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The Smooth Decomposition as a nonlinear modal analysis tool
Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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The Smooth Decomposition as a nonlinear modal analysis tool Sergio Bellizzi a,n, Rubens Sampaio b a b
LMA, CNRS, UPR 7051, Centrale Marseille, Aix-Marseille Univ, F-13402 Marseille Cedex 20, France PUC-Rio, Mechanical Engineering Department, Rua Marquês de São Vicente, 225, Gávea, Rio de Janeiro, RJ, CEP: 22451-900, Brazil
a r t i c l e i n f o
abstract
Article history: Received 16 September 2014 Received in revised form 23 February 2015 Accepted 16 April 2015
The Smooth Decomposition (SD) is a statistical analysis technique for finding structures in an ensemble of spatially distributed data such that the vector directions not only keep the maximum possible variance but also the motions, along the vector directions, are as smooth in time as possible. In this paper, the notion of the dual smooth modes is introduced and used in the framework of oblique projection to expand a random response of a system. The dual modes define a tool that transforms the SD in an efficient modal analysis tool. The main properties of the SD are discussed and some new optimality properties of the expansion are deduced. The parameters of the SD give access to modal parameters of a linear system (mode shapes, resonance frequencies and modal energy participations). In case of nonlinear systems, a richer picture of the evolution of the modes versus energy can be obtained analyzing the responses under several excitation levels. This novel analysis of a nonlinear system is illustrated by an example. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Decomposition POD Nonlinear Vibration Modal analysis Energy analysis
1. Introduction The Karhunen–Loève Decomposition (KLD) method, also named Proper Orthogonal Decomposition (POD), has been extensively used as a tool for analyzing random fields. The KLD is a statistical analysis technique for finding the coherent structures in an ensemble of spatially distributed data which defines an optimum basis in terms of energy. It has been advantageously used in different domains as, for example, the stochastic finite elements method [1,2], the simulation of random fields [3], the modal analysis of linear and nonlinear systems [4,5], and construction of reduced-order models [6,7]. A modified decomposition, that is not orthogonal in the Euclidean sense, named Smooth Decomposition (SD), is considered here. The SD can be viewed as a projection of an ensemble of spatially distributed data such that the vector directions of the projection not only keep the maximum possible variance but also the motions resulting along these vector directions are as smooth in time as possible. These vector directions (or structures, or smooth modes) are defined as the eigenvectors of the eigenproblem defined from the correlation matrices of the random field and of the associated time derivative. The basic idea of this decomposition derives from the optimal tracking approach proposed in [8]. SD was formulated as a multivariate data analysis in [9] and used as a modal analysis tool. Modal analysis of randomly excited system was considered in [10]. The SD approach was developed in cases of time-continuous stationary random vector processes in [11],
n
Corresponding author: Tel.: þ33 491164238; fax.: þ33 491164080. E-mail addresses: [email protected] (S. Bellizzi), [email protected] (R. Sampaio).
http://dx.doi.org/10.1016/j.ymssp.2015.04.015 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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in case of time-continuous stationary random fields in [12] and extended to the time-continuous non-stationary random vector processes in [13]. Recently, SD was also considered to generate reduced bases for discrete nonlinear dynamic systems (see [14,15]) and used in [16] to extract the modal parameters of a vehicle suspension system. The use of data to extract modal parameters was considered in [17]. In this paper, the main properties of the SD are discussed. The new notion of the dual smooth modes is introduced and used in the framework of oblique projection to expand the random response given in the dual smooth expansion. Some optimality properties of this expansion are shown. The parameters of the SD (dual smooth modes and the smooth values) are interpreted in terms of normal modes and resonance frequencies resulting in the modal analysis of a linear system using output-only data. Also, the introduction of the dual modes gives a new picture of the dynamics in terms of energy of the modes. The two pictures, in terms of frequencies and of energies, allows a richer interpretation of the dynamics. This approach overcomes some limitations of the POD. A novel modal analysis of nonlinear system is also proposed. 2. Smooth Decomposition 2.1. Decomposition principle Let fUðtÞ; t A Rg be a Rn valued random process indexed by R. We assume fUðtÞ; t A Rg to be a zero-mean second-order _ stationary ergodic process that admits a time derivative process fUðtÞ; t A Rg which is also a second-order stationary ergodic _ þ τÞT UðtÞÞ _ process. We take RU ðτÞ ¼ EðUðt þ τÞT UðtÞÞ and R U_ ðτÞ ¼ EðUðt to denote the covariance matrix function of fUðtÞ; t A Rg _ _ and fUðtÞ; t A Rg respectively. We assume that the covariance matrices (of UðtÞ and UðtÞ) RU ð0Þ and RU_ ð0Þ are symmetric positive-definite matrices. As described in [13], the SD of fUðtÞ; t A Rg is designed to obtain the most characteristic deterministic vectors Γ ð A Rn Þ maximizing the ratio between the ensemble average of the inner product between UðtÞ and Γ and to the inner product _ between UðtÞ and Γ maxn J Γ
ΓAR
EððUðtÞT ΓÞ2 Þ ΓT RU ð0ÞΓ ¼ : with J Γ ¼ _ T ΓÞ2 Þ ΓT R _ ð0ÞΓ EððUðtÞ
ð1Þ
U
This maximization problem is equivalent to the conditional extreme value problem max Γ R U ð0ÞΓ T
Γ A Rn
subject to Γ RU_ ð0ÞΓ ¼ 1: T
ð2Þ
The objective function JðΓÞ significantly differs from that used to define the KLD (see for example [18]). Here the denominator of the objective function takes the covariance matrix of the time-derivative process into account. The numerator and the denominator which seem to be time-independent can be related to the time evolution of trajectories of _ the random processes fUðtÞ; t A Rg and fUðtÞ; t A Rg over a long time following ergodic property as Z T 1 E ðUðtÞT ΓÞ2 ¼ lim ðUðsÞT ΓÞ2 ds ð3Þ T 4 12T T Z T _ T ΓÞ2 ¼ lim 1 _ T ΓÞ2 ds E ðUðtÞ ðUðsÞ T 4 12T T
ð4Þ
_ where UðsÞ and UðsÞ in the right-hand-side of the equations denote one of the trajectories and not a random variable as in _ the left-hand-side (for UðtÞ and UðtÞ). From the ergodic point of view, maximizing JðΓÞ corresponds to find a structure Γ which captures the maximum possible variance in terms of time average of the time displacement field, simultaneously with the minimum possible variance of the time velocity field (in accordance with the drift tracking algorithm proposed in [8]). The condition for local optimality is given by the gradient of the objective function JðΓÞ or by the Lagrange multipliers method applied to (2) and reduces to the following generalized eigenproblem: RU ð0ÞΓk ¼ λk R U_ ð0ÞΓk :
ð5Þ
Due to the properties of the covariance matrices (which are symmetric positive-definite matrices), (5) admits n real positive eigenvalues ðλk ) and the associated eigenvectors satisfy the following properties:
ΓTk RU ð0ÞΓl ¼ ΓTk R U_ ð0ÞΓl ¼ 0 for k a l and ΓTk R U ð0ÞΓk ¼ λk ΓTk RU_ ð0ÞΓk : In the sequel we will assume that the eigenvalues satisfy the constraint condition
λk are sorted in descending order and the eigenvectors are scaled to
ΓT RU_ ð0ÞΓ ¼ I n ;
ð6Þ
where Γ ¼ ½Γ1 Γ2 …Γn and I n denotes the identity matrix. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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Now we introduce the matrix Ψ as
Ψ ¼ RU_ Γ
ð7Þ
The column vectors of Ψ, (Ψ1 ; Ψ2 ; …; Ψn ) (with Ψk ¼ R U_ Γk Þ constitute a basis which is the dual-basis of Γ in the framework of the oblique projection (see Appendix A). The eigenvalue λk, named Smooth Value (SV), the corresponding normalized (with (6)) eigenvector Γk , named Smooth Mode (SM), and the dual eigenvector Ψk (defined in (7)), named Dual Smooth Mode (DSM), define the characteristics of the mode number k of SD of fUðtÞ; t A Rg. 2.2. Smooth expansion In [11], the Γbasis has been used to express the random process fUðtÞ; t A Rg as the expansion UðtÞ ¼
n X
ξk ðtÞΓk
ð8Þ
k¼1
where fξk ðtÞ; t A Rg are scalar random process defined by (using (6))
ξk ðtÞ ¼ ΓTk R U_ ð0ÞUðtÞ ¼ ΨTk UðtÞ:
ð9Þ
The expansion (8) was named Smooth Expansion of fUðtÞ; t A Rg associated to the Dual Smooth Components (DSCs) ξk ðtÞ. The DSCs are scalar random processes satisfying Eðξk ðtÞξl ðtÞÞ ¼ Γk RU_ ð0ÞRU ð0ÞRU_ ð0ÞΓl : T
ð10Þ
The major disadvantage of using the expansion (8) is that the basis vectors are not mutually orthogonal and the components are not mutually uncorrelated. A better representation is given in the following. 2.3. Dual smooth expansion In the framework of oblique projection, the dual-basis Ψ can be used to express the random process fUðtÞ; t A Rg as the expansion UðtÞ ¼
n X
ζ k ðtÞΨk
ð11Þ
k¼1
where fζ k ðtÞ; t A Rg are scalar random process defined by (see Appendix A and (6))
ζ k ðtÞ ¼ ΓTk UðtÞ:
ð12Þ
The expansion (11) will be named Dual Smooth Expansion of fUðtÞ; t A Rg associated to the Smooth Components (SCs) ζ k ðtÞ. The SCs are scalar random processes satisfying Eðζ k ðtÞζ l ðtÞÞ ¼ 0
for k a l and Eðζ k ðtÞ2 Þ ¼ Γk RU ð0ÞΓk ¼ λk : T
ð13Þ
Considering the time derivative of ζ k ðtÞ, ζ_ k ðtÞ, the similar properties hold for the set of random variables ζ_ k ðtÞ (for k ¼ 1; …; n) i.e. Eðζ_ k ðtÞζ_ l ðtÞÞ ¼ 0
for k a l and Eðζ_ k ðtÞ2 Þ ¼ Γk RU_ ð0ÞΓk ¼ 1: T
ð14Þ
The major advantage of using the expansion (11) instead of expansion (8) is that the components are mutually uncorrelated. Hence, it can be shown that the “energy” of UðtÞ in terms of Euclidean norm ( J X J 2 ¼ XT X) takes the following form: Eð‖UðtÞ‖2 Þ ¼
n X
λk ‖Ψk ‖2
ð15Þ
k¼1
and the truncation error satisfies ! p X 2 E ‖UðtÞ ζ k ðtÞΨk ‖ ¼ k¼1
n X
λk ‖Ψk ‖2 :
ð16Þ
k ¼ pþ1
Note that the dual smooth basis is not an optimal basis in terms of truncation error (see Section 2.5). From Eqs. (15) and (16), it is clear that the quantity
λk ‖Ψk ‖2
ð17Þ
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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characterizes the energy of the mode number k and the ratio
λk ‖Ψk ‖2 2 l ¼ 1 λl ‖Ψl ‖
Pn
ð18Þ
defines the fraction of the total energy captured by the mode number k. This equation will allow, as in the linear case for the normal modes, to see how the energy is distributed among the modes. 2.4. Some properties of the characteristics of the SD 2.4.1. Linear transformation Let fVðtÞ; t A Rg be a random process defined as linear transformation of fUðtÞ; t A Rg as VðtÞ ¼ AUðtÞ where A is an invertible matrix. The following relationships between the SD characteristics of fVðtÞ; t A Rg and the SD characteristics fUðtÞ; t A Rg hold: ðiÞ
λVk ¼ λUk
ðiiÞ
ΓVk ¼ A T ΓUk
ðiiiÞ
If AAT ¼ In ; ξk ðtÞ ¼ ξk ðtÞ
ðivÞ
ΨVk ¼ AΨUk
ðvÞ
ζ Vk ðtÞ ¼ ζ Uk ðtÞ
U
V
ð19Þ
Items (i)–(iii) were demonstrated in [9,11]. Items (iv) and (v) show that the Dual Smooth Expansion (11) is preserved by linear transformation. 2.4.2. Gaussian random processes Let fUðtÞ; t A Rg be a Rn valued Gaussian random process, then the SCs and the DSCs are also Gaussian. Moreover, for fixed t, the SCs ζ k ðtÞ for k ¼ 1; …; n are mutually independent (as random variables). Of course, the result is not generally true for a set of non-Gaussian random processes fζ k ðtÞ; t A Rg. 2.5. Dual smooth expansion as KL expansion ~ Let LU_ be the square root matrix of R U_ ð0Þð ¼ LU_ LU_ ) and define the random processes fUðtÞ; t A Rg and its time derivative as the linear transformation ~ UðtÞ ¼ LU_ 1 UðtÞ
and
1 _ ~_ ðtÞ ¼ L U _ UðtÞ: U
ð20Þ
The associated covariance matrices are then given by 1 RU~ ð0Þ ¼ LU_ T RU ð0ÞLU _
and
R U~_ ð0Þ ¼ In :
ð21Þ
~ It follows that the SD of fUðtÞ; t A Rg coincides with its KLD. The associated characteristics are related to the characteristics of fUðtÞ; t A Rg with the linear transformation properties given in Section 2.4. Finally, considering the RU_ ð0Þ 1 norm (‖X‖2R ð0Þ−1 ð0Þ ¼ XT RU_ ð0Þ−1 X), it can be shown that U_
Eð‖UðtÞ‖2R _ ð0Þ 1 Þ ¼ U
n X
λk
ð22Þ
k¼1
and also that the following optimal relationship holds for the DS expansion of fUðtÞ; t A Rg i.e. 0 1 0 1 2 2 p p n X X X A¼ A E@UðtÞ ζ k ðtÞΨk λk r E@UðtÞ ζ k ðtÞΨ k 1 1 k¼1
RU_ ð0Þ
k ¼ pþ1
k¼1
ð23Þ
R U_ ð0Þ
for any arbitrary R U_ ð0Þ 1 orthogonal basis Ψ . This result shows that the dual smooth expansion can be interpreted as a Karhunen–Loève expansion considering the R U_ ð0Þ 1 inner product. 2.6. SD in practice If the matrices R U ð0Þ and R U_ ð0Þ are known, SD is computed solving the generalized eigenvalue problem (5). On the other hand, if only numerical or experimental data are available, SD can be approximated estimating first R U ð0Þ and RU_ ð0Þ and solving a generalized eigenvalue problem. SD can be obtained also solving a generalized singular value decomposition problem. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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3. Physical interpretation of the SD Let us consider a mechanical system with n degrees of freedom governed by the equation of motion € þ CUðtÞ _ þ KUðtÞ þ GðUðtÞÞ ¼ FðtÞ MUðtÞ
ð24Þ
where M, C, and K are n n symmetric matrices, and the nonlinearity is in the vector function GðUÞ. The vector FðtÞ, describing the external random forces, is supposed to be a zero-mean white-noise random excitation with matrix intensity SF (i.e., RF ðτÞ ¼ EðFðt þ τÞFT ðtÞÞ ¼ SF δðτÞ). 3.1. Linear case G ¼0 Let ðωk ; Φk Þ be the normal modes of the linear system (KΦ ¼ MΦΩ with Φ ¼ ½Φ1 Φ2 …Φn and Ω ¼ diagðωk ÞÞ. As shown in [11], SD of the stationary response of (24) can be related to the modal parameters of the system (24). The result reads as: if T T the system is such that Φ CΦ and Φ SF Φ are diagonal matrices then the following relationships hold: 2
ðiÞ
Φ ¼ Ψ;
ðiiÞ
ω2k ¼
1
λk
for k ¼ 1; …; n:
ð25Þ
The mode shapes coincide with the dual smooth modes and the square of the resonance frequencies is equal to the inverse of the corresponding smooth values. Moreover, the dual smooth expansion of UðtÞ coincides with the classical modal expansion. Note that the matrix Ψ can be easily obtained from (7). There is hence no need to invert the matrix Γ. 3.2. Nonlinear case SD can be applied to random nonlinear responses and interpreted in the framework of linearized systems using the statistical linearization method (as defined in [19]). For a given external random force (in our case for a given matrix intensity), the statistical linearization method gives access to a linear system (named equivalent linear system) such that its stationary covariance matrix coincides with the stationary covariance matrix of the nonlinear response. Following the results discussed in Section 3.1, the SMs, DSMs and SMVs can be considered as the modal parameters of the equivalent linear system. 3.3. About the smooth mode ordering Following the results discussed in Section 3.1, ordering the SVs, λk, in descending order is equivalent to the classical ordering of the resonance frequencies (ascending order). This ordering used Section 2.1 to number the SD modes which will be named frequency ordering. Following the property (15) of the dual smooth expansion (11), the quantities λk ‖Ψk ‖2 ranking in descending order can also be used to number the SD modes. This ordering will be named energy ordering. The energy ordering is in line with the ordering used in the POD method. In the frequency ordering one considers only 1=λk and in the energy ordering one considers λk ‖Ψk ‖2 . With this convention the first mode (lowest frequency) may not be the most energetic. 4. Illustrative example In this section, the SD is used as an output-only analysis tool to characterize a nonlinear system. The system is depicted in Fig. 1. It is composed of a linear chain of M identical linear oscillators connected at the left end to a fixed wall and connected at the right end to a nonlinear oscillator, also named nonlinear end-attachment. The equations of motion read as (observe that the equations are ordered from right to left with respect to Fig. 1) mN v€ þ λN ðv_ u_ 1 Þ þ f N ðv u1 Þ ¼ 0;
ð26Þ
u€ 1 þ λg u_ 1 þkg u1 þ λc ðu_ 1 u_ 2 Þ þ kc ðu1 u2 Þ λN ðv_ u_ 1 Þ f N ðv u1 Þ ¼ 0;
ð27Þ
Fig. 1. Schema of the nonlinear system.
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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u€ m þ λg u_ m þkg um þ λc ð2u_ m u_ m 1 u_ m þ 1 Þ þ kc ð2um um 1 um þ 1 Þ ¼ 0; ð2 r m r M 1Þ
ð28Þ
u€ M þ λg u_ M þ kg uM þ λc ðu_ M u_ M 1 Þ þ kc ðuM uM 1 Þ ¼ f ðtÞ
ð29Þ
where v (respectively um) denotes the displacement of the nonlinear end-attachment (respectively the mth mass of the linear chain). The nonlinear end-attachment is constituted of a mass mN, a linear damper with coefficient λN , and a nonlinear spring characterized by the restoring force ( C 1 x þ C 3þ x3 if x Z0 ð30Þ f N ðxÞ ¼ C 1 x þ C 3 x5 if x r0 where C1 C 3þ and C 3 are real positive scalars. Note that C1 will be chosen so small that maxðC 3þ ; C 3 Þ corresponds to an essentially nonlinear function. An external force f(t) is applied to the mass number M, the leftmost mass. We assume that it is of the form f ðtÞ ¼ s0 WðtÞ
ð31Þ
where fWðtÞ; t A Rg is a Gaussian white-noise scalar process with intensity one and s0 denotes the excitation level. The Monte–Carlo method was used to estimate the covariance matrices of the stationary responses (displacement and velocity) of the system. From a given excitation level, the response time history was obtained from a time history of excitation (31) by solving Eqs. (26)–(29) over the time interval [0, tf] numerically using the Newmark method with a time-step of size Δt ¼ 1=f s (where fs denotes the sampling frequency). Zero initial displacement and velocity were assumed. The time histories of W(t) (a Gaussian white-noise scalar process with intensity one) were generated using the procedure described in [20]. Assuming ergodicity, the last-half points of the displacement and velocity time histories were used to approximate (as time averages) the covariance matrices RU ð0Þ and RU_ ð0Þ with U ¼ ðv; u1 ; u2 ; …; uM ÞT . The Smooth Decomposition analysis is then obtained solving the eigenvalue problem (5). The following numerical parameter values were used to simulate the system (26)–(29): M¼9 (that is a 10-DOF system including 9-DOF from the linear chain and 1-DOF from the nonlinear end-attachment), λc ¼ 0:005, kc ¼1, λg ¼ 0:005, kg ¼ 1, mN ¼0.05, λN ¼ 0:001, kN ¼0.0001, C 1 ¼ 0:01, C 3þ ¼ 1, C 3 ¼ 0:0123. The equations of motion were numerically solved with the following time-discretization parameter values Δ ¼ 0:143 s (i.e. fs ¼7 Hz) and tf ¼74942 s (corresponding to 524 286 instants). The excitation levels s0 have been used as the parameter of analysis. Forty-eight linearly equally spaced points between 0.0025 and 0.12 have been considered. 4.1. Linear case as low excitation response The stationary response associated to the excitation level s0 ¼ 0:0025 is considered here. The first six dual smooth modes (using frequency ordering) are reported in Fig. 2 and compared to the normal modes of the underlying linear system. The SD correctly estimates the normal modes (as predicted by the theoretical results), four of them not shown here. This is also the case for the resonance frequencies. The resonance frequencies estimated from (25) (0.0068, 0.1613, 0.1773, 0.204, 0.2359, 0.2680, 0.2973, 0.3220, 0.3405 and 0.3519) coincide (numerically) with the exact resonance frequencies of the underlying linear system (0.0071, 0.1613, 0.1773, 0.2042, 0.2359, 0.2680, 0.2974, 0.3220, 0.3405 and 0.3520). This result confirms that for the selected excitation level the nonlinear stiffness of the end-attachment does not affect the response. The modal motion associated to the first normal mode (corresponding to the smaller resonance frequency) is localized on the nonlinear attachment whereas the modal motions associated to the other modes have the linear chain also moving and coincide with the normal modes of linear chain. 4.2. Nonlinear case The SD approach has been applied to the stationary responses associated to the chosen 48 excitation levels s0 and the results are now discussed. Fig. 3(a) shows the fraction of energy captured by the modes (as defined in Eq. (18)) for the 48 excitation level values. The red line marks the most energetic modes, the first mode considering the energy ordering. For 0:0025 o s0 r 0:045, the mode number 1 is the most energetic. For 0:045 r s0 r 0:0575, the most energetic is the mode number 3. For 0:0575 rs0 r0:08, the mode number 4 is the most energetic. Finally, for s0 Z0:08, there is no dominant mode, the energy is distributed, principally, among the first five modes. Note that, in this problem, the mode number 2 does not appear as a dominant mode. Fig. 3(b) shows the resonance frequencies estimated from the SD versus the excitation level. As in Fig. 3(a), the red line corresponds to the most energetic modes. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
Mode 1
1
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 3
1
Mode 4
1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 5
1
Mode 2
1
0.5
7
Mode 6
1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Fig. 2. SD analysis for s0 ¼ 0:0025: dual smooth modes (cross markers, blue), and normal modes of the underlying linear system (red line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
100
Modal energy (%)
80 70 60 50 40 30 20
0.3 0.25 0.2 0.15 0.1 0.05
10 0
0.35
Resonance frequency (Hz)
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 10
90
0.02
0.04
0.06
s0
0.08
0.1
0.12
0.02
0.04
0.06
0.08
0.1
0.12
s0
Fig. 3. SD analysis: (a) Fraction of energy captured by the modes versus excitation level s0. (b) Resonance frequencies estimated from (25) versus excitation level. The red line corresponds to the most energetic modes. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
At s0 ¼0.0025, the estimated resonance frequencies coincide with the resonance frequencies of the underlying linear system as already mentioned in the previous section. For s0 4 0:0025, the behaviors of the resonance frequencies with respect to s0 differ. The resonance frequency associated to the mode number 1 (cross markers) first increases when the excitation level increases and for s0 large (approximatively greater than 0.047), it becomes constant with a numerical value near, but smaller than, the second resonance frequency of the underlying linear system. The resonance frequency associated to the mode number 2 (star markers) remains first constant for small values of s0, then slowly increases and for large s0 (approximatively greater than 0.055), it becomes constant with a numerical value near, but smaller than, the third resonance frequency of the underlying linear system. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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The same behavior can be noted for the resonance frequency associated to the mode number 2–5 (see Fig. 3(b) circle markers, square markers and diamond markers). These results illustrate the hardening behavior of the system. It is also interesting to note that the fraction of energy associated to the current most energetic mode is large with respect to the other modes (see Fig. 3(a)), the corresponding resonance frequency increases with the excitation level (see the red line Fig. 3(b)). Finally, the results shown in Fig. 3(a) and (b) can be combined given the frequency–energy dependence of the modal oscillations of the nonlinear system. For a given mode number, the corresponding resonance frequency and the associated modal energy depend on the input excitation level (or equivalently to the input energy) and can be represented in a frequency–energy plot. The result is reported in Fig. 4 where for each mode, the resonance frequency (as defined by (25)) is plotted versus the energy (as defined by (17)). A log scale has been used in the modal energy axis. The frequency behaviors of the modes with respect to the energy are similar to the frequency behaviors of the same modes with respect to the excitation s0. For a given mode, the frequency starts from the corresponding value of the resonance frequency of the normal
Frequency (Hz)
0.25
0.2
0.15
0.1
0.05
Mode Mode Mode Mode Mode Mode
10
1 2 3 4 5 6
10
10
10
10
Energy Fig. 4. SD analysis: Frequency–energy plot of each mode.
1
Mode 1
1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 3
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 5
Mode 4
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 2
Mode 6
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Fig. 5. SD analysis for s0 ¼ 0:022: dual smooth modes (cross markers, blue), and normal modes of the underlying linear system (red line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
1
Mode 1
1
Mode 2
1
Mode 1
9
1
0.5
0.5
0.5
0.5
0
0
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 3
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 4
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 3
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
0.5
0.5
0.5
0.5
0
0
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 5
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 6
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 5
1
0.5
0.5
0.5
0
0
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 4
v u1 u2 u3 u4 u5 u6 u7 u8 u9
0.5
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 2
Mode 6
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Fig. 6. SD analysis for (a) s0 ¼ 0:05 and (b) s0 ¼ 0:07: dual smooth modes (cross markers) and normal modes of the underlying linear system (red line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
mode of the underlying linear system. The energy and the frequency first increase, next the frequency increases whereas the energy decreases and finally the frequency becomes constant with increasing energy. Note that this representation is in line with the frequency–energy plot used to represent the nonlinear normal modes as described in [21,22] and will be completed in the following with modal shape information. Figs. 5 and 6 show the shapes of the first six dual smooth modes obtained, respectively, at the excitation levels s0 ¼ 0:022; 0:05, and 0.07. These three values have been chosen in the intervals where the mode number 1, 3 and 4 are, respectively, the most energetic (see Fig. 3(a)). The shapes of the normal modes of the underlying linear system are also reported on the same plot. For s0 ¼0.022, only the second, third and fourth dual smooth modes are weakly affected by the nonlinearity (see Fig. 5). For s0 ¼0.05 and s0 ¼0.07, significant differences appear between the first fourth dual smooth modes and the corresponding normal modes. The visualization of the shapes of the dual smooth modes can be combined with the frequency–energy plot as shown in Fig. 7 where, for the first four modes, the corresponding curve of frequency–energy is depicted together with the corresponding shape of the dual modes (realized as in Figs. 2, 5 and 6). This visualization gives now, as the frequency– energy plot used to represent the nonlinear normal modes in [21,22], information in terms of frequency, shape of the mode and energy. As already mentioned, for each mode (here mode 1 to mode 4), the mode shape of the dual smooth mode obtained at low energy level (i.e at low frequency) coincides with the mode shape of the associated normal mode of the underlying linear system (see the plots (1) in Fig. 7(a)–(d)). Moreover, at high energy level (i.e. at high frequency), the mode shape of the dual smooth mode shows a non-localized motion (see the plots (7) in Fig. 7(a)–(d)). For each mode, these two “limit” shapes are well-marked. The transition between these two “limit” shapes is marked by a region (see the plots (3) and (4) in Figs. 7(a)–(d)) where frequency and energy simultaneously increase and mode shapes get closer to the shape of the first normal mode of the underlying linear system corresponding to the localized motion on the nonlinear attachment degree of freedom (see Fig. 2, Mode 1). This behavior can be interpreted as a resonance capture occurring between a mode localized on the linear chain and a mode localized on the nonlinear attachment. Note that, classically, the term resonance capture is used to characterize dynamics phenomena with respect to time. Here we are observing a phenomenon which occurs varying the excitation level. The last point to consider is to compare the shape of the dual smooth modes of the most energetic modes. For s0 ¼0.022 (respectively 0.05 and 0.07), the most energetic mode is the mode number 1 in Fig. 5 (respectively number 3 in Fig. 6(a) and number 4 in Fig. 6(b)). The shapes of these three modes are very close and similar to the shape of the first normal mode of the underlying linear system. This observation is also true as well as the current first dual mode is dominant (in terms of energy (see Fig. 3)) that is for 0:005 r s0 r 0:08. These shapes are compared in Fig. 8. These mode shapes define motions
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
10
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Fig. 7. Frequency–energy plot of the first four modes obtained by the SD approach: (a) mode number 1, (b) mode number 2, (c) mode number 3 and (d) mode number (4).
1 0.8
Dual smooth mode 1
0.6 0.4 0.2 0
v
u1
u2
u3
u4
u5
u6
u7
u8
u9
Fig. 8. Shapes of the dual smooth modes of the most energetic modes for 0:005 r s0 r 0:08. First normal mode of the underlying linear system (red line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
localized on the nonlinear attachment (v component), so the form of the most energetic mode does not change. For s0 Z0:08, the most energetic dual smooth modes are no more localized at the nonlinear attachment (not shown here). The last comment concerns the effect on the behavior of the system of these localized modes with dominant energy. Fig. 9 shows the fractions of the energy dissipated by the linear chain and the nonlinear attachment versus the excitation level. When the excitation level increases, the fraction of energy dissipated by the nonlinear attachment increases, becomes greater than the energy dissipated by the linear chain and finally decreases. The range where the nonlinear attachment is effective Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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11
100
Dissipated energy (%)
90 80 70 60 50 40 30 20 10 0
0.02
0.04
0.06
0.08
0.1
0.12
s0 Fig. 9. Fraction of dissipated energy by the linear chain (circle markers) and by the nonlinear attachment (square markers) versus excitation level s0.
(with respect to the linear chain) to dissipate energy is localized in the interval 0:035 r s0 r0:08. In the effective range, the shape of the most energetic modes corresponds to motions localized on the nonlinear attachment which of course privilege the dissipation of energy in the damper of the nonlinear attachment. Note that in the range 0:005 rs0 r 0:035, the shape of the most energetic modes corresponds also to motions localized on the nonlinear attachment.
5. Conclusion In this paper, the SD of a vector valued random process has been discussed starting from the classical properties of the SD. The concept of dual smooth modes has been introduced and used to expand a random process. The dual smooth expansion was interpreted as a Karhunen–Loève expansion choosing a convenient inner product. Two orderings of the modes were introduced, one in line with the classical ordering of the resonance frequencies, the other in line with the modal energy content. The two orderings do not necessarily coincide, and give two different observations of the system under study. The SD was also interpreted in terms of modal analysis in case of linear and nonlinear mechanical systems. For nonlinear systems it constitutes a novel way to analyze the system using linear analysis tools which was not considered in[23]. The efficiency of this approach as a tool to understand nonlinear behavior was demonstrated. Future work concerns the study of the relations between the nonlinear normal modes and the modes defined by the SD analysis.
Acknowledgments This work was supported by CAPES (Brazil) and COFECUB (France) agencies (under CAPES/COFECUB project No. 672/10) and by CNPq and Faperj.
Appendix A. Oblique projection Let us consider a p-dimensional subspace Ep with p r n in Rn with Rn ¼ Ep En p . ~ ;Φ ~ ; …; Φ ~ Þ be a basis of E and ðΦ ; Φ ; …; Φ Þ be a basis of ET Let ðΦ p p p 1 2 1 2 n p the orthogonal space of E n p such that ~ ¼I ΦT Φ p
ðA:1Þ
~ ¼ ½Φ ~ Φ ~ , Φ ¼ ½Φ Φ ⋯Φ and I denotes the p p identity matrix. ~ ⋯Φ where Φ p p p 1 2 1 2 The oblique projection Π of Rn into Ep along the subspace En p is defined by the matrix ~ T: Π ¼ ΦΦ
ðA:2Þ
The projection appears as a simple truncation in the appropriated basis as ~ TU ¼ Π U ¼ ΦΦ
p X
T ~ : ðΦk UÞΦ k
ðA:3Þ
k¼1
~ TΦ ~ comes from the p vectors of an orthonormal basis (Φ ~ ¼ I ) of Rn then (A.3) corresponds to the classical If Φ p orthogonal projection. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
12
~ can be considered as the dual basis of Φ in the sense that If p¼n, the basis Φ U¼
n X
~ : ðΦk UÞΦ k T
ðA:4Þ
k¼1
References [1] R. Ghanen, P. Spanos, Stochastic Finite Elements: A Spectral Approach, Dover, New York, 2003. [2] E.S. de Cursi, R. Sampaio, Uncertainty Quantification and Stochastic Modeling with Matlab, Edited by Elsevier, ISTE, John Wiley, 2015 ISBN: 978178540058. [3] S. Sakamoto, R. Ghanem, Simulation of multi-dimensional non-Gaussian non-stationary random field, Probab. Eng. Mech. 17 (2002) 167–176. [4] B. Feeny, R. Kappagantu, On the physical interpretation of proper orthogonal modes in vibrations, J. Sound Vib. 211 (4) (1998) 607–616. [5] G. Kerschen, J.-C. Golinval, A. Vakakis, L. Bergman, The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview, Non-linear Dyn. 41 (2005) 147–169. (Special issue on “Reduced Order Models: Methods and Applications” Applications”). [6] X. Ma, M. Azeez, A. Vakakis, Non-linear normal modes and non-parametric system identification of non-linear oscillators, Mech. Syst. Signal Process. 14 (1) (2000) 37–48. [7] M. Trindade, C. Wolter, R. Sampaio, Karhunen–Loève decomposition of coupled axial/bending vibrations of beams subject to impacts, J. Sound Vib. 279 (2005) 1015–1036. [8] A. Chatterjee, J. Cusumano, D. Chelidze, Optimal tracking of parameter drift in a chaotic system: experiment and theory, J. Sound Vib 250 (5) (2002) 877–901. [9] D. Chelidze, W. Zhou, Smooth orthogonal decomposition-based vibration mode identification, J. Sound Vib. 292 (2006) 461–473. [10] U. Farooq, B. Feeny, Smooth orthogonal decomposition for modal analysis of randomly excited systems, J. Sound Vib. 316 (2008) 137–146. [11] S. Bellizzi, R. Sampaio, Analysis of stationary random vibrating systems using smooth decomposition, Shock Vib. 20 (2013) 493–502. [12] S. Bellizzi, R. Sampaio, Smooth decomposition of random fields, J. Sound Vib. 331 (2012) 3509–3520. [13] R. Sampaio, S. Bellizzi, Analysis of nonstationary random processes using smooth decomposition, J. Mech. Mater. Struct. 6 (2011) 1137–1152. [14] F.A. Lülf, D.-M. Tran, R. Ohayon, Reduced bases for nonlinear structural dynamic systems: a comparative study, J. Sound Vib. 332 (2013) 3897–3921. [15] M.P. Mignolet, A. Przekop, S.A. Rizzi, S.M. Spottswood, A review of in direct/non-intrusive reduced order modeling of nonlinear geometric structures, J. Sound Vib 332 (2013) 2437–2460. [16] M. Rezaee, V. Shaterian-Alghalandis, A. Banan-Nojavani, Development of the smooth orthogonal decomposition method to derive the modal parameters of vehicle suspension system, J. Sound Vib. 332 (2013) 1829–1842. [17] R. Caldwell, B.F. Feeny, Output-only modal identification of a nonuniform beam by using decomposition methods, J. Vib. Acoust. 136 (4) (2014) 041010–041010-10. [18] S. Bellizzi, R. Sampaio, POMs analysis of randomly vibrating systems obtained from Karhunen–Loève expansion, J. Sound Vib. 297 (2006) 774–793. [19] F. Kozin, The method of statistical linearization for nonlinear stochastic vibrations, in: Nonlinear stochastic dynamic engineering systems, in: IUTAM Symposium, Innsbruck, 1987, Springer, Berlin, 1988, pp. 45–56. [20] F. Poirion, C. Soize, Simulation numérique des champs stochastiques gaussiens homogènes et non homogènes, La Recherche Aérospatiale 1 (1989) 41–61. [21] G. Kerschen, M. Peeters, J. Golinval, A. Vakakis, Nonlinear normal modes. Part I: a useful framework for the structural dynamicist, Mech. Syst. Signal Process. 23 (1) (2009) 170–194. [22] M. Peeters, R. Viguié, G. Sérandour, G. Kerschen, J.-C. Golinval, Nonlinear normal modes. Part II: toward a practical computation using numerical continuation techniques, Mech. Syst. Signal Process. 23 (2008) 195–216. [23] G. Kerschen, K. Worden, A. Vakakis, J.C. Golinval, Past, present and future of nonlinear system identification in structural dynamics, Mech. Syst. Signal Process 20 (2006) 505–592.
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
The Smooth Decomposition as a nonlinear modal analysis tool Sergio Bellizzi a,n, Rubens Sampaio b a b
LMA, CNRS, UPR 7051, Centrale Marseille, Aix-Marseille Univ, F-13402 Marseille Cedex 20, France PUC-Rio, Mechanical Engineering Department, Rua Marquês de São Vicente, 225, Gávea, Rio de Janeiro, RJ, CEP: 22451-900, Brazil
a r t i c l e i n f o
abstract
Article history: Received 16 September 2014 Received in revised form 23 February 2015 Accepted 16 April 2015
The Smooth Decomposition (SD) is a statistical analysis technique for finding structures in an ensemble of spatially distributed data such that the vector directions not only keep the maximum possible variance but also the motions, along the vector directions, are as smooth in time as possible. In this paper, the notion of the dual smooth modes is introduced and used in the framework of oblique projection to expand a random response of a system. The dual modes define a tool that transforms the SD in an efficient modal analysis tool. The main properties of the SD are discussed and some new optimality properties of the expansion are deduced. The parameters of the SD give access to modal parameters of a linear system (mode shapes, resonance frequencies and modal energy participations). In case of nonlinear systems, a richer picture of the evolution of the modes versus energy can be obtained analyzing the responses under several excitation levels. This novel analysis of a nonlinear system is illustrated by an example. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Decomposition POD Nonlinear Vibration Modal analysis Energy analysis
1. Introduction The Karhunen–Loève Decomposition (KLD) method, also named Proper Orthogonal Decomposition (POD), has been extensively used as a tool for analyzing random fields. The KLD is a statistical analysis technique for finding the coherent structures in an ensemble of spatially distributed data which defines an optimum basis in terms of energy. It has been advantageously used in different domains as, for example, the stochastic finite elements method [1,2], the simulation of random fields [3], the modal analysis of linear and nonlinear systems [4,5], and construction of reduced-order models [6,7]. A modified decomposition, that is not orthogonal in the Euclidean sense, named Smooth Decomposition (SD), is considered here. The SD can be viewed as a projection of an ensemble of spatially distributed data such that the vector directions of the projection not only keep the maximum possible variance but also the motions resulting along these vector directions are as smooth in time as possible. These vector directions (or structures, or smooth modes) are defined as the eigenvectors of the eigenproblem defined from the correlation matrices of the random field and of the associated time derivative. The basic idea of this decomposition derives from the optimal tracking approach proposed in [8]. SD was formulated as a multivariate data analysis in [9] and used as a modal analysis tool. Modal analysis of randomly excited system was considered in [10]. The SD approach was developed in cases of time-continuous stationary random vector processes in [11],
n
Corresponding author: Tel.: þ33 491164238; fax.: þ33 491164080. E-mail addresses: [email protected] (S. Bellizzi), [email protected] (R. Sampaio).
http://dx.doi.org/10.1016/j.ymssp.2015.04.015 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
2
in case of time-continuous stationary random fields in [12] and extended to the time-continuous non-stationary random vector processes in [13]. Recently, SD was also considered to generate reduced bases for discrete nonlinear dynamic systems (see [14,15]) and used in [16] to extract the modal parameters of a vehicle suspension system. The use of data to extract modal parameters was considered in [17]. In this paper, the main properties of the SD are discussed. The new notion of the dual smooth modes is introduced and used in the framework of oblique projection to expand the random response given in the dual smooth expansion. Some optimality properties of this expansion are shown. The parameters of the SD (dual smooth modes and the smooth values) are interpreted in terms of normal modes and resonance frequencies resulting in the modal analysis of a linear system using output-only data. Also, the introduction of the dual modes gives a new picture of the dynamics in terms of energy of the modes. The two pictures, in terms of frequencies and of energies, allows a richer interpretation of the dynamics. This approach overcomes some limitations of the POD. A novel modal analysis of nonlinear system is also proposed. 2. Smooth Decomposition 2.1. Decomposition principle Let fUðtÞ; t A Rg be a Rn valued random process indexed by R. We assume fUðtÞ; t A Rg to be a zero-mean second-order _ stationary ergodic process that admits a time derivative process fUðtÞ; t A Rg which is also a second-order stationary ergodic _ þ τÞT UðtÞÞ _ process. We take RU ðτÞ ¼ EðUðt þ τÞT UðtÞÞ and R U_ ðτÞ ¼ EðUðt to denote the covariance matrix function of fUðtÞ; t A Rg _ _ and fUðtÞ; t A Rg respectively. We assume that the covariance matrices (of UðtÞ and UðtÞ) RU ð0Þ and RU_ ð0Þ are symmetric positive-definite matrices. As described in [13], the SD of fUðtÞ; t A Rg is designed to obtain the most characteristic deterministic vectors Γ ð A Rn Þ maximizing the ratio between the ensemble average of the inner product between UðtÞ and Γ and to the inner product _ between UðtÞ and Γ maxn J Γ
ΓAR
EððUðtÞT ΓÞ2 Þ ΓT RU ð0ÞΓ ¼ : with J Γ ¼ _ T ΓÞ2 Þ ΓT R _ ð0ÞΓ EððUðtÞ
ð1Þ
U
This maximization problem is equivalent to the conditional extreme value problem max Γ R U ð0ÞΓ T
Γ A Rn
subject to Γ RU_ ð0ÞΓ ¼ 1: T
ð2Þ
The objective function JðΓÞ significantly differs from that used to define the KLD (see for example [18]). Here the denominator of the objective function takes the covariance matrix of the time-derivative process into account. The numerator and the denominator which seem to be time-independent can be related to the time evolution of trajectories of _ the random processes fUðtÞ; t A Rg and fUðtÞ; t A Rg over a long time following ergodic property as Z T 1 E ðUðtÞT ΓÞ2 ¼ lim ðUðsÞT ΓÞ2 ds ð3Þ T 4 12T T Z T _ T ΓÞ2 ¼ lim 1 _ T ΓÞ2 ds E ðUðtÞ ðUðsÞ T 4 12T T
ð4Þ
_ where UðsÞ and UðsÞ in the right-hand-side of the equations denote one of the trajectories and not a random variable as in _ the left-hand-side (for UðtÞ and UðtÞ). From the ergodic point of view, maximizing JðΓÞ corresponds to find a structure Γ which captures the maximum possible variance in terms of time average of the time displacement field, simultaneously with the minimum possible variance of the time velocity field (in accordance with the drift tracking algorithm proposed in [8]). The condition for local optimality is given by the gradient of the objective function JðΓÞ or by the Lagrange multipliers method applied to (2) and reduces to the following generalized eigenproblem: RU ð0ÞΓk ¼ λk R U_ ð0ÞΓk :
ð5Þ
Due to the properties of the covariance matrices (which are symmetric positive-definite matrices), (5) admits n real positive eigenvalues ðλk ) and the associated eigenvectors satisfy the following properties:
ΓTk RU ð0ÞΓl ¼ ΓTk R U_ ð0ÞΓl ¼ 0 for k a l and ΓTk R U ð0ÞΓk ¼ λk ΓTk RU_ ð0ÞΓk : In the sequel we will assume that the eigenvalues satisfy the constraint condition
λk are sorted in descending order and the eigenvectors are scaled to
ΓT RU_ ð0ÞΓ ¼ I n ;
ð6Þ
where Γ ¼ ½Γ1 Γ2 …Γn and I n denotes the identity matrix. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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3
Now we introduce the matrix Ψ as
Ψ ¼ RU_ Γ
ð7Þ
The column vectors of Ψ, (Ψ1 ; Ψ2 ; …; Ψn ) (with Ψk ¼ R U_ Γk Þ constitute a basis which is the dual-basis of Γ in the framework of the oblique projection (see Appendix A). The eigenvalue λk, named Smooth Value (SV), the corresponding normalized (with (6)) eigenvector Γk , named Smooth Mode (SM), and the dual eigenvector Ψk (defined in (7)), named Dual Smooth Mode (DSM), define the characteristics of the mode number k of SD of fUðtÞ; t A Rg. 2.2. Smooth expansion In [11], the Γbasis has been used to express the random process fUðtÞ; t A Rg as the expansion UðtÞ ¼
n X
ξk ðtÞΓk
ð8Þ
k¼1
where fξk ðtÞ; t A Rg are scalar random process defined by (using (6))
ξk ðtÞ ¼ ΓTk R U_ ð0ÞUðtÞ ¼ ΨTk UðtÞ:
ð9Þ
The expansion (8) was named Smooth Expansion of fUðtÞ; t A Rg associated to the Dual Smooth Components (DSCs) ξk ðtÞ. The DSCs are scalar random processes satisfying Eðξk ðtÞξl ðtÞÞ ¼ Γk RU_ ð0ÞRU ð0ÞRU_ ð0ÞΓl : T
ð10Þ
The major disadvantage of using the expansion (8) is that the basis vectors are not mutually orthogonal and the components are not mutually uncorrelated. A better representation is given in the following. 2.3. Dual smooth expansion In the framework of oblique projection, the dual-basis Ψ can be used to express the random process fUðtÞ; t A Rg as the expansion UðtÞ ¼
n X
ζ k ðtÞΨk
ð11Þ
k¼1
where fζ k ðtÞ; t A Rg are scalar random process defined by (see Appendix A and (6))
ζ k ðtÞ ¼ ΓTk UðtÞ:
ð12Þ
The expansion (11) will be named Dual Smooth Expansion of fUðtÞ; t A Rg associated to the Smooth Components (SCs) ζ k ðtÞ. The SCs are scalar random processes satisfying Eðζ k ðtÞζ l ðtÞÞ ¼ 0
for k a l and Eðζ k ðtÞ2 Þ ¼ Γk RU ð0ÞΓk ¼ λk : T
ð13Þ
Considering the time derivative of ζ k ðtÞ, ζ_ k ðtÞ, the similar properties hold for the set of random variables ζ_ k ðtÞ (for k ¼ 1; …; n) i.e. Eðζ_ k ðtÞζ_ l ðtÞÞ ¼ 0
for k a l and Eðζ_ k ðtÞ2 Þ ¼ Γk RU_ ð0ÞΓk ¼ 1: T
ð14Þ
The major advantage of using the expansion (11) instead of expansion (8) is that the components are mutually uncorrelated. Hence, it can be shown that the “energy” of UðtÞ in terms of Euclidean norm ( J X J 2 ¼ XT X) takes the following form: Eð‖UðtÞ‖2 Þ ¼
n X
λk ‖Ψk ‖2
ð15Þ
k¼1
and the truncation error satisfies ! p X 2 E ‖UðtÞ ζ k ðtÞΨk ‖ ¼ k¼1
n X
λk ‖Ψk ‖2 :
ð16Þ
k ¼ pþ1
Note that the dual smooth basis is not an optimal basis in terms of truncation error (see Section 2.5). From Eqs. (15) and (16), it is clear that the quantity
λk ‖Ψk ‖2
ð17Þ
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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4
characterizes the energy of the mode number k and the ratio
λk ‖Ψk ‖2 2 l ¼ 1 λl ‖Ψl ‖
Pn
ð18Þ
defines the fraction of the total energy captured by the mode number k. This equation will allow, as in the linear case for the normal modes, to see how the energy is distributed among the modes. 2.4. Some properties of the characteristics of the SD 2.4.1. Linear transformation Let fVðtÞ; t A Rg be a random process defined as linear transformation of fUðtÞ; t A Rg as VðtÞ ¼ AUðtÞ where A is an invertible matrix. The following relationships between the SD characteristics of fVðtÞ; t A Rg and the SD characteristics fUðtÞ; t A Rg hold: ðiÞ
λVk ¼ λUk
ðiiÞ
ΓVk ¼ A T ΓUk
ðiiiÞ
If AAT ¼ In ; ξk ðtÞ ¼ ξk ðtÞ
ðivÞ
ΨVk ¼ AΨUk
ðvÞ
ζ Vk ðtÞ ¼ ζ Uk ðtÞ
U
V
ð19Þ
Items (i)–(iii) were demonstrated in [9,11]. Items (iv) and (v) show that the Dual Smooth Expansion (11) is preserved by linear transformation. 2.4.2. Gaussian random processes Let fUðtÞ; t A Rg be a Rn valued Gaussian random process, then the SCs and the DSCs are also Gaussian. Moreover, for fixed t, the SCs ζ k ðtÞ for k ¼ 1; …; n are mutually independent (as random variables). Of course, the result is not generally true for a set of non-Gaussian random processes fζ k ðtÞ; t A Rg. 2.5. Dual smooth expansion as KL expansion ~ Let LU_ be the square root matrix of R U_ ð0Þð ¼ LU_ LU_ ) and define the random processes fUðtÞ; t A Rg and its time derivative as the linear transformation ~ UðtÞ ¼ LU_ 1 UðtÞ
and
1 _ ~_ ðtÞ ¼ L U _ UðtÞ: U
ð20Þ
The associated covariance matrices are then given by 1 RU~ ð0Þ ¼ LU_ T RU ð0ÞLU _
and
R U~_ ð0Þ ¼ In :
ð21Þ
~ It follows that the SD of fUðtÞ; t A Rg coincides with its KLD. The associated characteristics are related to the characteristics of fUðtÞ; t A Rg with the linear transformation properties given in Section 2.4. Finally, considering the RU_ ð0Þ 1 norm (‖X‖2R ð0Þ−1 ð0Þ ¼ XT RU_ ð0Þ−1 X), it can be shown that U_
Eð‖UðtÞ‖2R _ ð0Þ 1 Þ ¼ U
n X
λk
ð22Þ
k¼1
and also that the following optimal relationship holds for the DS expansion of fUðtÞ; t A Rg i.e. 0 1 0 1 2 2 p p n X X X A¼ A E@UðtÞ ζ k ðtÞΨk λk r E@UðtÞ ζ k ðtÞΨ k 1 1 k¼1
RU_ ð0Þ
k ¼ pþ1
k¼1
ð23Þ
R U_ ð0Þ
for any arbitrary R U_ ð0Þ 1 orthogonal basis Ψ . This result shows that the dual smooth expansion can be interpreted as a Karhunen–Loève expansion considering the R U_ ð0Þ 1 inner product. 2.6. SD in practice If the matrices R U ð0Þ and R U_ ð0Þ are known, SD is computed solving the generalized eigenvalue problem (5). On the other hand, if only numerical or experimental data are available, SD can be approximated estimating first R U ð0Þ and RU_ ð0Þ and solving a generalized eigenvalue problem. SD can be obtained also solving a generalized singular value decomposition problem. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
5
3. Physical interpretation of the SD Let us consider a mechanical system with n degrees of freedom governed by the equation of motion € þ CUðtÞ _ þ KUðtÞ þ GðUðtÞÞ ¼ FðtÞ MUðtÞ
ð24Þ
where M, C, and K are n n symmetric matrices, and the nonlinearity is in the vector function GðUÞ. The vector FðtÞ, describing the external random forces, is supposed to be a zero-mean white-noise random excitation with matrix intensity SF (i.e., RF ðτÞ ¼ EðFðt þ τÞFT ðtÞÞ ¼ SF δðτÞ). 3.1. Linear case G ¼0 Let ðωk ; Φk Þ be the normal modes of the linear system (KΦ ¼ MΦΩ with Φ ¼ ½Φ1 Φ2 …Φn and Ω ¼ diagðωk ÞÞ. As shown in [11], SD of the stationary response of (24) can be related to the modal parameters of the system (24). The result reads as: if T T the system is such that Φ CΦ and Φ SF Φ are diagonal matrices then the following relationships hold: 2
ðiÞ
Φ ¼ Ψ;
ðiiÞ
ω2k ¼
1
λk
for k ¼ 1; …; n:
ð25Þ
The mode shapes coincide with the dual smooth modes and the square of the resonance frequencies is equal to the inverse of the corresponding smooth values. Moreover, the dual smooth expansion of UðtÞ coincides with the classical modal expansion. Note that the matrix Ψ can be easily obtained from (7). There is hence no need to invert the matrix Γ. 3.2. Nonlinear case SD can be applied to random nonlinear responses and interpreted in the framework of linearized systems using the statistical linearization method (as defined in [19]). For a given external random force (in our case for a given matrix intensity), the statistical linearization method gives access to a linear system (named equivalent linear system) such that its stationary covariance matrix coincides with the stationary covariance matrix of the nonlinear response. Following the results discussed in Section 3.1, the SMs, DSMs and SMVs can be considered as the modal parameters of the equivalent linear system. 3.3. About the smooth mode ordering Following the results discussed in Section 3.1, ordering the SVs, λk, in descending order is equivalent to the classical ordering of the resonance frequencies (ascending order). This ordering used Section 2.1 to number the SD modes which will be named frequency ordering. Following the property (15) of the dual smooth expansion (11), the quantities λk ‖Ψk ‖2 ranking in descending order can also be used to number the SD modes. This ordering will be named energy ordering. The energy ordering is in line with the ordering used in the POD method. In the frequency ordering one considers only 1=λk and in the energy ordering one considers λk ‖Ψk ‖2 . With this convention the first mode (lowest frequency) may not be the most energetic. 4. Illustrative example In this section, the SD is used as an output-only analysis tool to characterize a nonlinear system. The system is depicted in Fig. 1. It is composed of a linear chain of M identical linear oscillators connected at the left end to a fixed wall and connected at the right end to a nonlinear oscillator, also named nonlinear end-attachment. The equations of motion read as (observe that the equations are ordered from right to left with respect to Fig. 1) mN v€ þ λN ðv_ u_ 1 Þ þ f N ðv u1 Þ ¼ 0;
ð26Þ
u€ 1 þ λg u_ 1 þkg u1 þ λc ðu_ 1 u_ 2 Þ þ kc ðu1 u2 Þ λN ðv_ u_ 1 Þ f N ðv u1 Þ ¼ 0;
ð27Þ
Fig. 1. Schema of the nonlinear system.
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
6
u€ m þ λg u_ m þkg um þ λc ð2u_ m u_ m 1 u_ m þ 1 Þ þ kc ð2um um 1 um þ 1 Þ ¼ 0; ð2 r m r M 1Þ
ð28Þ
u€ M þ λg u_ M þ kg uM þ λc ðu_ M u_ M 1 Þ þ kc ðuM uM 1 Þ ¼ f ðtÞ
ð29Þ
where v (respectively um) denotes the displacement of the nonlinear end-attachment (respectively the mth mass of the linear chain). The nonlinear end-attachment is constituted of a mass mN, a linear damper with coefficient λN , and a nonlinear spring characterized by the restoring force ( C 1 x þ C 3þ x3 if x Z0 ð30Þ f N ðxÞ ¼ C 1 x þ C 3 x5 if x r0 where C1 C 3þ and C 3 are real positive scalars. Note that C1 will be chosen so small that maxðC 3þ ; C 3 Þ corresponds to an essentially nonlinear function. An external force f(t) is applied to the mass number M, the leftmost mass. We assume that it is of the form f ðtÞ ¼ s0 WðtÞ
ð31Þ
where fWðtÞ; t A Rg is a Gaussian white-noise scalar process with intensity one and s0 denotes the excitation level. The Monte–Carlo method was used to estimate the covariance matrices of the stationary responses (displacement and velocity) of the system. From a given excitation level, the response time history was obtained from a time history of excitation (31) by solving Eqs. (26)–(29) over the time interval [0, tf] numerically using the Newmark method with a time-step of size Δt ¼ 1=f s (where fs denotes the sampling frequency). Zero initial displacement and velocity were assumed. The time histories of W(t) (a Gaussian white-noise scalar process with intensity one) were generated using the procedure described in [20]. Assuming ergodicity, the last-half points of the displacement and velocity time histories were used to approximate (as time averages) the covariance matrices RU ð0Þ and RU_ ð0Þ with U ¼ ðv; u1 ; u2 ; …; uM ÞT . The Smooth Decomposition analysis is then obtained solving the eigenvalue problem (5). The following numerical parameter values were used to simulate the system (26)–(29): M¼9 (that is a 10-DOF system including 9-DOF from the linear chain and 1-DOF from the nonlinear end-attachment), λc ¼ 0:005, kc ¼1, λg ¼ 0:005, kg ¼ 1, mN ¼0.05, λN ¼ 0:001, kN ¼0.0001, C 1 ¼ 0:01, C 3þ ¼ 1, C 3 ¼ 0:0123. The equations of motion were numerically solved with the following time-discretization parameter values Δ ¼ 0:143 s (i.e. fs ¼7 Hz) and tf ¼74942 s (corresponding to 524 286 instants). The excitation levels s0 have been used as the parameter of analysis. Forty-eight linearly equally spaced points between 0.0025 and 0.12 have been considered. 4.1. Linear case as low excitation response The stationary response associated to the excitation level s0 ¼ 0:0025 is considered here. The first six dual smooth modes (using frequency ordering) are reported in Fig. 2 and compared to the normal modes of the underlying linear system. The SD correctly estimates the normal modes (as predicted by the theoretical results), four of them not shown here. This is also the case for the resonance frequencies. The resonance frequencies estimated from (25) (0.0068, 0.1613, 0.1773, 0.204, 0.2359, 0.2680, 0.2973, 0.3220, 0.3405 and 0.3519) coincide (numerically) with the exact resonance frequencies of the underlying linear system (0.0071, 0.1613, 0.1773, 0.2042, 0.2359, 0.2680, 0.2974, 0.3220, 0.3405 and 0.3520). This result confirms that for the selected excitation level the nonlinear stiffness of the end-attachment does not affect the response. The modal motion associated to the first normal mode (corresponding to the smaller resonance frequency) is localized on the nonlinear attachment whereas the modal motions associated to the other modes have the linear chain also moving and coincide with the normal modes of linear chain. 4.2. Nonlinear case The SD approach has been applied to the stationary responses associated to the chosen 48 excitation levels s0 and the results are now discussed. Fig. 3(a) shows the fraction of energy captured by the modes (as defined in Eq. (18)) for the 48 excitation level values. The red line marks the most energetic modes, the first mode considering the energy ordering. For 0:0025 o s0 r 0:045, the mode number 1 is the most energetic. For 0:045 r s0 r 0:0575, the most energetic is the mode number 3. For 0:0575 rs0 r0:08, the mode number 4 is the most energetic. Finally, for s0 Z0:08, there is no dominant mode, the energy is distributed, principally, among the first five modes. Note that, in this problem, the mode number 2 does not appear as a dominant mode. Fig. 3(b) shows the resonance frequencies estimated from the SD versus the excitation level. As in Fig. 3(a), the red line corresponds to the most energetic modes. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
Mode 1
1
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 3
1
Mode 4
1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 5
1
Mode 2
1
0.5
7
Mode 6
1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Fig. 2. SD analysis for s0 ¼ 0:0025: dual smooth modes (cross markers, blue), and normal modes of the underlying linear system (red line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
100
Modal energy (%)
80 70 60 50 40 30 20
0.3 0.25 0.2 0.15 0.1 0.05
10 0
0.35
Resonance frequency (Hz)
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Mode 8 Mode 9 Mode 10
90
0.02
0.04
0.06
s0
0.08
0.1
0.12
0.02
0.04
0.06
0.08
0.1
0.12
s0
Fig. 3. SD analysis: (a) Fraction of energy captured by the modes versus excitation level s0. (b) Resonance frequencies estimated from (25) versus excitation level. The red line corresponds to the most energetic modes. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
At s0 ¼0.0025, the estimated resonance frequencies coincide with the resonance frequencies of the underlying linear system as already mentioned in the previous section. For s0 4 0:0025, the behaviors of the resonance frequencies with respect to s0 differ. The resonance frequency associated to the mode number 1 (cross markers) first increases when the excitation level increases and for s0 large (approximatively greater than 0.047), it becomes constant with a numerical value near, but smaller than, the second resonance frequency of the underlying linear system. The resonance frequency associated to the mode number 2 (star markers) remains first constant for small values of s0, then slowly increases and for large s0 (approximatively greater than 0.055), it becomes constant with a numerical value near, but smaller than, the third resonance frequency of the underlying linear system. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
8
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
The same behavior can be noted for the resonance frequency associated to the mode number 2–5 (see Fig. 3(b) circle markers, square markers and diamond markers). These results illustrate the hardening behavior of the system. It is also interesting to note that the fraction of energy associated to the current most energetic mode is large with respect to the other modes (see Fig. 3(a)), the corresponding resonance frequency increases with the excitation level (see the red line Fig. 3(b)). Finally, the results shown in Fig. 3(a) and (b) can be combined given the frequency–energy dependence of the modal oscillations of the nonlinear system. For a given mode number, the corresponding resonance frequency and the associated modal energy depend on the input excitation level (or equivalently to the input energy) and can be represented in a frequency–energy plot. The result is reported in Fig. 4 where for each mode, the resonance frequency (as defined by (25)) is plotted versus the energy (as defined by (17)). A log scale has been used in the modal energy axis. The frequency behaviors of the modes with respect to the energy are similar to the frequency behaviors of the same modes with respect to the excitation s0. For a given mode, the frequency starts from the corresponding value of the resonance frequency of the normal
Frequency (Hz)
0.25
0.2
0.15
0.1
0.05
Mode Mode Mode Mode Mode Mode
10
1 2 3 4 5 6
10
10
10
10
Energy Fig. 4. SD analysis: Frequency–energy plot of each mode.
1
Mode 1
1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 3
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 5
Mode 4
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
0.5
0.5
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 2
Mode 6
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Fig. 5. SD analysis for s0 ¼ 0:022: dual smooth modes (cross markers, blue), and normal modes of the underlying linear system (red line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
1
Mode 1
1
Mode 2
1
Mode 1
9
1
0.5
0.5
0.5
0.5
0
0
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 3
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 4
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 3
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
0.5
0.5
0.5
0.5
0
0
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 5
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 6
v u1 u2 u3 u4 u5 u6 u7 u8 u9 1
Mode 5
1
0.5
0.5
0.5
0
0
0
0
v u1 u2 u3 u4 u5 u6 u7 u8 u9
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 4
v u1 u2 u3 u4 u5 u6 u7 u8 u9
0.5
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Mode 2
Mode 6
v u1 u2 u3 u4 u5 u6 u7 u8 u9
Fig. 6. SD analysis for (a) s0 ¼ 0:05 and (b) s0 ¼ 0:07: dual smooth modes (cross markers) and normal modes of the underlying linear system (red line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
mode of the underlying linear system. The energy and the frequency first increase, next the frequency increases whereas the energy decreases and finally the frequency becomes constant with increasing energy. Note that this representation is in line with the frequency–energy plot used to represent the nonlinear normal modes as described in [21,22] and will be completed in the following with modal shape information. Figs. 5 and 6 show the shapes of the first six dual smooth modes obtained, respectively, at the excitation levels s0 ¼ 0:022; 0:05, and 0.07. These three values have been chosen in the intervals where the mode number 1, 3 and 4 are, respectively, the most energetic (see Fig. 3(a)). The shapes of the normal modes of the underlying linear system are also reported on the same plot. For s0 ¼0.022, only the second, third and fourth dual smooth modes are weakly affected by the nonlinearity (see Fig. 5). For s0 ¼0.05 and s0 ¼0.07, significant differences appear between the first fourth dual smooth modes and the corresponding normal modes. The visualization of the shapes of the dual smooth modes can be combined with the frequency–energy plot as shown in Fig. 7 where, for the first four modes, the corresponding curve of frequency–energy is depicted together with the corresponding shape of the dual modes (realized as in Figs. 2, 5 and 6). This visualization gives now, as the frequency– energy plot used to represent the nonlinear normal modes in [21,22], information in terms of frequency, shape of the mode and energy. As already mentioned, for each mode (here mode 1 to mode 4), the mode shape of the dual smooth mode obtained at low energy level (i.e at low frequency) coincides with the mode shape of the associated normal mode of the underlying linear system (see the plots (1) in Fig. 7(a)–(d)). Moreover, at high energy level (i.e. at high frequency), the mode shape of the dual smooth mode shows a non-localized motion (see the plots (7) in Fig. 7(a)–(d)). For each mode, these two “limit” shapes are well-marked. The transition between these two “limit” shapes is marked by a region (see the plots (3) and (4) in Figs. 7(a)–(d)) where frequency and energy simultaneously increase and mode shapes get closer to the shape of the first normal mode of the underlying linear system corresponding to the localized motion on the nonlinear attachment degree of freedom (see Fig. 2, Mode 1). This behavior can be interpreted as a resonance capture occurring between a mode localized on the linear chain and a mode localized on the nonlinear attachment. Note that, classically, the term resonance capture is used to characterize dynamics phenomena with respect to time. Here we are observing a phenomenon which occurs varying the excitation level. The last point to consider is to compare the shape of the dual smooth modes of the most energetic modes. For s0 ¼0.022 (respectively 0.05 and 0.07), the most energetic mode is the mode number 1 in Fig. 5 (respectively number 3 in Fig. 6(a) and number 4 in Fig. 6(b)). The shapes of these three modes are very close and similar to the shape of the first normal mode of the underlying linear system. This observation is also true as well as the current first dual mode is dominant (in terms of energy (see Fig. 3)) that is for 0:005 r s0 r 0:08. These shapes are compared in Fig. 8. These mode shapes define motions
Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
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S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
Fig. 7. Frequency–energy plot of the first four modes obtained by the SD approach: (a) mode number 1, (b) mode number 2, (c) mode number 3 and (d) mode number (4).
1 0.8
Dual smooth mode 1
0.6 0.4 0.2 0
v
u1
u2
u3
u4
u5
u6
u7
u8
u9
Fig. 8. Shapes of the dual smooth modes of the most energetic modes for 0:005 r s0 r 0:08. First normal mode of the underlying linear system (red line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
localized on the nonlinear attachment (v component), so the form of the most energetic mode does not change. For s0 Z0:08, the most energetic dual smooth modes are no more localized at the nonlinear attachment (not shown here). The last comment concerns the effect on the behavior of the system of these localized modes with dominant energy. Fig. 9 shows the fractions of the energy dissipated by the linear chain and the nonlinear attachment versus the excitation level. When the excitation level increases, the fraction of energy dissipated by the nonlinear attachment increases, becomes greater than the energy dissipated by the linear chain and finally decreases. The range where the nonlinear attachment is effective Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
11
100
Dissipated energy (%)
90 80 70 60 50 40 30 20 10 0
0.02
0.04
0.06
0.08
0.1
0.12
s0 Fig. 9. Fraction of dissipated energy by the linear chain (circle markers) and by the nonlinear attachment (square markers) versus excitation level s0.
(with respect to the linear chain) to dissipate energy is localized in the interval 0:035 r s0 r0:08. In the effective range, the shape of the most energetic modes corresponds to motions localized on the nonlinear attachment which of course privilege the dissipation of energy in the damper of the nonlinear attachment. Note that in the range 0:005 rs0 r 0:035, the shape of the most energetic modes corresponds also to motions localized on the nonlinear attachment.
5. Conclusion In this paper, the SD of a vector valued random process has been discussed starting from the classical properties of the SD. The concept of dual smooth modes has been introduced and used to expand a random process. The dual smooth expansion was interpreted as a Karhunen–Loève expansion choosing a convenient inner product. Two orderings of the modes were introduced, one in line with the classical ordering of the resonance frequencies, the other in line with the modal energy content. The two orderings do not necessarily coincide, and give two different observations of the system under study. The SD was also interpreted in terms of modal analysis in case of linear and nonlinear mechanical systems. For nonlinear systems it constitutes a novel way to analyze the system using linear analysis tools which was not considered in[23]. The efficiency of this approach as a tool to understand nonlinear behavior was demonstrated. Future work concerns the study of the relations between the nonlinear normal modes and the modes defined by the SD analysis.
Acknowledgments This work was supported by CAPES (Brazil) and COFECUB (France) agencies (under CAPES/COFECUB project No. 672/10) and by CNPq and Faperj.
Appendix A. Oblique projection Let us consider a p-dimensional subspace Ep with p r n in Rn with Rn ¼ Ep En p . ~ ;Φ ~ ; …; Φ ~ Þ be a basis of E and ðΦ ; Φ ; …; Φ Þ be a basis of ET Let ðΦ p p p 1 2 1 2 n p the orthogonal space of E n p such that ~ ¼I ΦT Φ p
ðA:1Þ
~ ¼ ½Φ ~ Φ ~ , Φ ¼ ½Φ Φ ⋯Φ and I denotes the p p identity matrix. ~ ⋯Φ where Φ p p p 1 2 1 2 The oblique projection Π of Rn into Ep along the subspace En p is defined by the matrix ~ T: Π ¼ ΦΦ
ðA:2Þ
The projection appears as a simple truncation in the appropriated basis as ~ TU ¼ Π U ¼ ΦΦ
p X
T ~ : ðΦk UÞΦ k
ðA:3Þ
k¼1
~ TΦ ~ comes from the p vectors of an orthonormal basis (Φ ~ ¼ I ) of Rn then (A.3) corresponds to the classical If Φ p orthogonal projection. Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i
S. Bellizzi, R. Sampaio / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
12
~ can be considered as the dual basis of Φ in the sense that If p¼n, the basis Φ U¼
n X
~ : ðΦk UÞΦ k T
ðA:4Þ
k¼1
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Please cite this article as: S. Bellizzi, R. Sampaio, The Smooth Decomposition as a nonlinear modal analysis tool, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.04.015i