Structural control design and defective systems

Structural control design and defective systems

Vincenzo Gattulli & Francesco Potenza

Continuum Mechanics and Thermodynamics ISSN 0935-1175 Continuum Mech. Thermodyn. DOI 10.1007/s00161-014-0410-5

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Author's personal copy Continuum Mech. Thermodyn. DOI 10.1007/s00161-014-0410-5

O R I G I NA L A RT I C L E

Vincenzo Gattulli · Francesco Potenza

Structural control design and defective systems

Received: 7 February 2014 / Accepted: 23 December 2014 © Springer-Verlag Berlin Heidelberg 2015

Abstract The intersection between the two concepts of structural control and defectiveness is discussed. Two simple oscillators differently connected by serial spring-dashpot arrangement are used to simply simulate technically relevant cases: dissipatively coupled adjacent free-standing structures, structures equipped by TMD and base-isolated structures. Eigensolution loci of the two classes of systems are tracked against one or more significant parameters to determine the potential benefits realized by different combinations of stiffness and viscosity. In both studied cases, codimension-two manifolds in the four-parameter space corresponding to coalescing eigenvalues are determined by analytical expressions. Conditions to discern semi-simple eigenvalues from defective ones confirm that the latter is the generic case laying in a two-parameter space while the former span a one-parameter subspace. The knowledge of the location of the defective systems in the parameter space permits to determine regions with specific dynamical properties useful for control design purpose. Keywords Structural control · Damping · Coalescence · Defective system · Coupled oscillators · Base isolation

1 Introduction Since the seventies of the last century, great research efforts have been focused on reducing environmentalinduced vibrations through control devices purposely designed to assign precisely selected stiffness and damping properties of engineering structures [19]. Thus, in the development of structural control technologies, the design of highly energy-dissipative or/and base-isolated structures has taken into consideration more and more the general case of non-proportional viscous damping as basic assumption. Understanding how the damping distribution affects the modal characteristics has considered crucial to the design of efficient damping devices when applied to flexible structures [10]. The parameter dependence of the eigenvalue as a design criterion has recently been emphasized in several fields for passive control purposes [17,20]. An approximate solution for the complex eigenproblem associated with the free vibration of a discrete system having several viscous dampers has been developed in [9,16]. Damping design affects the eigenproperties of the associated linear dynamical system. In particular, tracking the eigensolution loci against one or more significant parameters has theoretical and practical interest in stability analysis, design optimization, model updating, structural identification, and vibration control. Doing that, regions corresponding to intersection, or closeness, between two or more eigenvalue loci appear and are worth particular attention. In order to simplify the analysis, due to the smallness of damping occurring in Communicated by Francesco dell’Isola. V. Gattulli (B) · F. Potenza Department of Civil, Construction-Architectural and Environmental Engineering, University of Aquila, Nucleo industriale di Pile 67100 L’Aquila, Italy E-mail: [email protected]

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mostly physical structure, the associated terms are neglected making the system derivable from Hamiltonian principle. Hamiltonian N-dimensional systems with a generic number of close eigenvalues have been objective of eigensolution sensitivity analysis by a two-step multi-parameter perturbative procedure in the critical parameter region characterized by several close frequencies belonging to non-defective systems [12]. Moreover, for general viscously damped systems, the eigenvalue analysis is generally performed in state space through a non-Hermitian eigenvalue problem, and for systems with distinct eigenvalues, the corresponding eigenvectors diagonalize the state-space matrices. However, with general viscous damping, the mechanical parameters can be selected such that the system presents repeated complex eigenvalues with associated insufficient linearly independent complex eigenvectors. These systems are named defective, and the repeated roots allow modes that are irreducibly complex; hence, the defectiveness is impossible when the damping is classical [1]. A matrix perturbation method has been proposed to evaluate the influence between the change of structural parameters and the dynamic characteristics of defective systems [2]. In the proximity of resonant defective system, eigensolution sensitivity analysis has been performed for non-symmetric matrices with multiple eigenvalues through a perturbation method based on series expansion of noninteger powers of the perturbation parameter due to the rapid modification of the eigenvectors for small change of the parameter values [13]. Approximating eigenvalues and eigenvectors of nearly defective (mistuned) systems have been found by starting the perturbation expansion from an unknown exactly defective (tuned) system, determined by solving an inverse problem, close to the mistuned one [14]. Recently, a fast method for sensitivity analysis of non-defective and defective systems based on the generalized eigenvector and adjoint matrix has been presented in which the expansion coefficients can be obtained by a recursion algorithm [18]. Furthermore, looking at the problem from a different point of view, a procedure has been presented to generate low-rank modifications of the classical damping matrix and determined whether, if the modified system possess pairs of repeated complex eigenvalues, the resulting system is defective [4]. The important conclusion is that defective systems maybe much more common in practice than originally thought. Indeed, if a modified system has repeated eigenvalues and non-proportional damping, then it is very likely to be defective. Besides that, it has shown that defective system may play an important role in the structural control design of tuned mass dampers devoted to wind-induced vibration suppressions. In particular, even if the effects of a tuned mass damper on galloping oscillations, can been analyzed through the Multiple Scale Method starting from the critical scenario of simple-Hopf and a non-resonant double-Hopf bifurcations [5], the overall critical and postcritical scenario is completely described centering the attention on the resonant double-Hopf bifurcations, which starts from a codimension-three critical manifold [6]. Due to the coalescence of its eigenvalues, the system is defective at the criticality and therefore admits an incomplete set of eigenvectors but this condition maximize the critical wind velocity making the all family of defective system in some sense “optimal.” Starting from such system, steady solutions representing limit cycles for the original mechanical system have been expressed in analytical form through the Multiple Scale Method. Thus, the regions of existence of such limit cycles have been studied in the space of the control parameters. Sections of the critical manifold of defective system have been presented, successively, enlarging the description of the dependence of the limit cycle amplitudes even to a class of nonlinear internal damping, concluding that these one are uneffective in the reduction of the limit cycle amplitude if the starting linear system is the optimal perfectly tuned one [7]. Nevertheless, if the comparisons are conducted in a region sufficiently far from the optimal linear case, a better nonlinear TMD exists with respect to the obtainable limit cycle amplitude, although this is triggered at lower flow velocity. More recently, the coalescence of two eigenvalues has been studied in non-proportionally damped system, far away from the critical conditions of dynamic instabilities, focusing the attention on the peculiar characteristics of the damping performance of two oscillators coupled by a dissipative connection described by Kelvin–Voigt or Maxwell models [8]. A discussion on the dynamic of the coalescing systems, selected as design points, in forced vibrations induced by an imposed motion at the base is analyzed evaluating their performance through different indicators. In the present paper, a discussion is presented on the potential role of non-proportional damped system with coalescing eigenvalues in structural control design problem. Two simple oscillators opportunely connected by serial spring-dashpot arrangement are used to simply simulate technically relevant cases: adjacent free-standing structures and base-isolated structures. Eigensolution loci of the two classes of systems are tracked against one or more significant parameters to determine the potential benefits realized by different combinations of stiffness and viscosity. In both studied cases, codimension-two manifolds in the four-parameter space corresponding to coalescing eigenvalues are determined by analytical expressions. Opportune contractions of the parameter space permit to discuss all the relevant simple structural control models, as previously cited. Non-defectiveness conditions are used to demonstrate that systems with coalescing eigenvalues are generally

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defective. The knowledge of the location of the defective systems in the parameter space permits to determine regions with specific dynamical properties useful for control design purpose.

2 Equations of motion Consider two simple linear oscillators with mass M j and stiffness K j , ( j = 1, 2), coupled by a stiffness K . Denoting U1 and U2 as the relative horizontal displacements, and F as the viscous force applied by a damper differently located in the system as depicted in Fig. 1a, b, the free dynamic response of the two-degrees-offreedom (2dofs) system is governed by the equations M1 U¨ 1 + K 1 U1 − K (U2 − U1 ) + s1 F = 0 M2 U¨ 2 + K 2 U2 + K (U2 − U1 ) + s2 F = 0

(1)

where dot indicates the derivative with respect to time t and si (i = 1, 2) are two location boolean variables used to simulate the damper location. Denoting L as a convenient reference length, the following dimensionless variables and parameters can be introduced uj =

Uj Kj ω2 M2 K2 K F ; ω2j = ; β= ; ρ= ; κ= ; η= 2 u= 2 ; τ = ω1 t L Mj ω1 M1 K1 ω1 M 1 ω1 M 1 L

(2)

where the dimensionless force u is understood as the control variable, and the relevant parameters ρ and β stand for the mass and frequency ratios between the two uncoupled oscillators, respectively, while κ = ρβ 2 is introduced as a dependent parameter useful only for the purpose of further system classification. The equations of motion can be rewritten in the synthetic form ˙ =0 Mu¨ + Ku + su(u)

(3)

where u is the displacement vector, M and K are the mass and stiffness matrices, s is the location vector of the control force  u=

       1+η −η u1 s 1 0 , K= , s= 1 , M= 0 ρ u2 s2 −η ρβ 2 + η

(4)

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2.1 Damping in relevant structural control problems Introducing the simple rheological model of a purely viscous damper in the two cases depicted in Fig. 1, by ˙ relating the control force to the velocity vector the most relevant cases in structural the constitutive law u(u), control may be described. Indeed, the damper behavior described by a linear dashpot with viscous constant C can be considered introducing the dimensionless parameter C γ = (5) 2ω1 M1 permitting to describe the different location of the dashpot in case a (Fig. 1a) and case b (Fig. 1b) through the ˙ and the location vector s as follows specification of the constitutive law u(u) • case a: u = 2γ (u˙ 2 − u˙ 1 ); s = {−1, 1} • case b: u = 2γ u˙ 1 ; s = {1, 0} Consequently, the damping matrix Ci (i = a, b) can be introduced for the two cases, as     2γ −2γ 2γ 0 , Cb = (6) su = Ci u˙ with Ca = −2γ 2γ 0 0 ˙  } , the second-order Adopting a state-space representation, with the use of the state vector x = {u , (u) equations (3) defined by the mass, damping and stiffness matrices M, C and K may be rewritten as Ax + B˙x = 0 where the state matrix A and B are, respectively     −K 0 0 K , B= A= 0 M K C

(7)

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It is worthy to note that relevant simple models are contained in the two selected cases. In particular, in case a for η → 0 the simple model of two adjacent oscillators coupled only by a purely viscous dashpot is described, while corresponding to κ → 0 the second larger mass M2 (note ρ ≥ 1) is not externally connected (K 2 → 0) and it is only connected with the smaller mass M1 through the dissipative internal connection. More relevant for κ → ∞ (K 1 → 0) the larger mass M2 is externally connected, and the smaller mass is attached to it through the internal device realizing the simple basic model of a primary structure equipped with a TMD-type damper (see Fig. 2a). Differently, in case b for κ → 0 (which means β → 0 due to the assumption ρ ≥ 1), the second larger mass M2 is not externally connected and can be assumed to describe an heavier super-structure, while the smaller mass M1 is attached to the ground by means of a dissipative connection (described by the nondimensional parameters η and γ ) potentially representing a base isolation system (see Fig. 2b).

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3 Modal analysis and defectiveness The linear response to a generic loading time history can be described through input–output transfer functions, directly related to the system spectral properties. Therefore, analyzing the variation of the system modal properties due to different linear couplings may be a matter of theoretical and technical interest. In particular, attention can be focused on the sensitivity of the eigenvalues to wide-range variations of the parameter set µ = (ρ, β, η, γ ). Without restrictions on the proportional damping condition, the natural frequencies ω and modal damping coefficients ξ of the system are determined from the complex roots of the characteristic polynomial p(λ) := detA + λB = λ4 + c1 (µ)λ3 + c2 (µ)λ2 + c3 (µ)λ + c4 (µ) = 0 (9) Since the generic complex root can be expressed as λ j (µ) = a j (µ) + ib j (µ), where a j ∈ R, b j ∈ R and i is the imaginary unit, the equations can be rearranged employing polynomial factorization p(λ) = (λ − λ1 (µ)) (λ − λ¯ 1 (µ))(λ − λ2 (µ))(λ − λ¯ 2 (µ)) = 0

(10)

where the bar indicates a complex conjugate. Therefore, expanding the factorized polynomials, equating coefficients of the same λ-power and solving the resulting system of nonlinear equations, it is possible, in principle, to carry out the dependence of the real and imaginary part of the conjugate eigenvalues on the parameters, i.e., to determine the implicit relations a1,2 (µ) and b1,2 (µ). The real and imaginary part of the eigenvalues, ai and bi , are directly related to the damping ratio with respect to the critical one ξi and the damped and undamped circular frequency ωdi and ωi as follows  ai = −ξi ωi bi = ωdi ωdi = ωi 1 − ξi2 (11) Moreover, the parameter combinations can be obtained by imposing selected relations among the roots of the characteristic polynomial determining a possible design criterion. In the applications, it is convenient to state the design problem as follows: given the oscillator pair, described by the structural properties (ρ, β), determine the damper characteristics (γ , η) according to a certain design strategy. 3.1 Parametric analysis and coalescing eigenvalues Here, the selected design strategy, for both internal and external dissipative devices, is based on eigenvalue parametric analyses, that is, on the loci of the two complex eigenvalues λ1 (µ), λ2 (µ) for varying (γ , η)-values, and depicted for example in the Argand plane (Fig. 3), where the imaginary part accounts for the damped frequency, and the real part is proportional to the modal damping ratio. A system with selected structural data (ρ = 5, β = 2) is selected to describe the effects of damper size and location. The eigenvalues of the uncoupled oscillators are marked with white points on the imaginary axis. The figure shows several iso-η eigenvalue loci for the two cases of internal and external coupling. The eigenvalue loci are tracked by increasing the viscosity parameter, starting from the initial value γ = 0, corresponding to purely imaginary eigenvalues. Physically, this initial condition corresponds to the undamped system with purely elastically coupled oscillators. Consequently, the starting points of the eigenvalue loci on the imaginary axis (white points in Fig. 3a, b) move upward for increasing η-values in both case a and b, with the lower eigenvalue that saturates for high η-values and the higher eigenvalue that grows indefinitely. From the comparison of the eigenvalue loci, some analogies and differences can be highlighted between the two models. In particular, it appears clear that the damping location strongly affects the root loci; however, a discussion of the results may be done using the coupling stiffness η as distinguished parameter. Indeed, for low coupling stiffness values and increasing viscous coefficients in case a, i.e., following the arrows along the green (light gray) iso-η curves in Fig. 3a, it can be observed that the eigenvalue corresponding to the higher frequency moves toward the left semi-plane, but its negative real part increases to a minimum value. This result, common to both cases of internal and external dissipative connection (Fig. 3b), means that an increment of the viscous damper coefficients cannot indefinitely increase the modal damping of the stiffer mode. Indeed, after the minimum, further γ -increments let the eigenvalue turn back, tending again to the imaginary axis for limiting (infinitely high) γ -values, physically corresponding to a perfectly rigid connection between the oscillators. In both cases, the semi-circular, or semi-elliptic eigenvalue loci are tracked counterclockwise

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Fig. 3 Iso-η eigenvalue loci in the Argand plane for different dissipative system: a internal coupling, b external coupling (color figure online)

for increasing viscous coefficients. From a physical viewpoint, this means that increasing the damper viscosity causes a systematic reduction of the damped frequency. The second eigenvalue, corresponding to the lower frequency, equally moves toward the left semi-plane. In both cases, its imaginary part systematically reduces up to vanishing, letting the eigenvalue collapse on the real axes. This result means that an increment of the damper viscosity ends up over-damping the softer mode. Thus, it can be concluded that the typical frequency reduction due to an increasing viscous coupling between the oscillators is realized for soft elastic connections. The qualitative scenario characterizing low-stiffness dampers is found to drastically change for a certain stiffness value ηd , corresponding to the red (black gray) curve in Fig. 3. Despite the quantitative difference (ηd = 0.498 for internal dissipative coupling, and ηd = 5.0) for external dissipative coupling, the value ηd qualitatively corresponds to a pair of intersecting eigenvalue loci, which are found to approach each other up to collision (points P in Fig. 3a and P1 in Fig. 3b) for a certain γ -value (γd = 2.34 for the internal coupling, and γd = 1.00 for the external coupling). For the parameter pair (ηd , γd ), the coalescing eigenvalues have the same real and imaginary part, and one of the two modal damping ratios reaches its maximum. Looking at the corresponding eigenspace, the point P turns out to be defective, that is, the double eigenvalue (algebraic multiplicity 2) corresponds to a single eigenvector (geometric multiplicity 1). Defective systems are known to possess strong eigenvalue sensitivity to small variations of the parameters [13–15]. This property clearly explains the abrupt change of direction experienced by the eigenvalue loci after their collision in the defective point, which gives rise to the well-known pitchfork-shaped intersection of the corresponding red (black) curves in Fig. 3. A discussion on the features of such peculiar points can be found in the next sections. However, if the damper stiffness is increased over the value ηd , the two complex eigenvalues tend to exchange the respective qualitative behavior. Indeed, looking first at case a for high stiffness values and increasing viscous coefficients, i.e., following the arrows along the blue (dark gray) iso-η curves in Fig. 3a, the eigenvalue which initially corresponds to the stiffer mode reduces its imaginary part (becoming progressively softer), collapsing on the real axes; on the contrary, the eigenvalue which initially corresponds to the softer mode increases its imaginary part (becoming gradually stiffer), turning back to the imaginary axes. According to this behavior, the two modes exchange the order of the corresponding frequencies for sufficiently high γ -value. It can be concluded that the typical frequency reduction due to an increasing viscous coupling between the oscillators is realized for the higher frequency mode in the internal dissipative coupling for stiff elastic connections, while lower frequency mode become stiffer increasing damping. This behavior is strongly enriched in the case of external dissipative connection, indeed here, the inversion of the damping injection mechanism produced by the overcoming of the threshold ηd is accompanied by a peculiar phenomenon, which augment the potentiality of inserting damping in the system as evidenced by the behavior of the eigenvalue which initially corresponds to the softer mode. Indeed, the increase of the coupling stiffness (ηd > 5) permits to reach eigenvalue position in the Argand plane with larger distance from the imaginary axis (ai = 0) and consequently larger modal damping ratios. The mechanism of exchanging the order of frequency for high internal stiffness in case b is

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more evident. Indeed, the eigenvalue associated with the lower frequency mode, in presence of sufficiently high stiffness (ηd > 5) strongly increases its imaginary part (becoming stiffer and damped), turning back to imaginary axis. Doing this, a second coalescence between two eigenvalues is generated as it appears in point P2 (ηd = 60, γd = 6.7) which has the same imaginary part of the P1 -point in Fig. 3b for which the achievable damping through the external connection is much higher than the one found in the internal case. 3.2 Parameter loci in codimension-two manifolds of coalescing eigenvalues A class of coalescing system can be located in the parameter space for both internal and external damping cases by enforcing the characteristic equation to have a double root (i.e., a1 (µ) = a2 (µ), b1 (µ) = b2 (µ)), corresponding to coincident eigenvalues. Analytical expression of codimension-two manifold can be found following the steps reported in the “Appendix”. For the internal dissipative coupling, case a, assigned the structural data (ρ, β), a closed form solution can be achieved for the coalescing coupling parameter (γd , ηd ), as follows   1/2 ρ(1 − β 2 )(1 − ρ 2 β 2 ) ρ ηd ηd 1/2 2 2 (12) , γd = 1 + ηd + β + − 2 β (1 + ηd ) + ηd = (1 + ρ)2 (1 + ρβ 2 ) 1+ρ ρ ρ while for the external dissipative coupling, case b, assigned the structural data (ρ, β), the two expressions are 1  ρ 1 + β 2 (ρ − 2) ± (1 + 2β 2 ρ + β 4 ρ(ρ − 4))1/2 2 

 1/2 2 ηd + ρβ (1 + ηd ) ηd + ρ + β 2 ρ + ηd ρ − 2ρ β 2 (1 + ηd ) + γd = ± ηd + ρβ 2

ηd =

ρ ηd

1/2 1/2 (13)

The physical systems corresponding to the defective point and the nearly defective systems in the surrounding parameter region are strongly characterized by the coincidence or closeness of their frequencies, which may involve significant energy transfer between the resonant modes. However, the interest in the complete description of the surfaces describing the systems for which eigenvalue coalescence occurs, should be searched in the possibility of determining from such peculiar point a classification of the studied systems in terms of structural control performances, as it will be viewed in the following section. Here, imposing the two coalescing condition, the four parameters space is restricted to a two-dimensional manifold, described by surfaces, on which are lying all systems with coalescing eigenvalues. It should be noted that some restrictions and classifications can be imposed on these surfaces, as it will be delineated in the following. Focusing attention, on the case a, the condition of the P-point in which for a given design parameter pair ηd , γd the two real and the two imaginary parts coincide, has permitted to evaluate the loci of these design points with respect to all the possible coupled systems, described by the parameters (ρ, β). These loci are represented by the two bidimensional surfaces ηd (ρ, β) and γd (ρ, β) in both 3D views (Fig. 5a, d) and sections (Fig. 5b, c, e, f). It can be noted that the nondimensional design stiffness value ηd of the coupling device increases with the increment of the frequency ratio β following a fractional function (Eq. 121 ). The dependence shows that for larger values of the mass ratio ρ the increment of ηd is faster (Fig. 5e). Moreover, the design parameters of the coupling device for the limit system obtained for β → 0 (due to κ → 0) should be characterized by the limit 3 values ηd → ρ/(1+ρ)2 and γd → 2ρ/(1+ρ) 2 . Increasing β in the range 0 ≤ β ≤ 1, the design parameter ηd decreases from the limit value assumed for β → 0 to the plane ηd = 0, the intersection between the manifold and this plane determines a curve described by the equation β = 1/ρ, which limits the space of the possible oscillator pair possessing the capacity to reach equal eigenvalues through a dissipative coupling as clearly shown in Fig. 5b. Behind this border curve, the solution requires negative stiffness which can be potentially induced through active dissipation devices up to the successive limiting curve formed again by the intersection of the manifold with the zero-plane along the line described by the equation β = 1. For value of β < 1, the nondimensional design stiffness grows faster for larger values of ρ increasing β. The nondimensional viscous damping parameter γd in the space of the parameters describing all possible oscillator pairs is described only for positive values. However, the domain in which ηd is negative produces a restriction of the admissible domain also for the positive-valued manifold γd . It is worth noting that, along the limiting domain curve β = 1/ρ, the viscous parameter assumes the value γd = (ρ − 1)/(ρ + 1), determining the equation which describes

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the locus of the oscillator pairs coupled through a purely viscous device, indeed ηd = 0 (see Fig. 5b). This occurrence evidences that in case a the codimension-two manifold in the four-dimensional parameter space describes also the coalescence in a contracted three-dimensional parameter space in the case of two oscillators with purely internal viscous coupling, already presented in [15]. From a mechanical point of view, the results illustrated clearly show a different behavior in the design goal of reaching, through an internal dissipative coupling the condition that a pair of oscillators may have equal eigenvalues in the coupled configuration. Indeed, if the second oscillator with larger mass is highly stiffer (ρ > 1, κ > 1) than the first one and the second frequency is also greater (β > 1) than the first one, then is always possible to find a coupling device that lets frequency of the coupled system coincide. Differently, if the first oscillator, with smaller mass, is highly stiffer than the second one up to a ratio that is inversely proportional to the mass ratio (up to the curve β = 1/ρ which means κ = 1/ρ), it is possible to realize a coupling device that satisfy the design requirements. Instead, if the system is characterized by the stiffness ratio within the range 1/ρ ≤ κ ≤ ρ, it is impossible to realize a passive coupling devices that produces a system with coincident frequencies. It is worth noting that even the admissible region of the existence of coupling passive device which produces a system with complex conjugate coalescing eigenvalues is still limited by the fact that in the case of first oscillator, with smaller mass and highly stiffer than the second one, for the possible coupling stiffness ηd (see Fig. 5a), the increasing of damping is limited by the fact that up reached a certain value, signed as dotted line, the coalescing complex conjugate eigenvalues collide on the real axis (critical damping) and with the increasing of damping they separate going in opposite directions (overcritical damping). The richer scenario of the case b, already evidenced looking at the root loci, is confirmed studying the codimension-two manifold in the parameter space for which the oscillator couple with an external damping device has two coalescing eigenvalues. Indeed, the corresponding parameter loci appears with a double surface such that for a given system (ρ, β) are possible two design P1 and P2 -points associated with different ηd , γd values. Imposing the eigenvalue coalescence permits to individuate two bidimensional surface loci ηd (ρ, β) and γd (ρ, β) in both 3D views (Fig. 6a, d) and sections (Fig. 6b, c, e, f). It can be noted that the nondimensional design stiffness value ηd has two branches which both increase with the increment of the frequency ratio β following a radical function (Eq. 131 ). The dependence shows that for larger values of the mass ratio ρ the increment of ηd is faster in the branch related to the P2 -point with respect to the lower branch related to the P1 -point (Fig. 6b). Besides that, the higher branch saturates faster than the lower one for the collision of the eigenvalue pair associated with the P2 -point with the real axis (critical damping condition). This fact justifies the absence of a double curve for high ρ-values (Fig. 6b). This behavior is more evident looking to the sections of the manifold with β-planes, as depicted in (Fig. 6e), where the higher branch after the turning point (at ρ 4) saturates in the blue (dark) dots for ρ-values approximately similar (at ρ 6) evidencing the border in which the coalescing eigenvalues lying on the manifold change from being complex conjugates and become real. The loci of turning points depicted as green (light gray) curves in both manifolds ηd (ρ, β) and γd (ρ, β) as shown in Fig. 7a, b can be determined by the following equation 1 + 2β 2 ρ + β 4 ρ(ρ − 4) = 0

(14)

which is obtained imposing that the root of Eq. 131 ∈ R. This condition generates also a loci of turning points in the surface γd for which the expression is more complicated and it is omitted for sake of brevity (see Eq. 132 ). Figure 7 represents the codimension manifolds of coalescing eigenvalues in the range 0 < β ≤ 2 and ρ > 1. In both surfaces, the dotted blues (gray) curves represent the border on the manifolds which separates coalescing eigenvalues ∈ C with the ones ∈ R. Moreover, the loci β = 1 delimitates a portion of the lower branch of the surface in which for β < 1 the Eq. 131 furnishes value ηd < 0. Consequently, also in the case b, it can be made the consideration, already given for the previous case, on the possibility of realizing a system with an internal negative stiffness. Moreover, the design parameters of the coupling device for the limit system obtained for β → 0 (due to κ → 0 which means κ2 → 0) should √ be characterized by a pair of limiting loci ηd → ρ and ηd → 0 corresponding, respectively, to γd → ρ and γd → 1. Increasing β in the range 0 ≤ β ≤ 1, the design parameter ηd has two surfaces. In the upper one for ρ = 1, the ηd decreases from the limit values ηd = 1 (for 1 β = 0 and ρ = 1) along the curve ηd = 21 (1 − β 2 + (1 + 2β 2 − 3β 4 ) 2 ), to the value ηd = 0 (for β = 1 and ρ = 1. In the lower one, ηd assumes negative values. The nondimensional viscous damping parameter γd in the space of the parameters describing all possible oscillator pairs is described only for positive values. However, the domain in which ηd is negative produces a restriction of the admissible domain also for the

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surface γd > 0 and this is indicated in gray on Fig. 7b. It is worth noting that, along the limiting domain curve β = 0, √ the stiffness parameter assumes the value ηd = ρ while the viscous parameter assumes the value γd = ρ, determining the equation which describes the locus of the oscillator pairs representing the relevant case of a simple system simulating a base-isolated structures (see Fig. 2b), possessing two coincident eigenvalues. This occurrence evidences that the case b codimension-two manifold in the four-dimensional parameter space describes also the coalescence in a contracted three-dimensional parameter space modeling base-isolated systems. 3.3 Semi-simple and defective systems in codimension-two manifolds In order to define the dynamical behavior of systems with repeated complex eigenvalues, as the ones described above, it is necessary to discern if they belongs to semi-simple or defective systems. If λi is a root of the characteristic polynomial p(λ) of a n-dimensional system than the smallest number n A = n A (λi ) ∈ N for which d n A p(λ) |λ=λi  = 0 (15) d(λ)n A is called the algebraic multiplicity of λi and the natural number n G (λi ) = n G := n − rank (A + λi B)

(16)

is called the geometric multiplicity of the eigenvalue λi . Generally, the geometric multiplicity is less or equal than the algebric one, n G ≤ n A . If n G < n A for a given root λi then λi is called a defective root; otherwise, the system is termed simple (or not defective) [11]. In the general case, defectiveness is consequence of a coalescence of two or more eigenvalues [13], accompanied by an incomplete set of eigenvectors. If suitable geometrical symmetries exist or energy conservation holds [12], the coalescing eigenvalues may be not defective and with the associated system are called semi-simple. Hence, it is interesting to determine whether the studied systems described by the parameter loci associated with codimension-two manifolds of

Author's personal copy V. Gattulli, F. Potenza

Fig. 5 3D views of the loci of the internal dissipative design parameters ηd , γd for 0 ≤ β ≤ 2: a existing solutions ηd > 0 limited by bold curves, b locus of design purely viscous connection ηd = 0, γd  = 0

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coalescing eigenvalues are defective (generic case) or if it exists a subdomain of these parameter loci associated with non-simple systems. In this respect, if we evaluate the eigenvalues associated with the following matrix  = (A + λic (µ)B) H (A + λic (µ)B)

(17)

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Fig. 7 3D views of the loci of the external dissipative design parameters ηd , γd for 0 ≤ β ≤ 2: a existing solutions ηd > 0 limited by bold curves, b locus of design purely viscous connection ηd = 0, γd  = 0 (color figure online)

will furnish the roots σi ∈ C with i = 1, . . . , 4, which can be ordered as σ42 ≥ σ32 ≥ σ22 ≥ σ12 . It can be demonstrated that in the generic case the system is defective when σ12 = 0 for system with coalescing eigenvalues, while the system is semi-simple when σ12 = σ22 = 0 [3]. Imposing such conditions on the codimension-two manifold of coalescing systems, it has possible to demonstrate that the parameter loci can be opportunely classified. Indeed, the semi-simple not-generic case is associated with system described by the equation β = 1 in the (ρ, β)-plane in both studied cases, while the defective generic case holds in the remaining surfaces described by Eqs. 12–13 and depicted in Figs. 4, 5, 6 and 7.

4 Defectiveness and structural control design Among several methods proposed in the literature to define the design parameters characterizing fluid-viscous or viscous-elastic dampers for vibration mitigation as summarized in [8], in this section, a discussion on the dynamic performances of defective systems is presented in view of structural control optimization. The methodology discussed in this section is driven by the leading idea of using the surfaces describing defective system in the parameter space as reference system around which different control performance indexes are evaluated in order to discern which are the peculiarities of such systems with respect the nearly defective ones. In particular, initially, it is supposed that most of the information needed to orient the design can be derived from the parameter-dependent behavior of the system damping ratio, observed through the eigenvalue evolution in the complex plane. Similar approaches have already been used in the case of vibration control of string and beam adding a supplemental device [9], in discrete structural systems [16], in selecting the parameters of a TMD for maximization of the critical wind velocity in aeroelastic oscillators [6] or even for active and semi-active damping injection [8,17]. Following this idea, we would like briefly show how the location of defective damped systems in the space of the relevant parameter of the studied system may play a role in structural control design in comparison with the dynamics of the majority of the available system which can be viewed as nearly defective ones. Several different dynamical performance criteria will be evaluated and discussed for a few selected cases to give an initial insight to the problem.

4.1 Modal damping maximization The design parameter pair (η, γ ) selected in order to have coalescing eigenvalues permits to realize an absolute maximum of the modal damping of a target mode in the case a (as clearly shown in [8]) or two relative maxima of the modal damping in the two coalescing systems present in the case b. To look deeper into the problem, in addition to the classical Argand plane, the positions of the real and imaginary part of the system eigenvalues

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have been determined within the complete four-dimensional manifold through identification of the (γ , η, β, ρ)coordinates for both cases. For internal dissipative coupling, two-dimensional sections of the real and imaginary parts of the eigenvalues laying on the manifolds, plotted in the (γ , η)-space, respectively, for fixed values of the (β, ρ)-coordinates are presented in [8]. The real parts of the two eigenvalues in the relevant (γ , η)-space confirm the observation made previously, evidencing that the real part (Re1 ) of the eigenvalue associated with the lower eigenfrequency (Im1 ) is generally higher than the other one (Re2 ). However, it can be observed that the two real parts lay on two surfaces which cross each other along a curve in the (γ , η)-space which starts from the origin and terminates in the design point P, after which there are no (γ , η)-values at which the two real parts coincide. The peculiar point P is also characterized by the fact that the real part (Re2 ) associated with the higher imaginary part (Im2 ) here reaches its maximum value. Extracting the modal damping ξ2 and the natural circular frequency ω2 from these values through Eq. 11, it is confirmed that P is a point for which ξ2 is maximized. More complex is the topology of the manifolds for the case of internal damping, for which the twodimensional sections in the control parameter space are depicted in Fig. 8. These sections of the manifolds in the (γ , η)-plane on which are laying the real and imaginary part of the two eigenvalues are briefly discussed. In Fig. 8, the black gray (red) solid lines represent the intersection of the manifold with the plane containing the P1 -point. Indeed, it can be noted that for a specific pair (ηd , γd ) of the control parameters both the real and imaginary parts of the two eigenvalues intersect each other. In the other cases (light gray lines), the intersections of the real part of the eigenvalue loci happen at different parameter values with respect to the intersection of the imaginary parts. It is worth to note that differently from the case of internal damping, in this case there is a second couple of planes which intersections with the manifold evidences the coalescing eigenvalues, namely the P2 -point. It is evident that in both cases the (Re)-part of one of the two eigenvalues is locally maximized. Therefore, around such defective system, potentially the dynamics involving mainly the mode which have associated such high damping is maximally damped.

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4.2 System performance in free oscillations The performance of the studied systems is evaluated in free oscillations investigating the variations of the response with respect to the variation of the system parameters around the selected defective system. In particular, analytical solutions and direct numerical integration of the equation of motion have been performed. For nearly defective systems, the analytical solution of Eq. 7 for given initial conditions x0 , has been expressed, for the undercritical case, as follows x(t) = 2

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(19)

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where c j eiθ = wHj x0 with w j the left eigenvectors belonging to the matrix W = V−1 in which V is the modal matrix composed by the right eigenvectors bi-orthonormal to the left ones. Consequently, the system  displacement components can be expressed through the modal amplitudes as u i = 2j=1 qi j . Hence, the system orbit in the (u 1 , u 2 )-plane is considered to evidence how the system behavior changes around the defective system. In Fig. 9 are depicted the modal amplitudes of nearly defective systems selected around the defective P1 -point. In particular two types of parameter variations have been considered. In the first case, only a small perturbation of the γd parameter has been used to select the nearly defective system for which the damping parameter has been selected as γ = (1 ± )γd . In the second case, a small perturbation has imposed to the full set of parameters as µ = (1 ± )µd . For the nearly defective systems, the modal amplitudes orbits have been evaluated according to Eq. 19. Figure 9a, b shows the modal amplitude orbits for two nearly defective systems in which only the damping parameter γ has been changed ( = 0.1). The modal proprieties of the systems change fastly around the defective point evidencing a strong modification of the modal amplitude trajectories [15]. However, evaluating the displacement components and comparing directly the orbits of the defective systems and the nearly defective

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ones a smooth behavior is observed in the variation of the response moving around the defective system (see Fig. 9c). Similar results are observed when a full perturbation of the parameter vector is considered (Fig. 9d–f). Several examples have been considered to characterize the dependence of the observed behavior from the initial conditions and a few of them have been summarized in Fig. 10. In the illustrated cases, as in the previous ones, in the dynamics of nearly defective systems, strongly different orbits of the modal components can be recognized (Fig. 10a, b, d, e), which are then composed to evaluate the natural component trajectories producing orbits in the natural components very close to the ones of the defective system, for a general set of initial conditions (Fig. 10c, f). In order to evaluate the performance of the systems with external dissipative coupling (case b) in free oscillations, the following performance index has been introduced:  tf 2 u 1 (t) + u 22 (t) Ifo = dt (20) 2 2 t0 u 1d (t) + u 2d (t) which may represent the integral of the square of the distances of the system along the orbit with respect to the rest position. According to this rule, the performance of the nearly defective system is directly compared with the ones of the two defective system described by the P1 - and P2 -points. The results are shown in Fig. 11 for three selected initial conditions, evidencing that a smooth behavior of the performance around the defective point has been found despite the fact that eigenvalue properties change fastly around the defective point. Moreover, it is also clear that the selected performance index Ifo depends on initial conditions due to the fact that the system modal components are differently participating to the motion. Therefore, if the induced motion involves primary the mode in which the modal damping is maximized in the reference defective system, the performance index Ifo takes a minimum close the defective system γ = γd (Fig. 11a), while in other cases (Fig. 11b, c), the minimum is more distant and it is reached for γ > γd . The selected cases are associated with the same initial conditions used to evidence the orbit features in Figs. 9 and 10. Differently, the behavior of the performance index around the defective system P2 is more complicated because changing the initial conditions the minimum can be reached either for γ > γd that for γ < γd (Fig. 11d,e,f).

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5 Conclusions The investigations conducted on simple 2dofs models have shown the intersections between a potential methodology for assessing the basic mechanical parameters that commonly characterize the preliminary design stage of substructures interconnected by a dissipative devices and the peculiar eigensystem characteristics of nonproportionally damped structures with coalescing eigenvalues. The analyses have shown that peculiar points (eventually design points) in the parameter space exist in which eigenvalue coalescence occurs and one of the two damping ratios is maximized. This feature appears in a single eigenvalue coalescence for internal dissipative coupling while, unexpectedly, it is enriched in the case of external dissipative coupling showing that for a given system it is possible to have two different eigenvalue coalescences changing the design parameters. This occurrence depends on the paths in the Argand plane followed by the eigenvalues due to the influence of the stiffness and damping parameters differently characterizing the two studied systems. Due to the dimension of the eigenproblem in both cases, an analytical expression of the design parameters has permitted depicting the manifolds in the system parameter space where the design parameters are belonging. The loci of the design parameters are deeply discussed evidencing areas, in the analytically described surfaces, in which the systems posses undercritical, critical and overcritical damping. Furthermore, it has also been demonstrated that for certain parameter value characterizing the two simple oscillators, it is necessary to select negative value of the coupling stiffness parameter in order to reach eigenvalue coalescence. The eigenvalue coalescence has also been discussed in relation to defectiveness. Indeed, further conditions have been introduced to determine in the class of coalescing systems which are defective from the ones that are semi-simple. Codimension-three loci of semi-simple systems are then enucleated from the codimension-two manifolds of coalescing system evidencing clearly that defectiveness is the generic case. Finally, the implications of selecting defective systems as suitable controlled system have been discussed, particularly in the case of undercritical damping. In such case, it has been shown that if a defective system is designed as a target one, and for imperfections the design point is missed, the associated small variations in the parameters produce a nearly defective system. However, assigned the same initial conditions to the two systems, namely the defective and quasi-defective ones, the dynamics evolve along orbits in the natural

Author's personal copy V. Gattulli, F. Potenza

component space which are close to each other. Furthermore, it should be underlined that the rapid change of the eigenvalue properties around a defective system, produces natural component trajectories of the two oscillators, depicted as orbits, which are very sensitive to initial conditions, but appears to be smoothly changing from defective to nearly defective system in all the experienced cases. Smooth modification in the dynamical response varying the system parameters around defectiveness has been found in the case of defective and quasi-defective system subjected to external harmonic force [8]. Smoothness in the variations of free-oscillation response around the defective system is evidenced also in the present work by the evaluation of a proposed performance index, used to compare the width of the orbits in the natural components, which permits to evidence the role played by modal damping maximization in the defective system. Acknowledgments This work has been partially supported by the Italian Department of Civil Protection under the programs RELUIS-DPC 2014, RELUIS-DPC 2010 project task 2.3.2, “Development of new technology for seismic retrofitting” and by the Italian Ministry of Education, Universities and Research (MIUR) through the PRIN funded program “Dynamics, Stability and Control of Flexible Structures” [Grant No. 2010MBJK5B].

Appendix The characteristic polynomial, expressed by Eq. (18), in the case of coalescing eigenvalues can be generically re-written as follows: pc (λ) := (λ − (a + ib))2 (λ − (a − ib))2 = λ4 −4aλ3 +(6a 2 +2b2 )λ2 −4a(a 2 +b2 )λ+(a 2 +b2 )2 = 0 (21) For the problem of internal dissipative coupling, case a, evaluating the expression of the coefficients ci (µ) of the characteristic polynomial, and equalizing the coefficients of each term with an equal exponent with the ones expressed by Eq. (21), the following system of four equations can be found −4a = 2γ +

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while for the problem of external dissipative coupling, case b, the system is the following one −4a = 2γ 6a + 2b2 = 1 + β 2 + η + 2

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In both cases, eliminating the coefficients a and b from the above equations it is possible to find two equations in the four parameters. Such equations can be rearranged in order to give the expressions of the main structural control design parameters ηd and γd in function of the system parameters ρ and β, as it is presented in Sect. 3.1. References 1. Bernal, D.: Closely spaced roots and defectiveness in second-order systems. ASCE J. Eng. Mech. 131, 276–281 (2005) 2. Chen, S.H., Xu, T., Han, W.Z.: Matrix perturbation for linear vibration defective systems. Acta Mech. Sin. 24, 747–753 (1992) 3. Dieci, L., Pugliese, N.: Two-parameter SVD: coalescing singular values and periodicity. SIAM J. Matrix Anal. Appl. 31, 375– 403 (2009)

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4. Friswell, M.I., Prells, U., Garvey, S.D.: Low-rank fast sensitivity analysis of defective system. J. Sound Vib. 279, 757– 774 (2005) 5. Gattulli, V., Di Fabio, F., Luongo, A.: Simple and double Hopf bifurcations in aeroelastic oscillators with tuned mass dampers. J Frankl. Inst. 338, 187–201 (2001) 6. Gattulli, V., Di Fabio, F., Luongo, A.: One to one double Hopf bifurcation in aeroelastic oscillators with tuned mass dampers. J. Sound Vib. 262, 201–217 (2003) 7. Gattulli, V., Di Fabio, F., Luongo, A.: Nonlinear tuned mass damper for self-excited oscillations. Wind Struct. 7, 251– 264 (2004) 8. Gattulli, V., Potenza, F., Lepidi, M.: Damping performance of two simple oscillators coupled by a visco-elastic connection. J. Sound Vib. 332, 6934–6948 (2013) 9. Krenk, S.: Complex modes and frequencies in damped structural vibrations. J. Sound Vib. 270, 981–996 (2004) 10. Krenk, S.: Dampers on Flexible Structures, Lectures Presented at “Semi-active Vibration Suppression—The Best from Active and Passive Technologies CISM, Udine, October 1–5 (2007) 11. Lancaster, P.: Lambda-Matrices and Vibrating Systems. Pergamon Press, Oxford (1966) (reprinted by Dover Publications Inc., New York, 2002) 12. Lepidi, M.: Multi-parameter perturbation methods for the eigensolution sensitivity analysis of nearly-resonant non-defective multi-degree-of-freedom systems. J. Sound Vib. 332, 1011–1032 (2013) 13. Luongo, A.: Eigensolutions sensitivity for nonsymmetric matrices with repeated eigenvalues. AIAA J. 31, 1321–1328 (1993) 14. Luongo, A.: Eigensolutions of perturbed nearly defective matrices. J. Sound Vib. 185, 377–395 (1995) 15. Luongo, A.: Free vibrations and sensitivity analysis of a defective two degree-of-freedom system. AIAA J. 33, 120–127 (1995) 16. Main, J., Krenk, S.: Efficiency and tuning of viscous dampers on discrete systems. J. Sound Vib. 286, 97–122 (2005) 17. Premount, A.: Vibration Control of Active Structures, 3rd edn. Springer, Berlin (2011) 18. Xu, T., Xu, T., Zuo, W., Hao, L.: Fast sensitivity analysis of defective system. Appl. Math. Comput. 217, 3248–3256 (2010) 19. Yao, J.T.P.: Concept of structural control. J. Struct. Div. 98(7), 1567–1574 (1972) 20. Zare, A.R., Ahmadizadeh, M.: Design of viscous fluid passive structural control systems using pole assignment algorithm. Struct. Control Health Monit (2013). doi:10.1002/stc.1633