An algorithm for the optimal design of passive vibration controllers for flexible systems

A gradient algorithm is developed for the optimal design of discrete passive dampers in the vibration control of a class of flexible (distributed parameter) systems. A complete mathematical development is presented for slender beams in flexural vibration. The algorithm systematically seeks to make the modal damping and natural frequencies of the system reach a set of preassigned values. Single damper and the two damper control examples indicate that the proposed algorithm converges faster than the Davison method used in reference [1] for those cases.     

Receptances of non-proportionally and continuously damped plates—Reduced dampers method

This paper presents a method for the dynamic analysis of continuously and non-proportionally damped plates in bending modes. The damping can be in the form of constrained or unconstrained layers. The method is an extension of the equivalent dampers method discussed in a previous paper, in which the damping matrix of a discretized plate is replaced by a diagonal equivalent damping matrix. Each diagonal element represents an equivalent damper inserted between the structure and ground. In this method the number of equivalent dampers is reduced so that the receptance matrix of the damped structure can be obtained economically by a direct matrix method. The receptances of two different partially coated plates in transverse directions are computed by the method suggested. The verification of the results is demonstrated by comparison with the experimental values and also with the theoretical values obtained by the equivalent dampers method. The method presented can also be applied to the transverse vibration analysis of plates with discrete damping inserts.     

Optimisation of dynamic vibration absorbers to minimise kinetic energy and maximise internal power dissipation

The tuning of a dynamic vibration absorber is considered such that either the kinetic energy of the host structure is minimised or the power dissipation within the absorber is maximised. If the host structure is approximated as a damped single degree of freedom, the optimal values for the ratio of the absorber's natural frequency to the host structure and the optimal damping ratio of the absorber are shown to be the same whether the kinetic energy of the host structure is minimised or the power dissipation of the absorber is maximised. It is also demonstrated that the total power input into the system does not depend on the two parameters but only on the host structure's mass.     

Optimum vibration absorber (tuned mass damper) design for linear damped systems subjected to random loads

Optimum design of dynamic vibration absorbers (DVAs) installed on linear damped systems that are subjected to random loads is studied and closed-form design formulas are provided. Three cases are considered in the optimization process: Minimizing the variance of the displacement, velocity and acceleration of the main mass. Exact optimum design parameters for the velocity case, which to the best knowledge of the author do not exist in the literature, are derived for the first time. Exact solutions are found to be directly applicable for practical use with no simplification needed. For displacement and acceleration cases, a solution for the optimum absorber frequency ratio is obtained as a function of optimum absorber damping ratio. Numerical simulations indicate that optimum absorber damping ratio is not significantly related to the structural damping, especially when the displacement variance is minimized. Therefore, optimum damping ratio derived for undamped systems is proposed for damped systems for the displacement case. When acceleration variance is minimized, however, the optimum damping ratio derived for undamped systems is found not as accurate for damped systems. Therefore, a more accurate approximate expression is derived. Numerical comparisons with published approximate expressions at the same level of complexity indicated that proposed design formula yield more accurate estimates. Another important finding of the paper is that for specific applications where all of the response parameters are desired to be minimized simultaneously, DVAs designed per velocity criteria provide the best overall performance with the least complexity in the design equations.     

On eigenproblem solution of damped vibrations associated with gyroscopic moments

A new efficient approach is presented for solving the quadratic eigenvalue problem of weakly, nonproportionally damped vibration systems. In the analysis of these systems, gyroscopic moments and external damping are both considered. Traditional restriction of symmetry of inertia, damping and stiffness matrices is slightly relaxed. A second-order perturbation theory is developed such that the perturbed solution is based on the eigensolution of an unperturbed subproblem. This subproblem considers the unperturbed system in two different forms: (i) a conservative, gyroscopic part of an original problem, or (ii) a nonconservative, gyroscopic part of an original problem that is proportionally damped. To cope with asymmetry of the system matrices, a Duncan's like state formulation is used to bring these matrices into a suitable form for perturbations. Two numerical examples are introduced for explaining the detailed implementation of the presented approach. Additionally, a practical problem of rotor supported by two tilting pad-bearings is investigated. The eigensolutions obtained by the current approach match, to a great extent, other solutions obtained by time-consuming exact methods. The investigation procedure given here gives a framework to handle vibration problems of weakly nonproportional damping and/or weakly asymmetric inertia, damping and stiffness matrices.     

Passive-active vibration isolation systems to produce zero or infinite dynamic modulus: theoretical and conceptual design strategies

The application of mechanical springs connected in parallel and/or in series with active springs can produce dynamical systems characterised by infinite or zero value stiffness. This mathematical model is extended to more general cases by examining the dynamic modulus associated with damping, stiffness and mass effects. This produces a theoretical basis on which to design an isolation system with infinite or zero dynamic modulus, such that stiffness and damping may have infinite or zero values. Several theoretical designs using a mixture of passive and active systems connected in parallel and/or in series are proposed to overcome limitations of feedback gain experienced in practice to achieve an infinite or zero dynamic modulus. It is shown that such systems can be developed to reduce the weight supported by active actuators as demonstrated, for example, by examining suspension systems of very low natural frequency or with a very large supporting stiffness or with a viscous damper or a self-excited vibration oscillator. A more general system is created by combining these individual systems allowing adjustment of the supporting stiffness and damping using both displacement and velocity feedback controls. Frequency response curves show the effects of active feedback control on the dynamical behaviour of these systems. The theoretical design strategies presented can be applied to design feasible hybrid vibration control systems displaying increased control performance.     

Optimum design of Lanchester damper for a viscously damped single degree of freedom system by using minimum force transmissibility criterion

The problem of optimum design of a Lanchester damper for minimum force transmission from a viscously damped single degree of freedom system subjected to harmonic excitation is investigated. Explicit expressions are developed for determining the optimum absorber parameters. It is shown that for the particular case of the undamped single degree of freedom system the results reduce to the classical ones obtained by using the concept of a fixed point on the transmissibility curves.     

Estimation of resonant peak amplitudes

A simple method is proposed for estimating the resonant peak response levels of damped structures, by using an analysis of undamped vibration. The method is based on a simple identity which is exact for a single degree of freedom system and approximate for more general systems. However, since the level of damping in practical structures can seldom be described with any precision, it is argued that a simple approximate method for predicting resonant vibration levels is the most appropriate. A number of examples and applications are described.     

Modeling of axially moving wire with damping: Eigenfunctions, orthogonality and applications in slurry wiresaws

The classical modal analysis is applied to derive the analytical solution and to obtain the free vibration response of damped axially moving wire in this paper. The corresponding eigenvalues, eigenfunctions, and orthogonal relationship are presented. The orthogonality property and closed-form solution of free vibration response with damping are the main contributions of this study. In addition, the analytical modal analysis, with damping factor removed, shows agreement with those in existing research literature of moving wire without damping. The specific relevance of this general solution is discussed with respect to the moving wire in a slurry wiresaw. The theoretical definition of the damping factor of the slurry wiresaw system is also provided.     

A damping mechanics model and a beam finite element for the free-vibration of laminated composite strips under in-plane loading

A theoretical framework is presented for predicting the nonlinear damping and damped vibration of laminated composite strips due to large in-plane forces. Nonlinear Green-Lagrange axial strains are introduced in the governing equations of a viscoelastic composite and new nonlinear damping and stiffness matrices are formulated including initial stress effects. Building upon the nonlinear laminate mechanics, a damped beam finite element is developed. Finite element stiffness and damping matrices are synthesized and the static equilibrium is predicted using a Newton-Raphson solver. The corresponding linearized damped free-vibration response is predicted and modal frequencies and damping of the in-plane deflected strip are calculated. Numerical results quantify the nonlinear effect of in-plane loads on structural modal damping of various laminated composite strips. The modal loss-factors and natural frequencies of cross-ply Glass/Epoxy beams subject to in-plane loading are measured and correlated with numerical results.     

Autoparametric vibration absorber effect to reduce the first symmetric mode vibration of a curved beam/panel

This paper presents the implementation of autoparametric phenomena to reduce the symmetrical vibration of a curved beam/panel under external harmonic excitation. The internal energy transfer of a first symmetric mode into first anti-symmetric mode in a curved panel is one example of autoparametric vibration absorber effect. This is similar to the vibration energy transfer from the resonance of a primary structure to the resonance of a secondary spring–mass (tuned mass damper). The nonlinear response of a curved beam is analyzed using an equation with two modes, and a shaker test. The effect of different configurations of the curve beam/panel, including damping ratios and excitation levels, on the energy transfer of the first symmetric mode to the first anti-symmetric mode was studied.The conventional tuned mass damper (TMD) can reduce the resonance response by energy transfer using damping dissipation, whereas an autoparametric vibration absorber (AVA) can reduce the resonance response by energy transfer using parametric interaction. The results indicate that there is a non-absorption region in which vibration is amplified. For the AVA, the non-absorption region can be minimized by tuning the resonance frequency of the first anti-symmetric mode to half of the first symmetric mode resonance frequency using additional mass. No additional damping material is required for achieving sufficient vibration reduction. The AVA can maintain reliable performance in hot and corrosive environments where damping material cannot perform effectively. This paper presents the first successful experimental results of an autoparametric vibration absorption mechanism in a curved beam.     

Bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin-Voigt damping

Utilizing the Timoshenko beam theory and applying Hamilton's principle, the bending vibration equations of an axially loaded beam with locally distributed internal damping of the Kelvin-Voigt type are established. The partial differential equations of motion are then discretized into linear second-order ordinary differential equations based on a finite element method. A quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams. The effects of the internal damping, sizes and locations of damped segment, axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated. It is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics.     

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.     

Theoretical prediction of the damping of a railway wheel with sandwich-type dampers

This paper presents a procedure for predicting the damping added to a railway wheel when sandwich-type dampers are installed. Although there are different ways to reduce the noise generated by a railway wheel, most devices are based on the mechanism of increasing wheel damping. This is why modal damping ratios are a clear indicator of the efficiency of the damping device and essential when a vibro-acoustic study of a railway wheel is carried out. Based on a number of output variables extracted from the wheel and damper models, the strategy explained herein provides the final damping ratios of the damped wheel. Several different configurations are designed and experimentally tested. Theoretical and experimental results agree adequately, and it is demonstrated that this procedure is a good tool for qualitative comparison between different solutions in the design stages.     

The decoupling of damped linear systems in free or forced vibration

The purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non-oscillatory free vibration or in forced vibration. Based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations, the concept of damped modes of imaginary vibration is introduced. By phase synchronization of these real and physically excitable modes, a time-varying transformation is constructed to decouple non-oscillatory free vibration. When time drifts caused by viscous damping and by external excitation are both accounted for, a time-varying decoupling transformation for forced vibration is derived. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the second and final part of a solution to the “classical decoupling problem.” Together with an earlier paper, a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system.     

一种机载光电设备减振系统设计及稳定精度试验研究

以光电设备减振系统为研究对象,从光电设备轻型化、小型化设计角度出发,针对现有内置橡胶减振系统的诸多不足,采用金属减振器,设计出一款适用于机载光电设备的外置型金属减振系统。通过对该减振系统的振动和冲击试验,测得固有频率、最大传递率及冲击最大位移值。通过对比光电设备在两种减振系统下的稳定精度, 得出以下结论:外置金属减振系统和内置橡胶减振系统均能使光电设备的稳定精度满足不大于25μrad的使用要求,但外置金属减振系统能提高光电设备内部有效空间,同时,外置金属减振系统作为整体可更换单元,降低了减振系统的维修难度,提高了光电设备整体维修性,为其他光电传感器的集成和实现机载光电设备轻型化、小型化提供了可能。     

Vibration suppression of a cantilever beam using magnetically tuned-mass-damper

Eddy currents are induced by the movement of a conductor through a stationary magnetic field or a time varying magnetic field through a stationary conductor. These currents circulate in the conductive material and are dissipated, causing a repulsive force between the magnet and the conductor. These electromagnetic forces can be used to suppress the vibrations of a flexible structure. A tuned mass damper is a device mounted in structures to reduce the amplitude of mechanical vibrations and is one of the effective vibration suppression methods. In the present study, an improved concept of this tuned mass damper for the vibration suppression of structures is introduced. This concept consists of the classical tuned mass damper and an eddy current damping. The important advantages of this magnetically tuned mass damper are that it is relatively simple to apply, it does not require any electronic devices and external power, and it is effective on the vibration suppression. The proposed concept is designed for a cantilever beam and the analytical studies on the eddy current damping and its effects on the vibration suppression. To show the effectiveness of the proposed concept and verify the eddy current damping model, experiments on a cantilever beam are performed. It is found that the proposed concept could significantly increase the damping effect of the tuned mass damper even if not adequately tuned.     

Damping uncertainty due to noise and exponential windowing

Estimation of damping levels in structures is required in many applications and the theoretical damping predictions in many cases are either difficult or not very reliable. Therefore, determination of damping levels based on experimental data is employed quite often. However, in some cases, experimental approach may not be able to provide definite answers either in the sense that the identified damping level may exhibit high level of uncertainty. In experimental approach, the most widely used methods for damping determination require vibration spectrums or the Frequency Response Function(s) which are obtained by Fourier transformation of the time-domain data. During this process, it is often necessary, especially for lightly damped structures, to modify the time-domain data by using exponential windowing so as to minimise the leakage effect in the spectrum. The so-called numerical damping artificially added by this process can be subtracted later in order to obtain the correct damping level. However, for lightly damped structures, the artificially introduced numerical damping can be significantly greater than the actual damping level. This inevitably brings the accuracy and the reliability issues, especially when the data are contaminated by noise. This paper addresses damping uncertainty in frequency-domain estimation when (i) the data are contaminated by noise and (ii) numerical damping via exponential windowing is introduced during the signal processing phase of the spectrum estimation. Some numerical simulations are performed first in order to assess the adverse effects of noise on damping estimations and resulting damping uncertainty is examined as a function of noise level in the data. Then, damping uncertainty due to the use of exponential windowing is investigated using experimental data. A relationship between damping uncertainty and the level of added numerical damping is presented when the so-called Line-Fit method is used for damping estimation from measured Frequency Response Functions.     

A comparison of two nonlinear damping mechanisms in a vibration isolator

The vibration transmissibility characteristics of a single-degree-of-freedom (SDOF) passive vibration isolation system with different nonlinear dampers are investigated in this paper. In one configuration, the damper is assumed to be linear and viscous, and is connected to the mass so that it is perpendicular to the spring (horizontal damper). The vibration is in the direction of the spring. The second configuration is one in which the damper is in parallel with the spring but the damping force is proportional to the cube of the relative velocity across the damper (cubic damping). Both configurations are studied for small amplitudes of excitation, when some analysis can be conducted based on analytical expressions, and for large amplitudes of excitation, where the analysis is based on numerical simulations. It is found that the two nonlinear systems can outperform the linear system when force transmissibility is considered. However, for displacement transmissibility, the system with the horizontal damper exhibits some desirable properties, but the system with cubic damping does not.     

Parameter estimation and multivariable model building for the non-destructive, on-line determination of eggshell strength

In the non-destructive quality assessment of agro-products using vibration analysis, the resonant frequency and the damping of the vibration are the main interest. Those parameters are usually calculated starting from the frequency spectrum, obtained after a fast Fourier transformation (FFT) of the time signal. However, this method faces several drawbacks when applied to short-time signals, as in the case of impact testing of highly damped specimen. An alternative to the FFT method is used for the high-resolution estimation of both resonant frequency and damping. Furthermore, the mass-spring model that is used in the literature for non-destructive quality assessment of various agro-products is extended with the incorporation of the damping and a shape characteristic. As a practical example, eggshell stiffness was estimated using vibration measurements. A data set consisting of 229 eggs was measured. It is shown that both the damping and the shape characteristics are of major importance to explain eggshell strength. This paper makes clear that a univariable model, as is mostly used in the literature, is not always satisfactory to describe the vibration behaviour of biological products.