Abstract
Perturbation methods, already used in the first part of the book to analyze the sample system, are here applied to general structures. The metamodel introduced in Sect. 5.2 is considered, for which several nonlinear problems, of different nature, are investigated. The static perturbation method is used: (i) in elastostatics (in which the stiffness operator is non-singular), and (ii) in buckling (in which the stiffness operator is instead singular). The Multiple Scale Method is employed to tackle dynamic problems, i.e.: (i) the external resonance phenomena (primary, sub-harmonic and super-harmonic); (ii) the parametric excitation of conservative systems with periodically time-variant properties; (iii) the dynamic bifurcation of nonconservative systems. Here, only the bifurcation equations are derived, since their study requires performing the same steps illustrated in Chaps. 2–4.
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Notes
- 1.
The multiplier \(\alpha \) itself could be used as perturbation parameter, but the procedure illustrated here is preferred, since systematic.
- 2.
It has been exploited, that \(\boldsymbol{n}_{2}\left( \textbf{u}+\textbf{v},\textbf{u}+\textbf{v}\right) =\boldsymbol{n}_{2}\left( \textbf{u},\textbf{u}\right) +2\boldsymbol{n}_{2}\left( \textbf{u},\textbf{v}\right) +\boldsymbol{n}_{2}\left( \textbf{v},\textbf{v}\right) \), where \(\boldsymbol{n}_{2}\left( \textbf{u},\textbf{v}\right) =\boldsymbol{n}_{2}\left( \textbf{v},\textbf{u}\right) \) (see also Note 3 in the Appendix B). As a scalar example, if \(n_{2}(u,u):=uu'\), it follows that \(n_{2}(u+v,u+v)=(u+v)(u+v)'=uu'+(u'v+uv')+vv'\), from which:
$$ n_{2}(u,v)=\frac{1}{2}(u'v+uv'). $$Similarly, for cubic operators, it is \(\boldsymbol{n}_{3}\left( \textbf{u}+\textbf{v},\textbf{u}+\textbf{v},\textbf{u}+\textbf{v}\right) =\boldsymbol{n}_{3}\left( \textbf{u},\textbf{u},\textbf{u}\right) +3\boldsymbol{n}_{3}\left( \textbf{u},\textbf{u},\textbf{v}\right) +3\boldsymbol{n}_{3}\left( \textbf{u},\textbf{v},\textbf{v}\right) +\boldsymbol{n}_{3}\left( \textbf{v},\textbf{v},\textbf{v}\right) \). For example, if \(n_{3}(u,u,u)=:uu'u''\), by the same reasoning, it follows:
$$ \begin{aligned}n_{3}(u,u,v) & =\frac{1}{3}(uu'v''+uu''v'+u'u''v),\\ n_{3}(u,v,v) & =\frac{1}{3}(uv'v''+u'vv''+u''vv'). \end{aligned} $$.
- 3.
If this is not the case, a change of variable \(\textbf{u}=\textbf{u}^{f}\left( \textbf{s};\mu \right) +\textbf{v}\left( \textbf{s}\right) \), with \(\textbf{u}^{f}\) the known non-trivial path, permits to reduce the problem to the simpler case, although the dependence on \(\mu \) of the total stiffness matrix \(\boldsymbol{\mathcal {K}}\) is no longer linear (see, e.g., [10]).
- 4.
The vector \(\boldsymbol{n}_{2}\) has been expanded with respect to the load as \(\boldsymbol{n}_{2}\left( \textbf{u},\textbf{u};\mu \right) =\boldsymbol{n}_{2}\left( \textbf{u},\textbf{u};\mu _{0}\right) +\epsilon \mu _{1}\boldsymbol{n}_{2,\mu }\left( \textbf{u},\textbf{u};\mu _{0}\right) +\cdots \), where \(\boldsymbol{n}_{2,\mu }:=\frac{\partial \boldsymbol{n}_{2}}{\partial \mu }\).
- 5.
It is assumed here that the operator possesses a one-dimensional kernel.
- 6.
Usually, the interest is on \(\mu >0\), for example, on compression loads acting on the Euler beam.
- 7.
The structure, therefore, behaves as a single degree of freedom system, but its configurations does not span a subspace (i.e., the kernel of the stiffness operator), but rather a manifold, having the same dimension and tangent to it. The analogy with the Center Manifold Theory [4, 5, 18, 19] (mentioned in the Remark 4.3), relevant to dynamical systems, should be noticed.
- 8.
An operator \(\boldsymbol{\mathcal {L}}\), with its boundary conditions \(\boldsymbol{\mathcal {B}}\), is self-adjoint when, for any vectors \(\boldsymbol{\phi }\), \(\boldsymbol{\psi }\) defined in the function space on which \(\boldsymbol{\mathcal {L}}\) and \(\boldsymbol{\mathcal {B}}\) operate, the following Extended Green Identity holds:
$$ \intop _{\Omega }\!\!\boldsymbol{\psi }^{T}\boldsymbol{\mathcal {L}}\boldsymbol{\phi }\,\textrm{d}\Omega +\intop _{\Gamma }\!\!\boldsymbol{\psi }^{T}\boldsymbol{\mathcal {B}}\boldsymbol{\phi }\,\textrm{d}\Gamma =\intop _{\Omega }\!\!\boldsymbol{\phi }^{T}\boldsymbol{\mathcal {L}}\boldsymbol{\psi }\,\textrm{d}\Omega +\intop _{\Gamma }\!\!\boldsymbol{\phi }^{T}\boldsymbol{\mathcal {B}}\boldsymbol{\psi }\,\textrm{d}\Gamma . $$.
- 9.
Namely, for any h, k:
$$ \intop _{\Omega }\!\!\boldsymbol{\phi }_{h}^{T}\boldsymbol{\mathcal {K}}\boldsymbol{\phi }_{k}\,\textrm{d}\Omega +\intop _{\Gamma }\!\!\boldsymbol{\phi }_{h}^{T}\boldsymbol{\mathcal {B}}\boldsymbol{\phi }_{k}\,\textrm{d}\Gamma =\delta _{hk}\omega _{k}^{2}, $$where \(\delta _{hk}\) is the Kronecker operator, and the (arbitrary) normalization \(\intop _{\varOmega }\boldsymbol{\phi }_{k}^{T}\boldsymbol{\mathcal {M}}\boldsymbol{\phi }_{k}\,\textrm{d}\Omega =1\) has been used.
- 10.
Rescaling is performed in such a way the load and the dam** forces are made of the same order of \(\left| \textbf{u}\right| ^{3}\), for the reasons which will appear clear soon.
- 11.
Internal resonances [11] are excluded, i.e., the natural frequencies are assumed to be incommensurable.
- 12.
If \(\textbf{x}=\textbf{a}+\bar{\textbf{a}},\textbf{y}=\textbf{b}+\bar{\textbf{b}}\), then \(\boldsymbol{n}_{2}\left( \textbf{x},\textbf{x}\right) =\boldsymbol{n}_{2}\left( \textbf{a},\textbf{a}\right) +2\boldsymbol{n}_{2}\left( \textbf{a},\bar{\textbf{a}}\right) +\boldsymbol{n}_{2}\left( \bar{\textbf{a}},\bar{\textbf{a}}\right) =\boldsymbol{n}_{2}\left( \textbf{a},\textbf{a}\right) +\boldsymbol{n}_{2}\left( \textbf{a},\bar{\textbf{a}}\right) +\mathrm {c.c.}\), the overbar indicating complex conjugate. Similarly, \(\boldsymbol{n}_{2}\left( \textbf{x,y}\right) =\boldsymbol{n}_{2}\left( \textbf{a,b}\right) +\boldsymbol{n}_{2}\left( \textbf{a},\bar{\textbf{b}}\right) +\mathrm {c.c.}\) and \(\boldsymbol{n}_{3}\left( \textbf{x,x},\textbf{x}\right) =\boldsymbol{n}_{3}\left( \textbf{a,a},\textbf{a}\right) +3\boldsymbol{n}_{3}\left( \textbf{a,a},\bar{\textbf{a}}\right) +\mathrm {c.c.}\), to be used later.
- 13.
Indeed, considered the problem:
$$ \begin{aligned}\boldsymbol{\mathcal {M}}\ddot{\textbf{u}}+\boldsymbol{\mathcal {K}}\textbf{u} & =\textbf{p}e^{i\omega _{r}t},\qquad \textrm{in}\,\Omega ,\\ \boldsymbol{\mathcal {B}}\textbf{u} & =\textbf{q}e^{i\omega _{r}t},\qquad \textrm{on}\,\Gamma , \end{aligned} $$and taken the solution as \(\textbf{u}=\hat{\textbf{u}}e^{i\omega _{r}t}\), a space-differential problem follows:
$$ \begin{aligned}\left( \boldsymbol{\mathcal {K}}-\omega _{r}^{2}\boldsymbol{\mathcal {M}}\right) \hat{\textbf{u}} & =\textbf{p},\\ \boldsymbol{\mathcal {B}}\hat{\textbf{u}} & =\textbf{q}. \end{aligned} $$Since the operator \(\boldsymbol{\mathcal {K}}-\omega _{r}^{2}\boldsymbol{\mathcal {M}}\) is singular, whose kernel is spanned by \(\boldsymbol{\phi }_{r}\), the known terms \(\textbf{p},\textbf{q}\) must be orthogonal to \(\boldsymbol{\phi }_{r}\) (i.e., the forces must spend zero virtual work on the ‘floppy’ mode admitted by the operator), namely:
$$ \intop _{\Omega }\!\!\boldsymbol{\phi }_{r}^{T}\textbf{p}\,\textrm{d}\Omega +\intop _{\Gamma }\!\!\boldsymbol{\phi }_{r}^{T}\textbf{q}\,\textrm{d}\Gamma =0. $$.
- 14.
The motion, therefore, takes place on a family of two dimensional manifolds, parametrized by \(\alpha \) and \(\sigma _{r}\).
- 15.
This rescaling is useful for symmetric systems, to make \(\left| \textbf{u}\right| ^{3}=O\left( \epsilon \left| \textbf{u}\right| \right) \).
- 16.
The adjective ‘principal’ refers to the fact that the resonance is the first one encountered in the perturbation scheme, and therefore more important, since of lowest order. ‘Secondary’ resonances would appear to higher orders, not investigated here (see [11]).
- 17.
The motion, therefore, develops on a family of two dimensional manifolds, parametrized by \(\mu \) and \(\sigma _{r}\).
- 18.
They are typical of the aeroelastic forces [10], not addressed in this book.
- 19.
They have no qualitative influence on dynamic bifurcations [10].
- 20.
This operation is called splitting of the parameter, which is something different from expansion (which contains an infinite number of terms).
- 21.
Terms as \(\boldsymbol{n}_{2,\mu }\left( \textbf{u},\textbf{u};\mu _{0}\right) \), mentioned in the Note 4 of this chapter by dealing with buckling, are here of higher order, since the neighborhood of the bifurcation point explored is now smaller, of order \(\epsilon ^{2}\).
- 22.
Equation 6.67 is formally identical to Eq. 6.29, relevant to the external resonance of a conservative system, but now the mode \(\boldsymbol{\phi }_{c}\) is complex, entailing:
$$ \textbf{u}_{0}=a\left( t_{2}\right) \left\{ \mathrm {Re\left( \boldsymbol{\phi }_{c}\right) }\cos \left[ \omega _{c}t_{0}+\varphi \left( t_{2}\right) \right] -\mathrm {Im\left( \boldsymbol{\phi }_{c}\right) }\sin \left[ \omega _{c}t_{0}+\varphi \left( t_{2}\right) \right] \right\} , $$as \(A(t_{2})=\frac{1}{2}a(t_{2})e^{i\varphi (t_{2})}\).
- 23.
The left eigenvalue satisfies the adjoint of Eqs. 6.66, i.e.:
$$ \begin{aligned}\left( \boldsymbol{\mathcal {K}}_{e}+\mu _{c}\boldsymbol{\mathcal {K}}_{g}^{*}-i\omega _{c}\mathcal {\boldsymbol{\mathcal {C}}}_{s}-\omega _{c}^{2}\boldsymbol{\mathcal {M}}\right) \boldsymbol{\psi }_{c} & =\boldsymbol{0},\\ \left( \boldsymbol{\mathcal {B}}_{s}+\mu _{0}\boldsymbol{\mathcal {B}}_{g}^{*}-i\omega _{c}\boldsymbol{\mathcal {D}}_{s}\right) \boldsymbol{\psi }_{c} & =\textbf{0}, \end{aligned} $$where \(\boldsymbol{\mathcal {K}}_{g}^{*}\) is the adjoint stiffness operator and \(\boldsymbol{\mathcal {B}}_{g}^{*}\) the adjoint boundary conditions, which are drawn from the Extended Green Identity:
$$ \intop _{\Omega }\!\!\bar{\boldsymbol{\psi }}^{T}\boldsymbol{\mathcal {L}}\boldsymbol{\phi }\,\textrm{d}\Omega +\intop _{\Gamma }\!\!\bar{\boldsymbol{\psi }}^{T}\boldsymbol{\mathcal {B}}\boldsymbol{\phi }\,\textrm{d}\Gamma =\intop _{\Omega }\!\!\bar{\boldsymbol{\phi }}^{T}\mathcal {L^{*}}\boldsymbol{\psi }\,\textrm{d}\Omega +\intop _{\Gamma }\!\!\bar{\boldsymbol{\phi }}^{T}\boldsymbol{\mathcal {B}}^{*}\boldsymbol{\psi }\,\textrm{d}\Gamma . $$From the same identity, the compatibility condition is derived (see the Appendix A, Sect. A.4).
- 24.
The motion, therefore, takes place on an (invariant) two dimensional manifold, according to the Center Manifold Theory.
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Luongo, A., Zulli, D., Ferretti, M., D’Annibale, F. (2024). Perturbation Methods for a Continuum Metamodel. In: Perturbation Methods and Nonlinear Phenomena. Synthesis Lectures on Engineering, Science, and Technology. Springer, Cham. https://doi.org/10.1007/978-3-031-49397-3_6
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