Synthesis of adequate mathematical description of the main mechanical line of rolling mill are considered. The four-mass model with weightless elastic connections is chosen as mathematical model of dynamic system of the main mechanical line of the rolling mill. The problem was reduced to solution of integral equation of the first kind (to unsteady problem). The methods of obtaining of the steady solutions are suggested. Synthesis of the adequate mathematical descriptions with unitary model of external load are suggested. The metal rolling was executed with using of real experimental measurements as an example.

In this paper, the problem of identification of the characteristics of the rotor unbalance on two supports is investigated as the inverse problem of measurement. The vibration of rotor supports in two mutually perpendicular directions used as the initial information. The inverse problem is considered, taking into account the error of the mathematical description of rotor-bearings system. To obtain estimates of real unbalance characteristics, the hypothesis as to the exact solutions is applied. The method of Tikhonov regularization is used to obtain stable results. Test calculations are given to illustrate the proposed approach.

In this chapter, adequacy estimation criteria for mathematical descriptions in the form of ordinary differential equations were proposed. Adequate mathematical descriptions can increase the objectivity of the results of mathematical modeling for future use. These descriptions make it possibly reasonable to use the results of mathematical modeling to optimize and predict the behavior of physical processes. Interrelations between criteria are considered. The proposed criteria are easily transferred on mathematical descriptions in algebraic form.

We consider the problem of determining the center of mass of an unknown gravitational body, using the disturbances in the motion of observed celestial bodies. In this paper an universal approach to obtain the approximate and stable estimate of problem solution is suggested. This approach can be used in other fields of Science. For example, it can be applied for investigation of interactions between fields of forces and elementary particles using known trajectories of elementary particles motions.

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