Control of fluid-containing rotating rigid bodies by Anatoly A Gurchenkov

By Anatoly A Gurchenkov

"This ebook is dedicated to the examine of the dynamics of rotating our bodies with cavities containing liquid. easy sessions of movement are analyzed: rotation and libration. situations of entire and partial filling of cavities with excellent liquid and entire filling with viscous liquid are handled. a mode is gifted for acquiring relatives among angular velocities perpendicular to major rotation and exterior force Read more...

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The index n ranges over all combinations of numbers of the longitudinal and transverse harmonics l and p. The quantity ξlp is the pth root of the equation J1 (ξlp ) + iσlp J1 (ξlp ) · Y1 (ξlp r0 ) + Klp Y1 (ξlp r0 ) − J1 (ξlp r0 ) + Klp J1 (ξlp r0 ) · Y1 (ξlp ) + iσlp Y1 (ξlp ) = 0, Klp = 3 2 iσlp − 4(1 + σlp ) 2 r0 σlp . 7, respectively. Similarly, in Figs. 5, the graphs of χ0p are shown. 7. 8. 7. 49 50 Control of fluid-containing rotating rigid bodies are orthogonal in the domain Q, that is, ∗ (un , um )dQ = − Q 1 r0 ∗ ϕn ϕm d =0 for n = m.

Consider the scalar function (the Bellman function) f (x, t) = f (x1 , . . , xn , t) ⎡ ⎛ T = min ⎣g ⎝x + ⎞ T K(τ)M(τ)dτ ⎠ + γ M(t)∈U t ⎤ F(M(τ))dτ ⎦. 3) t We shall use the time grid with points ti = i∆ and ti+1 = ti + i∆ on the interval [0, T]; the set of i is determined by the value of the time interval under 32 Control of fluid-containing rotating rigid bodies consideration. 3) satisfies the recursion relation ⎡ ⎛ ⎞ T T f (x, ti ) = min ⎣g ⎝x + K(τ)M(τ)dτ ⎠ + γ M(t)∈U ti ⎡ ⎛ T K(τ)M(τ)dτ + M(t)∈U T ⎢ ⎜ = min ⎣g ⎝x + M(t)∈U T ⎞ ⎤ ⎥ F(M(τ))dτ ⎦ ti ⎟ K(τ)M(τ)dτ + K(ti )M(ti )∆⎠ ti +∆ ti +∆ +γ ⎞ ⎟ K(τ)M(τ)dτ ⎠ + γ ti +∆ ti ⎡ ⎛ F(M(τ))dτ ⎦ ti ti +∆ ⎢ ⎜ = min ⎣g ⎝x + ⎤ T F(M(τ))dτ + γ ⎤ ⎥ F(M(τ))dτ ⎦ ti +∆ ti = min [γF(M(ti )) + f (x + K(ti )M(ti )∆, ti + ∆)], M(t)∈U or f (x, ti ) = min [γF(M(ti )) + f (x + K(ti )M(ti )∆, ti + ∆)].

On . 1) Here S denotes the wetted surface of the cavity walls, is the free surface of the fluid, v is the relative velocity of fluid particles in the system Ox1 x2 x3 , ρ is the density of the fluid, and U is the potential of the mass forces (for the gravitational force, we have U = −gx3 ). The equation (x1 , x2 , x3 , t) = 0 described the free surface of the fluid. 1) are satisfied if v = 0, P=ρ U+ 1 (ω × r)2 + C0 + P0 , 2 1 2 2 2 0 (x1 , x2 , x3 ) = −U − ω0 x1 + x2 − C0 . 3) Control of a rotating rigid body containing a fluid with free surface 41 where D is the volume bounded by the surfaces of the cavity walls and by the free surface 0 = 0.

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