Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations

In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude–frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here, a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. This model identification utilizes Taken’s delay-embedding theorem, as well as a least square fit to the Taylor expansion of the sampling map associated with that embedding. The SSMs are then constructed for the sampling map using the parametrization method for invariant manifolds, which assumes that the manifold is an embedding of, rather than a graph over, a spectral subspace. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.

A Proof of Theorem 1 (main text) we need to solve the algebraic equation for the unknown Taylor coefficients w s j and r s j . We carry this out step by step for increasing powers of z: O (|z|): Since the Taylor series of G starts with second-order terms, the first-order monomials of z arising from substitution into (2) satisfy ΛW = W • R, which simplifies to ΛW = W Λ, because the linear part of R is equal to the diag {µ ,μ } block of the diagonal matrix Λ. Consequently, (2) can be written at leading order as whose simplest solution is with δ j denoting the Kronecker delta. This proves the formulas for the first-order coefficients in Theorem 2 (in the main text). We note that w (1,0) and w (0,1) +1 are only determined up to a constant, which we have chosen to be equal to 1.
O |z| 2 : Since r s j = 0 for |s| = 2 by (??), the quadratic terms in |z| on the right-hand side of (2) only arise from the substitution of linear terms of R(z) into the quadratic terms of W (z). As a consequence, equating the coefficients of O |z| 2 terms on both sides of (2) gives the equation µ j w (s 1 ,s 2 ) j + g (s 1 @ ,s 2 @( +1)) j = µ s 1 μ s 2 w (s 1 ,s 2 ) j , |s| = 2, whose solution for w proving the formulas for the second-order coefficients w in the statement of Theorem 2 (main text). Note that the denominator in (6) is guaranteed to be nonzero by the nonresonance condition (equation (17), main text) O |z| 3 : We write out the j th coordinates in the three terms of eq. (2) in detail to obtain the following cubic terms: We now write out the individual terms in eq. (9). For y = W (z), we have +O |z| 2 , |z| 4 , thus, for the first sum in (9), we obtain |m|=2 For the second sum in (9), we have Working out these expressions in detail, we find that Substituting the expressions for A, B and C into (12), then substituting (12) and (11) into the invariance condition (2), we equate equal powers of z to obtain the following linear equations for the cubic coefficients of the mapping W and of the mapping R: From the first and last equation in (14), we obtain We select j = and assume that there is no first-order near-resonance (or exact resonance) involving the eigenvalues µ and µ j (stated as µ j ≈ µ under the assumptions of the theorem).
Recalling w (1,0) j = δ j , we then obtain from the second equation of (14) that whenever δ j = 0. Similarly, selecting j = + 1, assuming no first-order near-resonance (or exact resonance) involving the eigenvalues µ +1 and µ j (i.e., µ j ≈ µ +1 ), and recalling w (0,1) j = δ j( +1) , we obtain from the third equation of (14) that whenever δ j( +1) = 0. Next we select j = in the second equation of (14), and select j = + 1 in the third equation of (14). These choices force us to select in these equations to avoid small denominators arising from the near-resonances. Then the second equation of (14) with j = gives the solution But equations (15)-(19) prove the formulas for the cubic coefficients of W j and β in the statement of Theorem 2 (main text).

B Analogous results for continuous dynamical systems
Here we discuss spectral submanifolds, backbone curves and their leading-order computation for continuous dynamical systems. The formulas we derive are useful for benchmarking our databased SSM and backbone-curve approach on exactly known mechanical models. The concepts and formulas derived here, however, are also of independent interest in computing the dynamics on SSMs in analytically defined mechanical models. We start with the continuous analogue of the the complex mapping which is a complex differential equation of the forṁ y = Λy + G(y), y ∈ C 2ν , Λ = diag(λ 1 , . . . , λ 2ν ), λ 2l−1 =λ 2l , l = 1, . . . , ν, (21) G(y) = O |y| 2 , with a fixed point at y = 0, and with a class C r function G. The eigenvalues of Λ are ordered so that and hence y = 0 is asymptotically stable. If (21) is the equivalent first-order complexified form of a mechanical system of the form then we specifically have ν = n. If, furthermore, the mechanical system has linear and weak proportional damping, then we can write with ζ l and ω l denoting Lehr's damping ratio and undamped natural frequency, respectively, for the l th mode of the linearised system at the q = 0 equilibrium. Finally, we assume that E is a two-dimensional spectral subspace (eigenspace) of the operator Λ, corresponding to the complex pair of simple eigenvalues λ = λ +1 for some ∈ [1, 2ν − 1].

B.1 Existence and uniqueness of SSMs
Following [2], we address this issue via the following definition: Definition 1. A spectral submanifold (SSM) W (E) corresponding to a spectral subspace E of Λ is (i) an invariant manifold of the dynamical system (21) that is tangent to E at y = 0 and has the same dimension as E; (ii) strictly smoother than any other invariant manifold of (21) satisfying (i).
We now recall from Haller and Ponsioen [2] the specific existence and uniqueness result pertaining to two-dimensional SSMs, deducible from the more general results of Cabré et al. [1]. The relative spectral quotient of E is now defined as the positive integer whose meaning is the same as pointed out after formula (25) for mappings. In case of a proportionally damped mechanical system, one may use the formulas (24) and conclude that Remark 1 continues to provide the correct specific form of σ(E) in this case. We again assume that and that no resonance relationships up to order σ(E) hold between the eigenvalues λ , λ +1 and the rest of the spectrum of Λ, i.e., The alternative form of this nonresonance condition given in Remark 1 again applies whenever formulas (25) hold.
Theorem 1. Assume that conditions (26)-(27) are satisfied. Then following statements hold: (i) There exists an SSM, W (E) , for the dynamical system (21) that is tangent to the invariant subspace E at the y = 0 fixed point.
(ii) The invariant manifold W (E) is class C r smooth and unique among all two-dimensional, class C σ(E)+1 invariant manifolds of (21) that are tangent to E at y = 0.
(iii) The SSM W (E) can be viewed as a C r immersion of an open set U ⊂ C 2 into the phase space C 2ν of system (21) via a map (iv) There exists C r polynomial function R : U → U such that i.e., the dynamics on the SSM, expressed in coordinates z = (z ,z ) ∈ U, is governed by the polynomial ODEż = R(z), DR(0) = diag(λ ,λ ), whose right-hand side has only terms up to order O |z| σ(E) .
(v) Under the further internal non-resonance assumption within E, the mapping W in 28 can be selected such that the j th coordinate component R j of R does not contain the term (z s 1 ,z s 2 ).
Proof. This is merely the re-statement of the main theorem of Haller and Ponsioen [2] (deduced from Cabré et al. [1]) for the case of a two-dimensional SSM corresponding to a simple pair of complex eigenvalues with negative real parts.

B.2 Backbone curves and their computation
When the spectral subspace E of (21) is lightly damped (|Reλ | 1), the low-order near-resonance relationships 2λ +λ ≈ λ , λ + 2λ ≈λ always hold. As in the case of mappings, this prompts us to seek the polynomial dynamics on the SSM (cf. statement (iv) of Theorem 1) in the forṁ z =R(z) = λ z + β z 2 z λ z +β z z 2 .
Introducing polar coordinates z = re iθ , we can further transform (31) to the real amplitudephase equationsρ Equation (33) gives instantaneous frequency of nonlinear oscillations as whereas as instantaneous amplitude Amp(ρ) of the vibration can be calculated as where W is the mapping featured in statement (iii) of Theorem 1, and V is the linear mapping that transform the original, first-order dynamical system to its standard complex form (21). With the quantities defined in (34) and (35), the definition of a backbone curve B given in Definition 2 carries over without change to our present context. Again, to compute the backbone curve we need to find expressions for the complex coefficient β and the mapping W (z), as the eigenvalue λ is assumed to be known.
To this end, we seek the Taylor series coefficients of the j th coordinate functions, W j (z) ∈ C, j = 1, . . . , 2ν, of the mapping W (z), and the third-order Taylor coefficient β ∈ C of the polynomial function R(z) defined in (31). These unknowns will again be expressed as functions of the j th coordinate functions G j (y) ∈ C, j = 1, . . . , 2ν, of the nonlinear part G(y) of the right-hand side of the dynamical system (21). Using the same notation as in Theorem (2), we obtain the following expressions for the required Taylor coefficients.
Theorem 2. Suppose that the assumptions of Theorem 1 hold but with the strengthened version of the external non-resonance condition (27). Then, for any j ∈ [1, 2ν], the j th coordinate function W j of the mapping W in (28) and the cubic Taylor coefficient β of the conjugate map R in (29) are given by the following formulas: Proof. The algebraic equation (29) is similar to the equation (1), which we have solved in detail up to cubic order in the proof of Theorem 2 (main text). The first difference between the two equations is that the term ΛW in (1) has the j th component Substituting formulas (38)-(39) into (1), and using the expression for (W • G) j from the proof of Theorem 2 (main text), we obtain the formulas in the statement of Theorem (2) after comparing equal powers of z up to cubic order.