Linear Damping Coefficients Research Articles

Acoustic nonlinearities in a quasi 1-D duct with arbitrary mean properties and mean flow

Nonlinear temporal dynamics of acoustic oscillations in a quasi one-dimensional (1-D) duct are investigated using both numerical and analytical methods. The spatiotemporal nonlinear wave equation is derived for pressure oscillations in a quasi 1-D duct with axially varying cross-section and spatially inhomogeneous mean properties such as the velocity, temperature, density and pressure. Using the finite element method with quadratic interpolation functions, the linear Helmholtz equation is solved for the modal shapes and frequencies. With the modal shape as the weighting function, the standard Galerkin method is applied to transform the spatiotemporal wave equation into a second-order nonlinear ordinary differential equation (ODE) governing the time evolution of modal amplitudes. The limit-cycle amplitude and frequency of pressure oscillations are quantified analytically using the Lindstedt–Poincaré perturbation method. Furthermore, to capture the transient evolution to the limit cycle, the method of averaging is applied to the nonlinear temporal ODE for the modal amplitude. From the Lindstedt–Poincaré method, it is seen that a limit cycle exists when the linear damping coefficient μ in the nonlinear ODE is of the opposite sign as the quantity S that is a function of the coefficients of the quadratic and cubic terms in the ODE. For a given S, limit cycle is created or destroyed as μ changes sign, a scenario referred to as the “Hopf bifurcation.” The phase portraits and limit-cycle amplitudes obtained from numerical solution are in excellent agreement with those derived from the analytical techniques.

An investigation into the nonlinear effects in the roll motion of 2-D bodies by SPH method

Nonlinear effects in the roll motion of a 2–D body in the free surface are investigated by utilizing the Smoothed Particle Hydrodynamics (SPH) method. The continuity and Navier–Stokes equations are solved by employing the Weakly Compressible SPH (WCSPH) approach and our in-house code, which relies on the accumulated development process through the authors’ previous works including the fluid–solid coupling modeling by viscous penalty technique, (Tofighi et al. (2015)), and a hybrid Velocity-Variance Free Surface (VFS) and Artificial Particle Displacement (APD) algorithm (Ozbulut et al., 2014, 2018, 2020; Kolukisa et al., 2020). In this work, the GPU parallelization of the code is performed, to mitigate the computational burdens of the relatively high number of particles due to the long wavelengths generated by the roll motion of the cylinder. Presently, an alternative and exact approach in reducing the equation of roll motion with quadratic damping term is introduced. In parallel to this approach, a quadratic regression equation approximating the hydrodynamic roll moment is also employed. Aside from hydrodynamic coefficients, with linear and quadratic damping coefficients, vortex flow characteristics are also presented, qualitatively and quantitatively. It is understood from the results that, with the present capability, the WCSPH approach introduced is able to disclose nonlinearities inherently exist in the roll motion of oscillatory bodies in the free surface.

Integral Resonant Controller to Suppress the Nonlinear Oscillations of a Two-Degree-of-Freedom Rotor Active Magnetic Bearing System

Within this article, the nonlinear vibration control of the rotor active magnetic bearings system is tackled utilizing the integral resonant controller for the first time. Two integral resonant controllers are proposed to mitigate the system lateral oscillations in the horizontal and vertical directions. Based on the suggested control technique, the whole system dynamical model is derived as a two-degree-of-freedom nonlinear system (i.e., rotor system) coupled linearly to two first-order filters (i.e., the integral resonant controllers). The nonlinear autonomous system that governs the oscillation amplitudes of the controlled system as a function of the control parameters is extracted by applying perturbation analysis. The obtained autonomous system showed that the linear damping coefficients of the rotor system are functions of the control gains, feedback gains, and internal loop feedback gains of the coupled controller. Accordingly, the sensitivity of the rotor oscillation amplitudes to the different control parameters is explored. The stability margins and the optimal control gains are reported via plotting the different stability charts in two-dimensional space. The main acquired results demonstrated that the vibration suppression efficiency of the proposed controller is proportional to the product of both the control and feedback signal gains, and inversely proportional to the square of the internal loop feedback gains. In addition, the analytical investigations confirmed that the proposed integral resonant control method can force the rotor system to respond as a linear one with a single periodic attractor when the control parameters are designed properly. Finally, numerical simulations are performed that have illustrated the excellent correspondence with the obtained analytical results.

OC6 Phase Ia: CFD Simulations of the Free-Decay Motion of the DeepCwind Semisubmersible

Currently, the design of floating offshore wind systems is primarily based on mid-fidelity models with empirical drag forces. The tuning of the model coefficients requires data from either experiments or high-fidelity simulations. As part of the OC6 (Offshore Code Comparison Collaboration, Continued, with Correlation, and unCertainty (OC6) is a project under the International Energy Agency Wind Task 30 framework) project, the present investigation explores the latter option. A verification and validation study of computational fluid dynamics (CFD) models of the DeepCwind semisubmersible undergoing free-decay motion is performed. Several institutions provided CFD results for validation against the OC6 experimental campaign. The objective is to evaluate whether the CFD setups of the participants can provide valid estimates of the hydrodynamic damping coefficients needed by mid-fidelity models. The linear and quadratic damping coefficients and the equivalent damping ratio are chosen as metrics for validation. Large numerical uncertainties are estimated for the linear and quadratic damping coefficients; however, the equivalent damping ratios are more consistently predicted with lower uncertainty. Some difference is observed between the experimental and CFD surge-decay motion, which is caused by mechanical damping not considered in the simulations that likely originated from the mooring setup, including a Coulomb-friction-type force. Overall, the simulations and the experiment show reasonable agreement, thus demonstrating the feasibility of using CFD simulations to tune mid-fidelity models.

The non‐conservative forced Toda oscillator

AbstractThe vibration theory has become a hot topic in both mathematics and engineering. This work is an attempt to reduce the complexity of the Toda model and to exploit the advantages of annihilator operators. The reducing rank method is proposed for the outcoming jerk oscillator model. The modified homotopy perturbation method (HPM) is presented to solve the resulting model. Besides, to simplify the model, the outstanding features of the resonance reaction are preserved. It is declared that the obtained results are robust as they can exist in a wide range of oscillators. It is found that the amplitude of oscillation greatly affects its periodic nature and has an exponential form. The dynamics of a non‐autonomous oscillator in which the frequency of the external force depends on the dynamical variable is studied. It is also revealed that the linear damping coefficient plays a dual role in the stability behavior, while the nonlinear counterpart exerts a destabilizing influence on the resonance response.

Linear and quadratic damping coefficients of a single module of a very large floating structure over variable bathymetry: Physical and numerical free-decay experiments

The linearity assumption is widely used when acquiring the hydrodynamic coefficients of a floating structure. However, the linear damping is frequently underestimated, especially for the natural frequency. To investigate the sloping seafloor effects on the damping terms of a single module of a semi-submersible Very Large Floating Structure (VLFS), this paper revisits the conventional formulation and further proposes the direct integration method for obtaining the linear and quadratic damping coefficients from free-decay tests. Numerical free-decay simulations of the single module over variable bathymetry are carried out by the CFD numerical tank. Corresponding model tests are also implemented to verify and validate against the numerical solutions. The effects of the sloping seafloor, as well as the water depth, on the hydrodynamic coefficients are investigated based on the validated CFD modeling. Both numerical and experimental results indicate that the acquisition of the linear and quadratic damping coefficients is sensitive to the data-processing and identification approaches. For the case studied in present paper, the identification errors introduced by the conventional method are 1.5% while they are 0.5% using the direct integration method. The quadratic damping coefficient for heave mode decreases about 10.4% when the sloping angle increases from 0 to 6 deg.

Verification and validation of CFD simulations for semi-submersible floating offshore wind turbine under pitch free-decay motion

In this paper a detailed verification and validation study is performed to evaluate the accuracy of CFD simulations for a semi-submersible floating offshore wind turbine in pitch free-decay motion. The simulations are carried out with both linear and dynamic mooring models. The linear and quadratic damping coefficients of the semi-submersible are derived from the pitch motion data. In the verification study, a least-squares method using the observed order of convergence in combination with a data quality measure for different spatial and temporal refinement is applied to estimate the discretization errors. The numerical uncertainty is quantified by applying a safety factor to the summation of numerical errors. The numerical solutions are validated against the experimental measurements obtained from wave tank test of OC5-DeepCWind semi-submersible. Validation is achieved in the pitch natural period and the linear damping coefficient within the level of validation uncertainty. While the numerical simulation under-predicts the quadratic damping coefficient. A guideline is provided for verification and validation study of a CFD code application for floating marine structures under decay tests.

Two dimensionless parameters and a mechanical analogue for the HKB model of motor coordination

The widely cited Haken–Kelso–Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue. We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model. The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations. This observation has implications for the development of virtual partner interaction and the human dynamic clamp, and potentially for the HKB model itself.

Experimental study of the influence of the rope material on mooring fatigue damage and point absorber response

The present work investigates the influence of mooring material on the hydrodynamic responses of a point absorber by experimental tests. Three taut mooring systems are considered with different materials, namely wire, nylon and polyester. In order to study the influence of mooring material on point absorber hydrodynamic responses, the other factors that may influence system dynamics are kept as unchanged as possible. During the model tests, the pretension of the three mooring systems are set to be similar and the parameters of wave conditions are identical. Decay tests are conducted to determine the system natural periods and linear damping coefficients in different modes. Totally seventeen regular wave cases and two irregular wave conditions are tested to study the motion responses of the buoy as well as mooring dynamics. The response amplitude operators of motion responses and mooring tensions are obtained by the regular wave model tests. Three wideband spectral methods as well as two simplified spectral methods are applied to study mooring fatigue damage in the random wave conditions, and the results are compared with the estimations of rainflow counting approach. It is shown that the α0.75 method provides the most precise estimations of fatigue damage in most mooring cases among spectral methods discussed this study.

A Simplified Design Strategy of Nonlinear Fluid Viscous Dampers for MDOF Structures

Energy dissipation devices are increasingly used as vibration mitigation systems owing to their efficiency. Therefore, many attempts were made to facilitate their design process. The design of fluid viscous dampers is summarized in the selection of suitable damping values and their location in the structure. This paper presents a strategy that estimates the damping coefficient value for nonlinear fluid viscous dampers in a straightforward manner. This minimizes the design time, and thus, more flexibility is offered for further design considerations. Combining a methodology that predicts a linear damping coefficient based on the concept of target-damping ratio, then, applying an equivalent energy-consumption approach to determine the nonlinear damping coefficient. This research gives a description of the method and its implementation by studying a structure equipped with a set of additional fluid viscous dampers. The results illustrate the efficiency and simplicity of the method.

Identification of an Autonomous Underwater Vehicle hydrodynamic model using three Kalman filters

The controllability and maneuverability of an Autonomous Underwater Vehicle (AUV) in practical applications need to be properly validated and assessed before the prototype is finalized for manufacturing. With regard to mathematical system model of the AUV, hydrodynamic coefficients have a dominant effect on the quality of vehicle pre-testing and evaluation, which is crucial to be estimated with adequate accuracy to curb the uncertainty from modeling simplifications. The standard time domain discrete Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) algorithms are two promising numerical approximation approaches for hydrodynamic identification technique with favourable computational complexity and acceptable estimation precision. In this paper, a study with respect to time domain discrete optimized UKF (OUKF) algorithm based on recursive tuning rule and update gradient descent lemma for a typical prototype known as NPS AUV II will be proposed to enhance adaptability and prediction performance of the identification approach with appropriate verification. In addition, Auto Regressive Moving Average (ARMA) noisy model needs to be included into three Kalman Filter (KF) algorithms to further improve the estimation precision and perturbations inhibiting performance. Pre-processing numerical model validation and non-dimensional viscous linear damping coefficient identification based on three KF algorithms will be implemented respectively to provide an assistant pre-assessment for the vehicle. In accordance with comparable outputs and estimation-experiment errors, the OUKF identification algorithm is certified to be more precise and superior compared with EKF and UKF approaches in the presence of ARMA noisy model.

Equivalent Beam Model for Spatial Repetitive Lattice Structures with Hysteretic Nonlinear Joints

This study presented an equivalent modeling method for analysis of the fundamental harmonic response of the spatial repetitive lattice structures with hysteretic nonlinear joints. Firstly, the joint was modelled as a spatial nonlinear spring-damper system with bilinear hysteresis, the equivalent linear stiffness and viscous damping coefficients of the hysteretic joint were derived using the describing function method. Subsequently, high-precision displacement and rotation fields that consider the warping and distortion of the cross section of the lattice structure were presented for the repeating element and the equivalent continuum modeling method based on the principle of energy equivalence was adopted to obtain an 8-DOFs spatial beam element for the repeating element of the lattice structure. Afterwards, the equations of motion of the equivalent beam model for the lattice structure were established in the frequency domain and solved by using the Newton-Raphson method. In the numerical studies, the joints with axial nonlinearity and rotational nonlinearity were considered, and the influences of joint parameters and excitation amplitude on the dynamic response of the lattice structure were investigated. The correctness of the presented modeling method was verified by comparison with the nonlinear finite element model of the original structure.

Mooring Analysis of a Floating OWC Wave Energy Converter

This investigation focuses on the modelling of a floating oscillating water column (FOWC) wave energy converter with a numerical code (ANSYS AQWA) based on potential flow theory. Free-floating motions predicted by the numerical model were validated against experimental data extrapolated from a 1:36 scale model device in regular and irregular sea states. Upon validation, an assessment of the device’s motions when dynamically coupled with a four-line catenary mooring arrangement was conducted at different incident wave angles and sea states ranging from operational to survivable conditions, including the simulation of the failure of a single mooring line. The lack of viscosity in the numerical modelling led to overpredicted motions in the vicinity of the resonant frequencies; however, the addition of an external linear damping coefficient was shown to be an acceptable method of mitigating these discrepancies. The incident wave angle was found to have a limited influence on the magnitudes of heave, pitch, and surge motions. Furthermore, the obtained results indicated that the mooring restoring force is controlled by the forward mooring lines under the tested conditions.

Nonlinear Primary Frequency Response Analysis of Self-Sustaining Nanobeam Considering Surface Elasticity

This paper is focused to investigate the effects of nonlinear sources, including viscoelastic foundation and geometrical nonlinearity along with the surface elasticity and residual surface stress effects on the primary frequency response of a harmonically excited nanoscale Bernoulli-Euler beam. Due to large surface-area-to-volume ratio, the theory of surface elasticity as well as residual surface stress effects are taken into account within the beam models. The Galerkin approach accompanied by trigonometric shape functions is utilized to reduce the governing PDEs of the system to ODEs. The multiple scales perturbation method theory is applied to compute the nonlinear frequency response of nanobeam. The effects of linear and nonlinear viscoelastic damping coefficient of the medium, crystallographic directions of [100] and [111] of anodic alumina, geometrical nonlinear term and geometrical property on the nonlinear primary frequency response of nanoscale beam are investigated. The results show that theses parameters have a significant effect on the nonlinear frequency response of nanobeams in the case of primary resonance.

Very large amplitude vibrations of flexible structures: Experimental identification and validation of a quadratic drag damping model

This article presents a theoretical, numerical and experimental study of resonant structures undergoing very large amplitude vibrations. The purpose of this work is to validate a model for the damping due to the action of the air on a structure’s single-mode response in the steady-state. Experiments are performed on cantilever beams and beam assemblies of various sizes, from centimetric to micrometric, under harmonic base excitation. Dimensionless linear and nonlinear modal damping coefficients are simultaneously identified by means of frequency-domain identification techniques. These measurements demonstrate the pertinence of the presented model.

Damped Tailgate System Drop Dynamics and Evaluation of Strut Force for Durability

<div class="section abstract"><div class="htmlview paragraph">This paper describes a simplified CAE simulation for the calculation of peak strut force in a damped tailgate system. Tailgate systems in the past included a torsion rod for lift assist, but did not include damping in the opening/drop direction. Newer tailgate systems include a strut to provide damping during drop and possibly lift assist. These tailgates may also be designed to experience a free-fall for a few degrees after being unlatched and before the strut engages. When the latch is released, the tailgate accelerates as it falls freely and then encounters a rapidly increasing damping force as the strut engages. The deceleration of the gate depends upon the damping force, yet the damping force depends upon the deceleration of the tailgate, making a precise calculation of the damping force difficult. Damping characteristics of the strut depend on the volume, the viscosity of oil and its flow restrictions across the piston. These parameters are not known early in the design process.</div><div class="htmlview paragraph">This study presents a method to back calculate the damping force on a tailgate solely based on the desired drop time, and does not require physical measurements of the strut. An ABAQUS explicit analysis with beams and connectors is defined to simulate the tailgate drop event. The simulation calculates the damping force, based on a linear damping coefficient and the velocity of the strut at each incremental drop position. Peak strut force is found to occur mid-way through the drop event and is greater than the force of the fully compressed strut. The simulation shows good correlation to measured time history for strut force, tailgate drop angle and strut linear motion.</div></div>

Optimum suspension design for non-linear half vehicle model using particle swarm optimization (PSO) algorithm

This paper is considered with a non-linear suspension design for half vehicle model by using particle swarm optimization technique. To analyze the ride comfort, a five-degree of freedom system is built, and it is integrated with the Particle Swarm Optimization (PSO) for optimizing the vehicle vibrations. A multi-objective function is proposed as the sum of the minimum seat and vehicle body acceleration, the minimum suspension deflection and the constraints and the design variables of the optimization problem are selected as the spring and damping coefficients of the front and rear suspension and the non-linear spring and the linear damping coefficients of the seat. The simulations are carried out and the results are compared with the non-optimized values. It is demonstrated that the vehicle vibration is decreased significantly with the help of the optimum values of the suspension parameters.

An Empirical Formulation for the Damping Ratio of Shape Memory Alloy for Base-Isolated Structures

Shape Memory Alloys (SMAs) are now widely used as a damping element into the isolation systems. The pre-stressed SMAs exhibit hysteretic damping through a nonlinear flag-shaped hysteresis loop. Many nonlinear models of the SMA are available to depict such behavior. The nonlinear models require a lot of effort and computational time for the analysis of base-isolated structures. Therefore, the codes recommend that a nonlinear model can be replaced by an equivalent linear model in the analysis. Linearization is a method to convert the nonlinearity of a system into a system with analogues linear parameters. This paper proposes an empirical equation for a damping ratio to get a linear damping coefficient of the SMAs which can be used in the seismic analysis of base-isolated structures. The evaluation of any damping ratio using the traditional system identification method does not give precise solutions due to variation in hysteretic parameters and the unpredictable nature of an earthquake. The empirical equation is proposed using a set of optimal statistical data obtained from the seismic analysis of a base isolated structure. Moreover, analysis of the base isolated structure using the newly modified equivalent elastic-viscous SMA model gives comparable and conservative results with a nonlinear SMA model as compared to the existing elastic-viscous SMA model. Since the hysteresis parameters are used to derive the empirical equation for the damping ratio, this equation is also applicable for any type of structure.

Hydrodynamic derivative determination based on CFD and motion simulation for a tow-fish

A new methodology to determine hydrodynamic derivatives of a tow-fish underwater vehicle using computational fluid dynamics (CFD) was presented. Hydrodynamic derivatives were derived from a second-order modulus expansion and consist of added mass, linear damping, and nonlinear damping coefficients. Added mass coefficients were analytically obtained using strip theory. All components of linear and nonlinear damping coefficients were determined by CFD simulations without any pre-elimination. The determination procedure for damping coefficients is as follows. First, a series of simulations was prepared for various situations reflecting the actual operating conditions of the tow-fish. Then, actual CFD simulations were performed for each of the cases, and 6 degree-of-freedom (6-DOF) forces and moments acting on the tow-fish were obtained. Finally, all linear and nonlinear damping coefficients were determined by optimized fitting of the CFD results using the least-squares procedure. The effectiveness of the determined hydrodynamic derivatives was demonstrated by reproducing the 6-DOF forces and moments and comparing them with those obtained from the CFD simulations. The relative significance of each coefficient was estimated by a global comparative view of normalized coefficients. To demonstrate the applicability of the current approach, 6-DOF simulations of the tow-fish for three different scenarios (L -, U -, and S -turn scenarios) at various towing speeds were conducted. The efficacy of the current methodology was strengthened by graphical and physical analyses of the virtual simulation results.

A “direct five-step procedure” for the preliminary seismic design of buildings with added viscous dampers

In the present work a direct procedure for the preliminary seismic design of building structures with added dampers is described which represents the simplification of the so-called “five-step procedure” originally developed in 2010 by some of the authors. The procedure is applicable to yielding frame structures with a generic along-the-height distribution of inter-storey viscous dampers. It is aimed at guiding the structural engineer through the sizing of both viscous dampers and structural elements making use of an equivalent static analysis approach. First, the peak structural response under earthquake excitation is reduced by imposing an overall reduction factor accounting for both the ductility demand and the viscous damping provided by the added dampers. Second, linear damping coefficients are calculated in order to reduce the structural response according to the selected target damping ratio. Then, analytical formulas allow the estimation of peak velocities and forces in the dissipative devices, and an energy criterion is used to identify the non-linear mechanical characteristics of the actual manufactured viscous dampers. Finally, the internal actions in the structural elements are estimated through the envelope of two equivalent static analyses (ESA). At this initial stage of the research, the procedure appears suitable for the preliminary design phase, while correction factors for the higher modes contributions need to be applied to improve its accuracy, especially for high-rise buildings. A numerical verification of the final behaviour of the system by means of non-linear time-history analyses is recommended. An applicative example is finally developed to highlight the soundness of the procedure.