Linear Hysteretic Damping and the Hilbert Transform

Inaudi, José A.; Kelly, James M.

doi:10.1061/(asce)0733-9399(1995)121:5(626)

Cited by 134 publications

(63 citation statements)

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“…Following Hardin's pioneering work it has become a practice to use the following simple expression of the dynamic modulus of soil materials (8) where and are frequency independent quantities and is the frequency variable. The signum function is appended at the imaginary part of (8) The main flaw of this model that is due to its noncausality is well known to the literature (Caughey 1962;Crandall 1963Crandall , 1970Inaudi and Kelly 1995) and has been rigorously addressed by Makris (1997) and Makris and Zhang (2000). Despite its noncausal behavior the constant hys-…”

Section: The Shear Beam Approximationmentioning

confidence: 99%

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Kinematic response functions and dynamic stiffnesses of bridge embankments

Zhang

Makris

2002

Earthq Engng Struct Dyn

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SUMMARYRecognizing that soil-structure interaction affects appreciably the earthquake response of highway overcrossings, this paper compares approximate analytical solutions and finite element results to conclude on a simple procedure that allows for the estimation of the kinematic response functions and dynamic stiffnesses of approach embankments. It is shown that the shear-wedge model yields realistic estimates for the amplification functions of typical embankments and reveals the appropriate levels of dynamic strains which are subsequently used to estimate the stiffness and damping coefficients of embankments. The shear-wedge model is extended to a twodimensional model in order to calculate the transverse static stiffness of an approach embankment loaded at one end. The formulation leads to a sound closed-form expression for the critical length, , that is the ratio of the transverse static stiffness of an approach embankment and the transverse static stiffness of a unit-width wedge. It is shown through two case studies that the transverse dynamic stiffness ("spring" and "dashpot") of the approach embankment can be estimated with confidence by multiplying the dynamic stiffness of the unit-width wedge with the critical length,. The paper concludes that the values obtained for the transverse kinematic response function and dynamic stiffness can also be used with confidence to represent the longitudinal kinematic response function and dynamic stiffness respectively.KEY WORDS: bridge embankment, seismic response, kinematic response function, dynamic stiffness.

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Section: The Shear Beam Approximationmentioning

confidence: 99%

“…Similarly, the vertical static stiffness due to a distributed load at the crest is given by (20) For the case of a symmetric embankment equations (19) and (20) reduce to equations (16) and (17), since . …”

Section: The Shear Beam Approximationmentioning

confidence: 99%

Kinematic response functions and dynamic stiffnesses of bridge embankments

Zhang

Makris

2002

Earthq Engng Struct Dyn

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“…For the hysteretic damped model loaded by a harmonic force F = F 0 e jωt , which is similarly not real and not causal, the equation of motion may be expressed, in a mnemonic form, as [11][12][13] …”

Section: Root Locimentioning

confidence: 99%

Complex‐Damped Dynamic Systems in the Time and Frequency Domains

Bonisoli

Mottershead

2004

Shock and Vibration

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This paper is dedicated to the memory of BrunoAbstract. The fact that a complex-damped model may represent the dynamic behaviour of elasto-mechanical systems when acted upon by a magnetic field was brought to the attention of the structural dynamics community very recently by Professor Bruno A. D. Piombo and his colleagues at the Politecnico di Torino. In this paper a thorough analysis of the single degree-of-freedom complex-damped mass-spring system is presented. The analysis includes the root locus, the (non-causal) impulse response, the frequency response and the transmissibility. Regions of different behaviour in the frequency response and transmissibility are described in detail. The stiffening behaviour observed in Prof. Piombo's experiments and known as the "phantom effect" is demonstrated by the complex-damped model.

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“…It is also very well known that the constant hysteretic damping model is non-causal (see, for instance [8,9,11,12]). In fact, as reminded by Inaudi and Kelly [10] and discussed in [13], it is not possible to formulate a causal model that has simultaneously a storage modulus and a loss modulus independent of the frequency. Back in 1958, Biot [14] proposed a viscoelastic damping model whose variation with frequency is very weak with respect to the loss modulus, and that can therefore come close -in practice -to the constant hysteretic model, although it leads to a storage modulus that increases with frequency.…”

Section: Introductionmentioning

confidence: 99%

“…From a physical point of view, it does not make any sense indeed. For instance, Inaudi and Kelly [10] state that the force k (1 + iη) x(t) in Eq. (4) is clearly incorrect, as it would mean that a real response implies a complex force.…”

Section: Introductionmentioning

confidence: 99%

Reflections on the Hysteretic Damping Model

Maia

2009

Shock and Vibration

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Abstract. This paper presents a reflection on a recently proposed solution to the problem of the free vibration response with the constant hysteretic damping model, that has been presented in some conferences in recent years, by the author himself and some of his colleagues. On the one hand, as expected, the subject has been received with natural criticism, mainly due to the well-known non-causal behaviour of the model in free vibration. On the other hand, it was not easy to understand what could be wrong in that proposal, as apparently everything was perfect from a mathematical point of view. The author decided that this subject deserved a more careful and detailed analysis and -in this kind of tutorial paper -the issue seems to have been clarified. It is concluded that the proposed solution involving the constant hysteretic damping corresponds in fact to an equivalent viscously damped model; it is therefore concluded that the application of the constant hysteretic damping to model the free vibration of practical engineering problems should be considered only in the perspective of an equivalent viscously damped model.

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Linear Hysteretic Damping and the Hilbert Transform

Cited by 134 publications

References 5 publications

Kinematic response functions and dynamic stiffnesses of bridge embankments

Kinematic response functions and dynamic stiffnesses of bridge embankments

Complex‐Damped Dynamic Systems in the Time and Frequency Domains

Reflections on the Hysteretic Damping Model

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