2.1 Viscous and structural damping
In this section one analyses the properties of classical models with viscous and structural damping of the form
where matrices are assumed to be constant.
This type of models does not allow a correct representation of the local behavior of damping treatments (constitutive laws detailed in the preceding chapter). At the level of a complete structure, it is however often possible to represent the effect of various damping mechanisms by a viscous or structural model. One then uses a global behavior model. It does not necessarily have a local mechanical meaning, but this does not lower its usefulness.
The main results introduced in this section are
- properties of the damped oscillator;
- the conditions of validity of the modal damping assumption which leads to models where the response is decomposed in a sum of independent oscillators;
- techniques used to estimate equivalent viscous damping models.
2.1.1 Properties of the damped 1 DOF oscillator
This section illustrates the properties of the single degree of freedom oscillator with a viscoelastic stiffness shown in figure ??.
| Figure 2.1: Oscillator with a viscoelastic stiffness |
For a viscous damping K(s) = k + c s, the load to displacement transfer is given by
whose poles (root of the transfer denominator) λ are
For structural damping K(s) = k (1 + i η) , the load to displacement transfer is given by
its pole with a positive imaginary part is identical to that of the viscous model for
which leads to a difference Hη−Hvisc that is zero at resonance ωn. The pole with a negative imaginary part is unstable (positive real part) which is a classical limitation of the structural damping model.
Damping is also defined by a quality factor which can be measured in a shaking table as the ratio between the acceleration of the mass at resonance and the acceleration of its base. The value is approximately
Under the assumption that the strain energy is sufficiently uniform to be represented as a spring, Q−1 the inverse of the quality factor corresponds to a loss factor.
For a damper following the 3 parameter law of a standard viscoelastic solid (??), the system has a pair of complex poles λ,λ and one real pole β
and the model characteristics depend on those poles as follows
For low damping of the conjugate pair of poles (that is ζ ≪ 1) and β and ω in the same frequency range, p and z are close which leads to a small maximum loss factor and a response that is very similar to that of the oscillator with viscous damping (??).
Figure ?? shows that for viscous, structural and viscoelastic dynamic stiffness models for the oscillator (c), the dynamic flexibilities (a-b) are almost exactly overlaid. This is linked to the fact that the real parts of the dynamic stiffness coincide naturally since they are they not (viscous or structural) or little (viscoelastic) influenced by the damping model and the imaginary parts (d) are equal at resonance.
This equivalence principle is the basis for the Modal Strain Energy (MSE) method that will be detailed in section ??.
| Figure 2.2: Low sensitivity of the dynamic flexibility to the damping model (× viscous, o structural, + standard viscoelastic) |
2.1.2 Real modes and modal damping
For an elastic model, normal modes are solution of the eigenvalue problem (see ref [] for more details)
associated with elastic properties (sometimes called the underlying conservative problem). They verify two orthogonality conditions in mass
and stiffness
There are different standard scaling for normal modes, and one will assumed that they are scale so as to obtain µj=1 which greatly simplifies equation writing. The other standard scaling, often used in experimental modal analysis, sets one DOF (node, direction) of φj to unity, µj is then called the generalized mass at this DOF.
The basis of normal modes is classically used to build reduced model by congruent transformation (??) with {q}=[T]{qr}=[φ1 ... φNM]{qr}. In the resulting coordinates, called principal coordinates, the mass and stiffness matrices are diagonal ((??)-(??) conditions). But this is not the case for the viscous and hysteretic damping matrices TTCT and TTDT.
The modal damping assumption (also called Basile's hypothesis in French terminology) consists in an approximation of the response where the off-diagonal terms of the damping matrices in principal coordinates are neglected. In practice, one even further restricts the model to viscous damping in principal coordinates since the result can then be integrated in the time domain. On thus has
where the approximation is linked to the fact of neglecting coupling terms described by
Damping is exactly modal (Fd=0) if the matrices C and/or D are linear combinations of products of M and K []
Rayleigh damping [], typically called proportional damping, where
is thus often used. This is a behavior model, that can be easily adjust to predict correct damping levels for two modes since
Over a wide frequency band, it is however very unrealistic since 1/2ωj gives high damping ratio for low frequencies ωj and ωj/2 for high frequencies. Rayleigh damping is thus really inappropriate to accurate damping studies.
When one starts from a local description of dissipation in the materials or interfaces, modal damping is an approximation whose validity needs to be understood. To do so, one should analyze whether the coupling force (??) can be neglected. To validate the domain of validity of this hypothesis, one considers a two DOF system []
from which one determines an expression of the response of mode 1 given by
with
and
The Γ matrix being positive definite, one has γ12γ21/(γ11γ22)<1. Thus, the e1 term can be close to 1 and coupling be significant, if and only if both factors in the denominator are small simultaneously, that is
Similarly, the e2 ter, is only significant in special cases of closely spaced modes or such that b1 ≪ b2.
In practice, the validity of the modal damping assumption is thus linked to the frequency separation criterion (??) which is more easily verified for small damping levels.
In cases where modal damping is not a good approximation (that is criterion (??) is not verified), a generalization of the modal damping model is the use of a viscous damping matrix that is non diagonal in principal coordinates. One the generally talks of non proportional damping since condition (??) is not verified. The use of non proportionally damped models will be discussed in section ??.
For groups of modes that do no verify condition (??), one can still use an assumption of modal damping by block [] where the off diagonal terms of φjCφk are considered for j and k in the same group of modes. Between modes of different groups, the reasoning developed above remains valid and the error induced by neglecting damping coupling terms is small.
2.1.3 Selection of modal damping coefficients
Modal Strain Energy method
The Modal Strain Energy method (MSE []) is a classical approximation base on the choice of an equivalent viscous damping coefficient chosen by evaluating the loss factor for a cycle of forced response along a particular real mode shape. As we saw for the particular case of section ??, this is an appropriate choice since it leads to a near perfect superposition of the transfer function for an isolated mode.
A general approach, that can also be considered for a weakly non linear system, is to compute the ratio, called loss factor, of energy dissipated over an enforced motion cycle qj(t)={φj} cos(ω t)divided by 2π times the maximum elastic energy associated to that deformation
and to impose at each resonance frequency ωj of the elastic problem, the equality of this loss factor with that of the model with modal damping (??)
For a model where a viscous and/or structural damping model is associated with each component/element (m), the loss factor of mode j is thus obtained as a weighted sum of loss factors in each component
For non linearities, there exists classical results of equivalent damping for dry friction [], dissipation associated with drag in a viscous fluid [], a plastic spring [], or small impacts [].
While the modal strain energy method is typically associated with the modal damping assumption, it can be easily extended to account for frequency dependent non diagonal (one says non-proportional) damping matrices. Thus using {q}=[T]{qr}=[φ1 ... φNM]{qr}, leads to a model of the form
This is typically referred to as a modal solution. In NASTRAN for example, you will find modal complex eigenvalue (SOL110), frequency response (SOL111), transient (SOL112).
For more general viscoelastic models, it is always possible to define pseudo normal modes solutions of
to normalize the using a condition similar to (??) and to define an equivalent damping ratio at resonance by
Experimental and design damping ratio
The other classical approach is to use modal damping ratio determined experimentally. Identification techniques of experimental modal analysis [] give methods to determine these ratios.
For correlated modes (when a one to one match between test and analysis is established), one thus uses a damping ratio ζjTest while typically preserving the analysis frequency.
For uncorrelated modes, one uses values determined as design criteria.
ζ=10−3 for a pure metallic component, 10−2 for an assembled metallic structure, a few percent in the medium frequency range or a civil engineering structure. Each industry typically has rules for how to set these values (for example [] for nuclear plants).
©1991-2024 by SDTools
PDF Index
