Impact-induced vibrations of a simple viscoelastic column model

A two-degree-of-freedom model for an almost-axially impacted viscoelastic cantilever column is analyzed. The impact load is produced by a mass striking the free end of the column. Under the assumption of small displacements two second-order non-linear ordinary differential equations for the coupled longitudinal and transverse vibrations of the column are derived. In the absence of damping these equations of motion are reduced to Mathieu's equation through the use of a perturbation method. The excitation parameters are (i) the natural frequency of small amplitude transverse vibrations of the undamped column and (ii) the initial velocity of the end of the column. The boundaries of two unstable regions are obtained. In the stable regions the solution of Mathieu's equation for the transverse displacement is close to that of the original non-linear equations of motion. In the first unstable region there is agreement only for early time. With increasing damping the peak of the maximum transverse displacement in the first unstable region decreases or even vanishes.     

Influence of the alignment of load and oscillation on the frictional shakedown of an elastic rolling contact with Coulomb friction

We examine frictional shakedown of a three dimensional elastic rolling contact. Slight oscillatory rolling of one contacting body varies the normal pressure distribution. In turn this causes incremental sliding processes and a macroscopic rigid body motion. We consider two settings: tangential force and rolling direction aligned parallel and perpendicular to each other. In both cases, the slip ceases after the first few periods and a safe shakedown occurs if the oscillation is sufficiently small. Otherwise ratcheting occurs and the accumulated slip leads to a continuing rigid body motion.Numerical simulations with Kalker’s and Vollebregt’s software CONTACT show that the rolling direction leads to differences in the contact region and the traction distribution. Using the method of dimensionality reduction we derive the analytical shakedown limits for the tangential load and the oscillation amplitude. The results show strong agreement with experimental data and allow the accurate prediction of the shakedown displacement and the maximum tangential load capacity in the shakedown state. It shows that a perpendicular alignment of force and rolling direction increases the final displacement in case of shakedown as well as the incremental shift in case of ratcheting.     

DYNAMIC ANALYSIS OF A SPIRAL BEVEL-GEARED ROTOR-BEARING SYSTEM

Spiral bevel gears can transmit motion between two rotors, which are commonly perpendicular to each other. In this paper, the dynamic analysis of a spiral bevel-geared rotor-bearing system is studied. Firstly, the constraint equation describing the relationship between the generalized displacements of spiral bevel gear pairs is derived briefly. Then the modelling of coupled axial-lateral-torsional vibration of the rotor system geared by spiral bevel gears is discussed. Finally, the mechanism of coupled vibration of the spiral bevel-geared rotor system is analyzed theoretically and the dynamic behavior of the system is investigated numerically. The conclusions are characterized as follows. The influences of the critical speeds in rigid journal supports, stability threshold speed and unbalanced responses in hydrodynamic journal bearings are not remarkable in comparison with the spur bevel-geared system under the same conditions. However, the critical speeds and stability threshold speed are essentially affected by boundary conditions such as the torsional stiffness, and meanwhile the effect of the unbalanced responses is not prominent under the concerned rotating speeds except that around the resonance peaks. The steady state response due to torsional excitation is also analyzed, and the results show that it cannot be neglected either in the torsional direction or in the lateral and axial directions in the spiral bevel-geared rotor system.     

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

COUPLED TRANSVERSE AND AXIAL MOTION OF A COMPLIANT STRUCTURE IN RESPONSE TO VORTEX-SHEDDING LOADS

A compliant offshore tower is modelled as a structure having extensional and transverse degrees of freedom. McIver's extension of Hamilton's principle is applied to the system, resulting in coupled non-linear equations of motion, due to the assumption of small strain and moderate rotation. The equations of motion are discretized using finite differences and solved numerically. Several forms for the vortex-shedding load are tested, as well as experimental force data. To better understand the model response, Monte Carlo simulations are performed. The results show the feasibility of the present model for representing the response of a compliant structure subject to transverse loading.     

Vibration control of beams on elastic foundation under a moving vehicle and random lateral excitations

The formulation of three-dimensional dynamic behavior of a Beam On Elastic Foundation (BOEF) under moving loads and a moving mass is considered. The weight of the vehicle is modeled as a moving point load, however the effect of the lateral excitation is considered by modeling: (case 1) a lateral moving load with random intensity for wind excitation and (case 2) a moving mass just in lateral direction of the beam for earthquake excitation. A Dirac-delta function is used to describe the position of the moving load and the moving mass along the beam. The beam foundations are considered as elastic Winkler-type in two perpendicular transverse directions. This model is proposed to investigate the bending response of the rails under the effect of traveling vehicle weight while a random excitation such as earthquake or wind takes place. The results showed the importance of considering the effect of earthquake/wind actions as in bending stress of the beam on elastic foundations. The effect of different regions (different support stiffness) and different velocities of the vehicle on the response of the beam are investigated in mentioned directions. At the end, a linear optimal control algorithm with displacement–velocity feedback is proposed as a solution to suppress the response of BOEFs. By the method of modal analyses and taking into account enough number of vibration modes, state-space equation is obtained, then sufficient number of actuators was chosen for each direction. Stochastic analyses were performed in lateral direction in order to illustrate a comprehensive view for the response of the beam under the random moving load in both controlled and uncontrolled systems. Furthermore, the efficiency of control algorithm on critical velocities is verified by parametric analyses in the vertical direction with the constant moving load for different regions.     

Heat generation in a torsional spring subjected to sinusoidal oscillations

The effect of thermomechanical coupling in a viscoelastic hollow cylinder subjected to sinusoidal shear stresses or shear displacements has been studied. The problem is a simple model of torsional springs made of rubberlike materials. The material is assumed to be thermorheologically simple. A non-linear boundary value problem is formulated for the almost steady mechanical oscillation coupled with a slowly varying temperature distribution. The wave motion is analyzed by the well-known WKB approximation. Corresponding temperature distributions are calculated numerically by iterative procedures. Although the applied stress and applied displacement are small, significant temperature rises are found. Different stress and temperature distributions are compared for various frequencies.     

Nonlinear analysis and enhancement of wing-based piezoaeroelastic energy harvesters

We investigate the level of harvested power from aeroelastic vibrations for an elastically mounted wing supported by nonlinear springs. The energy is harvested by attaching a piezoelectric transducer to the plunge degree of freedom. The considered wing has a low-aspect ratio and hence three dimensional aerodynamic effects cannot be neglected. To this end, the three dimensional unsteady vortex lattice method for the prediction of the unsteady aerodynamic loads is developed. A strong coupling scheme that is based on Hamming's fourth-order predictor–corrector method and accounts for the interaction between the aerodynamic loads and the motion of the wing is employed. The effects of the electrical load resistance, nonlinear torsional spring and eccentricity between the elastic axis and the gravity axis on the level of the harvested power, pitch and plunge amplitudes are investigated for a range of operating wind speeds. The results show that there is a specific wind speed beyond which the pitch motion does not pick any further energy from the incident flow. As such, the displacement in the plunge direction grows significantly and causes enhanced energy harvesting. The results also show that the nonlinear torsional spring plays an important role in enhancing the level of the harvested power. Furthermore, the harvested power can be increased by an order of magnitude by properly choosing the eccentricity and the load resistance. This analysis is helpful in designing piezoaeroelastic energy harvesters that can operate optimally at specific wind speeds.     

Non-linear vibrations of three-layer beams with viscoelastic cores,II: Experiment

Experimental results are presented for large amplitude, forced motion of damped, three-layer beams. The beams are constructed with a viscoelastic material constrained between stiff, elastic, outer layers. The sandwich beam is axially restrained; therefore large amplitude displacements cause non-linear response. When the beam is forced at one-half of the lateral vibration resonant frequency, superharmonic response occurs. The experiment is briefly described and frequency response characteristics, spatial shapes and a measure of superharmonic response are presented. The results are compared with predictions from a previously developed theory.     

The effects of large vibration amplitudes on the fundamental mode shape of a clamped-clamped uniform beam

The non-linear vibration of a clamped-clamped beam at large displacement amplitudes is examined in this work. Complementary theoretical and experimental studies have been carried out to examine the amplitude dependence of the fundamental mode shape and its derivatives and the spatially-dependent harmonic distortion of the transverse displacement which occurs at large deflections.     

Nonlinear dynamics of galloping-based piezoaeroelastic energy harvesters

The normal form is proposed as a tool to analyze the performance and reliability of galloping-based piezoaeroelastic energy harvesters. Two different harvesting systems are considered. The first system consists of a tip mass prismatic structure (isosceles 30° or square cross-section geometry) attached to a multilayered cantilever beam. The only source of nonlinearity in this system is the aerodynamic nonlinearity. The second system consists of an equilateral triangle cross-section bar attached to two cantilever beams. This system is designed to have structural and aerodynamic nonlinearities. The coupled governing equations for the structure’s transverse displacement and the generated voltage are derived and analyzed for both systems. The effects of the electrical load resistance and the type of harvester on the onset speed of galloping are quantified. The results show that the onset speed of galloping is strongly affected by the load resistance for both types of harvesters. The normal form of the dynamic system near the onset of galloping (Hopf bifurcation) is then derived. Based on the nonlinear normal form, it is demonstrated that smaller levels of generated voltage or power are obtained for higher absolute values of the effective nonlinearity. For the first harvesting system, the results show a supercritical Hopf bifurcation for both isosceles 30° or square cross-section geometries. The nonlinear normal form shows that the isosceles triangle section (30°) is more efficient than the square section. For the second harvesting system, the normal form is used to identify the values of the nonlinear torsional spring which changes the harvester’s instability. It is demonstrated that this critical value of the nonlinear torsional spring depends strongly on the load resistance.     

Fundamental frequency evolution in slender beams subjected to imposed axial displacements

The influence of applied axial loads on the fundamental vibration frequency is strictly connected with the stability analysis of elastic slender beams. For this reason, the correct evaluation of the fundamental frequency is of primary importance in designing new structures and components, as well as in monitoring existing ones. At the same time, if an internal axial load arises in a slender element as the consequence of an imposed (static) axial end displacement, then a different dynamic structural response is encountered respect to the case in which a beam end is free to slide, during transverse vibration, and a (constant) axial load is applied externally. This difference is due to the change in the axial boundary condition. Moreover, the presence of an initial curvature of the beam axis may significantly affect the aforesaid response. The experimental study proposed in the present paper investigates the dependence of the fundamental frequency on the axial load in slender beams subjected to imposed axial end displacements. The considered specimens presented different geometrical imperfections (initial curvatures), and were tested in two different constraint conditions (hinged–hinged and hinged–clamped). In addition, the behaviors observed during the experiments were reproduced by numerical simulations offering a valid confirmation for test results and contributing to understand the evolution of the fundamental frequency in the analyzed slender elements subjected to imposed axial end displacements.     

Free vibration analysis of a cracked shear deformable beam on a two-parameter elastic foundation using a lattice spring model

The free vibration of a shear deformable beam with multiple open edge cracks is studied using a lattice spring model (LSM). The beam is supported by a so-called two-parameter elastic foundation, where normal and shear foundation stiffnesses are considered. Through application of Timoshenko beam theory, the effects of transverse shear deformation and rotary inertia are taken into account. In the LSM, the beam is discretised into a one-dimensional assembly of segments interacting via rotational and shear springs. These springs represent the flexural and shear stiffnesses of the beam. The supporting action of the elastic foundation is described also by means of normal and shear springs acting on the centres of the segments. The relationship between stiffnesses of the springs and the elastic properties of the one-dimensional structure are identified by comparing the homogenised equations of motion of the discrete system and Timoshenko beam theory.     

DSPI strain measurement on an externally reinforced bending beam: A comparison of step-by-step addition and pixel shift correlation

A small-scale concrete beam reinforced with an adhesively bonded carbon fiber reinforced polymer (CFRP) plate was subjected to four-point bending. Finite element analyses (FEA) of the bending deformations were carried out to predict strain gradients near the end of the CFRP plate. In order to measure these strains, phase-stepping 3D-digital speckle pattern interferometry was employed. To avoid speckle decorrelation due to the inevitable rigid body motion of the specimen, the load was increased in small increments. Two evaluation schemes for the electronic speckle pattern interferometry phase maps are compared: summing up the measured displacement components load step-by-load step versus regain of the correlation by shifting the final image by an integer number of pixels. Measured strain values are evaluated using a polynomial fit to the measured in-plane displacements and are compared to the FE predicitions. It can be concluded that pixel shift correlation is preferable to summing up load steps for cases of large rigid body motion.     

Influence of physical non-linearity on the dynamic behavior of composite plates

Influence of the physically non-linear property is investigated for three types of motion in laminated plates, namely harmonic waves, free vibrations and shock waves. Only the non-linearity in the transverse shear deformation is included.     

Complex eigensolutions of coupled flexural and longitudinal modes in a beam with inclined elastic supports with non-proportional damping

Structure borne vibration and noise in an automobile are often explained by representing the full vehicle as a system of elastically coupled beam structures representing the body, engine cradle and body subframe where the engine is often connected to the chassis via inclined viscoelastic supports. To understand more clearly the interactions between a beam structure and isolators, this article examines the flexural and longitudinal motions in an elastic beam with intentionally inclined mounts (viscoelastic end supports). A new analytical solution is derived for the boundary coupled Euler beam and wave equations resulting in complex eigensolutions. This system is demonstrated to be self-adjoint when the support stiffness matrices are symmetric; thus, the modal analysis is used to decouple the equations of motion and solve for the steady state, damped harmonic response. Experimental validation and computational verifications confirm the validity of the proposed formulation. New and interesting phenomena are presented including coupled rigid motions, modal properties for ideal angled roller boundaries, and relationships between coupling and system modal loss factors. The ideal roller boundary conditions when inclined are seen as a limiting case of coupled longitudinal and flexural motions. In particular, the coupled rigid body motions illustrate the influence of support stiffness coupling on the eigenvalues and eigenfunctions. The relative modal strain energy concept is used to distinguish the contribution of longitudinal and flexural deformation modes. Since the beam is assumed to be undamped, the system damping is derived from the viscoelastic supports. The support damping (for a given loss factor) is shown to be redistributed between the system modes due to the inclined coupling mechanisms. Finally, this article provides valuable insight by highlighting some technical issues a real-life designer faces when balancing modeling assumptions such as rigid or elastic formulations, proportional or non-proportional damping, and coupling terms in multidimensional joint properties.     

用激光散斑法测盐水溶液浓度的改进

在利用激光散斑法测盐水溶液浓度时,光束透过溶液时会产生不同的纵向位移,但散斑计量对纵向位移不敏感.因此重新设计光路将纵向位移转化为横向位移,实现了溶液浓度的精确测量.     

Vibration characteristics of a rotating flexible arm with ACLD treatment

In this paper, the vibration behavior and control of a clamped–free rotating flexible cantilever arm with fully covered active constrained layer damping (ACLD) treatment are investigated. The arm is rotating in a horizontal plane in which the gravitational effect and rotary inertia are neglected. The stress–strain relationship for the viscoelastic material (VEM) is described by a complex shear modulus while the shear deformations in the two piezoelectric layers are neglected. Hamilton's principle in conjunction with finite element method (FEM) is used to derive the non-linear coupled differential equations of motion and the associated boundary conditions that describe the rigid hub angle rotation, the arm transverse displacement and the axial deformations of the three-layer composite. This refined model takes into account the effects of centrifugal stiffening due to the rotation of the beam and the potential energies of the VEM due to extension and bending. Active controllers are designed with PD for the piezosensor and actuator. The vibration frequencies and damping factors of the closed-loop beam/ACLD system are obtained after solving the characteristic complex eigenvalue problem numerically. The effects of different rotating speed, thickness ratio and loss factor of the VEM as well as different controller gain on the damped frequency and damping ratio are presented. The results of this study will be useful in the design of adaptive and smart structures for vibration suppression and control in rotating structures such as rotorcraft blades or robotic arms.     

Producing and testing binary amplitude gratings using a self-imaging and double-exposure technique

The influence of illuminating beam aberrations and master grating deviations on the performance of the technique of producing arbitrary opening ratio binary gratings using the self-imaging and double-exposure technique is studied analytically and experimentally. It is shown that due to the shear effect encountered in the direction perpendicular to the grating lines and due to the lateral displacement implemented between the two exposures the technique is sensitive to the first derivative of grating in-plane displacements and to the second derivative of the beam and grating out-of-plane errors.     

Non-linear coupled slosh dynamics of liquid-filled laminated composite containers: a two dimensional finite element approach

This paper brings into focus some of the interesting effects arising from the non-linear motion of the liquid free surface, due to sloshing, in a partially filled laminated composite container along with the associated coupling due to fluid-structure interaction effects. The finite element method based on two-dimensional fluid and structural elements is used for the numerical simulation of the problem. A numerical scheme is developed on the basis of a mixed Eulerian-Lagrangian approach, with velocity potential as the unknown nodal variable in the fluid domain and displacements as the unknowns in the structure domain. The FE formulation based on Galerkin weighted residual method along with an iterative solution procedure are explained in detail followed by a few numerical examples. Numerical results obtained by the present investigation for the rigid containers are first compared with the existing solutions to validate the code for non-linear sloshing without fluid-structure coupling. Thereafter the computational procedures are advanced to obtain the coupled interaction effect of non-linear sloshing in laminated composite containers.