fDfPfunctions | R Documentation |
Computes the thrust requirement dependency factor for drag and power factors in flapping flight based on reduced frequency (kf
) and strokeplane angle (phi
).
fD.ind(kf, phi)
fD.pro0(kf, phi)
fD.pro2(kf, phi)
fP.ind(kf, phi)
fP.pro0(kf, phi)
fP.pro2(kf, phi)
Using f
for wingbeat frequency, b
for wingspan, and U
for air speed:
kf |
reduced frequency ( |
phi |
strokeplane angle in radians; valid range between 0 and 0.87 rad (50 deg) |
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 = k_D D^\prime
. The factor k_D
depends on reduced frequency k_f
, strokeplane angle \phi
and the thrust-to-lift ratio T/L
: k_D = 1 + f_D(k_f,\phi) \frac{T}{L}
. Functions fD.ind
,fD.pro0
and fD.pro2
compute f_D(k_f,\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 D^\prime U
, again with k_P = 1 + f_P(k_f,\phi) \frac{T}{L}
. Functions fP.ind
,fP.pro0
and fP.pro2
compute f_P(k_f,\phi)
for induced power, zero lift profile power and lift dependent profile power, respectively.
Numeric value
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 f_D
.
Marco Klein Heerenbrink
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 \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1098/rspa.2014.0952")}
computeFlappingPower
## 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
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