Evolution of Motions of a Rigid Body About its Center of Mass by Felix L. Chernousko Leonid D. Akulenko & Dmytro D. Leshchenko

Author:Felix L. Chernousko, Leonid D. Akulenko & Dmytro D. Leshchenko
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5.86)

The first equality in (5.83) assumes the form:

(5.87)

Differentiating equality (5.87), we conclude that the k-th derivative ω (k) will be, up to the terms of the order of ε 2, a homogeneous polynomial of degree k + 1 of the components of vector ω, where k = 0 , 1 , ….

In the first example (R 0 ≡ 0), we conclude from the first equality (5.85) that are homogeneous polynomials of ω of the second degree having the order of m 0 lω 2. Here, m 0 is the characteristic mass of points P i , and l is the characteristic linear dimension of the order of ρ i .

Then it follows from (5.80) that q is the sum of homogeneous polynomials of ω of the second and third degrees having the order of m 0 lc −1 ω 2 and m 0 lc −2 bω 3, respectively. Here, c is the characteristic stiffness of elastic couplings (of the order of elements of matrix C), and b is the characteristic dissipation coefficient (of the order of elements of matrix B). The vectors r i , i = 1 ,  …  , N, defined by formula (5.54), have the same structure. Note that, according to (5.87), each differ entiation increases the degrees of polynomials of ω by one. Therefore, vector μ from (5.85) for R 0 ≡ 0 is the sum of homogeneous polynomials of the fourth and fifth degrees of components ω j of vector ω, namely:

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