Mechanical Systems and Signal Processing 36 (2013) 540–548

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Effect of damping on the nonlinear modal characteristics of a piecewice-smooth system through harmonic forced response Paolo Casini n, Oliviero Giannini, Fabrizio Vestroni Dipartimento di Ingegneria Strutturale e Geotecnica (DISG), Universita degli Studi di Roma ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Roma, Italy

a r t i c l e in f o

abstract

Article history: Received 16 January 2012 Received in revised form 26 September 2012 Accepted 3 October 2012 Available online 21 December 2012

Several engineering systems present piecewise-linear characteristics, among them, the damaged beams with breathing cracks are of particular interest. The dynamics of such systems exhibits bifurcations at internal resonances characterized by the onset of superabundant nonlinear normal modes with their individual modal shapes. In this paper, a 2-DOF system with piecewise linear stiffness, representative of a damaged system with a breathing crack, is analyzed by means of a numerical and theoretical investigation. The oscillator is forced by a harmonic base excitation and the role of damping on the modification of the nonlinear modal characteristics is investigated. The outcomes are compared with the reference behavior of the undamped system which allows for semi-analytical solution. It is found that the damping, on one hand, softens the abrupt transitions from one behavior to another, typical of undamped systems, on the other, affects the bifurcations of the nonlinear modes causing some of them to completely disappear and leaving others largely unaffected. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Piecewise linear systems Damaged systems Forced response Damping Nonlinear normal modes

1. Introduction The study of the forced response of engineering systems is of primary importance because in the real world their mechanical and dynamic properties are commonly extracted from the measured response under time varying loads. To this aim, in nonlinear dynamics, the harmonic base excitation is the most suitable for its simplicity and because it easily permits to highlight the nonlinear characteristics of the system. The harmonic response of a linear system is governed by its modal characteristics, the resonance peaks occur in the vicinity of natural frequencies, the operating deformed shape at that frequency resembles accurately enough the corresponding modal shape. From an engineering point of view, the damping plays an important role at resonance providing a lowering and smoothing effect, but has a qualitative small effect on the frequency and modal shapes. The Nonlinear Normal Modes (NNMs) defined as any periodic motion of the undamped autonomous system [1,2] are the counterpart of the linear modes for nonlinear systems but their contribution to the forced response becomes more complex and not always straightforward: in particular, even low levels of damping, can significantly modify the bifurcation scenarios and, therefore, the number of expected resonance peaks associated to the NNMs. These phenomena, spotted by the authors during the theoretical and experimental investigations of a particular piecewise-smooth system [3–5], are thoroughly investigated in this paper.

n

Corresponding author. E-mail address: [email protected] (P. Casini).

0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2012.10.001

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Piecewise-smooth mechanical systems are governed by sets of ordinary differential equations which are smooth in regions of phase space, smoothness is lost as trajectories cross the boundaries between adjacent regions (discontinuity boundaries). Much effort in science and engineering has focused on the analysis of the bifurcation scenarios and of the modal characteristics of this class of systems [6–10]: typical mechanical applications include stick-slip mechanical systems with friction and systems colliding with a deformable or rigid stop; moreover, particular interest is devoted to the study of piecewise-linear system, modelling beams with breathing cracks [11–13]. The study of the forced response of these nonlinear systems is important on a two-fold point of view: on one hand unexpected forced resonances, which are essentially nonlinear and have no analogies in linear theory, arise since the number of normal modes of a nonlinear system may exceed its degrees of freedom (superabundant NNMs) [14], on the other hand, it is possible to obtain information on the system’s characteristics from the measured response that is an important goal of this work. In these systems a linearization of the system will not be possible, or might not provide all the possible resonances that can be experienced. For safety reasons, the first point is important for engineering applications; in fact, piecewise linear systems may experience a resonant behavior at far more forcing frequencies than the corresponding linear system, since each nonlinear mode may induce large responses also at frequencies having an integer ratio with the resonant ones. Because of the ever-presence of damping in real structures and in any experimental environment, this paper is focused on its role on the forced response with the aim of highlighting how the information about the nonlinear dynamic characteristics can be suitably retrieved through its forced response. Numerical simulations are carried out for different level of damping and nonlinearity, and the effects on the various types of NNMs of this system and on its forced response are examined. 2. Model description and equations of motion The investigated system, Fig. 1a, consists of a 2-DOF oscillator: two masses, m1 and m2, are connected by two piecewise linear springs of stiffness k1 and k2, and reduced stiffness (1 e1)k1 and (1  e2)k2 (0r ei o1), where e1 and e2 govern the stiffness reduction and are denoted as damage parameters; the relevant restoring forces exhibit the bilinear behavior shown in Fig. 1b. The system is subjected to a harmonic base excitation ag(t)¼ z€ ðtÞ ¼Asin(Ot) where A is the amplitude of the base acceleration, while O is the driving frequency With reference to Fig. 1a, by assuming as Lagrangian coordinates the relative displacements between the masses and the base, x1 ¼z1  z and x2 ¼z2  z, the stiffness of the nonlinear springs can be represented, Fig. 1b: ( 1 Zi Z 0     i ¼ 1,2 ð1Þ kbil,i ¼ ki ð1H Zi ei Þ, H Zi ¼ 0 Zi o 0 H is the Heaviside function, Z1 ¼x1 and Z2 ¼x2  x1. Therefore in the configuration plane (x1, x2), Fig. 1c, four regions with different stiffness properties can be observed: in regions I and III the system exhibits only one spring with reduced stiffness at a time, whereas in regions II and IV the springs are both damaged or both undamaged, respectively. Denoting by q the vector of Lagrangian coordinates, the following equations of motion in time domain are found:      M q€ þ Dq_ þ K 0 e1 H Z1 K 1 e2 H Z2 K 2 Þq ¼ p sinðOtÞ ð2Þ where M is the mass matrix, D is the damping, K0 is the stiffness matrix of the undamaged system and p collects the forcing amplitudes associated to the imposed acceleration: " # " # " #   m1 0 k1 þk2 k2 k2 k2 k1 0 M¼ , K2 ¼ , K0 ¼ , K1 ¼ ð3Þ 0 m2 k2 k2 k2 k2 0 0 " D¼

c1 þc2

c2

c2

c2

# ,

( q¼

x1 x2

) ,

( p¼

m1 A

)

m2 A

Fig. 1. (a) System model; (b) piecewise restoring forces (Z1 ¼ x1; Z2 ¼ x2  x1); (c) discontinuity boundaries in the physical plane.

ð4Þ

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In the following, mass and stiffness ratios will be denoted: m2 ¼ a, m1

k2 ¼b k1

ð5Þ

The nonlinear normal modes will be described in the following section by considering the autonomous undamped system: D ¼0 and p¼ 0 in Eq. (2). 3. Free response and nonlinear normal modes (NNMs) The original definition of nonlinear normal mode given by Rosenberg [1] has been extended over the years [13–23] to deal with: (i) systems affected by damping and gyroscopic forces, (ii) systems in internal resonance, where the coordinates do not vibrate in unison, (iii) non-smooth systems. In the scientific literature, the formulation of nonlinear normal modes based on invariant manifolds applicable to piecewise and damped systems [16,19,20] is available; however, in this paper it is adopted the extended definition that considers any periodic motion of the undamped and autonomous system in which all generalized coordinates vibrate, not necessarily, in a synchronous way and pass through the zeros, not necessarily, simultaneously. In fact, such a definition, which allows for manageable semi analytic solution [3], is the most suitable to be compared to the steady-state response of the forced systems. As shown in [13,24], since the nonlinearities are concentrated at the origin, the NNM frequencies are independent of the energy level as in linear systems. Furthermore, due to the positive homogeneity of degree 1 of the solution with respect to the initial conditions, the trajectories of NNMs with the same frequency but at different energy levels form a set of homothetic curves. Therefore the following investigations will be performed by fixing a certain level of the total energy; in this way initial conditions can be expressed by only one independent parameter, i.e., the periodic free solutions are sought in a one dimensional space of initial conditions. For most nonlinear systems the number of NNMs is at least equal to the number of normal modes of the underlying linear system: we denote these as fundamental NNMs. Depending on the nonlinearity, additional NNMs (superabundant), usually generated by bifurcation mechanisms, are observed. The existence of these modes causes the occurrence of extrapeaks in the frequency response function leading to important nonlinear phenomena such as unexpected resonances, localization and energy transfer due to internal resonances [3,14,25,26]. With reference to Eq. (2), by neglecting the effect of damping and by considering the autonomous system (c1 ¼c2 ¼0, p¼0), when e1 and/or e2 vary, first and second fundamental NNMs may interact: this occurs when the nonlinear frequencies of the system o1 and o2 are nearly commensurate i.e.,: no1 ffi mo2 (n,m integers) generating (n:m) internal resonances. Each internal resonance produces a specific structural change in Poincare´ maps generally leading to new superabundant modes [3]; it is possible to give a closed-form approximated expression for the ratio r between the two fundamental frequencies as a nonlinear function of the damage parameters ei. In fact, the value of r is well approximated by the ratio of the equivalent bilinear frequencies oeqi, obtained by considering that during half period the springs have both reduced stiffness and during the other one are both undamaged:

oeq1 ¼

2o01 oe1 2o02 oe2 , oeq2 ¼ o01 þ oe1 o02 þ oe2

ð6Þ

where o0i and oei are the natural frequencies of the system when both springs are assumed undamaged and damaged, respectively. Moreover, the frequency ratio r(a,b,0,0) of the undamaged system is denoted by r0: r ða, b, e1 , e2 Þ ¼

o2 oeq2 ffi , r ða, b,0,0Þ ¼ r 0 o1 oeq1

ð7Þ

In general, it is possible to approximately characterize the values of e1 and e2 around which a bifurcation, caused by an internal resonance, is expected. To this end, given a and b, we simply consider the level curves of surfaces in Eq. (7) associated to integer or rational values of r: these curves lying on the (e1, e2) plane are denoted as resonant lines. Fig. 2 illustrates the frequency-damage plot and the bifurcations occurring in (n:1) internal resonances for varying e1 and e2 ¼0. The frequency o1 of the first fundamental NNM significantly decreases with e1, while o2 remains almost constant leading to an increase of r which, starting from r0 ¼2.62, can attain integer values 3–5. At internal resonances there is a sequence of higher periodic superabundant NNMs bifurcating from the backbone of the first NNM: these tongues are qualitatively similar to the ones revealed in various smooth nonlinear systems with energy-dependent frequency [22,23]. Each tongue takes place in the neighborhood of (n:1) internal resonances and determines a qualitative change in the trajectory of the first NNM: the evolution of these trajectories (thick lines) is described in Fig. 2b where some characteristic modal shapes (denoted by letters A, B, C, D) are depicted together with the maximum equipotential boundaries (thin closed lines) in the plane (x1, x2). It is worth noticing that, along the backbone of the first mode, a jump after each tongue arises generating an increasing number of inflection points in the modal curves. Some other examples of NNMs obtained in the study of the undamped autonomous systems are the superabundant NNMs, E and F, generated by a (7:2) internal resonance (Fig. 3a): in this case, the first mode loses its stability and bifurcates in a NNM (E) with a period doubling; furthermore a second NNM (F) appears and approaches the unstable NNM as long as the first mode recovers its stability. The bifurcating stable modes E and F exhibit similar shapes but start from initial

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Fig. 2. Evolution of the first fundamental frequency for varying e1 ¼ e; (a) branching behavior along the o1 backbone; (b) relevant modal curves: shape A for e ¼ 0.5, shape B for e ¼ 0.55, shape C for e ¼ 0.82, shape D for e ¼0.85.

Fig. 3. Examples of modal shapes: (a) superabundant NNMs generated by a (7:2) internal resonance; (b) period doubling of the second mode.

conditions with opposite signs. Finally Fig. 3b reports the case of a (2:1) internal resonance: here the second mode (G) experiences a period doubling bifurcation becoming unstable; consequently a stable NNM (H) with double period arises. The influence of damping on these bifurcations and consequently on the forced response will be detailed in the next section.

4. Effect of damping on the system dynamics By forcing a linear damped system with harmonic force with an exciting frequency close to a natural frequency of the system, the trajectories of the steady-state response in the configuration plane resemble very closely the corresponding linear normal mode. When dealing with nonlinear systems, the scenario becomes far more complicated, the correspondence between the number of modes and degrees of freedom is lost, and it is well known [2] that resonance frequencies increase in number and spread over the spectrum because: (a) superabundant modes arise with their own frequencies; (b) each nonlinear normal mode can be also excited at sub and superharmonic frequencies. The damping is modelled as viscous dissipation and, for each set of numerical simulations a damage-independent viscous damping is considered and measured by the modal damping of the undamaged system d. As the damage increases, modal damping loses its physical meaning, but it still gives a tangible measure of its relative amplitude useful for understanding its influence on the phenomena presented hereafter. In fact, within the same range of damping values, nonlinear normal modes can behave very differently, from being unaffected up to completely disappear. The forced response of the system is computed by numerically integrating the equations of motion, Eq. (3). The integration scheme is the 4th order Runge–Kutta with constant time step that is set to one order of magnitude lower than that for which independence of the solution on the step size is reached and numerical instabilities are avoided. Fig. 4a shows the response amplitude of x2 when the system is excited by harmonic base excitation for a varying level of the reduced stiffness and for a small viscous damping (0.001 Ns/m). Considering the undamaged system with r0 ¼2.62, the dissipation of 0.001 Ns/m amounts to a modal damping d1 ¼0.03% for the first mode. In Fig. 4, the x-axis reports the damage e1 whereas the y-axis refers to the excitation frequency O; each colour refers to the amplitude x2 of the steadystate solutions starting from natural initial conditions scaled to an unitary maximum. The qualitative inspection of the figure reveals that the response produces the branches which highlight the structure of the underlying undamped system as becomes evident by comparing Fig. 4a to Fig. 2a. By considering the effect of larger damping in the system (c1 ¼c2 ¼0.05 Ns/m, d1 ¼1.5%), the frequency-damage plot becomes that of Fig. 4b. The maximum amplitude of the response is reduced by a factor of about 50 and a widening of the high responding zone is observed; moreover, in correspondence of each bifurcation occurring at the internal resonances, a soft transition takes the place of the discontinuity shown in Fig. 4a. Moreover, in order to detail the different effects that damping has on the dynamics and

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Fig. 4. Response amplitude x2 as a function of damage parameter e and forcing frequency (r0 ¼2.62); (a) low damping c¼ 0.01 Ns/m; (b) high damping, c ¼0.05 Ns/m.

Fig. 5. Forced response around the (3:1) internal resonance; (a) damping c¼ 0.001 Ns/m; (b) c ¼0.01 Ns/m; (c) c ¼0.05 Ns/m; (d)–(f) operating modal shapes.

the harmonic forced response of the system, specific examples are reported in the following, recalling the other bifurcations presented for the undamped autonomous system at the end of Section 3.

4.1. Damping-induced smooth transitions In analogy with linear systems, the damping smoothes the system response avoiding abrupt changes in the observed dynamics and it causes the forced response to loose the synchronous behaviour; therefore the trajectories of the steady state response in the configuration plane, called operating modal shapes, exhibit loops. Fig. 5 presents the results of the numerical simulation when the frequency ratio of the linear system is r0 ¼ 2.62; the damage parameter e1 ranges between 0.45 and 0.75, e2 ¼0, and the forcing frequency sweeps the range between 0.49 rad/s and 0.53 rad/s, that corresponds to the range around o10. As the damage e1 reaches the value 0.54, o1 approaches o2/3 and the internal resonance (3:1), characterized by a cyclic fold bifurcation, generates a superabundant NNM, meaning that the corresponding undamped system presents another periodic stable solution in addition to the first and second NNMs. By inspecting Fig. 5a, characterized by a very low damping (0.001 Ns/m, d1 ¼0.03%), the overall system behaviour strictly reflects the underlying nonlinear dynamics of the undamped system. Moreover, by plotting in the configuration plane the steady state solution of the system for points along the resonance traces (e.g., the points A and B in Fig. 5a), one obtains trajectories that resemble very closely the corresponding NNM A and B of the undamped system (Fig. 2).

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By increasing the damping to 0.01 Ns/m (d1 ¼0. 3%) the overall system behaviour does not show any qualitative changes and the bifurcation can still be detected by the discontinuity in the color map shown in Fig. 5b. At this level of damping, the operating modal shapes maintain their resemblance with the corresponding NNM, but synchronicity is lost and loops appear (Fig. 5d and e). In Fig. 5c, with larger damping (0.05 Ns/m, d1 ¼1.5%), the tongue bifurcating from the main branch fades, and the transition between the forced mode shape A to mode shape B becomes smooth. In particular, moving along the resonance trace, the shape A gradually morphs into the shape B through the trajectory A þB. This behaviour has been observed at every (n:1) internal resonance along the main branch of the first NNM of the system and, together with other peculiar characteristics, at the (n:2) internal resonances. 4.2. Damping-independent bifurcations The system at hand, when the initial frequency ratio is r0 ¼1.95, exhibits, for a large range of e1 and e2, the destabilization of the second mode through a period doubling occurring at the internal resonance (2:1). The damping has a peculiar effect on this kind of bifurcation that is detailed through the particular case characterized by 0.1o e1 o0.7 and e2 ¼ 0. The harmonic base excitation has a frequency that ranges between 1.1 rad/s and 1.35 rad/s that corresponds to the range around the second frequency of the undamaged system, 0.86o20–1.05o20. As the damage increases, 2o1 approaches o2 and the second mode destabilizes. This is qualitatively detected by a discontinuity along the resonance trace in Fig. 6a. The corresponding operating modal shapes, before and after this discontinuity for a very low level of damping 0.001 Ns/m (d2 ¼0.08%), resemble very closely the corresponding NNM G and H of the undamped system (Fig. 6a). By increasing the damping level to 0.01 Ns/m (d2 ¼0.8%) and then to 0.05 Ns/m (d2 ¼4%) (Fig. 6b and c, respectively), only small qualitative changes can be detected. In particular, the stability loss of the second mode continues to occur, even for large value of the damping coefficient. Consequently, discontinuity along the resonance trace is still present, characterized by a sudden change in modal shape. The damping however, still causes a loss of synchronicity; therefore the damped operating modal shapes Gn and Hn show the characteristic loops. As it can be seen also from the colormap plot in Fig. 6c, even for large damping, a discontinuity occurs in the frequency-damage plane: the boundary between the two dynamic behaviours of this system is not affected by the increase in the damping level. 4.3. Damping-dependent bifurcations A different damping dependence is exhibited by the NNMs bifurcating at the (n:2) internal resonances. These NNMs, for low level of damping, are excitable by harmonic forces, but, as the damping increases, the underlying bifurcation ceases to exist and consequently the superabundant NNMs generated by the bifurcation disappear from the forced response. This

Fig. 6. Forced response around the (2:1) internal resonance; (a) damping c¼ 0.001 Ns/m; (b) c¼ 0.01 Ns/m; (c) c ¼0.05Ns/m; (d), (e) operating modal shapes.

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Fig. 7. The (7:2) internal resonance; (a) forced response for the damping c ¼0.001 Ns/m; (b) operating modal shape E þF.

Fig. 8. Forced response at the (7:2) internal resonance; (a) damping c ¼0.005 Ns/m; (b) c¼ 0.01 Ns/m; (c) c¼ 0. 05 Ns/m.

is illustrated by the particular case of the internal resonance (7:2), when the initial frequency ratio is r0 ¼2.62 and the damage parameter e1 ranges between 0.750 and 0.774 (e2 ¼0). For the autonomous undamped system, when e1 approaches 0.75, the bilinear frequencies o1 and o2 are close to the 7:2 ratio: the first NNM B (Fig. 2) becomes unstable, and a superabundant NNM E (Fig. 3a) arises with frequency o1/2. By further increasing e1, a second superabundant mode F (Fig. 3a) occurs with frequency o2/7. The NNM E disappears when its frequency equals o2/7 and mode F coalesces with the first mode when its frequency equals o1/2. Fig. 7a shows the forced counterpart of the autonomous branching when a small amount of damping is considered, 0.001 Ns/m. In particular, by exciting the systems at o1 frequency, the forced behavior shows the destabilization of the NNM C, according to the autonomous dynamics of the undamped system. By following the fundamental resonance trace, it is possible to detect the discontinuities in the response amplitude. Across each discontinuity, by plotting the steady state solution in the configuration plane, it is possible to detect a sudden change of the operating modal shape from one resembling NNM C to one similar to NNM E. By increasing the damage, the boundary between the NNM E and the NNM F is reached and, by further increasing the damage parameter e1, the NNM C regains its stability while the NNM F fades. It is here important to notice that, while the boundaries between C and E and between F and C, due to the destabilization of mode C, are independent of the initial conditions, the boundary between E and F changes as the initial conditions change because, in this frequency range, both E and F are persistent NNMs, and their basins of attraction are competing. Fig. 8 shows the strong influence of damping on this bifurcation: by increasing c from 0.001 Ns/m (d1 ¼0.03%) up to 0.005 Ns/m (d1 ¼0.15%), Fig. 8a, the behavior remains qualitatively unchanged, it is still possible to detect along the o1 resonance trace the transition from C to E, then to F and back to C, however, loops appear in the operating modal shapes because of the damping. Similarly to the case of the (n:1) internal resonance, once the damping level reaches 0.01 Ns/m (d1 ¼0.3%), it is no longer possible to excite the E and F modes, Fig. 8b, they merge presenting the E þF shape, Fig. 7b. The peculiar behavior of the (n:2) internal resonance appears by further increasing the damping level; when c 40.03 Ns/m (d1 40.9%) which is still a moderately low damping for civil and mechanical structures, both mode E and F disappear from the forced response, because mode C remains stable and avoids mode E and F to be directly excited, Fig. 8c.

4.4. General case In order to present a general overview and to verify the consistency of the observed nonlinear behaviours in the forced response of the system, a general numerical exploration in the (e1,e2) plane is performed. For each point (e1,e2), the frequency of the first NNM is computed analytically, and a numerical integration of the system harmonically excited at

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Fig. 9. Forced response as both e1 and e2 varies. The exciting force follows the first bilinear natural frequency; (a) theoretical resonance lines; (b) damping c¼ 0.005 Ns/m; (c) c ¼ 0.05 Ns/m; d) c¼ 0.08 Ns/m.

that frequency starting from natural initial condition is carried out. The amplitude of the steady-state solution of the x2 Lagrangian coordinate is displayed as colormap for varying damping levels, Fig. 9b–d. This analysis is able to detect the occurrence of each bifurcation departing from the main branch of first NNM that is detected by the corresponding change in response amplitude. These plots are also amenable for comparison with the resonant lines, obtained from the semi-analytical model [3] and recalled in Fig. 9a, that trace on the (e1,e2) plane all the occurrences of superabundant modes branching from the first natural frequency. By comparing the resonant lines of Fig. 9a with the forced response in Fig. 9b, obtained for a small damping level c¼ 0.001 Ns/m (d1 ¼0.03%), it is clearly possible to detect the presence of the (n:1) resonance lines, as well as each (n:2) resonance. It is here interesting to recall that, also in the case of very small damping, no detection is possible for the general (n:m) resonant lines when m42. In fact, these are ghost NNMs and therefore do not appear in the forced harmonic response [5]. As the damping level increases, even the conditionally-persistent NNMs disappear; in particular, once c 40.03 Ns/m (d1 40.9%) the (7:2) resonant lines cease to exist, while the (5:2) ones vanish when c exceeds 0.08 Ns/m (d1 ¼2.4%) leaving only the (n:1) resonant lines.

5. Conclusions The forced response to harmonic base excitation of a two degrees of freedom piecewise linear system is investigated by means of numerical simulations. The capability of obtaining the nonlinear characteristics of the system from the forced response is investigated as it would occur in the real world and compared with the theoretical results of the corresponding autonomous undamped system that allows for semi-analytic solutions. Within this ambit, the role played by the damping is addressed reporting a variety of effects which may limit the possibility of a complete description of the system dynamics from the forced response. It is important to point out that the level of damping considered in this study is quite low, in the range between 0.03% and 1%, meaning that the dynamics of the corresponding linear system shows frequency response functions with sharp

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peaks at the natural frequencies. On the contrary, in the same range of damping values, the bifurcations of the nonlinear system can be largely affected by the dissipation, possibly suppressing the corresponding NNMs. Similarly to linear systems, the damping lowers the amplitude of the forced response and widens the resonance peaks. By looking at the operating modal shapes in the configuration plane, the damping also causes the modal lines to create loops. In correspondence of the bifurcations, the general effect of the damping, for values around 0.05%, is a smoothing of the transition from one dynamic behaviour to another that occurs with a continuous morphing of the modal lines and, consequently, discontinuities disappear from the forced response of the system. This kind of behaviour is common to all the (n:1) and (n:2) bifurcations of the bilinear system at hand, with the only exception of the (2:1) internal resonance. This bifurcation is characterized by the destabilization of the second mode and by a sudden change in the modal shape which occurs even for large damping values thus, for this mode, the damping does not provide the expected smoothing effect. The NNMs that occur at (n:2) resonances exhibit, beyond the smoothing effect typical of the (n:1), a stronger dependence on the damping. In fact, as a specific value is reached the resonant peaks due to these modes disappear from the forced response as it is revealed by the disappearing of the resonant lines in the damage plane. In any case, regardless the level of damping it is not possible to excite all the other superabundant NNMs of the system and thus the corresponding resonant lines do not appear in the forced response. In conclusion, while the undamped autonomous system exhibits a very rich scenario of its dynamic properties, a small damping strongly reduces this peculiar behavior, suppressing some bifurcations and, consequently, regions of resonance.

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