Mathematical formulation and numerical solutions of continual damaged deformable bodies are discussed for different external factors simulation. Pipes damaged due to creep and stress corrosion cracking is considered as example.

A problem of obtaining Rosenberg nonlinear normal modes for piecewise linear systems is considered in the paper. Numerical methods for computation of such modes are proposed. Efficiency of those methods is demonstrated using a model of torsion oscillations of a power transmission in a triplex transport engine.

The procedure of originals construction of the first kind Bessel functions with rooted imaginary argument has been presented. The necessity of these transformants inversion occurs at solution of thin round plates deformation non-stationary problems.

The construction which considered from a strip and an elastic half plane has been considered. The compressing load has been applied to the upper boundary of the strip. The method of the determination of contact zone and contact stresses in the strip which layered on an elastic half-plane has been proposed in the article. Fourier's transformation has been used. The singular integral equation of the problem has been obtained. The influence of the thickness and the coefficient of elasticity on the contact zone has researched partly.

The spaces $V_n$ of finite linear combinations of shifts of infinitely differentiable functions $v_n(x)$ are considered. It is proved, that $V_n\subset V_{n+1}$. It is proved, that there exist compactly supported wavelet-function $w_n(x)$, such that the system of shifts of this function forms an algebraic basis of wavelet space $W_n=\{\varphi\in V_n:\ \varphi\perp V_{n-1}\}$.