Bond Graph Representation of the Coaxial Contra-Rotating Propeller (with the Bond-Graph)

A contra rotating propeller, with five gears and seven shafts.

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.
bs1	Bearing resistance on shaft number one.
Js1	JS1=Js’1+Jg1+Jr.
Js1	Total inertia for shaft number one.
Js’1	Inertia of shaft number one.
Jg1	Inertia of gear number one.
Jr	Inertia for dc motor rotor.
Ks1	Stiffness of shaft number one.
Ms1	Ms1=Ms’1+Mm+Mg1.
Ms1	Total mass for the shaft number one.
Ms’1	Mass for the shaft number one.
Mm	Mass for the dc motor.
Mg1	Mass for the gear number one.
m1	Distance from the center of mass of the Ms1 to the shaft number one; moment arm to transform the torque in to force to be applied to the Ms1.
TF	Transformer, mechanical transformer in this study, spur gear mesh.
Rs12	Transformer Modulus, transmission ratio between shaft number one and two.
bs2	Bearing resistance on shaft number two.
Js2	JS2=Js’2+ Jg2+Jg6.
Js2	Total inertia for shaft number two.
Js’2	Inertia of shaft number two.
Jg2	Inertia of gear number two.
Jg6	Inertia of gear number six.
Ks2	Stiffness of shaft number two.
Ms2	Ms2=Ms’2+Mg6+Mg2.
Ms2	Total mass for the shaft number two.
Ms’2	Mass for the shaft number two.
Mg6	Mass for the gear number six.
Mg2	Mass for the gear number two.
m2	Distance from the center of mass of the Ms2 to the shaft number two; moment arm to transform the torque in to force to be applied to the Ms2.
Rs23	Transformer Modulus, transmission ratio between shaft number two and three.
bs3	Bearing resistance on shaft number three.
Js3	JS3=Js’3+ Jg3+Jg4.
Js3	Total inertia for shaft number three.
Js’3	Inertia of shaft number three.
Jg3	Inertia of gear number three.
Jg4	Inertia of gear number four.
Ks3	Stiffness of shaft number three.
Ms3	Ms2=Ms’2+Mg3+Mg4.
Ms3	Total mass for the shaft number one.
Ms’3	Mass for the shaft number three.
Mg3	Mass for the gear number three.
Mg4	Mass for the gear number four.
m3	Distance from the center of mass of the Ms3 to the shaft number three; moment arm to transform the torque in to force to be applied to the Ms3.
Rs34	Transformer Modulus, transmission ratio between shaft number three and four.
bs4	Bearing resistance on shaft number four.
Js4	JS4=Js’4+ Jg5+Jp1.
Js4	Total inertia for shaft number four.
Js’4	Inertia of shaft number four.
Jg5	Inertia of gear number five.
Jp1	Inertia of propeller number one.
Ks4	Stiffness of shaft number three
Ms4	Ms4=Ms’4+Mg5+Mp1.
Ms4	Total mass for the shaft number four.
Ms’4	Mass for the shaft number four.
Mg5	Mass for the gear number five.
Mp1	Mass for the propeller number one.
m4	Distance from the center of mass of the Ms4 to the shaft number four; moment arm to transform the torque in to force to be applied to the Ms4.
Rs25	Transformer Modulus, transmission ratio between shaft number two and five.
bs5	Bearing resistance on shaft number five.
Js5	JS5=Js’5+ Jg7+Jp2.
Js5	Total inertia for shaft number five.
Js’5	Inertia of shaft number five.
Jg7	Inertia of gear number seven.
Jp2	Inertia of propeller number two.
Ks5	Stiffness of shaft number five.
Ms5	Ms4=Ms’5+Mg5+Mp2.
Ms5	Total mass for the shaft number four.
Ms’5	Mass for the shaft number five.
Mg7	Mass for the gear number seven.
Mp2	Mass for the propeller number two.
m5	Distance from the center of mass of the Ms5 to the shaft number five; moment arm to transform the torque in to force to be applied to the Ms5.
RPS	RPS=KT1/(KQ1*D1)
RPS	Transformer Modulus, for propeller number one.
KT1	Coefficient of thrust for propeller number one.
KQ1	Coefficient of torque for propeller number one.
D1	Diameter for propeller number one.
KT1	KT1=Tp1/(Rho*n12*D14)
KQ1	KQ1=Qp1/(Rho*n12*D15)
Tp1	Thrust from propeller number one.
n1	Speed for propeller number one, in rev/s.
Qp1	Torque required to rotate propeller number one.
P1	Propeller number one, mounted on the shaft number four.
RPH	RPH=KT2/(KQ2*D2)
RPH	Transformer Modulus, for propeller number 2.
KT2	Coefficient of thrust for propeller number two.
KQ2	Coefficient of torque for propeller number two.
D2	Diameter for propeller number two.
KT2	KT=Tp2/(Rho*n22*D24)
KQ2	KQ=Qp2/(Rho*n22*D25)
Tp2	Thrust from propeller number two.
Rho	Density of the fluid, air in this study.
n2	Speed for propeller number two, in rev/s.
Qp2	Torque required to rotate propeller number two.
P2	Propeller number two, mounted on the hollow shaft; only of the system, shaft number five.
mc1	Distance from the center of mass of the chassis to the shaft number one; moment arm to transform torque in to force to be applied to the chassis.
mc2	Distance from the center of mass of the chassis to the shaft number two; moment arm to transform torque in to force to be applied to the chassis.
mc3	Distance from the center of mass of the chassis to the shaft number three; moment arm to transform torque in to force to be applied to the chassis.
mc4	Distance from the center of mass of the chassis to the shaft number four; moment arm to transform torque in to force to be applied to the chassis.
mc5	Distance from the center of mass of the chassis to the shaft number five; moment arm to transform torque in to force to be applied to the chassis.
Mc	Mass of the chassis.

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.

Bond Graph Representation of the Coaxial Contra-Rotating Propeller

Abstract of my research in to the bond graph modeling for a coaxial contra-rotating propeller is given below.

A system of propellers formed when two propellers rotate about the same axis but in opposite direction is called coaxial contra-rotating propeller (Noriyuki Sasaki, 1998). A well-designed contra-rotating propeller has no rotational air flow, a maximum amount of air is pushed uniformly through the propeller disk resulting in an increase in performance and low induced energy losses. This study focuses on obtaining the system of equations needed to represent a contra-rotating propeller to be designed using seven spur gears mounted on five shafts. This study was completed using a bond graph model with a desire of obtaining a clearer understanding of this multi-domain , nonlinear and nonhomogeneous  system. A bond graph is a graphical representation of a physical dynamic system. The study resulted in a total of thirteen differential equations; needed to represent the system.