By Ilya Blekhman
Vibrational mechanics is a brand new, intensively constructing portion of nonlinear dynamics and of the idea of nonlinear oscillations. It provides a normal method of the learn of the results of vibration on nonlinear platforms. This technique is characterised through simplicity of software and via actual clearness. lately a few new, crucial effects were acquired either at the improvement of the mathematical gear of vibrational mechanics and at the resolution of definite utilized difficulties. This paintings goals to mirror these effects in the course of the presentation of the authors - recognized scientists from Germany, Denmark and Russia. For the benefit of readers, the most content material is preceded by means of a short description of the most theses of vibrational mechanics.
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Additional resources for Selected Topics in Vibrational Mechanics (Series on Stability, Vibration and Control of Systems: Series A, Vol. 11)
Sample text
31) are being considered. Then in the first 24 2. Mathematical Apparatus. 32) at the fixed (^'frozen") X and t. 35) satisfy the conditions of Bogolyubov's second theorem, then the asymptotically stable positions of the equilibrium X = 0, X = X*of the system, defined by Eq. 29). 34) for the vibrational force V should be understood as the product of the matrix 3 0 j /dX multiplied by the vector Yj . We will also note that another approach to the substantiation of the presented approximate method in the special case under consideration was suggested by Kirghetov on the basis of the concept of u-stability introduced by him [16].
In other words, a prognosis is made about the fulfillment of the main assumption of vibrational mechanics. 2. 8) is composed, based on the initial equations of motion. 3. e. the supposition about the n in the relation Y ~ e " X , and the values of the parameters and of groups of summands are estimated, and large and small parameters and groups of summands are selected. On account of that supposition and those estimations, due simplifications are made in the equations. 11. Summary 33 4. Xand t are sought as a rule approximately.
11 Summary: On the Procedure of the Practical Use of the Method The use of any method, including even the strict mathematical method or result, when solving an applied problem comprises, as a rule, a number of stages which do not belong to those mathematically strict, but can be referred to the methods realized at the so-called rational level of rigour. This circumstance and some other specific features of applied mathematical research are discussed in detail in the book [5]. This certainly refers to the method of direct separation of motions as well.