Hysteretic Damping Model Research Articles

Numerical Methods for Time-Domain Responses of the Frequency-Independent Damped System

The complex damping model can only be used to calculate the steady-state responses, while the transient responses are divergent. Based on the complex damping model, the Hilbert transform is introduced to establish a hysteretic damping model, eliminating the divergence phenomenon. However, with the increase of the loss factor, the damped natural frequency also increases. To overcome this shortcoming, a frequency-independent damping model is proposed based on the hysteretic damping model. However, traditional time-domain methods are no longer applicable to frequency-independent damping models. Therefore, the transient response and steady-state response are separated, and the assumption of external excitation acceleration is introduced. Time-domain methods-based linear polynomial assumption, quadratic polynomial assumption and trigonometric function assumption are proposed, respectively. Numerical examples show that the time-domain methods based on linear polynomial assumption and quadratic polynomial assumption have high computational efficiency. But these two methods cannot take into account the vibration frequency of external excitation acceleration. Hence, the computational accuracy is low. Compared with them, the time-domain method based on trigonometric function assumption has the lowest computational efficiency and the highest computational accuracy.

Complex mode superposition method of non-classically damped systems based on hysteretic damping model with frequency-dependent loss factors

For non-classically damped systems, it is easy to construct a damping matrix based on a hysteretic damping model. However, the damped natural frequency is non-physical, and the frequency response function is non-causal. These two drawbacks can be overcome by the hysteretic damping model with frequency-dependent loss factors. In this paper, a complex mode superposition method for non-classically damped systems based on the hysteretic damping model with frequency-dependent loss factors is proposed. Frequency-dependent loss factors of the natural frequencies are considered in the transient vibration response, while frequency-dependent loss factors of the external excitation frequencies are considered in the steady state vibration response. By decoupling complex modes in state space and physical space, non-proportional damping characteristics of hybrid structures can be effectively considered. Therefore, the proposed method can be used to calculate the time-domain dynamic responses of hybrid structures, whose loss factors are sensitive to frequency variation and have strong non-proportional damping characteristics. The time-domain dynamic responses of numerical examples are compared based on different mode superposition methods. Shaking table tests are performed on cantilever plates with different damping characteristics. The numerical and test results validate the proposed method. Moreover, the proposed method has a wide range of applications.

The Effect of Landing Gear Dimension Variation on the Static Strength and Dynamic Response of Unmanned Aerial Vehicle (UAV)

This research discusses the static and dynamic analysis of the landing gear structure of an unmanned aerial vehicle (UAV). The dimensional study is conducted to investigate the effect of landing gear dimension variation on UAVs’ static strength and dynamic response. Static analysis was performed with Finite Element Method (FEM) software. The dynamic response of the UAV is analyzed using a single-degree-of-freedom vibration model. Based on the static analysis results, the landing gear stiffness and strength can be increased by increasing the width and decreasing the height, radius, and length of the landing gear structure. The energy dissipation in the dynamic analysis is described by hysteresis and viscous damping model. The dynamic response simulation results show that the increase in the stiffness of the landing gear leads to an increase in force transmission and acceleration of the UAV. Furthermore, the UAV response using the viscous damping model can accurately predict the system’s response with the hysteretic damping model for small damping conditions. However, the deviation was observed for large damping conditions.

A Complex Mode Superposition Method Based on Hysteretic Damping Model for Mode Static Correction

Due to modal truncation, the dynamic responses calculated by the mode superposition method have some errors. In this paper, the source of truncation error of the complex mode superposition method for hysteretic damped systems with non-proportional characteristics is analyzed. It includes truncation errors of external excitation load and Hilbert transform term. With the increase of mode order, the ratio of excitation frequency to natural frequency gradually approaches zero. Combined with the expression of structural modal response caused by harmonic wave, the corresponding approximate expression of high-order modal response is obtained. Through quasi-static analysis, a complex mode superposition method of non-proportionally damped systems for mode static correction is proposed. Numerical examples show that when the ratios of the domain frequencies of the external excitation to the natural frequency of some high-order modes are small, the complex mode superposition method for mode static correction has a good effect.

Practical negative stiffness device with viscoelastic damper in parallel or series configuration for cable damping improvement

Negative stiffness mechanism has been found able to improve damping performance of dampers on a stay cable which otherwise is limited by the damper installation distance from a cable end. This study provides a practical negative stiffness device (NSD) with adjustable negative stiffness and experiments are performed to validate the negative stiffness effect. The NSD is then combined with a viscoelastic damper in parallel or series for cable damping improvement. Explicit design formulas are derived for optimal design with a target enhancement effect in damping considering the damper described respectively using the Kelvin model and the linear hysteretic damping model. The formulas are verified by analytical and numerical solutions. Parametric analyses show damping enhancement effects of the NSD and it is found more efficient when combined with a damper in series because both deformation amplitudes of the damper and the NSD are further increased in this configuration. Subsequently, case studies are carried out based on two cables of the Sutong Bridge respectively with a shear-type viscous damper and a high damping rubber damper. The results show that the designed NSD can fulfill practical requirements. Particularly, a 100% increase in damping can be achieved by the presented NSD when combined with the damper installed on a cable of 546.9 m long.

The Magnetoelastic Contribution to the Steel Internal Damping.

In this paper, the steel internal damping due to both the thermoelastic and the magnetoelastic phenomena has been investigated through a formulation based on thermodynamical potential joints with a hysteretic damping model. With the aim of focusing on the temperature transient in the solid, a first configuration has been considered, which is characterized by a steel rod with an imposed alternating pure shear strain in which only the thermoelastic contribution was studied. The magnetoelastic contribution was then introduced in a further configuration, in which a steel rod in free motion was subjected to torsion on its ends in the presence of a constant magnetic field. A quantitative assessment of the influence of the magnetoelastic dissipation in steel has been computed according to the Sablik-Jiles model by giving a comparison between the thermoelastic and the prevailing magnetoelastic damping coefficients.

Comparative Analysis of Viscous Damping Model and Hysteretic Damping Model

A damping model is one of the key factors in dynamic analysis. Viscous damping and hysteretic damping models are commonly used in structural damping models. In this study, transient and steady responses are analyzed for a single degree of freedom system based on the two damping models. The attenuation coefficient and damped natural frequency are important parameters of the transient response. In addition, the vibration amplitude is an important parameter of the steady response. When the relative errors of the parameters for the two damping models are less than 10%, the threshold of the damping ratio is selected as 0.1736 and the threshold of the loss factor is 0.3472. The numerical examples show that the dynamic responses based on the viscous damping model are approximately equal to those based on the hysteretic damping model in small damping cases. With the increase in the damping ratio, the difference between the dynamic responses calculated by the two damping models gradually increases. In large damping cases, the two damping models must be distinguished, and the choice of the damping model depends on the characteristic of dissipate energy.

A Calculation Method for Determining the Number of Truncated Modes Based on Hysteretic Damping Model

A Calculation Method for Determining the Number of Truncated Modes Based on Hysteretic Damping Model

The numerical calculation method based on equivalent frequency-independent time-domain damping model

In order to realize the time-domain analysis based on hysteretic damping model, the frequency-independent time-domain damping model of single degree of freedom (SDOF) system is constructed. Based on the assumed relationship of vibration responses, the equivalent frequency-independent time-domain damping model in complex domain and real domain are proposed. The characteristic that the dissipated energy in each cycle is not related to the vibration frequency of external excitation is retained for the two equivalent damping models. Combined with Newmark- β method, the corresponding numerical methods are obtained. The numerical examples show that the free vibration responses are stably convergent based on equivalent damping models. The numerical results of vibration responses of SDOF system due to earthquake wave have high calculation accuracy. Compared with equivalent frequency-independent time-domain damping model in real domain, the computational accuracy of equivalent frequency-independent time-domain damping model in complex domain is higher, and the computational efficiency is lower.

Dynamics of plates resting on layered transversely isotropic poroelastic media under moving loads

This paper investigates the dynamic interaction between a plate and multi-layered transversely isotropic (TI) hysteretically damped poroelastic media under a moving rectangular load. Following Kirchhoff's thin plate theory, the plate deflection is derived utilizing the Fourier series expansion and the integral transform technique. The governing equation for the layered TI hysteretically damped poroelastic media is formulated through the standard hysteretic damping model and Biot's theory. The solution for the plate-soil dynamic interaction in the wavenumber domain considering the plate-soil interface condition is obtained by the extended precise integration method (PIM) for the layered TI saturated porous media. This solution can be further converted into the spatial domain by the inverse transform technique. We have verified the efficiency and accuracy of the proposed method. Several numerical examples are presented to investigate the influences of the plate thickness, plate modulus ratio, surface layer, speed of the moving load, and hysteretic damping of the solid skeleton on the dynamic behaviors of the plate on layered TI poroelastic media.

Hysteresis and damping properties of steel and polypropylene fiber reinforced recycled aggregate concrete under uniaxial low-cycle loadings

To investigate the hysteresis and damping properties of fiber-reinforced recycled aggregate concrete (FRAC), a series of cyclic compressive tests of SF-reinforced natural aggregate concrete (SF-R-NAC), SF-reinforced RAC (SF-R-RAC), and PPF-reinforced RAC (PPF-R-RAC) with different fiber contents were carried out. The development law of residual strain was explored, and the relationship between residual strain and unloading strain was suggested. Also, a modified constitutive model of the stress–strain relationship was proposed. Variation of the hysteretic characteristics of FRAC including hysteretic strain energy and residual strain energy under cyclic compressive loadings was analyzed. An equivalent hysteretic viscous damping ratio was suggested and analyzed by using based on Jacobsen’s approach. Furthermore, a new hysteretic viscous damping model represented by residual strain was proposed accounting for the fiber content. Based on the modified constitutive model, the hysteretic viscous damping model is predicted and analyzed to verify the effectiveness and accuracy of the model.

The dynamic contact of a viscoelastic coated half-plane under a rigid flat punch

This paper investigates the dynamic contact response of a viscoelastic coated half-plane under a forced rigid punch. This punch, acting on the coating surface, is subjected to a concentrated vertical harmonic force. The coating and half-plane are assumed as viscoelastic materials which can be described by the linear hysteretic damping model. Using the Fourier integral transform technique and the asymptotic method, the dynamic contact problem is reduced to a Cauchy singular integral equation. By using appropriate numerical method, the dynamic contact stress and vertical impedance for the flat punch are calculated. It is found that the dynamic contact behaviors are not only affected by the external excitation frequency but also dependent on the material properties of the coated half-plane.

Comparison of damping parameters based on the half-power bandwidth methods of viscous and hysteretic damping models

The half-power bandwidth method is usually used to calculate structural damping parameters by frequency response function (FRF). In this note, the half-power bandwidth methods for the displacement FRF, the velocity FRF, and the acceleration FRF are proposed based on viscous and hysteretic damping models, respectively. Comparison results show that the application conditions of half-power bandwidth methods for the displacement and acceleration FRFs are limited. They can only be used to calculate the small damping ratio/loss factor. The application condition of half-power bandwidth method for the velocity FRF is not limited. It can be used to calculate the large or small damping ratio/loss factor, which should be the first choice for calculating damping parameters. Besides, when the damping ratio is less than 0.2546 or the loss factor is less than 0.5658, the relative difference between the loss factor and twice the damping ratio is less than 10%. With the increase of the damping ratio or loss factor, the relative difference will increase rapidly, and the approximate relationship is no longer applicable.

Inerter dampers with linear hysteretic damping for cable vibration control

Long stay cables in cable-stayed bridges have low and close-spaced modal frequencies and are subjected to multimode vibrations, e.g., rain–wind vibrations and vortex-induced vibrations. Many types of dampers used in practice for cable vibration control can be described using a linear hysteretic damping model, e.g., high-damping rubber dampers and viscous-shear dampers. Such dampers are able to provide frequency-independent damping effects while the maximal achievable damping is limited due to their intrinsic stiffness effect. Therefore, this study investigates dampers with linear hysteretic damping enhanced by inerters for cable vibration control. A general inerter damper, consisting of a spring with complex stiffness and an inerter in parallel which is then connected to another inerter in series, is attached to a cable for dynamic analyses. Complex modal analysis is performed to appreciate cable damping. The optimal damping effect is discussed with reference to dynamic properties of the damper with respect to frequency, with comparison to the system of a cable with an inerter damper with viscous damping. It is found that two inerters, respectively in parallel and in series with a damper of linear hysteretic damping, can achieve a large improvement on multimode damping of a cable. The inertance of the inerter in series with the damper needs to be large for optimal performance while a small inertance of the other inerter is preferable. Furthermore, a case study based on a cable attached with a viscous-shear damper on the Sutong Bridge is conducted to show the feasibility of the inerter dampers.

A new hysteretic damping model and application for the combined finite-discrete element method (FDEM)

In the field of mechanical numerical simulation, damping is always a vital parameter. However, in the combined finite-discrete element method (FDEM), the current application of damping is insufficient. Therefore, in this paper, a new hysteretic damping model and its corresponding critical damping coefficient acquisition method are proposed. Furthermore, simulations of quasi-static tunnel excavation and quasi-dynamic rock slope slip are employed to investigate the influence of different hysteretic damping ratios on the numerical results. The study results show that the new hysteretic damping model, which is independent on the calculation time step, considers the effects of the element size, Young's modulus and material density and can obtain the critical damping coefficient by comprehensively using cantilever beam vibration numerical modelling and theoretical equations. In addition, the sensitivity of quasi-static tunnel excavation to the hysteretic damping ratio is very weak; however, for quasi-dynamic rock slope slip simulation, the damping ratio η has a substantial influence on the simulation results. The new hysteretic damping model proposed in this paper is important for improving the accuracy of FDEM numerical simulation results, especially for quasi-dynamics simulation.

Structural damping models for passive aeroelastic control

Aeroelastic qualification requirements are typically met by sizing aircraft to achieve adequate stability margins and keep peak gust responses below specified thresholds. A possible alternative approach is delaying flutter and alleviating gust response by embedding dissipative materials into structural components. This approach requires accurate damping models applicable to the analysis of complex configurations. In this paper, the effect of damping models in the evaluation of flutter boundaries and gust response of an aeroelastic test-bed is studied. In particular, three damping models completely different in frequency are considered to model the presence of skin patches for passive aeroelastic control: viscous damping, hysteretic damping and Biot damping models. In order to make them comparable the three models are tuned in order to dissipate the same amount of power at the flutter frequency by means of the introduction and definition of a generalized loss factor. The damping models are compared by evaluating their effect on flutter suppression and gust load alleviation. Finally, the sensitiveness of the stability and response aeroelastic analyses to the considered damping models are outlined and the obtained results are compared to provide modeling recommendations for passive flutter suppression and gust alleviation studies. The proposed methodology is applicable to any linear material model for which the related complex stiffness can be expressed in the frequency domain meeting the Hilbert restriction.

Finite Element Modelling and Simulation of the Hysteretic Behaviour of Single- and Bi-metal Cantilever Beams using a Modified Non-linear Beta-damping Model

This paper explores a novel non-linear hysteresis model obtained from the modification of the conventional Kelvin-Voigt model, to produce a non-viscous hysteretic behaviour that is closer to metal damping. Two case studies are carried out for a vibrating cantilever beam under tip loading (bending), the first considering a single uniform material and the second considering a bimetallic structure. The damping behaviour is studied in the frequency domain (constant damping ratio model vs. Kelvin-Voigt/ beta damping model) and time-domain (proposed modified hysteresis model vs. Kelvin-Voigt/ beta damping model). In the frequency domain, it was found that the Kelvin-Voigt model essentially damps out the displacement response of the modes more than the constant damping ratio model does. In the transient analysis, the Kelvin-Voigt model likewise produced unnaturally rapid damping of the oscillations for both the single- and bi-metal beam, compared to the modified hysteretic damping model, which produced a damping behaviour closer to actual metal behaviour. This was consistent with results obtained in the frequency domain.

Hysteretic damping model for laterally loaded piles

Fatigue assessment is a critical design aspect for many offshore structures. Soil-foundation interaction has a direct impact on the system dynamic response of these structures. While the stiffness of the soil-foundation interaction influences the system's natural frequency, the damping influences the amplification of the structural response to environmental excitations. This paper presents a simplified model for estimating the soil damping due to nonlinear soil response for pile foundations, which have wide applications in the offshore industry, such as for supporting jacket platforms, wind turbines and wellhead facilities. The proposed model is fundamentally linked to the damping response of the soil measured at element level therefore it offers design engineers an efficient and accurate way to estimate soil-pile interaction damping based on site-specific soil data. Approaches to include the suggested model for structural analysis are also proposed.

Damping Mechanisms in Cable-Harnessed Structures for Space Applications: Experimental Validation

Abstract In this technical brief, the experimental study and model validations for the damping mechanisms of cable-harnessed beam structures are presented. The structure consists of cables wrapped around a host beam in a periodic zigzag pattern. A special case of cable attached along the beam length over its centerline is also considered. First, material damping in the cables is characterized using dynamic tests and the relevant cable damping factors are calculated for both the Kelvin–Voigt and hysteretic damping models. Experimental modal testing is then performed on the fabricated cable-harnessed beams to obtain the frequency response functions (FRFs). Finally, the experimental FRFs are compared with the damped analytical models. The test and model results are shown to be in very good agreements in predicting the structural damping induced by the cables.

Complex mode superposition method of nonproportionally damped linear systems with hysteretic damping

The vibration response of a damped linear system is always calculated based on the mode superposition method. However, the construction of the damping matrix is difficult for the conventional mode superposition methods based on the viscous damping model, and this problem is much more serious for nonproportionally damped linear systems. The damping matrix based on the hysteretic damping model is easy to construct and unique, which is determined only by the structural stiffness and material loss factor. The time-domain motion equation of a multi-degree-of-freedom nonproportionally damped linear system is easily constructed based on the hysteretic damping model. According to the characteristics of external excitation, the general solution of the corresponding homogeneous equation and special solution of the corresponding nonhomogeneous equation can be solved. By the aid of the easiness of the damping matrix, a complex mode superposition method based on the hysteretic damping model is proposed for the nonproportionally damped linear system. Based on the proposed method, a user subroutine in ANSYS and MATLAB is developed to calculate vibration responses in time-domain dynamic analyses. A shaking table test of a cantilever plate composed of host and damping layers is conducted to validate the proposed method. The method proposed in this article is unconditionally convergent, and its convergence is independent of the time step of time-domain analyses. Compared with the common complex mode superposition method based on the viscous damping model, the simulation results of the proposed method are closer to the test results, and its accuracy and efficiency are higher. In addition, the calculation results of the proposed method are unique, which is irrelevant to the choice of vibration modes.