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

## 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 damping* has traditionally been used, in the last years the concept *nonviscous damping* is 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

where the dofs’ time-domain response is represented by

where *dynamic stiffness matrix*.

In this chapter, we will analyze Biot’s damping model with

where

The coefficients

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

In general, the damping matrix

where *nonviscous* eigenvalues. 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

## 2. Single degree-of-freedom systems

A single dof nonviscously damped vibrating system is dynamically characterized by a mass

where

where

Checking solutions of the form

Multiplying this expression by

## 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

The characteristic Eq. (11) now becomes

Reordering this equation, we can express it as

Let

equivalent to

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

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

As mentioned before, the characteristic polynomial can be obtained multiplying the above equation by

The method of Adhikari and Pascual [18] is based on the application of the first iteration of Newton’s method with

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

where

Under the hypothesis of light damping

#### 2.2.2. Lázaro’s method

Lázaro’s method [19, 20] is based on considering the

Now, multiplying this equation by

With this operation, the singularity associated to the

Eq. (24) explicitly defines

The value

The eigenvalue associated to the

where

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

The eigenvalues can be separated in

### 3.1. Mathematical characterization of eigenmodes

It is assumed that the damping matrix is not proportional, that is,

We define the following expressions for each nonviscous eigenmode:

These values can be interpreted as modal mass and stiffness, respectively, associated to the

We introduce functions

which can be interpreted as the dimensionless modal representation of the damping matrix at the

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

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

It turns out that the final size

where

where

so that the following relations are straightforward:

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

For our purposes, we need the time derivative

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

In the above expression, the vector

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

Now, in order to complete the extended linear system, we need to relate the variables

Premultiplying by matrix

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

where

In these expressions,

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

The complete solution of this problem allows to construct the spectral set of nonviscously damped systems. On one hand, we have

### 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

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

where

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

Let us define the matrix

and Eq. (57) can be written as

Since the damping is assumed to be light,

where

Substituting this result together with

where

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

From Eqs. (62) and (64),

and the

We highlight two interesting results from this method: (i) the computation of the nonviscous modes has been reduced to solve

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

where

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

### 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

The damping coefficients are

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

EIGENVALUES | ||||

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 | ||||

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) |

The rank of these matrices can easily be calculated obtaining

The number of nonviscous eigenvalues of this system is

## 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.