Coulomb Damping Research Articles

Generative quasi-zero stiffness paradigm for vibration isolation by constraining the constant force with hardening boundaries

Quasi-zero stiffness (QZS) isolators are widely used to mitigate the low-frequency vibration. This paper proposes a new QZS paradigm for vibration isolation utilizing constant force with hardening boundaries (CFHB). Different from the conventional parallel connection of positive and negative stiffness elements, the CFHB directly exhibits the QZS characteristic which is generated when the uniform magnetic field is converted to a non-uniform one. Therefore, it avoids complex assembly and material deformation, and eschews the reliability issues related to stiffness combination. Concretely, the radially magnetized shaft has a uniform area in the middle part which can be distorted by a coaxial sleeve made of high permeability material. A constant magnetic force is generated in this process. Two hardening boundaries formed by two pair of repulsive magnets are applied to constrain the QZS stroke and maintains the working position. The shaft's magnetic field is studied to clarify the filed distribution, and the distortion by the sleeve is simulated by finite element analysis (FEA). The force-displacement relationship of the case study is proven to be continuous and fitted by a polynomial form. The influence of dimension parameters on the performance of vibration isolator is systematically investigated to guide the design. Dynamic differential equation considering Coulomb damping and viscous damping is established and solved applying the Runge-Kutta Method (RKM) to obtain the vibration response. A prototype is fabricated to experimentally validate its performance, demonstrating the QZS properties in quasi-static calibration and the vibration isolation effect under the frequency sweep and periodic excitations. The generative QZS offers a new paradigm with a simple structure, suggesting its substantial potential for practical deployment.

Multi-freedom effects on a raft-type wave energy convertor- hydrodynamic response and energy absorption

This research delves into the effects of multiple degrees of freedom on a raft-type wave energy converter (WEC), using both experimental and computational methods. The WEC features two interconnected floating bodies linked by a specialized parallelogram joint, allowing for distinct heave, pitch, and surge motions. The power take-off (PTO) system, integrating Coulomb damping, is analysed for its impact on energy absorption. Despite varying damping amplitudes, peak energy absorption is consistent across different wave periods, as shown in scaled model tests. Computational simulations further investigate the multi-freedom effects by imposing constraints on various degrees of freedom, thus elucidating their individual and combined influences on energy absorption. Preliminary results indicate that the addition of degrees of freedom can enhance the WEC's efficiency, especially through coordinated release of the bodies' relative pitch and surge motions. The study also finds that the energy absorption efficiency from pitch motion depends on the release of surge motion. Moreover, the energy distribution among each degree of freedom is significantly affected by the device's multi-freedom characteristics. This research provides valuable insights and practical guidelines for the development of advanced multi-freedom WEC technologies.

A Two-Part Analytical Solution for the Shock Response Spectrum with Coulomb Damping

A Two-Part Analytical Solution for the Shock Response Spectrum with Coulomb Damping

Experimental and numerical analysis of power take-off control effects on the dynamic performance of a floating wind-wave combined system

The integration of wave energy converters (WEC) into floating offshore wind turbines (FOWT) is regarded as a promising approach for comprehensively harnessing deep-sea energy and reducing the levelized cost of energy. This study aims to investigate the WEC power take-off (PTO) control effects on the dynamic performance of a floating wind-wave combined system, wherein three heaving-type WECs are integrated into a semi-submersible FOWT. In particular, the hydraulic PTO is modelled as a Coulomb damping system to enable a more realistic analysis. The aero-hydro-servo-elastic-mooring coupled numerical simulations and 1:50 wave basin experimental data demonstrate good agreement. It is observed that wave power production varies significantly with different control settings, potentially reaching 24.0 % of the overall hybrid energy production with proper control parameters. Furthermore, platform pitch oscillation generally shows a decreasing trend with increasing damping forces under below-rated and rated conditions, while showing minimal influence when above-rated. Conversely, tower base damage equivalent load (DEL) tends to initially decrease and subsequently increase across all examined conditions. This consistent convexity indicates the potential use of DEL as a performance index for optimising PTO control. In summary, power increment, motion reduction, and load mitigation could be achieved concurrently with appropriate control design. For instance, an extra 0.48 MW wave power could be produced under the rated condition, while a 10.4 % reduction in tower base DEL and a 6.57 % mitigation in platform pitch oscillation could also be achieved at the same time.

How to model vibration damping in complex materials and structures

Damping is perhaps the most challenging aspect of any model where damping is relevant. For example, an accurate damping model is often important when modeling vibration of metamaterials, vehicles, and machinery. Mass density is easily and accurately calculated from simple measurements of weight and volume. Elastic moduli are easily and accurately calculated from simple measurements of stress and strain. Large databases of mass density and elastic moduli are readily available and accurate for most materials. But damping is elusive. Many models have been proposed in many different research communities. Published numerical values for the parameters are in short supply. But the choice of damping model may dramatically affect the predicted response. The damping model is likely to make the difference between a useful result and a useless result. This presentation seeks to collect and summarize knowledge related to damping to help us model damping more accurately and more efficiently. The three most common models of damping are reviewed. These are material damping, viscous damping, and Coulomb damping. For each, an analysis is presented that begins with the fundamental assumptions and ends with the inclusion of the damping mechanism in finite element models. [Work supported by ONR under Grant N00014-22-1-2785.]

Numerical and experimental investigation of a new model of friction damper in diagonal brace under cyclic loading

Owing to the performance of prefabricated friction dampers, which are rapidly increasing, the resistance of structural systems against earthquakes increases and depreciates the energies generated by earthquakes in the structure. It is important to use passive control systems to control the vibrations of a structure at a single level. However, in the design, they cannot depreciate the incoming energy from earthquakes at different levels. Friction dampers focus on displacement variables and are mostly used in steel structures according to the rules of a Coulomb damper or a friction brake that converts kinetic energy into heat through friction. In this study, a new type of friction damper, called a flat cylindrical friction damper, was designed using brake pads in a diagonal brace. The performance and seismic resistance of this system, as well as the amount of energy loss, were investigated. In this study, 18 laboratory tests were conducted using the universal device to check the research variables, their hysteresis diagrams were shown, and the configuration of the damper was designed such that grooved bolt connections and twin steel cables of different sizes and thicknesses were used. Moreover, the way the internal and external components are placed in the damper is such that innovation in tension and pressure has been created. In addition, the movement of the cylindrical element in the damper increased the amount of friction, owing to the presence of two types of brake pads with friction coefficients of 0.11 and 0.16 and the bolt connections. This demonstrates the damper performance under different sliding forces at different friction levels. The geometric dimensions, thickness of the brake pad, number and size of bolts, diameter of cables, sliding force, and location of the damper were among the variables investigated in this study. Brake pads, steel cables, and bolt connections are important and economically affordable. The seismic performance of the intended frame was investigated using 80 different damper models and the placement of the research variables. The desired optimal models were modelled using Abaqus software, analysed, and designed. This system aims to reduce the relative horizontal displacement of floors and increase the amount of energy absorption. An increase in theaxial-force created in multiple loading cycles always causes damage to damper components and frames. In this study, we used a special multi-level geometry to reduce the axial-force created in the damper, the relative displacement of the frame, and damages in the elements, as well as increase the ductility. The results show that the friction surfaces of steel plates and brake pads is very high due to the displacement and damping of the cables. Additionally, with the consumption of energy and its absorption by the damper in cyclic loads, displacement is easily controlled. The seismic response of the structures in terms of frame and damper displacement, base shear forces, and energy absorption is also observed. Numerical studies confirmed that the intended damper is an independent seismic-resistant member in critical building structures when high seismic performance or seismic resilience during moderate and strong earthquakes is desirable.

Экспериментальное определение нелинейной функции демпфирования механических систем

In this article, the coefficients of the nonlinear damping function of a mechanical system with one translational degree of freedom are determined from an experimentally obtained oscillogram of free vibrations. The function is modeled using three types of damping: coulomb damping, linear viscous, and nonlinear viscous damping. Numerical values of the damping coefficients are identified. The characteristic of the dissipative force as a function of displacement is obtained, and is used to find the amount of energy dissipated over a time period. An equivalent relative damping ratio is approximated using the energy balance method and then used to perform numerical integration of the equation of motion. A satisfactory match of the envelope curve and the phase of the vibrational process is demonstrated by comparing the calculated oscillogram to the experimental one. The damping function parameters can be further refined by approximating the experimental amplitudes. The obtained value of the relative damping coefficient can be used to solve nonlinear problems in the area of dynamics of weakly damped systems.

Damage hysteresis dynamic model of tangled metal wire based on higher-order dry friction (HDF)

Tangled metal wire (TMW) is a kind of porous hyperplastic high damping material, which can realize the vibration isolation of key equipment. However, the unclear mapping relationship between internal microscopic damage and overall macroscopic performance made it difficult to construct a damage hysteresis dynamic model. This limits the application and development of TMW vibration isolation. In this paper, a strategy to analyze the dynamic damage evolution of TMW was proposed. Firstly, a cross-scale relationship between internal damage in TMW and macroscopic performance degradation was established. Furthermore, a damage hysteresis dynamic model was constructed by considering the high-order dry friction (HDF). The complex spatial spiral structure and preparation process parameters were described by using the HDF. Then, the damage factors of each component force were built and analyzed: The damage factor of HDF could be used to characterize the damage evolution of the metal wire inside TMW and the cross-scale correlation between microscopic damage and macro performance degradation, whereas the degree of hysteresis characteristics of TMW could be characterized by the damage factor of hysteretic coulomb damper. Finally, the damage hysteresis dynamic model was evaluated through residual analysis, and it has high prediction accuracy in more than 80 % of the cycles R2>0.7. This paper provided a new idea for the modeling of TMW life prediction.

Prediction of energy dissipation by analytical solution to combined viscous and Coulomb damping

Damping in mechanical vibration systems primarily falls under (i) linear viscous and (ii) non-linear Coulomb damping. Practically, most systems possess characteristics of both linear viscous and non-linear Coulomb damping. In this work, the method of phase space coordinate averaging is used to obtain an approximate analytical solution for the system’s response when it is subjected to both viscous and Coulomb damping. The accuracy of the proposed solution is examined by comparing it with the numerical simulations and experimental results. The energy dissipated by both damping mechanisms is quantified using semi-analytical integrals. The analysis of energy dissipation quantifies the conditions for natural frequency, damping ratio, and coefficient of friction, at which the energy dissipated by viscous damping is equal to the energy dissipated by Coulomb damping. This has led to the introduction of a non-dimensional quantity called the dissipation number (D). With certainty, it can be stated that if D>1, friction is dominant, and if D<1, viscosity is dominant.

Limits on the Range and Rate of Change in Power Take-Off Load in Ocean Wave Energy Conversion: A Study Using Model Predictive Control

Previous work comparing power take-off (PTO) architectures for ocean wave-powered reverse osmosis suggests that variable displacement in the wave energy converter (WEC)-driven pump does not offer a significant performance advantage. A limitation of that study is that the WEC was subject to a constant load within a given sea state (“Coulomb damping”) and did not account for controlled, moment-to-moment variation of the PTO load enabled by a variable displacement pump. This study explores the potential performance advantage of a variable PTO load over Coulomb damping. Model predictive control is used to provide optimal load control with constraints on the PTO load. The constraints include minimum and maximum loads and a limit on the rate of load adjustment. Parameter studies on these constraints enable conclusions about PTO design requirements in addition to providing an estimated performance advantage over Coulomb damping. Numerical simulation of the Oyster 1 WEC is carried out with performance weighted by historical sea state data from Humboldt Bay, CA. The results show a performance advantage of up to 20% higher yearly-average power absorption over Coulomb damping. Additionally, the parameter studies suggest that the PTO load should be adjustable down to at least 25% of the maximum load and should be adjustable between the minimum and maximum loads within a few seconds.

Longitudinal displacement control of a long-span suspension bridge under random traffic loads using a polyimide friction damper

This paper proposes a novel method to control the cumulative displacement of the girder end of a long-span suspension bridge caused by traffic loads using friction dampers. Friction dampers have Coulomb damping characteristics and high initial stiffness, providing a significant damping force at low speed to suppress the longitudinal deformation of the girder end. The working mechanism of the friction damper is explained, and the performance of the proposed polyimide (PI) friction damper is tested. A random traffic load model of a long-span suspension bridge is established, and the longitudinal displacement, reaction force time history, longitudinal peak displacement, and cumulative longitudinal displacement (CLD) under different yield forces of the PI friction damper are analyzed. We propose the excess ratio δ of the yield displacement to determine the optimum yield force. The influences of different vehicle speeds and traffic flows on the performance of the PI friction damper are discussed. The results show that the PI friction damper substantially reduces the longitudinal displacement, longitudinal peak displacement, and CLD of the girder end; the reduction rate of the CLD exceeds 57% when the yield force of the PI friction damper is 1000 kN. The vehicle speed and traffic flow have negligible influences on the CLD of the long-span suspension bridges equipped with PI friction dampers in the normal working range.

Influence of coulomb damping on wave propagation behaviors of nonlinear nonconservative phononic chains/lattices

Fascinating nonlinearity-induced behavior of phononic crystals (PCs) has recently become a hot research topic in the community. However, due to the limitations in the analytical modelling of damping in dynamic systems, the study of damped PCs has not received proper attention. In this paper, the influence of Coulomb damping on the wave propagation behavior of cubically nonlinear monoatomic phononic chains is investigated. To do so, the nonlinear dispersion relation is obtained analytically using the well-established multiple scales method and the band structure of the damped nonlinear chains is compared to the ones corresponding to the linear and nonlinear undamped chains. Due to the coupling between the amplitude and the frequency, stemmed from the nonlinear nature of the chain, Coulomb damping can lead to lower dispersion frequencies in the chain. The formulation and results are then expanded to 2D nonlinear lattices. The present manuscript is the first attempt to capture the effect of Coulomb damping on the wave propagation behavior of nonlinear lattices and the results put us one step closer to developing a comprehensive analytical model for the behavior of damped PCs which can in turn lead to invaluable design concepts for nonlinear nonconservative wave-manipulation devices.

Design and modeling of a novel triple-magnet magnetic dynamic vibration absorber

A novel Triple-magnet Magnetic Dynamic Vibration Absorber (TMDVA) with Eddy Current Damping (ECD) and Coulomb damping is proposed in this paper. A multi-storey structure is selected as the vibration reduction object. First, the equivalent dynamics model of the TMDVA multi-storey structure vibration reduction system is established. Then the nonlinear magnetic force model is built according to the magnetizing current theory. Next, an ECD accurate calculation model is developed and derived based on Faraday’s law of electromagnetic induction. Because of the model, the ECD coefficient can be designed and adjusted arbitrarily. The full text indicates that adjusting the ECD coefficient properly can not only improve the vibration reduction effect of the TMDVA, but also reduce the sensitivity of the vibration reduction effect to the excitation amplitude as well as enhance the broadband vibration reduction performance and the robustness of the TMDVA. With its simple structure, the TMDVA is easy to be implemented and miniaturized. And the low frequency vibration reduction effect of the TMDVA is better than other existing magnetic dynamic vibration absorbers. Taken together, the TMDVA has potential application value in passive vibration reduction of engineering structures.

Linear stability and nonlinear rotor responses of the gas foil bearing with multiple sliding beams

This article reports the stability and dynamic characteristics of a passive gas foil bearing (MSGFB) in which multiple sliding beams provide the elastic support and Coulomb damping. The linear and nonlinear dynamic models of rotor-MSGFB system are established, both considering the complex contact constraints inside foil structures. The instability mechanism of rotor-foil bearing system is revealed from the equivalent dynamic stiffness and the critical whirl speed. The half speed whirl and the frequency locked vibrations are both observed in the simulations, indicating the nonlinearities of gas film and foil structure respectively. The results also show that larger rotor preload of MSGFB help suppress the half speed whirl vibrations and assist improving high speed rotor stability especially under the friction effect.

Investigation of pendulum damping using an angle sensor and video analysis: combination of viscous and dry friction

Viscous and Coulomb’s friction combined contribute to the damping of a pendulum; however, they are usually treated as a silo effect. In this study, we investigate the combination of viscous and Coulomb damping in compound pendulum oscillation, employing two modern instruments that are cheap and handy to quantify the experiments, which are video analysis and a microcontroller angle sensor. From the model fitting, variables and parameters of the motion such as angular frequency, period, moment of inertia, and viscous and frictional parameters are obtained with high accuracy and consistency between angle sensor and video analysis data. This study contributes to the use of technology in physics education, specifically in physics experiments. It is relevant for high school physics and university introductory physics as the basis to advance study in the damped oscillation of the pendulum.

Coulomb friction effect on the forced vibration of damped mass–spring systems

This paper aims at assessing the effect of dry friction on the dynamic behaviour of a damped mechanical system subject to harmonic forcing. Previous work on friction damped systems highlighted that not including other forms of damping in the dynamic analysis can lead to unrealistic results such as the presence of infinite resonances. In this contribution, an exact solution is derived for the continuous steady-state response of multi-degree-of-freedom systems with a contact between one of the masses and an external wall, using Coulomb’s law to model the friction force and a modal damping model to account for the system’s damping. Closed-form expressions are also derived for the amplitude and phase of the continuous responses, while stick–slip responses are investigated by using an ad-hoc numerical approach. In addition, analytical and numerical results are used for exploring the features and the motion regimes of the dynamic response, leading to the following conclusions: (i) system’s damping has a limited effect on low- and high-frequency behaviours, on the presence of invariant points and inversions across the transmissibility curves and can therefore be neglected, in non-resonant conditions, in the analysis of structures where dry friction is the main source of dissipation; (ii) when the damping of the system is accounted for in the mechanical models along with Coulomb damping, finite resonant peaks are also obtained in continuous sliding regime and their amplitude decreases linearly with the friction force generated in the contact.

Damping parameter estimation using topological signal processing

Energy is dissipated in engineering systems through a variety of different, typically complex, damping mechanisms. While there are several common damping models with physical interpretations, it is often necessary to examine noisy, experimental data in order to identify and fit the damping parameters. One class of methods fits the damping parameters based on the decay envelope of the signal’s peaks based on some assumed damping mechanism. While there exist results in the literature, especially for viscous damping, to guide the selection of the spacing between the peaks for the optimum damping ratio identification, these methods often overlook the difficult and classic problem of identifying the signal’s “true” peaks in the presence of noise. Therefore, in this work we utilize tools from Topological Data Analysis (TDA) to address the problem of robust and automatic power-law damping identification from decay or free-response function. The decay function can be obtained from initial excitation, or using random decrement method thus making our approach a viable tool for operational modal analysis. We present our approach using one-dimensional examples, and describe extensions to multiple degree of freedom systems. Our damping identification framework uses the persistent homology of zero dimensional (0D) sub-level sets, one of the main tools of TDA, to separate the true topological features of the signal (i.e. the peaks and valleys) from measurement noise. Using persistent homology we are able to automatically represent the damped response in a noise-robust, two-dimensional summary of the peak–valley pairs called the persistence diagram. The persistence diagram is then analyzed using two methods: (1) a theoretical analysis of the significant persistence pairs and (2) function fitting to the persistence space. We present theoretical results that establish the class of general power-law damping terms where our approach applies, and develop specific tools to identify damping parameters for viscous, coulomb, and quadratic damping. We show that our approach is computationally fast, operates on the raw signal itself without requiring any pre-processing, and reduces the number of decisions needed by the user to perform the needed calculations. The results are validated using a combination of numerical and experimental data.

Identification and parameter estimation of non-polynomial forms of damping nonlinearity in dynamic systems

Damping is often assumed to viscous and linear in modelling a dynamic system. However, damping is inherently nonlinear and often is in non-polynomial forms such as Coulomb damping, bilinear, or quadratic damping. Recently, non-polynomial forms of damping have found wide applications in vibration isolators and absorbers. However, not much study is available for the identification of these non-polynomial forms of damping. The present work attempts to develop an identification and parameter estimation procedure for two non-polynomial damping forms, i.e., bilinear damping and quadratic damping. First, response harmonic amplitudes are formulating using Volterra series and equivalent linearised damping; and then they are compared with representative polynomial forms, i.e., square damping and cubic damping. Bilinear damping was identified employing a method of harmonic probing and studying first and second harmonic amplitude characteristics. Similar way, quadratic damping is identifying using first and third harmonic characteristics. Finally, a parameter estimation procedure is developed as a step-by-step algorithm and demonstrated through numerical simulation and error analysis. It is shown that reasonable accuracy can be obtained in estimation of the nonlinear parameters with proper selection of excitation frequencies and excitation levels.

A tunable hybrid damper with Coulomb friction and electromagnetic shunt damping

Vibration dampers with tunable properties are of practical significance to accommodate the need of vibration control under different circumstances. Among major parameters, tunable damping to deliver required accuracy within a given frequency range is difficult to achieve in practice. This paper proposes a hybrid tunable damper by combining the advantages of both Coulomb friction damping and electromagnetic shunt damping. The tunable friction damper (FD) allows for coarse tuning with a larger tunable range, while the tunable electromagnetic shunt damper (EMSD) functions as a fine tuner with fine precision. The end result is a significantly increased tunable frequency range with high precision. The proposed method allows for simple and low cost adjustment of the damping forces in both dampers. A prototype is fabricated and experimentally tested in a dynamic vibration absorber (DVA) system. The predicted tenability is experimentally verified and H∞ optimal damping of the DVA is readily achieved by using the proposed hybrid damper.

Latching control of a raft-type wave energy converter with a hydraulic power take-off system

The raft-type wave energy converter (WEC), consisting of several hinged rafts and hydraulic power take-off (PTO) systems, captures wave power in a line absorber manner, whose scattered and radiated wave pattern is different from that of a point absorber. This paper focuses on the latching control of a raft-type WEC consisting of two rafts and a hydraulic PTO system. To reduce the complexity of finding the optimal latching duration, the hydraulic PTO system is modelled as a Coulomb damping and the optimal latching control law is derived from a Coulomb damping-based model using the optimal control theory. The effectiveness of latching control on improving power capture ability of the raft-type WEC with a hydraulic PTO system and the benefit introduced by latching control in terms of power improvement are investigated. Results show that the optimal latching control law derived from the Coulomb damping-based model can effectively improve the power capture ability of the raft-type WEC with a hydraulic PTO system. The benefit due to latching control can reach a peak value of 3.04, which is 2 times that introduced by load control. Moreover, latching control can further benefit the raft-type WEC where load control has been performed.