Linear Damping Coefficients Research Articles

Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation

The nonstationary probability densities of system response of a single-degree-of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then, the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise is given as an example to illustrate the proposed procedures. The effects of the system parameters, such as the linear damping coefficient and the nonlinear stiffness coefficient, on the system response are discussed.

VIBRATION OF CIRCULAR BLADED DISK WITH IMPERFECTIONS

The steady state response of a model of circular bladed disk with imperfection is investigated. Disk imperfection results from additional two groups of damping heads fixed on opposite ends of one diameter. These damping heads are introduced into the computing model as additional point mass, damping and stiffness. Such type of imperfection causes the bifurcation of double eigenfrequencies into pairs of close eigenfrequencies. The effect of imperfection is examined both numerically on three-dimensional nonrotating FE-model and analytically on a simplified split 2DOF model of rotating disk excited by single point harmonic force. Nonlinear friction connection is analyzed and equivalent linear damping coefficient is derived and used in the calculation procedure. It is shown that nonproportional distribution of damping strongly influences the high of resonance peaks. Some examples of response curves illustrate the dynamic properties of stationary and rotating disks with mass-damping-stiffness imperfection.

A Study of the Roll Motion by Means of a Free Decay Test

In this study, a method for the extraction of damping by tracing free roll decay is presented. For this purpose, in calm waters, a bulk carrier model is given a large initial roll angle and then released. Consequently, the roll motion is recorded. Restoring coefficients and virtual moments of inertia for the model are determined by means of an inclining test and recording the damped period, respectively. The linear damping coefficient is evaluated by using the damping ratio. Four different forms of combinations of restoring moment and damping coefficient are assumed in order to determine the nonlinear form of the roll motion. These equations are numerically solved for various damping coefficients and results are compared with the experimental data. By virtue of this comparison, the damping coefficients are determined for each case. It may be concluded that the use of the nonlinear restoring moment, which is an odd polynomial of the fifth order, and the cubic form for the nonlinear damping moment best fits the roll behavior for the ship model. The amount of energy dissipated by the damping moments is also calculated in the time domain. The energy method also confirms that the nonlinear form of restoring force in conjunction with the cubic form of the damping force is the best solution of the roll motion for small to large angles.

Effective linear damping and stiffness coefficients of nonlinear systems for design spectrum based analysis

A stochastic approach for obtaining reliable estimates of the peak response of nonlinear systems to excitations specified via a design seismic spectrum is proposed. This is achieved in an efficient manner without resorting to numerical integration of the governing nonlinear equations of motion. First, a numerical scheme is utilized to derive a power spectrum which is compatible in a stochastic sense with a given design spectrum. This power spectrum is then treated as the excitation spectrum to determine effective damping and stiffness coefficients corresponding to an equivalent linear system (ELS) via a statistical linearization scheme. Further, the obtained coefficients are used in conjunction with the (linear) design spectrum to estimate the peak response of the original nonlinear systems. The cases of systems with piecewise linear stiffness nonlinearity, along with bilinear hysteretic systems are considered. The seismic severity is specified by the elastic design spectrum prescribed by the European aseismic code provisions (EC8). Monte Carlo simulations pertaining to an ensemble of nonstationary EC8 design spectrum compatible accelerograms are conducted to confirm that the average peak response of the nonlinear systems compare reasonably well with that of the ELS, within the known level of accuracy furnished by the statistical linearization method. In this manner, the proposed approach yields ELS which can replace the original nonlinear systems in carrying out computationally efficient analyses in the initial stages of the aseismic design of structures under severe seismic excitations specified in terms of a design spectrum.

Vibration reduction of a nonlinear spring pendulum under multi-parametric excitations via a longitudinal absorber

The vibration of a ship is a disadvantage for safety operation. It should be reduced to the minimum. In this paper, the vibration of a ship pitch-roll motion described by a nonlinear spring pendulum (three degrees of freedom) subjected to multi-parametric excitation forces can be reduced using a longitudinal absorber. The theoretical results are obtained by applying the multiple scale perturbation technique (MSPT). The stability of the system near the simultaneous sub-harmonic resonance is studied by using frequency response equations. The numerical solution needs extensive investigations to illustrate the effects of the absorber and some system parameters at selected values on the vibrating system. Optimum working conditions of the system are extracted applying passive control methods. A threshold value of the linear damping coefficient can be used directly for vibration suppression of both vibration modes. Comparison with the available published work is reported.

Influence of excitation amplitude on the characteristics of nonlinear butyl rubber isolators

The purpose of this study is to explore the advantages and characteristics of nonlinear butyl rubber (type IIR) isolators in vibratory shear by comparison with linear isolators. It is known that the mechanical properties of viscoelastic materials exhibit significant frequency and temperature dependence, and in some cases, nonlinear dynamic behavior as well. Nonlinear characteristics in shear deformation are reflected in mechanical properties such as stiffness and damping. Furthermore, even when the excitation amplitude is small the response amplitude may often be large enough that nonlinearities cannot be ignored. The treatment involves developing phenomenological models of the effective storage modulus and effective loss factor of a rubber isolator material as a function of excitation amplitude. The transmissibility of a nonlinear viscoelastic isolator is compared with that of a linear isolator using an equivalent linear damping coefficient. Forced resonance vibration and impedance tests are used to characterize nonlinear parameters and to measure the normalized transmissibility. It is found that as the excitation amplitude of the nonlinear viscoelastic isolator increases, the response amplitude decreases and the transmissibility is improved over that of the linear isolator for excitation frequency that exceeds a particular value governed by the temperature and excitation amplitude. The method of multiple scales and numerical simulations are used to predict the response characteristics of the isolator based on the phenomenological modeling under different values of system parameters.

THE REGULAR SOLUTIONS OF THE ISENTROPIC EULER EQUATIONS WITH DEGENERATE LINEAR DAMPING

The regular solutions of the isentropic Euler equations with degenerate linear damping for a perfect gas are studied in this paper. And a critical degenerate linear damping coefficient is found, such that if the degenerate linear damping coefficient is larger than it and the gas lies in a compact domain initially, then the regular solution will blow up in finite time; if the degenerate linear damping coefficient is less than it, then under some hypotheses on the initial data, the regular solution exists globally.

Dynamic Characteristics and Stability Analysis of a Wavy Thrust Bearing

A numerical procedure to analyze wavy thrust bearings is described. The numerical model is developed by assuming that two circular plates rotate relative to each other. The upper plate is assumed to be flat and rotating, whereas the lower plate is assumed to be stationary and wavy in surface geometry. A Reynolds-equation-based procedure is used to simulate the dynamics engendered by various wavy geometries and loading conditions. The equilibrium position of the journal results from the equilibrium between the forces generated by the fluid-film pressures and the externally applied loads. A numerical small perturbation technique is applied to calculate the linear stiffness and damping characteristics of the bearing at the equilibrium position. Using a three-degrees-of-freedom system with one axial and two rotational displacements, nine linear stiffness coefficients (three principal and six cross-coupled coefficients) and nine linear damping coefficients are calculated. These linear coefficients are then used to calculate the eigenvalues of the system by solving the homogeneous equations of motion. The stability of the bearing system is then expressed using the lowest logarithmic decrement obtained from these eigenvalues. Using this procedure, a parametric study is carried out to examine the effects of external load, location of the applied load, bearing number, and bearing wave amplitude on journal equilibrium position, bearing linear stiffness, damping characteristics, and bearing stability.

Damping Of Pedestrian‐Induced Bridge Vibrations By Tuned Liquid Column Dampers

AbstractThe excessive lateral vibrations of Londons Millenium Bridge and the Toda Park Bridge in Japan due to a large number of crossing pedestrians have raised an unexpected problem in footbridge constructions. Secondary tuned structures, like the conventional tuned mass damper (TMD) or the tuned liquid damper (TLD) were installed to the bridge in order to suppress these vibrations. In the present investigation it is proposed to apply the more efficient and more economic tuned liquid column damper (TLCD), which relies on the motion of a liquid mass in a sealed tube to counteract the external motion, while a built‐in orifice plate induces turbulent damping forces that dissipate kinetic energy. For optimal tuning of TLCDs the natural frequency and equivalent linear damping coefficient have to be chosen suitable, likewise to the conventional TMDs, as given in Den Hartog [1]. The advantages of TLCDs are: simple tuning of natural frequency and damping, low cost of design and maintenance and a simple construction. A mathematical model of a three degree‐of‐freedom (DOF) bridge coupled with an optimal tuned TLCD is derived and analyzed numerically. Furthermore, a small scale experimental model set‐up has been constructed in the laboratory of the TU‐Insitute. The experimental results are in good agreement with the theoretical predictions and indicate that TLCDs are effective damping devices for the undesired pedestrian induced footbridge vibrations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear Dynamic Analysis of an Orthotropic Composite Rotor Blade

This work presents the nonlinear dynamic analysis of orthotropic composite rotor blade using differential quadrature method (DQM). It can be found that the same efficient and procedure of DQM for both uniform composite beam and pre-twisted non-uniform composite beam. In this approach, only nine sample points are needed to achieve convergence. Dynamic responses of orthotropic composite rotor blade with various rotor rotation speeds are determined. Both linear and nonlinear dynamic responses have been obtained in order to study the significance of the nonlinear effect. The transient responses of the derived systems are calculated by using Newmark method. The bending-torsion coupled beam model is used to characterize the composite rotor blade. Kelvin-Voigt internal and linear external damping coefficients are employed and determined for the orthotropic composite rotor blade. The DQM is used to transform the partial differential equations of a composite rotor blade into a discrete eigenvalue problem. In this study, the effects of the fiber orientation, internal damping, external damping, pre-twisted angle and the rotation speed on the dynamic behavior for an orthotropic composite beam are investigated. The effect of the number of sample points on the accuracy of the calculated natural frequencies is also discussed. The integrity and computational efficiency of the DQM in this problem will be derived in this paper.

The effect of fiber angle on the natural frequencies of orthotropic composite pre-twisted blades

The eigenvalue problem of a tapered pre-twisted orthotropic composite blade is formulated by employing the differential quadrature method (DQM). The Euler–Bernoulli beam model is used to characterize the pre-twisted orthotropic composite blade. The Kelvin–Voigt internal and linear external damping coefficients are employed and determined for the orthotropic composite blade. The DQM is used to transform the partial differential equations of a tapered pre-twisted orthotropic composite blade into a discrete eigenvalue problem. The Chebyshev–Gauss–Lobatto sample point equation is used to select the sample points. In this study, the effects of the fiber orientation, internal damping, external damping, inclined angle and the rotation speed on the natural frequencies for a tapered pre-twisted orthotropic composite blade are investigated. The effect of the number of sample points on the accuracy of the calculated natural frequencies is also discussed. The integrity and computational efficiency of the DQM in this problem will be demonstrated through a number of case studies. Numerical results indicated that the DQM is valid and efficient for a pre-twisted orthotropic composite blade formulation.

A Study on Sensitivity of Maneuverability Performance on the Hydrodynamic Coefficients for Submerged Bodies

Based on a constant-coefficient dynamic model, a study was made to determine the influence of various hydrodynamic coefficients on the predicted maneuverability quality of submerged bodies. Two types of geometries were considered, a submarine and an axisymmetric slender geometry. For the submarine, the equations of motion used were the revised standard submarine equations (Feldman 1979) while for the latter geometry a dynamic model was developed. From computer simulation of a few selected definitive maneuvers based on these two different dynamic models for the two geometries, the sensitivity of the simulated trajectory on changes in different coefficients was found. The results quantified in form of sensitivity values are presented. It is found that the typical measures from the maneuvers do not depend significantly on most of the nonlinear coefficients. The coefficients having significant effects on the trajectories are found to be the linear damping coefficients for the submarine and the linear inertial force coefficients for the axisymmetric body.

Analysis on Wave Transformations and Wave Forces about an Angular Body Considering Wave Energy Dissipations

Concerning wave energy dissipations caused by flow separations and resultant vortex formations from sharp edges of an angular body, a twodimensional analysis on wave transformations and wave forces on an angular body has been developed. In the analysis, a damping wave model was newly proposed to account for the wave energy dissipation approximately. An equivalent linear damping coefficient of fluid and a spatial range of the damping fluid are necessary in the analysis. In order to figure out these factors, wave tank tests were carried out. Two different structural models, i.e., a curtain-walled breakwater and a semi-submerged rectangular body, were used in the experiment. It was confirmed that the numerical analysis developed here is useful for estimating wave transformations and wave forces about an angular body including the effects of wave energy dissipation due to vortex formations around the body.

On the effectiveness of constant coefficients roll motion equation

As known, the rolling motion characteristics, amplitudes and accelerations, greatly influence the ability of a ship to operate and survive in bad weather. On the other hand, traditional computer codes for seakeeping calculations fail the forecasting of large amplitude rolling. There is a great need of using semi-empirical damping models and coefficients. This stresses the importance of campaigns of measurements as described in the paper, to get a deeper insight into the physical-mathematical modelling of the different contributions to rolling equation. Experimental tests on nonlinear rolling in a regular beam sea of a Ro-Ro ship model have been conducted by varying both the wave steepness and the wave frequency. The use of a parameter estimation technique, based on the least squares fitting of the stationary numerical solution of the nonlinear rolling motion differential equation, allowed to obtain informations on the damping model and on the linear and nonlinear damping coefficients. These exhibit a quite strong dependence on frequency that reduces the efficiency of constant coefficients rolling equation to simulate large amplitude nonlinear rolling. The results indicate that a good quality prediction model of nonlinear rolling cannot be based on constant coefficients time domain simulations. These can infact lead to incorrect estimates of rolling amplitudes even when the parameters have been obtained through high level parameter estimation procedures based on experimental data. The analysis indicates also a marked dependence of the effective wave slope coefficient on wave amplitude. The introduction of both these dependences on the rolling equation allows to reproduce the experimental results with great accuracy even at large amplitudes.

Identification of Single-Degree-of-Freedom Nonlinear Vibrating System by Piecewise Linear Approximation.

A new method for parameter identification of single-degree-of-freedom nonlinear vibrating systems is proposed in the time domain. The nonlinearity is approximated by piecewise linear characteristics. The nonlinear characteristics of restoring force and damping can be obtained from the linear spring constants and linear viscous damping coefficients identified on each piecewise linear section. Two experimental procedures, that is, the impact excitation method and the additional mass method without excitation, are performed. The response and excitation data for the impact excitation are required in the impact excitation method and the free vibration responses of the system with and without additional mass are required in the additional mass method. System parameters of the vibrating systems with various nonlinear characteristics, Duffing-type system and clearance system, can be accurately identified by both the procedures. It is shown that the additional mass method is valid for torsional nonlinear systems on which the measurement of excitation force is difficult.

Shock-Absorbing Mechanism of Human Leg.

The shock-absorbing mechanism to minimize the vertical acceleration when jumping down onto a hard surface was investigated. The dynamic model of a human leg was built with a link joint. The muscle model consisted of a contractile element, a elastic element and a viscous element. The equivalent linear spring constant and equivalent linear damping coefficient decreased upon lowering of the hip by varying the bending angles of the ankle and knee. The numerical results showed that shock could be absorbed passively by deep bending, and to a greater degree actively in the same bending position.

Experimental investigation of the lift component of roll damping

The lift force and turning moment acting on a model towed obliquely to the direction of motion have been measured. Two models were used; one of them was tested fitted with and without a rudder. These measurements were used to determine the magnitude of the lift coefficient and the point of application of the transverse force acting on the model. The data were then used to determine the lift component of the roll damping moment. It has been found that the equivalent linear damping coefficient due to lift is a nonlinear function of the forward speed of the ship.

Mechanical Model for Oscillating Water Column with Compressibility

Certain aspects of modeling of the oscillating water column (OWC) wave‐power device are examined, with particular reference to its behavior at representative off‐resonant frequencies. Although comparison is made with a particular study incorporating the use of compressible flow equations for the air column, this work uses a simple mechanical analog that dispenses with the thermodynamic equations and arrives at similar results. Bearing in mind that the compressible flow study was based on quadratic damping and this one is based on linear damping, the differences when the two results are compared on an energy basis are remarkably small. The compression is made possible by the calculation of an equivalent linear damping coefficient, for the model developed here, from the quadratic damping used elsewhere. The details of the conversion are also provided. For certain studies, it is therefore proposed that the mechanical analog may be used to simplify programming and facilitate evaluation.

Theoretical Condition for the Cxy=Cyx Relation in Fluid Film Journal Bearings

A computer program for calculating dynamic coefficients of journal bearings is necessary in designing fluid film journal bearings and an accuracy of the program is sometimes checked by the relation that the cross terms of linear damping coefficients of journal bearings are equal to each other, namely “Cxy = Cyx”. However, the condition for this relation has not been clear. This paper shows that the relation “Cxy = Cyx” holds in any type of finite width journal bearing when these are calculated under the following condition: (I) The governing Reynolds equation is linear in pressure or regarded as linear in numerical calculations; (II) Film thickness is given by h = c (1 + κcosθ); and (III) Boundary condition is homogeneous such as p=0 or dp/dn=0, where n denotes a normal to the boundary.

Damping and Phase Advance of the Tide in Western Hudson Bay by the Annual Ice Cover

Abstract Admittance analysts of yearlong current meter records and tidal height data shows that the annual ice cover affects the tidal currents and heights in Hudson Bay. Along the west coast of the bay, the semidiurnal tidal current and height are reduced by 10% and arrive twenty minutes earlier in winter than in summer. The diurnal tide shows similar characteristics but its weaker signal does not produce as reliable and consistent results. A simple theory of a rejecting Kelvin wave in a rectangular bay predicts that the extra damping by the ice cover causes the amphidromic points to move towards the shore along which the reflecting wave travels. Thus during the winter one sees relatively more of the incident wave than the reflected wave which causes a phase advance while the sum of their amplitudes decreases. Although further numerical modeling is required to determine the exact increase in linear damping coefficient caused by the ice cover, the simple analytical theory estimates it to be of the same or...