Open access peer-reviewed chapter

# Nonviscous Modes of Viscoelastically Damped Vibrating Systems

Written By

Mario Lázaro, César F. Casanova and Carlos Lázaro

Submitted: December 16th, 2015 Reviewed: May 12th, 2016 Published: September 21st, 2016

DOI: 10.5772/64205

From the Edited Volume

## Viscoelastic and Viscoplastic Materials

Edited by Mohamed Fathy El-Amin

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## Abstract

Nonviscously damped vibrating systems are characterized by dissipative mechanisms depending on the time history of the response velocity, introduced in the physical models using convolution integrals involving hereditary kernel functions. One of the most used damping viscoelastic models is Biot’s model, whose hereditary functions are assumed to be exponential kernels. The free-motion equations of these types of nonviscous systems lead to a nonlinear eigenvalue problem enclosing certain number of the so-called nonviscous modes with nonoscillatory nature. Traditionally, the nonviscous modes (eigenvalues and eigenvectors) for nonproportional systems have been computed using the state-space approach, computationally expensive. This number of real eigenvalues is directly related to the rank of the damping matrices associated with the exponential kernels. The state-space approach has traditionally been used up to now as the only method to compute the nonviscous modes for nonproportionally damped systems. Motivated by this open problem, we propose in this chapter to describe the available numerical methods for classically damped systems and present the recent methods for nonclassically damped systems. It is shown that the problem of finding the nonviscous modes can be reduced to solve as a set of linear eigenvalue problems. The presented methods are compared through a numerical example.

### Keywords

• vibrating systems
• nonviscous damping
• eigenvalues and eigenvectors
• nonproportional systems
• numerical methods

## 1. Introduction and background

It has been always very difficult to model the physical fundamentals of damping in structural dynamics. In general, the proposed models depend on several parameters, which must be fitted according to experimental results. The viscous model, proposed by Rayleigh [1], is the most used representation of dissipative forces for vibrating systems as it predicts an exponential decay rate of displacements, something that can be observed experimentally in a great variety of structural materials such as metals, concrete, wood, glass, or masonry. However, damping models need to be updated for the mathematical modeling of the real behavior of the so-called viscoelastic damping materials, widely used for vibration control and energy dissipation devices. Although the term viscoelastic dampinghas traditionally been used, in the last years the concept nonviscous dampingis also found in the bibliography, since this behavior can be considered as a generalization of the classic viscous damping. These materials, used in different areas of engineering as mechanical, civil, industrial, or aeronautics, are formed by polymer derivatives, rubbers, and rubber-like materials, and are characterized by a time-dependent constitutive model and by frequency-dependent Young and shear moduli.

Viscoelastic models of energy dissipation are introduced in the structure assuming that the damping forces are proportional to the history of the degrees-of-freedom (dof) velocities via kernel hereditary functions. These functions, also named damping functions, are the terms of the viscoelastic damping matrix in time domain, denoted by G(t)n×n. The dynamic balance of internal forces yields to the system of motion differential equations for a viscoelastically damped structure, with the form

Mu¨+0tG(tτ)u˙(τ)dτ+Ku=fe(t)E1

where the dofs’ time-domain response is represented by

are the mass and stiffness matrices. In general, we do not assume symmetry in these matrices although the mass matrix will be assumed to be non-singular. Under these conditions, the modes of the system can be obtained as the nontrivial solutions of the free-motion problem obtained considering fe(t)=v0=u0=0in Eq. (1). Thus, checking functions of the form u(t)=uestwe obtain

u˙(0)=v0,u(0)=u0E2

where D(s):CCn×nis the so-called dynamic stiffness matrix.

In this chapter, we will analyze Biot’s damping model with Nexponential kernels, a restriction commonly assumed in engineering applications. The expressions of the normalized damping functions in time and in frequency domain are, respectively

[s2M+sG(s)+K]uD(s)u=0E3
G(t)=k=1NCkμkeμktE4

where μk>0with 1kNare the relaxation or nonviscous parameters and CkRn×nare the damping matrices (in general asymmetric) of the limit viscous model, obtained if the relaxation parameters tend to infinite, that is

G(s)=L{G(t)}=k=1Nμks+μkCkE5

The coefficients μkcontrol the time and frequency dependence of the damping model while the spatial location and the level of damping are modeled via the matrices Ck. From this property, it is not strange to find them also as the damping coefficients of Biot’s model. The level of damping is closely related to the magnitude of these limit-damping matrices, while the relaxation parameters give information on how far is our nonviscous model from a viscous behavior [2]. It is also easy to demonstrate that the limit viscous damping and the time-domain kernel function are related by

k=1NCk=limμ1μNG(s)E6

Eqs. (2) and (4) clearly show the frequency dependence of the damping matrix, characteristic in this type of systems. This fact leads to a nonlinear eigenvalue problem whose eigenvalues are the roots of the equation

k=1NCk=0G(t)dtE7

In general, the damping matrix G(s)admits a rational representation, so that the polynomial of the denominator is at least of one order less than that of the numerator [2]. Thus, the determinant can be written as a polynomial, whose order is greater than 2nand, therefore, the total number of roots of Eq. (7) can be expressed as 2n+rand arranged as

det[D(s)]=0E8

where si,si*are ncomplex conjugate pairs and σ1,,σrare rnegative real numbers named nonviscouseigenvalues. The name is chosen precisely because they are characteristic of nonviscous or viscoelastic models. The number of these nonviscous eigenvalues will depend on the nature of the damping function, particularly on the number of hereditary exponential kernels. The complex conjugate pair forces the solution to be oscillatory, whereas the other eigenvalues are associated with overdamped, nonoscillatory modes. The latter modes decay rapidly and in general are not important for the system response.

The representation of the hereditary behavior was originally introduced by Boltzman [3] at the end of the nineteenth century. Its application to viscoelastic materials and to damping of vibrating systems was studied by different authors in the middle of the twentieth century. Among them, it is worth mentioning specially Biot [4, 5] whose multi-exponential hereditary model has widely been used for modeling viscoelastic damping materials. The fundamentals of viscoelasticity, a thorough study on the time-dependence constitutive models, and its application for modeling damping materials can be found in books such as Fluegge [6], Nashif [7], and Jones [8]. Although this chapter is closely related to Biot’s damping model, we must not forget the other viscoelastic models based on the fractional derivatives and widely used for representing the frequency-dependent behavior of damping materials. This model allows to use less parameters than exponential-based models [9], although the mathematical treatment is more difficult to implement, especially in the time domain, which is computationally more expensive [10].

This chapter is focused on the study of the rnonviscous modes of a nonviscously damped vibrating system. It is known that the effect of not considering these modes in the time-domain response is not important [2, 11]. Additionally, the exact calculation requires the use of the state-space approach, significantly increasing the computational effort and losing the physical insight of the involved internal variables [12, 13]. Maybe, these two reasons put together explain why they have not been analyzed in detail in the literature. Recently, Lázaro [14] derived a numerical approximated method to extract the nonviscous modes avoiding the state-space approach. In this chapter, we present a review of the nonviscous modes with nonoscillatory nature, giving their characteristics, mathematical properties, and the current available numerical methods for their computation.

## 2. Single degree-of-freedom systems

A single dof nonviscously damped vibrating system is dynamically characterized by a mass m, a linear stiffness k, and a nonviscous hereditary damping function G(t). The motion equation is

mu¨+tG(tτ)u˙(τ)dτ+ku=fe(t)E9

where

represents the degree of freedom and
represents the applied force in time domain. In this chapter, we analyze the nonviscous modes associated to Biot’s damping model with Nexponential kernels. The expressions of the normalized damping functions in time and in frequency domain for single dof are, respectively

G(t)=k=1Nckμkeμkt,G(s)=L{G(t)}=k=1Nckμks+μkE10

where μk>0with 1kNare the relaxation or nonviscous parameters. Eqs. (5) and (6) can be particularized for single dof resulting

limμ1μNG(s)=k=1Nck,0G(t)dt=k=1NckE11

Checking solutions of the form

, we can derive the characteristic equation

ms2+sG(s)+k=ms2+k=1Nsckμks+μk+k=0E12

Multiplying this expression by k=1N(s+μk), it results in a 2+N-order polynomial. If the system is lightly or moderately damped, the set of eigenvalues presents the form {s0,s0*,σ1,,σN}, where s0,s0*are a pair of conjugate-complex eigenvalues representing the modes of oscillatory nature. The rest Nroots are negative real numbers representing the nonviscous eigenvalues of nonoscillatory nature (overcritically damped). In this point, we focus on giving a mathematical characterization of these eigenvalues and to provide efficient methods to approximate the nonviscous eigenvalues avoiding solving the polynomial equation.

## 2.1. Mathematical characterization of eigenvalues

Let us see that the damping function evaluated at a nonviscous eigenvalue must always verify certain inequality related to the dynamic properties of the system, say mass mand stiffness k. Eq. (12) is rewritten in terms of the undamped natural frequency ωn=k/mand of a new dimensionless damping function denoted by

J(s)=G(s)2mωnE13

The characteristic Eq. (11) now becomes

s2+2sJ(s)ωn+ωn2=0E14

Reordering this equation, we can express it as

s2+2sJ(s)ωn+ωn2=[s+J(s)ωn]2+[1J2(s)]ωn2=0E15

Let

be any real nonviscous eigenvalue. Since
for all
it can be ensured that

1J2()=n+J()20E16

equivalent to |J(σ)|1. This inequality always holds for any real eigenvalue of Eq. (12). This result is a generalization of the well-known relationship between the dynamic parameters m, k, and cof a single dof viscously damped oscillator for having real eigenvalues: cccr=2mk(condition for critical damping).

As a direct consequence, we can define the following set:

B={sR:|G(s)|2mk}E17

assuring that every real eigenvalue of Eq. (12) lies inside

Lázaro and Pérez-Aparicio [15] derived the necessary condition expressed as |J(σ)|1and calculated approximate limits for the set
denoted as nonviscous set.

### 2.2. Numerical computation

It is known that the influence in the response of the nonviscous modes is much less important than that of the oscillatory complex modes [2, 16, 17]. For this reason, it is reasonable to look for closed-form approaches, avoiding the computational effort needed for solving the characteristic polynomial. Two methods based on the hypothesis of light damping can be found in the literature. They allow to approximate the nonviscous eigenvalues using closed-form formulas as function of the dynamic and damping parameters. The first one due to Adhikari and Pascual [18] approximates the nonviscous eigenvalues with the first iteration of Newton’s method applied to the characteristic polynomial. The second one, developed by Lázaro in his PhD Thesis [19] and published in the paper [20], is a perturbation-based approach. Both methods will be described in detail below and can be applied for both single dof systems and multiple dof systems with proportional (or classical) damping.

#### 2.2.1. Adhikari and Pascual’s method

Let us denote by ζk=ck/2mωnto the damping ratio associated to the jth exponential kernel. Introducing these parameters, the characteristic equation can be written as

s2+2sωnk=1Nζkμks+μk+ωn2=0E18

As mentioned before, the characteristic polynomial can be obtained multiplying the above equation by j=1N(s+μj), resulting the 2+N-order polynomial

P(s)=(s2+ωn2)j=1N(s+μj)+2sωnk=1Nζkμkj=1jkN(s+μj)E19

The method of Adhikari and Pascual [18] is based on the application of the first iteration of Newton’s method with s=μjas the initial point. Indeed, assuming that μj+Δjis close to the solution, Δjcan be explicitly calculated from the first-order expansion of P(s)around the initial point

0P(μj+Δj)P(μj)+P(μj)sΔjE20

After some simplifications, the expressions of Adhikari and Pascual published in Ref. [18] can be rewritten in terms of the current notation as

σjμjP(μj)P(μj)s=μj+μjζjpjζj(pjμjqj)rj+pj2ωnμj(μj2+ωn2)E21

where

pj=k=1kjN(μkμj),qj=k=1kjNr=1rj,kN(μrμj),rj=k=1kjNζkμkr=1rj,kN(μrμj)E22

Under the hypothesis of light damping ζj1, the nonviscous eigenvalue lies close to μj; therefore, it is expected that the solution from Eq. (21) accurately estimates the exact solution.

#### 2.2.2. Lázaro’s method

Lázaro’s method [19, 20] is based on considering the jth nonviscous eigenvalue σj, 1jNas a function of the jth associated damping ratio ζj. The damping ratio ζjcan be interpreted as a perturbation parameter of Eq. (18). Thus, we can write σj=σj(ζj)and Eq. (18) can be written for this eigenvalue as

σj2(ζj)+2σj(ζj)ωnζjμjσj(ζj)+μj+2σj(ζj)ωnk=1kjNζkμkσj(ζj)+μk+ωn2=0E23

Now, multiplying this equation by σj(ζj)+μj, we obtain

(σj(ζj)+μj)[σj2(ζj)+2σj(ζj)ωnZj(σj(ζj))+ωn2]+2σj(ζj)ζjωnμj=0E24

With this operation, the singularity associated to the jth nonviscous eigenvalue can be avoided. The function

introduced above is defined as

Zj(s)=j=1jkNζjμjs+μjE25

Eq. (24) explicitly defines σjas a function of ζj. Assuming light damping, we can expand σj(ζj)in terms of the damping parameter ζj, considering the latter as a perturbation parameter within the equation. Thus,

σj(ζj)=σj(0)+σj(0)ζj+σj(0)ζj22+E26

The value σj(0)can be obtained evaluating Eq. (24) at ζj=0

(σj(0)+μj)[σj2(0)+2σj(0)ωnZj(σj(0))+ωn2]=0E27

The eigenvalue associated to the kth hereditary kernel lies closely to the kth relaxation parameter [15, 21]. Therefore, we are interested in the real solution σj(0)=μj. The first-order derivative can be calculated solving for σj(0)after taking derivatives with respect to ζjin Eq. (24). The rest of higher-order derivatives σj(0),σj(0),are derived following the same procedure and using the previously calculated results. In general, it is sufficient to take up the second-order term since this approximation accurately estimates the nonviscous eigenvalues within a wide range of the damping ratios, including lightly and moderately damped structures [15, 20]. After obtaining the coefficients σj(0)and σj(0), the closed-form expression for σjremains as follows:

σjσj(0)+σj(0)ζjζj+2σj(0)ζj2ζj22E28

where

=μj+2μj2ωnζj(μj2+ωn22ωnμjη0j)+4μj3ωn2ζj2μj2ωn2+2ωnμj2μ1j[μj2+ωn22ωnμjη0j]3E29

Both Lázaro’s and Adhikari and Pascual’s methods are presented as closed-form expressions. On one hand, numerical computation of polynomial roots is avoided, and on the other hand the analytical expressions allow to explicitly observe the dependence of the nonviscous eigenvalues as functions of the rest of the parameters of the problem.

## 3. Multiple degrees-of-freedom systems

This section deals with the properties of the nonviscous modes in asymmetric nonproportional viscoelastically damped vibrating systems. A generalization of the mathematical characterization proved for single dof systems in the previous point will be derived. Regarding numerical analysis, the available methods for computing nonviscous modes will also be presented. As mentioned in the introduction, we consider an n-dof vibrating structure with mass and stiffness matrices denoted M,KRn×n. No restrictions with respect to the symmetry of these matrices are imposed and, additionally, it will be assumed that the mass matrix is not singular. The damping matrix G(t)Rn×ncontains the hereditary functions of the viscoelastic dissipative model. There are also no restrictions on the symmetry of the damping matrices Cj1jNand therefore it will be considered that CjCjT. Thus, the eigenvalues are the roots of the nonlinear equation

η0j=Zj(μj)=k=1kjNζkμkμkμj,η1j=Zjs|s=μj=k=1kjNζkμk(μkμj)2E30

The eigenvalues can be separated in nconjugate-complex pairs {sl,sl*}l=1nwith oscillatory nature and the rreal nonviscous eigenvalues {σj}j=1r. The number rof nonviscous eigenvalues and the range of the damping matrices Cjare directly related [12, 13]. In fact, Adhikari and Wagner [12] proved that r=j=1NCjin the context of the state-space approach, proof of which will be presented in this section.

### 3.1. Mathematical characterization of eigenmodes

It is assumed that the damping matrix is not proportional, that is, G(s)does not verify the necessary conditions to be diagonal in the modal space of the undamped problem [22]. As known, proportional damping matrices allow to reduce an n-dof system to nsingle dof systems due to the simultaneous decoupling capability. Thus, for these kinds of structures, the results of the previous point would apply. Assuming the nonproportionality, each nonviscous mode is characterized by a real eigenvalue

1jr, and both right and left real eigenvectors are denoted by ujand vj, respectively, so that

det[s2M+sG(s)+K]=0E31

We define the following expressions for each nonviscous eigenmode:

[σj2M+σjG(σj)+K]uj=D(σj)uj=0E32

These values can be interpreted as modal mass and stiffness, respectively, associated to the jthnonviscous mode. Using these new modal parameters, we can write that

vjT[σj2M+σjG(σj)+K]=vjTD(σj)=0,1jrE33

We introduce functions

defined as

Mj=vjTMuj,Kj=vjTKuj,Ωj=KjMj,E34

which can be interpreted as the dimensionless modal representation of the damping matrix at the jth real mode. Introducing this relationship in Eq. (33), we obtain

vjTD(σj)uj=Mjσj2+σjvjTG(σj)uj+KjE35

We can identify in this equality the same form as that of Eq. (14), derived for single dof oscillators. Therefore, and using identical mathematical manipulations, we can deduce that

or equivalently in terms of the damping matrix

Jj(s)=vjTG(s)uj2MjKjE36

expression of which represents the generalization for multiple dof systems of the necessary condition derived for single dof systems in the previous point, Eq. (16). Additionally, Eq. (36) can also be considered as a generalization of the result published by Lázaro and Pérez-Aparicio [15] for symmetric systems.

### 3.2. The state-space approach

In this section, the general state-space representation of the dynamic problem will be described. This methodology allows to transform the general n-dof system of integro-differential equations into a system of m>2nfirst-order differential equations through the introduction of internal variables. It was developed by Wagner and Adhikari [12] for symmetric systems and by Adhikari and Wagner [13] for asymmetric system.

It turns out that the final size mof the extended state-space formulation is directly related to the rank of the damping matrices Cj, 1jN. Because of that, it is appropriate to introduce the algebra associated to the matrix Cj. Let us assume that (Cj)=rjn, then there exist two matrices

, such that

σj2+2σjJj(σj)Ωj+Ωj2=0E37

where

is a diagonal block matrix with the nonzero eigenvalues of Cjand the blocks O1jand O2jare null matrices of size rj×(nrj)and (nrj)×(nrj), respectively. The columns of matrices Xjand Yjform two different bases of space Rn, hence both matrices can be written in the form

|vjTG(σj)uj|2MjKj,1jrE38

where xjkand yjkfor k=1,,rjare the right and left eigenvectors of the nonzero eigenvalues of Ck. These two bases have special relevance in the developments of the state-space method and it is convenient to group them in the two rectangular matrices

YjTCjXj=[djO1jO1jTO2j]E39

so that the following relations are straightforward:

Xj=[xj1xj,rjxj,rj+1xj,n]E40

Let us return now to the system of integro-differential equations presented in Eq. (1) written in terms of the dof

, and let us introduce a set of N+1internal variables denoted by
and wj(t),1jNand defined as

Yj=[yj1yj,rjyj,rj+1yi,n]E41

For our purposes, we need the time derivative w˙j, which can be calculated using Leibniz’s rule for differentiation of an integral, yielding

Rj=[xj1xj,rj]Rn×rjE42

With these new variables, Eq. (1) can be expressed as

Lj=[yj1yj,rj]Rn×rjE43

In the above expression, the vector

represents the image via the linear mapping defined by the matrix Cj. The kernel of this mapping is a subspace of Rnwith dimension nrjand characterized by Ker
The vectors xjk,k=rj+1,,nare a basis of this subspace. Therefore, only the rjprojections of wjonto the rest of eigenvectors, say xjk,k=1,,rj, are representative. Consequently, we can defined the rjinternal variables
from the rectangular transformation matrix Rj

LjTCjRj=djE44

Introducing this transformation into Eq. (43) and premultiplying by M1

v(t)=u˙(t)andwj(t)=0tμjeμj(tτ)u˙(τ)dτ,1jNE45

Now, in order to complete the extended linear system, we need to relate the variables wj(t)and their time derivatives. For that, let us combine Eq. (44) with (42) resulting in

w˙j=μj20teμj(tτ)u˙(τ)dτ+μju˙(t)=μjwj(t)+μjv(t)E46

Premultiplying by matrix LjTand denoting by Tj=[LjTRj]1LjT, we can write after some operations

Mv˙+j=1NCjwj+Ku=fe(t)E47

Eqs. (45) and (47) and the direct relations

can be put in order in the following extended linear system of ordinary differential equations:

wj(t)=Rjwj(t)E48

where

v˙=j=1NM1CjRjwj(t)M1Ku+M1fe(t)E49
Rjw˙j=μjRjwj(t)+μjv(t)E50

In these expressions,

and
represent the null matrix and vector (in column) of their respective spaces and Irjthe identity matrix of
. Since Tjrepresents a full-rank matrix of order rj, then the total order of the system is

m=2n+j=1nrjE51

showing that the extra order of the state-space formulation of a nonviscously damped vibrating system is governed by the rank of the damping matrices. Hence, the total number of nonviscous eigenvalues is given by r=j=1nrank(Cj). As known, checking solutions of the form z(t)=zestin the free-motion equations

leads to the linear eigenvalue problem

z˙=Az+r(t)E52

The complete solution of this problem allows to construct the spectral set of nonviscously damped systems. On one hand, we have 2ncomplex modes with oscillatory nature and, on the other hand, the rnonviscous eigenmodes with their respective eigenvectors. A detailed study of the eigenvalue problem of Eq. (52) has been described in the work of Adhikari and Wagner [13]. From a mathematical point of view, the problem of calculating the eigenmodes is totally solved. However, we can expose two reasons why it is worth to deepen in the numerical problem of the nonviscous modes: (a) to solve a linear eigenvalue problem as that shown in Eq. (52) requires in general

operations, something that affects the computational efficiency of the problem as we increase the number of hereditary kernels and the number of degrees of freedom. (b) The physical insight of the problem is somewhat lost with the introduction of new internal variables in the state-space method. Due to these two arguments, several numerical methods have been proposed in the bibliography to obtain the ncomplex modes with oscillatory nature (see for instance references [2, 18, 2327]). On the contrary, the nonviscous modes have not been studied with so much detail since, obviously, their influence in the response is much less important. Lázaro published a research focusing on nonviscous modes of symmetric systems [14] trying, on one hand, to reduce the computational complexity of computing the nonviscous modes and, on the other hand, to supply a physical interpretation of the significance of these kinds of modes, closely related to the properties of the damping model and to the matrices Cj, j=1,,N. In the next point, Lázaro’s method will be described including a generalization for asymmetric systems.

### 3.3. Approximate numerical method

As described above, we derive here the numerical method proposed by Lázaro [14] for the computation of nonviscous modes. We work under the generally accepted assumption of light damping, something that allows to predict that the nonviscous eigenvalues are close to the relaxation parameters {μj}j=1N. Let us consider the following decoupling of the damping matrix in the Laplace domain associated to the jth relaxation parameter, μj

G(s)=μjs+μjCj+k=1kjNμks+μkCkμjs+μjCj+Gj(s)E53

Something similar can be made for the dynamic stiffness matrix, yielding

D(s)=s2M+sG(s)+K=s2M+sGj(s)+K+sμjs+μjCjDj(s)+sμjs+μjCjE54

where Dj(s)=s2M+sGj(s)+KCn×ndenotes the dynamic stiffness matrix without the jth hereditary damping function. Note that under this manipulation, the function Dj(s)is now continuous and with continous derivatives at s=μj. Let us denote by

to any nonviscous eigenvalue associated to μjand by
the right and left eigenvectors associated to σj, respectively. The following relations hold:

m=2n+j=1nrjE55
(AsIm)z=0mE56

In order not to have to repeat every step for the right and left eigenvalues, the developments will be carried out only for Eq. (55). Thus, multiplying Eq. (55) by σj+μj

G(s)=μjs+μjCj+k=1kjNμks+μkCkμjs+μjCj+Gj(s)E57

Let us define the matrix

Aj(s)=(s+μj)Dj(s)E58

and Eq. (57) can be written as

D(σj)xj=[Dj(σj)+σjμjσj+μjCj]xj=0E59

Since the damping is assumed to be light, σjis close to μjand consequently there exists certain λjR, such that σj=μj+λj, with |λj/μj|1. Expanding the matrix Aj(σ)around λj=0and neglecting second-order terms, we obtain

DT(σj)yj=[DjT(σj)+σjμjσj+μjCjT]yj=0E60

where ()=()/s. From the definition of Aj(s), we have that

[(σj+μj)Dj(σj)+σjμjCj]xj=0E61

Substituting this result together with σj=μj+λjin Eq. (59) and rearranging

[CjλjBj]xj=0E62

where

Bj=Cjμj+Dj(μj)μj2Rn×nE63

Following the same steps for the left eigenvectors from Eq. (56), we obtain the following relation between λjand

is fulfilled:

[CjTλjBjT]yj=0E64

From Eqs. (62) and (64), λj, xj, and yjrepresent an eigensolution of the generalized linear asymmetric eigenvalue problem of matrix Cjwith respect to Bj. Denoting by rj=rank(Cj)n, then λj=0is eigenvalue of Eq. (26) with multiplicity nrj. Consequently, there exist other rjno-null eigenvalues, which will be named λj,1,,λj,rj. Hence, the complete spectral set of problem (64) can be listed as

{λj,1,,λj,rj,0,nrj,0}E65

and the rjnonviscous modes associated to the jth relaxation parameter can be denoted by

{μ+λj,k;xj,k;yj,k}k=1rjE66

We highlight two interesting results from this method: (i) the computation of the nonviscous modes has been reduced to solve Nlinear eigenvalue problems of order nand (ii) there is no need to previously calculate neither the modal space of the undamped model nor the eigenproblem of the matrices Cj. We find a limitation because a hypothesis of light damping has been used in the linearization (60). For vibrating problems under a higher level of damping, the method can be adapted just taking the second-order term in the expansion of Aj(μj+λj), that is

Aj(μj+λj)Aj(μj)+Aj(μj)λj+Aj(μj)λi22!=Dj(μj)λj+Dj(μj)λj2E67

Introducing this expression in Eq. (59) and after some manipulations the resulting right and left eigenvalue problems are

[HjλjFj]zj=0,[HjTλjFjT]z^j=0E68

where

Hj=[CjOnOnIn],Fj=[BjDj(μj)/μj2InOn],zj={xjλjxj},z^j={yjλjyj}E69

In general, the second-order approximation will lead to better approximations, although in this case a larger problem must be solved; this will be confirmed in the numerical example. The reader who wants to deepen in detail in higher-order approximations and their associated computational cost can refer to the work of Lázaro [14]. In this paper, it is proved that, from a computational point of view, it is profitable to increase the order of approximation up to certain limit order after which it is better to use the state-space approach. That limit value of the approximation order is

is the number of hereditary damping kernels.

### 3.4. Numerical example

In this numerical example, the presented computational methods to calculate the nonviscous modes will be compared. For that, we use a five-degree-of-freedom discrete system with viscoelastic dampers, shown in Figure 1. Each dof represents the displacement of a mass m=103kg. The linear stiffness between the masses is k=105N/m. Two nonviscous dashpots are located between ground and dof 1 and between dofs 3, 4, and 5, whose constitutive relationships are expressed as the sum of exponential kernels.

Ga(t)=ca(μ1eμ1t+μ2eμ2t),Gb(t)=cbμ3eμ3tE70

The damping coefficients are ca=600Nm1s and cb=200Nm1s and the relaxation parameters μj={10,25,45}rad/s. Since we have three relaxation parameters, the damping matrix in time domain yields

G(t)=C1μ1eμ1t+C2μ2eμ2t+C3μ3eμ3tE71

and according to the dashpots and rigidities distribution, the damping matrix coefficients and the stiffness matrix are

K=[2kk000k2kk000k2kk000k2kk000k2k],C1=C2=ca[1000000000000000000000000],C3=cb[0000000000001100012100011]E72
 EIGENVALUES μ1=10, r1=1 μ2=25, r2=1 μ3=45, r3=2 σ1,1 σ2,1 σ3,1 σ3,2 Exact −9,762536252 −24,539682264 −44,480104306 −44,817181343 1st order approx. −9,767255360 −24,552342078 −44,490259517 −44,818527825 (error, %) (0,04834) (0,05159) (0,02283) (0,00300) 2nd order approx. −9,762478350 −24,539554707 −44,480042888 −44,817178561 (error, %) (0,00059) (0,00052) (0,00014) (0,00001) EIGENVECTORS μ1=10, r1=1 μ2=25, r2=1 μ3=45, r3=2 x1,1 x2,1 x3,1 x3,2 Exact 0,920866633 0,991941304 0,000819770 0,001425705 0,359382674 0,125658720 0,018202462 0,032082433 0,140415976 0,016087932 0,395717303 0,707139978 0,053098476 0,001862474 −0,829159074 0,000761150 0,017681720 0,000209935 0,394424954 −0,706343512 1st order approx. 0,924263325 0,992396067 0,000786787 0,001405038 0,353130865 0,122132136 0,017832254 0,031844687 0,135129270 0,015194058 0,395980865 0,707139245 0,050069699 0,001700855 −0,828890371 0,000749666 0,016401313 0,000185500 0,394742065 −0,706355058 (error, %) (0,94544) (0,36701) (0,06167) (0,02392) 2nd order approx. 0,920823393 0,991936757 0,000819965 0,001425786 0,359460764 0,125693455 0,018204709 0,032082906 0,140482958 0,016096765 0,395715707 0,707139980 0,053137115 0,001864208 −0,829160702 0,000761173 0,017698131 0,000210211 0,394423029 −0,706343488 (error, %) (0,01192) (0,00362) (0,00037) (0,00005)

### Table 1.

Numerical example: results of nonviscous eigenvalues (rad/s) and eigenvectors.

The rank of these matrices can easily be calculated obtaining

r1=rank(C1)=1,r2=rank(C2)=1,r3=rank(C3)=2E73

The number of nonviscous eigenvalues of this system is r=r1+r2+r3=4. The results of the four nonviscous eigenvalues and eigenvectors are shown in Table 1. Exact solutions based on the state-space approach are shown in the first rows. Below, we find the approximated solutions calculated with Lázaro’s method using both the first- and the second-order approximation (see Eqs. (62) and (68), respectively). The relative error is also shown below each result (in brackets) for both eigenvalues and eigenvectors. For the latter, the relative error is calculated in terms of the vector norms. Note that in general, the eigenvalues are calculated more accurately than eigenvectors. Indeed, the relative error of the former is one order of magnitude lower than that of the latter. As expected, the second-order approximation improves notably the solution, decreasing the relative errors two or three orders of magnitude respect to those computed from the first-order approximation. In general, since the effect of the nonviscous modes in the response is not relevant, it is justified to use the first-or second-order approximations presented in this text, even for moderately or highly damped vibrating structures [14].

## 4. Conclusions

In this chapter, the mathematical modeling of damping materials has been presented. These materials are characterized by presenting dissipative forces depending on the history of degrees-of-freedom velocities via exponential kernel functions (or Biot’s model). The free-motion vibration of these structural systems leads to a nonlinear eigenvalue problem. There exist two types of eigensolutions: on one hand, the complex eigenmodes, with oscillatory nature and considered as perturbations of the undamped natural modes, on the other hand, the so-called nonviscous modes, overcritically damped modes (without oscillatory nature), characteristic of the type of damping model. These latter modes are the main objective of the research of the present chapter.

The nonviscous modes behind a viscoelastic exponential-damping-based system are closely related to the relaxation parameter of the exponential functions. In general, their influence in the response of the system is several orders of magnitude less important than that of the complex modes. In this paper, we try to summarize some of the most relevant properties of these modes, both from a theoretical and from a numerical point of view. Nonviscous modes for both single and multiple dof systems are studied. For both cases, a necessary condition of nonviscous modes relating to eigenvector, eigenvalue, and dynamic matrices is provided. Additionally, numerical methods to extract nonviscous eigenvalues and eigenvectors, assuming asymmetric and nonproportional dynamic matrices, are reviewed. The results have been compared with a numerical example.

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Written By

Mario Lázaro, César F. Casanova and Carlos Lázaro

Submitted: December 16th, 2015 Reviewed: May 12th, 2016 Published: September 21st, 2016