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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 = x0, n → ∞ (Fig. 1.1), while an asymptotic series is valid for...
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 = h0 + εh(x, y) (h0 = const, ε ≪ 1); contour of the circle plate slightly differs from a circle via relation r(θ) = r0 + ε cos nθ, etc. Although often the considered system does not...
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.
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...
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.
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...
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...
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.
In order to introduce to the problem we follow one dimensional example given in reference [154]: (10.1) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca...
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...
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: γn = γ + ΔγS0n (S0n is the Cronecker symbol). The following equations govern oscillations of the considered chain (12.1) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn...
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...
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.
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...
We consider pendulum oscillations with small mass governed by the equation (16.1) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMabm % iEayaadaGaey4kaSIabmiEayaacaGaey4kaSIaamiEaiabg2da9iaa...
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?
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