Z. Angew. Math. Phys. (2018) 69:7
2017 Springer International Publishing AG,
part of Springer Nature
published online December 12, 2017
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
The eﬀect of a weak nonlinearity on the lowest cut-oﬀ frequencies of a cylindrical shell
I. V. Andrianov, J. Kaplunov, A. K. Kudaibergenov and L. I. Manevitch
Abstract. The plane strain problem for a thin circular cylindrical shell is considered within the framework of the Sanders–
Koiter theory. The relative shell thickness and displacement amplitude are chosen to be of the same asymptotic order. The
leading nonlinear correction to the lowest cut-oﬀ frequencies is derived using the method of multiple scales. In contrast to
the traditional two-mode Galerkin expansions assuming inextensibility of the shell transverse cross section, the developed
fourth-order asymptotic scheme operates with ﬁve angular modes. The obtained results reveal asymptotic inconsistency of
previous approximate solutions to the problem.
Mathematics Subject Classiﬁcation. 41 Approximations and expansions, 74 Mechanics of deformable solids.
Keywords. Shell, Asymptotic, Nanotube, Nonlinear, Cut-oﬀ.
Recent trends in thin shell dynamics are inspired by modern advanced applications, including the tech-
nologies utilizing carbon nanotubes, e.g. see [1,2]. The latter are often characterized by a small inverse
aspect ratio specifying the relationship between their radius and length and therefore support speciﬁc low-
frequency vibration modes most strongly aﬀected by curvature. A nonlinear theory for a cylindrical shell
oriented to such modes was developed in [3,4]. The cited papers further simplify the popular Sanders–
Koiter theory by adapting kinematical hypotheses on mid-surface deformations similarly to the so-called
semi-membrane shell theory [5,6]. The linearized version of the Sanders–Koiter equations was subject to
asymptotic analysis near the lowest cut-oﬀ frequencies for both an isotropic and anisotropic cylindrical
shell [7,8]. It is remarkable that various aspects of near-cut-oﬀ behaviour in thin structures were initially
treated within the high-frequency range, where it is usually attributed to thickness resonance frequencies,
e.g. see [9–14] and references therein.
It is worth noting that the plane strain problem for a cylindrical shell is similar to the famous one-
dimensional problem for a ring, which has a number of important technical applications, including gyro-
scopes [15,16]. It was shown both theoretically and experimentally that nonlinearity of ring deformations
often has to be incorporated [17–19]. As for a thin shell, small-amplitude nonlinear vibrations of a ring
are governed by two main asymptotic parameters corresponding to its relative thickness and weak nonlin-
earity. Two basic sets of equations associated with the repeated limits in these parameters were derived
in . Both of them incorporate the asymptotically justiﬁed condition of ring inextensibility. At the
same time numerous publications assume inextensibility as ad hoc hypotheses, e.g. see [15,18,19,21,22].
In addition, we mention the papers [23–25] concerned with three-wave interaction for a ring and also
taking into consideration three-dimensional high-frequency phenomena.
In this paper we generalize the results of linear asymptotic treatment in  to a weekly nonlinear set-
up. For the sake of simplicity, we restrict ourselves to plane strain deformation of a thin circular cylinder.
The relative thickness and displacement amplitude are assumed to be of the same asymptotic order.
The leading-order asymptotic behaviour is established for the lowest cut-oﬀ frequencies by retaining ﬁve
angular modes within the framework of the method of multiple scales.