Torsional vibrations of a shaft with non-uniform cross section

A solution is presented for the equation governing the propagation of torsional waves in shafts with variable cross sections. The solution permits one to treat several types of profiles, and in particular to compute the natural frequencies of an axisymmetrical shaft with various kinds of end conditions. Computed frequencies for three kinds of end conditions are compared with those of an equivalent cylindrical shaft, particularly with respect to the dependence on the geometrical characteristics.     

Free vibrations and stability of stepped columns with cracks

In this paper, a solution to the free vibration problem of a stepped column with cracks is presented. The open cracks occur at step changes in the cross-section of the column or at the intermediate points of the uniform segments. The cracks in the column are represented by massless rotational springs. The frequency equation is obtained by using properties of the Green's functions corresponding to the uniform segments of the column. The approach pertains to the vibration of columns consisting of an arbitrary number of uniform segments. The stability of the column, which depends on the position and size of the cracks, is numerically investigated.     

Coupled bending and twisting of a timoshenko beam

Allowance is made for shear deflection and for rotary inertia of a non-uniform beam that executes coupled bending and twisting vibration. Principal modes are found, orthogonality conditions established and modal equations of forced motion derived.     

On the weakly damped vibrations of a vertical beam with a tip-mass

In this paper the wind-induced, horizontal vibrations of a weakly damped vertical Euler-Bernoulli beam with and without a tip-mass will be studied. The damping is assumed to be boundary damping and global Kelvin-Voigt damping. The boundary damping is assumed to be proportional to the velocity of the beam at the top. The horizontal vibrations of the beam can be described by an initial-boundary value problem. In this paper, the multiple-timescales perturbation method will be applied to construct approximations of the solutions of the problem. Also it will be shown that a combination of boundary damping and Kelvin-Voigt damping can be used to damp the wind-induced vibrations of a vertical beam with tip-mass uniformly.     

Coupled bending and torsional vibration of a beam with in-span and tip attachments

In this paper, the coupled flexural-torsional free and forced vibrations of a beam with tip and/or in-span attachments are studied. First, a mathematical model is established, which consists of a beam with several tip attachments, i.e, a tip mass of non-negligible dimensions, a linear spring grounding the tip mass, and a torsional spring connected at the end of the beam. The modal functions of this model and the orthogonality condition among them are derived. For the purpose of verification the properties of the tip attachments are changed, and the numerical results obtained are compared with those given in the relevant literature. Effects of tip mass and distributed mass in-span on natural frequencies and modes are investigated for two cantilever beams with different cross sections. An application of the orthogonality condition in the case of a beam with tip mass is also presented for a forced vibration example.     

Localized bending vibrations of piezoelectric plates

This paper investigates the existence and propagation of electro-elastic bending waves localized at the free edge of a piezoelectric plate. The problem is considered within the framework of the high-order refined plate theory introduced by Ambartsumian. The condition for existence of a localized bending wave is obtained, and the dispersion equation solved with respect to a dimensionless frequency. It is shown that the piezoelectric effect can increase the attenuation coefficient for a localized wave by up to 70% compared with that for a purely elastic plate, thus significantly decreasing the depth of penetration. The problem is also solved within the classical Kirchhoff theory. A comparison of results is carried out between two theories.     

On variable length induced vibrations of a vertical string

The purpose of this paper is to study the free lateral responses of vertically translating media with variable length, velocity and tension, subject to general initial conditions. The translating media are modeled as taut strings with fixed boundaries. The problem can be used as a simple model to describe the lateral vibrations of an elevator cable, for which the length changes linearly in time, or for which the length changes harmonically about a constant mean length. In this paper an initial-boundary value problem for a linear, axially moving string equation is formulated. In the given model a rigid body is attached to the lower end of the string, and the suspension of this rigid body against the guide rails is assumed to be rigid. For linearly length variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to car mass, and for the harmonically length variations small oscillation amplitudes are assumed and it is also assumed that the string mass is small compared to the total mass of the string and the car. A multiple-timescales perturbation method is used to construct formal asymptotic approximations of the solutions to show the complicated dynamical behavior of the string. For the linearly varying length analytic approximations of the exact solution are compared with numerical solution. For the harmonically varying length it will be shown that Galerkin?s truncation method cannot be applied in all cases to obtain approximations valid on long timescales.     

Three-dimensional vibrations of cylindrical elastic solids with V-notches and sharp radial cracks

This paper provides free vibration data for cylindrical elastic solids, specifically thick circular plates and cylinders with V-notches and sharp radial cracks, for which no extensive previously published database is known to exist. Bending moment and shear force singularities are known to exist at the sharp reentrant corner of a thick V-notched plate under transverse vibratory motion, and three-dimensional (3-D) normal and transverse shear stresses are known to exist at the sharp reentrant terminus edge of a V-notched cylindrical elastic solid under 3-D free vibration. A theoretical analysis is done in this work utilizing a variational Ritz procedure including these essential singularity effects. The procedure incorporates a complete set of admissible algebraic-trigonometric polynomials in conjunction with an admissible set of “edge functions” that explicitly model the 3-D stress singularities which exist along a reentrant terminus edge (i.e., α>180°) of the V-notch. The first set of polynomials guarantees convergence to exact frequencies, as sufficient terms are retained. The second set of edge functions—in addition to representing the corner stress singularities—substantially accelerates the convergence of frequency solutions. This is demonstrated through extensive convergence studies that have been carried out by the investigators. Numerical analysis has been carried out and the results have been given for cylindrical elastic solids with various V-notch angles and depths. The relative depth of the V-notch is defined as (1−c/a), and the notch angle is defined as (360°−α). For a very small notch angle (1° or less), the notch may be regarded as a “sharp radial crack.” Accurate (four significant figure) frequencies are presented for a wide spectrum of notch angles (360°−α), depths (1−c/a), and thickness ratios (a/h for plates and h/a for cylinders). An extended database of frequencies for completely free thick sectorial, semi-circular, and segmented plates and cylinders are also reported herein as interesting special cases. A generalization of the elasticity-based Ritz analysis and findings applicable here is an arbitrarily shaped V-notched cylindrical solid, being a surface traced out by a family of generatrix, which pass through the circumference of an arbitrarily shaped V-notched directrix curve, r(θ), several of which are described for future investigations and close extensions of this work.     

Short-time dynamics of a packing of polyhedral grains under horizontal vibrations

We analyze the dynamics of a 3D granular packing composed of particles of irregular polyhedral shape confined inside a rectangular box with a retaining wall subjected to horizontal harmonic forcing. The simulations are performed by means of the contact dynamics method for a broad set of loading parameters. We explore the vibrational dynamics of the packing, the evolution of solid fraction and the scaling of dynamics with the loading parameters. We show that the motion of the retaining wall is strongly anharmonic as a result of jamming and grain rearrangements. It is found that the mean particle displacement scales with inverse square of frequency, the inverse of the force amplitude and the square of gravity. The short-time compaction rate grows in proportion to frequency up to a characteristic frequency, corresponding to collective particle rearrangements between equilibrium states, and then it declines in inverse proportion to frequency.     

Non-linear control of torsional and bending vibrations of oilwell drillstrings

Drillstring dynamics is highly non-linear in nature and its model can only be described by a set of non-linear differential equations. In addition to this complexity, the drillstring dynamics are not linearly controllable and thus linear control methods are not suitable for suppressing the coupled torsional and lateral vibrations of a rotating drillstring. In this paper a non-linear dynamic inversion control design method is used to suppress the lateral and the torsional vibrations of a non-linear drillstring. It was found that the designed controller is effective in suppressing the torsional vibrations and reducing the lateral vibrations significantly.     

Forced vibrations of a non-uniform thickness rectangular plate with two free sides

A forced vibration of a rectangular plate with thickness varying linearly in one direction is studied. The plate is simply supported along two opposite sides, and is free along the other sides. Approximate formulae are proposed for estimating the maximum deflection and surface stresses of a rectangular plate subject to a uniformly distributed harmonic lateral load. The accuracy of the formulae is discussed. The approximate formulae show the possibility that one can obtain solutions for the forced vibration of a plate with non-uniform thickness from those for the static bending of the same plate when the plate is subjected to a uniformly distributed load.     

Scale effects in the bending vibrations of helicopter rotor blades

Scale effects and dynamic similarity in the bending vibrations of helicopter rotor blades are examined by expressing the first three modes of bending vibration of a uniform, conventional rotor blade by a series of Legendre polynomials as suggested by Wilde and others. The natural frequency ratios for these three modes are then determined as functions of a dynamic similarity parameter over the entire range of designs and operating conditions from very flexible, rapidly rotating blades to highly rigid, slowly turning conditions.     

External and internal coupling effects of rotor's bending and torsional vibrations under unbalances

External and internal bending–torsion coupling effects of a rotor system with comprehensive unbalances are studied by analytical analysis and numerical simulations. Based on Lagrangian approach, a full-degree-of-freedom dynamic model of a Jeffcott rotor is developed. The harmonic balance method and the Floquet theory are combined to analyze the stability of the system equations. Numerical simulations are conducted to observe the bending–torsion coupling effects. In the formulation of rotordynamic model, two bending–torsion coupling patterns, external coupling and internal coupling, are suggested. By analytical analysis, it is concluded that the periodic solution of the system is asymptotically stable. From numerical simulations, three bending–torsion coupling effects are observed in three cases. Under static unbalance, synchronous torsional response is observed, which is the result of external coupling under unbalanced force. Under dynamic unbalance, two-time synchronous frequency torsional response is observed, which is the result of internal coupling under unbalanced moment. Under comprehensive unbalance, synchronous and two-time synchronous frequency torsional components are observed, which are the results of both external and internal couplings under unbalanced force and moment. These observations agree with the analytical analysis. It is believed that these observed phenomena should make sense in the dynamical design and fault diagnostics of a rotor system.     

The electromagnetic fields of a vertical and a horizontal electric dipole located above a single-layer half-space

In the article we consider the secondary electromagnetic fields of elementary electric sources above a single-layer half-space. The results of calculations from approximate formulas obtained in the article are compared with the strict solution. The calculations by the strict formulas were carried out on a computer.     

Design of an active suspension to suppress the horizontal vibrations of a spray boom

Spray-boom vibrations are one of the main causes of a non-homogeneous distribution of agro-chemicals. Yawing and jolting motions of the boom are most critical. A horizontal active suspension, reducing yawing and jolting is designed. The use of special and consequently expensive equipment hampers the breakthrough of active suspensions. Therefore the proposed active suspension is built from standard hydraulic equipment, suited for mobile applications. Coulomb friction and asymmetric behaviour introduces considerable non-linearities, complicating the derivation of a linear model and controller design. It is explained how to identify a model, approximating in the best-possible way the general linear behaviour of the system. By the H^∞ methodology and an iterative procedure, introduced in this paper, non-linearities are taken into account in the controller design without increasing the controller dimensions. The resulting controller is stable and shows good performance.     

Vibration analysis of horizontal self-weighted beams and cables with bending stiffness subjected to thermal loads

This paper aims at proposing an analytical model for the vibration analysis of horizontal beams that are self-weighted and thermally stressed. Geometrical nonlinearities are taken into account on the basis of large displacement and small rotation. Natural frequencies are obtained from a linearization of equilibrium equations. Thermal force and thermal bending moment are both included in the analysis. Torsional and axial springs are considered at beam ends, allowing various boundary conditions. A dimensionless analysis is performed leading to only four parameters, respectively, related to the self-weight, thermal force, thermal bending moment and torsional spring stiffness. It is shown that the proposed model can be efficiently used for cable problems with small sag-to-span ratios (typically , as in Irvine's theory). For beam problems, the model is validated thanks to finite element solutions and a parametric study is conducted in order to highlight the combined effects of thermal loads and self-weight on natural frequencies. For cable problems, solutions are first compared with existing results in the literature obtained without thermal effects or bending stiffness. Good agreement is found. A parametric study combining the effects of sag-extensibility, thermal change and bending stiffness is finally given.     

Universal and non-universal aspects of wet granular matter under vertical vibrations

The phase diagram of vertically vibrated wet granular matter is investigated by both experiments and molecular dynamics type simulations, with a focus on the coexistence regime of the fluid and gas phases. Phase diagrams measured at various parameters including the rupture energy of liquid bridges, are presented. While for elastic grains, the transition from the fluid to the fluid-gas coexistence phase is found to be determined only by the rupture energy of liquid bridges [Fingerle et al. New J. Phys. 10, 053020 (2008)], inelasticity is found to introduce non-universal features into the phase diagram, which are also affected by the grain size.