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Bolotin did not give an exact definition of the concept of quasi-separation of variables. Intuitively, this means that the difference between solutions of boundary value problems with separated and quasi-separated variables is sufficient only near the boundaries. In other words, the energy accumulated in the EE zone is small compared to that accumulated in the inner zone. This allows us to not take into account DEE when expanding the natural mode of vibration during the calculation of forced oscillations. Bolotin’s conception of quasi-separation of variables (Bolotin 1961c, 1984) can be used in the theory of normal modes of nonlinear oscillations for continuous systems.

When studying linear oscillatory systems with a finite number of DOF, normal oscillation modes play a key role. Kauderer (1958) indicated the existence of solutions in a nonlinear system, which were, in a sense, similar to the normal modes of linear systems. He called these solutions the principal ones and showed how to construct their trajectories in the configuration space. Rosenberg (1962) defined normal vibrations of nonlinear systems with a finite number of DOF, formulated the problem in the configuration space and found several classes of nonlinear systems that allowed solutions with straight-line trajectories (for details, see Mikhlin and Avramov 2011; Avramov and Mikhlin 2013). Generalizations of this concept to continuous systems are related to the exact separation of spatial and time variables (Wah 1964; Avramov and Mikhlin 2013), i.e. to the possibility of representing the sought solutions in the form

The restriction of this approach is clear since the separation of variables only works for some boundary conditions. Based on Bolotin’s conception of the quasi-separation of variables, we can propose the following definition (Andrianov 2008): a function U(x,t) is called the normal mode of nonlinear oscillations of a continuous system if

where T(t) and Y(x, t) are the periodic and quasi-periodic functions in time, respectively; and function Y(x, t) is small compared to function X(x)T(t) in some energy norm. The last condition can be verified both a priori and a posteriori.