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.

Rotorcraft is a class of aircraft that uses large-diameter rotating wings to accomplish efficient vertical takeoff and landing. The class thus encompasses helicopters of numerous configurations, tilting proprotor aircraft, compound helicopters, and many other innovative concepts.

Defining “aeromechanics” is more difficult. Today's dictionaries do not capture what the term means for the rotorcraft community. The definitions are not broad enough, and they do not reflect the multidisciplinary facet of the word as applied to rotorcraft. In my 2010 Nikolsky Lecture for the American Helicopter Society, I proposed the following definition:

Aeromechanics covers much of what ther otorcraft engineer needs: performance, loads, vibration, stability, flight dynamics, and noise. These topics cover many of the key performance attributes and many of the often encountered problems in rotorcraft designs.

As with my previous book Helicopter Theory (written in 1976, published in 1980 by Princeton University Press, republished in 1994 by Dover Publications), this text is focused on analysis, with only occasional reference to test data to develop arguments or support results, and with nothing at all regarding the techniques of testing in wind tunnels or flight. Calculated results are presented to illustrate analysis characteristics and rotor behavior.

This chapter summarizes the principal nomenclature to be used in the text. The intention is to provide a reference for the later chapters and also to introduce the basic elements of the rotor and its analysis. Only the most fundamental parameters are included here; the definitions of the other quantities required are presented as the analysis is developed. A number of the basic dimensionless parameters of helicopter analysis are also introduced. An alphabetical listing of symbols is provided at the end of the chapter.

Dimensions

Generally the analyses in this text use dimensionless quantities. The natural reference length scale for the rotor is the blade radius R, and the natural reference time scale is the rotor rotational speed Ω(rad/sec). For a reference mass the air density ρ is chosen.

For typographical simplicity, no distinction is made between the symbols for the dimensional and dimensionless forms of a quantity when the latter are based on ρ, Ω, and R. New symbols are introduced for t hose dimensionless parameters normalized using other quantities.

Nomenclature

Physical Description of the Blade

R = the rotor radius; the length of the blade, measured from center of rotation to tip.

Ω = the rotor rotational speed or angular velocity (rad/sec).

ρ = air density.

Rotary-wing flow fields are as complex as any in aeronautics. The helicopter rotor in forward flight encounters three-dimensional, unsteady, transonic, viscous aerodynamic phenomena. Rotary-wing problems provide a stimulus for development and opportunities for application of the most advanced computational techniques.

Inviscid, potential aerodynamics is the starting point for many computational methods for rotors, allowing practical solutions of compressible and unsteady problems. Lifting-surface theory solves the linearized problem by using the result for a moving singularity, often of the acceleration potential. Panel methods use surface singularity distributions to solve problems with arbitrary geometry. Transonic rotor analyses use finite-difference techniques to solve the nonlinear flow equation.

The rotor wake is a factor in almost all helicopter problems. A major issue in advanced aerodynamic methods is how the wake can be included. Wake formation must at some level be considered a viscous phenomenon, and the helical geometry of the helicopter wake means that the detailed structure is important even at scales on the order of the rotor size. A useful rotor aerodynamic theory must account for the effects of viscosity, such as wake formation and blade stall, which are important for most operating conditions. Solution of Navier-Stokes equations for rotor flows is now common. Hybrid methods can be used for efficiency, typically using Navier-Stokes solutions near the blade and some vortex method for the rest of the flow field.

Sources for the derivations of the equations are Lamb (1932), Morse and Feshback (1953), Garrick (1957), A shley and Landahl (1965), and Batchelor (1967).

Helicopter Rotor Noise

The helicopter is the quietest VTOL aircraft, but its noise level can still be high enough to compromise its utility unless specific attention is given to designing for low noise. As the restrictions on aircraft noise increase, the rotor noise becomes an increasingly important factor in helicopter design. The complex aerodynamics of rotors lead to a number of significant noise mechanisms. Helicopter rotor noise tends to be concentrated at harmonics of the blade passage frequency NΩ, because of the periodic nature of the rotor as seen in the non-rotating frame. There is sound radiated because the mean thrust and drag forces rotate with the blades and because of the higher harmonic loading as well. The spectral lines are broadened at the higher harmonics because of the random character of the rotor flow, particularly variations in the wake-induced loads. The acoustic pressure signal is basically periodic in time (the period is 2π/NΩ), with sharp impulses due to localized aerodynamic phenomena such as compressibility effects and vortex-induced loads. Figure 14.1 illustrates the spectrum of rotor-generated sound. The contributions to helicopter rotor noise can be classified as vortex or broadband noise, rotational noise, and impulsive noise or blade slap. Although the distinction between these types of rotor noise is not as sharp as was once thought, the classification remains useful for purposes of exposition. Cox (1973), Burton, Schlinker, and Shenoy (1985), and Brentner and Farassat (1994, 2003) have presented summaries of helicopter rotor noise mechanisms and analysis.

The Helicopter Rotor in Forward Flight

Efficient hover capability is the fundamental characteristic of the helicopter, but without good forward flight performance the ability to hover has little value. During translational flight of the helicopter, the rotor disk is moving edgewise through the air, remaining nearly horizontal, generally with a small forward tilt to provide the propulsive force for the aircraft. A tiltrotor cruises with the rotors tilted to operate as propellers. A compound helicopter reduces the lift and propulsive force required of the rotor. Yet all rotorcraft configurations execute low-speed forward flight with the flapping rotor in edgewise flow, which is the subject of this chapter.

Thus in forward flight the rotor blade sees both a component of the helicopter forward velocity and the velocity due to its own rotation. On the advancing side of the disk the velocity of the blade is increased by the forward speed, whereas on the retreating side the velocity is decreased. For a constant angle-of-attack of the blade, the varying dynamic pressure of the rotor aerodynamic environment in forward flight would tend to produce more lift on the advancing side than on the retreating side; that is, a rolling moment on the rotor. If nothing were done to counter this moment, the helicopter would respond by rolling toward the retreating side of the rotor until equilibrium was achieved, with the rotor moment balanced by the gravitational force acting at the helicopter center-of-gravity. The rotor moment could possibly be so large that an equilibrium roll angle would not be achieved.

The aeroelastic equations of motion for the rotor were derived in Chapter 16. The present chapter examines the solutions of these equations for a number of fundamental stability problems in rotor dynamics. To obtain analytical solutions, each problem must be restricted to a small number of degrees of freedom and to only the fundamental blade motion. Rotorcraft engineering currently has the capability to routinely calculate the dynamic behavior for much more detailed and complex models of the rotor and airframe. Thus elementary analyses are less necessary for actual numerical solutions, but are even more important as the basis for understanding the rotor dynamics.

Pitch-Flap Flutter

Traditionally, the term “flutter” refers to an aeroelastic instability involving the coupled bending and torsion motion of a wing. For the rotary wing, flutter refers to the pitch-flap motion of the blade. Often the term is generalized to include any aeroelastic instability of the rotor or aircraft, but the subject of this section is the blade pitch-flap stability. The classical problem considers two degrees of freedom: the rigid flap and rigid pitch motion of an articulated rotor blade. Since the control system is usually the softest element in the torsion motion, the rigid pitch degree of freedom is a good representation of the blade dynamics. A general fundamental flap mode with natural frequency υβ is considered. A thorough analysis of the flutter of a hingeless rotor blade usually requires that the in-plane motion be modeled as well.