For studying the eigendynamic properties of the Lotte airship at specific flight regimes, the nominal flight mechanical model has been utilized for trimming and linearization facilities.
Although it is possible to express the characteristic modes in analytical form using stability derivatives, its form is too complicated to perform a comprehensive analysis. Therefore, the stability derivatives were determined numerically.
Figure 3.4 shows the pole distribution at different flight speeds, with the forward trim velocity ranging from to the maximal . For numerical evaluation, the nomi-nal flight mechanical model was assigned to a realistic buoyancy to gravity ratio . This ratio represents the static heaviness of the airship and can be alternatively expressed by . Usually the static heaviness varies from the “nearly neu-tral” to the “heavy” configurations .
In order to expand the analysis of the system dynamics, a contribution of a physical state to a particular eigenmode of the system using modal analysis will be performed. This involves a definition of a state space system in the modal state basis
, (3.29)
Longitudinal 3 2.0 Lateral-directional
4
Figure 3.4: Root locus of longitudinal dynamics at different trim velocity u0
4 m/s[ ] 12 m/s[ ]
where is the modal state vector, is the modal system matrix. By using the eigenvector matrix , the transforma-tion of the original system into the decoupled modal system is performed. Note, because the modal analysis is very sensitive on selection of the basis of the state vector, it is preferable to perform the analysis of a pre-scaled system (in the current case the analysis was performed in the following units, , , in [m/s], , , in [°/s], , , , , in [°], in [%] of ).
Longitudinal Dynamics
• Surge Mode: The surge mode is distinguished by a stable aperiodic dynamics with a large time constant. Its physical interpretation corresponds to the forward velocity damping and can be explained as a reminder of a well known phugoid mode, when the part due to potential energy is missing, which is a common property of lighter than air vehicles. This mode is always stable. Studying the eigenvector diagram, shown in Fig-ure 3.5, one also concludes that the surge mode is slightly coupled with perturbations of the pitch angle- and almost decoupled with the and states. This is mainly because of insignificant values of stability derivatives and .
• Heave Mode: The heave eigendynamic corresponds to a well dampened aperiodic motion with a comparatively small time constant. It is characterized by the vertical motion of the airship, incorporating the cross-flow aerodynamic effects. At low airship velocities , the coupling with the vertical velocity is dominant (see Figure 3.5).
This can be interpreted by the aerodynamic phenomena that acts in the vertical direc-tion, where the cross flow effects are dominant. As the forward trim velocity increases, the influence of the fins becomes stronger. This results in a stabilizing pitch-ing moment and, therefore, the couplpitch-ing with the and consequently with becomes apparent. At high trim velocities this mode is also referenced as pitch-subsid-ence mode [29]. From the pole map diagram given in Figure 3.4, it can be seen that the damping of the heave mode increases with increased trim velocity . The contribu-tion of the forward velocity perturbation is negligibly small.
• Longitudinal-Pendulum Mode: The longitudinal pendulum mode is the most critical eigenmode in the longitudinal dynamics. At very low velocities, where aerodynamic effects are negligible, the longitudinal pendulum mode appears as a slightly dampened low frequency oscillation in pitch. Physically this can be interpreted as a pendulum motion of the center of gravity “suspended” at the center of buoyancy [26].
As can be concluded from the eigenvector diagram (Figure 3.5), the states and are dominant. With increased trim velocity , the mode starts to interact with the heave mode throughout the variable. The stabilizing moment of the fins increases the damping ratio of the mode, as seen on the pole map at velocities of about . If the velocity is further increased, the unstable aerodynamic moment begins to dominate over the stabilizing moment produced by fins. At a certain velocity, the destabilizing aerodynamic moment becomes greater than the stabilizing moment due to gravity force, and the longitudinal pendulum mode becomes unstable (see Figure 3.4).
x˜ = S–1x A˜ = SAS–1 = diag(λ1, λ2, ,… λn)
Lateral-Directional Dynamics
• Sideslip-Subsidence Mode: In the studied configuration, the sideslip-subsidence mode resulted in unstable real pole on the complex plane. This instability arises from the static destabilizing aerodynamic yawing moment , which can be approximated by at small angles of attack. The states and play a dominant role in this unstable dynamics (see Figure 3.6). At higher trim velocities , this mode is also slightly coupled with perturbations of the roll angle due to centrifugal effects, which appear in the yaw motion due to low location. The time constant of the sideslip-subsidence mode is very sensitive to variations of the trim velocity . The terms and are negligibly small.
• Yaw-Subsidence Mode: This mode can be explained by similar physical phenomena, as was observed for the heave mode. Due to the dominant presence of the , per-turbations, concluded from Figure 3.6, the yaw-subsidence mode is strongly coupled with the unstable sideslip-subsidence mode. With increased speed , the mode increases its damping.
3 2
σ σ
jω jω
δu δu
δq δq
δw
δw
δθ δθ
u0 = 3[m/s] u0 = 11[m/s]
1
σ σ
jω jω
δu δq δu δq
δw δw
δθ δθ
u0 = 3[m/s] u0 = 11[m/s]
σ σ
jω jω
δu δu
δq δq
δw δθ δw δθ
u0 = 3[m/s] u0 = 11[m/s]
Figure 3.5: Eigenvector diagrams of longitudinal modes
NA
MA δv δr
u0 δφ
CG
u0 Yp Yφ
δv δr u0
• Roll-Oscillation Mode: The roll oscillation mode describes a lightly dampened pen-dulum motion around the axis of airship. Both, the aerodynamic and gravity phe-nomena contribute to this eigenmotion. The mode is mainly composed of the and states (see Figure 3.6). With increasing velocity , the rolling motion is better dampened due to aerodynamic roll damping produced by the fins.