# fDfPfunctions: Coefficient for thrust dependency of drag and power factors In afpt: Tools for Modelling of Animal Flight Performance

## Description

Computes the thrust requirement dependency factor for drag and power factors in flapping flight based on reduced frequency (`kf`) and strokeplane angle (`phi`).

## Usage

 ```1 2 3 4 5 6``` ```fD.ind(kf, phi) fD.pro0(kf, phi) fD.pro2(kf, phi) fP.ind(kf, phi) fP.pro0(kf, phi) fP.pro2(kf, phi) ```

## Arguments

 Using f for wingbeat frequency, b for wingspan, and U for air speed: `kf` reduced frequency (kf = 2 * π * f *b / U); valid range between 1 and 6 `phi` strokeplane angle in radians; valid range between 0 and 0.87 rad (50 deg)

## Details

Flapping of the wings alters the drag components on the wing. A drag component in flapping flight can be related to the drag component in non-flapping flight as D = kD * Dnf . The factor kD depends on reduced frequency kf, strokeplane angle phi and the thrust-to-lift ratio T/L: kD = 1 + fD(kf,phi) * T/L . Functions `fD.ind`,`fD.pro0` and `fD.pro2` compute fD(kf,phi) for induced drag, zero lift profile drag and lift dependent profile drag, respectively.

Similarly, the flapping power components can be computed as: P = k_P * Dnf * U, again with kP = 1 + fP(kf,phi) * T/L . Functions `fP.ind`,`fP.pro0` and `fP.pro2` compute fP(kf,phi) for induced power, zero lift profile power and lift dependent profile power, respectively.

Numeric value

## Note

Thrust requirement is the sum of all drag components in flapping flight divided by the lift. This means the thrust requirement itself is a function of the values of fD.

## Author(s)

Marco Klein Heerenbrink

## References

Klein Heerenbrink, M., Johansson, L. C. and Hedenström, A. 2015 Power of the wingbeat: modelling the effects of flapping wings in vertebrate flight. Proc. R. Soc. A 471, 2177 doi: 10.1098/rspa.2014.0952

`computeFlappingPower`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44``` ```## reduced frequency kf <- 2*pi*4/10 # 4 Hz at 10 m/s ## strokeplane angle phi <- 20*pi/180 # 20 degrees ## thrust ratio TL <- 0.2 ## induced drag factor: fDind <- fD.ind(kf,phi) kDind <- 1 + fDind*TL print(kDind) # [1] 1.623659 ## zero lift drag factor: fDpro0 <- fD.pro0(kf,phi) kDpro0 <- 1 + fDpro0*TL print(kDpro0) # [1] 1.014899 ## lift dependent profile drag factor: fDpro2 <- fD.pro2(kf,phi) kDpro2 <- 1 + fDpro2*TL print(kDpro2) # [1] 1.511107 ## induced power factor: fPind <- fP.ind(kf,phi) kPind <- 1 + fPind*TL print(kPind) # [1] 1.996891 ## zero lift power factor: fPpro0 <- fP.pro0(kf,phi) kPpro0 <- 1 + fPpro0*TL print(kPpro0) # [1] 1.076046 ## lift dependent profile power factor: fPpro2 <- fP.pro2(kf,phi) kPpro2 <- 1 + fPpro2*TL print(kPpro2) # [1] 1.811983 ```