The calculation of rotorcraft performance is largely a matter of determining the power required and power available over a range of flight conditions. The power information can then be translated into quantities such as payload, range, ceiling, speed, and climb rate, which define the operational capabilities of the aircraft. The rotor power required is divided into four parts: the induced power, required to produce the rotor thrust; the profile power, required to turn the rotor through the air; the parasite power, required to move the aircraft through the air; and the climb power, required to change the gravitational potential energy. The aircraft has additional contributions to power required, including accessory and transmission losses and perhaps anti-torque power. In hover there is no parasite power, and the induced power is 65% to 75% of the total. As the forward speed increases, the induced power decreases, the profile power increases slightly, and the parasite power increases until it is dominant at high speed. Thus the total power required is high at hover, because of the induced power with a low but reasonable disk loading. At first the total power decreases significantly with increasing speed, as the induced power decreases; then it increases again at high speed, because of the parasite power. Minimum power required occurs roughly in the middle of the helicopter speed range.

The task in rotorcraft performance analysis is the calculation of the rotor forces and power. Procedures to perform these calculations have been developed in the preceding chapters.

Beams and Rotor Blades

An adequate blade structural model is essential for the prediction of rotor loads and stability. Rotor blades almost universally have a high structural fineness ratio and thus are well idealized as beams. The complexities of rotation, and now multiple load paths and composite construction, have required extensive and continuing efforts to develop appropriate beam models for the solution of rotor problems. For exposition of beam theory, particularly relevant to rotor blade analyses, see Hodges (2006) and Bauchau (1985).

A beam is a structure that has small cross-section dimensions relative to an axial line. Based on the slender geometry, beam theory develops a one-dimensional model of the three-dimensional structure. The deflection of the structure is described as functions of the axial coordinate, obtained from ordinary differential equations (in the axial coordinate). The equations depend on cross-section properties, including two-dimensional elastic stiffnesses. The three-dimensional stress field is determined from the deflection variables. Beam theory combines kinematic equations relating strain measures to deflection variables, constitutive equations relating stress resultants to strain measures, and equilibrium equations relating stress resultants to applied loads. When inertial loads are included, the motion is described by partial differential equations, in time and the axial coordinate.

Rotor Configuration

The helicopter rotor type is largely determined by the construction of the blade root and its attachment to the hub. The blade root configuration has a fundamental influence on the blade flap and lag motion and hence on the helicopter handling qualities, vibration, loads, and aeroelastic stability. The basic distinction between rotor types is the presence or absence of flap and lag hinges, and thus whether the blade motion involves rigid-body rotation or bending at the blade root. A simple classification of rotor hubs has the categories articulated, teetering, hingeless, and bearingless, as sketched in Figures 8.1 to 8.4. With real designs (see Figure 1.2) the distinctions are not as clear as in these drawings.

An articulated rotor has its blades attached to the hub with both flap and lag hinges (Figure 8.1). The flap hinge is usually offset from the center of rotation because of mechanical constraints and to improve the helicopter handling qualities. The lag hinge must be offset for the shaft to transmit torque to the rotor. The purpose of the flap and lag hinges is to reduce the root blade loads (since the moments must be zero at the hinge) by allowing blade motion to relieve the bending moments that would otherwise arise at the blade root. With a lag hinge a mechanical lag damper is also needed to avoid a mechanical instability called ground resonance, involving the coupled motion of the rotor lag and hub in-plane displacement.

Hover is the operating state in which the lifting rotor has no velocity relative to the air, either vertical or horizontal. General vertical flight involves axial flow with respect to the rotor. Vertical flight implies axial symmetry of the rotor flow field, so the velocities and loads on the rotor blades are independent of the azimuth position. Axial symmetry greatly simplifies the dynamics and aerodynamics of the helicopter rotor, as is evident when forward flight is considered. The basic analyses of a rotor in axial flow originated in the 19th century with the design of marine propellers and were later applied to airplane propellers. The principal objectives of the analysis of the hovering rotor are to predict the forces generated and power required by the rotating blades and to design the most efficient rotor.

Momentum Theory

Momentum theory applies the basic conservation laws of fluid mechanics (conservation of mass, momentum, and energy) to the rotor and flow as a whole to estimate the rotor performance. The theory is a global analysis, relating the overall flow velocities to the total rotor thrust and power. Momentum theory was developed for marine propellers by W.J.M. Rankine in 1865 and R.E. Froude in 1885, and extended in 1920 by A. Betz to include the rotation of the slipstream; see Glauert (1935) for the history.

Vertical flight of the helicopter rotor at speed V includes the operating states of hover (V = 0), climb (V > 0), and descent (V < 0) and the special case of vertical autorotation (power-off descent). Between the hover and autorotation states, the helicopter is descending at reduced power. Beyond autorotation, the rotor is producing power for the helicopter. The principal subject of this chapter is the induced power of the rotor in vertical flight, including descent. The key physics are associated with the flow states of the rotor in axial flight. Axial flight of a rotor also encompasses the propeller in cruise (V > 0) and static (V = 0) operation, and a horizontal axis wind turbine (V < 0).

Induced Power in Vertical Flight

In Chapter 3, momentum theory was used to estimate the rotor induced power Pi for hover and vertical climb. Momentum theory gives a good power estimate if an empirical factor is included to account for additional induced losses, particularly tip losses and losses due to nonuniform inflow. In the present chapter these results are extended to include vertical descent. Momentum theory is not applicable for a range of descent rates because the assumed wake model is not correct. Indeed, the rotor wake in that range is so complex that no simple model is adequate. In autorotation, the operating state for power-off descent, the rotor is producing thrust with no net power absorption. The energy to produce the thrust (the induced power Pi) and turn the rotor (the profile power Po) comes from the change in gravitational potential energy as the helicopter descends. The range of descent rates where momentum theory is not applicable includes autorotation.

During translational motion of the helicopter, when the rotor is nearly horizontal, the rotor blades see a component of the forward velocity as well as the velocity due to their own rotation (Figure 5.1). In forward flight the rotor does not have axisymmetry as in hover and vertical flight; rather, the aerodynamic environment varies periodically as the blade rotates with respect to the direction of flight. The advancing blade has a velocity relative to the air higher than the rotational velocity, whereas the retreating blade has a lower velocity relative to the air. This lateral asymmetry has a major influence on the rotor and its analysis in forward flight. Thus the rotor blade loading and motion are periodic with a fundamental frequency equal to the rotor speed Ω. The analysis is more complicated than for hover because of the dependence of the loads and motion on the azimuth angle.

As a consequence of the axisymmetry, the analysis of the hovering rotor primarily involves a consideration of the aerodynamics. In forward flight, however, the lateral asymmetry in the basic aerodynamic environment produces a periodicmotion of the blade, which in turn influences the aerodynamic forces. The analysis in forward flight must therefore consider the blade dynamics as well as the aerodynamics. This chapter covers a number of aerodynamic topics that are familiar from the analysis of the rotor in vertical flight. In particular, we are concerned with the momentum theory treatment of the induced velocity and power in forward flight. Then the rotor blade motion and its behavior in forward flight are considered in Chapter 6.

Handling qualities are defined as “those qualities or characteristics of an aircraft that govern the ease and precision with which a pilot is able to perform the tasks required in support of an aircraft role” (Cooper and Harper (1969)). Generally the terms “flying qualities” and “handling qualities” are interchangeable, although the titles of specifications more often refer to flying qualities. Handling qualities involve the aircraft, the pilot, the tasks, and the environment (Padfield (1998)). Most of this chapter deals only with the aircraft flight dynamics or stability and control characteristics: the equations and fundamental behavior of the rotorcraft rigid-body motion. Simplifications and approximations are made to focus on the fundamental behavior of the aircraft. A more rigorous approach is needed to obtain models sufficient for rotorcraft flight control system design. Padfield (2007) covers rotorcraft flight dynamics and handling qualities in depth.

Control

Rotorcraft control requires the ability to produce moments and forces on the vehicle to establish equilibrium and thereby hold the aircraft in a desired trim state, and to produce accelerations and thereby change the aircraft velocity, position, and orientation. Like airplane control, rotorcraft control is accomplished primarily by producing moments about all three aircraft axes: pitch, roll, and yaw. The helicopter has in addition direct control over the vertical force on the aircraft, corresponding to its VTOL capability. This additional control variable is part of the versatility of the helicopter, but also makes the piloting task more difficult. The control task is eased by the use of a rotor speed governor to automatically manage the power.

The Helicopter

The helicopter is an aircraft that uses rotating wings to provide lift, propulsion, and control. Figure 1.1 shows t he principal helicopter configurations. The rotor blades rotate about a vertical axis, describing a disk in a horizontal or nearly horizontal plane. Aerodynamic forces are generated by the relative motion of a wing surface with respect to the air. The helicopter with its rotary wings can generate these forces even when the velocity of the vehicle is zero, in contrast to fixed-wing aircraft, which require at ranslational velocity to sustain flight. The helicopter therefore has the capability of vertical flight, including vertical take-off and landing. The efficient accomplishment of heavier-than-air hover and vertical flight is the fundamental characteristic of the helicopter rotor.

The rotor must supply a thrust force to support the helicopter weight. Efficient vertical flight means a high power loading (ratio of rotor thrust to rotor power required, T/P), because the installed power and fuel consumption of the aircraft are proportional to the power required. For a rotary wing, low disk loading (the ratio of rotor thrust to rotor disk area, T/A) is the key to a high power loading. Conservation of momentum requires that the rotor lift be obtained by accelerating air downward, because corresponding to the lift is an equal and opposite reaction of the rotating wings against the air.

The analysis of the wake is considerably simplified if the rotor is modeled as an actuator disk, which is a circular surface of zero thickness that can support a pressure difference and thus accelerate the air through the disk. The actuator disk neglects the discreteness in the rotor and wake associated with a finite number of blades, and it distributes the vorticity throughout the wake volume. The actuator disk model is the basis for momentum theory (sections 3.1.1 and 5.1.1). The simplest version of vortex theory uses an actuator disk model, which produces a tractable mathematical problem, at least for axial flight (section 3.7). In contrast to hover, the mathematical problem in forward flight is still not trivial, because of the skewed cylindrical geometry (section 5.2). Some results from actuator disk models were presented in section 5.2.1.

The focus of this chapter is the unsteady aerodynamics of the rotor associated with the three-dimensional wake. In particular, the dynamic inflow model is developed. This is a finite-state model, relating a set of inflow variables and loading variables by differential equations. Such a model is required for aeroelastic stability calculations and real time simulation. Vortex theory uses the Biot-Savart law for the velocity induced by the wake vorticity. Potential theory solves the fluid dynamic equations for the velocity potential or acceleration potential.

Vortex Theory

For the actuator disk in axial flow, the wake is a right circular cylinder (Figure 11.1). With uniform loading, the bound circulation is constant over the span, and the trailed vorticity is concentrated in root and tip vortices.

Two-Dimensional Unsteady Airfoil Theory

Since the aerodynamic environment of the rotor blade in forward flight or during transient motion is unsteady, lifting-line theory requires an analysis of the unsteady aerodynamics of a two-dimensional airfoil. Consider the problem of a two-dimensional airfoil undergoing unsteady motion in a uniform free stream. Linear, incompressible aerodynamic theory represents the airfoil and its wake by thin surfaces of vorticity (two-dimensional vortex sheets) in a straight line parallel to the free stream velocity. For the linear problem the solution for the thickness and camber loads can be separated from the loads due to angle-of-attack and unsteady motion. In the development of unsteady thin-airfoil theory, the foundation is constructed for a number of extensions of the analysis for rotary wings, which are presented in later sections of this chapter.

The airfoil and shed wake in unsteady thin-airfoil theory are modeled by planar sheets of vorticity, as shown in Figure 10.1. An airfoil of chord 2b is in a uniform free stream with velocity U. Since the bound circulation of the section varies with time, there is shed vorticity in the wake downstream of the airfoil.

To maintain low drag and high lift, the flow over an airfoil section must remain smooth and attached to the surface. This flow has a rapid acceleration around the nose of the airfoil to the point of maximum suction pressure, and then a slow deceleration along the remainder of the upper surface to the trailing edge. The deceleration must be gradual for the flow to remain attached to the surface. At a high enough angle-of-attack, stall occurs: the deceleration is too large for the boundary layer to support, and the flow separates from the airfoil surface. The maximum lift coefficient at stall is highly dependent on the Reynolds number, Mach number, and the airfoil shape. Figure 8.12 shows clmax values from 1.0 to 1.6 for various airfoils, corresponding to stall angles-of-attack of 10° to 16°. The unstalled airfoil has a low drag and a lift coefficient linear with angle-of-attack. The airfoil in stall at high angles-of-attack has high drag, a loss of lift, and an increased nose-down pitch moment caused by a rearward shift of the center of pressure. The aerodynamic flow field of an airfoil or wing in stall is complex, and for the rotary wing there are important three-dimensional and unsteady phenomena as well.

The digital computer programs that calculate the aeromechanical behavior of rotorcraft are called comprehensive analyses. Comprehensive analyses bring together the most advanced models of the geometry, structure, dynamics, and aerodynamics available in rotary-wing technology, subject to the requirements for accuracy and the constraints of economy. These computer programs calculate rotorcraft performance and trim, blade motion and airloading, structural loads, vibration, noise, aeroelastic stability, and flight dynamics. The multidisciplinary nature of rotorcraft problemsmeans that similarmodels are required for all of these jobs. Acomprehensive analysis performs these calculations with a consistent, balanced, yet high level of technology. Because the tasks require a similar level of technology and similar models, they are best performed with a single tool. The development of computer programs for rotorcraft started with the alternative approach of developing multiple codes separately for individual disciplines, such as performance, dynamics, and handling qualities. Often the range of application of a particular analysis was restricted, perhaps to improve efficiency, but more often for historical reasons. Such experience with early codes provided solid evidence of the resulting inefficient use of development and application resources and inevitable disparities in treatment of the various problems.

There are several implications of the word “comprehensive” in rotorcraft aeromechanics, all encompassed by the ideal analysis. Comprehensive refers to the need for a single tool to perform all computations, for all operating conditions and all rotorcraft configurations, at all stages of the design process. The technology is comprehensive, covering all disciplines with a high technology level.

The differential equations of motion for the rotor blade are derived in this chapter. First the focus is on the inertial and structural forces on the blade, with the aerodynamics represented by the net forces and moments on the blade section. Then the aerodynamic loads are analyzed in more detail to complete the equations. In subsequent chapters the equations are solved for a number of fundamental rotor problems, including flap response, aeroelastic stability, and aircraft flight dynamics. In Chapter 6 the flap and lag dynamics of an articulated rotor were analyzed for only the rigid motion of the blade, including hinge spring or offset. The present chapter extends the derivation of the equations of motion to include a hingeless rotor, higher blade bending modes, blade torsion, and pitch motion. The corresponding hub reactions and blade loads are derived, and the rotor shaft motion is included in the analysis.

The rotor blade equations of motion are derived using the Newtonian approach, with a normal mode representation of the blade motion. The chapter begins with a discussion of the other approaches by which the dynamics can be analyzed. Engineering beam theory is commonly used in helicopter blade analyses. The blade section is assumed to be rigid, so its motion is represented by the bending and rotation of a slender beam. This is normally a good model for the rotor blade, although a more detailed structural analysis is required to obtain the effective beam parameters for some portions of the blade, such as flexbeams and at the root.