Mode localization

When nonlinearities are present in the model, with energy input into the system is in- creased that energy will tend to confine to a part of the structure, hence those particular coordinates vibrate with more energy –amplitudes– than other parts of it.

Figure 2.17: In both NNMs shown, one of the dofs –the first mass– oscillates with a greater amplitude than the other dof –the second mass–.

2.5.3.2 Internal resonances

When the harmonics of a mode resonate with another NNM those harmonics can sustain an increase in energy and become dominant. This is called ’internal resonance’ and leads to the bifurcation of new NNMs branches or modal backbones. Internal resonance can also break the synchronicity of the oscillations of the degrees of freedom if the dominant harmonics are confined to a few system dofs.

Figure 2.18: One of the NNMs whose time series has been displayed on this figure has a dof oscillating with dominant frequency twice the fundamental frequency of the

2.5.3.3 Ramification of modes

The ramification of the NNMs branches is a result of internal resonances and symmetry- breaking bifurcations. New branches of NNMs –possible periodic solutions of the nonlinear system– could emerge from points on the main NNMs branches. Those particular points can be labeled ’BP’ in most bifurcation diagrams, which stands for ’branching point’.

Figure 2.19: This diagram shows two bifurcation points from one of the two main NNMs branches. A new modal backbone branches out from one of them and reengages

on the main curve on the following point.

All NNMs branches can be labeled with the following notation. It coincides with the convention taken in the vast majority of the bibliography consulted regarding NNMs, including [2], [12] and [13]:

Consider this template of a NNMs branch label: Smn+ The first letter represent the symmetry of the modes

• S: symmetrical mode. • U: unsymmetrical mode.

• m: number of half waves in the oscillation of the nonlinear attachment of the system between t = 0 and t = T₂.

• n: number of half waves in the oscillation of the linear part of the system between t = 0 and t = T₂.

The sign at the end can be a plus sign if the linear and nonlinear parts of the system oscillate in-phase, and a negative sign if they oscillate out-of-phase.

2.5.3.4 Turning points

In Bifurcation Theory, a turning point –also called ’fold’– is a point where a curve changes direction with respect to the curve parameters, i.e. a point where the curve steepness tends to infinity before it turns backwards. Turning points are very important in Bifurcation theory because they emerge for a particular type of bifurcation called ’tangent bifurca- tion’: they give rise to a pair of new solutions for certain values of the parameters. For our particular problem, turning points are those where the curve changes direction with respect to the total energy.

Figure 2.20: This figure shows some turning points tagged by blue arrows. On those points the curve reverses course with respect to the total energy in either direction.

2.5.3.5 Symmetry breaking

A nonlinear normal mode is said to be ’symmetrical’ if one half of the oscillating motion of a NNM through its period of minimum periodicity ’mirrors’ the motion of the other half of it, and ’asymmetrical’ in the other way round. Points on the MMNs branches where symmetry changes are called ’symmetry-breaking points’.

Figure 2.21: One of the NNM displayed on the time series plots is unsymmetrical, while the other one shows a symmetrical motion. More precisely, the whole branch of NNMs that sprouts from the main branch is one of unsymmetrical nonlinear normal

modes.

A symmetry-breaking point is a bifurcation point, thus involving a stability change.

2.5.3.6 Stability changes

Unlike linear modes, nonlinear normal modes could be stable or unstable. NNMs can be displayed on the FEP on different colours depending on their stability.

Points of change in stability on the modal backbones will always coincide with branching points, turning points and symmetry-breaking points.

NUMERICAL CONTINUATION

ALGORITHM FOR THE

COMPUTATION OF NONLINEAR

NORMAL MODES

From this chapter, the will divert our attention to numerical algorithm to compute the NNM branches and, consequently, the frequency-energy plots of conservative mechanical systems with nonlinear stiffnesses. This technique is based on ’sequencial continuation’, a method that is introduced in the next sections. The reader is presumed a knowledge of basic concepts of differential equations and dynamical autonomous –without an explicit dependence of time– systems.

The most important references considered to write this chapter are conference proceedings [21], [20] and [19], where the shooting algorithm with sequencial continuation is described in great detail.

3.1

Continuation methods: the concept

Briefly defined, ’continuation methods’ are a set of numerical algorithms with the potential to compute a sequence of solutions of equations that have a dependence on an indepen- dent parameter: f (z, λ) = 0. The independent parameter λ is usualy called ’homotopy variable’. The computation begins with an initial solution z0 for a particular parameter

value λ0 as the initial value.

Continuation methods originally emerged in the field of Nonlinear Dynamics and Non- linear Equations in order to compute how the position and stability of the equilibrium points of a dynamical system –or the solutions of an algebraic equation– varied with the value of a particular parameter. These techniques have the potential of approximating the branches of all the possible solutions of a system from a skeleton of sparsed discrete points in one single run of the algorithm. Other by-products from the computation are the points where the stability of the fixed points –if it is applied on a differential equation or SODE– change, or where new branches of solutions emerge. These points are called ’bi- furcation points’ and the results of the computation are usually displayed on plots called ’bifurcation diagrams’.

Another application is the resolution of nonlinear algebraic equations through ’homotopy’: a simpler system of equations with known solution is constructed, and a homotopy pa- rameter is introduced into the equations to connect the equations with known solution to the original system with unknown solution.

There are lots of bibliographic references dedicated to numerical continuation algorithms. Most of them describe it in the context of bifurcation analysis of dynamical systems or for algebraic equation solving with homotopy. A classic on numerical continuation can be found on [35], while [36] is a good introductory text to bifurcation and stability analysis, homotopy and continuation methods.