Asymptotical Mechanics of Thin-Walled Structures

Frontmatter

1. Asymptotic Approximations

Abstract

The series generated by a perturbation approach does not necessarily converge. Asymptotic methods use a mathematical apparatus of a somewhat peculiar nature — asymptotic series. They diverge but still approximate the functions in hand in a certain sense. Briefly, we can say that a convergent series represents a function at x = x ₀, n → ∞ (Fig. 1.1), while an asymptotic series is valid for n = n ₀, x → x ₀ (Fig. 1.2).

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

2. Regular Perturbations of Parameters

Abstract

A difference between real and idealized systems is very often reduced to perturbation of the input parameters. For instance, a thickness of a plate (or shell) is described via formula h = h ₀ + εh(x, y) (h ₀ = const, ε ≪ 1); contour of the circle plate slightly differs from a circle via relation r(θ) = r ₀ + ε cos nθ, etc. Although often the considered system does not follow Hook’s principle, but a difference is small. Non-linearity of many systems only slightly differs from linearity, and this system is said to be a quasi-linear one. The material of an object is weakly anisotropic, and so on. In all cited examples an influence of deviations (or perturbations) is small, and it can be estimated applying the method of regular perturbations. A being sought solution can be presented in the form of the following series

$$ f\left( {x,\varepsilon } \right) \sim \sum\limits_{n = 0}^\infty {{\delta _n}} \left( \varepsilon \right){f_n}\left( x \right) $$

, where δ _n (ε) is the asymptotic sequence depends upon the small parameter ε.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

3. Singular Perturbation Problems

Abstract

In this chapter the problems when the small parameter stands by a highest order derivatives are considered. Note that for ε = 0 a qualitative change of the system occurs since the system order of the analysed differential equation is decreased. The similar like asymptotics is called the singular one.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

4. Boundary Value Problems of Isotropic Cylindrical Shells

Abstract

We use as the governing equations of the general shells’ theory presented in reference [313].

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

5. Boundary Value Problems — Orthotropic Shells

Abstract

In this Chapter we consider a closed circle cylindrical shell supported in two principal directions. Supporting ribs are the one-dimensional elastic elements, situated uniformly with the same constant distance between them. The boundary value problems of the theory of closed circular cylindrical shells, eccentrically reinforced in the two principal directions, are investigated within the framework of the structurally orthotropic scheme. The supporting ribs are placed dense enough and we can homogenize their stiffness and mass characteristics. For the whole shell, the hypothesis about undeformable normal is valid. We assume that the ribs’ height is small in comparison with the curvature radius. There is no interaction between the two ribs lying in two directions.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

6. Composite Boundary Value Problems — Isotropic Shells

Abstract

The method of asymptotic analysis of the fundamental equation of the shells’ theory allows to reduce the problem to investigation of the limiting equations. These equations solve many practical problems analytically, but unfortunately they also posses some drawbacks. For different variation of the stress-strain state, we have to apply different approximate fromulas.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

7. Composite Boundary Value Problems — Orthotropic Shells

Abstract

The given in Chapter 5 simplified equations allow for a simple solution of a wide class of practically important problems. However, when a high number of the limiting simplified relations is needed, then the some problems with a practical application occur. Therefore, we propose a procedure to formulate the approximate equations guaranting the simplicity of the limiting equations of the asymptotic analysis. For every type of the shell’s support we include the terms playing the most important role for low and fast variations of the being searched solution.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

8. Averaging

Abstract

A wide spectrum of references has been devoted to the averaging method [180, 592, 706]. Although the roots of the method come from pioneering works of H. Poincaré and B. Van der Pol, the kernel has been developed by the works of N. Krylov and N.N. Bogolyubov. In general, the averaging method uses splitting of fast and slow solution components. Assume that a solution to a certain problem has the form shown in Figure 8.1.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

9. Continualization

Abstract

Although a change of a discrete medium by the continuous one can be considered as the particular case of the averaging method, it has the series of its own particularities. Therefore, the so called continualization process is further presented within the frame of the separated section.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

10. Homogenization

Abstract

In order to introduce to the problem we follow one dimensional example given in reference [154]:

$$\frac{d}{{dx}}\left[ {a(\frac{x}{\varepsilon })\frac{{du}}{{dx}}} \right] = q(x)$$

(10.1)

,

$$u = 0\,\,\,for\,\,\,x = 0,l$$

(10.2)

.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

11. Intermediate Asymptotics — Dynamical Edge Effect Method

Abstract

Bolotin [181, 182] proposed the asymptotic approach to estimate eigenfrequencies of oscillations of elastic systems, accuracy of which increases for higher number eigenfunctions. The key idea of the method is focused on a splitting of the input equations occupied domain in two groups: a) solution in interior zone of construction, and b) dynamical edge effect localized in a neighbourhood of boundaries or along the certain lines. Note that usually both dynamical edge effect as well as a solution within an interior zone are changed fastly with respect to spatial coordinates.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

12. Localization

Abstract

Consider a homogeneous linear chain with one inclusion, i.e. we assume that the elastic support number n = 0 possesses a stiffness which differs from other ones: γ _n = γ + ΔγS _0n (S _0n is the Cronecker symbol). The following equations govern oscillations of the considered chain

$${\mathop u\limits^{..} _n} - c\left( {{u_{n - 1}} - 2{u_n} + {u_{n + 1}}} \right) + \left( {\gamma + \Delta \gamma {\delta _{0n}}} \right){u_n} = 0$$

(12.1)

.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

13. Improvement of Perturbation Series

Abstract

The principal shortcoming of perturbation methods is the local nature of solutions based on them. Besides that, the following questions are very difficult for the theory: what values mayε be considered as small (large)? How can a solution for may ε be constructed if its behaviour is known for ε → 0 and ε → ∞? As the technique of asymptotic integration is well developed and widely used, such problems as elimination of the locality of expansion, evaluation of the convergence domain, construction of uniformly suitable solutions are very urgent.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

14. Matching of Limiting Asymptotic Expansions

Abstract

In many problems of mechanics the asymptotical series are very often obtained for various limiting values of the same parameter. An attempt to construct a uniformly suitable solution in whole interval of the parameter changes is not easy.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

15. Complex Variables in Nonlinear Dynamics

Abstract

When dealing with continualization of thin-walled structures with discrete reinforcing elements (see sections 10.3–10.5), a long wavelength asymptotics is usually implied. As this takes place, short wave length processes are excluded from consideration. Meantime, such processes can be important in many cases, especially for high frequency excitations. Similar situation was described in section 10.10 in application to a beam with concentrated masses and discrete supports.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

16. Other Asymptotical Approaches

Abstract

We consider pendulum oscillations with small mass governed by the equation

$$\varepsilon \ddot x + \dot x + x = 0,\varepsilon\ll 0$$

(16.1)

, with the following initial condditions

$$x(0) = 0,\dot x(0) = 1$$

(16.2)

.

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

Afterword

Abstract

The reader would probably ask the question: is it worth using asymptotic methods if computers have developed so much in recent time? Maybe it is simpler to program the input problem in full generality and solve it with universal numerical methods?

I. Andrianov, J. Awrejcewicz, L. I. Manevitch

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