Paradoxes in dynamic stability of mechanical systems: investigating the causes and detecting the nonlinear behaviors
 Angelo Luongo^{1}Email author,
 Manuel Ferretti^{1} and
 Francesco D’Annibale^{1}
Received: 2 September 2015
Accepted: 6 January 2016
Published: 21 January 2016
Abstract
A critical review of three paradoxical phenomena, occurring in the dynamic stability of finitedimensional autonomous mechanical systems, is carried out. In particular, the wellknown destabilization paradoxes of Ziegler, due to damping, and Nicolai, due to follower torque, and the less well known failure of the socalled ‘principle of similarity’, as a control strategy in piezoelectromechanical systems, are discussed. Some examples concerning the uncontrolled and controlled Ziegler column and the Nicolai beam are discussed, both in linear and nonlinear regimes. The paper aims to discuss in depth the reasons of paradoxes in the linear behavior, sometimes by looking at these problems in a new perspective with respect to the existing literature. Moreover, it represents a first attempt to investigate also the postcritical regime.
Keywords
Background
There exist several paradoxes in mechanics; the most amazing may occur in stability analysis. A celebrated problem is the ‘Ziegler Paradox’ (Ziegler 1952; Beck 1952; Bolotin 1963; Herrmann and Jong 1965; Herrmann 1967; Leipholz 1964; Plaut and Infante 1970; Plaut 1971; Walker 1973; Hagedorn 1970; Banichuk et al. 1989; Kounadis 1992), also known as the ‘destabilizing effect of damping’ , according to which a given small damping has a detrimental effect on the stability of linear circulatory systems. The Ziegler column, i.e. an upward double pendulum loaded at the tip by a follower force, represents a discrete mechanical prototype of this paradox. A geometrical explanation of the phenomenon, based on the existence of the ‘Whitney’s umbrella’ surface (Whitney 1943), was given by Bottema (1955, 1956). More recently, Seyranian and Kirillov, according to the singularity theory by Arnold (1983, 1992), gave a justification of the paradox for both general finitedimensional (Seyranian and Mailybaev 2003; Kirillov 2005, 2013; Kirillov and Seyranian 2005) and continuous (Kirillov and Seyranian 2005) systems, based on a perturbation analysis of the eigenvalues at the (known) double Hopf bifurcation point of the circulatory system. Differently from this latter approach, a perturbation algorithm was developed in Luongo and D’Annibale (2014, 2015) and Andreichikov and Yudovich (1974), where the starting point of the asymptotic analysis is an unknown, marginally stable, subcritical undamped system.
A less known, but equally surprising phenomenon, is the ‘Nicolai Paradox’ (Nicolai 1928, 1929; Bolotin 1963). It concerns a cantilever beam, of equal moments of inertia, loaded at the tip by a follower torque, which causes dynamic instability of the trivial equilibrium at a critical value equal to zero. The amazing phenomenon has been recently reconsidered by Seyranian and Mailybaev (2011), who, according to the singularity theory Arnold (1983), proved that this paradox is related to the bifurcation of a double semisimple eigenvalue, leading to a stability domain with a conic singularity. Moreover, the effects of the pretwist deformation, the damping, an axial dead load and the compressibility of the beam have been deeply analyzed in Seyranian et al. (2014), Luongo et al. (2014) and Seyranian and Glavardanov (2014). An extension of the problem to secondorder perturbations was also performed in Luongo and Ferretti (2014).
A third paradox, recently discovered by the authors of the present paper, concerns the stability of autonomous piezoelectromechanical (PEM) systems in the presence of nonconservative (positional) actions (D’Annibale et al. 2015). It has been proved that the socalled ‘similarity principle’ (see, e.g., Alessandroni et al. 2004, 2005; Andreaus et al. 2004; dell’Isola et al. 2003a, b, 2004; Maurini et al. 2004; Porfiri et al. 2004; Rosi and Pouget 2010; Alessandroni et al. 2002), which usually works in controlling vibrations of externally excited (i.e. non autonomous) systems, has instead detrimental effects on the occurrence of dynamic bifurcations. Said in other words, the connection of a similar piezoelectric system to a mechanical one, the former duplicating the whole spectrum of the eigenvalues of the latter, which would supply a complete protection from any excitation frequency, is indeed detrimental in terms of stability.
In spite of a wide literature existing on paradoxical linear systems, to the authors’ knowledge, really few studies concerning the postcritical analysis have been carried out (see, e.g., Hagedorn 1970; Thomsen 1995; O’Reilly et al. 1996). The main question to be answered is the following: does the paradoxical loss of stability predicted by the linear theory really lead to motions of large amplitude when nonlinearities are accounted for? In other terms, since the amplitudes of motions depend on nonlinearities, the question is to ascertain if nonlinearities are able or not to limit the amplitudes within values smaller than a certain tolerance. A comprehensive answer, of course, would require an indepth study which is beyond the scope of this work; anyway, a first attempt in this direction is made here by limiting ourselves to a numerical investigation.
The scope of this paper is twofold. First, we want to frame our previous results on linear stability analysis of paradoxical systems, so far developed independently, in a unique organic context; second, we want to illustrate some original, although so far limited, results concerning nonlinear behavior. To these ends, the above mentioned phenomena are reviewed for a class of finitedimensional mechanical systems. The reasons of the paradoxes are explained by recalling asymptotic expansions of the eigenvalues of the tangent operator previously developed in the literature. Preliminary results concerning the nonlinear behavior are obtained via numerical analyses, directly carried out on the equations of motion of prototype systems. The limitcycle which arises after the occurrence of (simple or semisimple) Hopf bifurcations, is determined, and the influence of the main parameters is studied. It is wished that the studies presented here will stimulate also experimental activity, necessary to validate the theoretical predictions.
The paper is organized as follows. In second section, the model of a class of finitedimensional mechanical system, suffering the paradoxes discussed above, is introduced. In third and fourth sections, the Ziegler and Nicolai paradoxes are investigated, respectively. In fifth section, the failure of the ‘similarity principle’ is discussed. In sixth section, some conclusions are drawn. Finally an “Appendix” furnishes details.
The model
Uncontrolled systems
Controlled systems
The trivial equilibrium position \({\mathbf{x}}={\mathbf{y}}=\mathbf{0}\) of system (3) can lose stability via the mechanism illustrated in Fig. 1c, as discussed ahead.
The Ziegler paradox
This section is devoted to recall the wellknown Ziegler paradox, with the aim to highlight some important aspects both in the linear and nonlinear regimes.
Linear analysis
Let us first consider the linearized equations (1), with the aim to discuss the bifurcation mechanism occurring in the paradox. When the system is undamped (also referred to as circulatory), i.e. \({\mathbf{C}}=\mathbf{0}\), and \(\mu =0\), the two pairs of complex conjugate eigenvalues lie on the imaginary axis so that the system is (marginally) stable. If \(\mu\) is increased from zero, the eigenvalues move on the imaginary axis, still remaining distinct (see Fig. 1a), until the load reaches a critical value, namely \(\mu =\mu _{c}\), at which they collide and a circulatory (or reversible) Hopf bifurcation takes place; if an infinitesimal increment \(\delta \mu >0\) is given, they separate and instability occurs. The load value \(\mu _{c}\) is the critical load of the circulatory system.
When the system is damped, namely \({\mathbf{C}}\) is positive definite, there exists a critical load \(\mu _{d}\), that is the smallest \(\mu\) at which an eigenvalue (together with its complex conjugate) crosses from the left the imaginary axis, (see Fig. 1b) and a simple Hopf bifurcation occurs. When the damping is sufficiently small, \(\mu _{d}<\mu _{c}\), as it has been show in several contributions in the literature (see, e.g., Ziegler 1952; Bolotin 1963; Herrmann and Jong 1965; Seyranian and Mailybaev 2003; Kirillov and Verhulst 2010; Kirillov 2005).
Finally, it is important to remark that the asymptotic procedure recalled above has not to be regarded as a mere perturbation algorithm, since it is able to explain the true essence of the paradox, that is: when a generic damping matrix is added to an undamped circulatory system in subcritical regime, modes that would be marginally stable can become incipiently unstable. In this perspective, no discontinuities appear in the damped system with respect to the undamped one; thus, the apparent discontinuity of the amazing paradox is a consequence of a wrong point of view, in which the damped system is compared with the unique critically loaded undamped system, instead that with the infinitely many subcritically loaded members of the undamped family.
Postcritical behavior

case study I: \(\xi _{1}=0.016, \xi _{2}=0.1\), entailing \(\mu _{d}\simeq 0.65\), for which damping has a strong destabilizing effect (−69 %);

case study II: \(\xi _{1}=0.081, \xi _{2}=0.06\), entailing \(\mu _{d}\simeq 1.63\) for which damping has a moderate destabilizing effect (−22 %).
It is observed that, even when the bifurcation parameter slightly exceeds the critical value \(\mu _{d}\), the column manifests large amplitude limitcycles. This is due to destabilizing effect of damping which persists also in the postcritical regime. As a matter of fact, if we consider, for example, an increment of the load with respect to the critical value, \(\delta \mu :=\mu \mu _{d}\) equal to \(\delta \mu = 0.1,\) we found: \(\max \left \vartheta _{1}\right \simeq 0.75\) rad, \(\max \left \vartheta _{2}\right \simeq 1.12\) rad in case I, and \(\max \left \vartheta _{1}\right \simeq 0.37\) rad, \(\max \left \vartheta _{2}\right \simeq 0.72\) rad in case II. Remarkably, we can conclude from this example that, for the same increment of the load, the higher the destabilizing effect on linear stability, the higher the amplitude of the limitcycle occurring in the postcritical regime.
Finally, the same Figs. 4 and 5 show the results furnished by the Harmonic Balance Method applied to the two case studies. It is observed that, when the first harmonic is considered (points represented by small circles in the figures), the approximation of the exact results is good only in the case study II, while it worsens when the destabilizing effect of damping is significant (case study I). When instead the first and the third harmonics are considered (points represented by small triangles in the figures) the approximation of the exact results is excellent in both the case studies.
The Nicolai paradox
It is important to remark that the continuous model from which the discretized equations of motion have been derived, has been formulated by modeling the Nicolai beam as a onedimensional polar continuum, geometrically nonlinear and internally constrained. In particular, the constraints are the unshearability, the inextensibility and the untwistability. The first two are commonly used in the modeling of beams while the third one is based on an analysis of the orders of magnitude of the energy contribution of the underlying elastic model, according to Luongo and Zulli (2013). Once the kinematics is established, the partial integrodifferential equations of motion are derived with the methods presented in Paolone et al. (2006) and Luongo and Zulli (2013) and expanded up to cubic terms.
Linear analysis
By using the discretized system introduced above, we want to highlight the bifurcation mechanisms which guides the Nicolai paradox. When the follower torque \(\mu\) is equal to zero (i.e. the system is Hamiltonian) and the two inertia moments are equal (i.e. \(J_{1}=J_{2}=1\)), the system admits a couple of purely imaginary coincident eigenvalues, see Fig. 1c; moreover, they are semisimple, since two independent (real) eigenvectors, describing the same modal shape in the two planes, are associated with each of them. When the Hamiltonian system is loaded by small nonconservative forces, the two coalescent eigenvalues split on opposite parts of the complex plane (Seyranian and Mailybaev 2011; Seyranian et al. 2014; Luongo et al. 2014; Seyranian and Glavardanov 2014), see Fig. 1c, thus entailing instability; the presence of a small asymmetry, that we label with a parameter \(\alpha\), is able to shift the critical load of a small amount only (Seyranian and Mailybaev 2011; Seyranian et al. 2014; Luongo et al. 2014; Seyranian and Glavardanov 2014).
Postcritical behavior
 1.
The system manifests a paradoxical behavior also in nonlinear regimes: indeed, starting with a couple of parameters \(\left( \alpha ,\mu \right)\) that belongs to the unstable zone (the white one in Fig. 7), the system reaches, after a transient motion, a circular trajectory in the space of configuration variables, i.e. \(\left( x_1\left( t \right) ,x_2\left( t \right) \right)\), whose amplitude is large and independent from the selected numerical parameters (see Fig. 8a); this circular motion is also unaffected by the choice of the initial conditions.
 2.
Once the large circular motion has been reached, the system increases its velocity unboundedly (see Fig. 8b). Therefore, the beam whirls with an increasing velocity, experiencing a conical motion.
 3.
The only effect played by the initial conditions and by the value of \(\left( \alpha ,\mu \right)\) concerns the time requested by the system to reach the circular motion.
The failure of the ‘similarity principle’
This section is devoted to detect the failure of the ‘similarity principle’ which occurs in linear and nonlinear behavior of PEM systems whose motion is described by equations in the form of Eq. (3). To this end, we will take as a prototype the piezoelectriccontrolled Ziegler column (Ziegler 1952) depicted in Fig. 2b. The mechanical system is that analyzed in “The Ziegler paradox” section with the reference to the discussion of the Ziegler paradox. Moreover, it is equipped with two piezoelectric devices indicated with the symbols \(\mathrm {Pz}_{1}\) and \(\mathrm {Pz_{2}}\) in the figure, respectively: these latter are of capacitances \(C_{1}^{P}:=2C_{P}, C_{2}^{P}:=C_{P}\), equal stiffnesses \(k_{1}^{P}=k_{2}^{P}:=k_{P}\) (in the following we will considered this stiffness negligible with respect to the stiffness of the springs), equal coupling coefficients \(g_{1}^{P}=g_{2}^{P}:=g\). Piezoelectric devices are placed in the correspondence of the ground and of the intermediate hinges, respectively, and each of them is connected to a joint of a twonodes active circuit (sketched in the Fig. 2b), of equal inductances \(L_{1}=L_{2}:=L\) and resistances \(R_{1}\) and \(R_{2}\), respectively, and to the ground.
Linear analysis
Let us first consider the linearized equations (3), with the aim to discuss the bifurcation mechanism occurring in this paradox. First, we neglect the electromechanical coupling, by letting \(\gamma =0\). The similarity principle then entails that the mechanical (primary) and electrical (secondary) subsystems possess the same spectrum of the eigenvalues: if an eigenvalue is simple for the primary (or the secondary) subsystem taken alone, it is semisimple for the whole PEM system. When the load \(\mu\) reaches a critical value \(\mu _{d}\), that is the smallest \(\mu\) at which a mechanical (or electrical) eigenvalue (together with its complex conjugate) crosses from the left the imaginary axis (see Fig. 1b), a simple Hopf bifurcation occurs for the primary (or the secondary) subsystem.
When the electromechanical coupling is accounted for, i.e. \(\gamma >0\), and the load is kept fixed at \(\mu =\mu _{d}\), the bifurcation mechanism occurring in the PEM system is analogous to that of Nicolai, namely, the semisimple eigenvalues split on the opposite part of the complex plane, thus entailing instability (Fig. 1c). What it is surprising in this paradox is that a vanishingly small gyroscopic coupling, which is introduced with the aim to control the mechanical system, i.e. to increase its critical load, produces instead instability (D’Annibale et al. 2015). Said in other words, the splitting, which represents the most valuable beneficial effect brought by an added device in the classical Den Hartog oscillator under external excitation, is, indeed, cause of instability in the autonomous nonconservative case.
It is possible to show the detrimental effect of the gyroscopic coupling also via a perturbation method, by following the lines of D’Annibale et al. (2015), in which the first sensitivity of the semisimple eigenvalue of a general continuous PEM system is determined, when a small coupling acts as a perturbation. By keeping fixed the load at \(\mu =\mu _{d},\) the semisimple eigenvalue \(\uplambda _{0}\) at bifurcation splits according to Eq. (11), namely \(\uplambda ^{\pm }=\uplambda _{0}\pm i\gamma \sqrt{b_{1}b_{2}}\), where \(b_1,\,b_2\) are coefficients depending on the right and left eigenvectors of the uncontrolled subsystems and on the gyroscopic matrix (see “Appendix” for details). It is apparent that (a) if the product \(b_{1}b_{2}\) is complex or real and negative, one of the two roots has positive real part, thus entailing instability; (b) if the product \(b_{1}b_{2}\) is real and positive, the two roots are purely imaginary, so that \(\gamma\) is neutral at the first order. It is concluded that a similar controller has a detrimental (or at most neutral) effect on stability.
Postcritical behavior
Preliminary results concerning the postcritical behavior of the controlled Ziegler column are discussed in this section, with the aim to investigate the effects of the controller on the large amplitude limitcycles occurring in the uncontrolled case (see “Postcritical behavior” section referred to the Ziegler paradox). To this end, we directly integrated the nonlinear equations of motion (3) in the case studies I and II (marked in Fig. 3b), i.e. we selected the same two damped systems discussed with the reference to the postcritical behavior of the uncontrolled Ziegler column.
A comparison between the time histories of the components of motion \(\vartheta _{1}\) and \(\vartheta _{2}\), in uncontrolled and controlled (\(\gamma =0.05\)) systems, relevant to the case study I, when \(\mu =0.68\), is presented in Fig. 10. It is seen that, for small increments of the load with respect the critical value, i.e. \(\delta \mu =0.03\) in the uncontrolled case and \(\delta \mu =0.05\) in the controlled one, the limitcycle of the PEM system is stable, even if its amplitude (displayed in light gray in Fig. 10) is larger than the amplitude of the uncontrolled column (displayed in dark gray in the figure). Thus, the similar controller increases the amplitude of the limitcycle of the uncontrolled Ziegler column, causing a detrimental effect.
An even more dangerous situation is illustrated in Fig. 11, in which the time histories of \(\vartheta _{1}\) and \(\vartheta _{2}\), in uncontrolled and controlled \((\gamma =0.01)\) systems, relevant to the case study II, when \(\mu =1.66\), are plotted. Indeed, for small increments of the load with respect the critical one (of the same order of the previous case, i.e. \(\delta \mu =0.03\) in the uncontrolled system and \(\delta \mu =0.1\) for the controlled one), the PEM system is unstable and the time histories of \(\vartheta _{1}\) and \(\vartheta _{2}\) diverge in time. Accordingly, the numerical integration has been truncated in the figures, just before this event. In contrast, the uncontrolled column experiences a stable limitcycle. Thus, in this case, the similar controller has a catastrophic effect in the postcritical regime of the controlled Ziegler column.
Conclusions
In this paper, some amazing paradoxical phenomena, well and lessknown in the literature, concerning linear dynamic stability of mechanical systems, have been studied referring to finitedimensional prototype systems. Paradoxes concern: (a) the destabilizing effect of damping, or Ziegler paradox; (b) the zero critical value of the load, or Nicolai paradox; (c) the failure of the similarity principle in controlling stability by piezoelectric devices. For all these problems, an explanation has been given, based on asymptotic expansions of the eigenvalues, started by simple or double and semisimple eigenvalues. In the Ziegler case, a procedure different from that usually adopted in literature (starting from a double and notsemisimple eigenvalue) has been followed, able to reveal the true essence of the paradox.
 1
The Ziegler column experiences stable largeamplitude limitcycles. The more destabilizing the damping, the larger the amplitude of the limitcycle. Therefore, the loss of stability (in the Lyapunov sense), is not a mere mathematical aspect of the problem, but a signal of an incoming dangerous phenomenon from an engineering point of view (more interested in the amplitude of the oscillations than in the quality of the equilibrium).
 2
The Nicolai beam also suffers largeamplitude circular motion in the space of configuration variables, even when a very small follower torque is applied, and, quite surprisingly, irrespectively of the chosen parameters and initial conditions. Even worse, this motion occurs at increasing velocity, diverging to infinite, representing a new paradoxical phenomenon existing in the nonlinear field. This unrealistic result is conjectured to depend on the absence of damping, so far not included in the model. Further investigations are therefore needed also considering the effects of damping, higher modes and twistability.
 3
The PEM Ziegler column, possesses double eigenvalues when a ‘similar’ control system is adopted, requiring an active circuit. This equipment, that previous studies have shown to be optimal in controlling external excitations, is instead detrimental in controlling stability. When the motion is analyzed in the postcritical range, both stable and unstable largeamplitudes limitcycles exist, the former close to bifurcation, the latter far from bifurcation. In both cases, however, the oscillations of the controlled system are larger than those of the uncontrolled system, so that the similar control is detrimental even in the nonlinear field.
Declarations
Authors’ contributions
AL conceived the scientific idea of this paper. FD developed perturbation algorithms relevant to Ziegler paradox and to the failure of the ‘similarity principle’ and MF those relevant to the Nicolai paradox. FD and MF carried out numerical simulations of the paradoxical phenomena. All authors contributed equally to this work. All authors read and approved the final manuscript.
Acknowledgements
This work was granted by the Italian Ministry of University and Research (MIUR), under the PRIN1011 program, Project No. 2010MBJK5B.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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