Vibration energy harvesting problems have strongly developed in recent years. However, many researchers just consider bending vibration models of energy harvesters. As a matter of fact, torsional vibration is also important and cannot be ignored in many cases. In this work, closed-form solutions of bending-torsion coupled forced vibrations of a piezoelectric energy harvester subjected to a fluid vortex are derived. Timoshenko beam model is used for modeling the energy harvester, and the extended Hamilton’s principle is used in the modeling process. Since piezoelectric effects in both bending and torsional directions are considered, two kinds of electric coupling effects appear in forced vibration equations, and a new model for the electric circuit equation is developed. Lamb–Oseen vortex model is considered in this study. Both the external aerodynamic force and moment are simple harmonic loads. Three damping coefficients are considered in the present model. Based on Green’s function method, closed-form solutions of the piezoelectric energy harvester subjected to the fluid vortex are derived. Some published results are used to verify the present solutions. It can be concluded through analysis that when torsional vibration is considered, the bandwidth of the high energy area of the voltage becomes large and the bending-torsion coupled vibration energy harvester can produce more power than a transverse vibration energy harvester.

In this paper, the mode coupling between bending, stretching, and torsional deformations is mainly studied by presenting an analytical model of a rotating cantilever beam with pre-twist angle and arbitrary cross section. Equations of motion of the beam are derived using Hamilton's principle. The Coriolis effect due to the coupling of the bending deformation and stretching deformation, the eccentricity caused by inconsistency between elastic center and centroid, spin softening effect, stress stiffening effect, shear deformation, and rotary inertia are included in the model. Equations of motion are solved by the Rayleigh-Ritz method. The natural frequencies obtained by the proposed analytical modal are in good agreement with those obtained by finite element method (FEM) which proved the accuracy of the analytical model. Finally, the coupling between different mode components is studied in detail based on a quantitative method. The transformation/conversion between different mode components is revealed, the influence of rotational speed, setting angle, and pre-twist angle on this conversion mode is studied. Results show that a specific mode shape is usually composed of multiple mode components. The essence of mode coupling is the coupling between different mode components. The influence of rotational speed, setting angle, and pre-twist angle on the mode coupling is that they cause the transformation/conversion between different mode components.

Many driveline systems are designed to accommodate angular misalignment by the use of flexible couplings or Universal Joints (U-Joints) which link individual shaft segments. The Sommerfeld effect is a nonlinear phenomenon observed in some rotor systems being driven through a critical speed when there is not enough power to accelerate the rotor through resonance. Previous studies have shown that rotor speed can become captured when transitioning through natural frequencies due to nonlinear interactions between a non-ideal driving input and rotor imbalance. This paper, for the first time, shows that this type of rotor speed capture phenomena can also be induced by driveline misalignment. During rotor spinup under constant motor torque, it is found that misalignment-induced rotor speed capture phenomena can occur as the shaft speed approaches ½ the first elastic torsional natural frequency. Depending on misalignment level and motor torque, the shaft speed will either dwell near this speed and then pass through, or the speed will become trapped. Here, a nonlinear rotordynamics model of a segmented driveshaft connected by two U-joints including effects of angular misalignment and load torque is developed for the study. This analysis also determines the minimum driveline misalignment angle for which the shaft speed capture phenomena will occur for a given motor torque and load torque condition.

For damping of fully coupled beam vibrations, a three-component tuned mass absorber is proposed, consisting of two translational and a single-rotational absorber, which together target a specific vibration mode. The representation of the interaction with residual nonresonant vibration modes is consistently included by two supplemental terms in the absorber equation, calibrated by frequency matching with the full structural model for vanishing absorber damping. The absorber tuning directly includes these residual terms without approximation when the underlying modal representation is based on the vibration form with absorber masses rigidly attached to the structure. It is demonstrated that the desired damping behavior requires nonhomogeneous absorber balancing, obtained by an approximate method for the relative sizing of each absorber mass.

In this paper, the authors present a discrete system model to study the coupled axial–torsional dynamics of a drill string. The model is developed taking into account state-dependent time delay and nonlinearities due to dry friction and loss of contact. Simulations are carried out by using a 32-segment model with 128 states. Bit bounce is observed through time histories of axial vibrations, while stick-slip phenomenon is noted in the torsion response. The normal strain contours of this spatial–temporal system demonstrate the existence of strain wave propagation along the drill string. The shear strain wave exhibits features of wave nodes and wave loops along the drill string, which indicate that the torsional motion has the properties of a standing wave. When the penetration rate is varied, qualitative changes are observed in the system response. The observed behavior includes chaotic and hyperchaotic dynamics. Stability analysis reveals a stable region for the degenerate one-segment model. This stable region becomes infinitesimally small, as the resolution of spatial discretization is increased. This finding suggests that drill-string motions have a high likelihood of being self-exited in practical drilling operations.

The purpose of this paper is to study the dynamics of an electromechanical system consisting of a torsion-bar or two mechanical pumps activated by an electromotor. Oscillatory solutions showing the jump and hysteresis phenomena are obtained using the harmonic balance method and direct numerical simulation. Chaotic behavior is presented via the bifurcation diagrams and corresponding Lyapunov exponent. Some implications of the results on the applications of the devices are discussed.

In this article, reduced-order models of a drill-string system are developed, the predictions of these models are studied, and qualitative comparisons are made with experimental studies. The reduced-order models allow for radial, bending, and torsion motions of a flexible drill string and stick-slip interactions between the drill string and an outer shell. Qualitative changes in the system motions are studied with respect to the rotation speed of the drill string and the friction coefficient between the drill string and the outer shell. The nonlinear motions predicted by the model are in agreement with the experimental observations, and these studies suggest that there may be preferred friction coefficient values for which most of the drill-string motions occur close to the center of a borehole. These modeling and experimental studies can serve as an important basis for furthering our understanding of drill-string dynamics and schemes for controlling them.

Embedded sensitivity analysis has proven to be a useful tool in finding optimum positions of structure reinforcements. However, it was not clear how sensitivities obtained from the embedded sensitivity method were related to the normal mode, or operational mode, associated to the frequency of interest. In this work, this relationship is studied based on a finite element of a slender sheet metal piece, with preponderant bending modes. It is shown that higher sensitivities always occur at nodes or antinodes of the vibrating system.

This paper presents a wave-based modeling and control approach for suppressing torsional vibrations of an elastic shaft driven by a motor at one end and an inertial load at the other end. A two-port network model representing the dynamics of torsional displacements is obtained starting from the partial differential equations governing a shaft in torsion. By incorporating appropriate boundary conditions, the infinite dimensional transfer function of the system is obtained. Furthermore, the system is represented by delay elements that can be used for simulation purposes. An inversion-based controller that can be used to suppress vibrations while rotating the shaft according to a specific trajectory is then developed. The performance of the controller is further studied using numerical simulations.

This article is devoted to the study of the local contact between the drillstring and the well during drilling operations. The study focuses on the Bottom-Hole-Assembly (BHA), which is subjected to compression. The work is motivated by the need to understand the complex behavior of such a system, in order to improve control over their constructive and destructive potentials. The contact zone is first determined using a global finite-element model obtained from a specific computer program. Contact, which is assumed to be located somewhere on the drill-collar or on stabilizers, is prejudicial and leads to premature abrasive wear of the drillstring, reduction of the Rate Of Penetration (ROP) of the tool into the rock and reduction of the Mean Time Between Failure (MTBF). The proposed mathematical model is expressed in terms of four independent degrees of freedom which are radial displacement, rotation of the section considered, bending along the tangential direction and torsion of the string. They include the effects of bending and torsion, the whirling motion of the drillstring as well as friction phenomena occurring between the drillstring and the well. The tangential effect is modeled by using Coulomb’s law of friction. The nonlinear equations of movement are derived using Lagrange equations and are solved numerically to obtain the response. Specific attention is paid to the study of friction and a consistent contact model capable of taking into account the rolling of the drillstring, both with and without slip, is included in the model. This paper also presents a parametric study on the influence of the initial position of the string and the friction coefficient of the contact on the dynamic behavior of the structure. The model is validated by an experimental set-up equipped with two opto-electronic devices.

In previous work, Mockensturm and Mote investigated the effects of twist on steady motions of an axially moving plate. It was found that twisting produces compressive stresses that increase with twist, aspect ratio, and initial, longitudinal tension. In this work, we study the effects of the stresses and non-flat equilibrium produced by twist on the free response. To accomplish this, the equations of motion are linearized about the equilibrium configuration, yielding a set of three, coupled, linear partial differential equations. The equations are discretized and the free response is predicted from the resulting discrete eigenvalue problem. As a function of twist angle, the natural frequencies first increase and then decrease rapidly to zero as the compressive lateral stresses become sufficiently large to cause wrinkling. The effects of thickness, aspect ratio, and initial tension on natural frequencies are also studied.

The dynamic behavior of a simply-supported spinning Timoshenko shaft with coupled bending and torsion is analyzed. This is accomplished by transforming the set of nonlinear partial differential equations of motion into a set of linear ordinary differential equations. This set of ordinary differential equations is a time-varying system and the solution is obtained analytically in terms of Chebyshev series. A beating phenomoenon is observed from the numerical simulations, which is not observed for shaft systems where only bending vibration is considered.

This paper focuses on laboratory tests concerned with the lateral behavior of a rod representative of part of a drill-string in the area of rotary oil drilling. The original experimental set-up takes into account the curvature of the rod, mud, stabilizers and rotation speed. The lateral behavior of the drill-string subjected to the axial excitations of the bit is governed by time varying parameter equations due to torsion-lateral and longitudinal-lateral couplings. The experimental results highlight the different kinds of lateral instabilities and they are compared either with existing experimental, or theoretical results. The experimental investigation described in this paper is included in a wide ranging study which also involves theory and the development of a computer code, both briefly presented here.

This paper presents a numerical study of bifurcation and modal interaction in a system of partial differential equations first proposed as a simplified model for bending-torsion vibrations of a thin elastic beam. A system of seven ordinary differential equations obtained using the first six bending and first torsional normal modes is studied, and Floquet theory is used to locate regions in the forcing frequency, forcing amplitude parameter plane where “planar” (i.e., zero torsion) motions are unstable. Numerical branch following and symmetry considerations show that the initial instability arises from a subcritical pitchfork bifurcation. The subsequent nonplanar chaotic attractor is part of a branch of 2-frequency quasiperiodic orbits which undergoes torus-doubling bifurcations. A new statistical technique which identifies interacting modes and the average stability properties of the associated subspaces is presented. The technique employs the Lyapunov vectors used in the calculation of the Lyapunov exponents. We show how this method can be used to split the modes into active and passive sets: active modes interact to contain the attractor, whereas passive modes behave like isolated driven oscillators. In particular, large amplitude modes may simply serve as conduits through which energy is supplied to the active modes.

Wake-body interactions for a two-dimensional structural angle member during stationary and vortex induced oscillatory conditions are studied using a conventional low turbulence wind tunnel. The response of an angle section with combined plunging and torsion indicates that the oscillations occur essentially in one of the two degrees-of-freedom. The measurements of frequency and phase substantiated this observation. The plunging resonance exhibits the familiar vortex capture phenomenon where the shedding frequency is controlled by the cylinder motion over a finite wind speed range. On the other hand, the torsional vibration shows a vortex control phenomenon where the vortex shedding governs the frequency of oscillation. The vortex-induced torsional resonance was found to be severe even at moderate damping levels. The results should prove useful in structural designs such as high-voltage transmission towers, antenna masts, bridges, etc., where angle sections are often used as secondary members.

The equations of motion for the small oscillations of a stranded, overhead transmission line are derived and linearized about the static equilibrium position. The influence of axial-torsional coupling on the natural frequencies is studied analytically and an expression is presented for the coupled natural frequency in torsion. In order to verify the analytical results, a finite element analysis of the linearized coupled differential equations is carried out. The results of each analysis are compared and show close agreement. The results of the zero coupling case also closely agree with previous work.

The effect of coupling with torsion on the unbalance response of flexible rotors, supported by isotropic flexible and damped bearings is investigated. Flexural vibrations of the shaft-disk system are coupled with torsional oscillations through mass eccentricity. The governing equations of motion of the continuous system are solved numerically with a modified Myklestad-Prohl method without the necessity of considering an equivalent lumped system. The cases of constant or harmonic torque applied to the disk are considered. Gyroscopic, rotary inertia, shear deformation, external and internal damping effects are taken into account.

In this paper a structural analysis of cemented lap joints between tubes under torsion is presented. The behavior of the joints has been evaluated using a linear viscoelastic model of the adhesive layer and for both steady state and general loading conditions. Analytical relations for the shear stress in the adhesive layer for the torsional compliance of the joints are proposed.

The low frequency reflection and transmission of vibrational energy at an L-junction of square section rods is investigated theoretically and experimentally. The theoretical description, in which thin rod theory is employed, is complete in that all modes of vibration are considered including compression, torsion, and bending both for the incident and generated waves. It is seen that bending vibration in the plane of the system is coupled to compressional waves and bending vibration normal to the plane of the system is coupled to torsional waves. The theory applies to junctions of semi-infinite rods and this condition was simulated experimentally by means of impulse response techniques in which the reflected waves were removed by truncating the time histories prior to frequency analysis. A novel combination of phase separation, time of flight methods, and filtering has allowed the various modes of vibration to be separated and the agreement between predicted and measured reflection and transmission loss, both for same mode and dissimilar mode coupling, is promising.

Theories have been developed for the torsion of nonhomogeneous thin-walled prismatic bars of open section and thin-walled tubes. The mechanical response of these members has been shown to be equivalent to the corresponding homogeneous case with the modulus of rigidity replaced by a suitable average of the variable rigidity modulus of the material. These average moduli, which are different for each geometry, are shown to be moments of the variable modulus of the nonhomogeneous material. Interpretations for the average moduli, which would be determined by torsion tests on nonhomogeneous materials, have been considered.