Hysteretic damping in rotordynamics: An equivalent formulation

Abstract

The hysteretic damping model cannot be applied to time domain dynamic simulations: this is a well-known feature that has been discussed in the literature since the time when analog computers were widespread. The constant equivalent damping often introduced to overcome this problem is also discussed, and its limitations are stated, in particular those linked with its application in rotordynamics to simulate rotating damping. An alternative model based on the nonviscous damping (NVD) model, but with a limited number of additional degrees of freedom, is proposed, and the relevant equations are derived. Some examples show applications to the rotordynamics field.

Introduction

The damping properties of structures are usually modeled by introducing a term that is linear with the velocity into the equation of motion. This model is usually referred to as viscous damping. When dealing with rotating structures or rotors this approach leads to two separate terms, one proportional to the deformation velocity and one to the rotational velocity and the displacements. The latter causes a circulatory matrix to be present in the linearized equation of motion and has a destabilizing effect [1].

An alternative is to use the so-called structural or hysteretic damping model [2]. It is essentially based on the observation that the energy losses in engineering materials undergoing cyclic loading are proportional to the square of the amplitude and almost independent of frequency, at least within a wide frequency range.

There has been much discussion in the last 60 years on the validity of such assumptions, but the fact remains that the dependence on the square of the amplitude allows for the introduction of linear models, and the independency of frequency (which amounts to saying that the phase lag between stresses and strains is essentially constant and independent of the frequency) allows for the definition of a constant complex stiffness, or complex Young's modulus when working at the level of material properties. Clearly, the last assumption cannot hold when the frequency of the hysteresis cycle tends to zero.

The hysteretic damping model is usually applied by defining a complex elastic modulus, if at the level of the characteristics of the material, or a complex stiffness, if at the level of the mechanical element. Under the assumptions above, the real and imaginary parts of the former (usually referred to as $E^{\prime }$ and $E^{″}$) and their ratio, the loss factor

$$\eta =\frac{E^{″}}{E\prime }\text{,}$$

are considered as constants and are characteristics of the material.

It is well known that the hysteretic damping model can be applied only to systems with harmonic motion [3], which means that it can be introduced into equations of motion referred to the frequency domain but not to the time domain [4]. The impossibility of using the hysteretic damping model in time domain formulations is a severe limitation, in particular because numerical simulation, based on the numerical integration in time of the equations of motion, is now a basic tool in the dynamic analysis of systems of all types.

Another critical issue is that hysteretic damping may overestimate the damping properties of materials when the vibration frequency is very low. Actually, if the vibration frequency tends to zero, it yields inconsistent results, which may not be an issue in structural dynamics, but it is in rotordynamics because synchronous whirling is seen by the material constituting the rotor as a vibration at vanishing frequency. This limitation has not prevented the use of the hysteretic damping model in rotordynamics [5], [6], but it has caused misunderstanding and incorrect interpretations [7].

The issue of looking for a model equivalent to hysteretic damping, but suitable for time domain formulations, was particularly felt in the 1950s and 1960s, in connection with the use of analog computers. Analog computers could be used only to study time-domain problems, and thus the usual hysteretic damping model could not be implemented on them.

Biot [8] and Caughey [9] showed that the model the former called ‘Voigt model of viscoelasticity’ and now often referred to as the ‘Maxwell–Weichert model’ [10] can be applied both in structural dynamics and in rotordynamics. This model can be applied in both frequency domain and time domain formulations and leads to results that, at least in a given frequency range, are very close to those obtained using the hysteretic damping model.

A different approach was introduced by Bagley and Torvick [11], [12]. It is based on a fractional derivative model and can model the damping of actual engineering materials in a wide frequency range with good precision.

The GHM (Golla, Hughes, McTavish) model [13], [14] is based on the Maxwell–Weichert model but is formulated in such a way that it is suitable to be used in the context of the FEM (finite element method).

More recently, the term ‘nonviscous damping’ has been used for this kind of energy dissipation, mainly by Adhikari [15], [16], [17], [18], [19], [20], who applied it to study in detail the behavior of damped vibrating systems of different types. The term nonviscous damping will be used throughout this paper. The aim of the present paper is to show that the nonviscous damping model can be easily applied to both hysteretic rotating and nonrotating damping in rotordynamics even in the context of the finite element modeling of complex rotors.