Generic Mathematical Model of a Propeller Driven Fixed-Wing Aircraft Rémi Borfigat – MSc Aeronautical Engineering - Supervised by Dr Douglas Thomson University of Glasgow, charity number SC004401 Flight simulation is extensively involved in the design of modern aircrafts and is less costly than wind tunnel and flight testing. It is also well established among the community of airline pilots for their training. The aerospace department wishes to have a generic model that can be adapted to different aircraft configurations. A first model was built last year that simulated the longitudinal dynamics of a North American/Ryan Navion (see picture [1]). [1] 05/08/2015, ‘http://1.usa.gov/1P82sgj’, web page. [2] Cook, Flight Dynamics Principles, 2013. [3] Thomson, Flight Dynamics 4, 2014. [4] Barry, Generic Mathematical Model of Fixed Wing Propeller-Driven Aircraft, 2014. [5] Sadraey, Aircraft Design, 2013. A 6-DOF model is fully built and can be trimmed for different flight conditions specified by the user (i.e. altitude, flight speed, climb angle, sideslip, flaps deflection). The aircraft behaviour is satisfactory in trim, although sensitive to c.g. displacement, which can be an issue when a different aeroplane is used. Unstable lateral response to disturbance is the main problem. Possible future work (except solving this issue): account for a stall angle, for trim tabs and for downwash/sidewash effects. The fin is responsible for the lateral motion in the plane. It also contributes to the directional and lateral trim. The forces are assumed to act at its aerodynamic centre (a.c.) (see Figure 3). A non-zero sideslip angle at the fin creates lift (coefficient ) and drag forces (coefficient ) in Figure 4: = sin − cos( ) − cos − sin( ) 0 The mathematical model must be tested to assess its accuracy and sensitivity. Trimming the aeroplane and studying its response to disturbance are two tools to test the model. No satisfactory response to a control/wind disturbance: unstable behaviour for and attitudes. The non-linear and the linear model (linearised about the trim states) give similar response to a same disturbance. User Inputs ( ,) Trim + Non- Linear Simulation Forces and Moments Wings Wing Data Flap Data Propeller Propeller Data Fuselage Horizontal Tail Tail Data Fin Fin Data Aircraft Model Simulation Figure 2: MATLAB code organisation The aircraft dynamics are governed by the equations of motion [3] relative to the body axis in Figure 1: + − = − sin Θ + − = + cos Θ sin Φ + − = + cos Θ cos Φ − + − − = + ( − )+ 2 − 2 = − + − + = Figure 1: Aircraft body axis system [2] The model of the aeroplane is made of different sub-systems (e.g. wings, tail); see Figure 2. Each component forces and moments are estimated [4]: = + + +⋯ ⋮ = + + +⋯ Build a 6-degree of freedom (DOF) model from the previous work (i.e. take into account the lateral dynamics). Trim the aeroplane for different flight conditions and assess the model accuracy and sensitivity. Simulate the response to control and wind disturbance. Figure 3: Vertical tail parameters [5] The total moment created by the fin is: = + 0 0 is the moment acting at the c.g. because of acting at ; with the vector position of relative to the c.g. = × is the yawing moment of the aerofoil at : = Figure 4: Vertical tail in sideslip [2] Introduction Objectives Frame of Work Lateral Dynamics Enhancement Testing References Conclusion and Future Work Trim (wings-level flight) at service ceiling 18,000 ft Response to Disturbance At lower speeds than ≅ 140 kts, a bigger is required to level the Navion (i.e. remains close to 0 ± 1.5 ° throughout the flight envelope). An aft c.g. gives more manoeuvrability: a smaller is required. This is the opposite for a forward c.g. c.g. displacements sensitive. counteracts the propeller induced rolling moment and thus increases with . is the consequence of this action. With = 0°, ≅ 0°. An aft c.g. requires a smaller deflection (i.e. more manoeuvrable aeroplane). Opposite behaviour for a forward c.g. • −1 • +1 • original
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Generic Mathematical Model of a Propeller Driven
Fixed-Wing AircraftRémi Borfigat – MSc Aeronautical Engineering - Supervised by Dr Douglas Thomson
University of Glasgow, charity number SC004401
Flight simulation is extensively involved in the design of modern aircrafts and is less costly than wind tunnel and flight testing.
It is also well established among the community of airline pilots for their training.
The aerospace department wishes to have a generic model that can be adapted to different aircraft configurations.
A first model was built last year that simulated the longitudinal dynamics of a North American/Ryan Navion (see picture [1]).
[1] 05/08/2015, ‘http://1.usa.gov/1P82sgj’, web page. [2] Cook, Flight Dynamics Principles, 2013. [3] Thomson, Flight Dynamics 4, 2014. [4]
Barry, Generic Mathematical Model of Fixed Wing Propeller-Driven Aircraft, 2014. [5] Sadraey, Aircraft Design, 2013.
A 6-DOF model is fully built and can be trimmed for different
flight conditions specified by the user (i.e. altitude, flight
speed, climb angle, sideslip, flaps deflection).
The aircraft behaviour is satisfactory in trim, although
sensitive to c.g. displacement, which can be an issue when a
different aeroplane is used.
Unstable lateral response to disturbance is the main problem.
Possible future work (except solving this issue): account for a
stall angle, for trim tabs and for downwash/sidewash effects.
The fin is responsible for the lateral motion in the 𝑥𝑦 plane. It also
contributes to the directional and lateral trim.
The forces are assumed to act at its aerodynamic centre (a.c.)
𝑎𝑐𝑣 (see Figure 3). A non-zero sideslip angle 𝛽𝑣𝑡 at the fin creates
lift (coefficient 𝐶𝐿𝑣𝑡) and drag forces (coefficient 𝐶𝐷𝑣𝑡) in Figure 4:
𝐹𝑣𝑡 = 𝑞𝑆𝑣𝑡
𝐶𝐿𝑣𝑡 sin 𝛽𝑣𝑡 − 𝐶𝐷𝑣𝑡cos(𝛽𝑣𝑡)
−𝐶𝐿𝑣𝑡 cos 𝛽𝑣𝑡 − 𝐶𝐷𝑣𝑡sin(𝛽𝑣𝑡)
0
The mathematical model must be tested to assess its
accuracy and sensitivity.
Trimming the aeroplane and studying its response to
disturbance are two tools to test the model.
No satisfactory response to a control/wind disturbance:
unstable behaviour for 𝜙 and 𝜓 attitudes.
The non-linear and the linear model (linearised about the trim
states) give similar response to a same disturbance.
User Inputs (𝑉𝑓 , 𝐻)
Trim + Non-Linear
Simulation
Forces and Moments
Wings
Wing Data
Flap Data
Propeller
Propeller Data
Fuselage Horizontal Tail
Tail Data
Fin
Fin Data
Aircraft Model
Simulation
Figure 2: MATLAB code organisation
The aircraft dynamics
are governed by the
equations of motion [3]
relative to the body axis
in Figure 1:
𝑚 𝑈 + 𝑄𝑊 − 𝑅𝑉 = 𝑋 −𝑚𝑔 sinΘ
𝑚 𝑉 + 𝑅𝑈 − 𝑃𝑊 = 𝑌 +𝑚𝑔 cosΘ sinΦ
𝑚 𝑊 + 𝑃𝑉 − 𝑄𝑈 = 𝑍 +𝑚𝑔 cosΘ cosΦ
𝐼𝑥𝑥 𝑃 − 𝐼𝑥𝑧 𝑅 + 𝑄𝑅 𝐼𝑧𝑧 − 𝐼𝑦𝑦 − 𝑃𝑄𝐼𝑥𝑧 = 𝐿
𝐼𝑦𝑦 𝑄 + 𝑅𝑃(𝐼𝑥𝑥 − 𝐼𝑧𝑧) + 𝑃2 − 𝑅2 𝐼𝑥𝑧 = 𝑀
𝐼𝑧𝑧 𝑅 − 𝐼𝑥𝑧 𝑃 + 𝑃𝑄 𝐼𝑦𝑦 − 𝐼𝑥𝑥 + 𝑄𝑅𝐼𝑥𝑧 = 𝑁Figure 1: Aircraft body axis system [2]
The model of the
aeroplane is made of
different sub-systems
(e.g. wings, tail); see
Figure 2. Each
component forces
and moments are
estimated [4]:
𝑋 = 𝑋𝑓𝑢𝑠 + 𝑋𝑝𝑟𝑜𝑝 + 𝑋𝑤𝑖𝑛𝑔𝑠𝑡𝑎𝑟 +⋯
⋮𝑁 = 𝑁𝑓𝑢𝑠 + 𝑁𝑝𝑟𝑜𝑝 + 𝑁𝑤𝑖𝑛𝑔𝑠𝑡𝑎𝑟 +⋯
Build a 6-degree of freedom (DOF) model from the previous work (i.e. take into account the lateral dynamics).
Trim the aeroplane for different flight conditions and assess the model accuracy and sensitivity.
Simulate the response to control and wind disturbance.
Figure 3: Vertical tail parameters [5]
The total moment
created by the fin is:
𝑀𝑣𝑡𝑡𝑜𝑡= 𝑀𝑣𝑡 +
00𝑁𝑣𝑡
𝑀𝑣𝑡 is the moment acting at the c.g. because of
𝐹𝑣𝑡 acting at 𝑎𝑐𝑣; with 𝑟𝑣𝑡𝑐𝑔 the vector position of
𝑎𝑐𝑣 relative to the c.g. 𝑀𝑣𝑡 = 𝑟𝑣𝑡𝑐𝑔 × 𝐹𝑣𝑡
𝑁𝑣𝑡 is the yawing moment of the aerofoil at 𝑎𝑐𝑣:
𝑁𝑣𝑡 = 𝑞𝑆𝑣𝑡𝐶𝑛𝑣𝑀𝐴𝐶𝑣 Figure 4: Vertical tail in sideslip [2]
Introduction
Objectives
Frame of Work
Lateral Dynamics Enhancement
Testing
References
Conclusion and Future Work
Trim (wings-level flight) at service ceiling 18,000 ft
Response to Disturbance
At lower speeds than
𝑉𝑓𝑐𝑟𝑢𝑖𝑠𝑒 ≅ 140 kts, a bigger
𝛿𝑒 is required to level the
Navion (i.e. 𝜃 remains close
to 0± 1.5 ° throughout the
flight envelope).
An aft c.g. gives more
manoeuvrability: a smaller
𝛿𝑒 is required. This is the
opposite for a forward c.g.
c.g. displacements sensitive.
𝛿𝑎 counteracts the propeller
induced rolling moment and
thus increases with 𝜃𝑝. 𝜙 is
the consequence of this
action.
With 𝛽 = 0°, 𝛿𝑟 ≅ 0°. An aft c.g. requires a smaller
𝛿𝑎 deflection (i.e. more
manoeuvrable aeroplane).
Opposite behaviour for a
forward c.g.
• 𝑥𝑐𝑔 − 1𝑚
• 𝑥𝑐𝑔 + 1𝑚
• 𝑥𝑐𝑔 original
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