Quadrotor Mathematical Model - work in progress

 WEIGH-IN    

Item	Weight per Piece (g)	Total Quantity	Total Weight Per Item
Center Body	180	1	180
Arm Tube	30	4	120
Motor	80	4	360
Motor Mount	11   7	4 4	44  28
ESC and Wire	20	4	80
Controller/IMU	120	1	120
RC Receiver	20	1	20
Main Battery	250	1	250
Controller Battery	100	1	100
Other Item
1. Cage	50	1	50
2. Vertical Legs
3. Landing Skid
4.
Total			1350g

CENTER OF MASS:

The center of mass is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the mean location of a distribution of mass in space. http://en.wikipedia.org/wiki/Center_of_mass

CENTER OF GRAVITY:

Center of gravity is the point in a body around which the resultant torque due to gravity forces vanish. Near the surface of the earth, where the gravity acts downward as a parallel force field, the center of gravity and the center of mass are the same.

The study of the dynamics of aircraft, vehicles and vessels assumes that the system moves in near-earth gravity, and therefore the terms center of gravity and center of mass are used interchangeably. http://en.wikipedia.org/wiki/Center_of_mass

Free Body Diagram on Level Flight

 

 

Example Scenario:

When section of Dualrotor by disturbance was tilted to right by about 10 deg from horizontal. Let's calculate the Thrust required by each motor to return it to positions while maintaining the original flying height. The total weight W is 1200 grams and the center of mass C.G. is located at Ccg=120mm from propellers center pressure C.P. 



Free Body Diagram @ C.P.

Initial premise:
1. To maintain hover elevation or flying height during tilt, the vertical thrust Fv must be the same as the weight W of the quad rotor at any incident/tilt angle.
2. The torque caused by displaced C.G. from C.P. must by counter by the motor.

Given: 

Ø      = 10 deg,        Tilt Angle
Rad   =  0.175 rad,   Tilt Angle in radians
Ccg   =   120 mm,    Torque arm measure from Center Pressure to C.G. of Quad
dm    =   600 mm,    Torque arm measure from Center-to-Center of propeller
W     = 1200 g,        Weight of Quadcopter acting at Center of Gravity
Fv      = 1200 g,       Total Vertical Thrust acting at props center pressure. C.P.
Fv1    =   600 g,       Thrust required for each motor i.e. F1=F2=Tm for Hover/level flight
Fv2    =   600 g,       Thrust required for each motor i.e. F1=F2=Tm for Hover/level flight
Mg   =    30 g/us      Motor thrust gain per microseconds (1us) of PWM (i.e. for each motor)

Variable:
d = 20.83 mm        Torque Arm cause by displacement of C.G. to Center Pressure of  
                                      the Propeller

Tg = 25,005.33 g-mm Torque generated from cause by shift of C.G.  from Center 
                                        Pressure generated by the Propeller thrust where Tg=W*d,
                                        note that summation of vertical forces yields Fv=W.

Tm=Tg ;                    Torque the two motor required to generate to counter the Tg, i.e. Tm = Tg
 
dFm = Tg/dm;            Differential thrust of the motor required to generate a torque (Tm) 
                                      equal to that Tg based on lever arm 'dm'.     

dFm = 41.67 g               Differential thrust of each motor required to generate counter torque

dPm  = 1.38 us               Differential pulse for each motor required to generate thrust 'Fm' (Derived from motor 
                                        thrust and pulse curve data or can be acquired through test)
                                        i.e dPm = dFm/Mg .

     

    Thrust Calculation Table:

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