Backbone curves and Nonlinear normal modes: a new identification tool A.Cammarano1a, T.L. Hill2, P.L. Green3 and S.A. Neild2

1University of Glasgow, Mechanical Engineering

JWS Building, University Avenue, G12 8QQ

Glasgow, UK,

2University of Bristol, Mechanical Engineering

Queens Building, University Walk, BS8 1TR

Bristol

3University of Liverpool, School of Engineering

Brownlow Hill, L69 3GH Liverpool

aandrea.cammarano@gla.ac.uk

Abstract. This work presents a new identification method that exploits second order nonlinear normal forms (soNNF) and the Bayesian inference to identify the shape and the coefficients of the equations that govern the dynamics of a structure. In recent works this authors highlighted the relation between backbone curves and the frequency response of a system with smooth nonlinearity. Using soNNF, algebraic equations can be found to describe the backbone curves. Such algebraic equations can be used in conjunction with a Bayesian framework to identify the parameter of the functional relations that relate the loading to the deformation of the structure.

Introduction

The design of all structures in mechanical, aerospace and civil engineer is done assuming linear response to external loading. Nevertheless, the general trend of all the engineering disciplines toward high performance and low carbon emission often translates in the need of lighter structures able to maintain the same safety levels and eventually to expand the operational envelope. For many structures of new generation, this means that the deformations exceed the linear range and nonlinear behaviours can be observed. Nonlinear vibrations in structures are very complex phenomena: frequency shift in the resonant peaks, isolas, localised high amplitude vibrations are only some of the characteristics that one can found in the frequency response of a nonlinear structure [1]. Understanding and predicting how a nonlinear structure responds is of the uttermost importance as high amplitude oscillations can cause reduction of the performances, inoperability or, in the worst case, failure of the structure. Many works in the last decades have been focussed on the correct analysis of the behaviour of nonlinear structures and all of them have contributed to explaining the mechanisms behind the complex behaviour of these structures [2-4]. Nonetheless, despite the grate advance in computing the response of nonlinear differential equations, how to build a model able to reproduce all the behaviour of an existing nonlinear structure is still an open quest. Several researchers have developed excellent techniques for nonlinear identification, but none of the techniques suggested is ready for complex structure. Some of the techniques reported in the literature are very elegant, but require very complicated and time demanding experimental tests, that make them unusable in industrial scenarios [2]. Others rely on the fact that the system can be described with a very limited numbers of degrees of freedom or on the fact that the shapes of the constitutive laws that govern the nonlinear structure are known. The method presented in this work adds to the existent literature and tries to provide an alternative method that identifies both the shape and the coefficients of the nonlinear equations necessary to describe the behaviour of the structure. The idea behind this method is that a structure that with a smooth nonlinearity, i.e. the nonlinear equations are smooth with their derivatives, at low amplitude of excitation behaves linearly. The linear system able to describe the dynamics of the system when the amplitude of the oscillations is small will be referred to in this work as the underlying linear system. The underlying linear system has all the characteristics of a linear system and can be identified with the most classical techniques of linear identification. This preliminary identification provides a coordinate frame onto the entire dynamics of the system can be projected: the modal space associated with the underlying linear system. The second order nonlinear normal forms (soNNF) based on the same idea: after the system is projected onto the modal space, a series of transformations simplify the nonlinear equations by retaining only the terms that provide a substantial contribution to the response of the system. Mode details on the method can be found in [6] and [7]. If the soNNF are applied to the structure when unforced and undamped, this technique provides algebraic equation for the backbone curves of the system. In previous work these author have proved how the backbone curves can be used to predict the response of the system [8].

Experimental techniques to compute the backbone curves of a system have been described in the literature []. Once the backbone curves are found experimentally, using the soNNF algebraic equations that interpolate the experimental data can be found. The method suggested in this work finds the best fitting between the backbone curves and the experimental data and also gives a confidence bounds for the model and its parameters. This is done using a Bayesian framework in conjunction with Markov Chain Monte Carlo to generate samples from the posterior.

Experimental techniques: procedure and limitations

The backbone curves can be obtained experimentally using the technique described in [5]. An initial sweep sine is used to identify the position of the resonances in the structures. Then the system is excited in the proximity of a resonant peak. Incrementing the frequency by small steps, responses at increasingly higher amplitude are induced. Once the maximum resonant response is reached, the forcing is deactivated so that the system response can decay due to the damping in the system. The backbone curve (solid black line in the plot in fig. 1), is computed by dividing the backbone curve in sub-blocks and relating the average amplitude in the block with the average instantaneous frequency of the signal in the block.

Fig 1: Schematic of the experimental technique (left). Numerical example used as a proof of concept (right) If the signal to noise ratio is particularly high, the error in the estimation of the backbone curve can be extremely high. This implies that the lowest part of the estimated backbone curve is less accurate. Also, since in proximity of the maximum response there is a folding point, the basin of attraction of the response in that region is very small, hence experimentally it is impossible to reach a solution that belongs to the backbone curve. The forcing, therefore, has to be deactivated before a solution on the backbone curve is reached. This results in the decay curve to pass through a transient before it finally sets on the backbone curve. For this reason it is recommended that the highest part of the computed backbone curve is not used for identification purposes. Identification

The identification technique is based on Bayes rule

𝑃 𝜃 𝔇,ℳ =𝑃(𝒟|𝜃,ℳ)𝑃(𝜃|ℳ)

𝑃(𝒟|ℳ) (1)

In particular, using eq. 1, one can evaluate the probability that the parameter vector 𝜃 assumes given a data distribution 𝒟 and a model ℳ. In this case the model is the equation for the backbone curves provided by the soNNF. A priori the model is unknown. This technique allows for multiple models to be tested. A general model from the soNNF has the form

( 𝝎𝒏𝟐 − 𝜴𝟐 )𝑼 + 𝒇(𝜽,𝑼) = 𝟎 (2)

Where 𝝎𝒏𝟐 is the diagonal matrix of the squares of the natural frequencies, 𝜴𝟐 is a diagonal matrix containing

the squares of the frequencies of the point of the measured backbone curve with amplitude 𝑼. Note that for each point of the backbone curve a vector 𝑼 exist. The components of this vector are the modal components necessary to describe the backbone curve. The vector 𝒇 contains the nonlinear functions that characterize a given model. For sake of simplicity in this work only one model is used. To obtain a distribution for the posterior a Marcov Chain Monte Carlo is used.

The results are shown in fig. 2

C

(K,) K K

P1 cos(t) P2 cos(t)

C2 C

x1 x2

M M

Figure 2: A schematic diagram of the nonlinear two-mass oscillator.

A schematic of the system is provided in figure 2. Both masses are connected to ground and betweeneach other with linear springs and viscous dampers. All the springs have the same sti↵ness value K of1 N/m, the external dampers (labeled C )have a damping constant of 1 103

Ns/m and the centraldampers (labeled C2) have a damping constant of 5 104

Ns/m. The spring between ground and thefirst mass has an additional cubic term with a nonlinear coecient of 0.5 N/m

3.In this example, the system has not been excited as previously described. Since the system presented

is simulated, it was possible to reverse the time flow and start the time decay from a solution in the closeproximity of zero. The advantage of this method is that when the amplitude of vibration approaches zero,the system behaves linearly and the initial displacement can be chosen so that it is compatible with themodeshape corresponding to the n

th mode of the underlying linear system. In this case, the eigenvectorsof the underlying linear system are [1, 1] and [1,1]. A simulation starting from [, ] is used to generatea time history from which the first backbone curve can be estimated. The same approach, starting froman initial displacement of [,] gives the time history to estimate the second backbone curve . Here with we mean a generic small displacement.

The backbone curve, estimated evaluating the instantaneous frequency at the zero crossing, are thenused as input data for the Bayesian technique illustrated in the previous section. For this test we hy-pothesized that the number of nonlinear springs and their position were known and that only the value ofthe nonlinearity was unknown. Using the nonlinear normal forms, analytical expressions for the backbonecurves of a system featuring two masses and three linear springs are derived:

S1 :

!

2n1 !

2r1

U1 +3

8m

h

(U1 + U2)3i

= 0, (12)

S2 :

!

2n2 !

2r2

U2 +3

8m

h

(U1 + U2)3i

= 0. (13)

In the model, the number of nonlinear springs is one and its position is known: this simplified the derivationof the backbone curves. Also, only one nonlinear parameter has to be identified. For the estimation of thenonlinear parameter a uniform distribution for the prior has been used. The limit of the distribution foreach parameter are listed in table 1

Parameter Lower Limit Upper limit 0 11 0 0.52 0 0.5

Table 1: Parameter limits for the prior distribution.

The parameter vector in this case is made of the nonlinear parameter and the variance of thedata around the value estimated by the model. To find the parameter vector 8000 posterior samples were

Fig 2: Projection of the backbone curves in the first (left) and the second (right) linear modal coordinates. The identified backbone curves are shown in red and intervals of confidence in blue.

References [1] Cammarano, A., Hill, T.L., Neild, S.A. and Wagg, D.J. 2013, Bifurcations of backbone curves for systems of coupled nonlinear two

mass oscillator Nonlinear Dynamics, pp. 1–10. [2] Worden, K. and Tomlinson, G.R., 2010,Nonlinearity in structural dynamics: detection, identification and modelling. CRC Press, pp.

286–376. [3] Josefsson, A., Magnevall, M. and Ahlin, K., 2007, On nonlinear parameter estimation with random noise signals. Proceedings of

IMAC XXV. [4] Carrella, A. and Ewins, D.J., 2011, Identifying and quantifying structural nonlinearities in engineering applications from measured

frequency response functions. Mechanical Systems and Signal Processing, 25(3) , pp. 1011–1027. [5] Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.,(2009) Nonlinear normal modes, Part I: A useful framework for the

structural dynamicist. Mech. Syst. Signal Pr. 23, pp 170–194 [6] Nayfeh, A.H.,(1993) Method of Normal Forms. Wiley, New York [7] Neild, S.A. and Wagg, D.J.,(2011) Applying the method of normal forms to second order nonlinear vibration problems. Proc. R.

Soc. A 467, pp. 1141–1163 [8] T.L. Hill, A. Cammarano, S.A. Neild, D.J.Wagg, Interpreting the forced responses of a two-degree-of-freedom nonlinear oscillator

using backbone curves, Journal of Sound and Vibration (under review)

1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

Ω

U1

Figure 4: Parameter distribution in the data generated by the MCMC.the fundamentals U1 and U2 respectively. This representation has been chosen to be consistent with thevariable used in the identification methods. Other representations are possible (for example in terms ofthe physical coordinates).

1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ω

U2

Figure 5: Parameter distribution in the data generated by the MCMC.

Conclusions

A procedure for identifying nonlinear system has been described. The method, based on the second ordernonlinear normal forms, uses a Bayesian framework to find the nonlinear parameters that minimize the error

1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

Ω

U1

Figure 4: Parameter distribution in the data generated by the MCMC.the fundamentals U1 and U2 respectively. This representation has been chosen to be consistent with thevariable used in the identification methods. Other representations are possible (for example in terms ofthe physical coordinates).

1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ω

U2

Figure 5: Parameter distribution in the data generated by the MCMC.

Conclusions

A procedure for identifying nonlinear system has been described. The method, based on the second ordernonlinear normal forms, uses a Bayesian framework to find the nonlinear parameters that minimize the error