The tiltrotor aircraft has often been proposed as a means to increase the maximum speed of the conventional helicopter. The tiltrotor aircraft consists of three primary flight modes that are the helicopter flight mode in low forward speed flight, airplane flight mode in high forward speed flight, and conversion flight mode. The aim of this paper is to develop a nonlinear flight dynamics mathematical modeling method of tiltrotor aircraft and investigate the dynamic stability characteristics of tiltrotor aircraft. First, a nonlinear tiltrotor aircraft flight dynamics model is developed. The trim and linearized results are present to verify the model. Then, using a numerical differentiation technique, the dynamic stability of the tiltrotor aircraft is assessed. The results show that the flight speed and nacelle angle would affect the magnitude and the trend of the aerodynamic derivatives. The damping of the pitch short period mode and the Dutch roll mode is insensitive to flight speed while they could be affected by nacelle angle. In all flight modes, as flight speed increases, the natural modes become more stable.
The maximum speed of a conventional helicopter is restricted due to aerodynamic limitations, installed engine power, and airframe drag [
Tiltrotor aircraft three flight modes.
Helicopter mode
Conversion mode
Airplane mode
The XV15 was a successful tiltrotor aircraft technology demonstrator. Reference [
This paper is organized as follows. In Section
In this section, a nonlinear flight dynamics model of the tiltrotor aircraft is established.
XV15 tiltrotor aircraft is chosen for this analysis. The basic parameters are shown in Table
Basic parameters of XV15 tiltrotor.
Weight ( 
5897 kg 
Rotor radius ( 
3.81 m 
Blade number ( 

Blade twist ( 
41° 
Rotor speed ( 
589 rpm (helicopter mode) 
517 rpm (airplane mode)  
Lock number ( 
3.83 
Solidity ( 
0.089 
Pitchflap coupling angle ( 
15 deg 
Flapping hinge restraint ( 
17476 Nm/rad 
Flap moment of inertia ( 
139 kg·m^{2} 
Nacelle length ( 
1.4 m 
Wing area ( 
16.82 m^{2} 
Horizontal tail areas ( 
4.67 m^{2} 
Vertical tail areas ( 
2.35 m^{2} 
The rotor model is the most important part of the tiltrotor aircraft model, and it is also the most complex component of the model due to the flapping motion and solution of induced velocity. As shown in Figure
Rotor aerodynamic calculation diagram.
The bladeroot collective pitch is given by
The XV15 tiltrotor has two 3bladed gimbaled rotors. The pitchflap coupling deltathree angle is 15 deg, which means decreasing the blade pitch for a flap increase. The feedback gain for pitchcoupling is given by [
The motion of the gimbaled hub relative to the shaft could be described by two degrees of freedom: the lateral tilt and longitudinal tilt angles. These two degrees of freedom correspond to the tippathplane tilt of an articulated rotor [
The hinge offset and flapping hinge restraint parameters are given in Table
Figure
Rotor blade flapping motion.
Consider incompressible, potential flow about a rotor, the induced inflow distribution at the rotor plane could be represented in terms of a set of harmonic and radial shape functions [
For engineering applications, truncations are required. The rule is to make the induced flow harmonic match the highest frequency we care about [
The wing aerodynamic coefficients are defined as functions of angle of attack, nacelle angle, and flap setting. Because part of the wing is disturbed by the rotor wake, it is necessary to take into account the rotor’s wake interference in wing modeling. The wing is divided into slipstream and freestream parts. The area of the wing influenced by rotor wake is given by [
Empennage aerodynamics are modeled similarly to the wing. The downwash at the empennage was assumed to be caused by the wing and the rotors [
The downwash angle at empennage due to the wing is also obtained from reference [
The XV15 tiltrotor aircraft uses both helicopter and airplane control strategies to control the aircraft. In helicopter flight mode, longitudinal cyclic, differential collective, and differential longitudinal cyclic are used to pitch, roll, yaw, and heave control, respectively. As the tiltrotor aircraft converts from helicopter flight mode to airplane flight mode, the helicopter rotor control surfaces are washed out as function of nacelle angle and flight speed, which is given by
The XV15 tiltrotor aircraft flight dynamics model could be given by
The state variables are given in the form of a vector
The trim calculation is to solve nonlinear differential equations, like equation (
In order to carry out trim validation, a comparison of the trim results with results from the GTRS is presented. Figures
Trim results in helicopter mode.
Trim results in nacelle 60 deg.
Trim results in nacelle 30 deg.
Trim results in airplane mode.
With the application of a linearization algorithm, the rotor and inflow modes residualized out via quasistatic reduction, then the nonlinear equation can be reduced to the form of
To linearize validation, a comparison of the eigenvalues for matrix A in equation (
Hover mode eigenvalue validation.
Hover mode  Natural mode  Calculated  Reference[ 
Flight test 

Longitudinal  Phugoid  
Pitch subsidence  1.0  0.5979  1.32  
Heave subsidence  0.17  0.2086  0.105  
Lateral  Dutch roll  
Spiral subsidence  0.1609  0.0728  0.102  
Roll subsidence  2.6539  −1.1247  1.23 
Airplane mode
Hover mode  Natural mode  Calculated  Reference [ 
GTRS 

Longitudinal  Phugoid  
Shortperiod pitch  
Lateral  Dutch roll  
Spiral subsidence  0.1384  0.0497  0.1226  
Roll subsidence  0.7128  0.9378  1.0649 
In conclusion, the calculated values in this paper including trim and eigenvalues are in good agreement with those in the GTRS and flight test. In brief, the XV15 tiltrotor flight dynamics model is proved to be valid. So we have enough confidence in the following analysis results.
Longitudinal static stability is also called speed stability, which defined the relationship between the longitudinal stick and the flight speed. ADS33EPRF handling quality requirements require the longitudinal static stability to meet positive speed stability for cyclic control [
In order to analyze the tiltrotor aircraft speed stability, longitudinal stick trim results of all flight modes are drawn on a single graph, as shown Figure
Longitudinal stick migration with respect to airspeed.
In order to study the speed stability in the conversion flight, a sample conversion path in conversion corridor is chosen. The conversion path begins at 80 knots in helicopter mode and passes through 120 knots in nacelle angle 60 deg and 140 knots in nacelle angle 30 deg, finally ends at 160 knots in airplane mode. The longitudinal stick is pushed forward with airspeed increasing before 120 knots. However, after 120 knots, longitudinal stick migrates after to gain airspeed. This results in apparent negative speed stability and mainly caused by the efficiency of the elevator that increases significantly with the flight speed. Reference [
In general, the speed stability for each flight mode is positive. However, when the tiltrotor aircraft covert from helicopter mode to airplane mode, the longitudinal stick will migrate backward as the speed increases, showing a typical negative speed stability.
There are 36 stability derivatives in the standard 6DoF set [
Figure
Variation of the derivative
In forward flight, the wing begins to contribute to the derivative and is given by
For all flight mode, this derivative decreases with the increase of speed. It is worth noting that in the same flight speed (140 kt), the derivative of airplane mode is the largest because of the absence of the rotor contribution.
Figure
Variation of the derivative
Two important derivatives,
Variation of the derivative
Figure
Variation of the derivative
Figure
Variation of the derivative
The contribution of the spring stiffness is minor with a numerical value of 0.2. The main reason is that the rotor stiffness number is small with a numerical value of 0.0752. The tailplane is the main source contribution to this derivative, which changes in a linear manner with flight speed.
The roll damping derivative
Variation of the derivative
Figure
Variation of the derivative
In the previous section, we analyzed the stability derivatives of the XV15 tiltrotor aircraft and find out their main contribution sources. For small amplitude stability analysis, the tiltrotor aircraft flight dynamics behavior can be described by a linear combination of natural modes.
Figures
Longitudinal eigenvalue movement in helicopter mode.
Lateral eigenvalue movement in helicopter mode.
Longitudinal eigenvalue movement in nacelle 60 deg.
Lateral eigenvalue movement in nacelle 60 deg.
Longitudinal eigenvalue movement in nacelle 30 deg.
Lateral eigenvalue movement in nacelle 30 deg.
Longitudinal eigenvalue movement in airplane mode.
Lateral eigenvalue movement in airplane mode.
Blake et al. used a groundbased flight simulator and developed a relationship between pilot rating and longitudinal stability. They find out that the pilot rating will have a rapid degradation when the maneuver margin,
As seen in Figure
The lateral/directional motion of tiltrotor aircraft in forward flight is composed of a Dutch roll oscillation and two nonoscillating modes which are called roll and spiral modes [
From the analysis in the previous section, we have known that the roll damping derivative
In terms of the Dutch roll mode, the frequency increases as the flight speed increases in all flight modes (see Figures
The XV15 tiltrotor aircraft stability derivatives discussed above are all stable; however, the nacelle angle will affect the contribution of the rotor and then affect the stability derivative magnitude and trend. In all flight modes, as flight speed increases, the natural modes become more stable. The damping of the pitch short period mode and the Dutch roll mode is insensitive to flight speed. In particular, the Dutch roll meets the ADS33EPRF handling quality specification level 1 requirement for allotherMTEs for airplane mode.
In this paper, a nonlinear flight dynamics mode is developed, and then the dynamic stability of the tiltrotor aircraft is assessed. The main conclusions from the current work are as follows:
The XV15 tiltrotor flight dynamics model developed in this paper is proved to be valid
The speed stability of the tiltrotor aircraft is positive in certain flight modes; however, when the tiltrotor aircraft covert from helicopter mode to airplane mode, it is apparent typical negative speed stability
The stability derivatives in all flight modes are stable, while the nacelle angle could affect their magnitude and trend. For example, the contribution of the rotors to the heave damping derivative
The natural modes become more stable with flight speed increases. The damping of the pitch short period mode and the Dutch roll mode is insensitive to flight speed, while they are sensitive to nacelle angle
Wing lift cure slope (1/rad)
Rotor blade lift cure slope (1/rad)
Acceleration due to gravity (m/s^{2})
Mast angle (rad)
Aircraft mass (kg)
Translational velocities (m
Angular velocities (rad
Induced velocity (m/s)
Time (s)
Blade radial coordinate nondimensionalized on
Blade radial coordinate (m)
System matrix
Blade area (m^{2})
Wing area (m^{2})
Control matrix
Flap moment of inertia (kg·m^{2})
Pitch moment of inertia (kg·m^{2})
Pitchflap coupling ratio,
Flapping hinge restraint (Nm/rad)
Roll damping derivative (1/s)
Dihedral effect (rad/s·m)
Speed static stability derivative (rad/s·m)
Incidence static stability derivative (rad/s·m)
Yaw damping derivative (1/s)
Weathercock stability (rad/s·m)
Number of blades
Rotor radius (m)
Stiffness number
Trim body
Drag damping derivative (1/s)
Side force damping derivative (1/s)
Heave damping derivative (1/s)
Forward flight velocity (m/s)
Euler angles (rad)
Induced inflow expansion coefficients
Rotor flap angle (rad)
Blade precone angle (rad)
Pitchflap coupling angle (rad)
Collective stick input (cm)
Lateral stick input (cm)
Longitudinal stick input (cm)
Pedal deflection (cm)
Bladeroot collective pitch (rad)
Blade pitch angle (rad)
Blade twist angle (rad)
Lateral cyclic pitch (rad)
Longitudinal cyclic pitch (rad)
Radial expansion shape function
Advance ratio
Pitchflap coupling angle
Lock number
Rotor inflow
Pitch short period eigenvalue
Flap frequency ratio
Pitch short period damping factor
Pitch short period frequency (rad/s)
Dutch roll damping factor
Dutch roll frequency (rad/s)
Rotor solidity
Rotor angular velocity (rad/s)
Air density (kg/m^{3})
Azimuthal location of reference blade (rad).
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there is no conflict of interest regarding the publication of this paper.
Ke Lu and Chunsheng Liu designed the research and wrote the paper; Chunhua Li and Renliang Chen helped perform the analysis with constructive discussions.
This work was supported by the Aeronautical Science Foundation of China (Grant Nos. 20175752045; 2016ZA02001) and the National Natural Science Foundation of China (Grant No. 11672128).