Nonlinear Normal Modes for nonlinear modal analysis

sis

Normal modes analysis is central to the understanding and the simulation of vibrating systems. In classical linear theory, Linear Normal Modes (LNMs) can be used to uncouple equations of motion and are closely associated to the property of linear superposition. Thanks to this, the LNMs are exploited for various purposes such as modal reduction, finite element model, experimental vibration analysis.

Linear modal theory is based on multiple properties of LNMs. The invariant property insure that modes shapes and frequency associated to the LNMs do not change in time. The orthogonality property allows any free or forced vibration to be expressed as a linear superposition of LNMs. Another interesting property of LNMs is their unconditional stability. However, the properties of superposition and unconditional stability are not conserved in nonlinear systems. Thus, the generalization of LNMs to NNMs is needed and several definitions of NNMs have been proposed in order to extend the concept of normal modes to the nonlinear equation of motions.

1.4.1 Definition of Nonlinear Normal modes (NNMs)

The concept of Nonlinear Normal Modes is the result of many attempts to extend the concept of Linear Normal Modes to nonlinear dynamics. The first step towards NNMs was made by Lyapunov by proving the existence of synchronous periodic solutions in a Hamil- tonian system next to equilibrium points. By using this theorem, Rosenberg [ROS 62] extended the definition of normal modes to define NNMs as synchronous periodic solutions. Recently the Rosenberg’s definition of NNMs was extended by Peeters et al. to take into account non necessarily synchronous periodic solutions [PEE 09]. The extension of Rosenberg’s definition allows internal resonances to occur. By using the theorem provided by Poincaré [POI 15] and Dulac [DUL 12], a second definition for NNMs based on normal form theory was presented by Jézéquel and Lamarque [JEZ 91]. Considering NNMs as non-necessarily synchronous periodic solutions or normal forms provides a di- rect extension of linear modes to nonlinear dynamics. Other theorems were used to yield new concepts for NNMs characterization. The center manifold theorem introduced by Carr [CAR 12] was used by Shaw and Pierre [SHA 91] to define a NNM as an invariant manifold in phase space. The definition of NNMs as invariant manifolds can be viewed as a new representation of NNMs. The extended definition of Rosenberg for NNMs is the one retained for the computation of NNMs in this Thesis.

Nonlinear Normal Modes for nonlinear modal analysis

1.4.2 Linear vs Nonlinear modal analysis

Nonlinearities appear frequently in real systems and trying to model such systems with linear modal theory without considering the nonlinearities leads to erroneous solutions. Therefore, a nonlinear modal analysis needs to be performed to take into account the complex behavior arising from the nonlinearities of the system. When generalizing the concept of normal modes to the nonlinear domain, the orthogonality and the superposition properties have been lost. Consequently, NNMs loss the main properties used for model reduction. On the other hand, NNMs are still invariant and present some interesting features:

Forced Resonances As in linear modal analysis, NNMs can be used to understand forced resonances. A forced resonance occurs when a NNM of the system is excited. For a nil amplitude of oscillation, the forced resonances happen as in linear theory. How- ever, with an increase of energy in the system, the frequency and the modal shape of the NNM change. The frequency of the modes can decrease leading to a softening effect, or increase leading to a hardening effect. Sometimes in MEMS applications, some cases of mixed behavior can be observed. Mixed behaviors in MEMS are characterized by an hardening effect for lower amplitude and a softening behavior for high amplitude, see Fig. 1.13. These effects can be seen as a dependency of the equivalent linear stiffness coefficient with the amplitude of oscillation.

24.2 24.25 24.3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Ω W max 24.11 24.12 24.13 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 24.05 24.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

Figure 1.13 – Frequency responses of a single MEMS resonator with hardening, mixte and softening configuration, after [NGU 13], superimposed with their corresponding NNMs

Changes of stability and bifurcations In nonlinear modal analysis, the concepts of stability and bifurcation points are additional features. Stability and bifurcation analysis are important parts of nonlinear modal analysis because they give interesting information about the local changes of the nonlinear dynamics of NNMs. In linear modal analysis, the stability of the modes cannot change, bifurcations cannot appears and the number of normal modes cannot exceed the number of Degree Of Freedom (DOF) of the system. However, in nonlinear modal analysis, NNMs can become unstable with the increase of the amplitude of excitation, leading to the establishment of phenomena such as hysteresis cycles and jumps. As in nonlinear frequency responses, NNMs can contain bifurcations. Bifurcation points are symptomatic of new behaviors such as new branches of periodic, quasi-periodic and static equilibrium, leading to a number of NNMs superior to the number of DOFs. Figure. 1.14 shows a bifurcation on the second NNM leading to two new stable NNMs and an unstable one. Some bifurcations are strongly related to symmetries present in the equation of motion and can exist only in these symmetric configurations. If ever these symmetries are to be broken, so do their associated bifurcations. The breaking of such bifurcations leads to the birth of isolated branches of NNMs (INNM). Therefore, a symmetry analysis of the equation of motion is important, because these symmetries determine the birth of INNMs.

0 0.5 1 1.5 2 2.5 3 1 1.2 1.4 1.6 1.8 2 Amplitu de [m] Frequency ω [rad/s] Stable NNM Unstable NNM BP NNM1 NNM2 NNM1+2 NNM1+2

Figure 1.14 – Stable and unstable NNMs with a single bifurcation from Subsection 3.5.2

Modal interactions The notion of modal interaction is not new and is already present in linear modal analysis and in rotor dynamics. In linear modal analysis, a modal interaction occurs when the frequencies of two modes are equal. Therefore, the two LNMs respond at the same frequency but are distinct from each other because of the orthogonality property. In rotor dynamics, the gyroscopic matrix is proportional to the speed of rotation and is superimposed to the damping matrix. However, the resulting equations of motions stay

Nonlinear Normal Modes for nonlinear modal analysis

linear with respect to the displacement. Therefore, in rotor dynamics modal interactions are more often encountered since modes can cross each other. Nevertheless, due to the orthogonality property the modes stay distinct. But in nonlinear modal analysis, due to the changes of frequency, of modal shape of the NNMs and due to the absence of the orthogonality property, NNMs can cross each other and be equal at some points; see for example the 3:1 modal interaction represented in Fig. 1.15.

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Figure 1.15 – Representation of NNMs with a 3:1 modal interaction, after [KER 09] One can see the advantage of using nonlinear modal analysis to observe the dynamics of nonlinear systems. Indeed, it brings important information on the dynamics that are not accessible with classical linear modal analysis.

1.4.3 Computation of conservative NNMs

The first investigations defined and computed NNMs as periodic solutions by using ana- lytical methods [MAN 72, JOH 79, RAN 92, NAY 94, VAK 01, GEN 04, MIK 10]. Then, with the appearance of continuation methods, NNMs started to get computed numerically. Pesheck et al. [PES 01a] used a numerical method based on invariant manifold theory to compute the NNMs of a rotor blade. Lee et al. [LEE 05] proposed to use Pilichuk work, see [PIL 85], to compute NNMs by solving an associated nonlinear boundary value problem with a shooting method. Then Arquier et al. [ARQ 06] used a shooting method coupled with the Asymptotic Numerical Method (ANM) to compute NNMs. Peeters et al. [PEE 09] proposed a shooting method to compute NNMs. Some others researchers,

such as Lewandowski [LEW 94, LEW 97a, LEW 97b], used HBM to compute NNMs with respect to the synchronous definition of NNMs. Since then, the same methodwas used to compute NNMs by Ribeiro [RIB 99a, RIB 99b], by Stoykov [STO 11] and by Arquier [ARQ 06]. Then, NNMs as non-necessarily synchronous periodic solutions were introduced. Sarrouy [SAR 11] used HBM to compute NNMs of a structure with cyclic symmetry. Krack [KRA 13] proposed a method to compute NNMs with distincts states using HBM. Moussi et al. [MOU 13] used HBM coupled with ANM to compute non- smooth NNMs of a system under impact forces. Renson et al. [REN 16] presented a review of methods for NNM computation in both time and frequency domain.

Phase condition and continuation method To compute NNMs as non-necessarily synchronous periodic solutions, a phase condition has to be introduced. Indeed, unlike periodic solutions of forced systems, NNMs are computed from an autonomous differential equation. Consequently, the phase of the NNM is unknown and needs to be imposed. To do so, an augmented system is considered. One simple way to eliminate the arbitrari- ness of the phase is to impose a component of the solution to zero at the beginning of a period. Rinzel [RIN 80] used this phase condition to compute periodic solutions of the Hodgkin-Huxely equation. Renson et al. [REN 16] used the velocity displacement vector obtained in the frequency domain set to zero as phase condition. However, not all NNM can be computed by using such a phase condition. Chua and Lin [CHU 75] presented an algorithm that automatically selects an adapted phase condition consisting in fixing a specific component of the periodic solution. This method fixes the problem but is still not optimal.

Other phases conditions can be found in the literature. Some phase conditions presented by Doedel [DOE 81] are based on the orthogonality between the vectors describ- ing the periodic solution and its velocity. Accordingly to the author, the phase condition representing the orthogonality law at a specific time is better used for theoretical purpose, whereas the phase condition based on the integral of the orthogonality law over one period is best suited for numerical methods. These two phase conditions are notably analyzed and used in the literature by Seydel et al. [SEY 09] and Beyn et al. [BEY 07] for the computation of periodic solutions computation. However, the resulting augmented system composed of the equation of motion and the phase condition is overconstrained. To perform continuation methods based on matrix inversions, various methods can be found in the literature. Renson et al. in [REN 16] used the Moore-Penrose pseudo inverse to perform the continuation of the overconstrained augmented system. An alternative method to the Moore Penrose inversion is proposed by Munoz-Almaraz in [MUN 03]. By introducing an parameter of relaxation into the equation of motion, this method permits a square jacobian to be generated in order to perform the continuation with standard inversion and the computation of linearized stability.

Stability and bifurcation analysis Linearized stability analysis of periodic solutions can be performed by means of Floquet theory. Nevertheless, during the computation of NNMs as non-necessarily synchronized solutions, the non-uniqueness coming from the

Nonlinear dynamics of resonant MEMS

undetermined phase generates a trivial singularity. One method to perform the stability analysis consists in considering the eigenvalues without the trivial singularity. After implementing this stability analysis method in AUTO, Doedel et al. [DOE 03] provided minimally extended systems characterizing both the trivial singularity and co-dimension 1 bifurcation points. However this method can lead to false detection of bifurcations points due to numerical errors. By using deflation techniques the trivial singularity can be re- moved. In [NET 15], Net and Sanchez used the shooting method combined with deflation techniques to obtain a modified jacobian characterizing bifurcations points.

1.4.4 Extension to non-conservative NNMs

Shaw and Pierre [SHA 93] used the invariant manifold definition of NNMs presented in [SHA 91] to compute non-conservative NNMs. Pesheck was the first to use the definition proposed by Shaw and Pierre. In [PES 01b], he used a Fourier-Galerkin method to compute the invariant manifold of a rotating beam. Then, the proposed method was used by Legrand et al. [LEG 04] to compute NNMs of a rotating shaft and by Laxalde and Thou- verez [LAX 09] to asses the non-conservative NNMs of turbomachinery bladings with dry friction. In [REN 14], the invariant manifold definition for NNMs wasused to provide a finite-element based method to compute non-conservative NNMs. However, all these descriptions consider non-conservative NNMs as oscillations that gradually loss energy.

In Section 3.2, an extension to non-conservative NNMs using the extended definitions of Rosenberg is proposed. The idea behind this extension is to compensate the energy decrease due to damping by introducing additional terms in the equation of motion while not modifying the original non-conservative NNM. To our knowledge, the definition proposed by Rosenberg has only been used by Krack [KRA 14] for the computation of non- conservative NNMs, see Fig. 1.16. He introduced artificial negative damping matrix to insure the energy balance of the problem. However, it was not proven that the introduction of the artificial damping matrix did not modify the non-conservative NNMs.

In document Modes nonlinéaires et continuation multiparamétrique de bifurcations : Application aux absorbeurs de vibrations et aux capteurs MEMS architecturés pour la détection de masse (Page 49-54)