Cessna 172 -- Linear Model

Graduate Research done by Jeff Scott
Advisor: Prof. Michael Selig
Applied Aerodynamics Group
Department of Aeronautical and Astronautical Engineering
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Last updated April 10, 2000

The linear aerodynamics model uses stability derivatives to approximate actual aircraft behavior. The Cessna 172 linear model is based on the following flight conditions:

Variable	Variable in Code	Description	Units	Value(s)	Notes
Flight Conditions
h	Altitude	altitude above sea level	ft	?	not available
	Alpha	angle of attack	deg	?	not available
V	V_true_kts	velocity (true air speed)	kts	?	not available
q	Dynamic_pressure	dynamic pressure	lb/ft²	?	not available
C.G.	-	center of gravity location	%	?	not available
C_L	CL	lift coefficient	-	?	not available
C_D	CD	drag coefficient	-	?	not available
C_m	Cm	pitching moment coefficient	-	?	not available

The data input into the simulator is listed in the following tables:

Variable	Variable in Code	Description	Units	Value(s)	Source
Geometry
b	bw	wingspan	ft	35.8	c172_aero.c
	cbar	wing mean aerodynamic chord	ft	4.9	c172_aero.c
S	Sw	wing surface area	ft²	174	c172_aero.c

Control Surface Geometry

Variable Variable
in Code Description Units Value(s) Source Notes

e_max demax maximum elevator deflection deg +28 c172_aero.c

e_min demin minimum elevator deflection deg -23 c172_aero.c

a_max damax maximum aileron deflection deg +20 c172_aero.c

a_min damin minimum aileron deflection deg -15 c172_aero.c

r_max drmax maximum rudder deflection deg +16 c172_aero.c

r_min drmin minimum rudder deflection deg -16 c172_aero.c

Control Surface Geometry
Variable	Variable in Code	Description	Units	Value(s)	Source	Notes
e_max	demax	maximum elevator deflection	deg	+28	c172_aero.c
e_min	demin	minimum elevator deflection	deg	-23	c172_aero.c
a_max	damax	maximum aileron deflection	deg	+20	c172_aero.c
a_min	damin	minimum aileron deflection	deg	-15	c172_aero.c
r_max	drmax	maximum rudder deflection	deg	+16	c172_aero.c
r_min	drmin	minimum rudder deflection	deg	-16	c172_aero.c

Mass Data

Variable Variable
in Code Description Units Value(s) Source Notes

GTOW Weight gross takeoff weight lb 2300 c172_init.c converted to Mass
[slug/ft³] in code

I_xx I_xx roll inertia slug ft² 948 c172_init.c

I_yy I_yy pitch inertia slug ft² 1346 c172_init.c

I_zz I_zz yaw inertia slug ft² 1967 c172_init.c

I_xz I_xz lateral cross inertia slug ft² 0 c172_init.c

Mass Data
Variable	Variable in Code	Description	Units	Value(s)	Source	Notes
GTOW	Weight	gross takeoff weight	lb	2300	c172_init.c	converted to Mass [slug/ft³] in code
I_xx	I_xx	roll inertia	slug ft²	948	c172_init.c
I_yy	I_yy	pitch inertia	slug ft²	1346	c172_init.c
I_zz	I_zz	yaw inertia	slug ft²	1967	c172_init.c
I_xz	I_xz	lateral cross inertia	slug ft²	0	c172_init.c

Propulsion Data

Variable Variable
in Code Description Units Value(s) Source Notes

T_max simpleSingleMaxThrust maximum thrust lb 350 c172_engine.c use c172 engine model in code

Propulsion Data
Variable	Variable in Code	Description	Units	Value(s)	Source	Notes
T_max	simpleSingleMaxThrust	maximum thrust	lb	350	c172_engine.c	use c172 engine model in code

Aerodynamic Data

Variable Variable
in Code Description Units Value(s) Source Notes

C_D₀ CDo drag coefficient at =0 - 0.031 c172_aero.c

C_D CD_a drag curve slope 1/rad 0.130 c172_aero.c not used in favor of CDK

K CDK induced drag constant, i.e. 1/(pi*e*AR) - 0.054 c172_aero.c

C_{D_e} CD_de drag due to elevator 1/rad 0.060 c172_aero.c e positive down

C_L₀ CLo lift coefficient at =0 - 0.310 c172_aero.c

C_L CL_a lift curve slope 1/rad 5.143 c172_aero.c

C_L CL_adot lift due to angle of attack rate 1/rad 1.7 c172_aero.c

C_{L_q} CL_q lift due to pitch rate 1/rad 3.9 c172_aero.c

C_{L_e} CL_de lift due to elevator 1/rad 0.430 c172_aero.c e positive down

C_m₀ Cmo pitch moment coefficient at =0 - -0.015 c172_aero.c

C_m Cm_a pitch moment due to angle of attack 1/rad -0.89 c172_aero.c

C_m Cm_adot pitch moment due to angle of attack rate 1/rad -5.2 c172_aero.c

C_{m_q} Cm_q pitch moment due to pitch rate 1/rad -12.4 c172_aero.c

C_{m_e} Cm_de pitching moment due to elevator 1/rad -1.28 c172_aero.c e positive down

C_Y CY_beta side force fue to sideslip angle 1/rad -0.31 c172_aero.c

C_{Y_p} CY_p side force due to roll rate 1/rad -0.037 c172_aero.c

C_{Y_r} CY_r side force due to yaw rate 1/rad 0.21 c172_aero.c

C_{Y_a} CY_da side force due to aileron 1/rad 0 c172_aero.c a positive
right aileron up

C_{Y_r} CY_dr side force due to rudder 1/rad 0.187 c172_aero.c r positive left

C_l Cl_beta dihedral effect 1/rad -0.089 c172_aero.c

C_{l_p} Cl_p roll damping 1/rad -0.47 c172_aero.c

C_{l_r} Cl_r roll moment due to yaw rate 1/rad 0.096 c172_aero.c

C_{l_a} Cl_da roll moment due to aileron 1/rad -0.178 c172_aero.c a positive
right aileron up

C_{l_r} Cl_dr roll moment due to rudder 1/rad 0.0147 c172_aero.c r positive left

C_n Cn_beta weathercock stability 1/rad 0.065 c172_aero.c

C_{n_p} Cn_p adverse yaw 1/rad -0.03 c172_aero.c

C_{n_r} Cn_r yaw damping 1/rad -0.099 c172_aero.c

C_{n_a} Cn_da yaw moment due to aileron 1/rad 0.053 c172_aero.c a positive
right aileron up

C_{n_r} Cn_dr yaw moment due to rudder 1/rad -0.0657 c172_aero.c r positive left

Variable	Variable in Code	Description	Units	Value(s)	Source	Notes
Aerodynamic Data
C_D₀	CDo	drag coefficient at =0	-	0.031	c172_aero.c
C_D	CD_a	drag curve slope	1/rad	0.130	c172_aero.c	not used in favor of CDK
K	CDK	induced drag constant, i.e. 1/(pieAR)	-	0.054	c172_aero.c
C_{D_e}	CD_de	drag due to elevator	1/rad	0.060	c172_aero.c	e positive down
C_L₀	CLo	lift coefficient at =0	-	0.310	c172_aero.c
C_L	CL_a	lift curve slope	1/rad	5.143	c172_aero.c
C_L	CL_adot	lift due to angle of attack rate	1/rad	1.7	c172_aero.c
C_{L_q}	CL_q	lift due to pitch rate	1/rad	3.9	c172_aero.c
C_{L_e}	CL_de	lift due to elevator	1/rad	0.430	c172_aero.c	e positive down
C_m₀	Cmo	pitch moment coefficient at =0	-	-0.015	c172_aero.c
C_m	Cm_a	pitch moment due to angle of attack	1/rad	-0.89	c172_aero.c
C_m	Cm_adot	pitch moment due to angle of attack rate	1/rad	-5.2	c172_aero.c
C_{m_q}	Cm_q	pitch moment due to pitch rate	1/rad	-12.4	c172_aero.c
C_{m_e}	Cm_de	pitching moment due to elevator	1/rad	-1.28	c172_aero.c	e positive down
C_Y	CY_beta	side force fue to sideslip angle	1/rad	-0.31	c172_aero.c
C_{Y_p}	CY_p	side force due to roll rate	1/rad	-0.037	c172_aero.c
C_{Y_r}	CY_r	side force due to yaw rate	1/rad	0.21	c172_aero.c
C_{Y_a}	CY_da	side force due to aileron	1/rad	0	c172_aero.c	a positive right aileron up
C_{Y_r}	CY_dr	side force due to rudder	1/rad	0.187	c172_aero.c	r positive left
C_l	Cl_beta	dihedral effect	1/rad	-0.089	c172_aero.c
C_{l_p}	Cl_p	roll damping	1/rad	-0.47	c172_aero.c
C_{l_r}	Cl_r	roll moment due to yaw rate	1/rad	0.096	c172_aero.c
C_{l_a}	Cl_da	roll moment due to aileron	1/rad	-0.178	c172_aero.c	a positive right aileron up
C_{l_r}	Cl_dr	roll moment due to rudder	1/rad	0.0147	c172_aero.c	r positive left
C_n	Cn_beta	weathercock stability	1/rad	0.065	c172_aero.c
C_{n_p}	Cn_p	adverse yaw	1/rad	-0.03	c172_aero.c
C_{n_r}	Cn_r	yaw damping	1/rad	-0.099	c172_aero.c
C_{n_a}	Cn_da	yaw moment due to aileron	1/rad	0.053	c172_aero.c	a positive right aileron up
C_{n_r}	Cn_dr	yaw moment due to rudder	1/rad	-0.0657	c172_aero.c	r positive left

Cessna 172 -- Linear Model

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