def dutch_roll(velocity, Cn_Beta, S_gross_w, density, span, I_z, Cn_r):
""" output = SUAVE.Methods.Flight_Dynamics.Dynamic_Stablity.Approximations.dutch_roll(velocity, Cn_Beta, S_gross_w, density, span, I_z, Cn_r)
Calculate the natural frequency and damping ratio for the approximate dutch roll characteristics
Inputs:
velocity - flight velocity at the condition being considered [meters/seconds]
Cn_Beta - coefficient for change in yawing moment due to sideslip [dimensionless]
S_gross_w - area of the wing [meters**2]
density - flight density at condition being considered [kg/meters**3]
span - wing span of the aircraft [meters]
I_z - moment of interia about the body z axis [kg * meters**2]
Cn_r - coefficient for change in yawing moment due to yawing velocity [dimensionless]
Outputs:
output - a data dictionary with fields:
dutch_w_n - natural frequency of the dutch roll mode [radian/second]
dutch_zeta - damping ratio of the dutch roll mode [dimensionless]
Assumptions:
Major effect of rudder deflection is the generation of the Dutch roll mode.
Dutch roll mode only consists of sideslip and yaw
Beta = -Psi
Phi and its derivatives are zero
consider only delta_r input and Theta = 0
Neglect Cy_r
X-Z axis is plane of symmetry
Constant mass of aircraft
Origin of axis system at c.g. of aircraft
Aircraft is a rigid body
Earth is inertial reference frame
Perturbations from equilibrium are small
Flow is Quasisteady
Source:
J.H. Blakelock, "Automatic Control of Aircraft and Missiles" Wiley & Sons, Inc. New York, 1991, p 132-134.
"""
#process
w_n = velocity * (Cn_Beta*S_gross_w*density*span/2./I_z)**0.5 # natural frequency
zeta = -Cn_r /8. * (2.*S_gross_w*density*span**3./I_z/Cn_Beta)**0.5 # damping ratio
output = Data()
output.dutch_w_n = w_n
output.dutch_zeta = zeta
return output
def lateral_directional(velocity, Cn_Beta, S_gross_w, density, span, I_z, Cn_r, I_x, Cl_p, J_xz, Cl_r, Cl_Beta, Cn_p, Cy_phi, Cy_psi, Cy_Beta, mass):
""" output = SUAVE.Methods.Flight_Dynamics.Dynamic_Stablity.Full_Linearized_Equations.lateral_directional(velocity, Cn_Beta, S_gross_w, density, span, I_z, Cn_r, I_x, Cl_p, J_xz, Cl_r, Cl_Beta, Cn_p, Cy_phi, Cy_psi, Cy_Beta)
Calculate the natural frequency and damping ratio for the full linearized dutch roll mode along with the time constants for the roll and spiral modes
Inputs:
velocity - flight velocity at the condition being considered [meters/seconds]
Cn_Beta - coefficient for change in yawing moment due to sideslip [dimensionless] (no simple relation)
S_gross_w - area of the wing [meters**2]
density - flight density at condition being considered [kg/meters**3]
span - wing span of the aircraft [meters]
I_z - moment of interia about the body z axis [kg * meters**2]
Cn_r - coefficient for change in yawing moment due to yawing velocity [dimensionless] ( - C_D(wing)/4 - 2 * Sv/S * (l_v/b)**2 * (dC_L/dalpha)(vert) * eta(vert))
I_x - moment of interia about the body x axis [kg * meters**2]
Cl_p - change in rolling moment due to the rolling velocity [dimensionless] (no simple relation for calculation)
J_xz - products of inertia in the x-z direction [kg * meters**2] (if X and Z lie in a plane of symmetry then equal to zero)
Cl_r - coefficient for change in rolling moment due to yawing velocity [dimensionless] (Usually equals C_L(wing)/4)
Cl_Beta - coefficient for change in rolling moment due to sideslip [dimensionless]
Cn_p - coefficient for the change in yawing moment due to rolling velocity [dimensionless] (-C_L(wing)/8*(1 - depsilon/dalpha)) (depsilon/dalpha = 2/pi/e/AspectRatio dC_L(wing)/dalpha)
Cy_phi - coefficient for change in sideforce due to aircraft roll [dimensionless] (Usually equals C_L)
Cy_psi - coefficient to account for gravity [dimensionless] (C_L * tan(Theta))
Cy_Beta - coefficient for change in Y force due to sideslip [dimensionless] (no simple relation)
mass - mass of the aircraft [kilograms]
Outputs:
output - a data dictionary with fields:
dutch_w_n - natural frequency of the dutch roll mode [radian/second]
dutch_zeta - damping ratio of the dutch roll mode [dimensionless]
roll_tau - approximation of the time constant of the roll mode of an aircraft [seconds] (positive values are bad)
spiral_tau - time constant for the spiral mode [seconds] (positive values are bad)
Assumptions:
X-Z axis is plane of symmetry
Constant mass of aircraft
Origin of axis system at c.g. of aircraft
Aircraft is a rigid body
Earth is inertial reference frame
Perturbations from equilibrium are small
Flow is Quasisteady
Zero initial conditions
Neglect Cy_p and Cy_r
Source:
J.H. Blakelock, "Automatic Control of Aircraft and Missiles" Wiley & Sons, Inc. New York, 1991, p 118-124.
"""
#process
# constructing matrix of coefficients
A = (0, -span * 0.5 / velocity * Cl_p, I_x/S_gross_w/(0.5*density*velocity**2)/span ) # L moment phi term
B = (0, -span * 0.5 / velocity * Cl_r, -J_xz / S_gross_w / (0.5 * density * velocity ** 2.) / span) # L moment psi term
C = (-Cl_Beta) # L moment Beta term
D = (0, - span * 0.5 / velocity * Cn_p, -J_xz / S_gross_w / (0.5 * density * velocity ** 2.) / span ) # N moment phi term
E = (0, - span * 0.5 / velocity * Cn_r, I_z / S_gross_w / (0.5 * density * velocity ** 2.) / span ) # N moment psi term
F = (-Cn_Beta) # N moment Beta term
G = (-Cy_phi) # Y force phi term
H = (-Cy_psi, mass * velocity / S_gross_w / (0.5 * density * velocity ** 2.))
I = (-Cy_Beta, mass * velocity / S_gross_w / (0.5 * density * velocity ** 2.))
# Taking the determinant of the matrix ([A, B, C],[D, E, F],[G, H, I])
EI = P.polymul(E,I)
FH = P.polymul(F,H)
part1 = P.polymul(A,P.polysub(EI,FH))
DI = P.polymul(D,I)
FG = P.polymul(F,G)
part2 = P.polymul(B,P.polysub(FG,DI))
DH = P.polymul(D,H)
GE = P.polymul(G,E)
part3 = P.polymul(C,P.polysub(DH,GE))
total = P.polyadd(part1,P.polyadd(part2,part3))
poly = total / total[5]
# Generate the time constant for the spiral and roll modes along with the damping and natural frequency for the dutch roll mode
root = np.roots(poly)
root = sorted(root,reverse=True)
spiral_tau = 1 * root[0].real
w_n = (root[1].imag**2 + root[1].real**2)**(-0.5)
zeta = -2*root[1].real/w_n
roll_tau = 1 * root [3].real
output = Data()
output.dutch_w_n = w_n
output.dutch_zeta = zeta
output.spiral_tau = spiral_tau
output.roll_tau = roll_tau
return output
