1. Introduction
Currently, there are many applications, from micro-scale systems to large structures, that require the control of vibration, sound, and wave propagation for proper operation. New techniques to optimize energy dissipation have been proposed in recent years, from the consolidation of time-dependent viscoelastic materials [
1,
2,
3,
4], to the emergent phenomenon of metadamping [
5,
6]. In part, the advances in new dissipation mechanisms have been carried out thanks to the constant increase in computing capacity, since the new discoveries are generally accompanied by more complex mathematical models. In the context of vibration control with nonviscously damped materials, it is especially important to know how to choose the characteristics of the dissipative devices or materials to ensure that the system becomes overcritically damped in one or more modes. In this article, we investigate criticality in nonviscously damped multiple-degrees-of-freedom (dof) systems, considering any number of exponential hereditary kernels.
Nonviscous damping is characterized by dissipative forces which depend on the past history of the velocity response via convolution integrals over hereditary kernel functions. Denoting, with
, the array with degrees of freedom, the equations of motion are expressed in integro-differential form, as [
1]:
where
are the mass and stiffness matrices assumed to be positive definite and positive semidefinite, respectively;
represents the time-domain damping matrix, assumed to be symmetric. Viscous damping arises just as a particular form of Equation (
1), with
, where
is the viscous damping matrix and
the Dirac’s delta function. Checking for solutions of the form
in Equation (
1) transforms the time-domain equations into a nonlinear eigenvalue problem with the form:
where
is the frequency-domain damping matrix,
is the dynamical stiffness matrix, and
is the Laplace parameter. The roots of the characteristic equation:
are the eigenvalues of Equation (
2). Viscoelastically damped structures are characterized as a frequency-dependent damping matrix. The time-domain response will be affected by the nature of the eigenvalues of Equation (
2). In turn, complex eigenvalues are associated to oscillatory motion; meanwhile, real negative eigenvalues lead to nonoscillatory modes. When the nonviscous damping model is based on hereditary exponential kernels, something that will be assumed in the current investigation, the number of eigenvalues will be invariably greater than
because there exist
p real nonviscous modes associated to hereditary exponential kernels [
7,
8]. Thus, undamped or lightly damped systems present
oscillatory modes with a relatively small real part in magnitude, together with
p real nonviscous eigenvalues. However, as the damping level increases, the complex eigenvalues move away from the undamped ones in the complex plane. For high-damping forces, some conjugate–complex pair of eigenvalues may drop into the real axis, vanishing the oscillatory nature. The root locus in the complex plane depends on the damping parameters presented in the matrix
. In general, viscoelastic models are mathematically defined by several parameters so that the set of all of them defines a multidimensional parametric domain. The nature of eigenvalues closely depends on where the multidimensional point of such parameters lies. Critical manifolds are sets in this parametric domain, limiting the undercritically and overcritically damped regions. Geometrically, one-dimensional manifolds represent critical curves, which depicts critical relationships between two parameters. Then, critical regions are 2D areas enclosed by such critical curves. Lázaro [
9] proved that critical manifolds of dynamical systems with viscoelastic damping can be found by eliminating the Laplace parameter
s from the two equations:
Purely viscous forces are characterized by being proportional to the velocity of the response. In the frequency domain, this fact simplifies the study of the critical damping conditions with respect to the use of nonviscous damping. In fact, Papargyri-Beskou and Beskos [
10] proved that, for viscous systems, approximations of critical curves may be derived by assuming that the critical eigenvalues are
, where
denotes the
jth natural undamped frequency. Bulatovic [
11] proved new necessary and sufficient conditions for critical damping based on the determinant of the system and on the minors of certain matrices, depending on the eigenvalues. Mathematically, the inclusion of hereditary exponential kernels in the damping model leads to an increase of the order of the characteristic polynomial. The first attempts to determine the conditions of overcritical damping were proposed by Muravyov and Hutton [
12] and Adhikari [
13]. These works were developed with just one hereditary kernel, addressing the problem by carrying out an exhaustive analysis of the nature of the root of the resulting third-order characteristic polynomial. Müller [
14] studied the nature of eigenvalues for single-dof systems based on a Zener three-parameter damping model. The critical oscillatory motion of nonviscous beams has been studied by Pierro [
15], solving the eigenvalues for one and two exponential kernels and discussing their nature (real or complex). Wang [
16] obtained fractional orders compatible with critical damping in fractional derivative-based, viscoelastic, classically damped structures. The problem of determining the critical manifolds of a single degree-of-freedom oscillator for any number of hereditary kernels has been analytically solved in exact form by Lázaro [
17], by transforming Equation (
4) into parametric closed-form expressions. For large multiple-dof systems, the general method proposed by Lázaro [
9], consisting of eliminating
s from Equation (
4), cannot be carried out since an analytical expression of the determinant is, in general, not available. Trying to overcome that, Lázaro [
18] proposed an approach for systems with multiple degrees of freedom, but the proposal was restricted to problems of one single hereditary kernel.
At present, the problem of finding approximate solutions to the critical curves of multiple-dof systems with multiple hereditary exponential kernels remains open. In this article, a novel approximate method that allows for obtaining these curves without limitations on the number of exponential kernels is proposed. In addition, the developments carried out deepen the problem from a theoretical point of view and improve some results already published, something that helps to consolidate the knowledge about this problem. These new theoretical results lead to a new equation, which is verified by critical eigenvalues. From critical curves arise, in parametric form, the solution of an eigenvalue problem whose nature depends on the type of curve to be solved. The proposed approach is validated by means of two numerical examples: a four-dof discrete lumped-mass system, and a continuous beam finite element model with viscoelastic supports.
2. New Results on Critical Damping of Structures with Viscoelastic Dampers
In this paper, nonviscous damping, based on Biot’s model [
19], will be considered. In general, the damping matrix can be written as the superposition of
N hereditary exponential functions, expressed both in the time and frequency domains as:
where
,
stand for the relaxation (or nonviscous) parameters, and
are the symmetric damping matrices of the limited viscous model, obtained as the relaxation parameters tend to infinite; that is,
The coefficients
control the time- (and frequency-) dependence of the damping model, while the spatial location and the level of damping are modeled via the matrices
. From now on, the matrix
will be assumed to depend on a set of damping parameters which monitor the dissipative behavior. Thus,
and
, from Equation (
5), depend at most on
-independent parameters:
N distinct non-viscous parameters, say
plus
matrix entries within each
:
where
is the
-entree of
, assumed to be symmetric. Real applications depend, in general, on less parameters, say
. Just for the shake of simplicity in our exposition, the array
is introduced to denote the set of independent damping parameters. Hence, both the viscoelastic and dynamical stiffness matrices can be written as functional arrays of
s and
, denoted as
and
, respectively, such that:
Let us consider the following system of
equations expressed in terms of the unknowns given by the
− tuple
:
where
is the entry
of the dynamical stiffness matrix
, and
is a real-valued function of the
n components
, which enables fixing the eigenvector
. Thus, some normalization forms can be used by the function
, for instance,
(unit vector),
(mass-normalized vector), or
(
jth unit component), where the above
denotes the conjugate–complex vector of
. Equation (
9) can be read as a system of
equations with
unknowns, say
, which, in turn, are functions of the
p parameters via
. If the actual state of the damping model, represented by
, induces light damping, then there exist
n conjugate–complex eigensolutions of the form
, and
r real nonviscous eigenmodes
, where
[
7]. As the damping level (through the variation of parameters
), some of the conjugate–complex eigensolutions may come close to the real axis. If one of these eigenvalues drops into the real axis, then the two pairs of conjugate–complex eigenvalues are transformed into a double real eigenvalue, and then
will lie exactly on a critical manifold. At this point, the resulting negative eigenvalue is critical and double; therefore, it will be the root, simultaneously, of both expressions in Equation (
4). If the level of damping continues to increase, the array
will be completely inside an overdamped region and the double root will be split into two overcritical (real and negative) eigenvalues with non-oscillatory natures. At this point, it may be useful to distinguish between
overdamped modes and
nonviscous modes. Both are properly non-oscillatory modes because they correspond to real and negative eigenvalues. However, while the presence of the former strongly depends on the level of damping, the amount of the latter,
, depends on the spatial distribution of the viscous matrices
, rather than the value of their coefficients. Therefore, the reader must be aware that nonviscous modes without an oscillatory nature will be present always in nonviscously damped structures based on kernels with exponential decay, no matter the damping level. Some works specifically devoted to the study of nonviscous modes can be found in the references [
8,
20,
21,
22], where, in particular, Mohammadi and Voss propose a mathematical characterization [
20] and study their distribution [
21]. However, in the context of the current investigation, overdamped modes are those whose non-oscillatory nature (as negative real numbers) can be affected by the damping level, so that they can be transformed into complex underdamped modes with oscillatory natures for low damping conditions.
Let us assume certain combination of damping parameters
, lying within the overdamped region and leading, consequently, to (at least) two overdamped modes. Let us denote, respectively, by
and by
, the eigenvalue and eigenvector of one such modes. The
− tuple
is, then, a particular solution of Equation (
9). Small variations of
around
lead to a functional dependence of
s and
with to
. Mathematically, it is said, then, that there exist two functions:
which are implicitly defined by the expression in Equation (
9) around the point
, holding that
and
. In addition, we will assume that:
The expressions in Equation (
9) can be written in a more compact form as:
where
is a vector field defined as:
with
and
. Sufficient conditions to guarantee the existence of the functions of Equation (
10) are provided by the implicit function theorem. Assuming that
is a continuously differentiable function in a neighborhood of
, where
and
, then if the Jacobian matrix
is invertible, it can be ensured that there exists an open set
around
and a unique continuously differentiable function
, such that
and
for all
. The existence of the function
is directly related to the location of
within the domain of the damping parameters. Thus, if
lies on a critical surface, the above functions are not well defined since small variations of the damping parameters lead to an indefinite state with two possible conjugate–complex solutions. An assessment of the conditions under which the Jacobian matrix becomes non-invertible will provide valuable information about critical damping. According to the definition of the Jacobian matrix, and after some straight operations, it yields:
This result shows that the Jacobian matrix is formed by four matrix blocks. In ref. [
18], it was demonstrated that if
(denoting
), then
, and therefore,
is non-invertible and the functions
and
are not well defined at that point. However, the inverse statement is not true in general; that is, eigensolutions holding
might lead to
. In the current paper, that result will be improved by finding the necessary and sufficient conditions for which Jacobian matrix is non-invertible, enabling a much more general characterization of the critical damping. These conditions are presented in the following theorem:
Theorem 1. Assume that and are a real eigensolution for a certain value of the damping parameters . In addition, it will be assumed that . Under these hypotheses, the Jacobian is non-invertible, i.e., , if and only if .
Proof. First, let us consider that
. Since we are free to choose
, it can be assumed that the last row of
, i.e., the vector
, is linearly independent of the previous
n rows; therefore,
. Then, there exist
n real coefficients,
, not all of which are zero, such that:
where
denotes the
jth column/row of the matrix
, due to the symmetry. Writting Equation (
15) in matrix form yields:
where
. Since
is an eigenvector,
, and consequently:
Inversely, assume that
. Then, the vector
belongs to the vector subspace:
It is straightforward that the
because
is a non-zero vector. Moreover, since
is an eigenvector associated to
s, the
n following relationships hold:
where
has already been defined above. It is clear that
, for
. Since
and
, then (without loss of generality) the set of vectors
is a basis of
. Thus, there exist
coefficients,
, such that:
Adding
to the previous set does not change the rank because, by hypothesis, it is
, yielding, then:
which leads to
and, therefore,
. □
The previous result provides a new theoretical characterization of critical eigenmodes as those modes whose functional dependency on the damping parameters is not well defined. According to such result, a combination of damping parameters is said to be critical if at least one mode
verifies the following relations:
The Equation (
22) results are of interest, from a theoretical point of view, but are not useful in practice since the critical eigenvectors are unknown. However, according to the following theorem, the location of critical eigenvalues along the real axis may be conditioned if the associated eigenvectors are close to the classical normal modes.
Theorem 2. Assume that is a critical eigenvalue with eigenvector of Equation (21), and are the n undamped natural frequencies. Then:
- (i)
;
- (ii)
Furthermore, if is close to a normal mode , in the sense that with and, in addition, if the inequality holds, then ;
- (iii)
(Ref. [23]) If the system is purely viscous, with , then
Proof. (i) Let us transform the relationship of Equation (
21) into a scalar equation by left multiplying by the eigenvector
. Both Equations (
21) and (
22) yield:
where
Dividing Equation (
23) by
s, and subtracting both equations, we obtain:
Since
verifies the conditions of Golla and Hughes [
24], thereby representing a strictly dissipative viscoelastic model, then:
After straight operations in Equation (
25):
which leads to the inequality
. Discarding positive solutions,
It is known [
25] that:
whence the inequality
holds.
(ii) Assume that the eigenvector
is close to the
jth normal mode
. Without loss of generality,
can be written as:
where
. Hence:
Using, to the modal orthogonality, relations
and
, the above expressions can be simplified, yielding:
Plugging these expressions into the Rayleigh quotient and rearranging the resulting expression:
According to the hypothesis of Theorem 2, it is
; therefore:
Finally, since , it leads to the searched inequality .
(iii) If the system is purely viscous, then
; therefore, it is
. Thus, from Equation (
25), it yields, straightforwardly:
The point (iii) of this theorem, proved here as a particular case of nonviscous systems, was already deduced by [
23] in the context of the decoupling of defective linear dynamical systems. □
According to the first part of Theorem 2, the presence of critical eigenvalues within the range can be discarded. Perhaps more important is the point (ii) of the theorem, which allows for bounding critical eigenvalues associated to each undamped mode, provided that the system can be considered as slightly nonproportional. In the next section, this result will be used to derive the proposed methodology.
3. Derivation of Critical Curves: The Modal Critical Equation
In the previous section, a new theoretical characterization of critical eigenmodes has been derived, considering critical modes as those whose functional dependency on the damping parameters is not well defined. According to one such result, a combination of damping parameters is said to be critical if it leads to (at least) one mode
verifying the following relations:
The aim of this section is to develop a numerical method based on the above equations to determine points of the critical curves. These curves graphically represent thresholds in the parametric domain where the oscillatory nature of a complex mode is lost, giving rise to two overdamped distinct real modes. Consider the two scalar relationships, Equations (
23) and (
24). Now, dividing both equations by
and
, respectively, and subtracting, yields:
After some straight operations, this equation can be rearranged in a more compact form:
The starting point of the developments is focused on this equation. For purely viscous systems, with
, Equation (
38) is reduced to
, as proved above. Papargyri-Beskou and Beskos pointed out [
10] that, for nonproportional systems, the approach:
can be assumed with accurate results. This affirmation can be justified directly from the approximation of Equation (
32), which states
with an error of the second order of magnitude in terms of the coefficients
of Equation (
29). These coefficients, in turn, are directly related to the nonproportionality of the system. Indeed, Adhikari [
1] proved that
is directly related to the off-diagonal terms of the modal damping matrix, i.e.:
provided that the matrix
fulfills the condition
, where:
and
denotes the Kronecker delta function. Equation (
38) depends on
on both sides. It is clear that if the system is proportional, then
, and such an equation will depend on
s and on the given mode
. Proportional systems admit closed-form analytical solutions for critical curves, as proved in reference [
26]. However, the presence of
in Equation (
38) requires some assumptions to reach approximate solutions. A more detailed inspection of the nature of equation Equation (
38) enables a determination of the dependence of the off-diagonal terms of the damping matrix. Indeed,
-notation will be considered to distinguish this order of magnitude. Thus, in general, a certain vector or scalar will be of order
h, say
, if its components depends on the
hth order of the entries
and their
s-derivatives,
. For instance, products of the type
or
are both considered to be of order
. If the system exhibits lightly nonproportional damping, it is expected that terms of a higher order involved in the equations could be neglected with respect to those ones of order zero. Based on the above considerations, we are interested in addressing the nonproportionality order defined above for each of the terms involved in Equation (
38) and, ultimately, the order of the entire equation. Thus, for
, it yields:
Similarly, and using the orthogonality relations, it is straightforward that:
Plugging the above relations into both sides of Equation (
38), it yields:
Encouraged by this result, we postulate that, in those cases where the eigenvector
is close to
, it can be stated that:
In the context of this article, this equation will be called the
Modal Critical Equation (MCE) associated to the
jth mode. This scalar equation, together with the eigenvalue problem
, makes up the system of equations for solving the critical curve. We find, in this equation, the two particular cases that have been already solved in the literature: (a) purely viscous damping, studied by Papargyri-Beskou and Beskos [
10], and (b) proportional nonviscous damping, investigated by Lázaro and García-Raffi [
26]. Both cases deserve some comments before addressing the proposed strategy to solve the general case of nonproportional nonviscously damped systems.
Mathematically, the purely viscous case is characterized by a frequency-independent damping matrix,
. Equation (
45) degenerates into the
n equations
for all modes
. The results obtained fit the actual overdamped regions, generally, quite well, as is reflected in ref. [
10]. One of the numerical examples of the current paper shows the outcomes for viscous systems, reinforcing those results.
The second particular case which can be derived from Equation (
45) is that of proportional damping. Indeed, if
, then it follows that Equation (
45) is no longer an approximation, but that both sides of the equation are irrefutably equal. Furthermore, as shown by [
26], the eigenvalue problem can be decoupled using the undamped modal space, and the critical curves arise as exact closed forms. The graphical realization of such curves matches perfectly with the overdamped regions, and overlappings represent regions with several overdamped modes.
The current developments, carried out for nonviscous nonproportional systems, should be consistent with those aforementioned (viscous damping and proportional nonviscous damping); thus, approximate critical surfaces arise by eliminating the Laplace parameter
s from both the characteristic equation and MCE, namely:
Symbolically manipulating determinants is not computationally efficient for moderate or large systems: on one hand, only in some cases is it possible to solve Equation (
47) analytically, because this equation could, ultimately, be expressed as a polynomial whose order increases with the number of hereditary kernels
N. On the other hand, even if analytical solutions are available, their substitution in Equation (
46) would lead to expressions that are unapproachable for moderate or large systems, due to the computational complexity behind the determination of analytical expressions of determinants.
In order to address these limitations, we will make use of the result obtained in Theorem 2(ii), which establishes, under the hypothesis of light nonproportional damping, the range
for any critical eigenvalue associated to the
jth normal mode. Hence, the following dimensionless parameter associated to mode
j is defined:
Therefore, translating the above limits to the new parameter, it is clear, then, that
. From now on, the variable
s, throughout the dimensionless parameter
, should not be read as an unknown but as an independent parameter. Both the eigenvalue problem,
, and the MCE can be expressed, in terms of this new parameter, as:
Considering the general form for matrix
consisting of
N hereditary kernels (see Equation (
5)), we have:
Plugging these expressions into Equations (
49) and (
50), and after further simplifications, it yields:
where:
All the parameters that control the dissipative model are included within Equations (
52) and (
53): indeed, the relaxation coefficients
are represented by the auxiliary variables
, and the viscous coefficients are represented as the entries of
. The strategy to build a critical curve starts by choosing two of these parameters of interest; let us denote them as
and
. Both can be either two viscous coefficients or two nonviscous coefficients, or even one of each type (possible combinations are listed below and in
Table 1). Setting a mode
j, then, by sweeping out the parameter
in the range
, the graph of a curve arises, solving the different solutions of Equations (
52) and (
53) in the unknowns
and
. Therefore, a family of
n curves may be plotted:
although, for large systems, the most representative modes can be taken, as shown later in Example 2. It turns out that Equations (
52) and (
53) constitute, themselves, an eigenvalue problem in the parameters
and
. In some cases, one of these parameters may be obtained as a linear or quadratic function of the other parameter from Equation (
53), and after plugging into Equation (
52), an one-parameter eigenvalue problem is obtained. In other cases, this is not possible, and both parameters must be solved simultaneously as part of a two-parameter eigenvalue problem. The different scenarios arise after combining the damping parameters in pairs, leading to four possible type of curves (see
Table 1). The details explaining how to transform Equations (
52) and (
53) into eigenvalue problems are explained in Example 1. Below, the four distinct types of curves which can be found are listed, together with a brief description of the algebraic numerical problem that arises:
- Type I.
Critical curves between two viscous coefficients
, where
and
are entries of
and
, respectively. If
, both coefficients belong to matrices associated to different kernels. Otherwise, if
, then both parameters are part of the same viscous matrix, a case which can be denoted as
. Since both Equations (
52) and (
53) are linear in the viscous parameters,
, one of these coefficients can be solved from Equation (
53) and plugged into Equation (
52), leading to a linear eigenvalue problem.
- Type II.
Critical curves that relate pairs of different nonviscous coefficients
corresponding to different hereditary kernels. The numerical method is based on considering the auxiliary variables
instead of
and
. Due to the mathematical structure of Equations (
52) and (
53), both parameters
and
must be solved simultaneously by a two-parameter linear eigenvalue problem.
- Type III.
Critical curves between a viscous and a nonviscous coefficient, both corresponding to the same
ith kernel,
, where
is an entree of
. Since both
and
belong to the same kernel, straight rearrangements of Equations (
52) and (
53) lead to a linear eigenvalue problem involving one parameter.
- Type IV.
Critical curves of a viscous coefficient of the
ith kernel, say
, with the
kth relaxation parameter
,
:
. Again, using Equation (
53) to solve for
, and after straight manipulations, the matrix equation (
52) can be transformed into a quadratic eigenvalue problem with one parameter. As known, any quadratic eigenvalue problem can be reduced to a double-sized linear problem.
The most important contributions, from a theoretical point of view, have been presented: (i) the new characterization of the critical modes given in Equation (
36); (ii) the derivation of the modal critical Equation (
45), and (iii) the development of a numerical model—summarized in both Equations (
52) and (
53)—which enables reducing the computation of any critical curve to solve an eigenvalue problem repeatedly along the interval
. In the following section, the aforementioned outcomes will be validated through two numerical examples, covering discrete and continuous systems.