LIGHTER-THAN-AIR VEHICLE/BLIMP DYNAMICS

Lighter-than-air-vehicles have several advantages and are finding increasing appli­cations in various tasks such as surveillance and advertising. Modern aerostats use helium for lift and consist of a power supply system. A two-way fiber-optic datalink is provided for sensor data and control. The low-pressure differential allows them to sustain for long. They are also difficult to detect with radar. Due to these features, the demand for the use of aerostats for surveillance is continually growing and, as they are less expensive, they offer a viable alternative to satellites for certain limited tasks.

In this section, we briefly discuss the mathematical model of an airship. As for the fixed-wing aircraft, the basic derivation of EOMs of an airship is based on rigid body dynamics [16]. One noticeable difference in developing the model equations for an aircraft and an airship is the consideration of buoyancy in the latter case. To account for the apparent mass effect due to the large volume of air mass displaced by an airship, EOMs are usually written about the center of buoyancy. In Ref. [17], effects of the change in the position of the CG, as a result of venting or intake of the ballast air, is also modeled. Reference [18] suggested simplification in the model equations by determining the change in the position of the CG and translating all the forces, moments, and inertias to CG to compute the airship dynamics. This results in EOMs that are similar to those discussed in Chapter 3 for the fixed-wing aircraft. In state-space form, these can be expressed as

X(t) = f [x(t), u(t), Q] (5.56)

where, for a 12th-order system, the state vector x consists of x = [u, v, w, p, q, r, f, U, С, X, Y, H]

The vector Q is a collection of the stability and control derivatives and u is the control input vector. Computing the center of mass location and translating the moment of inertias to the center of mass lead to decoupling the accelerations and writing EOMs in the conventional form. Since the variables X, Y, and С do not couple with the rest of the equations, mostly a ninth order linear model comprising the longitudinal states u, w, q, U, H and the lateral states comprising v, p, r, f is used to analyze the modal characteristics of an airship.

Due to the buoyancy and apparent mass of the airship, its climb performance is different from that of the conventional fixed-wing aircraft [18]. For an aircraft, we have

T — D — Wsing = 0 L — Wcos g = 0

where W is the weight, T is the engine thrust, D is the drag, L is the lift, and g is the flight path angle. Considering the buoyancy force B in the case of an airship, the force balance equations in climb are

T – D + (B – W) sing = 0 L — W cosg = 0

The first of the above equations describes the force balance in the direction of flight path while the second describes the force balance perpendicular to the flight path. The net buoyant force (B — W) for an airship in climb is positive. The velocity of the airship will undergo a change if the first equation is not balanced, and the airship flight path angle will undergo a change if the second equation is imbalanced.

Linearized EOMs provide the modal characteristics of the airship. The longitu­dinal modes of an airship generally consist of an oscillatory mode with low damping and a heave mode. Unlike a conventional aircraft, the airship has no mode equivalent to the phugoid mode as the coupling between the forward speed and heave mode is missing in the case of an airship. The lateral-directional modes comprise the DR mode, which is generally an oscillatory stable mode with high damping and a roll mode, which is an oscillatory stable mode with very low damping. Airship can also show sideslip subsidence and yaw subsidence modes. Typical aerodynamic deriva­tives for an aerostat/blimp configuration are given in Table 5.5 [18]. Reference [19] discusses the airship modal characteristics in greater details.