CAUSAL ANALYSISExample: Aquarium Air Pump Model simplification was based on three proposalsClaim #1The magnet-lever-bellows combination should resonate near 60 HzClaim #2Only magnet mass and bellows compliance are needed to portray resonanceClaim #3valve flow resistance is the only significant energy lossTask: evaluate these proposals AssumptionsElectrical sub-systemMechanical sub-systemFluid sub-system Fluid sub-assemblyCausal analysis Nonlinear check-valve resistors are causally constrained—flow out, effort in Mechanical & fluid sub-assemblytwo energy storage elementsno dissipation other than check valvesconsistent with claim #2 Causal analysismagnet inertia is a dependent storage elementonly one independent energy storage element — bellows complianceonly one state variablemodel cannot describe resonant oscillationThis is the problem we discovered previously—revealed through causal analysis—no need to develop equations Causal pathCasual path identifies which model element(s) determine magnet velocity—check-valve flow ratesIN PHYSICAL TERMS:Unlikely!—due to the assumption that air is incompressible.REJECT CLAIM #2Revise the model Causal analysisthree independent energy storage elements—magnet inertia, bellows compliance, air compressibilitymodel may describe resonant oscillationthird order system — three state variables Electro-mechanical-fluid system Causal analysismagnet inertia is again a dependent storage elementtwo independent energy storage elements— bellows compliance & air compressibilitybut the model cannot describe resonant oscillation Causal pathmagnet velocity depends on source voltagesolution: add an element between these twoone obvious candidate: the electrical resistance of the coil Reject Claim #3Revised modelCausal analysisthree independent energy storage elementsmodel may describe resonant oscillation CommentA bigger dissipator (resistor) usually means less oscillation—not always true.In this case:A smaller resistance means less oscillation.STATE AND OUTPUT EQUATIONSFirst choose state variables. Three independent energy storage elements, —three independent state variables.They define the energetic state of the system. Best choice depends on the particular modelIn this case, most elements are linear, so circuit variables may be appropriate.—a.k.a. power variables: efforts of independent capacitors, flows of independent inertias.In a nonlinear system, energy variables are usually a better choice.— displacements of independent capacitors, momenta of independent inertias. Usually easiest to visualize position & velocity of mechanical systems.spring displacement rather than force,magnet velocity rather than momentum. A reasonable choice:magnet velocity, vmagnetbellows displacement, xbellowsvolume of air that has flowed into the chamber, VairState equationsState equations may be written by inspection from the graph. Note that it is not necessary to assemble the equations by substitution. For numerical predictions, the un-assembled form of the equations makes for easier-reading computer code. Output equationsCAUSAL ANALYSIS Things should be made as simple as possible — but no simpler. Albert Einstein How simple is "as simple as possible"? Causal assignment provides considerable insight. EXAMPLE: AQUARIUM AIR PUMP coil60 Hz line voltageoscillatory motion in this directionmagnetleverpivotflow in this directionflexible rubber bellowscheck valves Schematic of a low-cost aquarium air pump. • Reciprocating pump • Electromagnet driven by 60 Hz line voltage • Drives a small permanent magnet attached to a lever • Lever drives a flexible rubber bellows on a chamber • Check valves prevent flow in one direction • Small flow resistance in the other Mod. Sim. Dyn. Sys. Air pump revisited page 1MODEL SIMPLIFICATION WAS BASED ON THREE PROPOSALS CLAIM #1 The magnet-lever-bellows combination should resonate near 60 Hz —to maximize oscillation amplitude and output flow rate CLAIM #2 Only magnet mass and bellows compliance are needed to portray resonance (other energy-storage effects are small by comparison) CLAIM #3 valve flow resistance is the only significant energy loss (other energy-dissipation effects are small by comparison) TASK: EVALUATE THESE PROPOSALS Mod. Sim. Dyn. Sys. Air pump revisited page 2ASSUMPTIONS Electrical sub-system AC power line — voltage source (the pump’s power load is unlikely to significantly affect line voltage) Electromagnetic coil — ideal gyrator By claims #2 & #3 all other electrical and magnetic effects are negligible. Mechanical sub-system Magnet — ideal translational inertia Its velocity is the same as that of the leftmost end of the lever — one junction Lever — ideal transformer relating two translational domains Bellows compliance — ideal translational spring (capacitor) (by claim #2) Fluid sub-system Check valves — ideal nonlinear resistors Chamber pressure proportional to bellows force — ideal transformer between translational and fluid domains Deflection of bellows compliance is the motion that pumps air — one junction Mod. Sim. Dyn. Sys. Air pump revisited page 3FLUID SUB-ASSEMBLY 110000R0RPininput from bellowsinΔPΔPoutoutPPcham ber::outQinQ Bond graph fragment for check valves and chamber CAUSAL ANALYSIS Check valve must have flow rate output for pressure input. Qvalve = ⎩⎪⎨⎪⎧0 if ∆Pvalve ≤ ε(∆Pvalve – ε)Rvalve if ∆Pvalve > ε QleftΔPleftRfluid1ε A well-defined function —with no definable inverse Mod. Sim. Dyn. Sys. Air pump revisited page 4Nonlinear check-valve resistors are causally constrained —flow out, effort in These constraints imply that the boundary conditions at the airflow inlet and outlet and the power input from the bellows must be efforts (pressures). 110000R0RPininput from bellowsinΔPΔPoutoutPPcham ber::outQinQ Bond graph fragment for check valves and chamber with causal constraints Mod. Sim. Dyn. Sys. Air pump revisited page 5MECHANICAL & FLUID SUB-ASSEMBLY (temporarily ignore the electromagnetic coil) ITF11100R0RC1TFSeSe:mmagnetPininΔPΔPoutoutPPcham ber::outQinQ:1/kvbello w s:Abml/lv two energy storage elements no dissipation other than check valves consistent with claim #2 Mod. Sim. Dyn. Sys. Air pump revisited page 6CAUSAL ANALYSIS ITF11100R0RC1TFSeSe:mmagnetPininΔPΔPoutoutPPcham ber::outQinQ:1/kvbello w s:Abml/lv magnet inertia is a dependent storage element only one independent energy