Geometric Damping Research Articles

Tailored nonlinear stiffness and geometric damping: Applied to a bistable vibration absorber

A novel device is proposed to obtain arbitrary restoring force characteristics and nonlinear geometric damping. It consists of linear springs and dampers that are compressed along a track. The obtained nonlinear damping and stiffness law depend on the track’s shape. The device is then applied to obtain a nonlinear energy sink (NES) to damp the vibrations of a host system. Both the case of transient vibrations, induced by shock loads, and sustained vibrations, induced by harmonic loads, are studied. For arbitrary stiffness and damping, slow flow dynamics and slow invariant manifolds (SIMs) are derived by applying harmonic balancing and multiple timescales techniques. Under transient vibrations, two performance measures are derived from the SIM, the relative residual energy and the pumping time, and are found for generic nonlinear spring force and nonlinear geometric damping. For sustained vibrations, a load-dependent frequency response (FR) is derived from the SIMs. This FR predicts the occurrence of the efficient strongly modulated responses but also the occurrence of the unfavorable detached responses called isolas, where the NES fails to mitigate and even amplifies vibrations.This research investigates the performance of a conventional cubic NES and a bistable NES (BNES) with nonlinear damping obtained from the proposed device. Especially the BNES with nonlinear damping shows attractive properties, with a high degree of robustness over a wide energy range and a low vibration threshold under transient loading. Under harmonic loads, the nonlinear damping helps reduce the effect of isolas. A novel tuning methodology is proposed to avoid isolas for a range of load magnitudes. The new device opens up a whole range of possibilities regarding energy dissipation through nonlinear damping.

Determining In-situ Frequency Independent Attenuation and Complex Modulus of Pavement Materials using Wavelet Power Density of Vibration Method

The features of vibration sources, site characteristics, the propagation of surface and body waves in the ground, and the reaction of structures are all important to consider when analyzing vibration-related issues, including those caused by traffic loading. Pavement vibration analysis can be used to determine the impact of damaging vibrations on the road and surrounding buildings or structures. Furthermore, this vibration can be utilized to calculate the dynamic properties of road pavement materials, such as the attenuation factor and damping ratio of the material, as well as the complex modulus of the pavement material. To minimize vibration damage, an environmental zone that is effective in reducing ground vibration amplitude is frequently used. However, estimating the degree of decrease in vibration amplitude at a given distance is challenging. In general, vibration attenuation with distance is made up of two components: geometric damping and material damping. Geometric damping is related to the qualities of the ground and the magnitude of the vibration, while material damping is dependent on the nature of the source of the vibration. The majority of ground vibrations are now monitored mainly at the ground surface, rather than in deep layers, and propagation patterns are not properly considered. The wavelet power density of vibration technique was used in this study to quantify the frequency-dependent attenuation of geometric damping at a variety of pavement structures in Malaysia and Indonesia. At each location, the energy in a seismic wave signal was calculated as a function of the Gaussian of the derivative wavelet in signal amplitude. The vibration source's propagating waves were described by examining the recorded particle motions and significant energy components in the time-frequency domain. The complex modulus of pavement materials was then calculated using the frequency-dependent attenuation factors. The results reveal that the wavelet power density of vibration method can properly define the physical properties of the dynamic pavement material, such as attenuation frequency, damping ratio, and complex pavement modulus. This approach has the advantages of being quick, cost-effective, and nondestructive.

From the Ravine Method to the Nesterov Method and Vice Versa: A Dynamical System Perspective

We revisit the Ravine method of Gelfand and Tsetlin from a dynamical system perspective, study its convergence properties, and highlight its similarities and differences with the Nesterov accelerated gradient method. The two methods are closely related. They can be deduced from each other by reversing the order of the extrapolation and gradient operations in their definitions. They benefit from similar fast convergence of values and convergence of iterates for general convex objective functions. We will also establish the high resolution ODE of the Ravine and Nesterov methods and reveal an additional geometric damping term driven by the Hessian for both methods. This will allow us to prove fast convergence toward zero of the gradients not only for the Ravine method but also for the Nesterov method for the first time. In the strongly convex case, we show linear convergence for the Ravine method at an optimal rate. We also highlight connections to other algorithms resulting from more subtle discretization schemes and finally describe a Ravine version of the proximal-gradient algorithms for general structured smooth + nonsmooth convex optimization problems.

Asymptotic behavior of Newton-like inertial dynamics involving the sum of potential and nonpotential terms

In a Hilbert space mathcal{H}, we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator A= nabla f +B , where ∇f is the gradient of a continuously differentiable convex function f and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as tto +infty of the generated trajectories towards the zeros of nabla f +B. The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open the door to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of the nonpotential term.

Experimental and numerical investigations on attenuation response of machine foundations under vertical excitation

ABSTRACT The attenuation response from a series of block vibration tests performed on a model square machine foundation at a site near IIT Kanpur, India, is reported in this paper. The dynamic response at different locations from the vibration source is measured for a wide range of frequencies. The observed attenuation response is compared with the analytical and the finite element (FE) solutions to bolster the experimental findings. A parametric study is conducted utilising the FE analysis to predict the surface wave mitigation characteristics in various soils. It can be observed that the surface waves dominate the attenuation characteristics at the far-field locations and attenuate at a faster rate in soft soils compared to stiff soils. The material and the geometric damping characteristics of the surface waves influence the attenuation characteristics of horizontal and vertical vibrations at the far-field pick-up points. The attenuation characteristics of horizontal and vertical amplitude responses are found to differ significantly. Curve fitting and regression analyses are also performed to develop simplified design expressions to predict the attenuation response of horizontal and vertical vibrations. The proposed design expressions compare well with the values reported in the literature and can be utilised by practicing engineers.

The Elliptic Harmonic Balance Method for the Performance Analysis of a Two-Stage Vibration Isolation System with Geometric Nonlinearity

This study develops a modified elliptic harmonic balance method (EHBM) and uses it to solve the force and displacement transmissibility of a two-stage geometrically nonlinear vibration isolation system. Geometric damping and stiffness nonlinearities are incorporated in both the upper and lower stages of the isolator. After using the relative displacement of the nonlinear isolator, we can numerically obtain the steady-state response using the first-order harmonic balance method (HBM1). The steady-state harmonic components of the stiffness and damping force are modified using the Jacobi elliptic functions. The developed EHBM can reduce the truncation error in the HBM1. Compared with the HBM1, the EHBM can improve the accuracy of the resonance regimes of the amplitude-frequency curve and transmissibility. The EHBM is simple and straightforward. It can maintain the same form as the balancing equations of the HBM1 but performs better than it.

Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling

<p style='text-indent:20px;'>In a Hilbert setting, we develop fast methods for convex unconstrained optimization. We rely on the asymptotic behavior of an inertial system combining geometric damping with temporal scaling. The convex function to minimize enters the dynamic via its gradient. The dynamic includes three coefficients varying with time, one is a viscous damping coefficient, the second is attached to the Hessian-driven damping, the third is a time scaling coefficient. We study the convergence rate of the values under general conditions involving the damping and the time scale coefficients. The obtained results are based on a new Lyapunov analysis and they encompass known results on the subject. We pay particular attention to the case of an asymptotically vanishing viscous damping, which is directly related to the accelerated gradient method of Nesterov. The Hessian-driven damping significantly reduces the oscillatory aspects. We obtain an exponential rate of convergence of values without assuming the strong convexity of the objective function. The temporal discretization of these dynamics opens the gate to a large class of inertial optimization algorithms.</p>

First-order optimization algorithms via inertial systems with Hessian driven damping

In a Hilbert space setting, for convex optimization, we analyze the convergence rate of a class of first-order algorithms involving inertial features. They can be interpreted as discrete time versions of inertial dynamics involving both viscous and Hessian-driven dampings. The geometrical damping driven by the Hessian intervenes in the dynamics in the form $$\nabla ^2 f (x(t)) \dot{x} (t)$$ . By treating this term as the time derivative of $$ \nabla f (x (t)) $$ , this gives, in discretized form, first-order algorithms in time and space. In addition to the convergence properties attached to Nesterov-type accelerated gradient methods, the algorithms thus obtained are new and show a rapid convergence towards zero of the gradients. On the basis of a regularization technique using the Moreau envelope, we extend these methods to non-smooth convex functions with extended real values. The introduction of time scale factors makes it possible to further accelerate these algorithms. We also report numerical results on structured problems to support our theoretical findings.

Application of Quarter Car Model for Assessment of Attenuation Characteristics of Soil at Low Strain

Due to increase in population, industrialisation and a lack of space in the urban areas of Surat city in Gujarat (India), numerous buildings are being constructed in the vicinity of the railway tracks. The perspective is further aggravated by the proposal for a high-speed corridor (≥ 200 kmph) connecting two key Indian cities, Mumbai and Ahmedabad. These structures need to be evaluated for their susceptibility to the dynamic stresses induced by such high-speed locomotives. However, before the assessment of such dynamic stresses, geometrical and material damping properties of the in situ soil must be determined as the stresses induced in the structure also depend on the attenuation characteristics of the subsoil. With the objective to determine attenuation characteristics of the soil due to high-speed trains, the current study uses modelling capabilities of SAP2000 to prepare a vehicle-track-soil interaction model for dynamic analysis of the site in Surat city (India). A quarter car model (lumped mass model) prepared in SAP2000 environment is used to replicate CHR3-type high-speed train. The passage of train over the track structure is represented as a sinusoidal harmonic load. The study establishes that the statistical analysis of the attenuation of the waves is exponentially related to the distance from the source and follows the Bornitz equation. The frequency-independent damping ratio is computed considering near-field and far-field effect of induced vibration in the soil. Finally, the damping ratio and transmissibility are used to derive the critical velocity for the given track near the proposed site.

Propagation and attenuation characteristics of the vibration in soft soil foundations induced by high-speed trains

The vibration caused by high-speed trains (HSTs) can result in reductions of strength and bearing capacity of the railway foundation, unexpected settlement, and other geotechnical engineering problems. Thus it is essential to estimate the vibration level at different depths of the foundation. This study investigates the propagation and attenuation characteristics of the vibration in the soft soil foundation induced by high-speed trains. A series of field experiments were conducted to measure the HST-induced vibration in the soft soil foundation in Shanghai, China. The measurement provides a unique dataset for numerical model validations. Based on the Bornitz equation, a new attenuation model is proposed for layered homogeneous media. The material damping coefficient and the geometric damping coefficient, two input parameters for the model, are thoroughly investigated using measurements in this study and from the existing literature. The proposed model was applied to estimate the vertical vibration levels at different depths in typical subgrade and foundation of HSTs in the soft area. Model performance is supported by the good agreement between predictions and observations. Our model provides an alternative tool to further investigate the effect of HST-induced vibration on foundation soils and adjacent underground structures.

In-situ measurement of railway-traffic induced vibrations nearby the liquid-storage tank

In this study, result of a field investigation of railway traffic-induced vibrations is provided to examine acceptability levels of ground vibration and to evaluate the serviceability of a liquid-storage tank. Free field attenuation of the amplitudes as a function of distance is derived by six accelerometers and compared with a well-known half-space Bornitz´s analytical solution which considers the loss of the amplitude of waves due to geometrical damping and material damping of Rayleigh. Bornitz´s solution tends to overlap vertical free field vibration compared with in-situ measured records. The vibrations of the liquid-storage tank were compared with the USA, Federal Transportation Railroad Administration (FTA) criteria for acceptable ground-borne vibrations and with the criteria in DIN 4150-3 German standard. Comparing the thresholds stated in DIN 4150-3, absolute peak particle velocities are within the safe limits, however according to FTA velocity level at the top of the water tank exceeds the allowable limits. Furthermore, it is intended to indicate experimentally the effect of the kinematic interaction caused by the foundation of the structure on the free-field vibrations.

Yüksek Hızlı Trenlerin Serbest Zemin Yüzeyinde Oluşturduğu Titreşimlerin Deneysel ve Analitik Olarak İncelenmesi

Normal 0 21 false false false EN-US X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Normal Tablo"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman",serif; mso-ansi-language:EN-US; mso-fareast-language:EN-US;} Railway platform which is vibrated with higher frequency dynamic loads induced by high speed train cars transfer these vibrations energies to surrounding ground and close by structures are effected by them. Strong ground motion not only can give damage to the nearby buildings and their footings but also effect the human comfort by undesired vibrations. Extensive in-situ research are still required for both recent ongoing construction projects in soft soil deposits (Eskişehir-Adapazarı-Izmit) and for planned construction projects on highly populated areas (Edirne-Istanbul-Ankara-Kars) in Turkey. The aim of the study can be summarized as to determine comfort disturbance and devastating environmental vibration created by high speed trains on free field. Strong ground motions induced by high speed train measured and recorded at the inspection points selected at the site close to high speed train line passing through highly populated regions. Environmental effects of these vibration data compared and evaluated. However, the applicability of a well know half space solutions based on Bornitz’s analytical approach which takes account the loss of the amplitude of waves due to geometrical damping and material damping of Rayleigh waves are investigated. Bornitz’s solution tends to overlap vertical free field vibration with in-situ measured records.

A numerical model for the dynamic analysis of inclined pile groups

SummaryThe paper presents a numerical model for the dynamic analysis of pile groups with inclined piles in horizontally layered soil deposits. Piles are modelled with Euler–Bernoulli beams, while the soil is supposed to be constituted by independent infinite viscoelastic horizontal layers. The pile–soil–pile interaction as well as the hysteretic and geometric damping is taken into account by means of two‐dimensional elastodynamic Green's functions. Piles cap is considered by introducing a rigid constraint; the condensation of the problem permits a consistent derivation of both the dynamic impedance matrix of the soil–foundation system and the foundation input motion. These quantities are those used to perform inertial soil–structure interaction analyses in the framework of the substructure approach. Furthermore, the model allows evaluating the kinematic stress resultants in piles resulting from waves propagating in the soil deposit, taking into account the pile–soil–pile interactions. The model validation is carried out by performing accuracy analyses and comparing results in terms of dynamic impedance functions, kinematic response parameters and pile stress resultants, with those furnished by 3D refined finite element models. To this purpose, classical elastodynamic solutions are adopted to define the soil–pile interaction problem. The model results in low computational demands without significant loss of precision, compared with more rigorous approaches or refined finite element models. Copyright © 2015 John Wiley & Sons, Ltd.

Characteristics of spherical stress wave propagation in the standard linear solid material

Based on the standard linear solid model, the solutions in Laplace domain, such as particle velocity v, particle displacement u, radial stress σr, tangential stress σθ, radial strain εr, tangential strain εθ, reduced velocity potential γ (RVP), and reduced displacement potential ψ (RDP), are derived from the spherical wave equations. The propagating characteristics of these physical quantities, as mentioned above, are calculated by using Crump algorithm for inverse Laplace transformation. The numerical inversion results reveal that the initial response to strong discontinuity spherical stress wave in viscoelastic material is purely elastic response. The strong discontinuities, such as σr, σθ, εr, εθ and v, contain geometrical attenuation and viscoelastic damping in the process of wave propagation. The variables, such as σr, σθ, εr, εθ,u and ψ, converge to steady values as time approaches to infinity. The peak values of RVP γ and RDP ψ, which are constant in a purely elastic material, are steadily reduced with the spreading distance increasing in viscoelastic material. The steady values of ψ are in inverse relation to the static shear modulus Ga, and directly proportional to the steady cavity pressure and the cube of the cavity radius r.

Stability of a flat arch subjected to deterministic and stochastic loads taking into account nonlocal damping

In the example of a flat arch, a problem of vibrational stability of systems subjected to deterministic and stochastic loads is considered taking into account the geometric nonlinearity and nonlocal damping of a material, which is characteristic of certain types of composites and nanomaterials. To study the stability of arch motion within the deterministic statement of the problem (stability in the sense of Lyapunov) and the stability in the almost certain stochastic statement, a method is used which is based on the calculation of the maximal Lyapunov’s exponent. The influence of damping and loading parameters on the degree of system stability is analyzed.

Influence zone around a closed-ended pile during vibratory driving

This paper presents a numerical study based on vibratory driving of closed-ended piles. A set of axisymmetric finite element models were used to replicate the wave propagation during vibratory pile driving and to investigate the effect of wave propagation on the surrounding ground. The numerical modelling technique adopted for the analysis takes into account the large soil deformations around the pile during driving and is based on the Arbitrary Lagrangian Eulerian technique. It has the ability to drive a pile few pile diameters below the initial position without mesh distortions and the numerical model was verified using field data available in the literature. Soil was modelled as an elastic-perfectly plastic material. A parametric study was performed to determine the influence of change in pile driving force on the far field using different operating frequencies and amplitudes of the driving force, and rigidity index and material damping of the surrounding soil. The parameters for the driving force were extracted from the published specifications of commercially available vibratory piling rigs. Finite element results were compared with ground vibration measurements of peak particle velocities during vibratory sheetpile driving found in the literature. These results show that the material damping is an important parameter contributing to wave attenuation around the driven pile in addition to the geometric damping. The impact on the far field is discussed comparing the peak particle velocity distributions with the specifications given by the American Association of State Highways and Transportation Officials (AASHTO), Swiss Standard SN640312 and Eurocode 3 for acceptable vibrations to avoid damages to existing nearby structures. Finally attenuation relationships, and upper and lower bounds for the peak particle velocity distributions around a driven closed-ended pile are presented to determine the influence zones for different types of nearby structures.

'Geometric' Viscous Damping Model for Nearly Constant Beam Modal Damping

This paper considers the damped transverse vibration of flexural structures. Viscous damping models available to date, such as proportional damping, suffer from the deficiency that the resulting modal damping is strongly frequency dependent, which is a situation not representative of experiments with built-up structures. The focus model addresses a viscous geometric damping term in which an internal resisting shear force is proportional to the time rate of change of the slope. Separation of variables does not lead directly to a solution of the governing partial-differential equation, although a boundary-value eigenvalue problem for free vibration can nevertheless be posed and solved. For small damping the method of weighted residuals provides an alternate approach to the development of approximate modal equations of motion and estimation of modal damping. In a discretized finite-element context the resulting damping matrix resembles the geometric stiffness matrix used to account for the effects of membrane loads on lateral stiffness. For beams having any combination of hinged and guided boundary conditions this model yields constant modal damping that is independent of frequency as well as real mode shapes. For more general boundary conditions modal damping varies somewhat, approaching the expected constant value with increasing mode number; furthermore, the mode shapes are complex. This viscous damping model should prove useful to researchers and engineers who need a simple time-domain damping model that exhibits more realistic variation of damping with frequency than the alternatives.

Deflagration-to-detonation transition by amplification of acoustic waves in type Ia supernovae

We study a new mechanism for deflagration to detonation transition in thermonuclear supernovae (SNe Ia), based on the formation of shocks by amplification of sound waves in the steep density gradients of white dwarfs envelopes. Given a large enough jump in density a small pressure and velocity perturbation, produced by the turbulent deflagration, turns into a shock down of the gradient, where it will dissipate and heat up the media. With the right frequency and amplitude the heating can be enough to initiate a detonation, which can propagate backward and up the density gradient. We studied planar and spherical geometry. In the planar case we made a parametric study of the frequency and amplitude. We found it possible to obtain a detonation for perturbations down to Mach number M=0.003. In the spherical case, geometrical damping makes it harder to initiate a detonation, but considering a small He atmosphere (<0.01 Msol) makes it possible again to obtain a detonation down to small perturbation (M=0.002). In the context of thermonuclear supernovae, this could be a mean to turn a turbulent flame producing sound waves to a detonation.

A second-order differential system with Hessian-driven damping; Application to non-elastic shock laws

We consider the second-order differential system with Hessian-driven damping ¨ u + α ˙ u +β∇ 2 Φ(u)˙ u +∇Φ(u )+ ∇Ψ(u )= 0, where H is a real Hilbert space, Φ,Ψ : H → R are scalar potentials, and α,β are positive parameters. An interesting property of this system is that, after introduction of an auxiliary variable y, it can be equivalently written as a first- order system involving only the time derivatives ˙ u, ˙ y and the gradient operators ∇Φ, ∇Ψ. This allows to extend our analysis to the case of a convex lower semicontinuous function Φ : H → R ∪{ +∞}, and so to introduce constraints in our model. When Φ = δK is the indicator function of a closed convex set K ⊆ H , the subdifferential operator ∂Φ takes account of the contact forces, while ∇Ψ takes account of the driving forces. In this setting, by playing with the geometrical damping parameter β , we can describe nonelastic shock laws with restitution coefficient. Taking advantage of the infinite dimensional framework, we introduce a nonlinear hyperbolic PDE describing a damped oscillating system with obstacle. The first-order system is dissipative; each trajectory weakly converges to a minimizer of Φ + Ψ, provided that Φ and Φ +Ψ are convex functions. Exponential stabilization is obtained under strong convexity assumptions.

Study on Seismic Response Characteristics of Double Cables Suspension Bridge

Damping in double Cables suspension bridge composed of steel reinforcement beams and reinforced concrete tower is non-classical, which leads to coupled equations of motion in main coordinate system. Based on the complex damping theory, the viscous damping ratio is solved, which can be used to describe energy dissipation characteristics of non-classical damping system approximately. Seismic response of double chains suspension bridge is analyzed through an example of double chains suspension bridge, considering the geometric nonlinearity and non-classical damping. And numerical calculation is presented for seismic response subjected to independent effect or combination effect of three orthogonal components of seismic wave. Single cable suspension bridge can be taken as a special case of double cable suspension bridge, after the main cable shape coefficient is introduced. The dynamic responses of double cable suspension bridge and single cable suspension bridge are compared to reveal the characteristics of Seismic Response of double cable suspension bridge. The study of the dynamic responses characteristics of double cable suspension bridge has a positive significance on structural form selection of such type bridge during designing, dynamic performance evaluation and reinforcement design has positive significance.