What nonlinear normal modes –NNMs– stand for

In the last section the dynamics of nonlinear mechanical systems have been discussed, performing first an heuristic computation of the forced response diagrams for 1dof and 2dofs oscillators and then overlaying on the diagrams a particular set of solutions or trajectories of the equivalent free undamped –conservative– system for each model. These sets of solutions had a fundamental property over the rest: the motion of the degrees of freedom in the configuration space was periodic. This particular choice was not made at random, these solutions of the free undamped model had properties that made them very useful. In the one-dimensional case these solutions were all possible solutions in the phase space because there are not other nodes to synchronize to, but in MDOF systems the distinction makes a mathematical sense.

For linear mechanical systems we could also obtain their frequency response curves, but on top of that the interest was on their free undamped response, neglecting damping and excitation. Free response was the linear combination of basis functions that were

for themselves solutions of the conservative model. These particular solutions were called ’normal modes’ and were the product of a vector called ’mode shape’ and a sine wave with a particular frequency called ’natural frequency’. Normal modes were particular solutions of the conservative unforced models but could help to understand the dynamics of both free and forced systems in a way that has been discussed earlier in the text.

From this moment, normal modes of linear systems will be called in this text ’linear normal modes’ or LNMs while normal modes derived from the dynamical analysis of nonlinear conservative systems will be refered to as ’nonlinear normal modes’ or ’NNMs’. In short, nonlinear normal modes in those systems are an analogous concept to linear normal modes in linear conservative systems, but they differ in many of their properties due to the nature of nonlinear dynamics. There are two main definitions for the nonlinear normal modes:

Rosenberg’s definition The first definition was purposed by Rosenberg –[29]–. He defined NNMs as a ’vibration is unison’ or ’synchronized vibration’ where all the nodes vibrate with the same frequencies and with a fixed displacement ratio. In other words, all dofs must reaches extreme values and pass through zero at the same time. In this sense, that synchronism would be a property shared with linear normal modes. However, this definition has two main restrictions: first, it cannot be extended to nonconservative systems –a case that is not a matter of discussion in this text, treating only conservative undamped systems–. Secondly, if two or more MMNs interact and internal resonance takes over in the system some nodes of it can vibrate with different dominant frequency than others, In that case, synchronicity is broken.

In the latter case Rosenberg’s definition can be extended to ’non-necessarily syn- chronous periodic motion of the system’, because in the case of internal resonances the motion is still periodic. This latter definition is the closest to what we had un- derstood as ’NNMs’ in the previous section: the syncronicity, which is characteristic of linear modes, may be lost, but their periodicity remains.

Shaw and Pierre’s definition Another definition that extended the notion of NNMs to damped systems was purposed by Shaw and Pierre –[30]–, this time based on a

geometric approach. They defined an NNM as an two-dimensional invariant mani- fold in the phase space. It is invariant because any orbit that starts in that manifold remains on it at any instant of time. This is a property shared with linear normal modes. The invariant manifold can be parametrized with two state variables called ’master coordinates’ such as the position and the velocity of a particular degree of freedom. The position of any point on that manifold could then be expressed as a function of those two parameters, making it possible to compute any orbits on it. This is the foundation of the ’invariant manifold method’ to approximate NNMs analytically –see section 2.4.3–.

NNM invariant manifolds can be regarded as an extension of LNMs, which are hyperplanes in the phase space due to their invariability and non-dependence on their energy –more in 2.4.2–. A graphical example showing the concept of a NNM invariant manifold along with its linear counterpart has been displayed on figure2.11. It can be noticed that for small energy levels LNMs and NNMs overlap due to the insignificance of nonlinearities for small vibration amplitudes. This property may spur us to define nonlinear normal modes as the nonlinear extensions of the linear normal modes of any system. In fact, the Lyapunov’s Theorem –1909– confirms that any N-dof conservative system with no internal resonances has at least N families of normal modes around the equilibrium point. Internal resonances between modes lead to bifurcations that increase the number of NNMs –see 2.5.3.2–.

A good introductory reference on nonlinear normal modes and nonlinear normal analysis can be found in –[13]–. This conference proceedings –[17] and [18]– and in this web- site: http://www.ltas-vis.ulg.ac.be/cmsms/index.php?page=sicon_tc5. It posts slideshows of lectures given in SICON courses by researchers of the University of Li`ege –Golinval, Vakakis, Kerschen et al– on the topic of Vibration and NNMs.

Figure 2.11: NNM invariant manifold in the phase space –in colour– and invariant manifold of the same linearized mode –in grey–. Both manifolds overlap near the origin, where nonlinearities can be neglected. The LNM manifold is an hyperplane, while the NNM manifold has a curvature in the phase space due to frequency-energy and shape-

energy dependence. This figure was originally shown on reference [17]