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 momentums, handled as regulate. This paintings might be of curiosity to execs and researchers at universities and laboratories focusing on difficulties of regulate for hybrid platforms and aerospace/mechanical engineering, in addition to to under-/postgraduates with this specialization"-- Read more...
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Extra info for Control of fluid-containing rotating rigid bodies
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.