## Calculation of Damping Moment Coefficient and Damping Ratio using OpenRocket

In the previous post I described a method of calculating the Corrective Moment Coefficient and the Natural Frequency of a rocket using variables available in OpenRocket (see Calculation of Natural Frequency in OpenRocket). In addition to the above coefficients two other important metrics can be calculated using OR. These are the Dumping Moment Coefficient and the Damping Ratio. Both are readily available in RockSim, but unfortunately OR does not provide them.

The Damping Moment Coefficient is calculated using the following formula:

C2=C2r + C2a (4)

Where:
C2- Damping Moment Coefficient
C2r – Propulsive Damping Moment Coefficient
C2a – Aerodynamic Damping Moment Coefficient

Propulsive C2r is calculated using the following formula:

C2r = m_dot * (Ln-W)^2 (5)

Where:
m_dot – mass expulsion rate, kg/sec
Ln – Distance to the nozzle throat from the tip of the nose cone, m
W - CG of the rocket from the nose cone tip, m

m_dot component can be approximately calculated as follows:

m_dot = propellant mass/burn time [kg/sec] (6)

The Aerodynamic Dumping Moment Coefficient C2r is calculated as follows:

C2r = (V*Aref*p/2)* SUM(Cnai*(Zi-W)^2) (7)

Where:
p- density of air, approx 1.24 kg/m^3
V- velocity of the rocket (Total velocity of rocket in OR), m/sec
Aref - reference area, m^2
Cnai - normal force coefficient of the individual components
Zi- Distance from the nose cone tip to CP of the component , m
W - CG of the rocket, m

V and Aref are available in OR export, however Cnai and Zi are not exported and have to be extracted manually. They can be found in menu Analyze/Component Analysis, tab Stability. There Cnai for each element is provided in the column CAN, and column CP provides Zi. Note that (7) requires CP to be in meters.

Finally, the Dumping Ratio is calculated as follows:

DR = C2/(2*SQRT(C1*IL))
Where:
C2 - Damping Moment Coefficient (4)
C1 - Corrective Moment Coefficient (2) (see Calculation of Natural Frequency in OpenRocket)
IL - Longitudinal Moment of Inertia, kg*m^2 (available in OR export)

Above and in the previous post I have shown that it is possible to bridge a gap existing between OR and RockSim with regard to availability in OR of the fundamentally important rocket’s dynamic characteristics [1]:
• Corrective Moment Coefficient
• Dumping Moment Coefficient
• Natural Frequency
• Damping Ratio

References:
1) Apogee Peak of Flight #192, #193, #195, #196, #197
2) Fundamentals of Dynamic Stability by Gordon K. Mandell; Model Rocketry #10-1968, #11-1968, #1-1969, #2-1969, #3-1969, #4-1969,