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me375_fa10_hwk09_soln

# me375_fa10_hwk09_soln - Problem 7.22 For the hydraulic...

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Unformatted text preview: Problem 7.22 For the hydraulic problem shown below, derive the dynamical equations for the heights h 1 and h 2 . Element (a) – accumulator: dp 1 dt = 1 C 1 q i- q 1 ( ) where C 1 = A 1 ρ g = A ρ g and q i = q mi / ρ (1) p 1 = p a + ρ gh 1 hydrostatic pressure ( ) (2) Element (b) - flow resistance: p 1- p a = R 1 q 1 ⇒ q 1 = p 1- p a R 1 (3) Combining (1)-(3): ρ g dh 1 dt = 1 C 1 q i- ρ g R 1 h 1 ⇒ dh 1 dt = 1 ρ gC 1 q i- 1 C 1 R 1 h 1 dh 1 dt = 1 ρ A 1 q mi- ρ g A 1 R 1 h 1 (4) p 2 q 2 p a p a q 2 ( c ) ( d ) p 1 q 1 p a q 1 p a ( a ) ( b ) Element (c) – accumulator: dp 2 dt = 1 C 2 q 1- q 2 ( ) where C 2 = A 2 ρ g = 4 A ρ g (5) p 2 = p a + ρ gh 2 hydrostatic pressure ( ) (6) Element (d) - flow resistance: p 2- p a = R 2 q 2 ⇒ q 2 = p 2- p a R 2 (7) Combining (2), (3) and (4)-(6): ρ g dh 2 dt = 1 C 2 ρ g R 1 h 1- ρ g R 2 h 2 ⇒ dh 2 dt = 1 C 2 R 1 h 1- 1 C 2 R 2 h 2 dh 2 dt = ρ g A 2 R 1 h 1- ρ g A 2 R 2 h 2 (8) Problem 7.23 Problem 7....
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me375_fa10_hwk09_soln - Problem 7.22 For the hydraulic...

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