Displacement-dependent Damping Research Articles

Displacement-Dependent Damping Inerter System for Seismic Response Control

Various inerter systems utilizing velocity-dependent damping for vibration control have been developed. However, a velocity-dependent damping element may exhibit relatively poor performance compared to a displacement-dependent damping element (DDE) of equivalent damping ratio, when the structural deformation is small in the early stage of the seismic response. To address this issue, the advantage of DDE in generating a larger control force in the early stage of excitation is promoted here and enhanced by a supplemental inerter-spring-system, thus realizing a proposed novel displacement-dependent damping inerter system (DDIS). First, the influence of various DDIS-parameters is carried out by resorting to the stochastic linearization method to handle non-linear terms. Then, seismic responses of the DDIS-controlled system are evaluated in the time domain taking the non-linearity into account, thus validating the accuracy of the stochastic dynamic analysis. Several design cases are considered, all of which demonstrated damping enhancement and timely control achieved by the DDIS. The results show that the energy dissipation as well as reduction of structural displacement and acceleration accomplished by the proposed system are significant. DDIS suppresses structural responses in a timely manner, as soon as the peak excitation occurs. In addition, it is demonstrated that interactions among the inerter, spring, and DDE, which constitute the damping-enhancement mechanism, lead to a higher energy-dissipation efficiency compared to the DDE alone.

Effect of the nonlinear displacement-dependent characteristics of a hydraulic damper on high-speed rail pantograph dynamics

A new simplified parametric model, which is more suitable for pantograph–catenary dynamics simulation, is proposed to describe the nonlinear displacement-dependent damping characteristics of a pantograph hydraulic damper and validated by the experimental results in this study. Then, a full mathematical model of the pantograph–catenary system, which incorporates the new damper model, is established to simulate the effect of the damping characteristics on the pantograph dynamics. The simulation results show that large $$F_{\mathrm{const}}$$ (saturation damping force of the damper during compression) and $$C_{\mathrm{0}}$$ (initial damping coefficient of the damper during extension) in the pantograph damper model can improve both the raising performance and contact quality of the pantograph, whereas a large $$C_{\mathrm{0}}$$ has no obvious effect on the lowering time of the pantograph; the nonlinear displacement-dependent damping characteristics described by the second item in the new damper model have dominating effects on the total lowering time, maximum acceleration and maximum impact acceleration of the pantograph. Thus, within the constraint of total lowering time, increasing the nonlinear displacement-dependent damping coefficient of the damper will improve the lowering performance of the pantograph and reduce excessive impact between the pantograph and its base frame. In addition, damping performance of the new damper model would vary with the vehicle speeds, when operating beyond the nominal-speed range of the vehicle, the damping performance would deteriorate obviously. The proposed concise pantograph hydraulic damper model appears to be more adaptive to working conditions of the pantograph, and more complete and accurate than the previous single-parameter linear model, so it is more useful in the context of pantograph–catenary dynamics simulation and further parameter optimizations. The obtained simulation results are also valuable and instructive for further optimal specification of railway pantograph hydraulic dampers.

Displacement-dependent nonlinear damping model in steel buildings with bolted joints

The stick–slip phenomenon is commonly found at structural connections in steel buildings. It is a major damping mechanism in a structure with bolted joints and makes a significant contribution to the total structural damping. This article reviews the stick–slip damping model of an elastic single-degree-of-freedom system with one stick–slip component. It is observed that the damping ratios of the system with the stick–slip mechanism first quickly increase when experiencing a very small displacement and then slowly decrease. After the number of activated slip surfaces is assumed to be a linear function related to the structural displacement, the equivalent damping ratios of a structural system with numerous stick–slip components are derived. However, this displacement-dependent damping model is very difficult to be used for a structural dynamic analysis due to its inherent complexity. Therefore, a new displacement-dependent damping model for a structural dynamic analysis is proposed based on the viscous damping. A high-rise steel moment resisting frame with bolted joints subjected to an earthquake ground motion is taken as an example to verify the proposed method.

Generalized Response Spectrum Analysis for Structures with Dampers

In this study, a computational seismic design routine is proposed based on a generalized response spectrum analysis for highly indeterminate structures with energy-dissipation members, such as viscous or elasto-plastic dampers. Complex stiffness terms are introduced to account for displacement-dependent damping, and a three-dimensional (3-D) element stiffness matrix with complex axial stiffness is proposed for elasto-plastic dampers. A modified complete quadratic combination method previously developed for real symmetric damped systems is extended to complex asymmetric damped systems, based on a theoretical analysis of eigenvalue equations. The response is evaluated by iteratively conducting complex eigenvalue analysis and modal combination. The accuracy is confirmed through comparison to nonlinear response history analysis of 2-D frame models. Finally, an example application is presented of a 3-D truss tower seismically retrofitted by replacing the braces with viscoelastic and then elasto-plastic dampers. The proposed design routine is used to rapidly identify novel and efficient damper arrangements and sizing distributions, avoiding computationally intensive nonlinear response history analysis.

Passive direction displacement dependent damping (D3) device

Viscous fluid damping has been used worldwide to provide energy dissipation to structures during earthquakes. Semi-active dissipation devices have also shown significant potential to re-shape structural hysteresis behaviour and thus provide significant response and damage reduction. However, semi-active devices are far more complex and costly than passive devices, and thus potentially less robust over time. Ideally, a passive device design would provide the unique response behaviour of a semi-active device, but in a far more robust and low-cost device. This study presents the design, development and characterization of a passive Direction and Displacement Dependent viscous damping (D3) device. It can provide viscous damping in any single quadrant of the force-displacement hysteresis loop and any two in combination. Previously, this behaviour could only be obtained with a semi-active device. The D3 device is developed from a typical viscous damper, which is tested to evaluate the baseline of orifice sizing, force levels and velocity dependence. This prototype viscous damper is then modified in clear steps to produce a device with the desired single quadrant hysteresis loop. The overall results provide the design approach, device characterization and validation for this novel device design.

Global attractor for the one dimensional wave equation with displacement dependent damping

We study the long-time behavior of solutions of the one dimensional wave equation with nonlinear damping coefficient. We prove that if the damping coefficient function is strictly positive near the origin then this equation possesses a global attractor.

Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent

In this paper the long time behaviour of the solutions of the 3-Dstrongly damped wave equation is studied. It is shown that thesemigroup generated by this equation possesses a global attractorin $H_{0}^{1}(\Omega )\times L_{2}(\Omega )$ and then it is provedthat this is also a global attractor in $(H^{2}(\Omega )\capH_{0}^{1}(\Omega ))\times H_{0}^{1}(\Omega )$.

A global attractor for the plate equation with displacement-dependent damping

We study the long-time behaviour of solutions of the plate equation with nonlinear damping coefficient. We prove under suitable conditions that this equation possesses a global attractor.

A strong global attractor for the 3D wave equation with displacement dependent damping

We prove that if the displacement coefficient of the damping of the 3D wave equation is a positive constant on the interval ( − l , l ) , for large enough l > 0 , then this equation has a strong global attractor in H 0 1 ( Ω ) × L 2 ( Ω ) . We also show that this attractor is a bounded subset of H 2 ( Ω ) ∩ H 0 1 ( Ω ) × H 0 1 ( Ω ) .

Global attractors for 2-D wave equations with displacement-dependent damping

We study the long-time behavior of solutions of the one dimensional wave equation with nonlinear damping coefficient. We prove that if the damping coefficient function is strictly positive near the origin then this equation possesses a global attractor.

Global and exponential attractors for 3‐D wave equations with displacement dependent damping

AbstractA weakly damped wave equation in the three‐dimensional (3‐D) space with a damping coefficient depending on the displacement is studied. This equation is shown to generate a dissipative semigroup in the energy phase space, which possesses finite‐dimensional global and exponential attractors in a slightly weaker topology. Copyright © 2006 John Wiley & Sons, Ltd.