H∞ optimization of fluid viscous dampers for reducing vibrations of high-speed railway bridges

This work deals with the optimization of fluid viscous damper systems (FVDs) to reduce the resonant dynamic structural response of high-speed railway bridges by algebraic and numerical approaches. The presented method chooses the objective function based on the _H∞$H_{\infty }$ norm over the frequency band of interest. This function allows taking into account structural damping properties and minimizing simultaneously the structural response associated with multiple modes. Especially, the proposed objective function may also be extended to nonlinear problems to determine optimal parameters of nonlinear fluid viscous dampers which may be an interesting solution in applications where high force levels are expected in the dampers. Finally, the proposed method is validated through numerical simulations. The simulation results show that the optimal FVD coefficients obtained by using the presented method are more exact than those by the previous method. Besides, the effectiveness of the method for solving the problems with the contribution of high modes and the consideration of nonlinear FVDs is also proved.     

Analytical study of nonlinear vibrations of a charged viscous liquid drop

The generatrix of a nonlinearly vibrating charged drop of a viscous incompressible conducting liquid is found by directly expanding the equilibrium spherical shape of the drop in the amplitude of initial multimode deformation up to second-order terms. A fact previously unknown in the theory of nonlinear interaction is discovered: the energy of an initially excited vibration mode of a low-viscosity liquid drop is gradually (within several vibrations periods) transferred to the mode excited by only nonlinear interaction. Irrespectively of the form of the initial deformation, an unstable viscous drop bearing a charge slightly exceeding the critical Rayleigh value takes the shape of a prolate spheroid because of viscous damping of all the modes (except for the fundamental one) for a characteristic time depending on the damping rates of the initially excited modes and the further evolution of the drop is governed by the fundamental mode. In a high-viscosity drop, the rate of rise of the unstable fundamental mode amplitude does not increase continuously with time, contrary to the predictions of nonlinear analysis in terms of the ideal liquid model: it first decreases to a value slightly differing from zero (which depends on the extent of supercriticality of the charge and viscosity of the liquid), remains small for a while (the unstable mode amplitude remains virtually time-independent), and then starts growing.     

Experimental determination of damping of plate vibrations in a viscous fluid

A method of determining the aerodynamic-drag coefficient of flat vibrating plates from the vibrogram of free damping vibrations of cantilever-fixed duralumin samples has been developed. From the results of our experiments, simple approximating formulas determining the decrement of damping vibrations and the aerodynamic-drag coefficient through the dimensionless vibration amplitude and the Stokes parameter are proposed. The approach developed in this study for determining the aerodynamic-drag coefficient of a vibrating plate can be a useful alternative to purely hydrodynamic methods of finding the drag of vibrating solids.     

On nonlinear stability of general undercompressive viscous shock waves

We study the nonlinear stability of general undercompressive viscous shock waves. Previously, the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation, using new pointwise methods. By examining time invariants associated with the linearized equations, we can now overcome a new difficulty in the general case, namely, nonlinear movement of the shock. We introduce a coordinate transformation suitable to treat this new aspect, and demonstrate our method by analyzing a model system of generic type. We obtain sharp pointwise bounds andL ^p behavior of the solution for allp, 1p.     

Cable connected active tuned mass dampers for control of in-plane vibrations of wind turbine blades

In-plane vibrations of wind turbine blades are of concern in modern multi-megawatt wind turbines. Today?s turbines with capacities of up to 7.5 MW have very large, flexible blades. As blades have grown longer the increasing flexibility has led to vibration problems. Vibration of blades can reduce the power produced by the turbine and decrease the fatigue life of the turbine. In this paper a new active control strategy is designed and implemented to control the in-plane vibration of large wind turbine blades which in general is not aerodynamically damped. A cable connected active tuned mass damper (CCATMD) system is proposed for the mitigation of in-plane blade vibration. An Euler–Lagrangian wind turbine model based on energy formulation has been developed for this purpose which considers the structural dynamics of the system and the interaction between in-plane and out-of-plane vibrations and also the interaction between the blades and the tower including the CCATMDs. The CCATMDs are located inside the blades and are controlled by an LQR controller. The turbine is subject to turbulent aerodynamic loading simulated using a modification to the classic Blade Element Momentum (BEM) theory with turbulence generated from rotationally sampled spectra. The turbine is also subject to gravity loading. The effect of centrifugal stiffening of the rotating blades has also been considered. Results show that the use of the proposed new active control scheme significantly reduces the in-plane vibration of large, flexible wind turbine blades.     

An alternative approach for the analysis of nonlinear vibrations of pipes conveying fluid

Based on a novel extended version of the Lagrange equations for systems containing non-material volumes, the nonlinear equations of motion for cantilever pipe systems conveying fluid are deduced. An alternative to existing methods utilizing Newtonian balance equations or Hamilton's principle is thus provided. The application of the extended Lagrange equations in combination with a Ritz method directly results in a set of nonlinear ordinary differential equations of motion, as opposed to the methods of derivation previously published, which result in partial differential equations. The pipe is modeled as a Euler elastica, where large deflections are considered without order-of-magnitude assumptions. For the equations of motion, a dimensional reduction with arbitrary order of approximation is introduced afterwards and compared with existing lower-order formulations from the literature. The effects of nonlinearities in the equations of motion are studied numerically. The numerical solutions of the extended Lagrange equations of the cantilever pipe system are compared with a second approach based on discrete masses and modeled in the framework of the multibody software HOTINT/MBS. Instability phenomena for an increasing number of discrete masses are presented and convergence towards the solution for pipes conveying fluid is shown.     

Optohydrodynamics of soft fluid interfaces: Optical and viscous nonlinear effects

Recent experimental developments showed that the use of the radiation pressure, induced by a continuous laser wave, to control fluid-fluid interface deformations at the microscale, represents a very promising alternative to electric or magnetic actuation. In this article, we solve numerically the dynamics and steady state of the fluid interface under the effects of buoyancy, capillarity, optical radiation pressure and viscous stress. A precise quantitative validation is shown by comparison with experimental data. New results due to the nonlinear dependence of the optical pressure on the angle of incidence are presented, showing different morphologies of the deformed interface going from needle-like to finger-like shapes, depending on the refractive index contrast. In the transient regime, we show that the viscosity ratio influences the time taken for the deformation to reach steady state.     

Granular dampers for the reduction of vibrations of an oscillatory saw

Instruments for surgical and dental application based on oscillatory mechanics submit unwanted vibrations to the operator’s hand. Frequently the weight of the instrument’s body is increased to dampen its vibration. Based on recent research regarding the optimization of granular damping we developed a prototype granular damper that attenuates the vibrations of an oscillatory saw twice as efficiently as a comparable solid mass.     

Whirling motion of a shallow cable with viscous dampers

The paper deals with the non-linear dynamic analysis of cables with a pair of viscous dampers close to one support. Such cables are characterized by a sag-to-chord-length ratio below 0.02, for which natural frequencies for the vertical and the horizontal vibrations are pair-wise close. Under resonance the non-linear coupling of pairs of modes may cause whirling harmonic motions around the chord line. Whirling motion may occur after bifurcation from single-mode response for harmonic loads in either vertical or horizontal direction. The non-linear features are included in the two coupled modes, while all other modes are treated as linear. The motion is discretized by expansion in terms of the damped complex eigenfunctions. The applied base functions fulfil the transition condition at the damper, leading to fast convergence of the expansion. It is demonstrated that the behaviour of the whirling motion is controlled primarily by the damper acting in the direction of the unloaded mode, whereas the magnitude of the damper in the loaded mode is less important. If the dampers in the vertical and horizontal direction are close to the optimal value of the corresponding taut cable case, substantial reduction of the vibration level of the whirling mode as well as the frequency interval of its occurrence is attained.     

The spectrum of natural vibrations in a medium composed of layers of elastic material and viscous fluid

In this study, we investigated the spectral properties of two homogenized models corresponding to transverse and longitudinal vibrations in a microinhomogeneous layered combined medium. It is established that investigation of the spectrum of each model is reduced to finding the roots of the corresponding cubic equations, and the qualitative difference between the spectra of these models is also revealed.     

On the principle of minimum kinetic energy dissipation in the nonlinear dynamics of viscous fluid

By the direct numerical integration of the complete set of the Navier-Stokes equations, it is found that the minimum kinetic energy dissipation principle, or the Helmholtz principle, is realized in some internal flows of a viscous fluid. Studies are conducted for the Reynolds numbers from 2 to 20. A class of problems where this principle takes place is considered.     

Properties of expansions in Legendre polynomial derivatives that show up in analysis of nonlinear vibrations of a viscous liquid drop

Analytical expressions for the coefficients of expansions of Legendre polynomial products in the first and second derivatives of the polynomials with respect to polar angles, as well as for the coefficients of expansions of derivative products in Legendre polynomials and their first derivatives, are derived. An interrelation between these expansion coefficients and a relationship between these coefficients and the Clebsch-Gordan coefficients are found. When axisymmetric nonlinear vibrations of a viscous liquid drop are investigated analytically, the toroidal component of the velocity field can be ignored, which significantly cuts the body of computation.     

Efficient modal control strategies for active control of vibrations

Some efficient strategies for the active control of vibrations of a beam structure using piezoelectric materials are described. The control algorithms have been implemented for a cantilever beam model developed using finite element formulation. The vibration response of the beam to an impulse excitation has been calculated numerically for the uncontrolled and the controlled cases. The essence of the method proposed is that a feedback force in different modes be applied according to the vibration amplitude in the respective modes i.e., modes having lesser vibration may receive lesser feedback. This weighting may be done on the basis of either displacement or energy present in different modes. This method is compared with existing methods of modal space control, namely the independent modal space control (IMSC), and modified independent modal space control (MIMSC). The method is in fact an extension of the modified independent space control with the addition that it proposes to use the sum of weighted multiple modal forces for control. The proposed method results in a simpler feedback, which is easy to implement on a controller. The procedure is illustrated for vibration control of a cantilever beam. The analytical results show that the maximum feedback control voltage required in the proposed method is further reduced as compared to existing methods of IMSC and MIMSC for similar vibration control. The limitations of the proposed method are discussed.     

Nonlinear finite amplitude vibrations of sharp-edged beams in viscous fluids

In this paper, we study flexural vibrations of a cantilever beam with thin rectangular cross section submerged in a quiescent viscous fluid and undergoing oscillations whose amplitude is comparable with its width. The structure is modeled using Euler–Bernoulli beam theory and the distributed hydrodynamic loading is described by a single complex-valued hydrodynamic function which accounts for added mass and fluid damping experienced by the structure. We perform a parametric 2D computational fluid dynamics analysis of an oscillating rigid lamina, representative of a generic beam cross section, to understand the dependence of the hydrodynamic function on the governing flow parameters. We find that increasing the frequency and amplitude of the vibration elicits vortex shedding and convection phenomena which are, in turn, responsible for nonlinear hydrodynamic damping. We establish a manageable nonlinear correction to the classical hydrodynamic function developed for small amplitude vibration and we derive a computationally efficient reduced order modal model for the beam nonlinear oscillations. Numerical and theoretical results are validated by comparison with ad hoc designed experiments on tapered beams and multimodal vibrations and with data available in the literature. Findings from this work are expected to find applications in the design of slender structures of interest in marine applications, such as biomimetic propulsion systems and energy harvesting devices.     

Collapse of a radiating viscous fluid

Recently a model of a classical collapsing radiating viscous fluid has been proposed by Lake. We study the history of the collapsing surface of the viscous fluid under the equations of state ζ = α?^v and p = λ? (ζ is the bulk viscosity coefficient, ? is the energy density, p is the pressure, α, v and λ are constants).     

Mitigation of edgewise vibrations in wind turbine blades by means of roller dampers

Edgewise vibrations in wind turbine blades are lightly damped, and large amplitude vibrations induced by the turbulence may significantly shorten the fatigue life of the blade. This paper investigates the performance of roller dampers for mitigation of edgewise vibrations in rotating wind turbine blades. Normally, the centrifugal acceleration of the rotating blade can reach to a magnitude of 7–8g, which makes it possible to use this kind of damper with a relatively small mass ratio for suppressing edgewise vibrations effectively. The parameters of the damper to be optimized are the mass ratio, the frequency ratio, the coefficient of rolling friction and the position of the damper in the blade. The optimization of these parameters has been carried out on a reduced 2-DOF nonlinear model of the rotating wind turbine blade equipped with a roller damper in terms of a ball or a cylinder, ignoring the coupling with other degrees of freedom of the wind turbine. The edgewise modal loading on the blade has been calculated from a more sophisticated 13-DOF aeroelastic wind turbine model with due consideration to the indicated couplings, the turbulence and the aerodynamic damping. Various turbulence intensities and mean wind speeds have been considered to evaluate the effectiveness of the roller damper in reducing edgewise vibrations when the working conditions of the wind turbine are changed. Further, the optimized roller damper is incorporated into the 13-DOF wind turbine model to verify the application of the decoupled optimization. The results indicate that the proposed damper can effectively improve the structural response of wind turbine blades.     

Problems on control for a steady-state magnetic-hydrodynamic model of a viscous heat-conducting fluid under mixed boundary conditions

The control problems for steady-state equations of magnetic hydrodynamics (MHD) for a viscous heat-conducting fluid considered under mixed boundary conditions for the magnetic field and temperature are investigated. Their solvability is proved, the optimality systems describing the necessary conditions of an extremum are derived, and the theorems of local uniqueness and stability of the optimum solutions for explicit quality functionals are formulated.     

Swirling of viscous fluid threads in microchannels

Viscous threads that are swept along in the flow of a less viscous miscible liquid can break up into viscous swirls. We experimentally investigate the evolution of miscible threads that flow off center in microchannels. Thin threads near the walls of a straight square channel become unstable to shear-induced disturbances. The amplification of the undulations transverse to the flow direction ultimately causes the threads to break up and form an array of individual viscous swirls, the miscible counterparts of droplets. This swirling instability provides a means for passively producing discrete diffusive microstructures in a continuous flow regime.     

Optimal design of passive viscous dampers for controlling the resonant response of orthotropic plates under high-speed moving loads

In this article, the resonant response of plates traversed by moving loads is addressed being its main application the dynamic performance of railway bridges under high-speed traffic. An innovative alternative to reduce the deck inadmissible oscillations that may appear in short simply supported structures in resonant conditions is proposed, based on artificially increasing the superstructure damping by retrofitting the deck with fluid viscous dampers. A particular auxiliary structure transforming the deck vertical deflection into relative movement within the devices is envisaged, being the main objectives of the study to optimise the retrofitting system parameters and to prove its efficiency under the action of railway vehicles. For these purposes, the retrofitted deck behaviour is first investigated using an orthotropic plate model under harmonic excitation. On the basis of an analytical approach, a dimensionless version of the equations of motion is presented, the governing parameters are extracted and an intensive sensitivity analysis of the plate response is performed. Finally, analytical closed-form expressions for the optimal dampers constants are derived and their adequacy is numerically evaluated. To this end, an existing bridge belonging to the Spanish Railway network is analysed using a three-dimensional finite element code specifically programmed by the authors for this application. In the end the controlling effect of the retrofitting system and the applicability of the optimal parameters analytical expressions are proven for a wide range of circulating velocities.