Bond Graph Representation of a Motor-Propeller Assembly (with the Bond-Graph)

A propeller mounted on a dc motor, connected to a Li-Po battery.

 

Se	Source of effort; voltage supplied by the power source; a Li-Po battery in this study.
Rw	Motor winding resistance.
Iw	Motor winding inductance.
GY	dc Motor; a Gyrator.
T	Gyrator Modulus.
bs	Bearing resistance on the shaft.
Js	JS=Js’+Jp+Jr.
Js	Total inertia for the shaft.
Js’	Inertia of the shaft.
Jp	Inertia of the propeller.
Jr	Inertia for dc motor rotor.
Ks	Stiffness of the shaft.
TF	A Transformer
m	Distance from the center of mass of the chassis to the shaft; moment arm to transform torque in to force to be applied to the chassis.
Mc	Mc=Ms’+Mm+Mp.
Mc	Total mass for the shaft.
Ms’	Mass of the shaft.
Mm	Mass of the dc motor.
Mp	Mass of the propeller.
RP	RP=KT/(KQ*D)
RP	Transformer Modulus, thrust to torque ratio for the propeller.
KT	Coefficient of thrust for the propeller.
KQ	Coefficient of torque for the propeller.
D	Diameter for the propeller.
KT	KT=Tp/(Rho*n2*D4)
KQ	KQ=Qp/(Rho*n2*D5)
Tp	Thrust from the propeller.
Rho	Density of the fluid, air in this study.
n	Speed for the propeller, in rev/s.
Qp	Torque required to rotate the propeller.
P	The propeller.

It was necessary that the propellers were modeled both as transformers and flow sources simultaneously. Propellers were modeled as sources of flow because they take effort from the system as torque is required to turn the propellers; that torque comes from the engine (Phillips, 2004, p. 133) or other power sources such as electric motor in this study. It is worth noting that opposite effect takes place in a wind turbine electric generator assembly; torque is taken by the system; electric generator; to produce electric power. As a propeller transforms torque into thrust; according to the relation mentioned in the Table; it was necessary to model the effects of this transformation in to the system as well.

The weights and inertias of the components cannot be ignored as their momentum plays an important role in the dynamics of the system; they may be ignored for scaled down models (such as RC toys) in which gears and shafts are very small; almost massless; hence simplifying the bond graphs and decreasing the number of differential equations in the system; but in full scale applications; ignoring such effects will be catastrophic because of high momentums and inertias. To explain the point further; consider an example of an RC aircraft’s propeller, the moment the operator powers on/off the system, the response is almost instantaneous, i.e. the propeller starts and/or stops to a zero velocity almost instantly; while in full size aircrafts, the components i.e. propellers/turbines etc. have large spooling up/down time.

If there is a mistake in it all, please point it out, with an explanation.

Thank you for reading.