The paper by Pevzner et al. (2000) claims that this modification is more straightforward and computationally faster, and the mode shapes derived are valid on a larger part of the plate.
1.6. Quasi-separation of variables and normal modes of nonlinear oscillations of continuous systems
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.
1.7. Short-wave (high-frequency) asymptotics. Possible generalizations of DEEM
DEEM can be considered a special case of short-wave (high-frequency) asymptotics. The corresponding algorithms are known as the method of geometric optics, the ray method, the semi-classical approximation, the WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) approach, the method of edge waves, the Keller–Rubinow method, etc. (Keller and Rubinow 1960; Maslov and Fedoryuk 1981; Babich et al. 1985; Babich and Buldyrev 1991; Chen et al. 1991, 1992; Chen and Zhou 1993; Bauer et al. 2015). They were independently developed in various fields of mathematics, mechanics and physics.
Note an interesting fact: Ufimtsev proposed the asymptotic method of edge waves (Ufimtsev 1962, 2003, 2014). According to Rich and Janos (1994) and Mitzner (2003), this theory played a critical role in the design of American stealth aircrafts F-117 and B-2. It is a fascinating example of the direct application of asymptotic formulas in engineering practice!
The key to short-wave asymptotics is the ansatz φ(x)exp(iε−1S (x)), in the nonlinear case – φ(x)Φ(iε−1S (x)), where
Let us show the generalization of DEEM based on the WKBJ approach using the toy problem – natural oscillation of a beam of variable cross-section (Bauer et al. 2015):
[1.55]
with clamped edges.
In dimensionless variables, we obtain
where
Functions φ1,φ2 are supposed to be smooth enough to avoid turning points.
The solution to equation [1.56] is sought in the form:
Substituting ansatz [1.58] into equation [1.56] and applying ε-splitting, we obtain a recurrent system of equations:
The eikonal equation [1.59] has the following solutions:
From the transport equation [1.60], we obtain
In the first approximation, the general solution of equation [1.56] can be written as follows:
In expression [1.61], function u0(ξ) is “frozen” at either end of the interval (i.e.
