Linear Viscous Damping Research Articles

Radial stiffness and damping of mountain bike tires subject to impact determined using the coefficient of restitution

Mountain bike tires are designed to encounter jumps, bumps, drops, and impacts. Despite their contribution to the vehicle suspension system, little has been published on their in-plane behaviour. In this work, a drop sled test bench is utilised to measure non-rolling dynamic radial stiffness and damping of numerous mountain bike tires across sizes, constructions, and usage categories. Each tire is treated as a classical, lumped Kelvin–Voigt model with a linear spring and viscous damper arranged in parallel. Identification of the system parameters is accomplished treating the tire as a bouncing ball. The coefficient of restitution from pre- and post-impact velocities is used to determine the dynamic stiffness and damping across two regimes of ‘impact’ and ‘in-contact’. Advantages of this approach over the classical logarithmic decrement are discussed. Measured first-bounce tire impact footprints exhibit a large ‘stadium’ shape compared to their elliptical static counterparts. A simple one degree-of-freedom, mass-spring-damper model with two pairs of dynamic stiffness and damping values is shown to faithfully reproduce the loaded, tire-only response for two of the inflation pressures tested. Measured dynamic radial stiffness and damping values are useful to compare tire behaviour and support future multibody simulation of the tire’s contribution to the suspension subsystem.

Comprehensive evaluation of seismic fragility in base isolated buildings enhanced with supplemental dampers considering pounding phenomenon

Base isolation is a proven technology effective in minimizing earthquake-induced damage to both the structural and non-structural elements (especially freestanding contents) of buildings. Nevertheless, it has been shown that allowing the flexible isolation layer to endure substantial demand displacements might subject the isolated structure to failure. This vulnerability arises from potential superstructure yielding due to increased deformations and reduced stiffness, such as those occurring through pounding against a moat wall during intense earthquakes. One alternative to controlling these large deformations is to use supplementary dampers at the isolation plane of the building, creating a hybrid base isolation system (HIS), which adds damping to the isolation system. In this research, the seismic performance of base isolated structures with lead-rubber bearing (LRB) are studied in conjunction with linear viscous dampers (LVD). Due to this aim, a base-isolated steel frame with 2 story X-type braces and a surrounding moat wall is modeled with two various configurations: LRB, and LRB with LVD. Fragility curves are developed for an ensemble of near-field (NF) pulse-like ground motions and for the maximum inter-story drift ratio (MIDR) as an engineering demand parameter (EDP). The incremental dynamic analysis (IDA) is conducted to calculate the damage probability of the structure by assuming different threshold values for damage states. It was found that adding viscous dampers to the isolation system balanced the behavior of the structure under near field pulse-type records; resulting in reduced isolation displacement while its adverse effects are insignificant and can be ignored. Moreover, the median spectral acceleration, namely the spectral acceleration associated to a probability of exceedance (POE) equal to 50 %, at the specific limit state, in the hybrid base isolation system is higher compared to the conventional one.

Amplitude‐dependent modal viscous damping for distributed stick–slip systems

AbstractAn accurate damping model serves as the foundation for reliable structural analysis. One of the primary sources of structural damping is the stick–slip mechanism caused by interfacial friction. Structural damping is usually represented by the modal damping ratio, and the modal equivalent viscous damping ratio (EVDR) of a stick–slip system is crucial in structural dynamic modeling. To date, an accurate yet simple prediction method for modal amplitude‐dependent EVDR that considers the distribution characteristics of the stick–slip mechanism remains unavailable. In the present study, the modal EVDRs of a distributed stick–slip system (DSSS) extended from a single degree‐of‐freedom stick–slip system were calculated using the modal energy equivalence method. The correlation between the stick–slip parameters and the modal damping behavior was investigated at various amplitude levels on the basis of the assumption of equal stick–slip parameters. A two‐parameter damping model represented by a single stick–slip damping model was developed to capture the amplitude dependence of the frictional EVDR. Furthermore, a practical prediction method based on the stick–slip parameters was proposed for the modal EVDR of the fundamental and higher‐order modes. The seismic analysis of two illustrative structures with nonuniform and uniform interstory stiffness demonstrated the effectiveness and applicability of the proposed method in predicting the modal EVDR of the DSSS. The proposed equivalent linear viscous damping system is suitable for the quick prediction of structural dynamic responses.

The Seismic Response Evaluation of an Existing Multi-span Reinforced Concrete Highway Bridge in the Presence of Linear and Nonlinear Viscous Dampers

AbstractIt’s crucial to keep bridges safe and operational all the time. Before or after an earthquake, they may require immediate seismic retrofitting. In this situation, adopting both isolation systems and viscous dampers could be used as a solution. Therefore, the seismic performance of a RC bridge with linear and nonlinear viscous dampers at both pier tops and abutments is investigated. The selected bridge model is the Incesu Bridge in Artvin province in north–eastern Turkiye. In the bridge model, elastomer bearings (EBs) and viscous dampers (VDs) are added to the abutments and the tops of the piers (i.e., the bridge deck connection points). The results revealed that the maximum bending moment of two piers could be significantly reduced compared with the case where the pier’s tops are fixed to its deck. However, a large level of a viscous damper coefficient added to a bridge could cause structural damage to its deck under a strong design earthquake.

The dynamic behaviour of flexible oscillators rocking and sliding on concentrated springs

AbstractThis study presents the Flexible Rocking Model on Concentrated Springs (FRMCS), developed to investigate 2D laterally flexible oscillators rocking and sliding on deformable support media during ground excitations. In this model, concentrated vertical springs and viscous dampers simulate the contact forces from support medium at the corners of the body; the tensionless vertical contact element is linear in compression. Horizontal concentrated springs and linear viscous dampers simulate the frictional behaviour at the corners; the constitutive law for the springs models elastic deformations and sliding (according to Coulomb's friction law). With these elements, FRMCS can model the response of a rocking body which can experience sliding and free‐flight phases of motion. The consideration of the flexibility of the support medium enables the evaluation of the forces exerted by the support medium on the structure during an impact. In this study, the FRMCS response is first compared to a previous model where the support medium deformability and the effects of sliding and free‐flight are ignored. Then, the responses of four configurations, which feature either stiff or soft lateral springs and stiff or soft high‐grip support media, are examined under the influence of pulse excitations. Finally, to understand the potential influence of sliding, a configuration with a low‐grip support medium is explored. The comparative influence of lateral flexibility and support medium deformability and sliding is quantified with stability diagrams and various response spectra, describing structural force and moment demands.

Comparison of Bracings and Fluid Viscous Dampers in 15-Storey Buildings across All Seismic Zones of India under Bi-Directional Excitation

Abstract: An earthquake is a natural phenomenon i:e, sudden horizontal movement of earth’s crust that causes sudden collapse of buildings and other structures such as large monuments, bridges etc. The damage of the building is based on the amount of magnitude and intensity of the earthquake. A building can withstand in lower intensity of earthquake. But higher intensity causes lots of damages on building or a building can suddenly collapse and causes increase in number of deaths. To mitigate these effects a control system is to be provided such as base isolations, dampers, bracings etc. The current topic is based on the behavior of linear and non linear fluid viscous damper provided at center and both side corners of 15 Storey RC building in diagonally bracing system. The building is modeled and analyzed by using ETABS software. Here IS 1983(part1): 2002 and medium soil type are to be considered for analysis with respect to all seismic zones (zone II - zone V) of India. Here linear static and response spectrum analysis are applied in each model. Results are to be drawn based on base shear, storey displacement and storey drift to understand how fluid viscous dampers helps to reduce the effects of earthquake.

Simultaneous analysis and design optimization for seismic retrofitting of hysteretic structures with fluid viscous dampers

Fluid viscous dampers, if properly sized and placed, can significantly improve the performance and safety of existing structures during an earthquake. This has motivated the development of several approaches to optimize the design of fluid viscous dampers. The optimization approaches available typically have an iterative nature, and in every iteration, expensive time-history analyses are performed. This paper presents a novel optimization-based approach for retrofitting hysteretic structures with fluid viscous dampers that does not require the computation of time-history analyses. The sizing and placement of the dampers are defined using a simultaneous analysis and design optimization approach based on nonlinear programming. In this approach, the equations of motion are treated as equality constraints of the optimization problem. As a consequence, design sensitivity analysis methods are not needed to calculate the gradients of the objective and constraint functions. Two design scenarios are considered: structures with hysteretic behavior retrofitted with linear or nonlinear fluid viscous dampers. The optimization variables include the damping coefficients of the dampers and the structural response over time. The manufacturing cost of the dampers is minimized, with constraints on the inter-story drifts. A continuation scheme on the design-driving inter-story drift constraints is proposed to effectively solve the class of problems considered. The numerical results demonstrate that the proposed approach can solve complex design optimization problems using reasonable computational resources and time. It is also shown that, thanks to the proposed approach, additional constraints can be incorporated in the problem formulation with minimal modeling effort, and without the need to develop additional sensitivity analyses for computing the gradients of the added constraints. The Julia codes used in this study are freely available on GitHub: https://github.com/pollinico/SAND_opt_nonlinFVDs.

SIMULATION OF FRICTION IN HINGED SUPPORTS DURING NON-STATIONARY VIBRATIONS OF A TIMOSHENKO BEAM

When carrying out research and technological processes in modern science and technology, the suppression or reduction of vibrations is often required for better operation of sensitive equipment and precision instruments. This paper presents a study on the damping of vibrations of a beam caused by the action of a dynamic load. The motion of an isotropic elastic beam is described using the model of S. P. Timoshenko. The beam has hinged supports at the edges. Damping moments located and acting at the anchoring points of the beam are used to damp the vibrations. As the beam moves in the hinges, a frictional moment with linear viscous damping occurs, which is proportional to the damping coefficient and the angular velocity of the beam in the hinge. In order to estimate the vibration damping, the solution of the direct problem of modeling the motion of the Timoshenko beam at zero ini- tial conditions is considered. The beam motion is modeled by a system of differential equations according to the model of S. P. Timoshenko. The re- quired functions are set in the form of Fourier series. Laplace integral transformation is used. The peculiarity of solution of a direct problem is that at this stage the friction moments in joints are unknown and are defined by solving corresponding inverse problem using Volterra integral equation the- ory. An analytical and numerical solution of the practical problem is obtained. Numerical results have been obtained in the form of graphs of beam point displacements and friction moments for different damping coefficients. A comparative evaluation of vibration reduction for different damping parameters has been carried out. The research results compare well with the results obtained by other authors.

Feasibility Study of a Novel Modal Decomposition Method for Low-Frequency Structure with Nonproportionally Distributed Rate-Independent Linear Damping

Compared with conventional linear viscous damping (LVD), rate-independent linear damping (RILD) mitigates floor response acceleration in low-frequency structures more effectively without compromising the control performance of displacement responses during long-period or low-frequency earthquakes. Although theoretical and experimental attempts have been made to overcome the noncausality of RILD devices and realize RILD using passive or semiactive devices, only a few studies have highlighted the impulse-response precursor and modal analysis method of multistory structures incorporated into RILD devices. This study investigated the impulse-response precursor of a noncausal RILD system and proposed a novel modal decomposition method for the structures equipped with nonproportionally distributed RILD devices. Additionally, real-time hybrid simulation was conducted to validate the effectiveness of the proposed modal analysis method and the feasibility of realizing ideal RILD using mechanical devices. This study is the first to demonstrate the differences between RILD and LVD devices in terms of controlling the modal responses of low-frequency structures and how RILD can lower the floor response acceleration more effectively compared to LVD.

Input Energy Reduction‐Oriented Control and Analytical Design of Inerter‐Enabled Isolators for Large‐Span Structures

Seismic isolation technologies for large‐span structures have rapidly developed alongside the popularization of the seismic resilience concept. To produce a high‐efficiency isolation technology with lower energy dissipation demands, this paper proposes a novel inerter‐enabled isolator (IeI) and a tailored input energy reduction‐oriented design method. The inerter‐based damper within the IeI is developed by combining the dashpot, tuning spring, and two inerters to facilitate the optimization of inerter distribution. Assuming the large‐span structure remains linear, the overall seismic input energy of the large‐span structure with IeIs and its allocation in the superstructure and additional damping are quantified using stochastic energy analysis. The advantages of the IeI over the conventional linear viscous damper (LVD) isolator are elucidated through dimensionless parametric analysis. Based on the results of parametric analysis, an input energy reduction‐oriented design method is proposed for the IeI, along with an easy‐to‐follow diagram that helps with preliminary design in practical applications. The effectiveness of the IeI and the proposed design method is validated through a design case study of a benchmark large‐span structure. The results demonstrate that the IeI reduces the seismic response of large‐span structures by simultaneously employing the input energy reduction effect of grounded inerters with the damping‐enhancing effect of inerter‐based dampers. The proposed design method effectively balances the performance of controlling the large‐span structure and the isolator displacement. Under consistent control performance and isolator displacement constraints, the IeI requires much less damping coefficient and energy dissipation capacity than the conventional LVD isolator. Moreover, leveraging the damping enhancement and input energy reduction effects, the IeI achieves comparable control performance to the conventional LVD isolator, even under stricter isolator displacement constraints.

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.

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.

Shaking table test of the shock absorbing performance of a new cable‐damper stopper for coal‐fired thermal power plants

SummaryCoal‐fired thermal power plants play important roles in daily life but are vulnerable to earthquakes. This paper presents a novel cable‐damper stopper with linear damping and carries out a series of shaking table tests to evaluate its performance on a 1/25‐scaled coal‐fired thermal power plant. The stopper consists of a cable, a linear viscous damper and a steel block attached to the boiler. The seismic responses of the structure equipped with different sets of cable‐damper stoppers which are of different damping coefficients and diameters of cables were studied and evaluated, where three seismic ground motions with two intensities were selected as input motions. The obtained acceleration and displacement responses of the support structure and the boiler reveal that the stopper cable effectively restrains the relative displacement between the boiler and the support structure. Furthermore, it can be found that the damping affects the seismic protection performance of the stopper depending on the intensity of the boiler vibration. In the paper, the working mechanism of the stopper is also analyzed based on the test data. It turns out that the stopper is more effective when it works under severe earthquakes.

Inerter-based Building Mass Damper: Optimization and experimental study

The Building Mass Damper is a design concept for vibration control of structures where the upper substructure effectively behaves as a Tuned Mass Damper (TMD) for the lower one. Unfortunately, its tuning usually requires softening or partial isolation of the upper substructure; limiting its applicability for retrofitting. The recently proposed Inerter-based Building Mass Damper (IBMD) is an in-parallel intervention on the upper substructure with an inerter and a damper. Thus, the inerter allows correct Tuning, the upper substructure provides the Mass, and the damper adds the Damping. Therefore, a very large mass-ratio TMD is obtained with marginal additional weight and minimal practical impact. In the present work, experimental tests supported by numerical simulations on a small two-story structure model quantitatively confirm the effectiveness of the IBMD. Besides, an innovative compliant mechanism design is introduced for the implementation of the translation-rotational converter that drives the inerter; which has minimal backlash and friction, still allowing large strokes. The experiments involved a frictional damper, which performed comparable to a linear viscous damper considered in the simulations.

Optimal design of tuned inerter negative stiffness dampers with linear viscous and nonlinear eddy current damping for structural effective damping ratio enhancement

Tuned inerter negative stiffness damper (TINSD) has demonstrated its effectiveness in controlling structural vibrations. However, the potential to improve the effective damping ratio of primary structures remains unexplored, and thus far, TINSD has solely featured linear viscous damping. In this study, we explored two damping types for TINSD: linear viscous damping (abbreviated as TINSD) and nonlinear eddy current damping (termed EC-TINSD). The optimal design of the TINSD was conducted based on the effective damping ratio enhancement (EDRE) effect. Our findings reveal that an increase in the absolute value of the negative stiffness does not result in a continuous improvement in the control efficiency and effectiveness of TINSD, which correspond to the EDRE effect and the structural effective damping ratio, respectively. Importantly, we observe that the proposed optimal TINSD solution outperforms the one derived from the fixed-point method, particularly when the inertance ratio is less than 0.15. For EC-TINSD, its optimal design was performed based on the EDRE effect using statistical linearization techniques that employ both force-based and energy-based equivalent criteria. The control efficiency and effectiveness of optimal EC-TINSD were then validated through Monte Carlo simulation. This analysis highlights the significance of a large critical velocity for EC-TINSD to achieve greater accuracy in vibration control. Numerical studies were further conducted to investigate the control efficiency and effectiveness of EC-TINSD under three types of real seismic excitations. The results demonstrate that both TINSD and EC-TINSD exhibit similar effectiveness in mitigating the maximum seismic responses of primary structures, and their efficiency surpasses that of viscous dampers and eddy current dampers with identical damping parameters. Additionally, the distinctive damping force limiting characteristic of EC-TINSD should be commended as it can protect the security and dependability of both the dampers and their connections to the structures.

Seismic control of base-isolated structures using rate-independent damping devices

Rate-independent linear damping (RILD) is a viable option for addressing excessive displacement challenges in low-frequency structures, particularly those subjected to ground motions containing many low-frequency components. RILD can mitigate the floor response acceleration of base-isolated structures more effectively than linear viscous damping (LVD) because of the larger damping force generated in the low-frequency region. However, RILD is an ideal element that exhibits noncausal characteristics. Many researchers have attempted to find approaches to realizing RILD, including both theoretical and mathematical investigations. Thus far, the physical realization of RILD remains challenging. In this study, a tuned inertoviscous-negative-stiffness Maxwell−Weichert (IvNsMW) model and a tuned inerter-viscous damper-negative-stiffness Maxwell−Weichert (IVdNsMW) model are proposed to promote progress in the application of RILD. These models differ in terms of the connection between the inerter and viscous damper. A direct design method is presented to determine the coefficients of these two devices. The efficiency of the proposed RILD-based devices for seismic control of low-frequency structures was examined using a multistory base-isolated structure. The results demonstrate that the proposed RILD devices are promising tools for both displacement and acceleration control of low-frequency structures.

On the use of frictional dampers for flutter mitigation of a highly flexible wing

The prospect of aeroelastic instability, leading to large-amplitude limit cycle oscillations, is of particular concern for the highly flexible wings of solar high-altitude long-endurance (HALE) aircraft. Among the different strategies of vibration control, passive mitigation through friction dampers stands out as a promising solution in this context. While this concept has been proposed previously, related studies systematically use simple two-degrees-of-freedom airfoil models. In this paper, a more realistic structural model involving a three-dimensional nonlinear beam description of a full wing is considered. A friction damper is installed at a specific location on the wing span, and the response to aerodynamic loads is investigated numerically. Comparisons of critical speed and limit cycle amplitudes with and without the damper suggest improved vibration mitigation with respect to a linear viscous damper.

Effect of damping on the seismic response of long-period structures considering crossover phenomena

This study systematically investigates the effect of damping on the seismic response of long-period structures. For long-period single-degree-of-freedom systems, the addition of damping would on average reduce the relative deformation and velocity, but increase the absolute acceleration. However, the crossover phenomena, i.e., the increase of the damping leads to the larger structural response, would occur in the spectral displacement or velocity of some certain ground motions, which is more pronounced with the increase of structural periods. The percentage of ground motions developing crossover phenomena attends approximately 100%, 40% and 20% for the acceleration, velocity and displacement, respectively, at very long structural periods. For long-period multiple-degree-of-freedom systems, illustrated by two cable-stayed bridges, the addition of linear viscous fluid dampers (VFDs) would averagely lead to the reduction of the displacement responses, but the crossover phenomena might also occur. It is recommended that VFDs be implemented to cable-stayed bridges as the deck longitudinal displacement is generally reduced and the deck longitudinal vibration is damped out during a short time compared to the uncontrolled case. However, the damping coefficient of the VFDs needs to be restricted such that the damping force is limited to a reasonable range, and significant response increments of some components are avoided.

Experimental evaluation of damping models for a nonlinear pendulum system

Popular damping models for a single pendulum system were experimentally evaluated in this study. The comparison between the numerical and experimental responses showed that neither linear viscous damping model nor squared velocity proportional viscous damping model was able to well capture the free vibration amplitudes of the pendulum in both large and small amplitude ranges. Specifically, the amplitudes of the numerical model with linear viscous damping decreased slower at the large amplitude vibration but faster at the small amplitude vibration than the experimental amplitudes did. The trend was reverse in the numerical model with the squared velocity proportional viscous damping model. Based on the experimental data and a numerical method combined with a fitting approach, a complete quadratic viscous damping model for the pendulum was derived. The model was able to confidently predict the free vibration amplitudes of the pendulum in both large and small amplitude ranges. This study provides analysts with the sense of the accuracy of some popular damping models for a nonlinear pendulum system. It also demonstrates the use of a fitting approach to retrieve physical properties of a system from experimental data. The contents of this paper are suitable for gifted high school students and undergraduate students.

Vibration Analysis of a Nonlinear Absorber Coupled to a Hand-Held Impact Machine

Abstract Exposure of human hands to harmful levels of vibration can lead to severe injuries. To attenuate these hand transmitted vibrations, a nonlinear vibration absorber inerter (NVAI) is proposed in this work. The proposed NVAI is attached to a coupled nonlinear system of a hand-held impact machine and a hand-arm system. The combined coupled nonlinear system is modeled as a lumped parameter model with a combination of cubic and linear stiffness components, linear viscous dampers, and lumped masses. The governing equations of motion are solved analytically using the method of harmonic balance and validated using numerical simulations. A numerical bifurcation diagram of the system reveals the existence of complex solutions such as quasi-periodic and chaotic attractors. The appearances of quasi-periodic and chaotic attractors are later confirmed by Lyapunov exponents. Further, we explore the ability of the proposed NVAI to decrease the area corresponding to unstable quasi-periodic and chaotic motion in the excitation amplitude–frequency space. This observation further implies the delay in the onset of quasi-periodic and chaotic motion for a range of forcing amplitude using Lyapunov exponents. Finally, parametric analyses are carried out to identify the critical design parameters of the NVAI. These analyses reveal that an increase in the damping, mass, and inertance of the absorber ameliorates the performance of the NVAI. Furthermore, the critical value of the external excitation, corresponding to a sudden change in the response of the system, can be controlled using an appropriate selection of absorber parameters.