading using an isogeometrical approach

One-dimensional elastic wave propagation under quadratic impact loading in a rod with a variable cross section and material distribution is the subject of this study. The material distribution of the problem under investigation as well as the variations of the geometry was elucidated with non-uniform rational B-spline (NURBS). The problem was analyzed employing the isogeometric approach in order to ensure precise modeling of the geometry. The effects of impact loading, cross sectional area, material distribution, and radial inertia on elastic waves were examined in this study. In addition, propagation, reflections, and propagation speed along the axis were investigated. It was observed that the speed was not constant along the axial direction. Also, the cross sectional area had more effect on the amplitude of the elastic wave than the radial inertia. Furthermore, it was concluded that the material distribution and radial inertia may influence the wave propagation speed.

Torsion of anisotropic and non-uniform cylindrical rods with elliptical section

In work the limit state of cylindrical and prismatic rods from anisotropic ideal rigid-plastic material is investigated under torsion for arbitrary condition of plasticity, and the torsion of anisotropic and non-uniform rods with elliptic section under the condition of Mises-Hill plasticity is considered. The integrals determining the stressed state of an anisotropic rod under arbitrary condition of plasticity are obtained, the field of characteristics of the basic ratios for anisotropic and composite rods under the condition of Mises-Hill plasticity is constructed, the ratios along characteristics are obtained, the envelopes of the family of characteristics and lines of tension rupture are found.

Vibration of non-uniform rod using Differential Transform Method

The present paper analyses the vibration characteristics of non-uniform rods using Differential Transform Method. The method is very effective in solving ordinary and partial differential equations. The governing equations are transformed into a set of polynomials and the solution of these algebraic equations gives the desired frequency. Method is implemented for fix-free and fix-fix end conditions of a non-uniform rod. It is seen that the lower modes are sensitive to taper ratio in case of fix-free rod and fix-fix case frequency is same as that of uniform rod for different taper ratios. The method has proved to be accurate, simple and effective for eigenvalue analysis.

Improved modal convergence using the assumed modes method for rods carrying various lumped elements

In this paper, the assumed modes method is used to determine the modes of vibration of an arbitrarily supported uniform and nonuniform rods carrying various lumped elements, including a lumped mass, a grounded spring, a grounded viscous damper, and an undamped and damped oscillator. In applying the assumed modes method, the set of trial functions used in the expansion can be arbitrary as long as they satisfy the geometric boundary conditions of the system. In practice, the trial functions are often selected to correspond to the eigenfunctions of the bare uniform rod. Numerical experiments show that while this set of trial functions converges to the exact results, the rate of convergence can be exceedingly slow. In order to expedite modal convergence, the eigenfunctions are augmented with piecewise linear functions that capture the slope discontinuities of the mode shapes at the attachment locations due to the presence of the lumped elements. The results obtained using the two sets of trial functions are compared with those obtained exactly. It is shown that including the piecewise linear functions significantly improves the accuracy of the modes of vibration of the system while drastically reducing the computational time and effort.

Exact Eigensolutions for a Family of Nonuniform Rods With End Point Masses

In this paper, new exact closed-form solutions for free longitudinal vibration of a one-parameter countable family of cantilever rods with one end tip mass are obtained. The analysis is based on the reduction of the equation governing the longitudinal vibration to the Sturm–Liouville canonical form and on the use of double Darboux transformations. The rods for which exact eigensolutions are provided are explicitly determined in terms of an initial rod with known closed-form eigensolutions. The method can be also extended to include longitudinally vibrating rods with tip mass at both ends.

Inverse spectral problem for a non-uniform rod with multiple cracks

An inverse spectral problem for a rod of variable cross-section containing a finite number of transverse cracks is considered. The cracks are simulated by translational springs. It is proved that the number of springs, their locations and flexibilities are uniquely reconstructed by two spectra corresponding to longitudinally vibrating rod with free-free and free-fixed end conditions. A constructive procedure for reconstructing unknown damage parameters is developed. Numerical examples are considered.

Heat propagation in a nonuniform rod of variable cross section

An integral formula is used to average a coupled problem of thermoelasticity for a nonuniform rod of variable cross section. Effective characteristics are found. It is shown that, in addition to the expected effective coefficients, there appear five independent coefficients characterizing the temperature change rate effect on the stresses in the rod, on the longitudinal heat flux, and on the entropy distribution along the length of the rod. A feature of these new coefficients is that they become equal to zero in the case of a uniform rod. The homogenization of the thermoelasticity equations for nonuniform rods allows one to propose a new theory of heat conduction in rods. This new theory differs from the classical one by the fact that some new terms are added to the Duhamel–Neumann law, to the Fourier heat conduction law, and to the entropy expression. These new terms are proportional to the temperature change rate with time. It is also shown that, in the new theory of heat conduction, the propagation velocity of harmonic heat perturbations is dependent on the oscillation frequency and is finite when the frequency tends to infinity.

Longitudinal vibrations of viscoelastic longitudinally nonuniform rod under power load distributed along its length

In this article, the dynamic behavior of a moving elongated object is simulated using the relations for a viscoelastic rod, which moves at a constant speed under the action of traction force and distributed along its length external forces of resistance. We investigate the change of displacements and internal forces after the sudden application in the rod section of the local force directed longitudinally. The correlations of the initial boundary value problem that describes the dynamic behavior of the rod are written down, and its solution is obtained in the form of a series expansion in terms of eigenfunctions. For a viscoelastic rod consisting of three connected uniform rods, the analysis of wave processes induced by the application to the rod of a sudden concentrated force that resists the motion is carried out. This affects the motion of the rod as a whole, and induces the wave processes, the propagation and reflection of waves on the inner surfaces of joints. The comparison is performed for the behavior of an elastic, piecewise nonuniform rod and a viscoelastic rod with different mechanical characteristics, where the waves during their propagation are damped and smoothed. Динамiчну поведiнку рухомого видовженого об’єкта змодельовано з використанням спiввiдношень для в’язко-пружного стрижня, котрий рухається зi сталою швидкiстю пiд дiєю сили тяги та розподiлених по його довжинi сил зовнiшнього опору. Дослiджено змiну перемiщень i внутрiшнiх сил пiсля раптового прикладання в його перерiзi локальної сили у поздовжньому напрямку. Записано спiввiдношення початково-крайової задачi, що описує динамiчну поведiнку стрижня, та отримано ї ї розв’язок у виглядi розвинення в ряд за власними функцiями. Для в’язко-пружного стрижня, що складається з трьох з’єднаних однорiдних стрижнiв, проведено аналiз хвильо-вих процесiв, викликаних раптовим прикладанням в областi стрижня зосередженої сили, що чинить опiр руховi. Це впливає на рух стрижня як цiлого, а також викликає хвильовi процеси, проходження i вiдбивання хвиль на внутрiшнiх поверхнях з’єднань. Приведено порiвняння поведiнки пружного кусково-неоднорiдного стрижня та в’язко-пружного стрижня з рiзними механiчними характеристиками, де хвилi пiд час поширення загасають i згладжуються.

Erratum for “Numerical Analysis of Free Longitudinal Vibration of Nonuniform Rods: Discrete Singular Convolution Approach” by Mohammad Shokrollahi and Amir Zayeri Baghlani Nejad

C-F −2.5 4.884 2.96 × 10−02 3.455 8.28 × 10−02 2.904 1.17 × 10−01 1.674 2.59 × 10−01 −0.5 3.233 −4.78 × 10−03 2.109 3.06 × 10−02 1.684 4.98 × 10−02 0.812 −5.66 × 10−03 0.5 2.350 −1.65 × 10−02 1.432 −5.79 × 10−02 1.100 1.42 × 10−02 0.470 1.39 × 10−03 2.5 0.864 2.99 × 10−03 0.424 2.87 × 10−02 0.295 5.03 × 10−03 0.101 4.05 × 10−02 C-C −2.5 5.286 5.77 × 10−02 4.486 9.52 × 10−02 4.316 1.02 × 10−01 4.353 8.18 × 10−02 −0.5 3.622 5.47 × 10−03 3.309 4.21 × 10−02 3.384 3.95 × 10−02 4.015 1.88 × 10−03 0.5 2.817 −3.78 × 10−03 2.906 −2.94 × 10−03 3.154 2.59 × 10−03 4.115 −2.03 × 10−03 2.5 2.030 3.89 × 10−02 3.074 5.56 × 10−02 3.598 3.99 × 10−02 5.012 2.38 × 10−03

Crack identification in non-uniform rods by two frequency data

We consider the inverse problem of identifying a single open crack in a longitudinally vibrating rod having non-uniform smooth profile. Without any a priori assumption on the smallness of the damage and assuming that the rod profile is symmetric with respect to the mid-point of the rod axis, we present a constructive diagnostic algorithm from minimal frequency data. We show that the crack can be uniquely identified, up to a symmetric position, from the first two positive natural frequencies of the rod under free–free end conditions. We also show that the non-uniqueness of the damage location can be removed by using as data the first positive resonant frequency of the free–free rod and the first antiresonant frequency of the driving-point frequency response evaluated at one end of the rod. The results of numerical simulations and of applications of the method to experimental data agree well with the theory.

Longitudinal vibrations of arbitrary non-uniform rods

Free and steady-state forced longitudinal vibrations of non-uniform rods are investigated by an iteration method, which results in a series solution. The series obtained are convergent and linearly independent. Its convergence is verified by convergence tests, its linear independence confirmed by the nonzero value of the corresponding Wronski determinant. Then, the solution obtained is an exact one reducible to a classical solution for the case of uniform rods. In order to verify the method, two examples are presented as an application of the proposed method. The results obtained are equivalent to the method in literature. In contrast to the proposed method capable of dealing with arbitrary non-uniform rods in principle, the method in literature is confined to work on special cases.