Identification of multi-degree of freedom non-linear systems using an extended modal space model

ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 23 (2009) 8–29 www.elsevier.com/locate/jnlabr/ymssp

Identification of multi-degree of freedom non-linear systems using an extended modal space model M.F. Plattena, J.R. Wrightb, G. Dimitriadisc,, J.E. Cooperd a

NVH Team, Romax Technology, Rutherford House, Nottingham Science and Technology Park, Nottingham NG7 2PZ, UK School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, P.O. Box 88, Manchester M20 4RL, UK c Aerospace and Mechanical Engineering Department (LTAS), University of Lie`ge, 1 Chemin des Chevreuils, B-4000 Lie`ge, Belgium d Department of Engineering, University of Liverpool, Chadwick Tower, Peach Street, Liverpool L69 7ZE, UK b

Received 12 May 2007; received in revised form 11 October 2007; accepted 17 November 2007 Available online 22 November 2007

Abstract The identification of non-linear dynamic systems is an increasingly important area of research, with potential application in many industries. Current non-linear identification methodologies are, in general, mostly suited to small systems with few degrees of freedom (DOF) and few non-linearities. In order to develop a practical identification approach for real engineering structures, the capability of such methods must be significantly extended. In this paper, it is shown that such an extension can be achieved using multi-exciter techniques in order to excite specific modes or DOF of the system under investigation. A novel identification method for large non-linear systems is presented, based on the use of a multi-exciter arrangement using appropriated excitation applied in bursts. This proposed non-linear resonant decay method is applied to a simulated system with 5 DOF and an experimental clamped panel structure. The technique is essentially a derivative of the restoring force surface method and involves a non-linear curve fit performed in modal space. The effectiveness of the resulting reduced order model in representing the non-linear characteristics of the system is demonstrated. The potential of the approach for the identification of large continuous non-linear systems is also discussed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Non-linear systems; Non-linear system identification; Resonant decay method; Force appropriation

1. Introduction The identification process for linear multi-degree of freedom (DOF) systems [1] is now mature and the methods, whilst operating in the time or frequency domains and using a variety of intermediate models, almost all yield a final model based upon modal parameters, viz. natural frequency, mode shape, modal damping and modal mass. For real, continuous (not lumped parameter) and complex structures, some methods identify effective mass, damping and stiffness matrices of the structure corresponding to measured positions, but usually only as a means of generating modal parameters. Corresponding author. Tel.: +32 4 366 9815.

E-mail address: [email protected] (G. Dimitriadis). 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.11.016

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In practice, most structures exhibit some non-linear behaviour and there is considerable interest in the identification of non-linear dynamic systems [2], particularly where they have multiple DOF. Identification involves a progression through detection, classification, estimation and finally validation of a model that represents the system dynamics adequately. However, the process of model identification for non-linear systems is not as well developed as for linear systems, and a range of different model forms is associated with the many different methods available. The emphasis of this paper is upon non-linear identification of the kind of complex, multi-mode, lightly damped, continuous systems that can presently be identified using linear modal testing methods but demonstrate some non-linear effects (e.g. aircraft). For these systems, the underlying linear behaviour may be represented by normal modes of vibration. It is considered that several of the models currently in use for nonlinear identification, whilst being powerful for lower order problems, are not likely to provide an equivalent to the linear methodology currently employed for higher order systems. For this reason, methods suitable for the identification of a model based in modal space will be considered, together with ideas for handling large complex non-linear systems [3,4]. The approaches presented here are based on the restoring force surface method [5] expressed in modal space, extended by the use of burst excitation, force appropriation and selective non-linear modal terms. The ideas will be illustrated upon a 5 DOF non-linear simulated system and an experimental clamped plate structure. 2. Features of an ideal non-linear system identification approach The authors consider that an ideal non-linear identification method for the kind of structures mentioned above will: (1) cater for continuous as well as lumped parameter systems; (2) allow for many modes, some that may be close or identical in frequency; (3) yield a multi-input/multi-output (MIMO) model, potentially for several excitation and many response positions; (4) identify the types of non-linearity present and the physical location and characteristics of any discrete nonlinear element(s); (5) be readily understandable by the practicing engineer. Current methodologies for non-linear systems fall a long way short of meeting all these criteria. Kerschen et al. [6] recently published a very thorough review of non-linear system identification methods in structural mechanics. They identified some methods such as conditioned reversed path [7,8] and non-linear identification with feedback of outputs [9], which could be capable of identifying large systems but have not been demonstrated on realistic large engineering structures as yet. Additionally complexities such as linear modes close in frequency have not been investigated with these methods either. Furthermore, Kerschen et al. stressed that non-linear system identification requires an accurate characterisation of the non-linear forces present in a system before any parameter estimation can be attempted. Therefore, reliable physical insight of the systems under investigation is essential for a successful identification. Consequently, existing non-linear system identification methodologies fail in terms of requirements (2) and (4) outlined above. 3. Review of current models Some of the main models currently in use for non-linear identification will now be summarised. 3.1. NAR(MA)X The non-linear auto-regressive moving average with exogenous inputs (NARMAX) model, first proposed by Billings [10], is based on discrete time and is a non-linear version of the discrete time ARMA model used in a number of linear methods. It permits the estimation of higher order FRFs by harmonic probing [11]. Most of the work so far carried out using this model has been based on single input/output data and it seems most

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suited to relatively low order complex non-linear systems. The model generated does not lend itself easily to obtaining a meaningful physical parametric model and large order MIMO systems would lead to an enormous number of terms. However, it is worth noting that an attempt to identify a modal space model using NARMAX has been proposed as a means of reducing the model order and catering for larger systems [12], so there could be a possibility of employing NARMAX identification in the approach proposed in this paper. Several authors have introduced improved model selection and parameter estimation methods NAR(MA)X models, such as the orthogonal estimator [13,14] and, more recently, genetic algorithms [15]. 3.2. Equations of motion in physical co-ordinates Second-order differential equations expressed in terms of physical co-ordinates with additional non-linear terms present are used to represent physical lumped parameter systems and are generated by a range of methods [16–19,7,20]. The main disadvantage of this model form lies in its unsuitability for representing continuous systems having a large number of DOF. For instance, if such methods are not suitable as a final form for continuous linear systems, it is unlikely that they will be for non-linear systems. 3.3. Equations of motion in modal co-ordinates A further parametric model based on second-order differential equations is expressed using linear modal coordinates. In this model, the modal mass and linear modal stiffness terms are uncoupled (for proportional damping) and the linear modes may be coupled non-linearly by additional terms in the system equations. Lin et al. [21] considered non-linear systems with complex modes and developed a method to identify non-linearity, which they attempted to extend to MDOF systems while minimising the effect of linear modes on their analysis. Nevertheless, methods employing modal models are primarily based upon the restoring force surface approach [5,22]. This model caters for continuous systems having a large number of DOF and for MIMO, is readily recognisable to the engineer and potentially satisfies many of the criteria for an ideal method. A variant of such models for continuous systems with discrete non-linearities [22] derives a mixed physical/modal space model in which the non-linearity is modelled compactly in the physical part of the model whereas the underlying linear behaviour of the continuous system is represented in modal space. Such an approach could assist in identifying the location of any non-linearity and hence in evaluating structural modification. 3.4. Non-linear normal modes Non-linear normal modes (NNMs) arise from a non-linear transformation applied to uncouple the known equations of motion and these modes assist in understanding the non-linear behaviour of systems [23,24]. They differ from the classical normal modes of the system and are difficult to interpret in terms of the physical system. Some methods to compute such non-linear modes from the equations of motion were developed during the 1990s, such as the multiple scales technique (e.g. [25]). Several new approaches have been proposed since, such as the Galerkin-based approach by Pesheck et al. [26] and the coupled nonlinear modes technique by Camillaci et al. [27]. The use of NNMs in non-linear system identification or characterisation has been theorised but not yet put in practice, although some work is currently under way [28]. 3.5. Higher order FRFs The higher order frequency response function (HFRF) may be derived in a number of ways and can provide a valuable understanding of the underlying non-linear mechanisms present in a system. In principle, the use of hypercurve fitting [29] may yield a second-order ODE in physical co-ordinates. However, it is considered that this approach is limited to relatively low order lumped parameter systems. The time domain equivalent of the HFRF is the Volterra Kernel [2] that allows the response of the system to be represented by a series of convolution integrals. However, the Volterra series is more normally used in defining the HFRF (see for example Tawfiq and Vinh [30], Chatterjee and Vyas [31] and others). The calculation of HFRFs of order higher than 3 is nearly computationally impossible. Vazquez Feijoo et al. [32,33] proposed an alternative

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method for estimating the HFRFs using associated linear equations. Additionally, Peyton Jones [34] introduced a simplified probing algorithm for the calculation of HFRFs. Nevertheless, the complete description of realistic dynamic systems with general non-linear functions using HFRF or Volterra-based approaches is not yet possible. The authors consider that the use of a model based upon modal (or possibly mixed physical/modal) space is arguably a powerful contender for being an identification algorithm with the potential of meeting the criteria set out above. The rest of this paper will consider an approach based on this model. 4. Non-linear modal model In this section, the mathematical form of an extended modal model for non-linear systems will be described. Consider a lumped parameter system or a continuous one that has been discretised as having N DOF, as for example using a finite element model. The equation of motion in physical space for a dynamic system including stiffness non-linearity only is ¯w ¯ ¯w ¯ þ k¯ NL ðwÞ ¼ fðtÞ, € þC _ þ Kw M (1) ¯ is the N  N damping matrix, K ¯ is the N  N mass matrix, C ¯ is the N  N linear stiffness matrix, where M ¯ is the N  1 vector of applied nodal forces and wðtÞ is k¯ NL ðwÞ is the N  1 vector of non-linear functions, fðtÞ the N  1 vector of physical displacements. The dots imply differentiation with respect to time. Notice that while Eq. (1) contains only stiffness non-linearities, some damping non-linearity terms could also be envisaged. The transformation between physical and modal space is defined by (2)

w ¼ Up,

where pðtÞ is the N R  1 vector of modal amplitudes, where N R is the number of retained modes and U is the N  N R modal matrix of the N R modes of the underlying linear system. Note that this is a transformation to a linear modal space and that the non-linearity will be accounted for approximately using cross-coupling terms. The number of retained modes, N R , depends on the frequency range of interest and, in general, will be much smaller than the number of physical DOF, so that N R  N; the modal approach is a form of model order reduction. Substituting the truncated modal expansion into the system equations of motion and premultiplying by UT yields ¯ p þ UT KUp ¯ ¯ p þ UT CU_ ¯ þ UT k¯ NL ðUpÞ ¼ UT fðtÞ. UT MU€

(3)

Using the orthogonality of the modes, this equation of motion in modal space becomes (4)

M€p þ C_p þ Kp þ kNL ðtÞ ¼ fðtÞ,

where the N R  N R modal mass matrix M and the N R  N R linear modal stiffness matrix K are diagonal and the N R  N R modal damping matrix C is diagonal for proportionally damped systems; f is the N R  1 applied modal force vector and kNL ðtÞ is the N R  1 vector of non-linear stiffness forces in modal space. The equation for a particular mode for a proportionally damped non-linear system is in the form of a single DOF system, namely mr p€ r þ cr p_ r þ kr pr þ kr;NL ðtÞ ¼ f r ðtÞ

for

r ¼ 1; 2; . . . ; N R ,

(5)

where pr is the rth modal displacement, mr , cr and kr are the rth mode modal mass, damping and stiffness and f r is the applied modal force. The modal mass and stiffness for the linear system are related by the undamped natural frequency from the equation kr ¼ o2r mr . Non-proportional damping would lead to the presence of modal damping coupling terms. The term kr;NL refers to the rth mode non-linear modal restoring force and in general is a function of several modal coordinates to allow for non-linear stiffness cross-coupling. It is assumed that the type of nonlinear functions present in the system is known and that these functions can be approximated using a set of basis functions gðpÞ, where g is a N b  1 vector of basis functions and N b can be any positive integer. Then, the non-linear force, kr;NL , in Eq. (5) can be approximated as kr;NL ¼

Nb X j¼1

aj; r gj ðbj; r  pÞ,

(6)

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where gj denotes the jth element of g, aj; r is the coefficient of the jth basis function approximating the rth nonlinear force, bj; r is a N R  1 vector whose elements are 0 or 1 and whose role is to determine which of the system’s modes are involved in this particular basis function; the notation bj; r  p is used to denote element by element multiplication of vectors. In fact, the inclusion of the bj; r vector is the basis of the non-linear system identification approach presented here as it demonstrates that only some of the modes are considered nonlinear and it is only these modes that are included in the non-linear curve-fit. As an example, consider a non-linear system described by two modes, p1 and p2 , and whose non-linear stiffness function can be approximated by sinusoidal basis functions. Then, g1 ðpÞ ¼ sin p1 þ sin p2 þ sinðp1 þ p2 Þ, g2 ðpÞ ¼ sin 2p1 þ sin 2p2 þ sin 2ðp1 þ p2 Þ, etc. Furthermore, if k1;NL ¼ sin p1 þ sin 2p2 and k2;NL ¼ sin p1 þ sinðp1 þ p2 Þ, the bj; r vectors must be given by b1;1 ¼ ½1 0, b2;1 ¼ ½0 1, b1;2 ¼ ½1 1 and b2;2 ¼ ½0 0. In cases where the most suitable basis functions are polynomial, Eq. (6) becomes kr;NL ¼

Nb X

aTj ðbj; r  pÞj ,

(7)

j¼2

where the notation xj is used here in a Kronecker sense, e.g. x2 ¼ x  x, x3 ¼ x2  x etc, and aj are vectors of coefficients of suitable dimensions. 4.1. Modal and physical coordinates The modal coordinates p of Eq. (2) will in practice be obtained from the measurements of N m measured physical coordinates wm , where usually N R oN m  N. Then, p ¼ ðUTm Um Þ1 UTm wm

(8)

where Um is the N M  N R modal matrix corresponding to the measured set of responses. The modal matrix will be identified using the data obtained at an excitation level where the system behaves in an essentially linear manner. An alternative and supplementary representation of the system is to use a mixed physical and modal model [22]. This may be appropriate if it is believed that a limited number of discrete non-linearities are present in the system and if the locations and parameters of the non-linear elements are sought. In this case the non-linear stiffness term is rewritten in terms of functions of physical displacements, namely kr;NL ¼

Nb X

a¯ j; r gj ðb¯ j; r  wm Þ,

(9)

j¼1

where a¯ j; r are the coefficients of the basis functions, and b¯ j; r is a N m  1 vector whose elements are 0 or 1. As with the purely modal model, the assumption is that only some of the physical coordinates are non-linear, as demonstrated by the inclusion of the b¯ j; r vector in Eq. (9). Note that some of the terms in Eq. (9) may be associated with the relative displacement of two points on the structure, and powers of it. For example, if the basis functions are polynomial and b¯ j; r ¼ ½1  1 0    0 then kr;NL can consist of terms of the form ðwm2  wm1 Þ2 , ðwm2  wm1 Þ3 , etc., where wm1 denotes the first element of wm . 5. Estimating the modal model of the underlying linear system While the modal and modal–physical models of the previous sections are very useful for describing large structures with few non-linear modes or states, they require prior knowledge of the modal matrix of the underlying linear system. For a linear structure, there are two common approaches to modal analysis: ‘phase separation’ methods seek to curve-fit a model containing all modes to the data whilst ‘phase resonance’ methods [35,36] aim to excite one mode at a time using normal mode tuning (so-called ‘force appropriation’). For a non-linear system, none of these approaches will work as long as the nonlinear elements are excited and contribute to the system response. However, for systems with a significant class of stiffness non-linearities, it is possible to obtain nearly linear responses as long as the excitation applied to the system (and hence the

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non-linearities) is of sufficiently low amplitude. Such non-linear function include polynomials, sine and piecewise linear functions. A general non-linear stiffness element whose extension is denoted by x and restoring force by f ðxÞ can be said to be nearly linear at small displacements if lim

x!xE

df ¼ c, dx

(10)

where c is a real constant and xE is an equilibrium point of the system. Applying excitation forces of low amplitude to systems containing such non-linearities allows for the identification of a modal model of the underlying nearly linear system. Examples of non-linearities that behave in the manner of Eq. (10) are polynomial stiffness and freeplay/bilinear stiffness [37]. 6. Types of modes The modal model approach has already been used in non-linear system identification in the past. Methods employing this form of model are primarily based upon the restoring force surface approach [5,22]. However, in these works all of the modes were considered as candidate non-linear modes, i.e. the non-linear functions were assumed to be functions of all the modes. This approach can cause difficulties for systems with many modes of interest and requires complex term-selection procedures to be employed in order to determine which of the modes really contribute to the non-linearity. As argued earlier, sophisticated mathematical procedures are no substitute for physical knowledge in non-linear system identification. It can be argued that the characteristics of many real-world multi-mode, non-linear and lightly damped structures, when represented in modal space, fall into one or more of the following categories [3]: (1) Linear proportionally damped modes, well separated in frequency. (2) Linear proportionally damped modes, very close or identical in frequency. (3) Linear non-proportionally damped modes (usually fairly close in frequency for damping coupling effects to be significant). (4) Modes influenced by non-linear effects but having no significant non-linear coupling to other modes. (5) Modes influenced by non-linear effects but where significant non-linear coupling to other modes (often close in frequency) is present. A modal model based on these categories would be far simpler than a fully coupled non-linear model because the number of coupling terms would be limited. Clearly, approximations would be needed in deciding which modes were in each category. However, if the model allowed the response of the system under various excitation cases to be replicated with reasonable accuracy, then it would suffice for practical purposes. The authors would argue that the majority of the modes of most complex non-linear structures (e.g. aircraft) would tend to fall into category (1) or (2) where reasonably accurate identification using classical linear methods is possible. The remaining modes will necessitate an identification process targeted at estimating key terms in the extended modal model, namely linear cross-coupling terms in the presence of nonproportional damping, direct non-linear terms and non-linear cross-coupling terms. The ideas presented in this paper aim to introduce, present and explore this identification concept. 7. Non-linear identification in modal space As noted above, classical restoring force surface identification of non-linear multi-DOF systems in modal space [5,22] would involve using data from a general excitation in which all modes, all direct non-linear terms and all non-linear cross-coupling terms were potentially present. Such an approach would lead to a very complex unknown non-linear model structure, with too many parameters to determine for a large number of modes. No guidance was given in [5,22] as to a strategy for dealing with this problem. An alternative philosophy would be to seek to reduce the order of each identification performed by aiming to use a multi-exciter force pattern to excite at any one time, (a) only a single mode or (b) a limited number (group) of modes. Both approaches are in essence based upon the idea of force appropriation which permits

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the determination of a monophase force vector, estimated from the measured FRF matrix [38,36]. This force vector will induce single mode behaviour when applied to a linear system at the relevant undamped natural frequency. A single mode may thus be ‘tuned’ and measured. (a) Exciting a single mode: One approach extends the classical linear force appropriation approach to nonlinear systems in order to reduce the scale of any identification performed. In the force appropriation of nonlinear systems (FANS) method [39], a force pattern that includes higher harmonic terms, whose parameters are determined using an optimisation approach, is used to excite the structure in the steady state at several different excitation levels. The excitation aims to prevent any response other than in the underlying linear mode shape of interest. Effectively, the additional harmonic force terms counteract the non-linear modal cross-coupling forces. The direct non-linear terms in the equation for this mode may then be identified for a single mode. Important non-linear modal cross-coupling terms could subsequently be identified using a combination of excitation vectors chosen to excite the coupled modes of interest—such an approach is described in category (b) below. This whole approach is currently rather slow and will not be explored further in this paper. (b) Exciting a group of modes: Another approach to reducing the effective model order is to extend the resonant decay method (RDM) [40] to the identification of a non-linear model in modal space. In its normal implementation for non-proportionally damped linear systems, the RDM uses force appropriation to enable small groups of modes coupled by damping forces to be excited. The appropriated force vector is applied as a ‘burst’ of a sine wave at the undamped natural frequency of a mode of interest. If the appropriation is perfect, then only this mode responds in the steady-state phase of the burst. However, during the decay, any modes coupled by damping forces to the mode of interest will also respond. A curve-fit to a limited subset of modes can then be performed to yield any significant modal damping coupling terms. Even if the appropriation is imperfect, most modes will be too distant in frequency to respond greatly at the excitation frequency, but any mode that does respond significantly may be included in the fit. The application and extension of this idea to non-linear systems will be explained below.

8. RDM extension for non-linear systems Here the extension of the RDM to non-linear systems (NL-RDM) is discussed. The practical identification process using an extension of the RDM for a large order system is summarised in the flow diagram in Fig. 1 and described below. To some extent, the approach may be regarded as being an addition to a standard ground vibration test where multiple exciters and normal mode tuning are involved. (a) Classify linear and non-linear modes: A modal test with multi-exciters and random (or multisine) excitation is performed at several excitation levels as a check on homogeneity. It is anticipated that some Test system at several force levels with multipoint excitation – identify modes behaving non-linearly

Using parameter estimation (or normal mode tuning) approach on FRF data at low excitation level, identify modal quantities for modes of underlying linear model

Calculate MMIF from low level FRF and determine appropriated force vectors for modes that behave nonlinearly (or may have significant non-proportional damping) – need to use suitable exciter positions

Apply force pattern as a ‘burst’ at the natural frequency for each these modes – use regression to obtain a modal model - include couplings to other modes as appropriate

Assemble and check final model as to ability to reproduce forced response of real system

Fig. 1. Flow diagram of identification process for high order systems.

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modal peaks will exhibit non-linear effects whereas other modes will be largely unaffected by an increase in excitation level. The modes may then be classified broadly as linear or non-linear. Referring to Eq. (5), if the lth mode is non-linear, then at least one of the non-linear functions kr;NL will depend on it, i.e. kr;NL ¼ kr;NL ðpl Þ. (b) Identify modal parameters of underlying linear system: An FRF matrix for a low excitation level is used to estimate the modal parameters for the so-called linear modes using a parameter estimation algorithm (e.g. least squares complex exponential). Provided the condition of Eq. (10) is satisfied, the FRF matrix represents the behaviour of the underlying linear model. Alternatively, the FRF matrix may be used to estimate the appropriated force vector for each mode. The mode shape may then be estimated by normal mode tuning, either on the structure itself at low level (‘hard’ tuning) or preferably using the FRF matrix (‘soft’ tuning). This approach is slower but should be used to obtain an improved mode shape estimate for any close modes or any fitted mode shapes that are not real and may therefore be non-proportionally damped. The aim of this stage is to establish a suitable modal matrix for transformation between the physical and modal spaces. Additionally, for any mode that was seen to behave linearly in the homogeneity test and was proportionally damped, the modal parameters estimated may be used in the final model. However, parameters for modes that behaved non-linearly will need to be identified separately using a process such as the modified RDM. (c) Mode by mode excitation: The multivariate mode indicator function (MMIF) [29] is then calculated from the low level FRF matrix and force vectors determined for each mode of interest (i.e. those affected by nonlinearity or non-proportional damping). It is important to use a suitable choice of number and location of exciter positions [41]. The appropriated force vector is then applied as a burst sine to each of these modes in turn, at a level great enough to excite any non-linear behaviour present. If the mode is non-linearly uncoupled, then the appropriated mode should dominate the response in the steady-state phase. If it is non-linearly coupled, other modes may also exhibit a significant response. During the decay, the presence of linear damping couplings as well as non-linear couplings between the modes will be apparent. (d) Mode by mode identification: A suitable curve-fit of Eq. (5) may then be used to identify the linear and non-linear modal parameters for each mode. For uncoupled linear or non-linear modes, Eq. (5) should only involve the modal response associated with that mode. For coupled linear or non-linear modes, Eq. (5) should include all modes coupled to the mode of interest and those excited significantly by virtue of imperfect appropriation. Whether the purely modal model of Eq. (6) or the combined physical and modal model of Eq. (9) is selected for the curve-fit, it is imperative that appropriate basis functions are chosen, that can represent the true non-linearity in the system. In essence, the burst appropriation excitation should allow a large model to be identified approximately mode by mode by curve-fitting a series of relatively small modal models. (e) Assemble and validate final model: The modal equations may now be assembled from the initial linear and non-linear test phases and the final model validate against suitable validation input–output data obtained from the actual system. It should be noted that the procedure described here will fail when applied to a system undergoing limit cycle or chaotic oscillations. In the presence of limit cycle or any other self-excited oscillations it will be impossible to obtain a decaying response from the system. Furthermore, if the system undergoes chaotic oscillations at particular amplitudes and frequencies of the excitation force, there will be several internal resonances that will most likely excite all the modes, even the ones that are not non-linearly coupled. 9. Illustration of the NL-RDM for a simple simulated non-linear system The application of such an approach will be illustrated at this stage using a five DOF non-linear system where some of the modes are in fact linear whilst others are non-linear [3]; the identification process will be exact because all the terms in the series may be included. A simple system is chosen initially to aid the understanding of the method. The system chosen is shown in Fig. 2. It is designed to be symmetric in its linear components so as to yield very simple mode shapes. The system has a hardening cubic stiffness non-linearity between Masses 2 and 4. If the displacements of mass 1, mass 2, etc. are denoted by w1 , w2 , etc., the unforced equations of

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motion can be written as 0 m 0 0 0 B B 0 2m 0 0 B B B0 0 3m 0 B B B0 0 0 2m @ 0 0 0 0 0 2k k 0 B B k 2k k B B þB B 0 k 2k B B 0 0 k @ 0 0 0

0

0

1

2c

0

c

0

0

1

C B C B c 2c c 0 0 C 0C C B C C B C B C 0 Cw € þ B 0 c 2c c 0 C Cw_ C B C B 0 0 c 2c c C 0C A @ A 0 0 0 c 2c m 1 1 0 0 0 0 C C B B bðw4  w2 Þ3 C 0 0 C C C B C C B C ¼ 0, B k 0 C 0 w þ C C B C C B 3 C C B 2k k A @ bðw4  w2 Þ A k 2k 0

ð11Þ

where w ¼ ½w1 w2 w3 w4 w5 T . The parameters used for this simulation are m ¼ 1 kg, c ¼ 4:8 Ns=m, k ¼ 4  103 N=m and b ¼ 5  109 Nm3 . The resulting natural frequencies and mode shapes of the undamped system are tabulated in Table 1.It can be seen that modes 4 and 5 are very close in frequency. Transformation of Eqs. (11) into linear modal space will yield a system of equations of the form 6:05p€1 þ 3:54p_1 þ 0:30  104 p1 ¼ 0, 5:07p€2 þ 15:43p_2 þ 1:29  104 p2 þ 1010 ðw4  w2 Þ3 ¼ 0, 6:29p€3 þ 34:24p_3 þ 2:85  104 p3 ¼ 0, 2:54p€4 þ 28:8p_4 þ 2:4  104 p4 þ 3:66  109 ðw4  w2 Þ3 ¼ 0, 2:75p€5 þ 31:78p_5 þ 2:65  104 p5 ¼ 0,

ð12Þ

which show that some of the modes are completely uncoupled while the modal equations for modes 2 and 4 contain non-linear contributions expressed in terms of physical coordinates. If the non-linear terms are

c

c

c Mass 2 2m

Mass 1 m

c Mass 3 3m

k

k

β

k

c Mass 4 2m

k

c Mass 5 m

k

k

Fig. 2. Five DOF System.

Table 1 Frequencies and mode shapes of 5 DOF simulated system Mode shapes Modal frequencies

1

2

3

4

5

3.51 8.01 10.72 15.48 15.63

0.44 0.82 1 0.82 0.44

0.73 1 0 1 0.73

0.81 0.7 1 0.7 0.81

1 0.37 0 0.37 1

1 0.41 0.16 0.41 1

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expressed in modal coordinates then the equations for modes 2 and 4 become 5:07p€2 þ 15:43p_2 þ 1:29  104 p2 þ 1010 ð0:73p4 þ 2p2 Þ3 ¼ 0, 2:54p€4 þ 28:8p_4 þ 2:4  104 p4 þ 3:66  109 ð0:73p4 þ 2p2 Þ3 ¼ 0.

ð13Þ

Initially, the process will be shown using a perfect appropriation with five exciters. The exact MMIF based on the known physical model is shown in Fig. 3 and the two close modes can be seen by the drop in two indicator functions around 15 Hz. The nearer the indicator function drop is to zero, the better will be the appropriation of that mode. The drops occur at the undamped natural frequencies. An MMIF based on a low-level random excitation yields very similar results. The eigenvectors corresponding to each drop are then used to determine the linear mode shapes by ‘soft tuning’, i.e. multiplying the imaginary part of the FRF matrix by the force vector at the relevant natural frequency. The resulting mode shapes are real and shown in Fig. 4 in phase scatter format—each arrow represents a displacement and a perfect normal mode has all arrows aligned vertically. In order to illustrate the burst principle for a linear mode, a burst is applied to excite mode 5. The result is shown in Fig. 5 where the modal force and response for all five modes are shown. Because mode 5 behaves linearly and the correct appropriated force vector is used, no modal force is inputted to the other modes, there is no coupling in modal space and only mode 5 responds. A similar result would be seen for modes 1 and 3. Thus, the model for modes 1, 3 and 5 may be written from the low level identification stage of the process without any further tests. It is modes 2 and 4 that are coupled non-linearly and need a non-linear model to be identified. As an example, consider a burst applied to mode 4 as shown in Fig. 6. Again there is only a modal force for mode 4 but now mode 2 responds due to the non-linear coupling. Modes 1, 3 and 5 are not excited because of the force appropriation that remains effective even though modes 2 and 4 are responding non-linearly. 1 0.9

MMIF Eigenvalue

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

5

10

15 20 Frequency (Hz)

25

30

Fig. 3. MMIF of 5 DOF system for five exciters using exact FRF.

120

90 1 0.5

150

120

60 30 0

180 210

330 240

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300

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330

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300

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60

0.5

150 180

120 30 0 330

210 240

270

300

90 1

60

0.5

150

30

180

0 330

210 240

Fig. 4. Scatter diagrams showing mode shapes of 5 DOF system determined by soft tuning exact FRF.

270

300

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-2

Modal Force (N)

Modal Acceleration (ms )

200 Mode 1

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9 200

Mode 5

20

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1

2

3

4

5

6

7

8

9

Time (s)

Time (s)

Fig. 5. Modal forces and responses to burst excitation of mode 5 of 5 DOF system using perfect appropriation.

In order to identify the parameters for mode 4, for example, a curve-fit is performed in modal space. This requires that velocity and acceleration modal responses are known. Because the burst is designed such that the signals are leakage free (i.e. they start at zero and decay to zero within the data window), the displacements can be differentiated in the time domain to yield the desired velocities and accelerations. In experimental application of the NL-RDM, accelerations are integrated to generate velocities and displacements. The integration process benefits greatly from the fact that the excitation and response signals are leakage-free. In most other non-linear system identification methods this is not taken advantage of. It is now possible to carry out the curve-fit for mode 4 using only the modal responses associated with modes 2 and 4, though the curve-fit process examines but rejects couplings with other modes too. The curve-fit of the rth modal equation, where r ¼ 2; 4, is of the form p€ r ¼ 

1 ðp_ p p3 p2 p p p2 p3 Þðcr kr a1; r a2; r a3; r a4; r ÞT . mr r r 4 4 2 4 2 2

(14)

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Modal Acceleration (ms-2 )

Modal Force (N) 200 Mode 1

50

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50

100

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2

3

4

5

6

7

8

9

Time (s)

Time (s)

Fig. 6. Modal forces and responses to burst excitation of mode 4 of 5 DOF system using perfect appropriation with five exciters.

Table 2 Identified and true non-linear coefficients of modal equation (4) of 5 DOF simulated system Parameter

Identified True

Non-linear cubic stiffness coefficients a1; 4

a2; 4

a3; 4

a4; 4

1:44  109 1:44  109

1:18  1010 1:18  1010

3:22  1010 3:22  1010

2:93  1010 2:93  1010

Note that mr can either be assumed to be known from the low level excitation stage of the identification process or set to be equal to 1. The resulting identified parameters are found to agree almost identically with the true linear and non-linear modal terms (e.g. see Table 2).

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20

As a further illustration of the method, this time with some degree of imperfection present, consider using three exciters at masses 1, 3 and 5 and generating an FRF matrix by simulating the response of the non-linear system to multiple uncorrelated random excitation at a low force level. The resulting indicator function is shown in Fig. 7. In this case, the choice of these three exciters is an excellent one but the errors introduced by estimating the FRF matrix affect the force vectors and hence the ‘soft tuned’ modes shown in Fig. 8. It may be seen that the modes show some degree of complexity and errors will be introduced into the resulting modal matrix. Interestingly, the errors on the force vectors are larger than the errors on the mode shapes, because of the sensitivity of the MMIF to errors on the FRFs where very close lightly damped modes are concerned. Note that the polyreference approach can be employed to estimate the underlying linear system’s modal parameters (including modal mass). The force vectors are now applied as a burst as before. As an example, consider the estimation of the modal parameters for mode 2, the mode that behaves most non-linearly. The set of modal forces and responses is shown in Fig. 9 and mode 4 is seen to respond due to the non-linear coupling. The errors in the force vectors and mode shapes mean that the modal force is not zero for mode 4 but this mode is coupled anyway to mode 2 so would be included in the non-linear curve-fit. Also, there is a modal force in modes 1 and 5 so a small response is present in these modes. The modal forces and responses for mode 5 appropriation are shown in Fig. 10. Here the imperfect appropriation leads to a modal force present in mode 1, together with a small response because this is distant in frequency. This small response is probably a linear feature but will be included in the curve-fit as possibly non-linear to illustrate the model reduction process. Note that the absolute values of modal force and response are not unique, as they depend upon the mode shape normalisation used.

1

MMIF Eigenvalue

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

5

10

15

20

25

30

Frequency (Hz)

Fig. 7. MMIF of 5 DOF System for 3 exciters using Estimated FRF

120

90 1 0.5

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120

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0 330

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300

Fig. 8. Scatter diagrams showing mode shapes of 5 DOF system determined by ‘soft tuning’.

240

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300

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Modal Acceleration (ms-2 )

Modal Force (N) 50 Mode 1

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14 50

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50 0

0

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2

4

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8

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12

14 50

Mode 5

50 0

0

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2

4

6

8

10

12

14

Time (s)

Time (s)

Fig. 9. Modal forces and responses to burst excitation of mode 2 of 5 DOF system using imperfect appropriation with three exciters.

Figs. 9 and 10 suggest that there is clear non-linearity in modes 2 and 4 and possibly in mode 1. As the nonlinear curve-fit of the modal equations must include all the suspected non-linear modes, Eqs. (14) changes to 1 ðp_ p p p p p2 p p2 p p2 p p2 p p2 p p2 p p3 p3 p3 Þ mr r r 1 2 4 1 2 1 4 2 1 2 4 4 1 4 2 1 2 4 ðcr kr a1; r a2; r a3; r a4; r a5; r a6; r a7; r a8; r a9; r a10; r ÞT

p€ r ¼ 

ð15Þ

for r ¼ 1; 2; 4. Parameters in Eqs. (15) must now be estimated using a model selection technique. Backwards elimination is used in this case, which removes all the terms involving p1 , thus accurately selecting the structure of the true system. The estimated modal parameters are a little different from the true values but still quite accurate. The comparison of the coefficients of the non-linear terms for mode 4 is shown in Table 3. The best check on the model is to compare the response in physical space with that for the true system at several force levels. The point FRF at mass 1 has been evaluated using stepped sine excitation at two force levels. The results are shown in Fig. 11and agreement is basically good except around the very close mode

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Modal Acceleration (ms-2 )

Modal Force (N)

Mode 1

50

200 100

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0

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2

3

4

5

6

7

8

9

Time (s)

Time (s)

Fig. 10. Modal forces and responses to burst excitation of mode 5 of 5 DOF system using imperfect appropriation with three exciters.

Table 3 Identified and true non-linear coefficients of modal equation (4) of 5 DOF simulated system - imperfect appropriation Parameter

Identified True

Non-linear cubic stiffness coefficients ra1; 4

a2; 4

a3; 4

a4; 4

1:39  109 1:44  109

1:15  1010 1:18  1010

3:19  1010 3:22  1010

2:93  1010 2:93  1010

pairs 4 and 5. The discrepancy is more likely due to inaccuracy in damping estimation. Fig. 11b was obtained for a high level of excitation amplitude; it can be clearly seen that mode 2 has been driven well into the nonlinear region with the classic ‘jump’ phenomena evident. The accuracy of the estimated model is good even at this high excitation level.

ARTICLE IN PRESS M.F. Platten et al. / Mechanical Systems and Signal Processing 23 (2009) 8–29

1

x 10

23

-3

True Identified

0.9

0.7

-1

Magnitude of H (mN )

0.8

0.6 0.5 0.4 0.3 0.2 0.1 0 2

4

6

8

10

12

14

16

18

20

Frequency (Hz)

1

x 10

-3

True Identified

0.9

Magnitude of H (mN-1)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2

4

6

8

10

12

14

16

18

20

Frequency (Hz)

Fig. 11. Point FRF at mass 1 for stepped sine excitation of 5 DOF system (a) Low amplitude excitation (b) High amplitude excitation.

10. Imperfect force appropriation and modal matrix Imperfect force appropriation implies that the force vector is unable to isolate the mode of interest perfectly. For a linear system, it can occur if too few exciters are used or if the exciters are not optimally placed. In practice, the number of exciters required should equal the effective number of modes around a frequency of interest; typically four exciters adequately placed should suffice. If the RDM is applied to a linear system and a burst applied to excite a given mode, the effect of imperfect appropriation will be to introduce a finite modal force into one or more other modes and thus cause them to respond. Response of modes close in frequency will be important but provided these modal responses are included in a model selection procedure, the imperfection will be catered for because all data of importance are included. No spurious modal coupling terms ought to be generated. The response of modes more distant in frequency will probably only be small and

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M.F. Platten et al. / Mechanical Systems and Signal Processing 23 (2009) 8–29

could be omitted from the curve-fit with little error being introduced. Thus, imperfect appropriation should not be a major problem provided the mode shapes are adequate. For non-linear systems, imperfect appropriation will also occur if a force vector, evaluated from an FRF generated at a low force level, behaves differently at a higher force level. Any changes due to excitation amplitude are expected to affect the modes responding at frequencies close to that of the non-linear mode. Thus, if the relevant modes are included in the regression, the identification should account for other modes being excited close in frequency, as for a linear system. Essentially, the force appropriation vector applied at a single frequency is intended to act as a filter to minimise the intrusion of too many other modes into the analysis; it need not be perfect. However, this issue needs to be demonstrated and explored on more complex systems. As regards using an imperfect modal matrix in the identification process, the effect will be that the modes of the underlying linear system will not be orthogonal so the transformed equations will be coupled on a linear basis. Inclusion of linear stiffness cross-coupling terms in the regression will provide some indication of the seriousness of any mode shape error but any re-orthogonalisation of the identified equations would not be ideal. This issue is probably more critical than the imperfect excitation and means that the low excitation level identification has to be of good quality and then any small linear coupling terms ignored. Use of a combination of curve-fitting and normal mode tuning would yield the best set of mode shapes, plus only a subset of responses needs to be employed in the transformation. Again, this issue needs further exploration. 11. Experimental application to a clamped plate Clearly what is of interest is whether the proposed method can be applied to real structures. Here, a clamped plate is used as a demonstration [4]. A fully clamped panel under transverse loading is known to display nonlinear geometric stiffening due to stretching of the plate centre line. However, it is extremely difficult to implement a clamping arrangement experimentally, and the behaviour of a test structure may be obscured somewhat by the contribution of the support structure. For this reason, it was decided to construct a test structure by machining a plate with integral edge support members from a solid piece of aluminium, and to test the whole structure in a free–free arrangement as shown in Fig. 12. The dimensions of the plate were 400 mm  300 mm  2 mm thick and the edge members were 50 mm square. In a sense, the structure is a single bay panel with sizeable edge stiffeners. The edge members do not provide a perfect clamp but they will partially restrict the in-plane deformation of the plate under transverse loading and thus some non-linear stiffening should result. For initial testing, the aim was to focus only on modes where nodal lines were parallel to the short edge of the panel. As shown in Fig. 13, the panel was instrumented with eight accelerometers, six along the centre line to measure these modes and two positioned off centre to allow information about the modes shapes to be known. The two Gearing and

Fig. 12. Non-linear panel structure under test.

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25

7

1

2

3

4

5

6

8

Fig. 13. Accelerometer positions for non-linear plate test—excitation is at positions 2 and 5.

Table 4 Modal parameters of symmetric modes of clamped plate Parameter

Mode 1

Mode 2

Mode 3

Frequency (Hz) Damping ratio (%)

113.1 1.76

180.3 1.11

321.2 0.94

Watson V4 exciters were positioned at 2 and 5, with applied forces measured using force gauges. Data were acquired and analysed using an LMS SCADAS II system. A dedicated user program was written for the burst acquisition. Initially, the FRFs were estimated using low level uncorrelated random excitation, followed by analysis using the polyreference time domain identification method. The FRFs were of good quality and demonstrated reasonable reciprocity, together with good coherence. The analysis yielded essentially real mode shapes for the modes of interest, with frequencies and damping ratios for the first three modes as shown in Table 4.Mode 1 has no nodal lines (i.e. is a simple bending mode) whereas modes 2 and 3 have, respectively, one and two nodal lines parallel to the short edge, as shown in Fig. 14. Other modes were exhibited but these had nodal lines parallel to the long edge and were obviously not well excited as the exciters were deliberately positioned on the centre line. At higher excitation levels the coherence deteriorated significantly due to non-linear effects, seen on all three modes. Next, an MMIF analysis was carried out to determine the appropriated force vectors needed to excite the modes of interest. The force ratios between the two forces were 0.99, 1:04 and 1.02, all very close to the unity values expected for a perfectly symmetric exciter placement on the centre line of a perfectly symmetric structure. Note, however, that essentially the same force pattern is needed for both modes 1 and 3, so when exciting mode 1 some response in mode 3 might be expected, though it is a lot higher in frequency. Note that the exciters were positioned relatively close to the edge members to avoid excessive force ‘drop-out’. Next, normal mode tuning was attempted for these three modes in order to determine the voltage patterns required to drive each mode in the burst appropriation stage. This is an issue because experimentally the forces are not in phase with the voltage. Tuning was not straightforward due to the severe shaker–structure interaction and the appearance of second harmonics in the force signal. For more substantial structures this is usually less of a problem. A burst excitation was then applied to each mode at a high force level by specifying the appropriate voltage pattern. Finally, the NL-RDM analysis was performed, after integrating the time signals and converting to modal space using a modal matrix comprising the polyreference derived mode shapes. The modal force and acceleration time histories used in the analysis for mode 1 are shown in Fig. 15.

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Mode 1

Mode 2

1

1

0.5

0

0 0.4

0.4

0.4

0.2 y

0

0.2 y

0.2 x

0

0.4 0

0

0.2 x

Mode 3

2 0

0.4 0.2 y

0.4 0

0.2 x

0

Fig. 14. First three mode shapes with nodal lines parallel to short edges.

Modal Displacement (mm)

Mode 3

Mode 2

Mode 1

Modal Force (N) 2

50

0

0 -50

-2 2

50 0

0

-50

-2 2

50 0

0

-50

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4

8

0

4

8

Time (s)

Fig. 15. Modal forces and response due to the burst appropriation of mode 1.

The curve-fit to mode 1 showed a significant cubic stiffness non-linearity, with small quadratic stiffness terms and some evidence of a relatively insignificant damping non-linearity at the level tested. When coupling to mode 3 was included, it was found to be insignificant so mode 1 is essentially a mode with direct non-linear terms only. The modal mass, stiffness and damping quantities indicated an undamped natural frequency of 112.1 Hz and a damping ratio of 1.47%, close to the parameters identified at low level (see Table 4). A nonlinear cubic model stiffness coefficient was obtained, so allowing the non-linear behaviour of the plate to be explored. The mode 1 restoring force surface derived from the experimental raw data of modal force and acceleration, using the fitted modal mass, is shown in Fig. 16a as a 3D surface, and in Fig. 16b as a section in the

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Fig. 16. Raw [black (blue in the online version)] and identified [gray (red in the online version)] restoring force surfaces for clamped panel—mode 1 (a) 3D surface. (b) Force–displacement plane.

force–displacement plane. The identified modal restoring force is overplotted at the same modal velocity and displacement values. The goodness of fit was above 0.99. Analysis of modes 2 and 3 did not yield any non-linear stiffening, presumably because the response levels were insufficiently high to exercise the non-linear effect. The inability of the shakers to produce the response levels required to make the non-linear behaviour of modes 2 and 3 apparent is due to a number of reasons. Force drop-out at resonance: This is caused by the large displacement response of the structure at resonance which affects the ability of the amplifier to provide the required current to the shaker. This can be minimised through the use of a constant current amplifier and careful choice of exciter positions. Shaker– structure interaction: The effect of this is to introduce further force drop-out due to the coil mass and this cannot be eliminated by the use of a constant current amplifier. The attachment of the shaker to the structure under test affects the dynamics of the test structure by adding mass (and to a degree, stiffness and damping). This is particularly apparent for lightweight structures where shakers with the smallest possible moving mass should be used. Shaker non-linearity: Studies have shown [42] that the position of the shaker voice coil in its magnetic field can lead to non-linear behaviour and the generation of second harmonics at high displacement amplitudes. Larger shakers with longer stroke may minimise this problem. However, this could lead to worse shaker–structure interaction problems due to increased moving mass. Nevertheless, these are problems associated with all modal testing procedures and will affect all non-linear system identification methods equally. An alternative to shakers would be the use of acoustic excitation (see for example Platten et al. [43]), some results from which have already been presented [44]. It was found that the acoustic exciter did not suffer from force drop-out and non-linear behaviour in modes 2 and 3 was identified, unlike the experiment with the shakers. 12. Conclusions In this paper a novel approach for the identification of complex, realistic non-linear structures has been presented. The method relies on the assumption that such structures are predominantly linear and feature a small number of non-linear modes. Thus, the curve-fitting process is simplified and rendered more robust since only a small set of non-linear basis functions is required.

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A non-linear form of the resonant decay method is developed in order to determine which modes are nonlinear. The resulting approach is in the form of a combination of force appropriation and the restoring force surface method. It is shown to have the potential to allow a multi-stage identification of high order continuous non-linear systems, although practical limitations such as imperfect appropriation and mode shapes can reduce accuracy. The potential of this approach is that the identification of practical systems with a large number of modes may be performed such that modes are treated individually or in small groups. A simple 5 DOF system has been identified as an illustration of the method. Additionally, the technique has been applied to an experimental clamped panel structure and was shown to yield an accurate identified modal restoring force for the most non-linear mode. While only stiffness non-linearities were considered in this paper, the proposed methodology can also accommodate damping non-linearities, simply by adding modal velocity terms to the non-linear curve-fits. Acknowledgements The authors would like to acknowledge the support received by the Engineering and Physical Sciences Research Council and BAE Systems.

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