Abstract
Model reduction via spectral submanifolds (SSMs) has displayed benefits such as the facilitation of nonlinear analysis and significant speed-up gains. One needs the sensitivity of the SSM-based model reduction to carry over these benefits to the settings of optimal design, modal updating, and uncertainty quantification of high-dimensional nonlinear mechanical systems. Here, we construct explicit third-order, SSM-based model reduction for general mechanical systems. We further derive the explicit sensitivity of the third-order SSM-based reduction using direct and adjoint methods. We demonstrate the effectiveness of the derived explicit sensitivity via a few examples with increasing complexity. We also show that the obtained sensitivity can be used to effectively construct perturbed SSMs, backbone curves, and forced response curves.
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1 Introduction
1.1 Spectral submanifold and model reduction
Spectral submanifolds (SSMs) have emerged as a powerful tool for constructing low-dimensional reduced-order models (ROMs) of high-dimensional nonlinear mechanical systems in the past several years [1,2,3]. An SSM is the unique smoothest nonlinear continuation of a spectral subspace (i.e., a direct sum of modal subspaces) of a mechanical system linearized at a fixed point [1]. Such a unique SSM is guaranteed to exist under proper non-resonant conditions. Moreover, it can be uniquely extended to a time-periodic invariant manifold under the addition of harmonic forcing [1], and a whisker attached to an invariant torus of a quasi-periodically forced system [1] or a Hamiltonian system with a dissipative effect proportional to velocity [4]. Importantly, slow SSMs attract nearby full-system trajectories and hence are attracting low-dimensional invariant manifolds in the full phase space of the system. Consequently, the internal dynamics on the SSM serve as a mathematically exact ROM for the full nonlinear system [1].
SSM-based model reduction enables effective analysis of nonlinear dynamics of mechanical systems. It provides an analytic extraction of backbone curves [5]. In addition, periodic orbits of harmonically excited high-dimensional systems are transformed as fixed points of the low-dimensional SSM-based ROMs [5, 6]. This transformation enables the analytic extraction of forced response curves (FRCs) and surfaces [7] and also isolas on FRCs for systems without internal resonance [8]. It also facilitates the detection of both local and global bifurcations of periodic orbits [9, 10]. Indeed, bifurcations of periodic orbits of the full systems can be predicted as the bifurcations of fixed points of the ROMs [9]. Meanwhile, quasi-periodic orbits of harmonically excited high-dimensional systems are transformed as limit cycles of the low-dimensional SSM-based ROMs [9]. Therefore, one can make predictions of bifurcations of the quasi-periodic orbits effectively via the bifurcation of the limit cycle. Recently, the SSM-based model reduction was combined with parameter continuation to efficiently extract ridges and trenches on a forced response surface without computing the surface [7]. Such ridges and trenches provide a compact yet informative characterization of the forced response under the variations in forcing frequency and amplitude.
The aforementioned facilitation to the analysis of nonlinear dynamics via SSM-based model reduction becomes feasible when one can efficiently construct the SSM-based ROMs. Parameterization methods [11, 12] have been used to compute such SSMs in an automated fashion [13, 14]. Along this line, an explicit third-order SSM-based ROM was derived in [15]. The implementation of the parameterization methods in [13,14,15], however, needs a complete eigenbasis of the linearized systems, which is out of reach for high-dimensional finite-element (FE) problems. To tackle this challenge, Jain and Haller [16] proposed a computational method of SSMs using only the knowledge of eigenvectors associated with the master modal subspace. This computational scheme has been recently extended to calculate high-order non-autonomous SSMs [17].
The implementation of the computational methods documented in [16, 17] has been made publicly available in SSMTool [18], a matlab toolbox that supports automated construction of SSM-based ROMs. Indeed, SSMTool has been successfully applied to extract FRCs of nonlinear FE problems with more than 10,000 degrees of freedom (DOF) [6, 16], where a significant speed-up gain of the SSM-based model reduction was demonstrated.
1.2 Sensitivity analysis in nonlinear dynamics
The computational gains of model reduction become even more significant in design settings [19,20,21], where system parameters are iterated and updated to achieve design goals of tuning nonlinear dynamics [22,23,24]. In this process, analysis needs to be performed when system parameters are updated, indicating that many analyses can be involved. Therefore, ROMs will be used multiple times, significantly reducing computational costs and making the design of high-dimensional systems’ nonlinear dynamics feasible [19, 20].
The update of system parameters (or design variables) in optimization design can be determined via the sensitivity of the objective function with respect to the design variables [25, 26]. Such a sensitivity plays a fundamental role in gradient-based optimization algorithms. For instance, the design variables are updated following a negative gradient direction to minimize the objective function in the steepest descent method [25]. Gradient-based optimization has been successfully applied to optimize the amplitude of transient responses [27], periodic responses [28], peak amplitude of FRC [29], and normal form coefficients [22].
Sensitivity can be used in other design tasks of nonlinear dynamics except for optimization. Finite element model updating is a process of adjusting certain parameters of the FE model to make corrections to the FE model based on vibration test data [30]. Sensitivity plays an essential role in the updating process. Indeed, the sensitivity method is probably the most successful approach for modal updating [30]. It has been applied successfully to large-scale industrial problems to match linear dynamics features such as natural frequencies, mode shapes, and forced response functions [30,31,32]. Model updating of nonlinear dynamics is much more challenging due to high computational costs. Meta-models such as surrogate models and reduced-order modeling were used to reduce the computational time of performing nonlinear model updating [33].
Another important application of sensitivity is uncertainty quantification of shape imperfections or geometry defects that may stem from the manufacturing process [34, 35]. Such defects can result in non-negligible modifications of the nonlinear response, as in the case of MEMS devices [36] and pipes conveying fluid [37, 38]. Taylor expansion via local derivatives is used to establish the dependence of nonlinear response on the uncertain parameters [39], from which one can quantify the propagation of uncertainty via either Monte Carlo method [35] or analytic derivations [40]. The perturbed FRCs under the small variations in the uncertainty parameters can also be obtained via the Taylor expansion, as demonstrated in [35, 40].
1.3 Our contributions
Motivated by the aforementioned various applications of sensitivity in nonlinear dynamics, we aim to establish the sensitivity of SSMs of mechanical systems. With such sensitivity, the benefits of SSM-based model reduction can be carried over to optimal design, model updating, and uncertainty quantification of high-dimensional nonlinear mechanics problems. We restrict our attention to two-dimensional SSMs applicable to mechanical systems without internal resonances. We leave the extension to higher-dimensional SSMs that are necessary for internally resonant mechanical systems [6, 9] for future study.
The remainder of this paper is organized as follows. In the next section, we list the setup of the mechanical systems. We allow for asymmetric damping and stiffness matrices and also velocity-dependent nonlinearities. We then present an explicit third-order SSM-based model reduction for the mechanical systems in Sect. 3. This explicit reduction is different from that of [15] because our explicit reduction follows [16] and needs only eigenvectors associated with the master subspace. Based on such an explicit model reduction, we derive explicit derivatives of the SSMs using the direct method in Sect. 4, where we also show how to construct perturbed SSMs, backbone curves, and limit cycles of unforced vibrations. We further extend the sensitivity analysis to forced vibration in Sect. 5, where we derive the sensitivity of periodic orbits and peaks on FRCs, and also discuss how to extract perturbed FRCs using the sensitivity. We demonstrate that adjoint method can also be used to derive the sensitivity in Sect. 6. Finally, we illustrate the effectiveness of explicit sensitivity in a few examples with increasing complexity before drawing conclusions.
2 Setup
Consider the following mechanical system
where \({\varvec{x}}\in \mathbb {R}^{{N}}\) is a vector of generalized displacements, \(\varvec{\mu }\in \mathbb {R}^q\) denotes a vector of system parameters, \(\varvec{M}\), \(\varvec{C}_\textrm{d}\), and \(\varvec{K}\) represent mass, damping and stiffness matrices, respectively, \(\varvec{f}_2\) and \(\varvec{f}_3\) give quadratic and cubic nonlinearities, and \(\varvec{z}=(\varvec{x},\dot{\varvec{x}})\) is a state vector. Here, we highlight the dependence on system parameters because we will establish the sensitivity of SSMs with respect to these parameters. We allow for velocity-dependent nonlinearities and general mass, damping, and stiffness matrices. In other words, these matrices are not necessarily symmetrical or positively definite.
The second-order equations of motion (1) can be rewritten in a first-order form below:
where
In the next section, we will use this first-order formulation to perform explicit third-order model reduction.
3 Explicit third-order model reduction
In this study, we consider the SSM constructed around a two-dimensional master spectral subspace \(\mathcal {E}={{\,\textrm{Span}\,}}\{{\varvec{v}},\bar{{\varvec{v}}}\}\), which is spanned by a pair of complex conjugate modes corresponding to the eigenvalues \(\lambda ,{\bar{\lambda }}\). Hence, we have \({{\,\textrm{spect}\,}}(\mathcal {E}) = \{\lambda ,{\bar{\lambda }}\}\). Let \(\varvec{p}=(p,{\bar{p}})\) be a vector of parameterization coordinates, we seek the autonomous SSM map \({\varvec{z}=}\varvec{W}(\varvec{p})\) and its associated reduced dynamics \(\dot{\varvec{p}}=\varvec{R}(\varvec{p})\) at \(\epsilon =0\). They are determined by solving an invariance equation, which requires that the response would be the same upon the substitution of the manifold map and the reduced dynamics. From now on, we ignore the explicit dependence of \(\varvec{A}, \varvec{B}, \varvec{Q}\) and \(\varvec{C}\) on the vector of system parameters \(\varvec{\mu }\) for compactness. The invariance equation for \(\epsilon =0\) is listed below
When \(\epsilon >0\), we solve for the time-periodic SSM \(\varvec{W}_\epsilon (\varvec{p},\phi )\) and its associated reduced dynamics \(\dot{\varvec{p}}=\varvec{R}_\epsilon (\varvec{p},\phi )\) and \({\dot{\phi }}=\varOmega \) from the following invariance equation
We note that these invariance equations are partial differential equations (PDEs). Following parameterization methods [11, 12, 16], one can approximate the unknown maps and vector fields using Taylor expansion in \(\varvec{p}\) and Fourier expansion in \(\phi \). Then, the unknown expansion coefficients can be determined by solving systems of linear equations. Next, we provide more details on the computation of these expansion coefficients (see Appendix A).
We first perform explicit third-order reduction for the general first-order system (2), where we do not use the structures shown in (3). Then, the obtained results are adapted to the second-order mechanical system (1). Specifically, we seek for the autonomous part of the 2D SSM of the following form
with \(\varvec{W}_{mn}=\bar{\varvec{W}}_{nm}\), and then make corrections to account for the harmonic excitation \(\epsilon \varvec{f}^\textrm{ext}\cos \varOmega t\) in (1). As we will see, our explicit third-order reduction is different from that of [15] because our derivation follows [16] and needs only eigenvectors associated with the master subspace, which makes it applicable for high-dimensional FE problems.
3.1 Computation for general first-order system
As detailed in Appendix A, we obtain the expansion coefficients for the SSM parameterization below:
where
and the definition of \(\hat{\varvec{Q}}\) and \(\hat{\varvec{C}}\) can be seen in (160) and (161) in the Appendix. Here \(\varvec{u}\) is the left eigenvector associated with the eigenvalue \(\lambda \). The corresponding third-order reduced dynamics is given by
Let \(p=\rho e^{\textrm{i}\vartheta }\), we obtain the reduced dynamics in polar coordinates below
3.2 Restriction to mechanical systems
Now we use (3) to adapt the derivations above to the second-order mechanical system (1). We have the following generalized eigenvalue problem
The right and left eigenvectors are obtained as below
The normalization condition asks that
As detailed in Appendix B, we obtain the expansion coefficients for the SSM parameterization below
where
with \({\tilde{\lambda }}=2\lambda +{\bar{\lambda }}\) and
where \(\varvec{a},\varvec{b}\) and \(\varvec{c}\) stand for the arguments of functions. We also the obtain the reduced dynamics (9) with \(\gamma = -\varvec{\psi }^\textrm{T}\varvec{f}_{21}\).
In the case that the quadratic and cubic nonlinearities are only displacement-dependent, we have
Thus \(\varvec{f}_2(\varvec{\mu },\varvec{v},{\varvec{v}})=\varvec{f}_2(\varvec{\mu },\varvec{\phi },{\varvec{\phi }})\) and
In the case that the system has positive definite mass and stiffness matrices and Rayleigh damping, we have
Then we have \(\varvec{\psi }=\kappa \varvec{\phi }\), where the coefficient \(\kappa \) is determined from the normalization condition
We further let \(\varvec{\phi }^\textrm{T}\varvec{C}_\textrm{d}\varvec{\phi }=2\zeta \omega \), where \(\zeta \) is a damping ratio, yielding
Substitution of \(\lambda \) and \(\varvec{\phi }^\textrm{T}\varvec{C}_\textrm{d}\varvec{\phi }=2\zeta \omega \) into (21) yields
4 Explicit derivatives of SSMs via direct computation
Now, we compute the derivatives of the SSMs and their associated reduced dynamics. In particular, we will calculate the derivatives of the expansion coefficients of the SSM parameterization and the reduced dynamics with respect to the vector \(\varvec{\mu }\) of system parameters.
4.1 Derivatives of eigenvalues and eigenvectors
We first present the derivatives of the master eigenvalue \(\lambda \) and its associated right and left eigenvectors. For general mechanical systems, e.g., systems with asymmetric damping or stiffness matrix, the relation \(\varvec{\psi }=\kappa \varvec{\phi }\) does not hold. Instead of using the normalization condition shown in (20), we add the normalization condition \(\varvec{l}_o^\textrm{T}\varvec{\phi }=1\) to uniquely determine the right eigenvector [41]. Here \(\varvec{l}_o=(2\lambda \varvec{M}+\varvec{C})^\textrm{T}\varvec{\psi }\) stands for a constant vector evaluated at a given parameter vector \(\varvec{p}=\varvec{p}_o\). It follows that the right eigenvector \(\varvec{\phi }\) and the eigenvalue \(\lambda \) can be uniquely solved from zero of the following function for \(\varvec{p}\approx \varvec{p}_o\):
Indeed, the Jacobian \(\partial \mathcal {F}\) evaluated at \(\varvec{p}=\varvec{p}_o\) is regular [41].
As detailed in Appendix C, the derivatives are obtained by solving the following two systems of linear equations:
where
and
Here and throughout this paper, the apex \(^{\prime }\) denotes the derivative with respect to one of the system parameters contained in the vector \(\varvec{\mu }\) introduced in (1). We note that the coefficient matrix in (25) is simply the Jacobian \(\partial \mathcal {F}\) evaluated at \(\varvec{p}=\varvec{p}_o\), which is invertible. Meanwhile, the coefficient matrix in (26) is the transpose of the Jacobian. Therefore, the solution to these linear equations is unique. Moreover, it turns out that we have \(\xi =0\) for the unique solution above [41].
For proportional damped mechanical systems, the relation \(\varvec{\psi }=\kappa \varvec{\phi }\) holds, and a much commonly used normalization for the mode vector \(\varvec{\phi }\) is shown in (20), namely, the normalization with respect to the mass matrix. Following this normalization, as detailed in Appendix C, we obtain
where \(\alpha \) and \(\beta \) are the coefficients in the Rayleigh damping \(\varvec{C}_\textrm{d}=\alpha \varvec{M}+\beta \varvec{K}\) that gives \(2\zeta \omega =\alpha +\beta \omega ^2\), and
Here, we can easily determine \(\varvec{\psi }^{\prime }=\kappa ^{\prime }\varvec{\phi }+\kappa \varvec{\phi }^{\prime }\), where we can solve for \(\kappa ^{\prime }\) from its expression shown in (23) along with \(\omega ^{\prime }\) and \(\zeta ^{\prime }\) listed above, namely,
4.2 Derivatives of quadratic expansion coefficients
As seen in (14), we have
Here and throughout the rest of the manuscript, the superscripts ‘(1)’ and ‘(2)’ denote the first and second half of the entries of a vector. We take the derivative of the first equation above, yielding
where
Here \(\varvec{v}^{\prime }=[\varvec{\phi }^{\prime },\lambda ^{\prime }\varvec{\phi }+\lambda \varvec{\phi }^{\prime }]^\textrm{T}\), which can be determined once \(\varvec{\phi }^{\prime }\) and \(\lambda ^{\prime }\) are given (see Sect. 4.1 for more details). Therefore, we obtain
As seen in (14), we have
We take the derivative of the first equation above, yielding
where
Therefore, we obtain
4.3 Derivatives of cubic expansion coefficients
As seen in (15), we have
We take the derivative of the first equation, yielding
where
and
Therefore, we obtain
We recall that the expansion coefficient of the cubic term of the reduced dynamics is given by \(\gamma =-\varvec{\psi }^\textrm{T}\varvec{f}_{21}\). It then follows that
where \(\varvec{f}_{21}\) has been defined in (16) and its full derivative is given below
As seen in (15), we have
where \(\varrho = {\gamma }/({2\textrm{Re}(\lambda )})\). We take the derivative of the first equation above, yielding
where
Therefore, we obtain
We take the derivative of the second system of equations shown in (47), yielding
consequently, we obtain
Remark 1
In the case that the quadratic and cubic nonlinearities are only displacement-dependent, as shown in (18), the nonlinear functions \(\varvec{f}_2\) and \(\varvec{f}_3\), and their derivatives only take the first half entries of their input vectors. For instance, \(\varvec{f}_2^{\prime }(\mu ,\varvec{v},\varvec{W}_{20})\) is reduced to \(\varvec{f}_2^{\prime }(\mu ,\varvec{\phi },\varvec{W}_{20}^{(1)})\). One can use these simplifications to reduce computational costs.
4.4 Perturbed SSM and backbone curve
With the substitution of \(p=\rho e^{\textrm{i}\vartheta }\) into (6), the SSM parameterization is given by
where \((\rho ,\vartheta )\) is a pair of parameterization coordinates. We note that the expansion coefficients characterize the geometry of the SSM. In particular, one can measure the curvature of an SSM via the quadratic expansion coefficients, as shown in [42], where a formal notion of scalar curvature based on graph-style parameterization was proposed. However, our derivation follows a normal-form-style parameterization, for which a formal scalar curvature definition is unavailable. For simplicity, we use the square of Euclidean norm of the quadratic expansion coefficients below to measure the curvature of the SSM [42]
Taking the derivative of the above equation yields
Then, we can use \(\mathcal {C}^{\prime }\) to characterize how the curvature of an SSM is affected by system parameters.
We recall that an SSM is parameterized via the map \({\varvec{z}}={\varvec{W}}(\varvec{p})\). In our case, we have \(\varvec{p}=(\rho e^{\textrm{i}\vartheta },\rho e^{-\textrm{i}\vartheta })\) and the associated map is given by (53). We note that the SSM can be visualized via a collection of mesh points \(\{{\varvec{z}}_{ij}\}\) with \(\varvec{z}_{ij}=\varvec{W}(\varvec{p}_{ij})\) and \(\varvec{p}_{ij}=(\rho _ie^{\textrm{i}\vartheta _j},\rho _ie^{-\textrm{i}\vartheta _j})\) for \(1\le i\le n_{\vartheta }\) and \(1\le j\le n_\rho \). In our case, we have \(\varvec{z}_{ij}=\varvec{z}(\rho _i,\vartheta _j)\) obtained from (53) and
Here, \(\rho _\textrm{max}\) denotes the radius of the domain of interest. With the mesh represented by \(\{\varvec{p}_{ij}\}\), we can further compute perturbed SSMs. When the SSM is perturbed, the map function is updated from \({\varvec{W}}\) to \(\varvec{W}^+\). Specifically, we consider the perturbation \(\mu _k\rightarrow \mu _k+\delta \mu _k\) and let \(\varvec{W}_{uv}^+\) be the perturbed expansion coefficients (cf. (53)), we have
where \(uv\in \{10,01,20,02,11,30,03,21,12\}\) throughout this paper and \(\varvec{W}_{uv,k}^{\prime }\) denotes the partial derivative of \(\varvec{W}_{uv}\) with respect to \(\mu _k\). Accordingly, we have \(\varvec{z}_{ij}^+=\varvec{W}^+(\varvec{p}_{ij})\), namely, the mesh node \(\varvec{z}_{ij}\) is perturbed as \(\varvec{z}_{ij}^+\) below
from which we can construct the perturbed SSM because the perturbed manifold passes through these perturbed mesh points. Therefore, we have the same set of mesh points \(\{\varvec{p}_{ij}\}\) in the reduced coordinates, yet their mappings in physical coordinates \(\{{\varvec{z}}_{ij}\}\) is updated as \(\{{\varvec{z}}_{ij}^+\}\) when the SSM parameterization changes, characterizing the perturbation of the SSM.
We recall that the reduced dynamics on the SSM is given by (10). It follows that the cubic term coefficient \(\gamma \) controls the nonlinear dynamics of the reduced dynamics. In particular, the sign of \(\textrm{Im}(\gamma )\) determines whether the backbone curve is softening or hardening because the backbone curve is given by [5]
Following [5, 43], for a point \((\varOmega ,\rho )\) on the backbone curve, we construct the corresponding periodic orbit \(p=\rho e^{\textrm{i}\varOmega t}\) for \(t\in [0,2\pi /\varOmega ]\), and further map this periodic orbit to physical coordinates via (53) with \(\vartheta =\varOmega t\). In other words, we obtain the periodic orbit below
from this, we can obtain \(\mathcal {L}^\infty \) norm-based amplitude of the periodic orbit [7], denoted as \(||\varvec{z}_\textrm{po}||_\infty \). We repeat the above procedure to construct the backbone curve \((||\varvec{z}_\textrm{po}||_\infty ,\varOmega )\) in physical coordinates with a collection of sampled \((\varOmega ,\rho )\).
Accordingly, the perturbed backbone curve in reduced coordinates is obtained as
where
In addition, we can further obtain the perturbed backbone curve in physical coordinates. Similarly, for a point \((\varOmega _+,\rho )\) on the backbone curve, we construct the corresponding periodic orbit \(p=\rho e^{\textrm{i}\varOmega _+ t}\) for \(t\in [0,2\pi /\varOmega _+]\), and further map this periodic orbit to physical coordinates via (58) with \(\vartheta =\varOmega _+t\). In other words, we obtain the perturbed periodic orbit below
Likewise, we can obtain the \(\mathcal {L}^\infty \) norm-based amplitude of the perturbed periodic orbit above, denoted as \(||\varvec{z}_\textrm{po}^+||_\infty \). We can then construct the perturbed backbone curve \((||\varvec{z}_\textrm{po}^+||_\infty ,\varOmega )\).
4.5 Perturbed limit cycles
We infer from the reduced dynamics (10) that it admits a limit cycle solution when \(\textrm{Re}(\lambda )\cdot \textrm{Re}(\gamma )<0\). In particular, the amplitude of the limit cycle in reduced coordinates is given by (letting \({\dot{\rho }}=0\))
The circular frequency of this limit cycle is obtained as
The limit cycle is obtained as \(p^*(t)=\rho ^*e^{\textrm{i}\omega ^*t}\). It follows that the corresponding limit cycle in physical coordinates is given by
Under the perturbation \(\mu _k\rightarrow \mu _k+\delta \mu _k\), we obtain a perturbed limit cycle with
Meanwhile, (66) is updated as
which gives the perturbed limit cycle in physical coordinates.
We note that the persistence of the limit cycle at higher-order expansions needs to be further checked. Indeed, the predicted limit cycle at \(\mathcal {O}(3)\) can be spurious and may disappear with increasing expansion orders. One can follow the procedure established in Ponsioen et al. [8] to check the persistence.
5 Forced vibration
5.1 Sensitivity of leading-order non-autonomous SSM
We take the external harmonic forcing \(\epsilon \varvec{f}^{\textrm{ext}}\cos \varOmega t\) into consideration. Then, the hyperbolic fixed point \(\varvec{z}=\varvec{0}\) is perturbed as a periodic orbit of the same stability type. Accordingly, the autonomous SSM is further perturbed into a time-periodic SSM tangent to resonant spectral subbundles of the periodic orbit. Specifically, the SSM parameterization (6) is updated as
where the last two terms on the right-hand side characterize the time periodicity. Here, \(\varvec{x}_0\) is the solution to the system of linear equations \( (\varvec{A}-\textrm{i}\varOmega \varvec{B})\varvec{x}_0=\varvec{B}\varvec{W}_\textbf{I}\varvec{s}_{0}^+ - \varvec{F}^\textrm{a}\) [6]. In our setting, the system of linear equations is simplified as
where
We obtain \(\varvec{x}_0=\varvec{A}_\varOmega ^{-1}\varvec{b}_\varOmega \), where
We note that \(-\varvec{D}\) is the dynamic stiffness matrix of the linear part of the full system (1). Therefore, (70) can be interpreted as an extended linear response analysis under harmonic excitation to account for the non-autonomous part of the SSM.
We then calculate the sensitivity of \(\varvec{x}_0\). Taking the derivative of (70) yields
where
with
5.2 Solving periodic orbit as fixed point
Let \(\vartheta =\theta +\varOmega t\), under the addition of the harmonic forcing, the reduced dynamics (10) is updated as
We note that a fixed point of the above vector field corresponds to a periodic orbit of the original system because \(p=\rho e^{\textrm{i}(\theta +\varOmega t)}\). In addition, the stability of the periodic orbit is the same as that of the fixed point, which is characterized by the spectrum of the Jacobian below [5]
where
Here, we have used the fact that \((\rho ,\theta )\) is a fixed point of the vector field (77) to simplify the expressions of \(J_{12}\), \(J_{21}\) and \(J_{22}\). Thus, the periodic orbit is stable when \(\textrm{trace}(\varvec{J})<0\) and \(\textrm{det}(\varvec{J})>0\). This fixed point can be further solved from the root of the following equation
For a given \(\epsilon \), the zero contour of the above equation gives the forced response curve (FRC) in reduced coordinates. This FRC can be further mapped to the FRC of periodic orbits of the original system via the map (69).
5.3 Sensitivity of periodic response
We derive the sensitivity of the periodic orbit for a given \(\varOmega \). As seen in (69), we have (cf. (53))
where \((\rho ,\theta )\) is a fixed point of the vector field (77) for the given \(\varOmega \), namely,
Since the derivatives of the SSM expansion coefficients are already derived in the previous sections, we can calculate \(\varvec{z}^{\prime }(t)\) using chain rules provided that we know \((\rho ^{\prime },\theta ^{\prime })\), which can be obtained by taking the derivatives of (82). In particular, we obtain
where \(J_{11}\) has been defined in (79) and
Then we can solve for \((\rho ^{\prime },\theta ^{\prime })\) from (83).
Since the amplitude of a periodic orbit gives an important characterization of the periodic response, we also derive an explicit sensitivity of the amplitude of the periodic response. Following [7], we let \(\varvec{Q}\) be a properly defined, symmetric weight matrix and take \(\mathcal {A}_{\mathcal {L}^2}(\varvec{z}(t))\) to characterize the response amplitude. In particular, we have [7]
where \(\mathcal {I}\subset \{1,\cdots ,2n\}\) is a set of indices representing the components of interest. We substitute (81) into (85), yielding
where
with
Here, we have used the fact that \(\varvec{W}_{10}=\bar{\varvec{W}}_{01}\), \(\varvec{W}_{21}=\bar{\varvec{W}}_{12}\), \(\varvec{W}_{20}=\bar{\varvec{W}}_{02}\), and \(\varvec{Q}^\textrm{T}=\varvec{Q}\) to simplify the derivations. Taking the derivative of (86) yields
from which we can calculate \(\mathcal {A}_{\mathcal {L}^2}^{\prime }\) that gives the sensitivity of the response amplitude (85). We have
where
5.4 Sensitivity of peak on FRC
We note that the last two terms in (86) are much smaller than the first term, which results in \(\mathcal {A}_{\mathcal {L}^2}^2(\varvec{z}(t))\approx \mathcal {A}_{\textrm{auto}}^2\). Remarkably, for a given FRC, we have
Consider a peak point \((\rho _\textrm{max},\varOmega _\textrm{max})\) on the FRC for \(\rho (\varOmega )\), we have \(d\rho /d\varOmega =0\) at the peak. Following (92), this peak is mapped exactly to the peak \((\mathcal {A}_{\textrm{auto,max}},\varOmega _\textrm{max})\) on the FRC in physical coordinates. Therefore, we can use the sensitivity of \((\rho _\textrm{max},\varOmega _\textrm{max})\) to characterize the sensitivity of the peak of FRC in physical coordinates.
Along the FRC in reduced coordinates, there is a peak for \(\rho _{\textrm{max}}\) at which [5]
We add the \(\textrm{sign}\) function to ensure that \(\rho _\textrm{max}\) is always positive. We can then calculate the sensitivity of \(\rho _{\textrm{max}}\) and \(\varOmega _{\textrm{max}}\) by taking the derivative of (93). Specifically, we have
5.5 Perturbed FRC
We conclude this section by presenting how to extract the whole perturbed FRC. Indeed, for a given \(\varOmega \), the dynamical system (1) can admit multiple periodic solutions. When the system parameter vector \(\varvec{\mu }\) is changed, the number of solutions can be changed as well for the given \(\varOmega \). Therefore, it is instructive to extract the whole perturbed FRC [35]. Unlike the implementation in [35], we will extract the perturbed FRC analytically; hence, the issues of choosing perturbation directions are not involved.
We recall that we can extract the whole FRC in reduced coordinates as the zero contour plot of (80). It follows that we can extract the perturbed FRC via (80) directly. Specifically, let \({\tilde{f}}_+={\tilde{f}}+{\tilde{f}}_k^{\prime }\delta \mu _k\), a perturbed FRC is obtained as the zero contour of below
or equivalently,
where \(\lambda _+\) and \(\gamma _+\) have been defined in (62), and \(\rho _+\) denotes the perturbed root.
We can then map this perturbed FRC in reduced coordinates back to the full system. In particular, we first solve for \(\theta _{+}\) as the fixed point of (77), yielding
where
We then update (81) as
where the superscript ‘+’ in the expansion coefficients stands for accordingly perturbed expansion coefficients (see (57)). Therefore, we have obtained a perturbed FRC in \(\varvec{z}_+\). We note that we can further predict the stability of the periodic solution from that of the fixed point \((\rho _+,\theta _+)\) via the spectrum of the Jacobian shown in (78).
Remark 2
We note that there is only one-time computation for the expansion coefficients of the autonomous part of SSM, i.e., \(\varvec{W}(\varvec{p})\), and also their derivatives, when we compute the FRC and its perturbed versions. However, \(\varvec{x}_0\) characterizing the leading-order non-autonomous SSM, depends on \(\varOmega \), as seen in (70). This \(\varOmega \)-dependence also holds for \(\varvec{x}^{\prime }_0\). Therefore, we need to solve for \(\varvec{x}_0\) and \(\varvec{x}^{\prime }_0\) for each sampled point on the FRC to obtain the FRC (and its perturbation) in physical coordinates. In practice, we should check whether it is necessary to include the contribution related to \(\epsilon \varvec{x}_0\) [7]. This check is conducted by comparing the FRCs with and without the contribution. If the FRC without the contribution matches well with the FRC with the contribution, we do not need to account for the contribution. We refer to Appendix F for more details about the check. If needed, we use a quadratic interpolation with three sampled \(\varOmega \), which are the two endpoints and the middle point of the frequency interval, to recover the contribution of the leading-order non-autonomous part of SSM. As we will see in later examples, this quadratic interpolation is sufficient to make accurate predictions.
6 Explicit derivatives of SSMs via adjoint method
The computational cost to perform the sensitivity analysis via the direct method is proportional to the number of design variables. In particular, the number of matrix inverse operations is proportional to the number of design variables (see (35) for instance). Therefore, the computational cost can be significant if the number of parameters is large. In this case, adjoint methods can be used to reduce the computational cost.
For an adjoint method, we need to specify an optimization objective, and then we can derive the derivatives of the objective function with respect to system parameters. As a demonstration, we take the cubic coefficient of the reduced dynamics, namely, \(\gamma \), as the objective function. As we discussed, this coefficient controls the hardening or softening behavior of the nonlinear dynamics.
Here, we focus on a proportionally damped system with only displacement-dependent nonlinearity for compactness. This is also the setting for most geometrically nonlinear mechanical systems. We note that the choice for the damping coefficients \(\alpha \) and \(\beta \) in the Rayleigh damping \(\varvec{C}_\textrm{d}=\alpha \varvec{M}+\beta \varvec{K}\) is not unique [44]. We consider the case that the damping ratio \(\zeta \) in \(2\zeta \omega =\alpha +\beta \omega ^2\) is specified. In other words, both \(\alpha \) and \(\beta \) can be functions of \(\omega \). Another interesting case is that \(\alpha \) and \(\beta \) are given constant. The latter case is easier, and the adjoint analysis presented below can be easily adapted to such an easier case.
We list all relevant constraints below
Now, we define a Lagrangian below
As detailed in Appendix D, the calculation of variation yields the solution to the Lagrange multipliers below
where
Let \(\varvec{e}_i\) denote the unit vector along i-th axis such that only the i-th entry of the vector is one and all the rest entries are trivial, here
where \(\varvec{a},\varvec{b}\) and \(\varvec{c}\) denote the arguments of functions.
Meanwhile, the variation of \(\mathcal {L}\) is left with
which gives the sensitivity of \(\gamma \) with respect to the vector of system parameters \(\varvec{\mu }\).
Remark 3
We see that the number of matrix inverse operations is independent of the number of design variables in the adjoint method. The computation of \(\varvec{F}_{i,j}\) is the bottleneck of the adjoint method. Indeed, the function handle needs to be evaluted N times, where N is the number of DOFs of the full system. It follows that these evaluations can be costly when N is large. However, one can speed up this process via either vectorization (or parallelization) of the code, or using the fact that only one entry of \(\varvec{e}_i\) is nonzero.
7 Examples
7.1 Two coupled nonlinear oscillators
7.1.1 Analytic SSM and its verification
Consider a system with two coupled nonlinear oscillators below
The first undamped natural frequency is \(\omega =1/\sqrt{m}\). We will compute the SSM associated with the first damped mode. In particular, we will provide analytic expressions for the SSM expansion coefficients and their sensitivity.
Since \(m\lambda ^2+2\zeta \sqrt{m}\lambda +1=0\), we have
As for the eigenvector, we have \(\varvec{\phi }=(1/\sqrt{m},0)^\textrm{T}\). Then we follow the results in Sects. 3.2 and 5.1 to obtain the analytic expansion coefficients below
where \({\tilde{\lambda }}=2\lambda +{\bar{\lambda }}\), \(\varrho =\gamma /(2\textrm{Re}(\lambda ))\), just as we have defined in Sect. 3.2, and
Recall that \(\gamma \) is the cubic term coefficient of the reduced dynamics (cf. (9)). Its expression for this example is listed below
As for the coefficient vector of the leading-order non-autonomous part of the SSM (see (70)), we obtain
The expression of \({\tilde{f}}\) in the reduced dynamics (77) is given by \({\tilde{f}}=\sqrt{m}\kappa /2\).
We first validate the above analytic expressions via a comparison with numerical results obtained directly from SSMTool. In the following computations, system parameters are chosen as \(m=0.5, \zeta =0.01, b_1=1\), and \(b_2=0.3\). To quantify the difference between the results obtained from the two methods, we use a relative error defined below
provided that \(||\varvec{y}_\textrm{ana}||\ne 0\). Here, the subscripts ‘ana’ and ‘num’ denote the results from the analytic prediction and that obtained via SSMTool. Computational results show that \(\textrm{RelTol}(\varvec{W}_{uv})\) is of magnitude \(10^{-13}\) or smaller, \(\textrm{RelTol}(\varvec{x}_0)=2.6\times 10^{-15}\), \(\textrm{RelTol}(\gamma )=2.5\times 10^{-16}\), and \(\textrm{RelTol}({\tilde{f}})=2.2\times 10^{-16}\). These tiny relative errors indicate the correctness of the derived analytic expressions.
7.1.2 Explicit derivatives and their validation
Next, we compute explicit derivatives of the coefficients above following the direct computation shown in Sects. 4 and 5.1. Meanwhile, since we have analytic expressions for the expansion coefficients, we derive their derivatives analytically in Appendix E. These analytical derivatives are used to validate the results from the direct computation.
Here, the vector of system parameters is given by \(\varvec{\mu }=(m,b_1,b_2)\). We list the derivatives of \(\gamma \) and \({\tilde{f}}\) in Table 1. As seen in the table, the results from the direct computation match well with that of the analytic derivatives. Indeed, their relative differences are of the magnitude \(10^{-14}\). We note that the direct computation gives that \(\partial _{b_2}\gamma =\partial _{b_1}{\tilde{f}}=\partial _{b_2}{\tilde{f}}=0\), which is consistent with the analytic derivatives shown in (229) and (234).
Here, we only list the detailed derivatives of \(\gamma \) and \({\tilde{f}}\) for compactness. We note that excellent agreement is also obtained for the derivatives of other expansion coefficients \(\varvec{W}_{uv}\) and \(\varvec{x}_0\). In particular, the direct computation results show that \(\partial _{b_1}\varvec{W}_{10}=\partial _{b_2}\varvec{W}_{10}=\varvec{0}\), \(\partial _{b_1}\varvec{W}_{20}=\partial _{b_2}\varvec{W}_{20}=\varvec{0}\), \(\partial _{b_1}\varvec{W}_{11}=\partial _{b_2}\varvec{W}_{11}=\varvec{0}\), \(\partial _{b_2}\varvec{W}_{30}=\partial _{b_2}\varvec{W}_{21}=\varvec{0}\), which is again consistent with the analytic results detailed in Appendix E. As for non-zero derivatives of \(\varvec{W}_{uv}\) and \(\varvec{x}_0\), their relative difference is of the magnitude \(10^{-12}\) or smaller. These findings validate the effectiveness of the direct method established in Sects. 4 and 5.1.
We note that we can also calculate the derivatives of \(\gamma \) via the adjoint method proposed in Sect. 6. The adjoint method correctly predicts \(\partial _{b_2}\gamma =0\). Moreover, as seen in Table 1, the results for \(\partial _m\gamma \) and \(\partial _{b_1}\gamma \) obtained from the adjoint method also match well with that of the analytic derivatives. Their relative difference is smaller than \(10^{-15}\). Therefore, the effectiveness of the adjoint method has also been validated.
7.1.3 Perturbed SSMs and backbone curves
We now compute perturbed SSMs following the procedures established in Sect. 4.4. We obtain the curvature of the SSM as \(\mathcal {C}=13\), and its derivatives are \(\partial _m\mathcal {C}=-68.02\) and \(\partial _{b_1}\mathcal {C}=\partial _{b_2}\mathcal {C}=0\). We plot the SSM along with its perturbed ones under the variations \(\pm 0.1m\) in Fig. 1. Since \(\partial _m\mathcal {C}<0\), the curvature is increased as \(m\rightarrow 0.9m\) and decreased as \(m\rightarrow 1.1m\). Indeed, as seen in Fig. 1, the mesh of the SSM is increasingly stretched as m decreases, which is consistent with the change of the curvature \(\mathcal {C}\).
Projection of SSM of the two coupled oscillators and its perturbations under the variations in m. (Color figure online)
We then present the backbone curve and its perturbations. Since the partial derivatives of \(\varvec{W}_{uv}\), \(\lambda \), and \(\gamma \) with respect to \(b_2\) are all zeros, the backbone curve is not affected by the coefficient \(b_2\). Therefore, we only plot the backbone curve and its perturbations under the variations in m and \(b_1\) in Fig. 2. We observe that the intersection point of the backbone curve with the \(\varOmega \) axis moves leftwards when m increases, yet it stays still when \(b_1\) is changed. Since this intersection point is characterized by \(\textrm{Im}(\lambda )\) (see (59) with \(\rho =0\)), this observation can be explained by \(\partial _m\textrm{Im}(\lambda )=-\textrm{Im}(\lambda )/(2m)<0\) and \(\partial _{b_1}\lambda =0\) (see (220)). Moreover, we find from the lower panel of Fig. 2 that the backbone curve becomes more hardening when \(b_1\) is decreased, which is associated with the fact that \(\partial _{b_1}\textrm{Im}(\gamma )<0\), as shown in Table 1. We note that \(\delta m=0.05m\) and \(\delta b_1=3b_1\) in Fig. 2. This indicates that the backbone curve is more sensitive to the mass m relative to \(b_1\).
Backbone curves depicting the instantaneous and periodic vibration amplitude of the degree of freedom \(x_1\) for the oscillator system (112). Here and throughout this paper, the perturbed backbone curves in solid lines are obtained using derivatives (see Sect. 4.4). The backbone curves represented via cross markers are obtained from the explicit third-order, SSM-based model reduction applied on modified dynamical systems with updated system parameters. The latter are presented to validate the former. The upper panel shows the perturbations under m with \(\delta m=0.05m\) and also \(4\delta m\), while the lower panel illustrates the perturbations under \(b_1\) with \(\delta b_1=3b_1\). In the upper panel, we see that the prediction via derivatives starts to deteriorate when the magnitude of the perturbations of m is quadrupled. Such a deterioration, however, is not observed in the lower panel even when \(\delta b_1\rightarrow 1000\delta b_1\)
We have further validated the correctness of the obtained perturbed backbone curves in Fig. 2. In particular, we consider modified systems with parameters updated accordingly and then extract the backbone curves of the modified systems directly via the explicit third-order, SSM-based model reduction detailed in Sect. 3. These backbone curves are denoted by cross markers, which agree with the ones obtained via the derivatives (in solid lines).
Here and throughout this paper, when we compute perturbed backbone curves or FRCs, we choose the magnitude of perturbations following two restrictions below: (1) the magnitude of perturbations should not be too small to see the effects of the perturbations in plots; (2) the magnitude of perturbations should not be enormous to ensure the predictions using the derivatives are reliable, namely, the coincidence between the solid lines and cross markers. Indeed, as seen in the upper panel of Fig. 2, the predictions via first-order derivatives start to deteriorate when the magnitude of perturbations is large. In this case, a higher-order Taylor expansion with higher-order derivatives can be used to improve the accuracy of the predictions [35]. We note that the extent of deterioration also depends on the choice of perturbed parameters. Numerical experiments show that the perfect match shown in the lower panel of Fig. 2 is still obtained when the perturbation \(\delta b_1\) is increased to \(\delta b_1\).
7.1.4 Forced periodic response
We now consider the forced vibration of the oscillators under harmonic excitation. Here and for the rest of this example, we set \(\epsilon =0.01\) in the computation. We set \(\varOmega =1.4\) and use the following metric to characterize the amplitude of the periodic response of the system (cf. (85))
Thus, we have \(\mathcal {I}=\{1,2,3,4\}\) and \(\varvec{Q}=\varvec{I}_4\). For this periodic orbit, we have \((\rho ,\theta )=(0.0476,-0.5672)\) and \(\mathcal {A}_{\mathcal {L}^2}=0.1650\). Following the derivations in Sect. 5.3, we obtain the derivatives of \(\mathcal {A}_{\mathcal {L}^2}\) in Table 2, from which we see that \(\mathcal {A}_{\mathcal {L}^2}\) is more sensitive to m relative to the other two parameters.
Forced response curves (FRCs) of the degree of freedom \(x_1\) (upper panels) and \(x_2\) (lower panels) for the oscillator system (112). Here and throughout this paper, the perturbed FRCs in lines are obtained using derivatives (see Sect. 5.5). Among them, the solid/dashed lines denote stable/unstable periodic orbits, respectively. The FRCs represented via cross markers are obtained from the explicit third-order, SSM-based model reduction applied on modified dynamical systems with updated system parameters. The latter are presented to validate the former. The left panels show the perturbations under m with \(\delta m=0.05m\), the middle panels illustrate the perturbations under \(b_1\) with \(\delta b_1=3b_1\), and the right panels show the perturbations under \(b_2\) with \(\delta b_2=10b_2\). We note that the upper-right panel has three lines and two sets of markers, just like the lower-right panel. However, these lines (markers) in the upper-right panel are indistinguishable
To validate the correctness of the obtained derivatives of \(\mathcal {A}_{\mathcal {L}^2}\), we use the central difference method to compute approximate derivatives. For instance, we have
We take \(\varDelta m=0.001m\) and calculate the approximated \(\partial _m\mathcal {A}_{\mathcal {L}^2}\). Likewise, we take \(\varDelta b_1=0.01b_1\) and \(\varDelta b_2=10^{-6}b_2\) to calculate approximated \(\partial _{b_1}\mathcal {A}_{\mathcal {L}^2}\) and \(\partial _{b_2}\mathcal {A}_{\mathcal {L}^2}\) respectively. These approximated derivatives are also listed in Table 2. They agree with the results of the direct method, which suggests its effectiveness.
7.1.5 Perturbed FRCs and their validation
We further present the FRC and its perturbations. We follow Sect. 5.5 to construct the perturbed FRCs. As detailed in Appendix F.1, we need to consider the contribution of the leading-order non-autonomous SSM, especially for the second oscillator \(x_2\). Therefore, we follow the quadratic interpolation introduced in Remark 2 to account for the leading-order contribution. The effectiveness of this quadratic interpolation scheme is also validated in Appendix F.1.
We present the perturbed FRCs for \(x_1\) and \(x_2\) under the variations in different parameters in Fig. 3. We observe from the left panels of Fig. 3 that the change of FRC for \(x_1\) under the variation in m is qualitatively the same as that of the FRC for \(x_2\). Specifically, the peak of FRC moves leftward, and the peak amplitude is increased when m is increased. By contrast, we observe from the middle (right) panels Fig. 3 that the evolution of FRC of \(x_1\) under the variation in \(b_1\) (\(b_2\)) is different from that of FRC of \(x_2\). In particular, the middle two panels show that the peak amplitude for \(x_1\) increases monotonically as \(b_1\) increases. Such a monotonic behavior does not hold for the peak amplitude of \(x_2\). Finally, the right two panels indicate that the FRC of \(x_1\) is insensitive to \(b_2\), which is not the case for the FRC of \(x_2\).
The aforementioned sharp difference in the right two panels can be deduced from (69). We recall that \(\partial _{b_2}\varvec{W}_{uv}=\varvec{0}\), \(\partial _{b_2}\lambda =\partial _{b_2}\gamma =\partial _{b_2}{\tilde{f}}=0\), yet \(\partial _{b_2}\varvec{x}_0\ne 0\). Thus, the term \(\varvec{W}(\varvec{p})\) in (69) is not affected by \(b_2\) while the terms \(\epsilon \varvec{x}_0e^{\textrm{i}\varOmega t}\) and \(\epsilon \bar{\varvec{x}}_0e^{-\textrm{i}\varOmega t}\) in (69) depend on \(b_2\). The first oscillator \(x_1\) is in external resonance, and hence the first term \(\varvec{W}(\varvec{p})\) dominates over the other two terms, as verified in Appendix F.1 (see the upper panel of Fig. 14). In contrast, the second oscillator \(x_2\) is not in resonance, and the contribution of the three terms is comparable, as demonstrated in Appendix F.1 (see the lower panel of Fig. 14). Thus, the FRC of \(x_1\) is insensitive to \(b_2\), which is not true for the FRC of \(x_2\).
We follow a similar approach to verifying the backbone curves in Fig. 2 to validate the effectiveness of the perturbed FRCs obtained via the explicit derivatives. Specifically, we consider appropriately modified systems with parameters updated according to the perturbations and then extract the FRCs via the explicit third-order, SSM-based model reduction detailed in Sect. 3. These FRCs are represented via cross markers, which again match well with that obtained via the derivatives (in lines), as shown in Fig. 3.
We conclude this example by presenting the sensitivity of the peak on FRC. As detailed in Sect. 5.4, we can use the peak of the FRC in the reduced coordinate \(\rho \) to represent the peak of FRC in physical coordinates provided that the contribution of the leading-order non-autonomous SSM is insignificant. Indeed, this is the case for the first oscillator \(x_1\). Thus, we list the derivatives of \((\varOmega _\textrm{max},\rho _\textrm{max})\) (see (93)) to quantify more precisely how the peak of FRC for \(x_1\) (cf. the upper panels of Fig. 3) is affected by the system parameters. The obtained derivatives are listed in Table 3 and also verified using the central difference approach that we have used in Table 2 (cf. 124). The sign of these derivatives is consistent with the change of peaks of FRCs on the upper panels of Fig. 3.
We note that we have used the third-order model reduction in this example. A natural question to ask is whether \(\mathcal {O}(3)\) approximation is accurate. As shown in Fig. 13 in Appendix F.1, this third-order expansion is sufficient to make accurate predictions.
A cantilevered pipe conveying fluid subjected to a uniformly distributed harmonic excitation
7.2 A cantilevered pipe conveying fluid
7.2.1 Example setup
Consider a geometrically nonlinear cantilevered pipe conveying fluid possibly subject to a uniformly distributed harmonic excitation \(\epsilon \cos \varOmega \tau \), as shown in Fig. 4. As detailed in [10], this pipe system is a general mechanical system because it consists of flow-induced gyroscopic and follower forces and velocity-dependent nonlinearities. Following the assumption that the pipe axis is inextensible and the pipe material is viscoelastic, we obtain the dimensionless governing equation of this pipe system as below [45, 46]
Here \(\eta \) and u are dimensionless transverse displacement and flow velocity, \(\alpha \) is a dimensionless damping coefficient, \(\beta \) denotes a mass ratio, \(()'=\partial ()/\partial \xi \) and \(\dot{()}=\partial ()/\partial \tau \), \(\epsilon \) and \(\varOmega \) are dimensionless excitation amplitude and frequency.
We apply a Galerkin approach to discretize the partial differential equation (125) into a system of ordinary differential equations (ODEs). Specifically, we substitute
into (125) and apply a Galerkin projection, yielding the ODEs below
where \(m_{ij}\), \(c_{ij}\), \(k_{ij}\), \(\alpha _{ijkl}\), \(\beta _{ijkl}\), \(\gamma _{ijkl}\) and \(g_{i}\) are computed from integrals with integrands that are related to the set of eigenfuncitons \(\{\phi _j\}\). These eigenfunctions are actually the mode functions of a cantilever beam (without moving fluid). Detailed expressions of these coefficients can be found in [10]. They are accessible in the open-source package SSMTool 2.5 [18]. In particular, we note that the damping and stiffness matrices are asymmetric because of the flow-induced gyroscopic and follower forces [10].
The pipe system can undergo a flutter bifurcation when the flow velocity exceeds a critical value, denoted as \(u_\textrm{f}\) [10]. This critical flutter speed \(u_\textrm{f}\) marks a Hopf bifurcation of the straight pipe static configuration. This configuration becomes unstable, and the pipe system undergoes a limit cycle motion when \(u>u_\textrm{f}\).
Here, we consider a four-mode truncation (\(n=4\)) because it is sufficient to generate converged solutions of interest [10]. In the following computations, system parameters are chosen as \(\alpha =0.001,\beta =0.2,u=5.75\). We have \(u_\textrm{f}=5.65<u\). The pipe system admits a two-dimensional unstable manifold, which is also an unstable SSM. We will perform explicit third-order model reduction on this unstable SSM and also conduct a sensitivity analysis of this SSM to predict the dynamics of the pipe system in the post-flutter regime.
7.2.2 Explicit SSM and its validation
We follow the explicit results in Sects. 3.2 and 5.1 to obtain the unstable SSM’s expansion coefficients and the reduced dynamics associated with the SSM. We again use the computational results obtained directly from SSMTool to validate the explicit results. We use the relative error metric defined in (122) to quantify the difference between the results of the two methods. In particular, \(\varvec{y}_\textrm{ana}\) in (122) is the explicit solution here.
Computational results show that \(\textrm{RelTol}(\varvec{W}_{10})=8.5\times 10^{-14}\), \(\textrm{RelTol}(\varvec{W}_{30})=1.2\times 10^{-15}\), \(\textrm{RelTol}(\varvec{W}_{21})=1.5\times 10^{-10}\), \(\textrm{RelTol}(\varvec{x}_{0})=6.9\times 10^{-12}\), \(\textrm{RelTol}(\gamma )=6.4\times 10^{-16}\), and \(\textrm{RelTol}({\tilde{f}})=2.2\times 10^{-17}\). We consistently observe that \(\varvec{W}_{20}=\varvec{W}_{11}\equiv \varvec{0}\), which is resulted from the fact that the pipe system does not consist of quadratic nonlinearities, as seen in (127). Thus, the relative error for the expansion coefficients of the quadratic terms is not listed here because they are not defined. These tiny relative errors suggest the correctness of the explicit solutions.
7.2.3 Explicit derivatives and their validation
We compute the explicit derivatives of the coefficients above following the direct computation shown in Sects. 4 and 5.1. Here, the vector of system parameters is given by \(\varvec{\mu }=(u,\beta ,\alpha )\). We list the derivatives of \(\lambda \), \(\gamma \) and \({\tilde{f}}\) in Table 4. The derivatives of the coefficients \(\varvec{W}_{uv}\) and \(\varvec{x}_0\) are not listed here for compactness. The explicit results correctly predict \(\varvec{W}_{20}'=\varvec{W}_{11}'=\varvec{0}\).
Unlike the previous example, we do not have analytical derivatives for these coefficients and hence cannot use analytic derivatives to validate the results from the direct computation. We use the central difference adopted in Tables 2 and 3 to obtain approximate derivatives. Specifically, we take \(\varDelta u=0.001u\), \(\varDelta \beta =0.001\beta \), and \(\varDelta \alpha =0.01\alpha \) (cf. (124)) to compute the approximate derivatives with respect to u, \(\beta \), and \(\alpha \), respectively. The obtained approximate derivatives of \(\lambda \), \(\gamma \), and \({\tilde{f}}\) via the central difference are also listed in Table 4. This table shows that the derivatives obtained from the two methods are in excellent agreement. Their relative difference is of magnitude \(10^{-5}\) or smaller. Moreover, the relative difference for other nonzero derivatives of coefficients \(\varvec{W}_{uv}\) and \(\varvec{x}_0\) is also of the magnitude \(10^{-5}\) or smaller. These validate the effectiveness of the direct method established in Sects. 4 and 5.1.
7.2.4 Perturbed SSMs and backbone curves
Since \(\varvec{W}_{20}=\varvec{W}_{11}\equiv \varvec{0}\), the curvature \(\mathcal {C}\) defined via the quadratic coefficients (see (54)) is precisely zero and does not depend on the system parameters. Indeed, as seen in Fig. 5, the SSM is close to flat. In addition, when the flow velocity changes, the SSM remains closely flat. In contrast, the orientation of the tangent plane of the SSM is changed considerably because the orientation is determined by \(\varvec{W}_{10}\), whose derivatives are nonzero.
Projection of SSM of the cantilevered pipe conveying fluid and its perturbations under the variations in u. (Color figure online)
We further present the backbone curve and its perturbations in Fig. 6, where \(\eta _\textrm{end}\) denotes the deflection at the free end of the pipe. From the left and right panels, we observe that the intersection point with the \(\varOmega \) axis moves leftwards as the flow velocity u or the damping coefficient \(\alpha \) increases. In contrast, the intersection point moves rightwards as the mass ratio \(\beta \) increases, as shown in the middle panel of Fig. 6. These observation are consistent with \(\partial _u\textrm{Im}(\lambda )<0\), \(\partial _\alpha \textrm{Im}(\lambda )<0\), and \(\partial _\beta \textrm{Im}(\lambda )>0\), as seen in Table 4.
We follow the procedure in Sect. 7.1.3 to validate the correctness of the obtained perturbed backbone curves in Fig. 6. As seen in Fig. 6, the backbone curves obtained from modified systems match well with those obtained via the explicit derivatives.
7.2.5 Limit cycle and its perturbations
We recall that the pipe system admits a limit cycle on the SSM. Here, we present the limit cycle and its perturbations predicted via the SSM-based model reduction and sensitivity analysis. More details on the prediction can be found in Sect. 4.5.
Backbone curves depicting the instantaneous and periodic vibration amplitude of tip deflection of the cantilevered pipe conveying fluid. The left panel shows the perturbations under u with \(\delta u=0.015u\), the middle panel illustrates the perturbations under \(\beta \) with \(\delta \beta =0.08\beta \), and the right panel shows the perturbations under \(\alpha \) with \(\delta \alpha =\alpha \). We add the green arrow in the middle panel to denote increasing \(\beta \) because the three lines are very close. (Color figure online)
The predicted limit cycle and its perturbations under the variations in flow velocity u, mass ratio \(\beta \), and damping coefficient \(\alpha \) are plotted in the left, middle, and right panels of Fig. 7, respectively. We find that the size of the limit cycle is increased with increasing u, but is decreased with increasing \(\beta \) or \(\alpha \). We note that the size of the limit cycle can be characterized by \(\rho ^*\) defined in (64), which is the radius of the limit cycle in reduced coordinates. The derivative of \(\rho ^*\) is given as below
Therefore, it is the difference between the relative derivative \(\textrm{Re}(\gamma )'/\textrm{Re}(\gamma )\) and \(\textrm{Re}(\lambda )'/\textrm{Re}(\lambda )\) that determines whether \(\rho ^*\) is increased or not. Here we have \(\lambda =0.2220 +13.5352\textrm{i}\) and \(\gamma =-1.0275 + 0.7196\textrm{i}\). Along with their derivatives listed in Table 4, we compute the difference and find that \(\partial _u\rho ^*>0\), \(\partial _\beta \rho ^*<0\), and \(\partial _\alpha \rho ^*<0\), which is consistent with the observations in Fig. 7.
Limit cycles regarding the tip deflection of the cantilevered pipe conveying fluid. The left panel shows the perturbations under u with \(\delta u=0.015u\); the middle panel illustrates the perturbations under \(\beta \) with \(\delta \beta =0.04\beta \), and the right panel shows the perturbations under \(\alpha \) with \(\delta \alpha =\alpha \)
We have also followed a similar approach to validating perturbed FRCs to validate the correctness of the perturbed limit cycles. As seen in Fig. 7, the results based on modified dynamical systems, which are denoted by cross markers, match well with that of the predictions via derivatives (solid lines). This demonstrates the correctness of the perturbed limit cycles obtained via the approach established in Sect. 4.5
7.2.6 Forced periodic response
We consider the forced vibration of the pipe conveying fluid under the uniformly distributed harmonic excitation. Here and for the rest of this example, we set \(\epsilon =0.1\) in the computation. We set \(\varOmega =13.5\) and compute the periodic response amplitude at the free end of the pipe (cf. (85))
where \(\varvec{\varPhi }_\textrm{end}=[\phi _1(1),\cdots ,\phi _4(1)]^\textrm{T}\). In our notation, we have \(\varvec{Q}=\varvec{\varPhi }_\textrm{end}^\textrm{T}\varvec{\varPhi }_\textrm{end}\). For this periodic orbit, we have \((\rho ,\theta )=(0.1562,2.9911)\), and \(||\mathcal {\eta }_\textrm{end}||_{\mathcal {L}^2}=0.0274\).
We follow the derivations in Sect. 5.3 to obtain the derivatives of \(||\mathcal {\eta }_\textrm{end}||_{\mathcal {L}^2}\), which are listed in Table 5, from which we see that the response amplitude is more sensitive to \(\alpha \) relative to the other two parameters. We again use the central difference (cf. (124)) to obtain approximate derivatives. Here we take \(\varDelta u=10^{-4} u\), \(\varDelta \beta =10^{-4}\beta \), and \(\varDelta \alpha =0.01\alpha \) to perform the central difference computation. As shown in Table 5, the results of the two schemes match well, again suggesting the effectiveness of the direct method established in Sect. 5.3.
Forced response curves (FRCs) for the tip deflection of the pipe conveying fluid. The upper-left panel shows the perturbations under u with \(\delta u=0.002u\), the upper-right panel illustrates the perturbations under \(\beta \) with \(\delta \beta =0.006\beta \), and the lower panel shows the perturbations under \(\alpha \) with \(\delta \alpha =0.2\alpha \)
7.2.7 Perturbed FRCs and their validation
We follow Sect. 5.5 to present the FRC and its perturbations further. As seen in Fig. 16 in the Appendix F.2, we can safely ignore the contribution of the leading-order non-autonomous SSM. Thus, the quadratic interpolation introduced in Remark 2 is not involved here. We ignore the terms related to \(\varvec{x}_0\) and \(\bar{\varvec{x}}_0\) in (69).
We plot the FRC and its perturbations in Fig. 8. These FRCs consist of two branches: a main branch with a small response amplitude and an isolated branch with a large response amplitude. We observe that the isola tends to merge with the main branch when u is decreased (see the upper-left panel) and \(\beta \) or \(\alpha \) is increased (see the other two panels). In other words, the isola moves downwards with decreasing u, increasing \(\beta \) or \(\alpha \). Moreover, the isola becomes smaller when it moves upwards, marking the tendency to isola bifurcation, where the isola is reduced to a single point and then disappears. More details on this isola bifurcation can be found in [8, 10, 47].
Similar to the previous example, we further validate the perturbed FRCs by extracting them via modified dynamical systems, and these extracted FRCs are represented via markers. As seen in Fig. 8, the perturbed FRCs obtained from the derivatives (in lines) are in excellent agreement with that of the modified systems.
We conclude this example by discussing the effectiveness of the third-order SSM-based model reduction. As seen in Fig. 15 of Appendix F.2, although the SSM-based predictions do not converge exactly at \(\mathcal {O}(3)\) approximation for the isola of FRC, the \(\mathcal {O}(3)\) reduction makes predictions with sufficient accuracy overall.
7.3 A shallow shell structure
7.3.1 Example setup
As the last example, we consider the nonlinear vibrations of a shallow-arc structure [16], shown in Fig. 9. The geometric and material properties of this shell can be found in Table 5 of [16]. We use the same finite-element (FE) model as in [16]. The FE model has 400 elements and 1320 DOFs, and the natural frequency of the first mode is obtained as \(\omega _1=147.4552~\mathrm {rad/s}\). Further, an external forcing \(\epsilon \varvec{f}_0\cos \varOmega t\) is added to study forced vibration, where \(\varvec{f}_0\) represents a vector of concentrated load in z-direction with magnitude of 100 N at the mesh node located at \((x,y)=(0.25L,0.5H)\).
The schematic of a shallow shell structure [16]. Here, the shell is simply supported at the two opposite edges aligned along the \(y-\)axis
Here, we adopt Rayleigh damping \(\varvec{C}_\textrm{d}=\alpha \varvec{M}+\beta \varvec{K}\) with \(\alpha =0.40215304\) and \(\beta =8.63116746\times 10^{-6}\) such that the damping ratio of the first two modes is 0.002. We take material parameters Young’s modulus E and density \(\rho \) as system parameters, namely, \(\varvec{\mu }=(E,\rho )\). However, computational results show that the derivatives of the expansion coefficients with respect to E is of magnitude \(10^{-7}\) or smaller. In other words, the SSM is insensitive to the Young’s modulus. Therefore, unless otherwise stated, we will only report the sensitivity analysis with respect to the density \(\rho \) in the rest of this example.
7.3.2 Explicit SSM and its validation
Similar to Sect. 7.2.2, we compute the explicit third-order SSM (see Sects. 3.2 and 5.1), and then validate the obtained explicit results. Computational results show that \(\textrm{ReTol}(\varvec{W}_{uv})\) is of magnitude \(10^{-8}\) or smaller, \(\textrm{RelTol}(\varvec{x}_{0})=2.9\times 10^{-11}\), \(\textrm{RelTol}(\gamma )=2.1\times 10^{-13}\), and \(\textrm{RelTol}({\tilde{f}})=2.7\times 10^{-14}\). These tiny relative errors again suggest the correctness of the explicit solutions.
7.3.3 Explicit derivatives and their validation
Similar to Sect. 7.2.3, we compute the explicit derivatives of the expansion coefficients of SSM parameterization and the associated reduced dynamics and further validate them using central difference. The derivatives of \(\lambda \), \(\gamma \), and \({\tilde{f}}\) with respect to \(\rho \) are listed in Table 6. The derivatives of the coefficients \(\varvec{W}_{uv}\) and \(\varvec{x}_0\) are not listed here for compactness. We take \(\varDelta \rho =0.001\rho \) to perform central difference and list the obtained approximate derivatives of \(\lambda \), \(\gamma \), and \({\tilde{f}}\) in Table 6. This table shows that the derivatives obtained from the two methods are in excellent agreement. In addition, the relative difference for derivatives of coefficients \(\varvec{W}_{uv}\) and \(\varvec{x}_0\) is also of the magnitude \(10^{-5}\) or smaller. These validate again the effectiveness of the direct method established in Sects. 4 and 5.1.
7.3.4 Perturbed SSMs and backbone curves
We compute perturbed SSMs following the procedure in Sect. 4.4. The curvature of the SSM is \(\mathcal {C}=5.5\times 10^8\), and its derivative is \(\partial _\rho \mathcal {C}=-6.1\times 10^5\). We plot the SSM and its perturbations in Fig. 10. Similar to Fig. 1, the mesh of the SSM is increasingly stretched as \(\rho \) decreases, which is consistent with the change of the curvature because \(\partial \rho \mathcal {C}<0\).
We further present the backbone curves on the SSMs. As seen in Fig. 11, the intersection point of the backbone curve with the \(\varOmega \) axis moves leftwards as \(\rho \) increases, which is consistent with \(\partial _\rho \textrm{Im}(\lambda )<0\), as seen in Table 6. In addition, we find that the three backbone curves become closer to each other as \(\varOmega \) decreases, indicating that the backbone curve tends more softening as \(\rho \) increases, which is consistent with \(\partial _\rho \textrm{Im}(\gamma )>0\), as shown in Table 6. We follow the same approach as the previous two examples to validate the effectiveness of the perturbed backbone curves. As seen in Fig. 11, the cross markers match well with the solid lines, which validates the correctness of the perturbed backbone curves.
7.3.5 Forced periodic response
Now, we consider the forced vibration of the shell structure. We set \(\epsilon =0.1\), \(\varOmega =148\), and compute the amplitude of the periodic response at the point where the concentric load is applied. For consistency with ref. [16], we denote the deflection at this point as \(z_\textrm{out}\). We take a \(\mathcal {L}^2\)-norm based metric to represent the amplitude of this periodic response (cf. (85))
For this periodic orbit, we have \(\rho =3.871194\times 10^{-3}\), \(\theta =-2.764760\), and \(||z_\textrm{out}||_{\mathcal {L}^2}=1.014862\times 10^{-3}\).
Projection of SSM of the von Kármán shallow shell and its perturbations under the variations in \(\rho \). (Color figure online)
Backbone curves depicting the instantaneous and periodic vibration amplitude of tip deflection of the shallow shell structure. Here, the perturbations are under \(\rho \) with \(\delta \rho =0.02\rho \)
We follow the derivations in Sect. 5.3 to obtain its derivative as \(\partial _\rho ||z_\textrm{out}||_{\mathcal {L}^2}=-2.144516\times 10^{-5}\). We again use the central difference (cf. (124)) to obtain an approximate derivative. Here we take \(\varDelta \rho =0.001\rho \) to perform the central difference computation and obtain that \(\partial _\rho ||z_\textrm{out}||_{\mathcal {L}^2}=-2.143032\times 10^{-5}\), which is in a good agreement with the explicit derivative.
7.3.6 Perturbed FRCs and their validation
We follow Sect. 5.5 to further present the FRC and its perturbations. As detailed in Appendix F.3, we can safely ignore the contribution of the leading-order non-autonomous SSM. Thus, the quadratic interpolation introduced in Remark 2 is not involved here. We ignore the terms related to \(\varvec{x}_0\) and \(\bar{\varvec{x}}_0\) in (69).
We obtain the FRC and its perturbation under \(\delta \rho =0.02\rho \) in Fig. 12. The peak of the FRC moves leftwards as \(\rho \) increases, which is consistent with the evolution of backbone curves under the variation in \(\rho \). We further validate the effectiveness of these perturbed FRCs following the same approach we used in the previous examples. As seen in Fig. 12, the cross markers match well with the lines, which validates the effectiveness, as mentioned earlier.
Forced response curves (FRCs) of the shallow shell structure. Here, \(\delta \rho =0.02\rho \)
7.3.7 On adjoint method
We conclude this example by demonstrating the effectiveness of the adjoint method established in Sect. 6, which has been illustrated in the first example (see Table 1). We present another demonstration here to highlight that the adjoint method can also be applied to high-dimensional mechanical systems.
To apply the adjoint method in Sect. 6, the damping coefficients should satisfy \(\alpha +\beta \omega ^2=2\zeta \omega \). Here we take \(\alpha =\zeta \omega \) and \(\beta =\zeta /\omega \) with \(\zeta =0.02\) in our computations. As seen in Table 7, the results from the direct method and the adjoint scheme are consistent, indicating the adjoint method’s effectiveness. We also provide the central-difference-based approximate \(\partial _\rho \gamma \) in the table (\(\varDelta \rho =0.001\rho \)). As we can see, the approximate value is very close to the results from the adjoint method, which again demonstrates the effectiveness of the adjoint method. Here, the approximate \(\partial _E\gamma \) via central difference is not presented because it is tiny, and the rounding error is significant, resulting in an incorrect approximated value for the central difference method.
8 Conclusion
We have derived explicit third-order reduced-order models (ROMs) for mechanical systems using the theory of spectral submanifolds (SSMs). The derivation does not assume symmetric damping and stiffness matrices and allows for velocity-dependent nonlinearities. Moreover, such derivation is directly based on physical coordinates and uses only the knowledge of a pair of master modes.
We have further derived explicit sensitivity analysis of the third-order SSM-based ROMs. We have used the direct method to obtain explicit derivatives of the coefficients for SSM parameterization and associated reduced dynamics. We have also demonstrated that one can use the adjoint method to perform sensitivity analysis, which is a better candidate when the number of design variables is vast.
We have illustrated that the derived sensitivity can be used to construct perturbed SSMs, backbone curves, and self-excited limit cycles for unforced vibrations. In addition, we have derived explicit derivatives for the periodic orbits of forced vibration and the location of the peak point on a forced response curve (FRC). Further, we have discussed how to extract analytically perturbed FRCs using the sensitivity.
Our derivations assume two-dimensional SSMs and are inapplicable for internally resonant mechanical systems, where higher-dimensional SSMs are necessary [6, 9]. We leave the extension of our derivations to higher-dimensional SSMs for future study. Another limitation of our derivations is the third-order truncation, which can be insufficient for large amplitude vibrations [6]. Thus, it is important to develop a procedure that computes the expansion of derivatives up to any orders in an automated fashion, just as the computation of parameterization of SSMs [16]. Finally, extending the sensitivity analysis to constrained mechanical systems whose SSM-based model reduction has been established in [48] is instructive.
The explicit sensitivity analysis established in this study can be used to perform optimal design, model updating, and uncertainty quantification of high-dimensional nonlinear mechanical systems, as we discussed in the introductory section. These are ongoing projects, and the results will be reported elsewhere.
Data availability
The data used to generate the numerical results included in this paper are available from the corresponding author on reasonable request.
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Acknowledgements
The financial support of the National Natural Science Foundation of China (No. 12302014) and Shenzhen Science and Technology Innovation Commission (No. 20231115172355001) for this work is gratefully acknowledged.
Funding
This work was funded by National Natural Science Foundation of China (No. 12302014) and Shenzhen Science and Technology Innovation Commission (No. 20231115172355001).
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Appendices
A Explicit third-order reduction for first-order system
We look for the 2D SSM of the following form
with \(\varvec{W}_{mn}=\bar{\varvec{W}}_{nm}\). We note that
and hence
We also note that
and
1.1 A.1 Linear terms
Now we write down the equations derived from the balance of polynomials. In particular, we have the below for \(\mathcal {O}(p)\):
It follows that we have solution
We have the following for \(\mathcal {O}({\bar{p}})\):
It follows that we have solution
1.2 A.2 Quadratic terms
Now we move to quadratic terms \(\mathcal {O}(p^k{\bar{p}}^l)\) with \(k+l=2\). We have for \(\mathcal {O}(p^2)\)
We substitute \(\varvec{W}_{10}=\varvec{v}\), \(\varvec{W}_{01}=\bar{\varvec{v}}\), \(R_{10}^{(1)}=\lambda \) and \(R_{10}^{(2)}=0\) into the above equation and rearrange it, yielding
Following norm-form-style parameterization, we set \(R_{20}^{(1)}=R_{20}^{(2)}=0\) and hence
For \(\mathcal {O}({\bar{p}}^2)\), we have
We substitute \(\varvec{W}_{10}=\varvec{v}\), \(\varvec{W}_{01}=\bar{\varvec{v}}\), \(R_{01}^{(1)}=0\) and \(R_{01}^{(2)}={\bar{\lambda }}\) into the above equation and rearrange it, yielding
Following norm-form-style parameterization, we set \(R_{02}^{(1)}=R_{02}^{(2)}=0\) and hence
Likewise, we have for \(\mathcal {O}(p{\bar{p}})\)
Substitution of known solutions yields
The equation above can be rearranged as
Following norm-form-style parameterization, we set \(R_{11}^{(1)}=R_{11}^{(2)}=0\) and hence
1.3 A.3 Cubic terms
We now consider \(\mathcal {O}(p^3)\) terms. We have
Substitution of known solutions yields
The equation above can be rearranged as
Following norm-form-style parameterization, we set \(R_{30}^{(1)}=R_{30}^{(2)}=0\) and hence
We next consider \(\mathcal {O}({\bar{p}}^3)\) terms. We have
Substitution of known solutions yields
The equation above can be rearranged as
Following norm-form-style parameterization, we set \(R_{03}^{(1)}=R_{03}^{(2)}=0\) and hence
As for the \(\mathcal {O}(p^2{\bar{p}})\) terms, we have
Substitution of known solutions yields
Let
then the equation above can be rearranged as
where \(\varvec{Q}_{21}\) has been defined in (8). We note that \(2\lambda +{\bar{\lambda }}\approx \lambda \) when the system is weakly damped. So the coefficient matrix is nearly singular. To solve this system of linear equations, we ask for the right-hand side vector to be in the range of the coefficient matrix. We further note that the orthogonal complement of the range is equal to the kernel of the transpose of the coefficient matrix. So we ask for
where \(\varvec{u}\) gives the left eigenvector of the matrix pair corresponding to the eigenvalue \(\lambda \). Since \((\varvec{u})^*\varvec{B}{\varvec{v}}=1\) and \((\varvec{u})^*\varvec{B}\bar{\varvec{v}}=0\), we have
Following norm-form-style parameterization, we set \(R_{21}^{(2)}=0\) and hence
Finally, we move to \(\mathcal {O}(p{\bar{p}}^2)\) terms. We have
Substitution of known solutions yields
The equation above can be rearranged as
We note that \(2\lambda +{\bar{\lambda }}\approx {\bar{\lambda }}\) because the system is weakly damped. So, the coefficient matrix is nearly singular. To solve this system of linear equations, we ask for the right-hand side vector to be in the range of the coefficient matrix. So we have
where \(\bar{\varvec{u}}\) gives the left eigenvector of the matrix pair corresponding to the eigenvalue \({\bar{\lambda }}\). Since \((\bar{\varvec{u}})^*\varvec{B}{\varvec{v}}=0\) and \((\bar{\varvec{u}})^*\varvec{B}\bar{\varvec{v}}=1\), we have
Following norm-form-style parameterization, we set \(R_{12}^{(1)}=0\) and hence
1.4 A.4 Reduced dynamics
The reduced dynamics is given by
The second equation above is simply the complex conjugate of the first one. Let \(p=\rho e^{\textrm{i}\theta }\), we obtain
B Explicit third-order reduction for second-order system
For the linear term, we obtain
For the quadratic term, we obtain
where
which is invertible if \(2\lambda \) is not an eigenvalue, namely, the system does not admit 1:2 internal resonance. We also obtain
where
For the cubic term, we obtain
where
We also obtain
where
We also have
where \({\tilde{\lambda }}=2\lambda +{\bar{\lambda }}\) and
It thus follows that \(\varvec{A}_{21}\) is singular when \(\lambda =-{\bar{\lambda }}\), namely, the system is conservative. Since we work on dissipative systems, this singularity is not an issue. We note that
Since \(\varvec{A}\varvec{v}=\lambda \varvec{B}\varvec{v}\), we have \(\varvec{B}^{-1}\varvec{A}\varvec{v}=\lambda \varvec{v}\), namely \(\varvec{v}\) is an eigenvector of the matrix \(\varvec{B}^{-1}\varvec{A}\). We also note that
which is well defined because \(\textrm{Re}(\lambda )\ne 0\). Then we have
Consequently
C Derivation of derivatives of eigenvalue and eigenvectors
1.1 C.1 General mechanical systems
We take the full derivative of \(\mathcal {F}=\varvec{0}\) (see (24)), yielding
The above two equations can be rewritten in a more compact form as
where the coefficient matrix above is simply the Jacobian \(\partial \mathcal {F}\) evaluated at \(\varvec{p}=\varvec{p}_o\). Since this Jacobian matrix is regular [41], one can solve for the \(\varvec{\phi }'\) and \(\lambda '\) from the linear equations above.
We further compute the derivative of the left eigenvector. Taking the full derivative of the second equation of (11), yields
Taking the full derivative of the normalization condition (13) yields
We note that \(\varvec{\psi }'^{\textrm{T}}(2\lambda \varvec{M}+\varvec{C}_\textrm{d})\varvec{\phi }=\varvec{\phi }^{\textrm{T}}(2\lambda \varvec{M}+\varvec{C}_\textrm{d})^{\textrm{T}}\varvec{\psi }'\). Then one can solve for the following extended system of equations with unknowns \(\varvec{\psi }'\) and \(\xi \in \mathbb {C}\) [41]
where \(\varvec{G}\) and r has been given in (27)-(28).
1.2 C.2 Proportionally damped mechanical systems
We have a simpler version of the generalized eigenvalue problem (20). We take the derivative of the first equation in (20), yielding
We note that \(\textrm{ker}(\varvec{K}-\omega ^2\varvec{M})=\varvec{\phi }\), namely, \(\varvec{K}-\omega ^2\varvec{M}\) is singular. Since both \(\varvec{K}\) and \(\varvec{M}\) are symmetric, the solvability condition gives
We also take the derivative of the second equation in (20) and obtain
We premultiply the above equation by \(\varpi \varvec{M}\varvec{\phi }\) and then add the equation to (196), yielding
As long as \(\varpi \ne 0\), the coefficient matrix is invertible, so we obtain
In our computation, we take \(\varpi =\omega ^2\).
We still need the derivative of the eigenvalue \(\lambda \). For the Rayleigh damping \(\varvec{C}_\textrm{d}=\alpha \varvec{M}+\beta \varvec{K}\), we have
It follows that
We also have
Taking the full derivative of the above equation yields
D Adjoint computation
We take the variation of the Lagrangian defined in (101), yielding
where
and
Here we have
Similar formulas hold for \(2\delta \hat{\varvec{f}}_2(\varvec{\mu },{\varvec{\phi }},\varvec{W}_{11}^{(1)})\). We also have
Since \(\lambda =-\zeta \omega +\textrm{i}\omega \sqrt{1-\zeta ^2}\), we have \(\delta \lambda =(-\zeta +\textrm{i}\sqrt{1-\zeta ^2})\delta \omega \) and \(\delta \textrm{Re}(\lambda )=-\zeta \delta \omega \). Recall that \(\alpha \) and \(\beta \) are functions of \(\omega \), so we have
Now we write some adjoint equations:
It follows that \(\lambda _\gamma =-1\), \(\lambda _\omega =-\lambda _\gamma \varvec{\phi }^\textrm{T}\varvec{f}_{21}/\omega \), and \(\varvec{\lambda }_{21}=-\lambda _\gamma \kappa \varvec{\phi }\) (see (102)). As for the variation with respect to \(\delta \omega \), we have
Next, we write the adjoint equation corresponding to \(\delta \varvec{\phi }\). We note that
where \(\varvec{F}_{2,2}\) and \(\varvec{F}_{2,3}\) have been defined in (109). Similarly, we also obtain
where \(\varvec{F}_{3,2}\), \(\varvec{F}_{3,3}\), and \(\varvec{F}_{3,4}\) have been defined in (110). With the above notations, we obtain adjoint equations below
where \(\varvec{f}_{21}^{\varvec{\phi }}\) has been defined in (108).
We also obtain
From which we solve for \(\varvec{\lambda }_{20}\) and \(\varvec{\lambda }_{11}\) as functions of \(\varvec{\lambda }_{21}\), as seen in Eqs. (103) and (104).
We solve for \(\varvec{\lambda }_{\varvec{\phi }}\) and \(\lambda _{\textrm{norm}}\) from (212) and (215), namely,
where \(\varvec{b}_{\varvec{\phi }}\) has been defined in (106) and \(b_{\textrm{norm}}\) has been defined in (107).
Consequently, we have (see (111))
which characterizes the sensitivity of \(\gamma \) with respect to \(\varvec{\mu }\).
E Analytic derivatives for the example of two coupled oscillators
The derivatives of the eigenvalue are below
The derivatives of the linear expansion coefficients are given by
The derivatives of the quadratic expansion coefficients are listed below
where
The derivatives of the cubic expansion coefficient vector \(\varvec{W}_{30}\) are given by
where
The derivatives of the cubic expansion coefficient vector \(\varvec{W}_{21}\) are given by
and
where
The derivatives of the cubic coefficient \(\gamma \) are listed below
where
The derivatives of the leading-order non-autonomous part of the SSM are given by
where
and
Finally, we list the derivatives of the added term coefficient \({\tilde{f}}\) below
F Supplementary analysis
1.1 F.1 Example 7.1
We first show that the third-order SSM-based model reduction is sufficient to make accurate predictions. As seen in Fig. 13, the FRC of \(x_1\) at higher-order reductions matches well with that of the FRC at \(\mathcal {O}(3)\), which suggests the effectiveness of the third-order reduction.
Forced response curve for the first oscillator \(x_1\) of the system (112). These FRCs are obtained from SSM-based model reduction at various approximation orders
We then check whether it is necessary to account for the leading-order non-autonomous part of SSM, i.e., \(\epsilon {\varvec{x}}_0e^{\textrm{i}\varOmega t}\) and its complex conjugate in (69). we refer to the SSM solution as time-varying (TV) if (69) is used and as time-independent (TI) when the leading-order non-autonomous part of SSM is ignored [7]. We plot the FRCs of \(x_1\) and \(x_2\) in the upper and lower panels of Fig. 14, from which we see that the TI and TV solutions are very close for \(x_1\), yet considerable difference is observed between the TI and TV solutions for \(x_2\). Indeed, the first oscillator \(x_1\) is in resonance, and the autonomous part \(\varvec{W}(\varvec{p})\) dominates the non-autonomous part, while the second oscillator is not in resonance, and hence both two parts are significant. Therefore, we need to account for the leading-order non-autonomous part of SSM if we are interested in both oscillators’ forced vibration.
Forced response curve for the first oscillator \(x_1\) (upper panel) and the second oscillator \(x_2\) (lower panel) of the system (112). These FRCs are obtained from third-order, SSM-based model reduction. TV-SSM and TI-SSM correspond to time-varying and time-independent SSM solutions, respectively. QI stands for the quadratic interpolation introduced in Remark 2
As discussed in Remark 2, the computation of the TV-solutions can be costly if the number of sampled points is significant. This is because we need to update \(\varvec{x}_0\) when \(\varOmega \) is changed. We have proposed a quadratic interpolation scheme to capture the \(\varOmega \)-dependence of \(\varvec{x}_0\), where only three times of computation is needed to construct the whole FRC. Indeed, as shown in the lower panel of Fig. 14, this quadratic interpolation (QI) scheme gives excellent construction of the FRC.
Limit cycle of free vibration (upper panel) and forced response curve (lower panel) for the cantilevered pipe conveying fluid. These results are obtained from SSM-based model reduction at various approximation orders
1.2 F.2 Example 7.2
Again, we first show that the third-order SSM-based model reduction is sufficient to make accurate predictions. As seen in the upper panel of Fig. 15, the limit cycle of the unforced vibration of the pipe already converges at \(\mathcal {O}(3)\) expansion. From the lower panel of Fig. 15, we see that the isola converges at \(\mathcal {O}(5)\), but the isola produced by \(\mathcal {O}(3)\) approximation is very to that of \(\mathcal {O}(5)\). Meanwhile, the main branch converges again at \(\mathcal {O}(3)\). All these together suggest the effectiveness of the third-order reduction.
We further check whether accounting for the leading-order non-autonomous part of SSM is necessary. As seen in Fig. 16, the TI-based FRC, including the isolated and the main branches, matches that of the TV-based FRC. This suggests that TI-based solutions are sufficient to make accurate predictions. Therefore, we take TI-based FRCs in this example.
Forced response curves for the cantilevered pipe conveying fluid. These FRCs are obtained from third-order, SSM-based model reduction. TV-SSM and TI-SSM correspond to time-varying and time-independent SSM solutions, respectively
Forced response curve for the von Kármán shallow shell structure. These results are obtained from SSM-based model reduction at various approximation orders
1.3 F.3 Example 7.3
First, we check whether the third-order truncation is sufficient to make accurate predictions. We extract the FRC of the shallow shell structure via SSM computation at different expansion orders. As seen in Fig. 17, the FRC converges at \(\mathcal {O}(5)\) expansion. The FRC from \(\mathcal {O}(3)\) computation also converges at low response amplitude and also nearly converges at high response amplitude. Therefore, the third-order reduction makes predictions that are sufficiently accurate overall.
We further check whether accounting for the leading-order non-autonomous part of SSM is necessary. As seen in Fig. 18, the TV-based FRC matches well with the TI-based one, suggesting that we can take TI-based FRCs in this example.
Forced response curves for the von Kármán shallow shell structure. These FRCs are obtained from third-order, SSM-based model reduction. TV-SSM and TI-SSM correspond to time-varying and time-independent SSM solutions, respectively
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Li, M. Explicit sensitivity analysis of spectral submanifolds of mechanical systems. Nonlinear Dyn 112, 16733–16770 (2024). https://doi.org/10.1007/s11071-024-09947-4
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DOI: https://doi.org/10.1007/s11071-024-09947-4