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# 23 - Pliny’s fountain The spherical pendulum...

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Unformatted text preview: Pliny’s fountain The spherical pendulum Configuration of rods in a plane More Examples Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) More Examples MATH 2070U 1 / 23 Pliny’s fountain The spherical pendulum Configuration of rods in a plane More Examples 1 Pliny’s intermittent fountain: an ODE with a boolean switch 2 The spherical pendulum 3 Configuration of rods in a plane c D. Aruliah (UOIT) More Examples MATH 2070U 2 / 23 Pliny’s fountain The spherical pendulum Configuration of rods in a plane Pliny’s intermittent fountain y = y ( t ) = height of water inside tank Siphon switches on when y ≥ y high Siphon switches off when y ≤ y low Torricelli’s law determines rate of outflow: Q out = C π r 2 p 2 gy c D. Aruliah (UOIT) More Examples MATH 2070U 4 / 23 Pliny’s fountain The spherical pendulum Configuration of rods in a plane Pliny’s intermittent fountain: model Understand the problem : model the fountain for 0 ≤ t ≤ 100 Goal: to compute height y = y ( t ) for all t values from 0 s to 100 s starting from y ( ) = Fixed parameters we know: y low = 0.025 m, R T = tank radius = 0.05 m, y high = 0.1 m, r = siphon radius = 0.007 m, g = 9.81 m · s- 2 , Q in = 50 × 10- 6 m 3 · s- 1 , C = 0.6 Basic conservation principle: dy dt = Q in- Q out π R 2 T Q out switches on or off at certain events c D. Aruliah (UOIT) More Examples MATH 2070U 5 / 23 Pliny’s fountain The spherical pendulum Configuration of rods in a plane Pliny’s intermittent fountain: model Final model is dy dt = Q in- [ siphon ] C π R 2 T p 2 gy π R 2 T y ( ) = where [siphon] ∈ { 0, 1 } (boolean/logical variable) Right-hand side is discontinuous due to [siphon] (This IVP actually can be solved analytically) c D. Aruliah (UOIT) More Examples MATH 2070U 6 / 23 Pliny’s fountain The spherical pendulum Configuration of rods in a plane Implementation from Chapra Plinyode.m function dy = Plinyode(t,y) global siphon % Boolean variable for state of siphon % Fix all the constant parameters for this problem Rt = 0.05; r = 0.007; yhi = 0.1; ylo = 0.025; C = 0.6; g = 9.81; Qin = 0.00005; if y(1) <= ylo siphon = 0; % Switch off when level too low elseif y(1) >= yhi siphon = 1; % Switch on when level too high end Qout = siphon * C*pi*r^2 * sqrt(2*g*y(1)); dy = (Qin-Qout) / (pi*Rt^2); end c D. Aruliah (UOIT) More Examples MATH 2070U 7 / 23 Pliny’s fountain...
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23 - Pliny’s fountain The spherical pendulum...

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