Normal form theory and nonlinear normal modes: Theoretical settings and applications

TL;DR: In this article, the relationship between normal form theory and nonlinear normal modes (NNMs) is discussed for the specific case of vibratory systems displaying polynomial type nonlinearities, and the development of reduced-order models based on NNMs expressed asymptotically with the formalism of real normal form is deeply presented.

Abstract: These lecture notes are related to the CISM course on ”Modal Analysis of nonlinear Mechanical systems”, held at Udine, Italy, from June 25 to 29, 2012. The key concept at the core of all the lessons given during this week is the notion of Nonlinear Normal Mode (NNM), a theoretical tool allowing one to extend, through some well-chosen assumptions and limitations, the linear modes of vibratory systems, to nonlinear regimes. More precisely concerning these notes, they are intended to show the explicit link between Normal Form theory and NNMs, for the specific case of vibratory systems displaying polynomial type nonlinearities. After a brief introduction reviewing the main concepts for deriving the normal form for a given dynamical system, the relationship between normal form theory and nonlinear normal modes (NNMs) will be the core of the developments. Once the main results presented, application of NNMs to vibration problem where geometric nonlinearity is present, will be highlighted. In particular, the developments of reduced-order models based on NNMs expressed asymptotically with the formalism of real normal form, will be deeply presented.

Summary

1 Introduction and notations

• These lecture notes are related to the CISM1 course on ”Modal Analysis of nonlinear Mechanical systems”, held at Udine, Italy, from June 25 to 29, 2012.
• After a brief introduction reviewingthe main concepts for deriving the normal form for a given dynamical system, the relationship between normal form theory and nonlinear normal modes (NNMs) will be the core of the developments.
• For generic dynamical systems,X will denote the state vector,X ∈ E , whereE is the phase space, of dimensionn.
• For the particular case of vibratory system, geometric nonli earity is considered, so that only quadratic and cubic type nonlinearities are present in the equations of motions.
• NNMs and normal form are used to derive theROMs, and examples on different shells are shown: a doubly-curved (hyperbolic paraboloid) panel illustrates a case without internal resonance, while a closed circular cylindrical shell allows illustrating a more complicated case with a 1:1 internal resonance.

2 Normal form theory

• Normal form theory is a classical tool in the analysis of dynamical systems, and general introductions can be found in many textbooks, see.g. [20, 21, 57, 29].
• Here however, the general theory will be usedfor another purpose: defining a nonlinear change of coordinates allowing one to express the dynamics in an invariant-based span of the phase space, where the generating axis are the invariantm ifolds arising from the linear eigenspaces, i.e. the NNMs of the system.
• Normal form theory is based on two major theorems, due to Poincaré and Poincaré-Dulac, which have been demonstrated in the beginning of the XXth century [39, 13].
• The presentation will begin with a reverse problematic, in order to understand better the main issue: starting from a simple, linear problem, the authors will show how a nonlinear change of coordinate can make it appear as complicated.
• The ”reverse” illustrative problematic is borrowed from [30], and the presentation of normal form used follows closely that shown in [30, 29].

2.1 Problematic

• Before entering the complicated calculations for tacklingthe general case, let us first introduce an illustrative example, allowing us to properly define the goal pursued, which is defined as trying tosimplify as far as possible, in the vicinity of a particular solution (e.g. fixed point or periodic orbit), a given dynamical system.
• Maybe that with a great intuition and a bit of luck, one could have find the solution which, in their case, is known by construction, and is simply given by: X(t) = exp(Y0 cos t)−.
• Without inspired intuition, one could at least try an asymptotic, power series expansion, in order to get an idea of the sought nonlinear change of coordinates.
• The common sense would state that this problem withp= 6 is ”more nonlinear” than the same withp=2.

2.2 Example study

• 2.1 HereY is the new variable, and the goal of the transformation is to obtain a dynamical system for the new unknownY that is simpler than the original one.
• In any case the authors can conclude that: The objective is fulfilled: nonlinearities have been repelled up to fourth order.
• The first two conditions have already been encoutered in the precedent subsection in dimensionn = 1: once again, they are the consequence of the assumption ofhyperbolicityof the fixed point, stating that the authors are not at a bifurcation point.
• In the analysis,the resulting normal form keeps the monom and the dynamics is different from a linearizable case.
• When no resonance condition exist between the eigenvalues, then the system isquivalent to a linear one (Poincaré).

2.4 Application to vibratory systems, undamped case

• The case of vibratory systems displaying quadratic and cubic nonlinearities, is now examined, following the general notation stated in the introduction, Eq. (2).
• A second family of resonance relationships may arise frominternal resonancesbetween the eigenfrequencies of the system.
• The following comments are worth mentionable: Even though the last Eq. (40) appears longer on the page than the original one (2), the reader must be convinced that it is simpler.

2.5 NNMs and Normal form

• Nonlinear Normal Modes (NNMs) is the core of this CISM courseon Modal analysis of nonlinear systems.
• As expressed with the modal coordinates, he equations of motion are written in a phase space which is spanned by the linear eigenspaces (two-dimensional planes parameterized by (Xp, Yp)).
• The nonlinear transform cancels the invariant-breaking terms.
• Once the system expressed in its normal form, the truncations can be realized as one is ascertained of an accurate and meaningful result thanks to the invariance property.

2.6 Single-mode motion

• In order to restrict the dynamics to a single NNM, one has justto proceed as usual with thenormal coordinates.
• Detailed comparisons with other computations led by different authors, have been realized.
• In particular the invariant manifold procedur proposed by S. Shaw and C. Pierre in [42] and its asymptotic development for solving out the Partial Differential Equation defining the geometry resolved in [37, 38] gives exactly the same expressions.
• This point will be key for the remainder of the presentation.
• Note that the normal form theory provides the most complete picture for expr ssing the NNMs of a system, as the nonlinear change of coordinate (39) is a complete change, from the phase space into itself.

2.7 Classification of nonlinear terms, case of internal resonance

• From all the results obtained in the previous subsection, one is now able to draw out a classification of the nonlinear coupling terms appearing in the equations of motion, in order to get a clear physical understanding of their meaning on the dynamics.
• The following classification can be derived from the precedent results: .
• Note that when no internal resonance of order two are present,all the quadratic terms can be cancelled by the normal transform. .
• This example shows that the treatment of internal resonanceis not made too difficult with the formalism of normal form, contrary to the huge complexities involved in other methods (invariant manifold, multiple scales) to adapt their treatments to the case of internal resonance.

2.8 Damped systems

• All the developments presented in the previous section havebeen obtained under the assumption of a conservative system.
• When modal damping is added to each linear contribution, the decay rate of all the linear modes that are gathered to create the selected NNM are somehow added, so that the decay rate onto the manifold is not as simple as the initial decay rate postulated for the linear modal coordinates.
• Hence all the calculations are led with the extra condition that when a coefficient in the normal form transform scales as1/ξp, then it must be cancelled, and the corresponding monom stayin the normal form.
• By doing so, only the trivially resonant terms are kept, and a continuity from undamped to damped real normal forms, is obtained.
• In Fig. 9(a), the two damping coefficients have the same values, so thatξ1=ξ2=ξ, andξ is increased from 0 to 0.4, so that a system that is more and more damped is studied.

2.9 Closing remarks

• The main theoretical results for deriving a normal form approach for nonlinear normal modes, has been shown in this section.
• In particular, the special case of structural system with quadratic and cubic nonlinearities, Eqs. (48), has been here assumed as it contains the great majority of applications to thin structures like beams, plates and shells, vibrating at large amplitudes.
• Another limitation could appear through the asymptotic develoments, systematically stopped at third order.
• Moreover, the legitimate question is to know if the game is worth the candle.
• Asymptotic expansions up to order five realized for computing the NNMs with the center manifold technique in [43] show that the gain in accuracy brought by the fifth-order is not significative, and in some cases can be poorest than the third-order.

3 Hardening/softening behaviour

• This section is entirely devoted to the correct prediction of the type of nonlinearity (hardening/softening behaviour) for the modes of an assembly ofN nonlinear oscillator equations as in Eq. (2).
• NNMs and normal form are used to derive a proper, easy-to-use and almost analytical method.
• The presentation in subsections 3.1 and 3.2 recalls the main results obtained in[55].
• The next subsection 3.3 with applications to shells gives the main results published in [53, 47],while subsection 3.4 gathers important results published in [50] on the influence of the damping.

3.1 Definition

• This dependence can be of two different types.
• Either ahardening behaviour is at hand, which implies that the oscillation frequ ncyincreaseswith the amplitude, or a softeningtype nonlinearity is present, which means that the oscillation frequencydecreaseswith the amplitude.
• Hence reducing a system ofN linear oscillator as Eq. (2) to a single linear mode, in orderto predict the type of nonlinearity of the selected mode, may lead to erroneous results.
• Note that for applying these formulae, the implicit conventio used throughout the notes:gpij = 0 when i > j, must be applied.
• The expressions in Eqs (61) shows that in case of internal resonance, a small denominator effect appears, leading to a divergence in the expressions ofappp andb p pp, and thus on the predicted type of nonlinearityΓp.

3.2 A two dofs example

• The two dofs system composed of a masse connnected to two nonlinear springs, whose equations of motions are given in Eqs. (42), is once again considered.
• This can be easily understood by comparing the quadratic and cubic coefficients of the oscillator-equation (62).
• One can notice for example the upper-left region, which is predicted to exhibit a hardening behaviour by the linear approximation, whereas the real behaviour is soft.
• In order to have a better picture of the behaviour of the type of non linearity versus the two parameters (ω1, ω2) of the system, Figure 11 shows two cuts in this two-dimensional parameter plane, namely for fixed ω2=2 andω1 variable, then for fixedω1= √ 0.5 andω2 variable,i.e. along the two lines indicated in Fig. 10.
• As already underlined, in the vicinity of this internal resonance, theconcept of the type of nonlinearity loses its meaning because the dynamics is essentially two-dimensional and cannot be reduced to a single NNM.

3.3 Application to shells

• The method shown previously for predicting accurately the type of nonlinearity, is now applied to the case of spherical-cap thin shallow shells with a varying radius of curvatureR.
• Flat plates are known to exhibit a hardening behaviour,as it has been shown both theoretically and experimentally (see e.g. [48, 58, 35, 44, 54, 46]), which means that the leading cubic coefficienthpppp is positive.
• Introducing a radius of curvatureR, going to infinity (perfect plate) to finite values (spherical- p shells) introduces an asymmetry in the restoring force, due to the loss of symmetryof the neutral plane of the shell.
• The model is based on von Kármán kinematical assumptions on the strain-displacement relationship, in order to take into account moderately large vibration amplitudes.
• The governing equations are first recalled, then the Galerkin method is briefly reviewed in order to explain how to pass from the PDE of motion to oscillator-equations having the form of Eqs. (2),then finally the type of nonlinearity for some eigenmodes of the structure, are given.

3.3.2 Linear analysis

• All the analysis is here performed for a free-edge boundary condition.
• They are not recalled here for the sake of brevity, theinterested reader can found the complete expressions in [47, 53].
• The results will be presented as functions ofκ, in order to set apart the material property which appear through the Poisson ratioν in the expression ofχ.
• On the contrary, the eigenfrequencies dependence on the aspect ratioκ, represented on Fig. 15, shows a different behaviour, which leads to classify the modes into two families.

3.3.3 Modal expansion

• The complete non-linear equations of motion (70) are projected onto the natural basis of the transverse eigenmodes.
• Now that the PDE has been reduced to nonlinear oscillator equations, the formalism of NNMs and normal form can be applied to derive the type of nonlinearityfor each mode of the shell.
• The nonlinear coefficientsgpij andh p ijk shows a very slight dependence on the curvature of the shell.
• This is the consequence of the slight dependence of the mode shapes with the aspect ratio, as the non-linear coefficients are computed from integrals involving the mode shape functions (Eqs (76-77)).
• The conditions for these quadratic coefficients to be non-zero are expressed in terms of the number of nodal diameterskl andkp of the l andp modes.

3.3.4 From circular plates to spherical-cap shells

• The type of nonlinearity is now computed for three differentmodes of the shell, representing each of the three families.
• Secondly, hardening behaviour is observed until the 2:1 resonance with mode (0,1), where softening behaviour settles down.
• The type of non-linearity tends to zero asκ tends to infinity.
• The case of an axisymmetric mode is now considered with mode (0,1).

3.5 Closing remarks

• A general strategy has been proposed for deriving the type ofnonlinearity of an assembly of nonlinear oscillator equations, based on normal form theory and reduction to a single NNM.
• The reason is the non invariance of the linear eigenspace, as shown in the previous section.
• The effect of the other modes on the type of nonlinearity, has been underlined, i particular the presence of 2:1 internal resonance.
• For the last section of this lecture, the focus is on the derivation of ROMs for structural systems in forced vibration.

4 Reduced-order models for resonantly forced response

• The goal of this section is to use the reduced-order modelling strategy based on NNMs and normal form theory, in order to compute the harmonically forced response of thin structures, vibrating at large amplitudes and excited in the vicinity of one of its eigenfrequency.
• Applications to shells will be specifically shown, and a comparison with the Proper Orthogonal Decomposition (POD) method will highlight the benefit of using NNMs in this case.
• In the first subsection, thederivation of the ROM is briefly reviewed and the advantage of using the normal form method including the damping is shown on a two-dofs example.
• The presentation in subsection 4.1 recalls some of the results published in [50].
• Subsection 4.2 selects one of the examples shown in [51] for illustration.

4.1 Derivation of the reduced-order model

• The previous developments have clearly highlighted the general method for deriving the ROM, shown schematically in Fig.
• 1 internal resonance is present and two NNMs must bere ained in the truncation, also known as Hence a 1.
• Secondly, the perturbation brought by the external force onto the normal form is at least a second-order effect [21].
• The second ROM, following the ”conservative NNM” method, isfound by subsituting in Eqs. (45) and (61) the coefficients by their expressions.
• The ROM constructed with the ”conservative NNM” method underestimates the damping in the system, and gives a maximum amplitude which is slightly larger that the reference solution.

4.2 Application : the case of a doubly-curved panel with in-plane inertia

• Frequency response curves, in the vicinity of its fundamental mode, will be computed for different ROMs.
• As a single NNM ROM is awaited to give good results, it is decidedto compare two ROMs having the same complexity (a single nonlinear oscillator equation).
• Fig. 26 shows the six main coordinates,i.e. the first four coordinates in transverse direction,w1,1, w3,1, w1,3 andw3,3, as well as the first two longitudinal coordinatesu1,1 andv1,1.
• Once again, the reference solution is compared to the two reduced models composed of a single linear and non-linear mode.

4.3 Application : the case of a closed circular cylindrical shell

• A water-filled perfect circular cylindrical shell, simply supported, and harmonically excited in the neighbourhood of the fundamental frequency, is selected in orderto derive a NNM-based ROM for a continuous structure.
• Mathematical expressions of boundary conditions are given in [6, 7].
• The case considered here (a perfect shell) does not produce new terms because of the perfect symmetry of the initial problem.
• One can see that all the dynamical features of the original system are recovered: the two branches are found as well as the nature of the bifurcations and the stability.
• For comparison, the sixth most important modal amplitudes are represented in Fig. 31, for the full-order model, and the“damped NNM” ROM.

4.4 Comparison with the Proper Orthogonal Decomposition method

• This last section is devoted to a comparison between reduced-order modeling as proposed with NNM via normal form theory, and the more popular Proper Orthogonal Decomposition (POD) method.
• Eq. (106) is used to solve the equations of motion of the shell, given in the beginning of the previous subsection, Eqs. (94) and (96), with the Galerkin method to find the equations of motion of the ROM.
• The coefficientsαm,n,i andβm,n,i are also meaningful in order to get a physical interpretation of the POMs in phase space.
• On the other hand, The NNM-based ROM is accurate with two NNMs only, as thebending of the phase space is taken into account in the nonlinear change of coordinates.

Figures

Figure 5: Trajectories of the nonlinear system, Eqs. (42). Motions initiated in the first linear mode with initial conditionsX1(t = 0) = 0.01, 0.025 and 0.05 (all other coordinates to zero), illustrating the non-invariance of the linear eigenspace for the nonlinear system.

Figure 26: Maximum amplitude of the response of 6 generalized coordinates versus excitation frequency, for an excitation amplitude of̃f = 4.37 N. Reference solution (thick line) is compared to the reduction to a single linear mode (LNM) and a single non-linear mode (NNM). (a : maximum ofw1,1. (b): maximum of w3,1. (c): maximum ofw1,3. (d): maximum ofw3,3. (e): maximum ofu1,1. (f): maximum ofv1,1.

Figure 19: Type of non-linearityΓ1, defined by Eq. (82), for increasing values ofξ2. The behaviour turns from hardening to softening type forξ2 = 0.081. Other selected values are:ω1 = 3, ω2 = 5.4, and ξ1 = 0.001.

Figure 14: Profiles of theoretical asymmetric (2,0) mode shape, for several values of a/R between 0 and 0.6: (left) transverse mode and (right) membrane mode.

Figure 13: Three representative mode shapes of a circular spherical thin shallow shell with free edge: axisymmetric mode (0,1), purely asymmetric mode (2,0) and mixed mode (1,1).

Figure 4: Trajectories (closed periodic orbits) of thelinear system, represented in space(X1, Y1, X2). Motions along the first linear mode with initial conditionsX1(t = 0) = 0.01, 0.025 and 0.05. The linear eigenspace is represented by the horizontal planeX2 = 0. The trajectories initiated along the linear mode stay within for al timet (invariance property).

Figure 3: Schematics of the system containing quadratic andcubic nonlinearities

Figure 29: Frequency-response curve for (a):f̃ = 2.84 N, and (b):f̃ = 6.62 N. Reference solution (ref) is compared to truncations with a single linear mode (LNM) and single non-linear mode (NNM).

Figure 25: Frequency-response curve for the HP panel, harmonically excited in the vicinity of the first eigenfrequencyω1. The reference solution is compared to the solution given bykeeping a single linear mode (LNM) or a single NNM. The excitation amplitude is̃f = 4.37 N. Point (A), withω = 1.3ω1, is used for time integration, see Fig. 27.

Figure 32: Maximum amplitude of transverse shell vibrationversus excitation frequency. (a): driven modeA1,n. (b): companion modeB1,n. Thick line: reference solution (unstable states (dashed tick line) and bifurcation points are indicated: BP : pitchfork,TR: Neimarck-Sacker (torus) bifuration). Blue thin line: ROM obtained by considering two NNMs. Green thickline: ROM obtained by considering three POMs. Points b and c denotes the dynamical responses used for building the POD ROM.

Table 1: Coefficientsαm,n,i andβm,n,i defined in Eq. (106), for the first three POD modes obtained with point c.

Figure 18: Type of nonlinearity for mode (1,1) as function ofthe geometrical parameterκ. N=1: truncation including only linear mode (1,1). N=17: converged result including modes (1,1), (2,0) to (2,4) (with both configurations), (0,1) to (0,5).

Figure 28: Maximum amplitude response ofw1,1 versus excitation frequency for̃f = 4.37 N, showing the convergence of the solution when increasingPLNM . Reference solution (22 basis functions) is compared to truncations with 1 linear mode, 5 LNM, 11 LNM and 15 LNM.

Figure 10: Regions of hardening/softening behaviour in theparameter plane(ω21, ω 2 2), for the two-dofs system. Yellow: hardening behaviour, blue: softening behaviour. Left: prediction given by the sign of Γ̃1, i.e. with truncation to the first linear mode. Right: prediction given by the sign ofΓ1, i.e. for oscillatory motions along the first NNM.

Figure 22: Frequency-response curves (maximum of the first coordinateX1 versus excitation frequency Ω). Thick solid line : reference solution. Thin blue solid line :”damped NNM”, Eq. (86). Dash-dotted green line : ”conservative NNM”, Eq. (85). Magenta dash-dotted thin line: linear mode truncation, Eq. (84). Selected values:ω1=2,ω2=4.5,ξ1=0.001,ξ2=0.01,F1=5.10−4.

Figure 16: Type of nonlinearity for mode (4,0) as function ofthe geometrical parameterκ. N=1: system truncate to linear mode (4,0). N=2: truncation including (4,0) and (0,1). N=3: (4,0), (0,1) and (0,2).

Figure 17: Type of nonlinearity for mode (0,1) as function ofthe geometrical parameterκ. N=1: truncation containing only mode (0,1). N=7: converged result withinclusion of the seventh first axisymmetric modes, from (0,1) to (0,7).

Figure 33: Poincaré section in (A(1,5), A(1,0)). Cloud of points: section of the orbits with the Poincaré plane, casesb andc. POD differs very little from the original axis, whereas thecut of the 4-dimensional invariant manifold (curved hyperbolic line) goes right in-between the points (NNMs).

Figure 1: (a) Time series of the transformed variableX (black solid line), compared to the originalY (thin blue line), for the problem defined through Eqs (3)-(7). (b): Fourier Transform ofX (black solid line) andY (thin blue line), showing their harmonic content.

Figure 31: Maximum amplitude of transverse shell vibrationversus excitation frequency for six slave modal coordinates: (a) and (b): driven and companion of asymmetric (1,2n) mode; (c) and (d): driven and companion of asymmetric (3,2n) mode; (e) and (f): axisymmetric (1,0) and (3,0) modes. Thick line: reference solution. Thin line : ROM computed from the two-dof “damped NNM” master coordinates.

Figure 8: Illustration of the nonlinear transform for derivation of reduced-order models.

Full text

HAL Id: hal-01134785
https://hal-ensta-paris.archives-ouvertes.fr//hal-01134785
Submitted on 24 Mar 2015
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Normal form theory and nonlinear normal modes:
Theoretical settings and applications
Cyril Touzé
To cite this version:
Cyril Touzé. Normal form theory and nonlinear normal modes: Theoretical settings and applications.
G. Kerschen. Modal Analysis of nonlinear Mechanical Systems„ 555, Springer, 2014, Springer Series
CISM courses and lectures, 978-3-7091-1790-2. �10.1007/978-3-7091-1791-0_3�. �hal-01134785�

Normal form theory and nonlinear normal modes:
Theoretical settings and applications
Cyril Touz´e
Unit´e de M´ecanique (UME),
ENSTA-ParisTech,
Chemin de la Huni`ere,
91761 Palaiseau Cedex, France.
cyril.touze@ensta-paristech.fr
http://www.ensta.fr/∼touze
June 26, 2012
Contents
1 Introduction and notations 2
2 Normal form theory 3
2.1 Problematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Example study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 The problem in dimension n = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 The problem in dimension n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 General case: Poincar´e and Poincar´e-Dulac theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Application to vibratory systems, undamped case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 NNMs and Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Single-mode motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Classification of nonlinear terms, case of internal resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.8 Damped systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.9 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Hardening/softening behaviour 21
3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 A two dofs example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Application to shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.1 Von K´arm´an model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.3 Modal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.4 From circular plates to spherical-cap shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Influence of the damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Reduced-order models for resonantly forced response 34
4.1 Derivation of the reduced-order model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Application : the case of a doubly-curved panel with in-plane inertia . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Application : the case of a closed circular cylindrical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 Comparison with the Proper Orthogonal Decomposition method . . . . . . . . . . . . . . . . . . . . . . . . . 46
A Expressions of the coefficients for the normal transform in the conservative case 56
A.1 Quadratic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
A.2 Cubic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1

1 Introduction and notations
These lecture notes are related to the CISM
1
course on ”Modal Analysis of nonlinear Mechanical sys-
tems”, held at Udine, Italy, from June 25 to 29, 2012. The key concept at the core of all the lessons given
during this week is the notion of Nonlinear Normal Mode (NNM), a theoretical tool allowing one to
extend, through some well-chosen assumptions and limitations, the linear modes of vibratory systems, to
nonlinear regimes. More precisely concerning these notes, they are intended to show the explicit link be-
tween Normal Form theory and NNMs, for the specific case of vibratory systems displaying polynomial
type nonlinearities. After a brief introduction reviewing the main concepts for deriving the normal form
for a given dynamical system, the relationship between normal form theory and nonlinear normal modes
(NNMs) will be the core of the developments. Once the main results presented, application of NNMs
to vibration problem where geometric nonlinearity is present, will be highlighted. In particular, the de-
velopments of reduced-order models based on NNMs expressed asymptotically with the formalism of
real normal form, will be deeply presented. Applications are devoted to thin structures vibrating at large
amplitudes, with a special emphasis on thin shells of different geometry (from plates to closed circular
cylindrical shells). Effective reduced-order models for the prediction of the type of nonlinearity (hard-
ening/softening behaviour), or the computation of complete bifurcation diagrams for the case of forced
vibrations, will be shown, for toy models including a small number of degrees-of-freedom (typicaly two
to three dofs), as well as for continuous models such as beams, plates and shells.
The following notations will be used throughout the lecture notes. For generic (nonlinear) dynamical
systems, X will denote the state vector, X ∈ E, where E is the phase space, of dimension n. Generally,
the simple choice E ≡ R
n
is retained. The dynamical system is denoted as:
˙
X = F(X), (1)
with F the vector field, which could depend on one or more parameters.
For the particular case of vibratory system, geometric nonlinearity is considered, so that only quadratic
and cubic type nonlinearities are present in the equations of motions. Hence for nonlinear vibration prob-
lems, the generic equations of motion consist in a set of N oscillator equations, denoted under the general
formulation:
∀p = 1, ..., N :
¨
X
p
+ ω
2
p
X
p
+
N
X
i=1
N
X
j≥i
g
p
ij
X
i
X
j
+
N
X
i=1
N
X
j≥i
N
X
k≥j
h
p
ijk
X
i
X
j
X
k
= 0, (2)
where g
p
ij
denotes the generic quadratic nonlinear coupling coefficient, and h
p
ijk
the cubic one. In this no-
tation, the upperscript p refers to the oscillator-equation considered, while the subscripts (i, j, k) denotes
the coupling monom X
i
X
j
X
k
. Note that damping is not considered in Eq. (2). Inclusion of dissipative
mechanism will be included in some parts of the notes. Note also that the linear part is diagonal, which
means that the variable X
p
is the modal amplitude of the p
th
linear normal mode. In cases where the
linear part is not given as diagonal, a linear change of coordinate can be performed to fit the framework
presented here. Finally, to recover the first-order dynamical system formalism, the velocity Y
p
=
˙
X
p
is used as complementary variable, so that, for a collection of N oscillators, the state variable writes:
X = [X
1
Y
1
X
2
Y
2
... X
N
Y
N
], so that dim(E) = 2N.
The first section is devoted to normal form theory. The main idea of introducing a nonlinear transform
in order to simplify as much as possible the equations of motion, is first introduced in an illustrative
1
CISM stands for ”Centre International des Sciences M´ecaniques” (in French), or ”International Centre for Mechanical
Sciences”, see www.cism.it
2

manner where the reverse problematic is considered. Very simple examples in dimension 1 and 2 allows
introducing the key concept of resonance. The core of the theory, the theorems of Poincar´e and Poincar´e-
Dulac, are then given in a general context, and the result is then specialized to the case of vibratory
problems. The main results are given for a conservative problem, and the link to NNMs, is illustrated,
allowing derivation of important ideas such as reduced-order modeling and classification of nonlinear
terms for physical interpretation. Finally, the case of damped mechanical systems is tackled.
The second section focus on the prediction of the type of nonlinearity (hardening/softening be-
haviour) for a system of oscillator of the form (2). It is shown that NNMs and normal form gives the
correct framework for an efficient and accurate prediction. Examples on two-dofs system, and spherical-
cap shells with varying radius of curvature (from the flat plate case to spherical shells), illustrates the
method. Finally the influence of the damping on the type of nonlinearity is discussed.
The third section tackles the problem of deriving accurate reduced-order models (ROMs) for thins
structures harmonically excited at resonance, in the vicinity of one of its eigenfrequency, and vibrating at
large amplitudes. NNMs and normal form are used to derive the ROMs, and examples on different shells
are shown: a doubly-curved (hyperbolic paraboloid) panel illustrates a case without internal resonance,
while a closed circular cylindrical shell allows illustrating a more complicated case with a 1:1 internal
resonance. the complete bifurcation diagram with different kind of bifurcation points are clearly recov-
ered by the ROM, and a comparison with the most routinely used Proper Orthogonal Decomposition
method (POD) is shown to conclude the notes.
2 Normal form theory
Normal form theory is a classical tool in the analysis of dynamical systems, and general introductions
can be found in many textbooks, see e.g. [20, 21, 57, 29]. It is generally used in bifurcation theory in
order to define the simplest form of dynamical systems generating classical bifurcations of increasing
co-dimension. Here however, the general theory will be used for another purpose: defining a nonlinear
change of coordinates allowing one to express the dynamics in an invariant-based span of the phase
space, where the (curved) generating axis are the invariant manifolds arising from the linear eigenspaces,
i.e. the NNMs of the system.
Normal form theory is based on two major theorems, due to Poincar´e and Poincar´e-Dulac, which
have been demonstrated in the beginning of the XX
th
century [39, 13]. The main idea is to simplify,
as far as possible, the equations of motion of a nonlinear dynamical systems, by means of nonlinear
change of coordinates. The presentation will begin with a reverse problematic, in order to understand
better the main issue: starting from a simple, linear problem, we will show how a nonlinear change of
coordinate can make it appear as complicated. Then the normal form will be introduced gradually with
two examples study, starting from the simplest cases with dimension 1 and 2. The ”reverse” illustrative
problematic is borrowed from [30], and the presentation of normal form used follows closely that shown
in [30, 29].
2.1 Problematic
Before entering the complicated calculations for tackling the general case, let us first introduce an illus-
trative example, allowing us to properly define the goal pursued, which is defined as trying to simplify
as far as possible, in the vicinity of a particular solution (e.g. fixed point or periodic orbit), a given
dynamical system.
To begin with, let us consider the simple initial-value problem:
d
2
Y
dt
2
+ Y = 0, (3a)
Y (t = 0) = Y
0
,
dY
dt
(t = 0) = 0, (3b)
3

where Y is a real coordinate depending on time t. This equation is that of a single oscillator, with
eigenfrequency equal to one. Its solution in time is known and reads:
Y (t) = Y
0
cos t. (4)
Now let us introduce the transformed variable:
X = exp(Y ) − 1. (5)
Differentiating Eq. (5) two times with respect to time t, and inserting in the original dynamical equation
(3a), one can show that the transformed variable X satisfy the following evolution equation:
(1 + X)
¨
X −
˙
X
2
+ (1 + X)
2
ln(1 + X) = 0, (6)
where ln is the natural logarithm. In the case where one would have to face a physical problem expressed
by Eq. (6), with appropriate initial conditions, then the solution would have been more difficult to find !
Maybe that with a great intuition and a bit of luck, one could have find the solution which, in our case, is
known by construction, and is simply given by:
X(t) = exp(Y
0
cos t) − 1. (7)
0 10 20 30 40
−1
−0.5
0
0.5
1
1.5
2
t
X,Y
0 2 4 6 8 10
10
−15
10
−10
10
−5
10
0
Frequency
FFT(X)
(a) (b)
Figure 1: (a) Time series of the transformed variable X (black solid line), compared to the original Y
(thin blue line), for the problem defined through Eqs (3)-(7). (b): Fourier Transform of X (black solid
line) and Y (thin blue line), showing their harmonic content.
The solution for X is represented in Fig. 1. As awaited, it is 2π-periodic, but contrary to the
initial solution for Y displaying only one harmonic component, the X solution shows an infinity of
harmonics, with exponential decay in amplitude, as revealed by the Fourier transform of X, Fig. 1(b).
This example shows that the apparent complexity of the problem for X only results from a nonlinear
change of coordinates. Looking at the things in a reverse manner, the question arising is naturally: for a
given problem, is it possible to find such a nonlinear transformation that could simplify, at best linearise,
the initial system ? The main idea of normal form theory is to give an answer to this question. Without
inspired intuition, one could at least try an asymptotic, power series expansion, in order to get an idea
of the sought nonlinear change of coordinates. This functional relationship has to be defined in the
vicinity of Y = 0 (or equivalently, X = 0), which is the fixed point of the original dynamical system
(equivalently, of the transformed system). Once again the result is known and reads, in our case:
X =
+∞
X
n=1
1
n!
Y
n
. (8)
4

Frequently asked questions

Q1. What are the contributions in "Normal form theory and nonlinear normal modes: theoretical settings and applications" ?

Touzé et al. this paper introduced the notion of Nonlinear Normal Mode ( NNM ), a theoretical tool allowing one to extend, through some well-chosen assumptions and limitations, the linear modes of vibratory systems to nonlinear regimes.