# Free vibrations of an elastica around its post-buckled configurations

Hasnaa Kibach, Alireza Ture Savadkoohi, Claude-Henri Lamarque

► To cite this version:

Hasnaa Kibach, Alireza Ture Savadkoohi, Claude-Henri Lamarque. Free vibrations of an elastica around its post-buckled configurations. International Journal of Non-Linear Mechanics, 2025, 10.1016/j.ijnonlinmec.2025.105250. hal-05239102

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Submitted on 5 Sep 2025

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# Free vibrations of an elastica around its post-buckled configurations☆

H. Kibach *, A. Ture Savadkoohi, C.-H. Lamarque

Univ Lyon, ENTPE, LTDS UMR CNRS 5513, Vaulx-en-Velin Cedex, 69518, France

## ARTICLE INFO

**Keywords:**

Buckling

Elastica

Free vibration analysis

Nonlinear vibrations

## ABSTRACT

The post-buckled free vibration problem of an elastica around one of its vibration modes is investigated. For this, different equilibrium states of the elastica are detected by analysing the dimensionless form of the governing equations. Then, the multiple scale method is employed to study the free vibration of the elastica in a selected post-buckled state around a specific vibration mode. At the micro-scale, the considered elastica can be supposed as an element of an architecture material with tailored mechanical behaviours.

## 1. Introduction

In recent decades, there has been a rise in the use of flexible elements and compliant mechanisms instead of rigid bodies, where elasticity and flexibility permit structures to transmit motions and actions or to absorb energy through the deformation of their structural elements [1–5]. The goal is to design architectured structures and materials that exhibit specific, tailored properties for energy management such as control and/or harvesting [6,7]. Exploiting the instabilities of structural elements such as beams beyond the domain of linear elasticity, through phenomena like flutter [8] and buckling [9,10], can lead to extraordinary responses, not only of single element itself, but also in the form of their assemblies for fabricating special materials and/or structures. These nonlinear behaviours enable a wide range of applications such as energy trapping under various loading conditions [11–14], energy focusing [15], energy and shock absorption [10, 16] and vibration isolation [17,18].

To design the behaviours of an architectured material at the macro-scale, it is a prerequisite to understand the responses of its components at the micro-scale, e.g. the large deformations [19] and post-buckling operations of beams [20]. Elastica theory [21], which models large nonlinear deformations [22] can be endowed to formalize the responses of elements beyond the linear elastic domain. This enables capturing complex buckling and post-buckling behaviours of elements [23–27] where traditional models such as Timoshenko beam model [28] halts to predict such behaviours.

There are quite intensive works which treat nonlinear static and dynamics problems of elastica and also beams in large deformation domains; some of them, as representative examples, are listed as it follows: Shoup and McLarnan [29] studied static behaviours of an end-loaded elastica reaching to an iterative procedure to trace its

solutions for the force or motion analysis. The buckling, postbuckling and stabilities of nonlinear elastic elements have been investigated in detail by Como and Grimaldi [30,31] while Caflisch and Maddocks [32] treated the dynamical responses of a slender rod via generalization of Euler’s elastica theory and consideration of the rotational inertia term. They investigated on the stability of equilibria in static and dynamic senses. Yamaki and Mori [33] investigated theoretically on the nonlinear vibration problem of a clamped beam with initial deflection under uniform periodic lateral excitation which was tested later on experimentally by Yamaki et al. [34]. Nayfeh et al. [35] detected vibration mode shapes and frequencies of an extensible beam with different boundary conditions which was validated via experimental studies on a fixed-fixed beam. Then, Nayfeh et al. [36] exploited the multiple scale method [37] to detect nonlinear normal modes of a fixed-fixed beam at its first buckled mode taking into account 1:1 and 3:1 internal resonances. Nayfeh and Emam [38] investigated on a nonlinear Euler–Bernoulli beam model [39,40] which was based on approximation of the curvature of the mid-plane by the small-strain and moderate-rotation hypotheses. They traced a closed-form solution for the post-buckling configuration of such beam. Zulli et al. [41,42] investigated on the dynamics of a geometrically non-linear beam with uniform initial curvature. Lenci and Kloda [43] studied vibration of a kinematically exact nonlinear beam via a multiple scale method [37] showing dependency on the natural frequency of the system and its initial curvature.

In this paper, the free vibration problem of a clamped elastica, with one end movable around one of its vibration modes at one of its arbitrary post-buckled configuration, is articulated via analytical developments accompanied by some numerical results. The main contribution of this article is the detection of nonlinear frequencies

<footnote>☆ This article is part of a Special issue entitled: 'Nonlinear Vibrations' published in International Journal of Non-Linear Mechanics.</footnote>

<footnote>* Corresponding author.
E-mail address: hasnaa.kibach@entpe.fr (H. Kibach).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. The elastica under gravitational field $\vec{g}$: it is clamped at both ends while the right end side is movable and is under prescribed force $F$ along its longitudinal axis or prescribed displacement $D$.</figcaption></figure>

of the elastica, i.e. its different backbone curves, around one of its post-buckled configurations without any assumption on the order of magnitude of the curvature, the rotation and also of the response.

The paper is organized as follows: the governing equations of the considered system are derived in Section 2 and then some numerical buckling mode shapes are presented. The multiple scale method is implemented in Section 4 to trace the free vibration characteristics of the element around one of its equilibrium states. Some numerical examples are provided in Section 5. Finally, conclusions are drawn in Section 7.

## 2. Presentation of the considered system

### 2.1. Considered system

An inextensible planar elastica which is depicted in Fig. 1 is considered. We assume that it is originally straight in its undeformed configuration under ideal support boundary conditions. It is inclined with an initial angle $\theta_0$ with respect to the horizontal axis, clamped on the left-side and can slide on the right side along the original axis of the elastica. On the sliding side of the elastica, it can be either under prescribed axial displacement $D$ (controlled end-shortening $D$) or axial loading $F$. It should be mentioned that prescribed axial displacement $D$ can be applied via controlled support motion. Any point along the elastica at the instant $T$, has the Cartesian coordinates as $X(S, T)$ and $Y(S, T)$ and a rotation $\Theta(S, T)$ with respect to the $X$-axis and where $S$ and $T$ stand for the arc length and the time, respectively. The geometry and material properties (bending stiffness $EI$ and weight per unit length $W$) remain constant along its entire length $L$.

In the next section, governing system equations will be derived.

### 2.2. Governing equations

The free-body diagram of the elastica is illustrated in Fig. 2. The geometric relations for a particle situated at the point $X(S, T)$ and $Y(S, T)$ in planar Cartesian coordinates and the curvilinear coordinate $S$ read:

$$ \frac{\partial X}{\partial S} = \cos(\Theta(S)); \quad \frac{\partial Y}{\partial S} = \sin(\Theta(S)); \qquad (1) $$

where $\Theta$ is the angle between the tangent of the beam at arc length $S$ and the $X$-axis.

Following exact relation for the elastica is retained:

$$ EI \frac{\partial \Theta}{\partial S} = M; \qquad (2) $$

$M$ is the bending moment at point $S$ while $E$ and $I$ stand for the Young's modulus and the second moment of area, respectively. Considering the free body diagram of the elastica illustrated in Fig. 2, the

<figure><img src="image_4.png" /><figcaption>Fig. 2. Free-body diagram of the elastica.</figcaption></figure>

kinematics and dynamics equilibrium equations after neglecting higher order terms yield to:

$$ \begin{aligned} \frac{\partial M}{\partial S} &= Q \cos(\Theta) - P \sin(\Theta); \\ \frac{\partial P}{\partial S} &= -W \sin(\Theta_0) - \frac{W}{g} \frac{\partial^2 X}{\partial T^2} - C \frac{\partial X}{\partial T}; \\ \frac{\partial Q}{\partial S} &= -W \cos(\Theta_0) - \frac{W}{g} \frac{\partial^2 Y}{\partial T^2} - C \frac{\partial X}{\partial T}; \end{aligned} \qquad (3) $$

where $P$ and $Q$ are the internal forces and $C$ the damping coefficient. Let us introduce the following dimensionless variables:

$$ \begin{aligned} w &= \frac{W L^3}{EI}; & x &= \frac{X}{L}; & y &= \frac{Y}{L}; & s &= \frac{S}{L}; \\ p &= \frac{PL^2}{EI}; & q &= \frac{QL^2}{EI}; & f &= \frac{FL^2}{EI}; & r &= \frac{RL^2}{EI}; \\ m &= \frac{ML}{EI}; & t &= (T/L^2) \sqrt{\frac{EIg}{W}}; & c &= CL^2 \sqrt{\frac{g}{WEI}}; & d &= \frac{D}{L} \end{aligned} \qquad (4) $$

where $R$ is the transverse loading component of the force, if the force were not to be applied axially. Then, Eqs. (1)–(3) in dimensionless form become:

$$ \begin{aligned} \frac{\partial x}{\partial s} &= \cos(\theta); \\ \frac{\partial y}{\partial s} &= \sin(\theta); \\ \frac{\partial \theta}{\partial s} &= m; \\ \frac{\partial m}{\partial s} &= q \cos(\theta) - p \sin(\theta); \\ \frac{\partial p}{\partial s} &= -w \sin(\theta_0) - \frac{\partial^2 x}{\partial t^2} - c \frac{\partial x}{\partial t}; \\ \frac{\partial q}{\partial s} &= -w \cos(\theta_0) - \frac{\partial^2 y}{\partial t^2} - c \frac{\partial x}{\partial t}; \end{aligned} \qquad (5) $$

While the corresponding boundary conditions of the system read:

$$ \begin{aligned} x(s = 0, t) &= 0 \\ \theta(s = 0, t) &= \theta(s = 1, t) = 0 \\ y(s = 0, t) &= y(s = 1, t) = 0 \end{aligned} \qquad (6) $$
According to the type of end loading, the described boundary conditions in Eq. (6) should be supplemented by those of one of the following items [44]:

**Case I.** If the system is under prescribed axial displacement *D*, then:

$$x(s = 1, t) = 1 - d \qquad (7)$$

and the dimensionless reaction force *f*(*t*) is unknown verifying *f*(*t*) = -*p*(1, *t*).

**Case II.** If the system is under imposed axial loading *F*, then:

$$p(s = 1, t) = -f \qquad (8)$$

and the displacement will be unknown and time dependent *d*(*t*).

In both cases, the reaction force *r*(*t*) is unknown, verifying *r*(*t*) = *q*(1, *t*).

For the remainder of the paper, we will consider **Case II**, whose boundary conditions translate to Eqs. (6) and (8).

In Section 3, the buckling mode shapes of the elastica will be detected for further investigation on its free vibration in a post-buckled configuration.

### 3. Buckling modes of the elastica

It is supposed that the external force *f* is applied to the system in a quasi-static manner. We seek for the equilibrium states of the system. To this end, we drop the time dependency of system variables in Eqs. (5) and (10). This leads to the following equations:

$$\begin{aligned} \frac{\partial x_e}{\partial s} &= \cos(\theta) \\ \frac{\partial y_e}{\partial s} &= \sin(\theta) \\ \frac{\partial \theta_e}{\partial s} &= m \\ \frac{\partial m_e}{\partial s} &= q \cos(\theta) - p \sin(\theta) \end{aligned} \qquad (9)$$

$$\begin{aligned} \frac{\partial p_e}{\partial s} &= -w \sin(\theta_0) \\ \frac{\partial q_e}{\partial s} &= -w \cos(\theta_0) \\ x_e(s = 0) &= 0 \\ x_e(s = 1) &= 0 \\ p_e(s = 1) &= -f \end{aligned} \qquad (10)$$

$$\begin{aligned} \theta_e(s = 0) &= \theta_e(s = 1) = 0 \\ y_e(s = 0) &= y_e(s = 1) = 0 \end{aligned}$$

The variables with the index *e* stand for equilibrium state of the elastica.

Eq. (9) can be represented in a closed-form and then its post-buckled configurations can be traced by exploiting the elliptic integrals [3,45, 46].

Moreover, the shooting technique [47] can be applied to Eq. (9) to trace buckling equilibrium configurations of the system. Fig. 3 illustrates the three first buckling mode shapes of the clamped-clamped with the right end moveable elastica obtained from the shooting technique. The mechanical and physical parameters of the system are provided in Table 1.

The free vibration problem of the elastica around one of its dynamical modes in one of its post-buckled configurations is articulated.

<table><caption>**Table 1**

Physical and mechanical parameters of the elastica. These values belong to a carbon nanotube.</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>Young's modulus [46]</td><td><i>E</i></td><td>1000</td><td>GPa</td></tr><tr><td>Length [46]</td><td><i>l</i></td><td>30</td><td>nm</td></tr><tr><td>Second moment of inertia</td><td><i>I</i></td><td>1.08 × 10<sup>-38</sup></td><td>m<sup>4</sup></td></tr><tr><td>Cross-section area</td><td><i>A</i></td><td>4.6 × 10<sup>-19</sup></td><td>m<sup>2</sup></td></tr><tr><td>Slenderness ratio</td><td><i>λ</i></td><td>99.5</td><td>unitless</td></tr><tr><td>Initial inclination</td><td><i>θ</i><sub>0</sub></td><td>10</td><td>deg</td></tr></tbody></table>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** The first three buckling modes of the elastica.</figcaption></figure>

### 4. Free vibration analysis of the post-buckled configuration

All variables of Eq. (5) are perturbed around values representing one of equilibrium positions of the clamped-clamped elastica. Then, the multiple scale method [39] is endowed with handling the equations.

Let us consider the post-buckled configuration of the elastica. The time *t* is embedded to different scales (*t*<sub>*i*</sub> = *e*<sup>*j*t<sub>*i*</sub></sup>, *j* ∈ N). Where the 0 < *e* ≪ 1 is a small bookkeeping parameter that couples different
timescales [37,48]. The system variables are described as it follows:

$$
\begin{align}
x(s,t) &= x_e(s) + \sum_{j=1}^{n} \epsilon^j x_j(s, t_0, t_1, t_2) \nonumber \\
y(s,t) &= y_e(s) + \sum_{j=1}^{n} \epsilon^j y_j(s, t_0, t_1, t_2) \nonumber \\
\theta(s,t) &= \theta_e(s) + \sum_{j=1}^{n} \epsilon^j \theta_j(s, t_0, t_1, t_2) \nonumber \\
p(s,t) &= p_e(s) + \sum_{j=1}^{n} \epsilon^j p_j(s, t_0, t_1, t_2) \nonumber \\
q(s,t) &= q_e(s) + \sum_{j=1}^{n} \epsilon^j q_j(s, t_0, t_1, t_2) \tag{11}
\end{align}
$$

For further investigations, we neglect the effects of the damping via
setting c = 0 in Eq. (5).

It is seen that $\mathcal{O}(\epsilon^0)$ of these equations are represented by the
equilibrium state of the variables. In the following, we will investigate
on different orders of $\epsilon$ of Eq. (5).

$$
\mathcal{O}(\epsilon^0)
$$

The following set of equations are obtained:

$$
\begin{align}
\frac{\partial x_e}{\partial s} &= \cos(\theta_e) \nonumber \\
\frac{\partial y_e}{\partial s} &= \sin(\theta_e) \nonumber \\
\frac{\partial p_e}{\partial s} &= -w \sin(\theta_0) \tag{12} \\
\frac{\partial q_e}{\partial s} &= -w \cos(\theta_0) \nonumber \\
\frac{\partial^2 \theta_e}{\partial s^2} &= q_e \cos(\theta_e) - p_e \sin(\theta_e) \nonumber
\end{align}
$$

It can be seen that this equation is the same as Eq. (10) which has
been already articulated for detection of different buckling modes of
the system.

$$
\mathcal{O}(\epsilon^1)
$$

Eq. (5) reads:

$$
\begin{align}
\frac{\partial x_1}{\partial s} &= -\theta_1 \sin(\theta_e) \nonumber \\
\frac{\partial y_1}{\partial s} &= \theta_1 \cos(\theta_e) \nonumber \\
\frac{\partial p_1}{\partial s} &= -\frac{\partial^2 x_1}{\partial t_0^2} \nonumber \\
\frac{\partial q_1}{\partial s} &= -\frac{\partial^2 y_1}{\partial t_0^2} \nonumber \\
\frac{\partial^2 \theta_1}{\partial s^2} &= (q_1 - p_e \theta_1) \cos(\theta_e) - (p_1 + q_e \theta_1) \sin(\theta_e) \tag{13}
\end{align}
$$

Applying defined boundary conditions in Eq. (6) $x(s = 0, t) = 0$ and $\theta(s = 1, t) = 0$, $x_1(s = 1, t_0, t_1, t_2) = 0$ and $p_1(s = 1, t_0, t_1, t_2) = 0$ can be determined. Deriving four times the equation of $y_1$ with respect to $s$ and using Eq. (13), following equation can be obtained:

$$
\begin{align}
\frac{\partial^4 y_1}{\partial s^4} - \beta_{e1} \frac{\partial^2 y_1}{\partial s^2} - \beta_{e2} \frac{\partial y_1}{\partial s} - \beta_{e3} \frac{\partial^2 y_1}{\partial t_0^2} + \beta_{e4} \int_0^s \left( \frac{\partial^2 y_1}{\partial t_0^2} \right) ds \nonumber \\
= \beta_{e5} \frac{\partial^2 x_1}{\partial t_0^2} + \beta_{e6} \int_0^s \left( -\frac{\partial^2 x_1}{\partial t_0^2} \right) ds \tag{14}
\end{align}
$$

where $\beta_{e_j}$, ($j = 1, 2, ..., 6$) are coefficients depending only on a chosen
post-buckled equilibrium configuration. Their explicit expressions are
provided in Appendix A.

Let us suppose that Ω<sub>n</sub> is the frequency of the n-th linear vibration
mode of the dimensionless system. Then, we represent the solution of
Eq. (14) as:

$$
y_1(s, t_0, t_1, t_2, \dots) = \Phi_n(s) (A_{1,n}(t_1, t_2, \dots) e^{i \Omega_n t_0} + A_{1,n}^*(t_1, t_2, \dots) e^{-i \Omega_n t_0}) \quad (15)
$$

$$
x_1(s, t_0, t_1, t_2, \dots) = \Phi_n(s) (B_{1,n}(t_1, t_2, \dots) e^{i \Omega_n t_0} + B_{1,n}^*(t_1, t_2, \dots) e^{-i \Omega_n t_0}) \quad (16)
$$

Injecting Eqs. (15) and (16) in Eq. (14), following system is obtained:

$$
\Phi_n''' - \beta_{e1} \Phi_n'' - \beta_{e2} \Phi_n' + (\beta_{e3} + \beta_{e5} \beta_{e7}) \Omega_n^2 \Phi_n + (\beta_{e4} + \beta_{e6} \beta_{e7}) \Omega_n^2 \\
\times \int_0^s \Phi_n ds = 0 \tag{17}
$$

Eq. (17) accompanied by boundary conditions of the system, defined in Eq. (10), can be solved numerically for detection of modal characteristics of the elastica i.e. dynamic mode shapes (Φ<sub>n</sub>) and corresponding frequencies (Ω<sub>n</sub>).

$$
\mathcal{O}(\epsilon^2)
$$

Following set of equations is obtained:

$$
\begin{align}
\frac{\partial x_2}{\partial s} &= -\sin(\theta_e)\theta_2 - \frac{1}{2}\cos(\theta_e)\theta_1^2 \nonumber \\
\frac{\partial y_2}{\partial s} &= \cos(\theta_e)\theta_2 - \frac{1}{2}\sin(\theta_e)\theta_1^2 \nonumber \\
\frac{\partial p_2}{\partial s} &= -\frac{\partial^2 x_2}{\partial t_0^2} - 2\frac{\partial^2 x_1}{\partial t_0 \partial t_1} \nonumber \\
\frac{\partial q_2}{\partial s} &= -\frac{\partial^2 y_2}{\partial t_0^2} - 2\frac{\partial^2 y_1}{\partial t_0 \partial t_1} \tag{18}
\end{align}
$$

$$
\frac{\partial^2 \theta_2}{\partial s^2} = \cos(\theta_e) \left( -\frac{1}{2} \frac{q_e}{\cos^2(\theta_e)} \frac{\partial y_1}{\partial s}^2 - \frac{p_e}{\cos(\theta_e)} \frac{\partial y_2}{\partial s} + q_2 \right) \\
+ \sin(\theta_e) \left( -\frac{q_e}{\cos(\theta_e)} \frac{\partial y_2}{\partial s} + \frac{1}{2} q_e \frac{\sin(\theta_e)}{\cos^3(\theta_e)} \frac{\partial y_1}{\partial s}^2 - p_2 \right)
$$

Deriving four times the equation in $y_2$ and using Eqs. (18) provide the
following equation in $y_2$:

$$
\begin{align}
\frac{\partial^4 y_2}{\partial s^4} - \beta_{v1} \frac{\partial^2 y_2}{\partial s^2} - \beta_{v2} \frac{\partial y_2}{\partial s} - \beta_{v3} \frac{\partial^2 y_2}{\partial t_0^2} - \beta_{v4} \int_0^s \left(-\frac{\partial^2 y_2}{\partial t_0^2}\right) ds \nonumber \\
= \beta_{v5} \frac{\partial^2 x_2}{\partial t_0^2} + \beta_{v6} \int_0^s \left(-\frac{\partial^2 x_2}{\partial t_0^2}\right) ds \nonumber \\
+ \beta_{v7} \left(\frac{\partial^2 y_1}{\partial s^2}\right)^2 + \beta_{v8} \left(\frac{\partial y_1}{\partial s}\right)^2 + \beta_{v9} \frac{\partial^2 y_1}{\partial s^2} \frac{\partial y_1}{\partial s} \nonumber \\
+ \frac{\partial^2 y_1}{\partial s^2} \left[ \beta_{v10} \int_0^s \left(-\frac{\partial^2 y_1}{\partial t_0^2}\right) ds + \beta_{v11} \int_0^s \left(-\frac{\partial^2 x_1}{\partial t_0^2}\right) ds \right] \nonumber \\
+ \frac{\partial y_1}{\partial s} \left[ \beta_{v12} \int_0^s \left(-\frac{\partial^2 y_1}{\partial t_0^2}\right) ds + \beta_{v13} \int_0^s \left(-\frac{\partial^2 x_1}{\partial t_0^2}\right) ds + \beta_{v14} \frac{\partial^2 y_1}{\partial t_0^2} + \beta_{v15} \frac{\partial^2 x_1}{\partial t_0^2} \right. \nonumber \\
\left. + \beta_{v14} \frac{\partial^2 y_1}{\partial t_0 \partial t_1} + \beta_{v15} \frac{\partial^2 x_1}{\partial t_0 \partial t_1} \right] \tag{19}
\end{align}
$$

where $\beta_{v_j}$, ($j = 1, 2, ..., 15$) are coefficients that depend only on a chosen post-buckled equilibrium configuration. Their explicit expressions are given in Appendix B. Similar to Eqs. (15) and (16), we suppose that $y_2$ and $x_2$ take following forms:
$$y_2(s, t_0, t_1, t_2, \dots) = \Phi_n(s) (A_{2,n}(t_1, t_2, \dots) e^{i\Omega_n t_0} + A_{2,n}^*(t_1, t_2, \dots) e^{-i\Omega_n t_0}) \quad (20)$$

$$x_2(s, t_0, t_1, t_2, \dots) = \Phi_n(s) (B_{2,n}(t_1, t_2, \dots) e^{i\Omega_n t_0} + B_{2,n}^*(t_1, t_2, \dots) e^{-i\Omega_n t_0}) \quad (21)$$

where (.)\* stands for complex conjugate of the variable (.). After injecting (20) and (21) into (19), removing the secular term yields to:

$$\frac{\partial A_{1,n}}{\partial t_1} = 0 \Rightarrow A_{1,n} = A_{1,n}(t_2, t_3, \dots) \quad (22)$$

$$\mathcal{O}(\epsilon^3)$$

We have:

$$\begin{aligned}
\frac{\partial x_3}{\partial s} &= -\theta_3 \sin(\theta_e) - \theta_1 \theta_2 \cos(\theta_e) + \frac{1}{6} \theta_1^3 \sin(\theta_e) \\
\frac{\partial y_3}{\partial s} &= \theta_3 \cos(\theta_e) - \theta_1 \theta_2 \sin(\theta_e) - \frac{1}{6} \theta_1^3 \cos(\theta_e) \\
\frac{\partial p_3}{\partial s} &= -\frac{\partial^2 x_3}{\partial t_0^2} - 2 \frac{\partial^2 x_2}{\partial t_0 \partial t_1} \\
\frac{\partial q_3}{\partial s} &= -\frac{\partial^2 y_3}{\partial t_0^2} - 2 \frac{\partial^2 y_2}{\partial t_0 \partial t_1}
\end{aligned}
\quad (23)$$

$$\begin{aligned}
\frac{\partial^2 \theta_3}{\partial s^2} &= \cos(\theta_e) \left[ q_3 - \frac{1}{2} q_1 \theta_1^2 - p_e \theta_3 - p_2 \theta_1 - p_1 \theta_2 - \theta_1 \theta_2 q_e + \frac{1}{6} p_e \theta_1^3 \right] \\
&\quad + \sin(\theta_e) \left[ -p_3 + \frac{1}{2} p_1 \theta_1^2 - q_e \theta_3 - q_2 \theta_1 - q_1 \theta_2 + \theta_1 \theta_2 q_e + \frac{1}{6} q_e \theta_1^3 \right]
\end{aligned}
$$

We also note the solutions for $x_2$, $p_2$ and $q_1$:

$$\begin{aligned}
x_2 &= \int_0^s -\frac{1}{2} y_1^{\prime 2} ds \\
p_2 &= \int_0^s \left(-\frac{\partial^2 x_2}{\partial t^2}\right) ds \\
q_1 &= \int_0^s \left(-\frac{\partial^2 y_1}{\partial t^2}\right) ds
\end{aligned}
\quad (24)$$

Deriving four times the governing equation of $y_3$ in Eq. (23), following system can be obtained:

$$\begin{aligned}
\frac{\partial^4 y_3}{\partial s^4} - \beta_{w_1} \frac{\partial^2 y_3}{\partial s^2} - \beta_{w_2} \frac{\partial y_3}{\partial s} - \beta_{w_3} \frac{\partial^2 y_3}{\partial t_0^2} - \beta_{w_4} \int_0^s \left(-\frac{\partial^2 y_3}{\partial t_0^2}\right) ds \\
= \beta_{w_5} \frac{\partial^2 x_3}{\partial t_0^2} + \beta_{w_6} \int_0^s \left(-\frac{\partial^2 x_3}{\partial t_0^2}\right) ds \\
+ H(y_1^{\prime 3}, y_1^{\prime 2}, y_1^{\prime\prime}, y_1^{\prime}, y_1^{\prime\prime\prime}, y_1^{\prime\prime}, y_1^{\prime\prime\prime\prime}, y_1^{\prime\prime\prime}, y_1^{\prime\prime\prime\prime\prime}, y_1^{\prime\prime\prime\prime\prime\prime})
\end{aligned}
\quad (25)$$

where $\beta_{w_j}$ ($j = 1, 2, \dots, 6$) are coefficients that depend only on a chosen post-buckled equilibrium configuration. Their explicit expressions are given in Appendix C. $H$ is a function of the different variables of the previous orders, coupled together, see Appendix C. Similar to Eqs. (15) and (16), we suppose that $y_3$ and $x_3$ take the following forms:

$$y_3(s, t_0, t_1, t_2, \dots) = \Phi_n(s) (A_{3,n}(t_1, t_2, \dots) e^{i\Omega_n t_0} + A_{3,n}^*(t_1, t_2, \dots) e^{-i\Omega_n t_0}) \quad (26)$$

$$x_3(s, t_0, t_1, t_2, \dots) = \Phi_n(s) (B_{3,n}(t_1, t_2, \dots) e^{i\Omega_n t_0} + B_{3,n}^*(t_1, t_2, \dots) e^{-i\Omega_n t_0}) \quad (27)$$

After injecting Eqs. (26) and (27) into Eq. (25), removing the secular term leads to:

$$\mathcal{J}(\Omega_n, \Phi_n, A_{1,n}) = 2i\Omega_n \frac{\partial A_{1,n}}{\partial t_2} \int_0^s \Phi_n^2 ds + A_n^2 A_n^* \int_0^s \Phi_n \mathcal{K}_n ds = 0 \quad (28)$$

where

$$\begin{aligned}
\mathcal{K}_n = & \frac{3}{2} \Omega_n^2 \Phi_n \Phi_n^{\prime 2} + 3 \Omega_n^2 q_0 \Phi_n^{\prime} \Phi_n^{\prime\prime} + 3 \Omega_n^2 \Phi_n^{\prime} \Phi_n^{\prime\prime} \int_0^s \Phi_n ds + 3 \Phi_n^{\prime\prime 3} \\
& + 9 \Phi_n^{\prime} \Phi_n^{\prime\prime} \Phi_n^{\prime\prime\prime} \\
& + \frac{3}{2} \Phi_n^{\prime 2} \Phi_n^{(\nu)} - 2 \Omega_n^2 \Phi_n^{\prime} \int_0^s \Phi_n^{\prime 2} ds - 2 \Omega_n^2 p_0 \Phi_n^{\prime\prime}
\end{aligned}
\quad (29)$$

Let us project Eq. (28) on the mode $\Phi_n$. This is carried out via:

$$\langle \mathcal{J}, \Phi_n \rangle = \int_0^1 \mathcal{J} \Phi_n ds \quad (30)$$

After expressing $A_{1,n}$ in the polar domain as:

$$A_{1,n}(t_2, t_3, \dots) = \frac{1}{2} a_{1,n}(t_2, t_3, \dots) e^{i \alpha_n(t_2, t_3, \dots)} \quad (31)$$

Following system is obtained from Eq. (30):

$$\left\{
\begin{array}{l}
\displaystyle \frac{\partial a_n}{\partial t_2} = 0 \Rightarrow a_n = a_n(t_3, t_4, \dots) \\[1em]
\displaystyle \frac{\partial \alpha_n}{\partial t_2} = \frac{1}{8} \mathcal{U}_n a_n^2(t_3, t_4, \dots) \Rightarrow \alpha_n = \frac{1}{8} \mathcal{U}_n a_n^2 t_2 + c(t_3, t_4, \dots)
\end{array}
\right.
\quad (32)$$

where the coefficient $c(t_3, t_4, \dots)$ is the constant of the integration, and

$$\mathcal{U}_n = \frac{\int_0^1 \int_0^s \Phi_n \mathcal{K}_n ds ds}{\Omega_n \int_0^1 \int_0^s \Phi_n^2 ds ds} \quad (33)$$

We can rewrite an expression of $y_1$ as follow:

$$y_1 = \Phi_n(s) \left( \frac{1}{2} a_n(t_2) e^{i \Omega_{N,L,n} t_0} + CC \right) \quad (34)$$

with:

$$\Omega_{N,L,n} = \frac{1}{8} \mathcal{U}_n a_n^2 e^2 + \Omega_n \quad (35)$$

Eq. (35) represents the variation of the amplitude-dependent nonlinear frequency of the elastica ($\Omega_{N,L,n}$) around its linear frequency $\Omega_n$ in one of its post-buckled configurations. It is seen that, this variation is a function of the considered vibration and buckled mode shapes of the elastica. Setting $\hat{a}_n = a_n \epsilon$, Eq. (35) can be represented as:

$$\frac{\Omega_{N,L,n}}{\Omega_n} = \frac{1}{8} \mathcal{U}_n \hat{a}_n^2 + 1 \quad (36)$$

And the transverse displacement $y(s, t_0, t_1, \dots)$ can finally be rewritten as:

$$y(s, t_0, t_1, \dots) = y_e(s) + \underbrace{\Phi_n \left( \frac{1}{2} \hat{a}_n(t_2) e^{i \Omega_{N,L,n} t_0} + CC \right)}_{\hat{y}_1 = e^{i\gamma_1}(s, t_0, t_1, \dots)} \quad (37)$$

## 5. Some numerical examples for analytical formulation

In this section, some numerical examples of backbone curves of the system in different post-buckled configurations and around different vibration modes will be presented and commented. The results will be presented in two distinct cases: Case (I) Backbone curves at a given post-buckled configuration around different vibration modes; Case (II) backbone curves around a given vibration mode at different post-buckled configurations.

### 5.1. Case (I)

Let us consider the elastica which its parameters are collected in Table 1. We would like to investigate its first post-buckled configuration by studying variations of the frequency of the system as a function of the amplitude of $\hat{y}_1(s, t_0, t_1, \dots)$ around its first vibration mode, i.e. $n = 1$ in Eq. (36). Fig. 4 shows variations of the dimensionless frequency, i.e. $\frac{\Omega_{N,L,n}}{\Omega_n}$ defined in Eq. (36), as a function of $|\hat{y}_1| = |\Phi_n| \hat{a}_n$ at $s = 0.5$. It is seen that the backbone curve presents a hardening behaviour.

Variations of the dimensionless nonlinear frequency around second and fourth modes are depicted in Figs. 5 and 6 respectively.
<figure><img src="image_8.png" /><figcaption>Fig. 4. The backbone curve of the system at its first post-buckled configurations around its first vibration mode ($\Omega_1$ = 123 rad/s).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. The backbone curve of the system at its first post-buckled configurations around its second vibration mode ($\Omega_2$ = 284.6 rad/s).</figcaption></figure>

The nature of system response changes around the fourth vibration mode via presenting the softening behaviour. This shows that for a given post-buckled configuration, the free response of the system can change its nature around different vibration modes postulating between hardening- and softening-types responses. The variation of the nature of the free response of the system for a given post-buckled configuration around its different vibration modes confirms that the type of nonlinearity changes for different vibration modes.

## 5.2. Case (II)

Here we would like to observe system responses about a given vibration mode, e.g. the second mode ($n = 2$), at different post-buckled configurations, namely the second, third and fourth ones. Results are collected in Figs. 7–9. It is seen around the second vibration mode of the elastica, the system presents softening behaviour at its second and third post-buckled configuration while its nature changes around the fourth post-buckled configuration showing hardening behaviour. Similar to the Case (I), the shift of the nature of the behaviour mirrors the change of the type of the nonlinearity at its different post-buckled configurations.

<figure><img src="image_10.png" /><figcaption>Fig. 6. The backbone curve of the system at its first post-buckled configurations around its fourth vibration mode ($\Omega_4$ = 294.4 rad/s).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7. The backbone curve of the system at its second post-buckled configurations around its second vibration mode ($\Omega_2$ = 284.6 rad/s).</figcaption></figure>

# 6. Some results from finite element modelling of the system

A finite element model of the described system in Fig. 1, is built using COMSOL Multiphysics®. The exploited Physics Interphase is solid mechanics. The elastica is modelled with clamped-clamped boundary conditions while the right end is axially movable. The analysis has been carried out via inclusion of geometric nonlinearities. Different vibration modes of the unbuckled element, i.e. their frequencies and mode shapes, are selected. Then, several buckling modes and associated critical buckling loads are evaluated. Let us suppose we would like to study free vibration problem of the beam around its *m*th buckling mode and its *n*th vibration mode. To reproduce the backbone curves of the system around its buckled modes, following steps are applied:

* The critical force of the *m*th buckling mode, namely $P_m$ is kept on the system.

* A uniform body force defined as $A(s) \sin(2f\pi t)$ with $t \in [0, t_1]$ is applied, where $t_1$ is kept very small compared to the time of the
<figure><img src="image_12.png" /><figcaption>Fig. 8. The backbone curve of the system at its third post-buckled configurations around its second vibration mode ($\Omega_2$ = 284.6 rad/s).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9. The backbone curve of the system at its fourth post-buckled configurations around its second vibration mode ($\Omega_2$ = 284.6 rad/s).</figcaption></figure>

analysis. Here, $A(s)$ and $f$ are the amplitude and frequency of the excitation. We set $A(s) = 1$ and sweep $f$ around $f_n$.

* The dynamic body excitation is stopped at $t = t_1$. Then, the calculation of the finite element modelling of the system is continued for $t \in [t_1, t_2]$. We select the maximum displacement of a point of interest, e.g. a mid-length point, along the elastica.

* The procedure is repeated for different frequencies of excitations around $f_n$.

* The maximum displacements collected at previous steps are plotted with respect to the sweeping frequency of the excitation around the one of the $m$th vibration mode.

The obtained results are added to the analytical backbone curves represented in Figs. 4–9. It is seen that the analytical and corresponding numerical results are qualitatively in good agreement. However, there are some discrepancies between two results when the amplitudes of responses increase. This can be due to the fact that one can study the response of the system in higher order terms of the perturbation parameter, which is out of the scope of current work. However, provided analytical tool for detection of backbone curves of the system around different buckling and vibration modes can provide necessary outlooks and design tools for tuning parameters of the elastica for posing desired responses. Moreover, these discrepancies between two results can be

due to some possible mismatching between the finite element model and the one of the elastica used in this paper.

## 7. Conclusions

Free vibration problem of an elastica around one of its arbitrary post-buckled configurations is articulated. To this end, after obtaining buckled modes of the elastica, its spatio-temporal variables are perturbed around their equilibria while the perturbation variables depend on different time scales. The method of multiple scale is endowed for handling system equations at different orders of the perturbation parameters around one of the vibration modes of the elastica. This led to detection of the backbone curve of the system, which depends on the considered buckled mode, and reveals amplitude-dependent variations of the nonlinear system frequency. For a given post-buckled configuration, variations of the nonlinear frequency of the system can present hardening or softening behaviours around different vibration modes, meaning that the type of nonlinearity changes for different vibration modes. Moreover, the nature of the behaviours of the system changes around a given vibration mode in its different post-buckled configurations. Analytically detected backbone curves are confronted with those obtained from analyses of the finite element model of the system around different buckled and vibration modes. The results are qualitatively in good agreements; however they depart from each other at high response amplitudes. This can be due to possible mismatching between finite element modelling of the system and the elastica model. Meanwhile, one can try to study the system at higher orders of the perturbation parameter to have a more precise analytical relationship for nonlinear frequency of the elastica.

The developed technique and detected nonlinear frequency relationship provide tools for tuning parameters of the elastica in order to present special types of behaviours around one of its buckling and vibration modes. The perspective of this work is to consider the forced vibration of the system in its post-buckled configuration.

### CRediT authorship contribution statement

**H. Kibach:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **A. Ture Savadkoohi:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **C.-H. Lamarque:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

The authors would like to thank the following organizations for supporting this research: (i) The “Ministère de la Transition écologique, de la Biodiversité, de la Forêt, de la Mer et de la Pêche” and (ii) LABEX CELYA (ANR-10-LABX-0060) of the “Université de Lyon” within the program “Investissement d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).
## Appendix A. Definition of the coefficients $\beta_{e_j}, (j = 1, 2, ..., 7)$ in Eq. (14)

We note: $\frac{\partial}{\partial s} = ()'$.

$$
\begin{align*}
\beta_{e1} &= -p_e \cos(\theta_e) + q_e \sin(\theta_e) - 3(\theta_e'' \tan(\theta_e) + \theta_e'^2) \\
\beta_{e2} &= 3p_e \theta_e' \sin(\theta_e) + q_e \frac{\theta_e'}{\cos(\theta_e)} [3 \sin(\theta_e)^2 - 1] - p_e' \cos \theta_e - q_e' \sin(\theta_e) \\
&\quad + \theta_e^3 \tan(\theta_e) - \theta_e''' \tan(\theta_e) + \theta_e' \tan(\theta_e) - 2\theta_e \theta_e' - \theta_e'' \theta_e'' \\
\beta_{e3} &= -\cos(\theta_e) \\
\beta_{e4} &= \theta_e' \sin(\theta_e)(1 + 3\cos(\theta_e)) \\
\beta_{e5} &= \sin(\theta_e) \\
\beta_{e6} &= \theta_e'(\cos(\theta_e) - 3\sin(\theta_e)^2) \\
\beta_{e7} &= \frac{y_e'}{x_e'}
\end{align*}
\tag{A.1}
$$

## Appendix B. Definition of the coefficients $\beta_{v_j}, (j = 1, 2, ..., 15)$ in Eq. (19)

We note: $\frac{\partial}{\partial s} = ()'$.

$$
\begin{align*}
\beta_{v1} &= [-p_e - q_e \tan(\theta_e)] \cos(\theta_e) - 3[\theta_e'' \tan(\theta_e) + \theta_e'^2] \\
\beta_{v2} &= 3\theta_e' \sin(\theta_e)[p_e + q_e \tan(\theta_e)] - \cos(\theta_e)[p_e' + q_e' \tan(\theta_e)] - q_e \frac{\theta_e'}{\cos(\theta_e)} \\
&\quad - 3\theta_e' \theta_e'' \tan^2(\theta_e) - 3\theta_e'^3 \tan(\theta_e) \\
\beta_{v3} &= -p_e \cos^2(\theta_e) \\
\beta_{v4} &= 2\theta_e' p_e \sin(\theta_e) \cos(\theta_e) - p_e' \cos^2(\theta_e) \\
\beta_{v5} &= -\cos(\theta_e) \sin(\theta_e) \\
\beta_{v6} &= \theta_e'(3 \sin^2(\theta_e) - \cos^2(\theta_e)) \\
\beta_{v7} &= -\frac{3\theta_e'}{\cos(\theta_e)}
\end{align*}
\tag{B.1}
$$

$$
\begin{align*}
\beta_{v8} &= 2\left(-\frac{1}{2}\sin(\theta_e)\right) \sec(\theta_e) \left(-p_e' - q_e' \tan(\theta_e) - q_e' \sec^2(\theta_e)\right) \\
&\quad + 6\left(-\frac{1}{2}\sin(\theta_e)\right) \left(\theta_e' \tan(\theta_e)\right) \left(-p_e - q_e \tan(\theta_e)\right) \\
&\quad + 6 \sec(\theta_e) \left(-\frac{1}{2}\sin(\theta_e)\right)' \left(-p_e - q_e \tan(\theta_e)\right) \\
&\quad + 3 \cos(\theta_e)' \left(\frac{1}{2}q_e (\tan^2(\theta_e) + 1) \sec(\theta_e)\right) \\
&\quad + \cos(\theta_e) \left(\frac{1}{2}q_e (\tan^2(\theta_e) + 1) \sec(\theta_e)\right)' \\
&\quad + 3 \cos(\theta_e)'' \left(\frac{1}{2}\theta_e' \sec^2(\theta_e) (3 \tan^2(\theta_e) + \sec^2(\theta_e))\right) \\
&\quad + 6 \sec(\theta_e) (\theta_e' \tan(\theta_e)) \left(-\frac{1}{2}\sin(\theta_e)\right)'' \\
&\quad + 6 \left(-\frac{1}{2}\sin(\theta_e)\right)' \left((\theta_e' \tan(\theta_e))\right)^2 + \left(-\frac{1}{2}\sin(\theta_e)\right)^{(3)} (\sec(\theta_e))^2 \\
&\quad + \cos(\theta_e)^{(3)} \left(\frac{1}{2}\tan(\theta_e) \sec^2(\theta_e)\right) \\
\beta_{v9} &= 4\tan(\theta_e)[p_e + q_e \tan(\theta_e)] + q_e[1 + \tan^2(\theta_e)] - 3\frac{\tan(\theta_e)}{\cos(\theta_e)}[\theta_e'' \tan(\theta_e) + \theta_e'^2] \\
&\quad + \frac{3}{\cos(\theta_e)}[\theta_e'^2 \tan(\theta_e) - \theta_e''^2] - 6\theta_e'^2 \tan(\theta_e)
\end{align*}
$$

$$
\begin{align*}
\beta_{v10} &= -3 \sin(\theta_e) \\
\beta_{v11} &= 3 \sin(\theta_e) \tan(\theta_e) \\
\beta_{v12} &= -4 \theta_e' \cos(\theta_e) - 3 \sin^2(\theta_e) \\
\beta_{v13} &= \theta_e' \sin(\theta_e) \left[3 \frac{\sin^2(\theta_e)}{\cos(\theta_e)} - 4\right] \\
\beta_{v14} &= -\sin(\theta_e) \\
\beta_{v15} &= -\sin(\theta_e) \tan(\theta_e)
\end{align*}
\tag{C.1}
$$

We note: $\frac{\partial}{\partial s} = ()'$.

$$
\begin{align*}
\beta_{v1} &= [-p_e - q_e \tan(\theta_e)] \cos(\theta_e) - 3[\theta_e'' \tan(\theta_e) + \theta_e'^2] \\
\beta_{v2} &= 3\theta_e' \sin(\theta_e)[p_e + q_e \tan(\theta_e)] - \cos(\theta_e)[p_e' + q_e' \tan(\theta_e)] - q_e \frac{\theta_e'}{\cos(\theta_e)} \\
&\quad - 3\theta_e' \theta_e'' \tan^2(\theta_e) - 3\theta_e'^3 \tan(\theta_e) \\
\beta_{v3} &= -p_e \cos^2(\theta_e) \\
\beta_{v4} &= 2\theta_e' p_e \sin(\theta_e) \cos(\theta_e) - p_e' \cos^2(\theta_e) \\
\beta_{v5} &= -\cos(\theta_e) \sin(\theta_e) \\
\beta_{v6} &= \theta_e'(3 \sin^2(\theta_e) - \cos^2(\theta_e)) \\
\beta_{v7} &= -\frac{3\theta_e'}{\cos(\theta_e)} \\
\beta_{v8} &= \frac{3}{2} \left[ \frac{q_e'}{\cos(\theta_e)} + \theta_e' \tan(\theta_e) \right] \\
\beta_{v9} &= 4\tan(\theta_e)[p_e + q_e \tan(\theta_e)] + q_e[1 + \tan^2(\theta_e)] \\
&\quad - 3 \frac{\tan(\theta_e)}{\cos(\theta_e)} [\theta_e'' \tan(\theta_e) + \theta_e'^2] \\
&\quad + \frac{3}{\cos(\theta_e)} [\theta_e'^2 \tan(\theta_e) - \theta_e''^2] - 6\theta_e'^2 \tan(\theta_e) \\
\beta_{v10} &= -3 \sin(\theta_e) \\
\beta_{v11} &= 3 \sin(\theta_e) \tan(\theta_e) \\
\beta_{v12} &= -4 \theta_e' \cos(\theta_e) - 3 \sin^2(\theta_e) \\
\beta_{v13} &= \theta_e' \sin(\theta_e) \left[3 \frac{\sin^2(\theta_e)}{\cos(\theta_e)} - 4\right] \\
\beta_{v14} &= -\sin(\theta_e) \\
\beta_{v15} &= -\sin(\theta_e) \tan(\theta_e)
\end{align*}
\tag{C.1}
$$

## Data availability

No data was used for the research described in the article.

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