Linear Hysteretic Damping Research Articles

Nonstationary response statistics of structures with hysteretic damping to evolutionary stochastic excitation

The damping of a structure has often been modeled as linear hysteretic damping (LHD), so its corresponding equation of motion (EOM) is an integro-differential equation that involves the Hilbert transform of response displacement. As a result, the system is non-causal in nature, and it is challenging to compute its nonstationary response statistics under evolutionary stochastic excitation. This article develops an efficient solution method to obtain closed-form solutions for various nonstationary response statistics, including the evolutionary power spectrum (EPS), correlation function and mean square values. The novel solution method utilizes the concept of causalization time to introduce a “causalized” impulse response function (IRF), by which causal response statistics are computed based on a pole-residue approach. This approach requires obtaining a pole-residue form of the transfer function (TF) from the frequency response function (FRF) of the system, which is readily obtained from the EOM. Subsequently, the desired response statistics are obtained by shifting the causal response statistics back to the original time. To obtain the pole-residue form of the TF, two steps are necessary: (1) taking the inverse Fourier transform of the FRF of the oscillator to obtain a discrete IRF and (2) using the Prony-SS method to decompose this discrete IRF to obtain the pole residues associated with the TF. The correctness of the proposed method is numerically verified by Monte Carlo simulations through examples of hysteretic damping and mixed viscous-hysteretic damping oscillators that are subjected to white noise, modulated white noise and modulated Kanai–Tajimi model random excitations.

Cross-ties with linear hysteretic damping for cable vibration mitigation: Theoretical analysis and full-scale experiment

This study investigates cross-ties with linear hysteretic damping (HDCs) for vibration mitigation of stay cables through theoretical analysis and full-scale experiments. The formed cable networks are modeled by considering cables as taut strings and the HDCs as mass-less elements of complex stiffness. Frequency equation of the system is formulated analytically and is solved numerically. Then, the influence of HDCs on system dynamics is discussed for typical cable networks and considering HDCs realized by a rigid tie in series with high damping rubber. The results indicate that the HDCs can realize high damping for multiple modes because of the frequency independency of hysteretic damping and no restriction on the installation location. Increasing the loss factor of the cross-tie is significant for improving system damping, but less effective for system frequency modification. Extending HDCs to the ground and using multiple HDCs are effective in further enhancing multi-mode frequency and damping of the system. Furthermore, full-scale experimental validations and investigations are conducted based on three cables of approximately 40 m. Experimental results have validated the effectiveness of HDC in enhancing system damping. This study demonstrates that the easily implementable HDCs have excellent performance in vibration mitigation of multi-mode cable vibrations.

Feasibility of an inerter-based causal rate-independent damping model for the protection of seismically isolated structures

Rate-independent linear damping (RILD), also referred to as linear hysteretic damping, is a linear model of structural or material damping in which the energy dissipation per cycle is independent of the frequency. The resistive force of an ideal RILD model is in phase with the velocity to dissipate energy, while its amplitude is proportional to the displacement, which is suitable for the direct control of seismic response displacement when incorporated into a structure. The tuned Maxwell–Wiechert (TMW) model is a viable option for approximating the damping characteristics of RILD. However, the undesirable stiffness produced by the TMW model compromises the flexibility of the isolator, and thereby fails to deliver the benefits of RILD. To overcome this challenge, the addition of two inerter elements to the TMW model in series and parallel arrangements is proposed in this study to eliminate the undesirable storage stiffness. The parameters of the proposed model are determined using particle swarm optimization. Analyses of a 10-story building structure mounted on linear and nonlinear isolation systems demonstrated that the proposed model achieved lower interstory drifts and approximately 40% reduction in floor response accelerations, with similar isolator displacements compared with LVD when subjected to ground motions dominated by high-frequency components. Moreover, floor response acceleration mitigation was attained at the slight expense of isolator displacement, even when the proposed system was incorporated into nonlinear isolation systems and subjected to low-frequency ground motions.

On Time-Domain Solutions to the Free Vibration of SDOF Systems with Linear Hysteretic Damping

This article focuses on the free vibration of single-degree-of-freedom (SDOF) systems with linear hysteretic damping and investigates the consistency and convergency of the solution to the free vibration in time domain. The criteria associated with the unit-impulse response function at [Formula: see text] are suggested to check the consistency of possible solutions. Exhaustive and systematic derivations are presented to clarify the confusing results in the literature. Subsequently, the integration in the Hilbert term proves to be nonconvergent when an exponential term ([Formula: see text]) is assumed to be the general form of displacement solutions. Hence, it can be concluded that the exact solution to the free vibration of SDOF systems with linear hysteretic damping is still an open question. At last, a numerical verification method is proposed to assess the accuracy of possible solutions.

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.

A novel bi-directional rail variable friction pendulum-tuned mass damper (BRVFP-TMD)

The shape of hysteretic loops for a friction pendulum system transforms when excited by bi-directional motion, which exhibits unexpected dynamic behavior compared with unidirectional motion. Therefore, a novel bidirectional rail variable friction pendulum-tuned mass damper (BRVFP-TMD) with two independent rails that satisfied the purpose of bidirectional structural response control and provided a full bidirectional friction force was proposed. The variable friction arrangement was applied to the rails so that the BRVFP-TMD could be accomplished with a linear hysteretic damping property that is contributed by the displacement-dependent variable friction force. Using unidirectional harmonic excitations with attack angle as bi-directional loading, illustrations were created to investigate the control effect of traditional pendulum friction TMDs with a spherical sliding surface. This demonstrated the collapse of hysteretic loops of traditional pendulum friction TMDs and superiority of BRVFP-TMD in terms of mechanism. For the BRVFP-TMD with a lightly damped structure, the optimizations of the fixed-point theory were accessed to obtain the closed-form solutions of the optimal parameters of the BRVFP-TMD. Numerical verifications based on bidirectional motions were conducted to evaluate the performance of the BRVFP-TMD. The bidirectional earthquake results indicated stable and excellent performance of the BRVFP-TMD in terms of either frequency response reduction or friction energy dissipation efficiency. Meanwhile, bidirectional wind-induced vibrations confirmed the correctness of the optimization and the stable robustness of the optimized BRVFP-TMD.

Laplace- and Frequency-Domain Methods on Computing Transient Responses of Oscillators with Hysteretic Dampings to Deterministic Loading

Efficiently computing the transient response of a linear hysteretic damping (LHD) system that is noncausal to deterministic loading that is regular or irregular is challenging. In this study, we developed a time-shifting causalization procedure and efficient solution methods to compute the transient response, conducted in the frequency or Laplace domain based on pole-residue operations. Because these operations require the causalized system’s transfer function to be in its pole-residue form, true poles were extracted by the Prony-SS method for the Laplace-domain method. In contrast, the frequency-domain method designates imaginary poles directly from the system’s frequency response function. Overall, the latter method was easier to implement and more computationally efficient but less capable of computing longer responses due to the imposition of the periodic oscillator. The proposed methods were used in numerical studies for oscillators with hysteretic damping and mixed viscous-hysteretic damping to earthquake-induced loading; correctness was verified by a time-domain method.

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.

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.

Using a Tuned-Inerto-Viscous-Hysteretic-Damper (TIVhD) for vibration suppression in multi-storey building structures

This paper explores the use of a novel tuned-inerto-viscous-hysteretic-damper (TIVhD) for reducing the seismic response of multi-storey building structures. The TIVhD is an inerter-based damper device consisting of a linear hysteretic damper connected in series with an inerto-viscous damper. The layout of TIVhD is similar to that of tuned-inerter-hysteretic-damper (TIhD) with an additional viscous damping element in parallel with an inerter. The design is motivated by the fact that most inerter designs cannot completely remove the parasitic damping due to friction, fluid compression, etc. Moreover, the use of linear hysteretic damping is considered to be a more realistic approach when material damping is present. In this paper, the TIVhD is installed between the ground and the first-storey and is tuned by firstly assumed the viscous damping coefficient to be zero. Then the other three parameters are optimised following the tuning procedure of the TIhD that is based on the fixed-point theory with additional fine-tuning procedure by targeting the first vibration mode of the multi-storey structure. The optimum TIVhD parameters are finally obtained using two scenarios: (1) amplifying its viscous damping coefficient and stiffness while keeping the inertance constant; (2) amplifying its inertance and stiffness while keeping the viscous damping constant. Both scenarios are aiming at the same reduction level of that given by the TIhD. Finally, the effectiveness of the TIVhD on reducing the structural response is demonstrated for both harmonic and seismic base excitation cases in the time domain. This has been made possible by a newly developed time domain response of linear hysteretic damping via the Hilbert transform and a time reversal technique.

Tuned inerter dampers with linear hysteretic damping

SummaryThis paper explores the influence of linear hysteretic damping on the performance of passive tuned‐inerter devices. An inerter is a device that produces a force proportional to the relative acceleration across its two terminals; devices incorporating inerters have received widespread attention in the earthquake engineering community, because they offer the ability to improve the seismic response of structures. However, the majority of this research has assumed that the damping components within the tuned‐inerter device exhibit viscous, rather than hysteretic, damping. This restriction imposes an essential question on how the hysteretic damping model will change the performance of the device compared with the viscous damping model. It is shown that the response of viscous and hysteretic inerter systems have significant differences in displacement amplitude due to the frequency dependency of the damping. Therefore, a new formulation for obtaining the optimum loss factor of the hysteretic damping in the inerter system is proposed. Next, the challenges associated with accurately predicting the time‐response of a hysteretically damped system are discussed. A numerical time‐integration method is extended to address these challenges, using a new formulation that has the benefit of being broadly applicable to multidegree‐of‐freedom hysteretic linear systems and nonstationary random signals. The results show that the earthquake responses from the hysteretic damping model can differ significantly from the ones obtained via the viscous model.

Time domain analysis of structures with hysteretic vibration suppression systems

The use of viscous damping terms to simplify the damping of a vibrating system has been well established for decades. For solid materials whose energy dissipated per cycle is frequency-independent, an equivalent viscous damping has often been used. However, this may give inaccurate results, especially at higher excitation frequencies. Alternatively, a complex stiffness term can be used. In this case, a challenge arises for the time domain analysis due to the unstable poles in the resulting model. Several methods have been proposed to deal with this issue. The use of an analytic signal along with Hilbert transform and a time reversal technique is one of the first introduced methods. In this paper, we extend the method so that it can be used for solving the system equations of motion using the numerical integration algorithm solvers that are available in MATLAB. We also present the application of this extended method to simulate a multi-degree-of-freedom (MDOF) structure with supplemental passive vibration suppression systems using linear hysteretic damping in the time domain.

Enhanced acoustic transmission into dissipative solid materials through the use of inhomogeneous plane waves

A number of applications, for instance ultrasonic imaging and nondestructive testing, involve the transmission of acoustic energy across fluid-solid interfaces into dissipative solids. However, such transmission is generally hindered by the large impedance mismatch at the interface. In order to address this problem, inhomogeneous plane waves were investigated in this work for the purpose of improving the acoustic energy transmission. To this end, under the assumption of linear hysteretic damping, models for fluid-structure interaction were developed that allow for both homogeneous and inhomogeneous incident waves. For low-loss solids, the results reveal that, at the Rayleigh angle, a unique value of the wave inhomogeneity can be found which minimizes the reflection coefficient, and consequently maximizes the transmission. The results also reveal that with sufficient dissipation levels in the solid material, homogeneous incident waves yield lower reflection values than inhomogeneous waves, due to the large degrees of inhomogeneity inherent in the transmitted waves. Analytical conditions have also been derived which predict the dependence of the optimal incident wave type on the dissipation level and wave speeds in the solid medium. Finally, implications related to the use of acoustic beams of limited spatial extent are discussed.

On Response of a Single-Degree-of-Freedom Oscillator with Constant Hysteretic Damping Under Arbitrary Excitation

The simple constant hysteretic damping model is known to be non-causal although it is used often in diverse branches of engineering. In this paper the response of a single degree of freedom oscillator having linear hysteretic damping under arbitrary force excitation has been studied after deriving the impulse response function of the system. Some shortcomings of the results available in literature have been pointed out. It has been shown that the damping model can be practically used for calculating the response of a physical system when the damping is small and the force has small duration.

Hysteretic damping and causality

It is proven that linear oscillatory systems with hysteretic damping in the form of complex stiffness and/or complex elastic moduli satisfy the causality principle: the response of such a system to an arbitrary external force cannot appear earlier than the onset of the force. The proof, based on a rigorous solution to the problem of forced oscillations, is presented in detail for an oscillator with a complex stiffness, as well as in a brief explanation for a system with N mass. It is also shown that these systems are Lyapunov-unstable. A comparison is made to other linear hysteretic damping models.

Dynamic Response of Machine Foundation on Layered Soil: Cone Model Versus Experiments

This paper presents the experimental validation of analytical solution based on cone model for machine foundation vibration analysis on layered soil. Impedance functions for a rigid massless circular foundation resting on a two layered soil system subjected to vertical harmonic excitation are found using cone model. Linear hysteretic material damping is introduced using correspondence principle. The frequency-amplitude response of a massive foundation is then computed using impedance functions. To verify the solution field experiments are conducted in two different layered soil systems such as gravel layer over in situ soil and gravel layer over concrete slab (rigid base). A total 72 numbers of vertical vibration tests on square model footing were conducted using Lazan type mechanical oscillator, varying the influencing parameters such as depth of top layer, static weight of foundation and dynamic force level. The frequency-amplitude response in general and in particular the resonant frequencies and resonant amplitudes predicted by cone model is compared with the results of experimental investigation, which shows a close agreement. Thus the cone model is reliable in its application to machine foundation vibration on layered soil.

Dynamic response of high-speed rail foundations using linear hysteretic damping and frequency domain substructuring

A proposed method for mitigating ground-borne vibration generated by high-speed trains is the usage of rubber-modified asphalt concrete as a ballast mat material. To gauge this material's influence in vibration control, a dynamic finite element code that models distinct damping properties in a large, nonhomogenous system was created. A linear hysteretic damping model is used to capture the dissipative mechanisms of each material; frequency domain substructuring is used for computational efficiency. Vibratory responses utilizing different ballast mat materials in a high-speed rail foundation are compared. It is shown that rubber-modified asphalt concrete results in a general reduction of motion, particularly in directions horizontally parallel and perpendicular to the train's passage. The described modeling procedure may be used for any dynamic analysis where the preservation of material damping characteristics is desired.

Dynamic response of a beam on a frequency-independent damped elastic foundation to moving load

The dynamic displacement response of an infinitely long beam on an elastic foundation with frequency-independent linear hysteretic damping subjected to a constant amplitude or a harmonic moving load was investigated. The advance velocity was assumed to be constant. Formulations were developed in the transformed field domain using (i) a Fourier transform in moving space for moving loads of constant amplitude, (ii) a double Fourier transform in time and moving space for moving loads of arbitrary amplitude variation or to include the transient due to the initial application of the load for moving harmonic loads, and (iii) a Fourier transform in moving space for the steady-state response to moving harmonic loads. The effects of velocity, damping, loaded length, and load frequency on the deflected shape and the maximum displacement were investigated. The critical (resonant) velocities and frequencies were obtained by analyses, and expressions to find them were suggested.Key words: beam on elastic foundation, damping, Fourier transform, frequency, harmonic load, moving load, transformed field, velocity.

Analysis of Hysteretic Damping Using Analytic Signals

The purpose of this technical note is to present a time-domain formulation for linear hysteretic damping. The integro-differential equations that govern the dynamics of structures with linear hysteretic damping are transformed into ordinary differential equations in analytic signals—that is, complex-valued signals in which the real and imaginary parts are a Hilbert transform pair. The poles of this type of system show radial symmetry in the complex plane, determining that for each stable pole in the left-hand half of the complex plane, there is an “unstable” pole in the right-hand half. The impulse response functions of these unstable poles are bounded but noncausal. To illustrate the formulation, the analytic impulse response of a single-degree-of-freedom oscillator with linear hysteretic damping is obtained. The response of the structure to any loading signal can be obtained using time convolution of this impulse response function and the corresponding analytic excitation signal.

Modal Coupling and Accuracy of Modal Strain Energy Method

The modal strain energy (MSE) method is a technique for approximate analysis of structures whose models in the frequency domain show frequency-dependent stiffness and damping matrices such as structures containing viscoelastic dampers. In this paper, the concept of modal coupling in the frequency domain is addressed and expressions for diagonalizable frequency-dependent stiffness and damping matrices are given. The MSE method is described and the accuracy of this technique is investigated by computing the exact and approximate mean square responses of structures subjected to random excitation. Closed-form expressions of the mean square error are obtained for single–degree-of-freedom systems with linear hysteretic and Maxwell-type damping and the accuracy of the MSE method in the estimation of response of multi–degree-of-freedom structures with nonclassical, hysteretic, and Maxwell-type damping is investigated using a simple base-isolated model. Finally, the use of the MSE method for preliminary sizing of viscoelastic dampers is illustrated using a moment-resistant steel frame.