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Demonstrate the main idea of DEEM on a spatially 1D problem, which can be reduced to a transcendental equation and solved numerically with any degree of accuracy (Weaver et al. 1990). Consider the natural oscillations of a beam of length L, described by the following PDE:

[1.1]

Here, w is the normal displacement, E is the Young modulus, F is the cross-sectional area of the beam, I is the axial inertia moment of the beam cross-section, and ρ is the density of the beam material.

Let us compare two versions of boundary conditions:

[1.2]

[1.3]

We use the following ansatz:

where ω is the eigenfrequency and W (x) is the eigenfunction.

The equation for eigenfunction W (x) has the form

[1.4]

The solution of the eigenvalue problem [1.4], [1.2] is given by

[1.5]

[1.6]

The eigenvalue problem [1.4], [1.3] does not allow separation of variables. However, if the eigenfunction oscillates rapidly along x (i.e. a rather high form of oscillations is considered), then we can hope that in this case a solution of the form [1.5] is also valid for the inner domain sufficiently distant from the boundaries (Figure 1.1). Such an expression does not satisfy the boundary conditions. However, if the solution that compensates the residuals at the boundary conditions and decays rapidly, then the approximate expressions for the eigenfunctions and eigenfrequencies can be obtained.

Figure 1.1. Curve 1 corresponds to the rapidly oscillating solution in the inner domain, and curve 2 corresponds to the sum of DEE and the rapidly oscillating solution

Let us suppose the solution of equation [1.4] in the form

[1.7]

The oscillation frequency ω is

[1.8]

Factorization of ODE [1.4] is (Vakhromeev and Kornev 1972)

[1.9]

The general solution of ODE [1.9] is given by

where functions W₀ and W_1,2 are the general solutions of the following equations:

[1.10]

[1.11]

For large frequencies (aω ≫1), the following estimates for the derivatives of functions W₀ and W_1,2 are obtained:

The behavior of these solution components is different: W₁ is the rapidly oscillating function, and W₂ is the sum of exponentials rapidly decreasing from the edges of the beam.

Therefore, the situation under consideration is fundamentally different from the case when the characteristic equation has small and large modulo roots, which is typical for boundary layer theory. In our case, we are talking about the separation of solutions, one of which oscillates at the same rate as the EE decays (i.e. the characteristic equation has large real and imaginary roots with moduli of the same order). The self-adjoint eigenvalue problem [1.1], [1.2] can be referred to as the boundary value problem with quasi-separable variables (Bolotin 1960a, 1960b, 1961a, 1961b, 1961c; Bolotin et al. 1950, 1961).

We proceed to the construction of the EE described by equation [1.11]. Taking into account the expression for the natural frequency [1.8], we obtain the following relations for EEs localized in the vicinity of the edges x = 0 and x = L, respectively:

[1.12]

[1.13]

We assume that the beam is so long such that EEs do not affect each other, i.e. exp(−πλ^-1L) ≪ 1.

Now, it remains to find the quantities x₀, λ and constants C₁, C₂ from the boundary conditions to determine the eigenmodes and eigenfrequencies:

[1.14]

[1.15]

Note that in original works by Bolotin, a slightly different matching procedure is used. Namely, the matching conditions [1.14], [1.15] are not given at the domain boundaries. They are set at some points, which are then determined from the solution of the system of transcendental equations along with other unknown constants.

Substituting expressions [1.7] and [1.12] into conditions [1.14], and expressions [1.7] and [1.13] into [1.15], we obtain

[1.16]

[1.17]

For λ and x₀, we have the following expressions:

Finally, the formula for the natural oscillation frequencies of the clamped beam is written as follows:

[1.18]

The same formula follows from the analysis of the transcendental equation (Weaver et al. 1990). Its difference from the numerical solution is less than 1% even for the first natural oscillation frequency.

Using DEEM to solve various problems proved its high accuracy even when calculating the first eigenfrequencies (Bolotin 1960a, 1960b, 1961a, 1961b, 1961c, 1961d; Gavrilov 1961a, 1961b; Bolotin 1963, 1970; Elishakoff and Wiener 1976; Elishakoff and Steinberg 1979; Emmerling 1979; Bolotin 1984; Elishakoff et al. 1993, 1994). It confirms “Crighton’s principle” (Crighton 1994):

All experience suggests that asymptotic solutions are useful numerically far beyond their nominal range of validity, and can often be used directly, at least at a preliminary product design stage, for example, saving the need for accurate computation until the final design stage where many variables have been restricted to narrow ranges.

The problem of natural oscillations was considered above. In the case of forced oscillations, the response is given as a series expansion in terms of normal modes of natural oscillations. It is important that one can use only the rapidly oscillating part of solution W₀, neglecting the dynamic edge effect (DEE) (Bolotin 1961b, 1961c, 1970, 1984).