Air Pump Model

MULTI-DOMAIN MODELING WHAT'S THE ISSUE? Why not just “write down the equations”? — standard formulations in different domains are often incompatible usually due to incompatible boundary conditions (choice of “inputs”) EXAMPLE: SIMPLE FLUID SYSTEM Scenario oscillatory motion in this direction coil pivot lever 60 Hz line voltage magnet flexible rubber bellows check valves flow in this direction A typical low-cost aquarium air-pump use “resonant” air pump design as a low-cost fuel pump does the pump really resonate? would it do so with fuel not air? Modeling goal The simplest model competent to portray resonance in this system. Assumptions • rigid-body motion (e.g., lever & pivot do not bend) • mass concentrated at c.g. of magnet • electro-magnet applies an oscillating force • small motion • mechanical elasticity only in the flexible bellows • no power dissipation except in the check valves (this assumption highlights any resonant behavior) • linear model of mechanical-to-fluid transduction • incompressible fluid • ideal check valves “Write down” the equations First define quantities: l1 m l2 y2 y1 A k Pcham ber Pl e f t Pr i g h t Mechanical Domain m ÿ2 := Fmagnet – Fspring – Fdissipation (symbol := denotes an assignment operator; right side is computed to evaluate left side.) l1 Fspring := k l y2 2 Fdissipation is due to fluid forces. Substituting: l1 l1 l1 m ÿ2 := Fmagnet – l k l y2 – l Fdissipation 2 2 2 This implies state equations: d dt y2 := ẏ 2 l1 l1 l1 ⎤ d 1 ⎡ ⎢ F – ẏ := k y – magnet 2 2 l2 Fdissipation⎥⎦ dt m⎣ l2 l2 Fluid Domain Left check valve: Qleft ∆Pleft := Pleft – Pchamber 0 if ∆Pleft ≤ ε ⎧ Qleft := ⎨(∆Pleft – ε) ⎩ Rfluid if ∆Pleft > ε Rfluid := g(∆Pleft) where ε is a small constant —flow is blocked until pressure is sufficient to overcome a preload Right check valve similar Couple the domains Relation: pressure = force times area Fdissipation := A Pchamber SNAG! 1 Pchamber is a fluid domain input, not output. ε ΔPleft Try manipulating the check-valve equations: Pchamber := Pleft – ∆Pleft But ∆Pleft ≠ f(Qleft) That is, the function ∆Pleft := f(Qleft) cannot be defined because the nonlinear function Qleft = g(∆Pleft) has no inverse. Try manipulating the mechanical equations: If Pchamber is a fluid domain input, Qchamber is a fluid domain output. 1 ẏ 1 := A Qchamber l2 ẏ 2 := l ẏ 1 1 d dt y2 := ẏ 2) (a differential equation for state variable y2 as before). Qchamber := Qleft – Qright Qleft := g(Pleft – Pchamber) Qright := g(Pchamber – Pright) 1 Pchamber := A Fdissipation l2 l1 l2 Fdissipation := l Fmagnet – k l y2 – l m ÿ2 1 2 1 But NOTE that ẏ 2 is not a state variable as before. Incompetent model! In fact, this is a first-order system of equations and hence is incapable of portraying resonance in the system. “Pseudo-code” for the sorted equations are as follows. d dt y2 := ẏ 2 l2 ẏ 2 := l ẏ 1 1 1 ẏ 1 := A Qchamber Qchamber := Qleft – Qright Qleft := IF Qleft := IF ∆Pleft > ε THEN (∆Pleft – ε)/Rfluid ELSE 0 Qright := IF ∆Pright > ε THEN (∆Pright – ε)/Rfluid ELSE 0 ∆Pleft := Pleft – Pchamber ∆Pright := Pchamber – Pright 1 Pchamber := A Fdissipation l2 l1 l2 Fdissipation := l Fmagnet – k l y2 – l m ÿ2 1 2 1 Note that only one constant of integration (i.e., initial condition) is required for the solution. t y2(t) := ⌠ ⌡ ẏ2 dt + y2(0) 0 Note also that a derivative operation is required. d ÿ2(t) := dt ẏ 2(t) DISCUSSION The model is well formulated The wisdom of the stated assumptions might be debated, but they are entirely consistent with the stated goal, to formulate the simplest model competent to portray resonant behavior. — the standard equations are not The “standard equations” for rigid-body mechanical motion implicitly assume that “force causes motion” — i.e., force is always an input to mass elements and their resulting motion is the output. But Newtonian physics only specifies a relation between force and acceleration, not an assignment of input and output. Similarly, the “standard equations” in the fluid domain assume that pressure gradient cause flows, but physics only specifies a relation. — the problem is due to boundary conditions The standard form for the mechanical sub-system in incompatible with the assumed fluid sub-system. “Just writing down” the equations of motion usually works for a single domain but often fails for multi-domain systems. “Standard” formulations in different domains implicitly assume specific “boundary conditions” (i.e., a particular choice of “inputs” and “outputs”). Unfortunately, these boundary conditions are often incompatible—as above. Network modeling approach Analysis of the relations between model domains prior to equation derivation reveals incompatibilities. clarifies boundary conditions identifies origins of conflicts defers writing equations until needed