**3.6. Lagrange equations**

In the dynamic analysis (while finding the frequency of natural vibrations [55]), the independent non-dimensional displacement and the load factor become a function dependent on time, and dynamic terms were added to equations describing postbuckling equilibrium path. Neglecting the forces associated with the inertia terms of prebuckling state and the second-order approximations, and taking into account the orthogonality conditions for the displacement field in the first ( )*<sup>i</sup> U* and second-order approximation ( ) *ij U* , the Lagrange equations can be written as [56]:

$$\frac{1}{2\alpha\_s^2}\ddot{\xi}\_s + \left(1 - \frac{\lambda}{\lambda\_s}\right)\ddot{\xi}\_s + a\_{ijs}\xi\_i\dot{\xi}\_j + b\_{ijks}\xi\_i\xi\_j\xi\_k = \xi\_s^\*\frac{\lambda}{\lambda\_s};\ \left(s = 1, 2, \dots, N\right) \tag{38}$$

where s is a natural frequency with mode corresponding to buckling mode; a*ijs* and *bijks* are the coefficients (34) describing the postbuckling behaviour of the structure (independent of time); however the parameters of load and the displacement are the functions of time *t*.

For the uncoupled buckling, i.e. the single-mode buckling (where index *s* = *N* = 1), the equations of motion may be written in the form:

$$\frac{1}{\alpha\_1^2} \ddot{\xi}\_1 + \left(1 - \frac{\lambda}{\lambda\_1}\right) \xi\_1 + a\_{111} \xi\_1^2 + b\_{1111} \xi\_1^3 = \xi\_1^\* \frac{\lambda}{\lambda\_1};\tag{39}$$

It is assumed that in the initial moment of time *t* = 0 the non-dimensional displacement , as well as the velocity of displacement are equal to zero, i.e.:

$$
\not\subset(t=0) = 0 \quad \text{and} \quad \dot{\not\subset}(t=0) = 0 \quad . \tag{40}
$$

The Runge-Kutta method [57] for solving the equation (39) requires the following substitutions:

$$\begin{aligned} \dot{\tilde{\xi}} &= \Gamma(t)\_{\prime} \\ \dot{\tilde{\Gamma}} &= -\alpha\_1^2 \left( 1 - \frac{\lambda(t)}{\lambda\_1} \right) \tilde{\xi} - \alpha\_1^2 b\_{111} \tilde{\xi}\_1^2 - \alpha\_1^2 b\_{1111} \tilde{\xi}\_1^3 + \alpha\_1^2 \frac{\lambda(t)}{\lambda\_1} \tilde{\xi}^\*, \end{aligned} \tag{41}$$

which lead to the system of two differential equations. ''Complete'' equations of motion (41) are solved with the numerical Runge–Kutta method of order 8 (5,3), thanks to Dormand and Price (with step-size control and density output).
