3.1. Trimming, Linearization, and Mode Analysis
The trim state refers to a steady level flight of the compound aircraft without relative motion between the constituent unit aircraft. Assuming that each unit aircraft in the compound aircraft can stay in level flight without deflection of the control surface at neutral pitch, then the nonlinear dynamics from Equation (
1) can be expressed in the form of Equation (
14) with
and
as state vectors and
as the control input.
Let
and solve Equation (
14); it is assumed that the AoAs of all unit aircraft remain neutral, while the velocity can vary. The solution of Equation (
14) provides results that indicate a symmetric configuration. The compound aircraft can be trimmed under both anhedral and dihedral configurations, as shown in
Figure 4.
Compared to the elevators and throttle, the ailerons play the most important role in trimming, allowing the aircraft to be balanced within a certain range of configurations between symmetrical anhedral and symmetrical dihedral, instead of just two trim points, which is the case without involving ailerons. The extension of the trimming range is related to the maneuverability of the aircraft’s ailerons and its maximum flight speed.
Table 1 shows a series of trimming configurations for the compound aircraft, ranging from dihedral to anhedral configurations. In this table, the bank angle is assigned while the speed and control surfaces deflections are calculated. A bank angle of −3 deg represents a dihedral configuration of 3 deg, whereas a bank angle of 3 deg represents an anhedral configuration of 3 deg.
A compound aircraft in a anhedral or dihedral configuration always operates at a higher airspeed compared to a planar configuration. The main reason for this is that the deflections of the unit aircraft on both sides decrease the projected wing area, requiring a higher airspeed to generate sufficient lift.
Trim states with downward deflection of the side unit aircraft (anhedral configuration) result in a stable configuration, while upward deflection (dihedral configuration) can lead to configuration divergence (positive eigenvalues of flapping mode 1). Although anhedral configurations can harm the lateral stability of the compound aircraft with a positive eigenvalue of the spiral mode, this configuration is chosen as the operational state to ensure a stable configuration of the compound aircraft.
An anhedral configuration without deflection of the ailerons is chosen here as the operation configuration, though other anhedral configuration make sense as well. In the chosen stable trimming state, the Jacobian matrices can be derived and the eigenvectors and eigenvalues can be obtain to investigate the flight modes and stability of the compound aircraft. The full-state dynamic equations contain sixteen variables, meaning that there should be sixteen eigenvalues; however, four of these are located at the origin, and as such can be ignored. The remaining twelve eigenvectors and eigenvalues, corresponding to seven modes of the compound aircraft, are listed in
Table 2 and
Table 3.
A comparison investigation of the eigenvectors carried out with the help of the MAC matrix leads to
By evaluating the dot product of two analyzed vectors, The MAC matrix can be applied to analyze of the degree of coupling between the eigenvectors (in this case, the flight modes) for greater simplicity and intuitiveness [
10]. Here,
values the correlation of the
and
vector
x, with
as the
eigenvector of the reference case and
as the
eigenvector of the currently considered case. The shading on the squares in the graph represent the strength of the correlation, with 1 indicating a perfect correlation and 0 indicating no correlation. An MAC matrix in diagonal form indicates low coupling between the investigated vectors.
Figure 5 shows the MAC matrix for the compound aircraft in anhedral equilibrium configuration. Additionally, the MAC matrix indicates a division of the modal groups of the full-state flight dynamics into a symmetric mode group and asymmetric mode group.
The relative roll motion (flapping modes) is the crucial mode of the compound aircraft. There are two types of flapping mode in the three-wingtip-docked compound aircraft, called flapping mode 1 and flapping mode 2, as shown in
Figure 6 and
Figure 7.
The division of longitudinal and lateral modes adapted in classic flight dynamics cannot be applied due to the addition of relative roll modes. The MAC matrix is basically presented in the form of a block diagonal matrix, indicating that the modal division of group 1 (symmetric motion) and group 2 (asymmetric motion) is reasonable. The symmetric mode group includes the short period, phugoidm and symmetric flapping (flapping mode 1) modes. The longitudinal motion (short period and phugoid) of the wingtip-docked compound aircraft analyzed in this study is not significantly different from that of a conventional aircraft [
6]. In symmetric mode, the reference aircraft should be free of rolling motion. Flapping mode 1 presents a motion in which the side aircraft engages in synchronous up and down flapping motions with the same amplitudes while the reference aircraft remains horizontal. Flapping mode 1 inspires an entirely vertical velocity variation and an induced pitch.
In anti-symmetric flapping mode, here flapping mode 2, the unit aircraft on both sides roll at the same angles while the reference aircraft rolls in the opposite direction. This behavior is coupled with the lateral motion of the entire aircraft in terms of lateral velocity, yaw rate, and roll rate. During anti-symmetric flapping motion, the direction of the resultant lift vector deviates from the plane of the wind axis.
Dutch roll mode, spiral mode, anti-symmetric flapping mode (flapping mode 2), and quasi-roll mode form the asymmetric mode group. Under the anhedral operating configuration, spiral mode is unstable. Spiral mode can be easily excited, inducing instability of the entire vehicle in terms of rigid body motion and relative roll under lateral disturbances such as a slide-slip. Quasi-roll mode is an evolutionary version of the roll mode of a single aircraft in a three-wingtip-docked compound aircraft. The coupling effect of the rigid body motion and relative roll motion are illustrated in detail in
Section 3.
Symmetrical motion does not stimulate asymmetrical motion; however, if asymmetrical motion is excited, symmetrical motion is excited as well. Strictly speaking, pure symmetrical motion does not exist in reality; if the initial condition is a symmetrical state, it can be easily disrupted in both numerical simulations and actual flight. When the roll attitude of the reference aircraft is not theoretically level enough, the vehicle enters a motion dominated by flapping mode 2 and dutch roll mode. In an anhedral configuration, all the flapping modes have an eigenvalue with a negative real part, guaranteeing the dynamic stability of relative roll motion.
Figure 5b,c indicates that the trimming configuration influences the degree of coupling between modes.
Figure 8 shows the change in modes as the trim state of the aircraft shifts from dihedral configuration to anhedral configuration.
The damping of the short period mode first increases and then decreases as the trim configuration changes. When the aircraft is completely planar, the damping of the short period mode is the largest. Under anhedral or dihedral configuration, the reduction of the projection area of the horizontal tail and the increase in the moment of inertia of the pitching axis cannot compensate, causing a decrease in damping of the longitudinal motion. The eigenvalues in phugoid mode vary on a scale of 10 × 10
, and it is trivial compare this to other modes. For brevity, the eigenvalues in phugoid mode are not included in
Figure 8.
Symmetric flapping motion varies from divergence to convergence as the configuration changes from dihedral to anhedral. This is because the aircraft on both sides tend to retract inward due to the torque generated by the lift with respect to the hinge point, resulting in configuration divergence. Symmetric flapping mode (flapping mode 1) changes from monotonic convergence to oscillatory convergence. This can be seen from
Figure 8; the two eigenvalues meet on the negative real axis, then symmetrically separate along the imaginary axis direction and become a pair of conjugate complex eigenvalues.
The stability of anti-symmetric flapping motion (flapping mode 2) increases as the configuration moves from dihedral to anhedral, which is reflected in the fact that the real part of the pair of conjugated eigenvalues becomes smaller. There is no significant change in the eigenvectors in flapping mode 2. As the bank angles of the unit aircraft on both sides increase, the roll motion of the compound aircraft causes larger lateral velocity on the unit aircraft on both sides, which excites the lateral mode of the unit aircraft. therefore, the coupling of the rigid body motion (mainly dutch roll mode and spiral mode) with the relative roll motion becomes exacerbated (represented as quasi-roll mode for the compound aircraft).
The characteristics of spiral mode for compound aircraft are similar to those of a rigid aircraft in either anhedral or dihedral configuration. This reveals a contradiction in the stability of the flapping modes and spiral mode, in that dihedral configurations lead to instability of flapping mode 1 while guaranteeing the stability of spiral mode, while in anhedral configurations the opposite is true.
The dutch roll mode is worth studying as well. When the anhedral and dihedral angles of the configuration are large enough, dutch roll mode is coupled with anti-symmetric flapping mode (flapping mode 2) due to the mismatch between the roll and yaw rates of each individual aircraft in rigid body motion (represented in
Figure 5a,c). Here, dutch roll mode no longer inherits the same meaning as it has for rigid aircraft; instead, it represents a motion that combines the relative roll motion and lateral flight mechanics. During a change in configuration, dutch roll mode changes from slight divergence to oscillatory convergence. This is because the equivalent
and
of the entire aircraft experience the process of sign change, with the absolute value of
increasing as well. In addition, it is reasonable to presume that the flapping modes have an impact on the flight mechanics when the gap between the frequencies of both modes becomes narrow.
3.2. Nonlinear Dynamic Response and Stability
Nonlinear dynamic analysis provides an overview of the dynamic response of the compound aircraft. Here, the dynamic characteristics of the compound aircraft under disturbance of the AoA, side-slip angle, symmetric relative roll angle, and asymmetric relative roll angle are provided. The nonlinear dynamic response of the compound aircraft reflects a different phenomenon from that of conventional rigid-body aircraft. This phenomenon and its mechanism are introduced and explained below.
For disturbance of the AoA, the response of the wingtip-connected compound multi-body aircraft is roughly the same as that of a single-body aircraft, which is due to the fact that the hinge of the wingtip around the x-axis has little effect on the longitudinal characteristics, while the AoA response of the aircraft on both sides increases the oscillation frequency due to the coupling of flapping motion, as shown in
Figure 9. How the flapping modes affect the short-period mode is explained in detail in the next chapter.
Lateral mode is considerably affected by the interaction of each unit aircraft, as shown in
Figure 10. The side-slip angle response of the compound multi-body aircraft is convergent at first, reflecting the same lateral stability as a single-unit aircraft. However, as time passes it begins to oscillate. The lateral aerodynamic coefficient
dominates the lateral stability by transferring side-slip motion to roll motion; meanwhile, the reference aircraft is forced to roll in the direction opposite the side aircraft under the same side-slip motion, stimulating anti-symmetric flapping mode in oscillation. Finally, unstable spiral mode increases the roll angle of the compound aircraft.
The results of our nonlinear simulation of the symmetric relative roll are shown in
Figure 11. Unit impulse disturbances of the symmetric relative roll angle were applied while the reference aircraft stayed in a level attitude;
Figure 11a,c shows the convergent results for anhedral and dihedral disturbance, respectively, while in
Figure 11b the relative roll angle response diverges rapidly under larger dihedral disturbance. This phenomenon can be seen more clearly in the phase diagram plot in
Figure 12.
Trim states are always in the symmetric anhedral or dihedral configuration. Thus, the X axis of the phrase diagram is the symmetric relative roll angle, with a positive value indicating an anhedral configuration and a negative value a dihedral configuration. The Y axis represents the corresponding deflection angular velocity.
The phase diagram shows that the two equilibrium configurations of the compound aircraft have different properties. The blue dot on the diagram represents the anhedral equilibrium configuration, which is a stable equilibrium point, while the brown dot represents the equilibria in dihedral configuration, which is a saddle point. This state represented by the brown dot is called the critical dihedral configuration, and the blue curve is the stability boundary dividing the attraction zone (in blue) and divergence area (in brown). A state point in the attraction area represents anhedral and slight dihedral configurations, both of which are stable in relative roll motion, while a state point outside the attraction area indicates that the relative roll angle of the compound aircraft will rapidly diverge; in the case of excessive upper deflection of the two aircraft on the sides, this means that the compound aircraft will be involved in a dangerous situation. This situation is recorded in flight experiments, shown in sequence from a to d in
Figure 13. The deflection angles of the aircraft on both sides diverge rapidly, while the reference aircraft stays level. This state corresponds to the red line labeled in the phase diagram. As illustrated by this example, compound aircraft should not operate outside the attraction zone.
Extension of the attraction zone can expand the operation boundary of the aircraft by introducing extra rolling aerodynamic torque, either by aileron deflection or by enhancing the spring stiffness of the hinge, corresponding to a leftward shift of the baseline. The dotted blue line beside the baseline in
Figure 12 represents a shift in the stability region boundary under different
K where
. In this case,
. As enhancing the spring stiffness of the hinge is contrary to the original intention of reducing the aircraft’s structural strength, aerodynamic design should instead be employed to add additional aerodynamic rolling torque to the aircraft on both sides, thereby extending the attraction zone.
During flight tests, it is often observed that the configuration of the compound aircraft diverges rapidly when it is disturbed by flying upwind, resulting in a situation similar to that shown in
Figure 13. This is because the attraction zone narrows (a rightward shift of the baseline in the phase diagram) under the influence of upwind disturbance. As a result, a state point that would normally be within the attraction zone may fall into the divergence area. This highlights the sensitivity of compound aircraft to gusts.